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Free Math Worksheets — Over 100k free practice problems on Khan Academy
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That’s because Khan Academy has over 100,000 free practice questions. And they’re even better than traditional math worksheets – more instantaneous, more interactive, and more fun!
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Kindergarten, basic geometry, pre-algebra, algebra basics, high school geometry.
Statistics and probability
High school statistics, ap®︎/college statistics, precalculus, differential calculus, integral calculus, ap®︎/college calculus ab, ap®︎/college calculus bc, multivariable calculus, differential equations, linear algebra.
- Addition and subtraction
- Place value (tens and hundreds)
- Addition and subtraction within 20
- Addition and subtraction within 100
- Addition and subtraction within 1000
- Measurement and data
- Counting and place value
- Measurement and geometry
- Place value
- Measurement, data, and geometry
- Add and subtract within 20
- Add and subtract within 100
- Add and subtract within 1,000
- Money and time
- Intro to multiplication
- 1-digit multiplication
- Addition, subtraction, and estimation
- Intro to division
- Understand fractions
- Equivalent fractions and comparing fractions
- More with multiplication and division
- Arithmetic patterns and problem solving
- Represent and interpret data
- Multiply by 1-digit numbers
- Multiply by 2-digit numbers
- Factors, multiples and patterns
- Add and subtract fractions
- Multiply fractions
- Understand decimals
- Plane figures
- Measuring angles
- Area and perimeter
- Units of measurement
- Decimal place value
- Add decimals
- Subtract decimals
- Multi-digit multiplication and division
- Divide fractions
- Multiply decimals
- Divide decimals
- Powers of ten
- Coordinate plane
- Algebraic thinking
- Converting units of measure
- Properties of shapes
- Ratios, rates, & percentages
- Arithmetic operations
- Negative numbers
- Properties of numbers
- Variables & expressions
- Equations & inequalities introduction
- Data and statistics
- Negative numbers: addition and subtraction
- Negative numbers: multiplication and division
- Fractions, decimals, & percentages
- Rates & proportional relationships
- Expressions, equations, & inequalities
- Numbers and operations
- Solving equations with one unknown
- Linear equations and functions
- Systems of equations
- Geometric transformations
- Data and modeling
- Volume and surface area
- Pythagorean theorem
- Transformations, congruence, and similarity
- Arithmetic properties
- Factors and multiples
- Reading and interpreting data
- Negative numbers and coordinate plane
- Ratios, rates, proportions
- Equations, expressions, and inequalities
- Exponents, radicals, and scientific notation
- Algebraic expressions
- Linear equations and inequalities
- Graphing lines and slope
- Expressions with exponents
- Quadratics and polynomials
- Equations and geometry
- Algebra foundations
- Solving equations & inequalities
- Working with units
- Linear equations & graphs
- Forms of linear equations
- Inequalities (systems & graphs)
- Absolute value & piecewise functions
- Exponents & radicals
- Exponential growth & decay
- Quadratics: Multiplying & factoring
- Quadratic functions & equations
- Irrational numbers
- Performing transformations
- Transformation properties and proofs
- Right triangles & trigonometry
- Non-right triangles & trigonometry (Advanced)
- Analytic geometry
- Conic sections
- Solid geometry
- Polynomial arithmetic
- Complex numbers
- Polynomial factorization
- Polynomial division
- Polynomial graphs
- Rational exponents and radicals
- Exponential models
- Transformations of functions
- Rational functions
- Trigonometric functions
- Non-right triangles & trigonometry
- Trigonometric equations and identities
- Analyzing categorical data
- Displaying and comparing quantitative data
- Summarizing quantitative data
- Modeling data distributions
- Exploring bivariate numerical data
- Study design
- Counting, permutations, and combinations
- Random variables
- Sampling distributions
- Confidence intervals
- Significance tests (hypothesis testing)
- Two-sample inference for the difference between groups
- Inference for categorical data (chi-square tests)
- Advanced regression (inference and transforming)
- Analysis of variance (ANOVA)
- Data distributions
- Two-way tables
- Binomial probability
- Normal distributions
- Displaying and describing quantitative data
- Inference comparing two groups or populations
- Chi-square tests for categorical data
- More on regression
- Prepare for the 2020 AP®︎ Statistics Exam
- AP®︎ Statistics Standards mappings
- Composite functions
- Probability and combinatorics
- Limits and continuity
- Derivatives: definition and basic rules
- Derivatives: chain rule and other advanced topics
- Applications of derivatives
- Analyzing functions
- Parametric equations, polar coordinates, and vector-valued functions
- Applications of integrals
- Differentiation: definition and basic derivative rules
- Differentiation: composite, implicit, and inverse functions
- Contextual applications of differentiation
- Applying derivatives to analyze functions
- Integration and accumulation of change
- Applications of integration
- AP Calculus AB solved free response questions from past exams
- AP®︎ Calculus AB Standards mappings
- Infinite sequences and series
- AP Calculus BC solved exams
- AP®︎ Calculus BC Standards mappings
- Integrals review
- Integration techniques
- Thinking about multivariable functions
- Derivatives of multivariable functions
- Applications of multivariable derivatives
- Integrating multivariable functions
- Green’s, Stokes’, and the divergence theorems
- First order differential equations
- Second order linear equations
- Laplace transform
- Vectors and spaces
- Matrix transformations
- Alternate coordinate systems (bases)
Frequently Asked Questions about Khan Academy and Math Worksheets
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120 Math Word Problems To Challenge Students Grades 1 to 8
Make solving math problems fun!
With Prodigy's assessment tools, you can engage your students that adapts for your curriculum, lesson and student needs.
- Teaching Tools
- Mixed operations
- Ordering and number sense
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You sit at your desk, ready to put a math quiz, test or activity together. The questions flow onto the document until you hit a section for word problems.
A jolt of creativity would help. But it doesn’t come.
Whether you’re a 3rd grade teacher or an 8th grade teacher preparing students for high school, translating math concepts into real world examples can certainly be a challenge.
This resource is your jolt of creativity. It provides examples and templates of math word problems for 1st to 8th grade classes.
There are 120 examples in total.
The list of examples is supplemented by tips to create engaging and challenging math word problems.
120 Math word problems, categorized by skill
Addition word problems.
Best for: 1st grade, 2nd grade
1. Adding to 10: Ariel was playing basketball. 1 of her shots went in the hoop. 2 of her shots did not go in the hoop. How many shots were there in total?
2. Adding to 20: Adrianna has 10 pieces of gum to share with her friends. There wasn’t enough gum for all her friends, so she went to the store to get 3 more pieces of gum. How many pieces of gum does Adrianna have now?
3. Adding to 100: Adrianna has 10 pieces of gum to share with her friends. There wasn’t enough gum for all her friends, so she went to the store and got 70 pieces of strawberry gum and 10 pieces of bubble gum. How many pieces of gum does Adrianna have now?
4. Adding Slightly over 100: The restaurant has 175 normal chairs and 20 chairs for babies. How many chairs does the restaurant have in total?
5. Adding to 1,000: How many cookies did you sell if you sold 320 chocolate cookies and 270 vanilla cookies?
6. Adding to and over 10,000: The hobby store normally sells 10,576 trading cards per month. In June, the hobby store sold 15,498 more trading cards than normal. In total, how many trading cards did the hobby store sell in June?
7. Adding 3 Numbers: Billy had 2 books at home. He went to the library to take out 2 more books. He then bought 1 book. How many books does Billy have now?
8. Adding 3 Numbers to and over 100: Ashley bought a big bag of candy. The bag had 102 blue candies, 100 red candies and 94 green candies. How many candies were there in total?
Subtraction word problems
Best for: 1st grade, second grade
9. Subtracting to 10: There were 3 pizzas in total at the pizza shop. A customer bought 1 pizza. How many pizzas are left?
10. Subtracting to 20: Your friend said she had 11 stickers. When you helped her clean her desk, she only had a total of 10 stickers. How many stickers are missing?
11. Subtracting to 100: Adrianna has 100 pieces of gum to share with her friends. When she went to the park, she shared 10 pieces of strawberry gum. When she left the park, Adrianna shared another 10 pieces of bubble gum. How many pieces of gum does Adrianna have now?
Practice math word problems with Prodigy Math
Join millions of teachers using Prodigy to make learning fun and differentiate instruction as they answer in-game questions, including math word problems from 1st to 8th grade!
12. Subtracting Slightly over 100: Your team scored a total of 123 points. 67 points were scored in the first half. How many were scored in the second half?
13. Subtracting to 1,000: Nathan has a big ant farm. He decided to sell some of his ants. He started with 965 ants. He sold 213. How many ants does he have now?
14. Subtracting to and over 10,000: The hobby store normally sells 10,576 trading cards per month. In July, the hobby store sold a total of 20,777 trading cards. How many more trading cards did the hobby store sell in July compared with a normal month?
15. Subtracting 3 Numbers: Charlene had a pack of 35 pencil crayons. She gave 6 to her friend Theresa. She gave 3 to her friend Mandy. How many pencil crayons does Charlene have left?
16. Subtracting 3 Numbers to and over 100: Ashley bought a big bag of candy to share with her friends. In total, there were 296 candies. She gave 105 candies to Marissa. She also gave 86 candies to Kayla. How many candies were left?
Multiplication word problems
Best for: 2nd grade, 3rd grade
17. Multiplying 1-Digit Integers: Adrianna needs to cut a pan of brownies into pieces. She cuts 6 even columns and 3 even rows into the pan. How many brownies does she have?
18. Multiplying 2-Digit Integers: A movie theatre has 25 rows of seats with 20 seats in each row. How many seats are there in total?
19. Multiplying Integers Ending with 0: A clothing company has 4 different kinds of sweatshirts. Each year, the company makes 60,000 of each kind of sweatshirt. How many sweatshirts does the company make each year?
20. Multiplying 3 Integers: A bricklayer stacks bricks in 2 rows, with 10 bricks in each row. On top of each row, there is a stack of 6 bricks. How many bricks are there in total?
21. Multiplying 4 Integers: Cayley earns $5 an hour by delivering newspapers. She delivers newspapers 3 days each week, for 4 hours at a time. After delivering newspapers for 8 weeks, how much money will Cayley earn?
Division word problems
Best for: 3rd grade, 4th grade, 5th grade
22. Dividing 1-Digit Integers: If you have 4 pieces of candy split evenly into 2 bags, how many pieces of candy are in each bag?
23. Dividing 2-Digit Integers: If you have 80 tickets for the fair and each ride costs 5 tickets, how many rides can you go on?
24. Dividing Numbers Ending with 0: The school has $20,000 to buy new computer equipment. If each piece of equipment costs $50, how many pieces can the school buy in total?
25. Dividing 3 Integers: Melissa buys 2 packs of tennis balls for $12 in total. All together, there are 6 tennis balls. How much does 1 pack of tennis balls cost? How much does 1 tennis ball cost?
26. Interpreting Remainders: An Italian restaurant receives a shipment of 86 veal cutlets. If it takes 3 cutlets to make a dish, how many cutlets will the restaurant have left over after making as many dishes as possible?
Mixed operations word problems
27. Mixing Addition and Subtraction: There are 235 books in a library. On Monday, 123 books are taken out. On Tuesday, 56 books are brought back. How many books are there now?
28. Mixing Multiplication and Division: There is a group of 10 people who are ordering pizza. If each person gets 2 slices and each pizza has 4 slices, how many pizzas should they order?
29. Mixing Multiplication, Addition and Subtraction: Lana has 2 bags with 2 marbles in each bag. Markus has 2 bags with 3 marbles in each bag. How many more marbles does Markus have?
30. Mixing Division, Addition and Subtraction: Lana has 3 bags with the same amount of marbles in them, totaling 12 marbles. Markus has 3 bags with the same amount of marbles in them, totaling 18 marbles. How many more marbles does Markus have in each bag?
Ordering and number sense word problems
31. Counting to Preview Multiplication: There are 2 chalkboards in your classroom. If each chalkboard needs 2 pieces of chalk, how many pieces do you need in total?
32. Counting to Preview Division: There are 3 chalkboards in your classroom. Each chalkboard has 2 pieces of chalk. This means there are 6 pieces of chalk in total. If you take 1 piece of chalk away from each chalkboard, how many will there be in total?
33. Composing Numbers: What number is 6 tens and 10 ones?
34. Guessing Numbers: I have a 7 in the tens place. I have an even number in the ones place. I am lower than 74. What number am I?
35. Finding the Order: In the hockey game, Mitchell scored more points than William but fewer points than Auston. Who scored the most points? Who scored the fewest points?
Fractions word problems
Best for: 3rd grade, 4th grade, 5th grade, 6th grade
36. Finding Fractions of a Group: Julia went to 10 houses on her street for Halloween. 5 of the houses gave her a chocolate bar. What fraction of houses on Julia’s street gave her a chocolate bar?
37. Finding Unit Fractions: Heather is painting a portrait of her best friend, Lisa. To make it easier, she divides the portrait into 6 equal parts. What fraction represents each part of the portrait?
38. Adding Fractions with Like Denominators: Noah walks ⅓ of a kilometre to school each day. He also walks ⅓ of a kilometre to get home after school. How many kilometres does he walk in total?
39. Subtracting Fractions with Like Denominators: Last week, Whitney counted the number of juice boxes she had for school lunches. She had ⅗ of a case. This week, it’s down to ⅕ of a case. How much of the case did Whitney drink?
40. Adding Whole Numbers and Fractions with Like Denominators: At lunchtime, an ice cream parlor served 6 ¼ scoops of chocolate ice cream, 5 ¾ scoops of vanilla and 2 ¾ scoops of strawberry. How many scoops of ice cream did the parlor serve in total?
41. Subtracting Whole Numbers and Fractions with Like Denominators: For a party, Jaime had 5 ⅓ bottles of cola for her friends to drink. She drank ⅓ of a bottle herself. Her friends drank 3 ⅓. How many bottles of cola does Jaime have left?
42. Adding Fractions with Unlike Denominators: Kevin completed ½ of an assignment at school. When he was home that evening, he completed ⅚ of another assignment. How many assignments did Kevin complete?
43. Subtracting Fractions with Unlike Denominators: Packing school lunches for her kids, Patty used ⅞ of a package of ham. She also used ½ of a package of turkey. How much more ham than turkey did Patty use?
44. Multiplying Fractions: During gym class on Wednesday, the students ran for ¼ of a kilometre. On Thursday, they ran ½ as many kilometres as on Wednesday. How many kilometres did the students run on Thursday? Write your answer as a fraction.
45. Dividing Fractions: A clothing manufacturer uses ⅕ of a bottle of colour dye to make one pair of pants. The manufacturer used ⅘ of a bottle yesterday. How many pairs of pants did the manufacturer make?
46. Multiplying Fractions with Whole Numbers: Mark drank ⅚ of a carton of milk this week. Frank drank 7 times more milk than Mark. How many cartons of milk did Frank drink? Write your answer as a fraction, or as a whole or mixed number.
Decimals word problems
Best for: 4th grade, 5th grade
47. Adding Decimals: You have 2.6 grams of yogurt in your bowl and you add another spoonful of 1.3 grams. How much yogurt do you have in total?
48. Subtracting Decimals: Gemma had 25.75 grams of frosting to make a cake. She decided to use only 15.5 grams of the frosting. How much frosting does Gemma have left?
49. Multiplying Decimals with Whole Numbers: Marshall walks a total of 0.9 kilometres to and from school each day. After 4 days, how many kilometres will he have walked?
50. Dividing Decimals by Whole Numbers: To make the Leaning Tower of Pisa from spaghetti, Mrs. Robinson bought 2.5 kilograms of spaghetti. Her students were able to make 10 leaning towers in total. How many kilograms of spaghetti does it take to make 1 leaning tower?
51. Mixing Addition and Subtraction of Decimals: Rocco has 1.5 litres of orange soda and 2.25 litres of grape soda in his fridge. Antonio has 1.15 litres of orange soda and 0.62 litres of grape soda. How much more soda does Rocco have than Angelo?
52. Mixing Multiplication and Division of Decimals: 4 days a week, Laura practices martial arts for 1.5 hours. Considering a week is 7 days, what is her average practice time per day each week?
Comparing and sequencing word problems
Best for: Kindergarten, 1st grade, 2nd grade
53. Comparing 1-Digit Integers: You have 3 apples and your friend has 5 apples. Who has more?
54. Comparing 2-Digit Integers: You have 50 candies and your friend has 75 candies. Who has more?
55. Comparing Different Variables: There are 5 basketballs on the playground. There are 7 footballs on the playground. Are there more basketballs or footballs?
56. Sequencing 1-Digit Integers: Erik has 0 stickers. Every day he gets 1 more sticker. How many days until he gets 3 stickers?
57. Skip-Counting by Odd Numbers: Natalie began at 5. She skip-counted by fives. Could she have said the number 20?
58. Skip-Counting by Even Numbers: Natasha began at 0. She skip-counted by eights. Could she have said the number 36?
59. Sequencing 2-Digit Numbers: Each month, Jeremy adds the same number of cards to his baseball card collection. In January, he had 36. 48 in February. 60 in March. How many baseball cards will Jeremy have in April?
Time word problems
66. Converting Hours into Minutes: Jeremy helped his mom for 1 hour. For how many minutes was he helping her?
69. Adding Time: If you wake up at 7:00 a.m. and it takes you 1 hour and 30 minutes to get ready and walk to school, at what time will you get to school?
70. Subtracting Time: If a train departs at 2:00 p.m. and arrives at 4:00 p.m., how long were passengers on the train for?
71. Finding Start and End Times: Rebecca left her dad’s store to go home at twenty to seven in the evening. Forty minutes later, she was home. What time was it when she arrived home?
Money word problems
Best for: 1st grade, 2nd grade, 3rd grade, 4th grade, 5th grade
60. Adding Money: Thomas and Matthew are saving up money to buy a video game together. Thomas has saved $30. Matthew has saved $35. How much money have they saved up together in total?
61. Subtracting Money: Thomas has $80 saved up. He uses his money to buy a video game. The video game costs $67. How much money does he have left?
62. Multiplying Money: Tim gets $5 for delivering the paper. How much money will he have after delivering the paper 3 times?
63. Dividing Money: Robert spent $184.59 to buy 3 hockey sticks. If each hockey stick was the same price, how much did 1 cost?
64. Adding Money with Decimals: You went to the store and bought gum for $1.25 and a sucker for $0.50. How much was your total?
65. Subtracting Money with Decimals: You went to the store with $5.50. You bought gum for $1.25, a chocolate bar for $1.15 and a sucker for $0.50. How much money do you have left?
67. Applying Proportional Relationships to Money: Jakob wants to invite 20 friends to his birthday, which will cost his parents $250. If he decides to invite 15 friends instead, how much money will it cost his parents? Assume the relationship is directly proportional.
68. Applying Percentages to Money: Retta put $100.00 in a bank account that gains 20% interest annually. How much interest will be accumulated in 1 year? And if she makes no withdrawals, how much money will be in the account after 1 year?
Physical measurement word problems
Best for: 1st grade, 2nd grade, 3rd grade, 4th grade
72. Comparing Measurements: Cassandra’s ruler is 22 centimetres long. April’s ruler is 30 centimetres long. How many centimetres longer is April’s ruler?
73. Contextualizing Measurements: Picture a school bus. Which unit of measurement would best describe the length of the bus? Centimetres, metres or kilometres?
74. Adding Measurements: Micha’s dad wants to try to save money on gas, so he has been tracking how much he uses. Last year, Micha’s dad used 100 litres of gas. This year, her dad used 90 litres of gas. How much gas did he use in total for the two years?
75. Subtracting Measurements: Micha’s dad wants to try to save money on gas, so he has been tracking how much he uses. Over the past two years, Micha’s dad used 200 litres of gas. This year, he used 100 litres of gas. How much gas did he use last year?
76. Multiplying Volume and Mass: Kiera wants to make sure she has strong bones, so she drinks 2 litres of milk every week. After 3 weeks, how many litres of milk will Kiera drink?
77. Dividing Volume and Mass: Lillian is doing some gardening, so she bought 1 kilogram of soil. She wants to spread the soil evenly between her 2 plants. How much will each plant get?
78. Converting Mass: Inger goes to the grocery store and buys 3 squashes that each weigh 500 grams. How many kilograms of squash did Inger buy?
79. Converting Volume: Shad has a lemonade stand and sold 20 cups of lemonade. Each cup was 500 millilitres. How many litres did Shad sell in total?
80. Converting Length: Stacy and Milda are comparing their heights. Stacy is 1.5 meters tall. Milda is 10 centimetres taller than Stacy. What is Milda’s height in centimetres?
81. Understanding Distance and Direction: A bus leaves the school to take students on a field trip. The bus travels 10 kilometres south, 10 kilometres west, another 5 kilometres south and 15 kilometres north. To return to the school, in which direction does the bus have to travel? How many kilometres must it travel in that direction?
Ratios and percentages word problems
Best for: 4th grade, 5th grade, 6th grade
82. Finding a Missing Number: The ratio of Jenny’s trophies to Meredith’s trophies is 7:4. Jenny has 28 trophies. How many does Meredith have?
83. Finding Missing Numbers: The ratio of Jenny’s trophies to Meredith’s trophies is 7:4. The difference between the numbers is 12. What are the numbers?
84. Comparing Ratios: The school’s junior band has 10 saxophone players and 20 trumpet players. The school’s senior band has 18 saxophone players and 29 trumpet players. Which band has the higher ratio of trumpet to saxophone players?
85. Determining Percentages: Mary surveyed students in her school to find out what their favourite sports were. Out of 1,200 students, 455 said hockey was their favourite sport. What percentage of students said hockey was their favourite sport?
86. Determining Percent of Change: A decade ago, Oakville’s population was 67,624 people. Now, it is 190% larger. What is Oakville’s current population?
87. Determining Percents of Numbers: At the ice skate rental stand, 60% of 120 skates are for boys. If the rest of the skates are for girls, how many are there?
88. Calculating Averages: For 4 weeks, William volunteered as a helper for swimming classes. The first week, he volunteered for 8 hours. He volunteered for 12 hours in the second week, and another 12 hours in the third week. The fourth week, he volunteered for 9 hours. For how many hours did he volunteer per week, on average?
Probability and data relationships word problems
Best for: 4th grade, 5th grade, 6th grade, 7th grade
89. Understanding the Premise of Probability: John wants to know his class’s favourite TV show, so he surveys all of the boys. Will the sample be representative or biased?
90. Understanding Tangible Probability: The faces on a fair number die are labelled 1, 2, 3, 4, 5 and 6. You roll the die 12 times. How many times should you expect to roll a 1?
91. Exploring Complementary Events: The numbers 1 to 50 are in a hat. If the probability of drawing an even number is 25/50, what is the probability of NOT drawing an even number? Express this probability as a fraction.
92. Exploring Experimental Probability: A pizza shop has recently sold 15 pizzas. 5 of those pizzas were pepperoni. Answering with a fraction, what is the experimental probability that he next pizza will be pepperoni?
93. Introducing Data Relationships: Maurita and Felice each take 4 tests. Here are the results of Maurita’s 4 tests: 4, 4, 4, 4. Here are the results for 3 of Felice’s 4 tests: 3, 3, 3. If Maurita’s mean for the 4 tests is 1 point higher than Felice’s, what’s the score of Felice’s 4th test?
94. Introducing Proportional Relationships: Store A is selling 7 pounds of bananas for $7.00. Store B is selling 3 pounds of bananas for $6.00. Which store has the better deal?
95. Writing Equations for Proportional Relationships: Lionel loves soccer, but has trouble motivating himself to practice. So, he incentivizes himself through video games. There is a proportional relationship between the amount of drills Lionel completes, in x , and for how many hours he plays video games, in y . When Lionel completes 10 drills, he plays video games for 30 minutes. Write the equation for the relationship between x and y .
Geometry word problems
Best for: 4th grade, 5th grade, 6th grade, 7th grade, 8th grade
96. Introducing Perimeter: The theatre has 4 chairs in a row. There are 5 rows. Using rows as your unit of measurement, what is the perimeter?
97. Introducing Area: The theatre has 4 chairs in a row. There are 5 rows. How many chairs are there in total?
98. Introducing Volume: Aaron wants to know how much candy his container can hold. The container is 20 centimetres tall, 10 centimetres long and 10 centimetres wide. What is the container’s volume?
99. Understanding 2D Shapes: Kevin draws a shape with 4 equal sides. What shape did he draw?
100. Finding the Perimeter of 2D Shapes: Mitchell wrote his homework questions on a piece of square paper. Each side of the paper is 8 centimetres. What is the perimeter?
101. Determining the Area of 2D Shapes: A single trading card is 9 centimetres long by 6 centimetres wide. What is its area?
102. Understanding 3D Shapes: Martha draws a shape that has 6 square faces. What shape did she draw?
103. Determining the Surface Area of 3D Shapes: What is the surface area of a cube that has a width of 2cm, height of 2 cm and length of 2 cm?
104. Determining the Volume of 3D Shapes: Aaron’s candy container is 20 centimetres tall, 10 centimetres long and 10 centimetres wide. Bruce’s container is 25 centimetres tall, 9 centimetres long and 9 centimetres wide. Find the volume of each container. Based on volume, whose container can hold more candy?
105. Identifying Right-Angled Triangles: A triangle has the following side lengths: 3 cm, 4 cm and 5 cm. Is this triangle a right-angled triangle?
106. Identifying Equilateral Triangles: A triangle has the following side lengths: 4 cm, 4 cm and 4 cm. What kind of triangle is it?
107. Identifying Isosceles Triangles: A triangle has the following side lengths: 4 cm, 5 cm and 5 cm. What kind of triangle is it?
108. Identifying Scalene Triangles: A triangle has the following side lengths: 4 cm, 5 cm and 6 cm. What kind of triangle is it?
109. Finding the Perimeter of Triangles: Luigi built a tent in the shape of an equilateral triangle. The perimeter is 21 metres. What is the length of each of the tent’s sides?
110. Determining the Area of Triangles: What is the area of a triangle with a base of 2 units and a height of 3 units?
111. Applying Pythagorean Theorem: A right triangle has one non-hypotenuse side length of 3 inches and the hypotenuse measures 5 inches. What is the length of the other non-hypotenuse side?
112. Finding a Circle’s Diameter: Jasmin bought a new round backpack. Its area is 370 square centimetres. What is the round backpack’s diameter?
113. Finding a Circle's Area: Captain America’s circular shield has a diameter of 76.2 centimetres. What is the area of his shield?
114. Finding a Circle’s Radius: Skylar lives on a farm, where his dad keeps a circular corn maze. The corn maze has a diameter of 2 kilometres. What is the maze’s radius?
Variables word problems
Best for: 6th grade, 7th grade, 8th grade
115. Identifying Independent and Dependent Variables: Victoria is baking muffins for her class. The number of muffins she makes is based on how many classmates she has. For this equation, m is the number of muffins and c is the number of classmates. Which variable is independent and which variable is dependent?
116. Writing Variable Expressions for Addition: Last soccer season, Trish scored g goals. Alexa scored 4 more goals than Trish. Write an expression that shows how many goals Alexa scored.
117. Writing Variable Expressions for Subtraction: Elizabeth eats a healthy, balanced breakfast b times a week. Madison sometimes skips breakfast. In total, Madison eats 3 fewer breakfasts a week than Elizabeth. Write an expression that shows how many times a week Madison eats breakfast.
118. Writing Variable Expressions for Multiplication: Last hockey season, Jack scored g goals. Patrik scored twice as many goals than Jack. Write an expression that shows how many goals Patrik scored.
119. Writing Variable Expressions for Division: Amanda has c chocolate bars. She wants to distribute the chocolate bars evenly among 3 friends. Write an expression that shows how many chocolate bars 1 of her friends will receive.
120. Solving Two-Variable Equations: This equation shows how the amount Lucas earns from his after-school job depends on how many hours he works: e = 12h . The variable h represents how many hours he works. The variable e represents how much money he earns. How much money will Lucas earn after working for 6 hours?
How to easily make your own math word problems & word problems worksheets
Armed with 120 examples to spark ideas, making your own math word problems can engage your students and ensure alignment with lessons. Do:
- Link to Student Interests: By framing your word problems with student interests, you’ll likely grab attention. For example, if most of your class loves American football, a measurement problem could involve the throwing distance of a famous quarterback.
- Make Questions Topical: Writing a word problem that reflects current events or issues can engage students by giving them a clear, tangible way to apply their knowledge.
- Include Student Names: Naming a question’s characters after your students is an easy way make subject matter relatable, helping them work through the problem.
- Be Explicit: Repeating keywords distills the question, helping students focus on the core problem.
- Test Reading Comprehension: Flowery word choice and long sentences can hide a question’s key elements. Instead, use concise phrasing and grade-level vocabulary.
- Focus on Similar Interests: Framing too many questions with related interests -- such as football and basketball -- can alienate or disengage some students.
- Feature Red Herrings: Including unnecessary information introduces another problem-solving element, overwhelming many elementary students.
A key to differentiated instruction , word problems that students can relate to and contextualize will capture interest more than generic and abstract ones.
Final thoughts about math word problems
You’ll likely get the most out of this resource by using the problems as templates, slightly modifying them by applying the above tips. In doing so, they’ll be more relevant to -- and engaging for -- your students.
Regardless, having 120 curriculum-aligned math word problems at your fingertips should help you deliver skill-building challenges and thought-provoking assessments.
A greater understanding of how your students process content and demonstrate understanding, informing your ongoing teaching approach.
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Open ended math questions and problems for elementary students.
Does your current math instruction involve only situations where there is one answer? Are students expected to solve problems following rigid procedures that do not require critical or creative thinking? These components are important; however, it’s time to take your math instruction to the next level! The answer: open ended math questions!
Open ended math questions, also known as open ended math problems, help learners grow into true mathematicians who use diverse problem solving strategies to explore mathematical situations where there isn’t necessarily one “right” answer. It equips them with the critical thinking skills they need to solve real world problems in the twenty-first century.
This blog post will answer the following questions:
- What is an open ended math question?
- What are the differences between open-ended and closed-ended problems in math?
- Why should I implement open ended questions in my classroom?
- What are the disadvantages of using open-ended math problems?
- How do I implement open ended math questions in my classroom?
- How do I create open ended math questions?
- What are some examples of open ended math problems?
- How do I grade open ended math tasks?
What is an Open Ended Math Question?
An open ended math question (which is known as an open ended math problem or open ended math task) is a real world math situation presented to students in a word problem format where there is more than one solution, approach, and representation. This instructional strategy is more than reciting a fact or repeating a procedure. It requires students to apply what they have learned while using their problem solving, reasoning, critical thinking, and communication skills to solve a given problem.
This strategy naturally allows for differentiation because of its open-ended nature. In addition, it is a valuable formative assessment tool that allows teachers to assess accuracy in computation and abilities to think of and flexibly apply more than one strategy. In addition to the teacher being able to learn about their students from this tool, the students can thoughtfully extend their learning and reflect on their own thinking through whole group discussions or partner talks.
What are the Differences Between Open-Ended and Closed-Ended Problems in Math?
The major difference between open-ended math problems and closed-ended math problems is that close-ended ones have one answer and open-ended ones have more than one answer. This simple difference creates a very different learning experience for elementary students when they work on solving the problem.
What are the Advantages and Disadvantages of Open Ended Math Problems?
Advantages of open ended math problems.
There are many benefits to using open ended math questions in your classroom. This list of advantages of open ended questions will help you understand their ability to transform your math block! Here are 8 advantages to using open ended math tasks:
- Provides valuable and specific information to the teacher about student understanding and application of learning
- Allows the teacher to assess accuracy in computation and abilities to think of and flexibly apply more than one strategy
- Permits the teacher to see flexibility in student thinking
- Gives students the opportunity to practice and fine tune their problems solving, reasoning, critical thinking, and communication skills
- Creates opportunities for real-world application of math
- Empowers students to extend their learning and reflect on their thinking
- Fosters creativity, collaboration, and engagement in students
- Facilitates a differentiated learning experience where all students can access the task
Disadvantages of Open Ended Math Problems
Although there are tons of benefits to using open ended math problems in your classroom, it is important to note that there are some disadvantages. Here are 3 disadvantages using open ended math tasks:
- Increases time in collecting data
- Provides a higher complexity of data
- Requires the implementation and practice of routines
3 Ways to Implement Open Ended Problems During Your Math Block
Here are 3 ways you can implement open ended problems in your elementary classroom:
- Start a lesson with an open-ended math problems for students to solve independently. Invite them to share their work and reasoning with a partner. Ask a few students to share their ideas with the whole class.
- Use the open-ended math problems for fast finishers . If a student or a group of students tend to finish independent work before the rest of the class, invite them to work on an open-ended math problem.
- Utilize open-ended math problems as a center during math workshop . You will not have to worry about students finishing that math center before it is time to switch to the next center.
3 Ways to Write Open Ended Math Questions with Examples
Here are 3 ways to create open ended math questions accompanied with easy-to-understand open ended math problems examples:
- Start with a Closed-Ended Question. For example, a closed-ended question could be: What is the sum of 10 plus 10? The related open-ended question would be: The sum is 20. What could the addends be? There are an infinite number of responses because students could use negative numbers.
- Ask Students to Explain, Prove, or Justify their Thinking. An example of this is, “Prove 5 + 6 = 11.” One possible student response could be that they know the sum is 11 because of the doubles + 1 rule. Another student may take out counters, while another draws a picture.
- Invite Students to Compare 2 Concepts. For example, ask students to identify the similarities and differences between 2D and 3D shapes. Some possible responses for similarities are that they are both geometry concepts and classifications of shapes. A difference they could say is that 2D shapes are flat, while a 3D shape is solid.
How do you Grade Open-Ended Math Questions?
Grading open-ended math tasks is not as clear cut as closed-ended questions. If you are using the task as a formative assessment for your own planning purposes, then you have flexibility on how you choose to evaluate students’ work. However, I recommend you use a rubric if you plan to use it as a summative assessment. Remember to share the rubric with students so that the expectations are clear.
Get Our Open-Ended Math Prompts
Math resources for 1st-5th grade teachers.
If you need printable and digital math resources for your classroom, then check out my time and money-saving math collections below!
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We would love for you to try these open ended question resources with your students. They offer students daily opportunities to practice solving open ended problems. You can download worksheets specific to your grade level (along with lots of other math freebies) in our free printable math resources bundle using this link: free printable math worksheets for elementary teachers .
Check out my daily open ended problem resources !
- 1st Grade Open Ended Math Question Problems
- 2nd Grade Open Ended Math Question Problems
- 3rd Grade Open Ended Math Question Problems
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Mathematical Word Problems
Math word problems is one of the most complex parts of the elementary math curriculum since translating text into symbolic math is required to solve the problem. Because the Wolfram Language has powerful symbolic computation ability, Wolfram|Alpha can interpret basic mathematical word problems and give descriptive results.
Solve a word problem and explore related facts.
Solve a word problem:
Once you know your basic operations ( addition , subtraction , multiplication , division ), you will encounter story problems, also known as word problems, which require you to read a problem and decide which operation to perform in order to get the answer. There are key words here that often indicate which operation you will use. We will give you a list of them, but remember that for many problems, there may not be a key word, and you’ll have to use your best judgment in order to figure out what to do!
Here are the key words:
In addition to, sum, total, more than, altogether, in all, combined, extra, raise, plus, both, additional
How many more, difference, how many less, fewer, left (sometimes, left over), change, lost, decreased (by), less, remain, take away
How many times, times, multiplied by, of, every, product, by, twice as much, three times as much (and so on), rate, at this rate, doubled, tripled (and so on), in all
How much/many will each receive?, divided among, split up between, per, ratio, percent, each, divide (or split) evenly, cut, average, share, quotient, equally (split, divided, etc)
For telling that something equals another amount:
In order to solve a story (word) problem successfully:
- Read the entire problem thoroughly
- Make a list of the numerical (number) information you’ll need. If the numbers have units attached (for example, 12 inches), make sure you attach units in your list so you don’t get confused.
- Write out the number equation you’ll need to solve.
- Complete the solving process carefully.
- When you get your answer, reread the problem and ask yourself, “Does this answer make sense?”
- Remember to label your answer with the correct units, if needed.
Example Story Problems
In this section, we’ll give you several examples of story (word) problems, starting with simple problems and working up towards more complex problems.
Nick had 8 toy trucks in his toy box. His friend Nathan brought over 3 more toy trucks. How many toy trucks did the boys have altogether?
What is the key word in this problem?
If you look back at the list of key words at the top of the page, you’ll find that altogether listed as a key word.
Altogether is our key word. Now, what operation will we have to perform to get the answer to this problem?
We know we’ll have to do addition, because altogether is a key word that means adding.
Now, what will our problem look like?
We know we’ll be adding together 8 + 3, because those were the two numbers mentioned in the problem.
What is 8 + 3?
Therefore, our answer is 11 toy trucks altogether.
Now, let’s try another one.
John had 15 books on his bookshelf. John’s dog, Buster, came in and slobbered all over four of them. How many books did John have left that were not slobbered on?
If you look back at the list of key words at the top of the page, you’ll find that left is listed as a key word.
Left is our key word. Now, what operation will we have to perform to get the answer to this problem?
We know we’ll have to do subtraction, because left is a key word that means subtract.
This problem is a subtraction problem. Now, let’s get it set up. How will this problem look?
We know we’ll be subtracting 15 – 4, because those were the two numbers mentioned in the problem.
Now, perform the subtraction. What is 15 – 4?
Our final answer is 11 books.
Now, let’s try a couple harder problems.
Dan is getting ready to go to a concert. He wants to figure out how many people will be there. He knows that there are 250 rows of seats, and each row has 40 seats in it. How many seats are there in the concert hall in all?
If you look back at the list of key words at the top of the page, you’ll find that in all is listed as a key word.
In all is our key word. Now, what operation will we have to perform to get the answer to this problem?
We choose multiplication because we see the keyword in all, but also because it makes sense. Essentially this may be seen as an addition problem, which is why the keyword is also in the addition section, but since the adding of the rows would all be the same, we can multiply to make the process faster.
This problem is a multiplication problem. Now, let’s get it set up. How will this problem look?
We would use 250 x 40 because we decided that this is a multiplication problem. Since we want to figure out a total number of seats in the hall, we’re going to multiply the two given numbers together, as if we were calculating area.
Now, perform the multiplication. What is 250 x 40?
Our final answer is that there are 10,000 seats at the concert Dan is attending.
Let’s look at one more example. Three friends go out to dinner. Near the end, they get the bill and they owe the restaurant $27.89. They want to split the bill evenly between the three of them. How much will each person pay?
If you look back at the list of key words at the top of the page, you’ll find that split evenly is listed as a key word.
Evenly is our key word. Now, what operation will we have to perform to get the answer to this problem?
We choose division because we see the keywords “split” and “evenly.” Also, division makes sense because they want to divide the bill between three people. Because they’re splitting it up, we would choose division.
This problem is a division problem. Now, let’s get it set up. What would our equation be?
We choose $27.89 / 3 because we know we have to split up the amount of money, $27.89, between the three friends, so we know we have to divide by three.
Now, perform the division. What is $27.89 / 3? (Round to the nearest cent)
When we divide, we get an answer of 9.2966666, with a repeating 6 at the end. We want to round it to the nearest cent, which is the hundredths place after the decimal. We see that that number is already a 9, and a 6 after it means round up. However, we can’t make one place value a ten, so we increase the tenths digit by one, turning the 2 into a 3. If this doesn’t make sense, please read rounding numbers . Thus, your final answer is $9.30 after rounding.
Now, let’s go through some harder story (word) problems. All of the story problems we’ve done have had only one step, and we’ve been able to easily decide if they are addition, subtraction, multiplication, or division. However, some story problems have more than one step, involving more than one key word and/or operation. We’ll show you a few of these now.
Carly is making a dress. She needs 1 yard of yellow fabric, 1.5 yards of purple fabric, and .5 yards of green fabric. Yellow fabric costs $5.95 per yard, purple fabric costs $3.95 per yard, and green fabric is on sale for $7.00 per yard. How much will she spend in all if she buys just enough fabric to make her dress? (Ignore tax in your calculations). Click Next Step for the first part of the solution.
First, we have to figure out how much Carly is spending on each amount of fabric; then, we can use the key word “in all” which tells us that we need to add the amounts together for a final total. In order to figure out how much each piece of fabric costs, we need to multiply the price by the amount she needs to get a total.
First, let’s calculate the yellow fabric cost. She needs one yard, and it costs $5.95 per yard, so she’ll be spending $5.95.
Now, let’s calculate the purple fabric cost. She needs 1.5 yards, and it costs $3.95 per yard; therefore we have to multiply 1.5 times $3.95, which comes out to be $5.93 (we round to the nearest cent).
Finally, let’s calculate the green fabric cost. The green fabric is on sale for $7.00 per yard, and she needs .5 yards of it, so we multiply $7.00 times .5 and get $3.50.
Now, we have three money amounts (one for each color fabric) that we can now add together to get a total amount that Carly will spend. We know that we have to add these amounts together, like this:
$5.95 + $5.93 + $3.50 = $15.38
Thus, our final answer is that Carly will spend $15.38 on fabric for her dress.
Now, we’ll give you one to practice on. John is planning to carpet three rooms in his house. One room is 15 by 12 feet, one room is 17 by 14 feet, and the last room is 10 by 12 feet. John has 130 square feet of carpeting already. How much more carpeting does he need in order to carpet all three rooms?
First, you have to figure out how many square feet he has to carpet overall. That means we need to figure out the area of each room, and add those together. We multiply the dimensions together as follows: 15 x 12 = 180 ft 2 17 x 14 = 238 ft 2 10 x 12 = 120 ft 2
Now, we have the area of each floor he has to carpet, so we can add these all together to find out the total amount of carpeting he needs.
180 + 238 + 120 = 538 ft 2 . This is the total amount John will need. However, the problem said that he already has 130 ft 2 of carpet, so we need to figure out how much more he needs. Therefore, we need to subtract 130 from 538, and we get 408 ft 2 leftover. This is how much more carpeting John will need to finish off his three rooms.
Final answer: 408 ft 2 .
The Smith family is going to take a vacation to Florida. They live in Illinois, and have figured out that the trip is 1,150 miles from their house to the hotel in Florida. They get 28 miles per gallon of gas, and plan on travelling at an average rate of 60 miles per hour. Gas costs about $2.89 per gallon.
a) How long will it take them to get to Florida? (in hours)
For this part, you divide the total miles (1,150) by the speed they’re travelling (60 mph) and you would get 19.16666 (repeating). You would round the answer to 19.2 hours.
Final answer: 19.2 hours.
b) How much money should they leave for gasoline (going one way)?
First, you would divide the total number of miles (1,150) by the amount of miles they get per gallon of gas (28); this gives you 41 gallons—the total amount needed for the trip. Then, you would multiply the number of gallons (41) by the cost per gallon of gas ($2.89) and round to the nearest cent, which gives you $118.49. This is the amount they should save for gas going one way.
Final answer: $118.49
- Belinda wants to invest $1,000. The table below shows the value of her investment under two different options for three different years
- A biologist is studying the growth of a particular species of algae. She writes the following equation to show the radius of the algae, f(d), in mm, after d days
- Chris decides to but 102 feet of 3 yards. What is the cost?
- Measure of a non-negative measurable function
Mathematical Reasoning & Problem Solving
In this lesson, we’ll discuss mathematical reasoning and methods of problem solving with an eye toward helping your students make the best use of their reasoning skills when it comes to tackling complex problems.
- Over the course of the previous lesson, we reviewed some basics about chance and probability, as well as some basics about sampling, surveys, etc. We also covered some ideas about data sets, how they’re represented, and how to interpret the results.
Approaches to Problem Solving
When solving a mathematical problem, it is very common for a student to feel overwhelmed by the information or lack a clear idea about how to get started.
To help the students with their problem-solving “problem,” let’s look at some examples of mathematical problems and some general methods for solving problems:
Identify the following four-digit number when presented with the following information:
- One of the four digits is a 1.
- The digit in the hundreds place is three times the digit in the thousands place.
- The digit in the ones place is four times the digit in the ten’s place.
- The sum of all four digits is 13.
- The digit 2 is in the thousands place.
Help your students identify and prioritize the information presented.
In this particular example, we want to look for concrete information. Clue #1 tells us that one digit is a 1, but we’re not sure of its location, so we see if we can find a clue with more concrete information.
We can see that clue #5 gives us that kind of information and is the only clue that does, so we start from there.
Because this clue tells us that the thousands place digit is 2, we search for clues relevant to this clue. Clue #2 tells us that the digit in the hundreds place is three times that of the thousands place digit, so it is 6.
So now we need to find the tens and ones place digits, and see that clue #3 tells us that the digit in the ones place is four times the digit in the tens place. But we remember that clue #1 tells us that there’s a one somewhere, and since one is not four times any digit, we see that the one must be in the tens place, which leads us to the conclusion that the digit in the ones place is four. So then we conclude that our number is:
If you were following closely, you would notice that clue #4 was never used. It is a nice way to check our answer, since the digits of 2614 do indeed add up to be thirteen, but we did not need this clue to solve the problem.
Recall that the clues’ relevance were identified and prioritized as follows:
- clue #3 and clue #1
By identifying and prioritizing information, we were able to make the information given in the problem seem less overwhelming. We ordered the clues by relevance, with the most relevant clue providing us with a starting point to solve the problem. This method also utilized the more general method of breaking a problem into smaller and simpler parts to make it easier to solve.
Now let’s look at another mathematical problem and another general problem-solving method to help us solve it:
Two trees with heights of 20 m and 30 m respectively have ropes running from the top of each tree to the bottom of the other tree. The trees are 40 meters apart. We’ll assume that the ropes are pulled tight enough that we can ignore any bending or drooping. How high above the ground do the ropes intersect?
Let’s solve this problem by representing it in a visual way , in this case, a diagram:
You can see that we have a much simpler problem on our hands after drawing the diagram. A, B, C, D, E, and F are vertices of the triangles in the diagram. Now also notice that:
b = the base of triangle EFA
h = the height of triangle EFA and the height above the ground at which the ropes intersect
If we had not drawn this diagram, it would have been very hard to solve this problem, since we need the triangles and their properties to solve for h. Also, this diagram allows us to see that triangle BCA is similar to triangle EFC, and triangle DCA is similar to triangle EFA. Solving for h shows that the ropes intersect twelve meters above the ground.
Students frequently complain that mathematics is too difficult for them, because it is too abstract and unapproachable. Explaining mathematical reasoning and problem solving by using a variety of methods , such as words, numbers, symbols, charts, graphs, tables, diagrams, and concrete models can help students understand the problem better by making it more concrete and approachable.
Let’s try another one.
Given a pickle jar filled with marbles, about how many marbles does the jar contain?
Problems like this one require the student to make and use estimations . In this case, an estimation is all that is required, although, in more complex problems, estimates may help the student arrive at the final answer.
How would a student do this? A good estimation can be found by counting how many marbles are on the base of the jar and multiplying that by the number of marbles that make up the height of the marbles in the jar.
Now to make sure that we understand when and how to use these methods, let’s solve a problem on our own:
How many more faces does a cube have than a square pyramid?
The answer is B. To see how many more faces a cube has than a square pyramid, it is best to draw a diagram of a square pyramid and a cube:
From the diagrams above, we can see that the square pyramid has five faces and the cube has six. Therefore, the cube has one more face, so the answer is B.
Before we start having the same problem our model student in the beginning did—that is, being overwhelmed with too much information—let’s have a quick review of all the problem-solving methods we’ve discussed so far:
- Sort and prioritize relevant and irrelevant information.
- Represent a problem in different ways, such as words, symbols, concrete models, and diagrams.
- Generate and use estimations to find solutions to mathematical problems.
Along with learning methods and tools for solving mathematical problems, it is important to recognize and avoid ways to make mathematical errors. This section will review some common errors.
These involve drawing a conclusion from a premise that is itself dependent on the conclusion. In other words, you are not actually proving anything. Circular reasoning often looks like deductive reasoning, but a quick examination will reveal that it’s far from it. Consider the following argument:
- Premise: Only an untrustworthy man would become an insurance salesman; the fact that insurance salesmen cannot be trusted is proof of this.
- Conclusion: Therefore, insurance salesmen cannot be trusted.
While this may be a simplistic example, you can see that there’s no logical procession in a circular argument.
Assuming the Truth of the Converse
Simply put: The fact that A implies B doesn’t not necessarily mean that B implies A. For example, “All dogs are mammals; therefore, all mammals are dogs.”
Assuming the Truth of the Inverse
Watch out for this one. You cannot automatically assume the inverse of a given statement is true. Consider the following true statement:
If you grew up in Minnesota , you’ve seen snow.
Now, notice that the inverse of this statement is not necessarily true:
If you didn’t grow up in Minnesota , you’ve never seen snow.
This mistake (also known as inductive fallacy) can take many forms, the most common being assuming a general rule based on a specific instance: (“Bridge is a hard game; therefore, all card games are difficult.”) Be aware of more subtle forms of faulty generalizations.
It’s a mistake to assume that because two things are alike in one respect that they are necessarily alike in other ways too. Consider the faulty analogy below:
People who absolutely have to have a cup of coffee in the morning to get going are as bad as alcoholics who can’t cope without drinking.
False (or tenuous) analogies are often used in persuasive arguments.
Now that we’ve gone over some common mathematical mistakes, let’s look at some correct and effective ways to use mathematical reasoning.
Let’s look at basic logic, its operations, some fundamental laws, and the rules of logic that help us prove statements and deduce the truth. First off, there are two different styles of proofs: direct and indirect .
Whether it’s a direct or indirect proof, the engine that drives the proof is the if-then structure of a logical statement. In formal logic, you’ll see the format using the letters p and q, representing statements, as in:
If p, then q
An arrow is used to indicate that q is derived from p, like this:
This would be the general form of many types of logical statements that would be similar to: “if Joe has 5 cents, then Joe has a nickel or Joe has 5 pennies “. Basically, a proof is a flow of implications starting with the statement p and ending with the statement q. The stepping stones we use to link these statements in a logical proof on the way are called axioms or postulates , which are accepted logical tools.
A direct proof will attempt to lay out the shortest number of steps between p and q.
The goal of an indirect proof is exactly the same—it wants to show that q follows from p; however, it goes about it in a different manner. An indirect proof also goes by the names “proof by contradiction” or reductio ad absurdum . This type of proof assumes that the opposite of what you want to prove is true, and then shows that this is untenable or absurd, so, in fact, your original statement must be true.
Let’s see how this works using the isosceles triangle below. The indirect proof assumption is in bold.
Given: Triangle ABC is isosceles with B marking the vertex
Prove: Angles A and C are congruent.
Now, let’s work through this, matching our statements with our reasons.
- Triangle ABC is isosceles . . . . . . . . . . . . Given
- Angle A is the vertex . . . . . . . . . . . . . . . . Given
- Angles A and C are not congruent . . Indirect proof assumption
- Line AB is equal to line BC . . . . . . . . . . . Legs of an isosceles triangle are congruent
- Angles A and C are congruent . . . . . . . . The angles opposite congruent sides of a triangle are congruent
- Contradiction . . . . . . . . . . . . . . . . . . . . . . Angles can’t be congruent and incongruent
- Angles A and C are indeed congruent . . . The indirect proof assumption (step 3) is wrong
- Therefore, if angles A and C are not incongruent, they are congruent.
“Always, Sometimes, and Never”
Some math problems work on the mechanics that statements are “always”, “sometimes” and “never” true.
Example: x < x 2 for all real numbers x
We may be tempted to say that this statement is “always” true, because by choosing different values of x, like -2 and 3, we see that:
Example: For all primes x ≥ 3, x is odd.
This statement is “always” true. The only prime that is not odd is two. If we had a prime x ≥ 3 that is not odd, it would be divisible by two, which would make x not prime.
- Know and be able to identify common mathematical errors, such as circular arguments, assuming the truth of the converse, assuming the truth of the inverse, making faulty generalizations, and faulty use of analogical reasoning.
- Be familiar with direct proofs and indirect proofs (proof by contradiction).
- Be able to work with problems to identify “always,” “sometimes,” and “never” statements.
K-5 Math Centers
K-5 math ideas, 3rd grade math, need help organizing your k-5 math block, 5 ways to include math problem solving activities in your classroom.
Are you looking for math problem solving activities that are not too easy and not too hard, but juuust right? I’ve got something just for you and your students.
Solve and Explain Problem Solving Tasks are open-ended math tasks that provide just the right amount of challenge for your kids. Here’s a little more about them.
Open-ended math problem solving tasks:
- promote multiple solution paths and/or multiple solutions
- boost critical thinking and math reasoning skills
- increase opportunities for developing perseverance
- provide opportunities to justify answer choices
- strengthen kids written and oral communication skills
What Makes These So Great?
- All Common Core Standards are covered for your grade level
- 180+ Quality questions that are rigorous yet engaging
- They are SUPER easy to assemble
- Provide opportunities for meaningful math discussions
- Perfect for developing a growth mindset
- Easily identify student misconceptions so you can provide assistance
- Very versatile (check out the different ways to use them below)
You can find out more details for your grade level by clicking on the buttons below.
I’m sure you really want to know how can you use these with your kids. Check out the top 5 ideas on how to use Solve and Explain Problem Solving Tasks in your classroom.
How and When Can I Use Them?
Solve and Explain Tasks Cards are very versatile. You can use them for:
- Math Centers – This is my favorite way to use these! Depending on your grade level, there are at least two (Kinder – 2nd) or three (3rd-5th) tasks types per Common Core standard. And each task type has 6 different questions. Print out each of the different tasks types on different color paper. Then, let students choose which one question from each task type they want to solve.
- Problem of the Day – Use them as a daily math journal prompt. Print out the recording sheet and project one of the problems on your white board or wall. Students solve the problem and then glue it in their spiral or composition notebooks.
- Early Finisher Activities -No more wondering what to do next!Create an early finishers notebook where students can grab a task and a recording sheet. Place the cards in sheet protectors and make copies of the Early Finisher Activity Check-Off card for your kids to fill out BEFORE they pull a card out to work on. We want to make sure kids are not rushing through there first assignment before moving on to an early finisher activity.
- Weekly Math Challenges – Kids LOVE challenges! Give students copies of one of the problems for homework. Then give them a week to complete it. Since many of the questions have multiple solutions and students have to explain how they got their answers, you can have a rich whole group discussion at the end of the week (even with your kindergarten and 1st grade students).
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- Formative Assessments – Give your students a problem to solve. Then use the Teacher Scoring Rubric to see how your kids are doing with each standard. Since they have to explain their thinking, this is a great way to catch any misconceptions and give feedback to individual students.
So this wraps up the top 5 ways that you can use problem solving tasks in your classroom. Click your grade level below to get Solve and Explain problem solving tasks for your classroom.
- Read more about: K-5 Math Ideas
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- 3.1 Use a Problem-Solving Strategy
- 1.1 Introduction to Whole Numbers
- 1.2 Use the Language of Algebra
- 1.3 Add and Subtract Integers
- 1.4 Multiply and Divide Integers
- 1.5 Visualize Fractions
- 1.6 Add and Subtract Fractions
- 1.7 Decimals
- 1.8 The Real Numbers
- 1.9 Properties of Real Numbers
- 1.10 Systems of Measurement
- Key Concepts
- Review Exercises
- Practice Test
- 2.1 Solve Equations Using the Subtraction and Addition Properties of Equality
- 2.2 Solve Equations using the Division and Multiplication Properties of Equality
- 2.3 Solve Equations with Variables and Constants on Both Sides
- 2.4 Use a General Strategy to Solve Linear Equations
- 2.5 Solve Equations with Fractions or Decimals
- 2.6 Solve a Formula for a Specific Variable
- 2.7 Solve Linear Inequalities
- 3.2 Solve Percent Applications
- 3.3 Solve Mixture Applications
- 3.4 Solve Geometry Applications: Triangles, Rectangles, and the Pythagorean Theorem
- 3.5 Solve Uniform Motion Applications
- 3.6 Solve Applications with Linear Inequalities
- 4.1 Use the Rectangular Coordinate System
- 4.2 Graph Linear Equations in Two Variables
- 4.3 Graph with Intercepts
- 4.4 Understand Slope of a Line
- 4.5 Use the Slope–Intercept Form of an Equation of a Line
- 4.6 Find the Equation of a Line
- 4.7 Graphs of Linear Inequalities
- 5.1 Solve Systems of Equations by Graphing
- 5.2 Solve Systems of Equations by Substitution
- 5.3 Solve Systems of Equations by Elimination
- 5.4 Solve Applications with Systems of Equations
- 5.5 Solve Mixture Applications with Systems of Equations
- 5.6 Graphing Systems of Linear Inequalities
- 6.1 Add and Subtract Polynomials
- 6.2 Use Multiplication Properties of Exponents
- 6.3 Multiply Polynomials
- 6.4 Special Products
- 6.5 Divide Monomials
- 6.6 Divide Polynomials
- 6.7 Integer Exponents and Scientific Notation
- 7.1 Greatest Common Factor and Factor by Grouping
- 7.2 Factor Quadratic Trinomials with Leading Coefficient 1
- 7.3 Factor Quadratic Trinomials with Leading Coefficient Other than 1
- 7.4 Factor Special Products
- 7.5 General Strategy for Factoring Polynomials
- 7.6 Quadratic Equations
- 8.1 Simplify Rational Expressions
- 8.2 Multiply and Divide Rational Expressions
- 8.3 Add and Subtract Rational Expressions with a Common Denominator
- 8.4 Add and Subtract Rational Expressions with Unlike Denominators
- 8.5 Simplify Complex Rational Expressions
- 8.6 Solve Rational Equations
- 8.7 Solve Proportion and Similar Figure Applications
- 8.8 Solve Uniform Motion and Work Applications
- 8.9 Use Direct and Inverse Variation
- 9.1 Simplify and Use Square Roots
- 9.2 Simplify Square Roots
- 9.3 Add and Subtract Square Roots
- 9.4 Multiply Square Roots
- 9.5 Divide Square Roots
- 9.6 Solve Equations with Square Roots
- 9.7 Higher Roots
- 9.8 Rational Exponents
- 10.1 Solve Quadratic Equations Using the Square Root Property
- 10.2 Solve Quadratic Equations by Completing the Square
- 10.3 Solve Quadratic Equations Using the Quadratic Formula
- 10.4 Solve Applications Modeled by Quadratic Equations
- 10.5 Graphing Quadratic Equations
By the end of this section, you will be able to:
- Approach word problems with a positive attitude
- Use a problem-solving strategy for word problems
- Solve number problems
Be Prepared 3.1
Before you get started, take this readiness quiz.
- Translate “6 less than twice x ” into an algebraic expression. If you missed this problem, review Example 1.26 .
- Solve: 2 3 x = 24 . 2 3 x = 24 . If you missed this problem, review Example 2.16 .
- Solve: 3 x + 8 = 14 . 3 x + 8 = 14 . If you missed this problem, review Example 2.27 .
Approach Word Problems with a Positive Attitude
“If you think you can… or think you can’t… you’re right.”—Henry Ford
The world is full of word problems! Will my income qualify me to rent that apartment? How much punch do I need to make for the party? What size diamond can I afford to buy my girlfriend? Should I fly or drive to my family reunion?
How much money do I need to fill the car with gas? How much tip should I leave at a restaurant? How many socks should I pack for vacation? What size turkey do I need to buy for Thanksgiving dinner, and then what time do I need to put it in the oven? If my sister and I buy our mother a present, how much does each of us pay?
Now that we can solve equations, we are ready to apply our new skills to word problems. Do you know anyone who has had negative experiences in the past with word problems? Have you ever had thoughts like the student below?
When we feel we have no control, and continue repeating negative thoughts, we set up barriers to success. We need to calm our fears and change our negative feelings.
Start with a fresh slate and begin to think positive thoughts. If we take control and believe we can be successful, we will be able to master word problems! Read the positive thoughts in Figure 3.3 and say them out loud.
Think of something, outside of school, that you can do now but couldn’t do 3 years ago. Is it driving a car? Snowboarding? Cooking a gourmet meal? Speaking a new language? Your past experiences with word problems happened when you were younger—now you’re older and ready to succeed!
Use a Problem-Solving Strategy for Word Problems
We have reviewed translating English phrases into algebraic expressions, using some basic mathematical vocabulary and symbols. We have also translated English sentences into algebraic equations and solved some word problems. The word problems applied math to everyday situations. We restated the situation in one sentence, assigned a variable, and then wrote an equation to solve the problem. This method works as long as the situation is familiar and the math is not too complicated.
Now, we’ll expand our strategy so we can use it to successfully solve any word problem. We’ll list the strategy here, and then we’ll use it to solve some problems. We summarize below an effective strategy for problem solving.
Use a Problem-Solving Strategy to Solve Word Problems.
- Step 1. Read the problem. Make sure all the words and ideas are understood.
- Step 2. Identify what we are looking for.
- Step 3. Name what we are looking for. Choose a variable to represent that quantity.
- Step 4. Translate into an equation. It may be helpful to restate the problem in one sentence with all the important information. Then, translate the English sentence into an algebraic equation.
- Step 5. Solve the equation using good algebra techniques.
- Step 6. Check the answer in the problem and make sure it makes sense.
- Step 7. Answer the question with a complete sentence.
Pilar bought a purse on sale for $18, which is one-half of the original price. What was the original price of the purse?
Step 1. Read the problem. Read the problem two or more times if necessary. Look up any unfamiliar words in a dictionary or on the internet.
- In this problem, is it clear what is being discussed? Is every word familiar?
Step 2. Identify what you are looking for. Did you ever go into your bedroom to get something and then forget what you were looking for? It’s hard to find something if you are not sure what it is! Read the problem again and look for words that tell you what you are looking for!
- In this problem, the words “what was the original price of the purse” tell us what we need to find.
Step 3. Name what we are looking for. Choose a variable to represent that quantity. We can use any letter for the variable, but choose one that makes it easy to remember what it represents.
- Let p = p = the original price of the purse.
Step 4. Translate into an equation. It may be helpful to restate the problem in one sentence with all the important information. Translate the English sentence into an algebraic equation.
Reread the problem carefully to see how the given information is related. Often, there is one sentence that gives this information, or it may help to write one sentence with all the important information. Look for clue words to help translate the sentence into algebra. Translate the sentence into an equation.
Step 5. Solve the equation using good algebraic techniques. Even if you know the solution right away, using good algebraic techniques here will better prepare you to solve problems that do not have obvious answers.
Step 6. Check the answer in the problem to make sure it makes sense. We solved the equation and found that p = 36 , p = 36 , which means “the original price” was $36.
- Does $36 make sense in the problem? Yes, because 18 is one-half of 36, and the purse was on sale at half the original price.
Step 7. Answer the question with a complete sentence. The problem asked “What was the original price of the purse?”
- The answer to the question is: “The original price of the purse was $36.”
If this were a homework exercise, our work might look like this:
Pilar bought a purse on sale for $18, which is one-half the original price. What was the original price of the purse?
Joaquin bought a bookcase on sale for $120, which was two-thirds of the original price. What was the original price of the bookcase?
Two-fifths of the songs in Mariel’s playlist are country. If there are 16 country songs, what is the total number of songs in the playlist?
Let’s try this approach with another example.
Ginny and her classmates formed a study group. The number of girls in the study group was three more than twice the number of boys. There were 11 girls in the study group. How many boys were in the study group?
Guillermo bought textbooks and notebooks at the bookstore. The number of textbooks was 3 more than twice the number of notebooks. He bought 7 textbooks. How many notebooks did he buy?
Gerry worked Sudoku puzzles and crossword puzzles this week. The number of Sudoku puzzles he completed is eight more than twice the number of crossword puzzles. He completed 22 Sudoku puzzles. How many crossword puzzles did he do?
Solve Number Problems
Now that we have a problem solving strategy, we will use it on several different types of word problems. The first type we will work on is “number problems.” Number problems give some clues about one or more numbers. We use these clues to write an equation. Number problems don’t usually arise on an everyday basis, but they provide a good introduction to practicing the problem solving strategy outlined above.
The difference of a number and six is 13. Find the number.
The difference of a number and eight is 17. Find the number.
The difference of a number and eleven is −7 . −7 . Find the number.
The sum of twice a number and seven is 15. Find the number.
Did you notice that we left out some of the steps as we solved this equation? If you’re not yet ready to leave out these steps, write down as many as you need.
The sum of four times a number and two is 14. Find the number.
The sum of three times a number and seven is 25. Find the number.
Some number word problems ask us to find two or more numbers. It may be tempting to name them all with different variables, but so far we have only solved equations with one variable. In order to avoid using more than one variable, we will define the numbers in terms of the same variable. Be sure to read the problem carefully to discover how all the numbers relate to each other.
One number is five more than another. The sum of the numbers is 21. Find the numbers.
One number is six more than another. The sum of the numbers is twenty-four. Find the numbers.
Try It 3.10
The sum of two numbers is fifty-eight. One number is four more than the other. Find the numbers.
The sum of two numbers is negative fourteen. One number is four less than the other. Find the numbers.
Try It 3.11
The sum of two numbers is negative twenty-three. One number is seven less than the other. Find the numbers.
Try It 3.12
The sum of two numbers is −18 . −18 . One number is 40 more than the other. Find the numbers.
One number is ten more than twice another. Their sum is one. Find the numbers.
Try It 3.13
One number is eight more than twice another. Their sum is negative four. Find the numbers.
Try It 3.14
One number is three more than three times another. Their sum is −5 . −5 . Find the numbers.
Some number problems involve consecutive integers. Consecutive integers are integers that immediately follow each other. Examples of consecutive integers are:
Notice that each number is one more than the number preceding it. So if we define the first integer as n , the next consecutive integer is n + 1 . n + 1 . The one after that is one more than n + 1 , n + 1 , so it is n + 1 + 1 , n + 1 + 1 , which is n + 2 . n + 2 .
The sum of two consecutive integers is 47. Find the numbers.
Try It 3.15
The sum of two consecutive integers is 95 . 95 . Find the numbers.
Try It 3.16
The sum of two consecutive integers is −31 . −31 . Find the numbers.
Find three consecutive integers whose sum is −42 . −42 .
Try It 3.17
Find three consecutive integers whose sum is −96 . −96 .
Try It 3.18
Find three consecutive integers whose sum is −36 . −36 .
Now that we have worked with consecutive integers, we will expand our work to include consecutive even integers and consecutive odd integers. Consecutive even integers are even integers that immediately follow one another. Examples of consecutive even integers are:
Notice each integer is 2 more than the number preceding it. If we call the first one n , then the next one is n + 2 . n + 2 . The next one would be n + 2 + 2 n + 2 + 2 or n + 4 . n + 4 .
Consecutive odd integers are odd integers that immediately follow one another. Consider the consecutive odd integers 77, 79, and 81.
Does it seem strange to add 2 (an even number) to get from one odd integer to the next? Do you get an odd number or an even number when we add 2 to 3? to 11? to 47?
Whether the problem asks for consecutive even numbers or odd numbers, you don’t have to do anything different. The pattern is still the same—to get from one odd or one even integer to the next, add 2.
Find three consecutive even integers whose sum is 84.
Try It 3.19
Find three consecutive even integers whose sum is 102.
Try It 3.20
Find three consecutive even integers whose sum is −24 . −24 .
A married couple together earns $110,000 a year. The wife earns $16,000 less than twice what her husband earns. What does the husband earn?
Try It 3.21
According to the National Automobile Dealers Association, the average cost of a car in 2014 was $28,500. This was $1,500 less than 6 times the cost in 1975. What was the average cost of a car in 1975?
Try It 3.22
U.S. Census data shows that the median price of new home in the United States in November 2014 was $280,900. This was $10,700 more than 14 times the price in November 1964. What was the median price of a new home in November 1964?
Section 3.1 Exercises
Practice makes perfect.
Use the Approach Word Problems with a Positive Attitude
In the following exercises, prepare the lists described.
List five positive thoughts you can say to yourself that will help you approach word problems with a positive attitude. You may want to copy them on a sheet of paper and put it in the front of your notebook, where you can read them often.
List five negative thoughts that you have said to yourself in the past that will hinder your progress on word problems. You may want to write each one on a small piece of paper and rip it up to symbolically destroy the negative thoughts.
In the following exercises, solve using the problem solving strategy for word problems. Remember to write a complete sentence to answer each question.
Two-thirds of the children in the fourth-grade class are girls. If there are 20 girls, what is the total number of children in the class?
Three-fifths of the members of the school choir are women. If there are 24 women, what is the total number of choir members?
Zachary has 25 country music CDs, which is one-fifth of his CD collection. How many CDs does Zachary have?
One-fourth of the candies in a bag of M&M’s are red. If there are 23 red candies, how many candies are in the bag?
There are 16 girls in a school club. The number of girls is four more than twice the number of boys. Find the number of boys.
There are 18 Cub Scouts in Pack 645. The number of scouts is three more than five times the number of adult leaders. Find the number of adult leaders.
Huong is organizing paperback and hardback books for her club’s used book sale. The number of paperbacks is 12 less than three times the number of hardbacks. Huong had 162 paperbacks. How many hardback books were there?
Jeff is lining up children’s and adult bicycles at the bike shop where he works. The number of children’s bicycles is nine less than three times the number of adult bicycles. There are 42 adult bicycles. How many children’s bicycles are there?
Philip pays $1,620 in rent every month. This amount is $120 more than twice what his brother Paul pays for rent. How much does Paul pay for rent?
Marc just bought an SUV for $54,000. This is $7,400 less than twice what his wife paid for her car last year. How much did his wife pay for her car?
Laurie has $46,000 invested in stocks and bonds. The amount invested in stocks is $8,000 less than three times the amount invested in bonds. How much does Laurie have invested in bonds?
Erica earned a total of $50,450 last year from her two jobs. The amount she earned from her job at the store was $1,250 more than three times the amount she earned from her job at the college. How much did she earn from her job at the college?
In the following exercises, solve each number word problem.
The sum of a number and eight is 12. Find the number.
The sum of a number and nine is 17. Find the number.
The difference of a number and 12 is three. Find the number.
The difference of a number and eight is four. Find the number.
The sum of three times a number and eight is 23. Find the number.
The sum of twice a number and six is 14. Find the number.
The difference of twice a number and seven is 17. Find the number.
The difference of four times a number and seven is 21. Find the number.
Three times the sum of a number and nine is 12. Find the number.
Six times the sum of a number and eight is 30. Find the number.
One number is six more than the other. Their sum is 42. Find the numbers.
One number is five more than the other. Their sum is 33. Find the numbers.
The sum of two numbers is 20. One number is four less than the other. Find the numbers.
The sum of two numbers is 27. One number is seven less than the other. Find the numbers.
The sum of two numbers is −45 . −45 . One number is nine more than the other. Find the numbers.
The sum of two numbers is −61 . −61 . One number is 35 more than the other. Find the numbers.
The sum of two numbers is −316 . −316 . One number is 94 less than the other. Find the numbers.
The sum of two numbers is −284 . −284 . One number is 62 less than the other. Find the numbers.
One number is 14 less than another. If their sum is increased by seven, the result is 85. Find the numbers.
One number is 11 less than another. If their sum is increased by eight, the result is 71. Find the numbers.
One number is five more than another. If their sum is increased by nine, the result is 60. Find the numbers.
One number is eight more than another. If their sum is increased by 17, the result is 95. Find the numbers.
One number is one more than twice another. Their sum is −5 . −5 . Find the numbers.
One number is six more than five times another. Their sum is six. Find the numbers.
The sum of two numbers is 14. One number is two less than three times the other. Find the numbers.
The sum of two numbers is zero. One number is nine less than twice the other. Find the numbers.
The sum of two consecutive integers is 77. Find the integers.
The sum of two consecutive integers is 89. Find the integers.
The sum of two consecutive integers is −23 . −23 . Find the integers.
The sum of two consecutive integers is −37 . −37 . Find the integers.
The sum of three consecutive integers is 78. Find the integers.
The sum of three consecutive integers is 60. Find the integers.
Find three consecutive integers whose sum is −3 . −3 .
Find three consecutive even integers whose sum is 258.
Find three consecutive even integers whose sum is 222.
Find three consecutive odd integers whose sum is 171.
Find three consecutive odd integers whose sum is 291.
Find three consecutive even integers whose sum is −36 . −36 .
Find three consecutive even integers whose sum is −84 . −84 .
Find three consecutive odd integers whose sum is −213 . −213 .
Find three consecutive odd integers whose sum is −267 . −267 .
Sale Price Patty paid $35 for a purse on sale for $10 off the original price. What was the original price of the purse?
Sale Price Travis bought a pair of boots on sale for $25 off the original price. He paid $60 for the boots. What was the original price of the boots?
Buying in Bulk Minh spent $6.25 on five sticker books to give his nephews. Find the cost of each sticker book.
Buying in Bulk Alicia bought a package of eight peaches for $3.20. Find the cost of each peach.
Price before Sales Tax Tom paid $1,166.40 for a new refrigerator, including $86.40 tax. What was the price of the refrigerator?
Price before Sales Tax Kenji paid $2,279 for a new living room set, including $129 tax. What was the price of the living room set?
What has been your past experience solving word problems?
When you start to solve a word problem, how do you decide what to let the variable represent?
What are consecutive odd integers? Name three consecutive odd integers between 50 and 60.
What are consecutive even integers? Name three consecutive even integers between −50 −50 and −40 . −40 .
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ If most of your checks were:
…confidently. Congratulations! You have achieved your goals in this section! Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific!
…with some help. This must be addressed quickly as topics you do not master become potholes in your road to success. Math is sequential—every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Who can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?
…no—I don’t get it! This is critical and you must not ignore it. You need to get help immediately or you will quickly be overwhelmed. See your instructor as soon as possible to discuss your situation. Together you can come up with a plan to get you the help you need.
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- Authors: Lynn Marecek, MaryAnne Anthony-Smith
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- Book title: Elementary Algebra
- Publication date: Feb 22, 2017
- Location: Houston, Texas
- Book URL: https://openstax.org/books/elementary-algebra/pages/1-introduction
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5 Problem-Solving Activities for Elementary Classrooms
Classroom problem-solving activities teach children how to engage problems rather than to become frustrated with them. Teachers have the opportunity to teach children the proper methods for dealing with stressful situations, complex problems, and fast decision-making. While a teacher is unlikely to actually put the child into a difficult or otherwise harmful situation, he or she can use activities to teach the child how to handle such situations later on in life.
Teach the problems
To solve any problem, students must go through a process to do so. The teacher can explore this process with students as a group. The first step is to fully understand the problem. To teach this, ask students to describe the problem in their own words. This ensures the student is able to comprehend and express the concern at hand. Then, they must describe and understand the barriers presented. At this point, it’s a good idea to provide ways for the student to find a solution. That’s where activities come into play.
The following are five activities elementary teachers can use to teach problem-solving to students. Teaching students to identify the possible solutions requires approaching the problem in various ways.
No. 1 – Create a visual image
One option is to teach children to create a visual image of the situation. Many times, this is an effective problem-solving skill. They are able to close their eyes and create a mind picture of the problem. For younger students, it may be helpful to draw out the problem they see on a piece of paper.
Ask the child to then discuss possible solutions to the problem. This could be done by visualizing what would happen if one action is taken or if another action is taken. By creating these mental images, the student is fully engaged and can map out any potential complications to their proposed solution.
No. 2 – Use manipulatives
Another activity that is ideal for children is to use manipulatives. In a situation where the problem is space-related, for example the children can move their desks around in various ways to create a pattern or to better visualize the problem. It’s also possible to use simple objects on a table, such as blocks, to create patterns or to set up a problem. This is an ideal way to teach problem-solving skills for math.
By doing this, it takes a problem, often a word problem that’s hard for some students to visualize, and places it in front of the student in a new way. The child is then able to organize the situation into something he or she understands.
No. 3 – Make a guess
Guessing is a very effective problem-solving skill. For those children who are unlikely to actually take action but are likely to sit and ponder until the right answer hits them, guessing is a critical step in problem-solving. This approach involves trial and error.
Rather than approaching guessing as a solution to problems (you do not want children to think they can always guess), teach that it is a way to gather more data. If, for example, they do not know enough about the situation to make a full decision, by guessing, they can gather more facts from the outcome and use that to find the right answer.
No. 4 – Patterns
No matter if the problem relates to social situations or if it is something that has to do with science, patterns are present. By teaching children to look for patterns, they can see what is happening more fully.
For example, define what a pattern is. Then, have the child look for any type of pattern in the context. If the children are solving a mystery, for example, they can look for patterns in time, place or people to better gather facts.
No. 5 – Making a list
Another effective tool is list making. Teach children how to make a list of all of the ideas they come up with right away. Brainstorming is a fun activity in any subject. Then, the child is able to work through the list to determine which options are problems or not.
Classroom problem-solving activities like these engage a group or a single student. They teach not what the answer is, but how the student can find that answer.
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Teaching Problem Solving in Math
- Freebies , Math , Planning
Every year my students can be fantastic at math…until they start to see math with words. For some reason, once math gets translated into reading, even my best readers start to panic. There is just something about word problems, or problem-solving, that causes children to think they don’t know how to complete them.
Every year in math, I start off by teaching my students problem-solving skills and strategies. Every year they moan and groan that they know them. Every year – paragraph one above. It was a vicious cycle. I needed something new.
I put together a problem-solving unit that would focus a bit more on strategies and steps in hopes that that would create problem-solving stars.
The Problem Solving Strategies
First, I wanted to make sure my students all learned the different strategies to solve problems, such as guess-and-check, using visuals (draw a picture, act it out, and modeling it), working backward, and organizational methods (tables, charts, and lists). In the past, I had used worksheet pages that would introduce one and provide the students with plenty of problems practicing that one strategy. I did like that because students could focus more on practicing the strategy itself, but I also wanted students to know when to use it, too, so I made sure they had both to practice.
I provided students with plenty of practice of the strategies, such as in this guess-and-check game.
There’s also this visuals strategy wheel practice.
I also provided them with paper dolls and a variety of clothing to create an organized list to determine just how many outfits their “friend” would have.
Then, as I said above, we practiced in a variety of ways to make sure we knew exactly when to use them. I really wanted to make sure they had this down!
Anyway, after I knew they had down the various strategies and when to use them, then we went into the actual problem-solving steps.
The Problem Solving Steps
I wanted students to understand that when they see a story problem, it isn’t scary. Really, it’s just the equation written out in words in a real-life situation. Then, I provided them with the “keys to success.”
S tep 1 – Understand the Problem. To help students understand the problem, I provided them with sample problems, and together we did five important things:
- read the problem carefully
- restated the problem in our own words
- crossed out unimportant information
- circled any important information
- stated the goal or question to be solved
We did this over and over with example problems.
Once I felt the students had it down, we practiced it in a game of problem-solving relay. Students raced one another to see how quickly they could get down to the nitty-gritty of the word problems. We weren’t solving the problems – yet.
Then, we were on to Step 2 – Make a Plan . We talked about how this was where we were going to choose which strategy we were going to use. We also discussed how this was where we were going to figure out what operation to use. I taught the students Sheila Melton’s operation concept map.
We talked about how if you know the total and know if it is equal or not, that will determine what operation you are doing. So, we took an example problem, such as:
Sheldon wants to make a cupcake for each of his 28 classmates. He can make 7 cupcakes with one box of cupcake mix. How many boxes will he need to buy?
We started off by asking ourselves, “Do we know the total?” We know there are a total of 28 classmates. So, yes, we are separating. Then, we ask, “Is it equal?” Yes, he wants to make a cupcake for EACH of his classmates. So, we are dividing: 28 divided by 7 = 4. He will need to buy 4 boxes. (I actually went ahead and solved it here – which is the next step, too.)
Step 3 – Solving the problem . We talked about how solving the problem involves the following:
- taking our time
- working the problem out
- showing all our work
- estimating the answer
- using thinking strategies
We talked specifically about thinking strategies. Just like in reading, there are thinking strategies in math. I wanted students to be aware that sometimes when we are working on a problem, a particular strategy may not be working, and we may need to switch strategies. We also discussed that sometimes we may need to rethink the problem, to think of related content, or to even start over. We discussed these thinking strategies:
- switch strategies or try a different one
- rethink the problem
- think of related content
- decide if you need to make changes
- check your work
- but most important…don’t give up!
To make sure they were getting in practice utilizing these thinking strategies, I gave each group chart paper with a letter from a fellow “student” (not a real student), and they had to give advice on how to help them solve their problem using the thinking strategies above.
Finally, Step 4 – Check It. This is the step that students often miss. I wanted to emphasize just how important it is! I went over it with them, discussing that when they check their problems, they should always look for these things:
- compare your answer to your estimate
- check for reasonableness
- check your calculations
- add the units
- restate the question in the answer
- explain how you solved the problem
Then, I gave students practice cards. I provided them with example cards of “students” who had completed their assignments already, and I wanted them to be the teacher. They needed to check the work and make sure it was completed correctly. If it wasn’t, then they needed to tell what they missed and correct it.
To demonstrate their understanding of the entire unit, we completed an adorable lap book (my first time ever putting together one or even creating one – I was surprised how well it turned out, actually). It was a great way to put everything we discussed in there.
Once we were all done, students were officially Problem Solving S.T.A.R.S. I just reminded students frequently of this acronym.
Stop – Don’t rush with any solution; just take your time and look everything over.
Think – Take your time to think about the problem and solution.
Act – Act on a strategy and try it out.
Review – Look it over and see if you got all the parts.
Wow, you are a true trooper sticking it out in this lengthy post! To sum up the majority of what I have written here, I have some problem-solving bookmarks FREE to help you remember and to help your students!
You can grab these problem-solving bookmarks for FREE by clicking here .
You can do any of these ideas without having to purchase anything. However, if you are looking to save some time and energy, then they are all found in my Math Workshop Problem Solving Unit . The unit is for grade three, but it may work for other grade levels. The practice problems are all for the early third-grade level.
- freebie , Math Workshop , Problem Solving
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Mathematics for Elementary Teachers
Michelle Manes, Honolulu, HI
Copyright Year: 2017
Publisher: University of Hawaii Manoa
Conditions of use.
Learn more about reviews.
Reviewed by Kevin Voogt, Assistant Professor, Grace College on 4/20/23
There seem to be subjects missing that are typical of the common core mathematics for elementary teachers texts (e.g., Ratios/Proportions, clear Partitive/Measurement division ideas, percentages, certain ideas in Geometry, Measurement). read more
Comprehensiveness rating: 4 see less
There seem to be subjects missing that are typical of the common core mathematics for elementary teachers texts (e.g., Ratios/Proportions, clear Partitive/Measurement division ideas, percentages, certain ideas in Geometry, Measurement).
Content Accuracy rating: 5
I did not find mathematical errors in the text during my review.
Relevance/Longevity rating: 3
I think there is need for quite a few updates to the text in regards to what is covered in elementary mathematics through the common core. The topics listed in my review of the Comprehensiveness above are just a start. I also see a need to add more activities to each section where prospective elementary teachers could do more exploration of the mathematics rather than what seems to be a more traditional approach of having the text explain it followed by problem sets alone.
Clarity rating: 5
The wording was quite clear and had nice explanations throughout.
Consistency rating: 5
It seems consistent throughout - with recurrent use of the same technical terms as needed.
Modularity rating: 4
There were a few issues with being able to assign the texts at different points within the course, as is the case for many math texts, in that many of the sections rely heavily on prior knowledge. If reorganization were to occur, there would be some need to re-structure how certain sections are taught.
Organization/Structure/Flow rating: 4
The text lacks much of the wonderful mathematical connections that could be made between ideas. While some connections are made, they seem a little outdated at times. I also think it would make more sense to have the properties of operations within their corresponding sections on operations rather than after all 4 operations are introduced.
Interface rating: 5
I did not see any issues with the interface. It was pretty user-friendly.
Grammatical Errors rating: 5
I did not notice any errors during my review.
Cultural Relevance rating: 5
I did not see anything insensitive or offensive in the text.
The text is just a small sampling of the many methods that could be used in teaching these mathematical ideas. I would have liked to see more activities for elementary teachers built into the lessons in each chapter as a means for learning and exploring ideas to facilitate more discussion as this text is used. There also are so many more connections that could be made between mathematical ideas that were lost a bit, especially with the general organization. On the whole, it is a nice resource and I could see it as useful for students studying for their certification exams to get some perspectives on the mathematical ideas they might encounter.
Reviewed by Sandra Zirkes, Teaching Professor, Bowling Green State University on 4/14/23
The text covers whole numbers, fractions, decimals, and operations well, and it provides some topics in geometry and algebraic thinking. However, the topics of ratio, proportion, and percent, as well as a more thorough coverage of geometry and... read more
The text covers whole numbers, fractions, decimals, and operations well, and it provides some topics in geometry and algebraic thinking. However, the topics of ratio, proportion, and percent, as well as a more thorough coverage of geometry and measurement are missing.
All information in the text is mathematically accurate and the writing and diagrams are error-free.
Relevance/Longevity rating: 4
While all of the information in the text is accurate and thought-provoking, some specific approaches are outdated with respect to the current standards and pedagogy. Approaching the concept of place value through the "Dots and Boxes" method, without reference to base ten blocks that are overwhelmingly used in the elementary math classroom, limits the coverage of this important topic. Similarly, approaching fractions using the "pies and kids" scenario is not consistent with the standards which emphasize the understanding of all fractions as iterations of unit fractions.
The text is written using clear and understandable prose that is both mathematically accurate and accessible to college level pre-service teachers.
The text has a clear organization and focus and uses consistent approaches and terminology throughout.
Much of the text is easily divisible into smaller subsections for student use. With respect to reorganization and realignment for a particular course, while some topics are revisited at appropriate points in the text, if those original topics were not covered in the course, revisiting the topic may not provide enough basis for the new topic. For example, the understanding of decimals is highly reliant on a student's understanding of the Dots and Boxes approach to place value earlier in the text.
Organization/Structure/Flow rating: 5
The topics in the text are organized in a logical way that is consistent with the structure of a typical mathematics education course.
Interface rating: 4
Navigating the text itself was seamless and intuitive. However, the videos that I viewed had poor visual quality and there was no audio.
The text is well written with no grammatical errors.
There is no apparent cultural insensitivity in the text.
This text has a problem solving focus and emphasizes deep thinking and reasoning about mathematics. Its approaches are clear and understandable. While its approaches are mathematically correct and thought-provoking, it is missing some key topics such as ratio, proportion, percent, and a more thorough coverage of geometry and measurement, as well as some standards-based approaches such as base ten blocks and understanding fractions as iterations of unit fractions.
Reviewed by Fred Coon, Assistant Professor, Anderson University on 2/16/23
The text covers all major points to help develop future teachers. read more
Comprehensiveness rating: 5 see less
The text covers all major points to help develop future teachers.
Text appears to be accurate.
The content is consist with concepts that elementary teachers should know. The methods are small in diversity.
Topics where well explained.
Text appears to use understandable and consistent terms.
Modularity rating: 5
Units appear to be mostly independent and can be used as stand alone units.
The topics are presented in a manner that build on each other but can be rearrange if desired.
Interface was useful and aided in navigating text.
I found no errors.
The text has no culturally insensitive or offensive items that I noticed.
I would like to have seen more diversity in methods discussed.
Reviewed by Perpetual Opoku Agyemang, Professor of Mathematics, Holyoke Community College on 6/17/21
The content in this text is built to help its readers, especially, pre-service elementary education majors learn to think like a mathematician in some very specific ways. The content addresses the subject framework in a complete yet concise... read more
The content in this text is built to help its readers, especially, pre-service elementary education majors learn to think like a mathematician in some very specific ways. The content addresses the subject framework in a complete yet concise manner. Although it does not provide an effective index/or glossary, LCD was not extensively tackled using factor tree, multiples or tables to express it, I still give props to the author since there are a lot of pictorial examples and a question bank for most of the various concepts. Furthermore, Dots and Boxes game on chapter 1 was very engaging and fun.
This text is very accurate and informative using a variety of felicitous examples to suit a diverse student population.
Conventional concepts are presented in a current and applied manner which allows for easier association with similar organized and retained information. This text could use some updated fraction problems and examples involving mixed numbers. Some of the YouTube videos have no sound at all.
Clarity rating: 4
Content material was presented in an easy to understand prose. Introduction of concepts and new terms were usually done by association or relevant previous knowledge. Some of the concepts like Multiplying Fractions, have YouTube videos embedded in the introductions.
Terminologies and framework are consistent throughout the text. The use of different notations were consistent throughout the various chapters and subunits.
This text has easily divisible content as stand alone subunits. However, numbering these chapters and subunits would have gone a long way to help its readers.
The topics in this text are organized from basic to complex concepts in a logical, clear fashion.
This text has an awesome interface (Online, PDF and XML). Moreover, it is untainted by distractions that may confuse its reader. Hyperlinks should have been included in the content.
I did not spot any grammatical errors in this text.
This content material contains no recognizable cultural insensitivity. It could use more examples involving modern affairs that are inclusive of diverse backgrounds.
I truly love the concise format of this text and how many different examples it uses to explain the concepts. The Geometry of Arts and Science and Tangrams were so informative with fun activities. It's easy to tell when one example ends and another begins, although index/or glossary and a system of links from the table of contents would be greatly appreciated. I did not see Points on a Coordinate Plane. Additionally, the number of exercises per section is too small. Of course this can be remedied by adding more. As with any textbook, the reader will need to supplement certain sections and clarify particular terms and concepts to best fit their situation. Pre-service elementary education majors could transition to this book fairly easily and successfully teach K-6 students in the United States in alignment with current Common Core Math Standards.
Reviewed by April Slack, Math Instructor, Aiken Technical College on 5/13/21
This text covers elementary mathematics strands including place value, numbers and operations, fractions, patterns, algebraic thinking, decimals, and geometry. Measurement and Data and Statistics strands are not included in this particular text. ... read more
This text covers elementary mathematics strands including place value, numbers and operations, fractions, patterns, algebraic thinking, decimals, and geometry. Measurement and Data and Statistics strands are not included in this particular text. The last chapter supplies the audience with problem-based learning approaches that include some measurement, but not in the detail of previous chapters of the book. It does incorporate problem solving strategies and pedagogical techniques teachers may use in the classroom. Examples with solutions and clarifying notes are provided throughout the text. The text does address Common Core Standards as well as the eight mathematical process standards. The textbook also provide teachers with a conceptual understanding of elementary mathematics along with appropriate mathematical terminology. The text does not offer an index or glossary.
The mathematics content provided in this text is accurate and provides thorough examples of teaching elementary mathematics for pre-service teachers. I found the text to build conceptual understanding and procedural fluency rather than just focus on basic algorithms to solve math problems. This is especially important for pre-service teachers, as they need to truly understand the "why" behind the math tricks that are often taught in early grades. The embedded links throughout the text are all in working order, as well.
The problem-solving approach to mathematics is especially relevant for elementary pre-service teachers; the intended audience. The book does expand beyond elementary mathematics, however, this is deemed extremely useful for all levels of mathematics teachers. Knowing the mathematical concepts beyond elementary strands allows teachers to know where there students are going and the mathematical purpose of content standards at each grade level. Many of the pedagogical techniques presented in the text are aligned with current research and instructional strategies for the elementary classroom.
This text provides explanations and defines mathematical terminology and has accessible prose. Beginning with the problem solving chapter before the specific content strands allows teachers to apply and consider strategies throughout the text. Often times, textbooks save problem solving for the end, but this text addresses strategies upfront and spirals nicely throughout the text. Some of the examples and visual representations are intended for an audience with mathematical background knowledge and strengths. A pre-service teacher may need help with content review prior to understanding the selection of particular problems highlighted in the text.
The text is well-organized and consistent with terminology throughout. The text is also consistent with provided examples that are used by mathematics teachers in everyday classrooms. There are multiple examples throughout each of the content chapters for pre-service teachers to reference and use in their own experiences.
This book is an easy read and may be easily broken up for weekly reading assignments and reflections. It seems as if mathematics teachers had a hand in writing this book. Bulleted and numbered lists are used throughout the text. The text also presents examples in clear, colored blocks. Visual models are clear and concise.
The book is well-organized with headings, subheadings, and the use of italics and boldface make this book extremely student friendly. The topics and content presented in this text are clear and in a logical order. Bulleted and numbered lists are reader friendly and easily understood. I found having the problem solving chapter appear first in the text stresses the importance and relevance of helping students become natural problem solvers. Often times texts and even worksheets save problem solving until the end, which poses a problem with students in the classroom.
This book is very easily navigated. The contents tab and drop down menu allows for the reader to quickly navigate to particular chapters and specific content. The previous and next buttons located at the bottom of the text allows readers to toggle between chapters very quickly. All embedded links work as they should and visual models are clear and understandable. There are no distractors present when trying to navigate the text. There is no index / glossary offered with this text.
The text is free from grammatical errors.
Cultural Relevance rating: 4
This text is not culturally offensive in any way. The final chapter of the text is dedicated to problem based learning and is centered around Voyaging on Hōkūle`a. The text provides embedded links to culturally relevant videos and models that help illustrate the cultural practices of Polynesians.
This textbook has a solid foundation and is well-organized for it's intended audience, the elementary mathematics pre-service teacher. This text will help build conceptual understanding of mathematics that will lead to procedural fluency for teachers. The text also provides clear examples of instructional strategies to be used in today's classrooms. Methods courses for pre-service teachers will find this text extremely useful and easy to incorporate in elementary mathematics methods instruction.
Reviewed by Kane Jessen, Math Instructor, Community College of Aurora on 8/13/20
This textbook is intended to cover the mathematics topics necessary to prepare pre-service elementary education majors to successfully teach K-6 students in the United States in alignment with current Common Core Math Standards. The textbook is a... read more
This textbook is intended to cover the mathematics topics necessary to prepare pre-service elementary education majors to successfully teach K-6 students in the United States in alignment with current Common Core Math Standards. The textbook is a mostly comprehensive collection of K-6 Common Core elementary math topics ranging from non-numerical problem solving through summative PBL assessments incorporating algebra, geometry and authentic problem solving. However, several topics related to K-6 CCSS Standards are not covered or minimally covered. CCSS topics with minimal coverage include set theory, logic, integers, probability, graphing and data analysis. At the beginning of the book, there is an effective and accessible table of contents with links included. However, sections and subsections are labeled only with names and page numbers. The text does not contain an index, glossary or appendices. Chapter summaries and links to previous concepts/problems are not included but would support student learning if included. More visuals and historical explorations would increase comprehensiveness.
Content was found to be accurate, error-free and unbiased.
Relevance/Longevity rating: 5
The language and examples of this text are written with a constructivist and meta-pedagogical voice that is both academic and accessible. The author immediately addresses the importance of CCSS and consistently utilizes the “Exploding Dots” curriculum. The “Exploding Dots” curriculum is a brave and differentiated approach to holistically teaching multi-base mathematics to K-12 students. “Exploding Dots” has been a core focus of K-12 Global Math Project and was pioneered by James Tanton . As future teachers, students can expect to teach “Exploding Dots” or similar CCSS curriculum sometime during their teaching career.
The language of the text is well-written, accessible and clear. Some sections and examples could be expanded for clarity/depth. Prior definitions/review concepts are not consistently linked.
The text is internally consistent in terms of its own terminology, framework and graphics. The “Exploding Dots” infusion helps maintain continuity throughout the text but is not present in all modules.
This text follows the common sequence that many “Mathematics for Elementary Teachers” textbooks commonly follow. The text is organized into eight modules. The text initially builds upon itself without being overly self-referential. The text’s sections, subsections, definitions, axioms and problem banks are all well delineated but lack sections/subsection numbers/identifiers and links to previous concepts/definitions
This textbook has a solid flow and follows a common sequence shared by most for profit “Mathematics for Elementary Teachers” texts. The text is well organized and builds upon itself.
Minimal issues involving interface were observed. Observed interface issues include, one broken video link and unnumbered sections. Definitions and review topics are not linked or referenced with page numbers/sections, however, this creates minimal usability issues. The text contains adequate procedural visuals and also cultural and historical visuals that enhance the student learning experience.
This text is largely free from grammatical errors. Grammatical errors that were observed were minor and non-persistent.
The text is not culturally insensitive or offensive in any way. It consistently uses examples that are inclusive of a variety of races, ethnicities, and backgrounds. Textbook examples often include references to Hawaiin culture. These references are easily understandable and could be readily adapted for students in other places. In an effort to increase relevance, further additions to the text could be made to provide a more equitable and historical focus on women, minorities and problem based learning cross-sectional explorations similar to the Hōkūleʻa section.
This textbook has a solid structure and great flow, I thoroughly enjoyed reviewing this textbook. I am genuinely excited to incorporate Michelle Manes ‘Mathematics for Elementary Teachers’ into my upcoming semester’s curriculum. With subsequent editions and revisions, this textbook will become a wonderful text for students majoring in primary education, especially those who are either lacking in basic math skills or math confidence.
Reviewed by Reina Ojiri, Assistant Professor, Leeward Community College on 7/27/20
The book begins with a reference to the Common Core State Standards (CCSS) for Mathematics and the eight “Mathematical Practices". Though not all states have adopted and/or are currently using the Common Core Standards, with its incorporation at... read more
Comprehensiveness rating: 2 see less
The book begins with a reference to the Common Core State Standards (CCSS) for Mathematics and the eight “Mathematical Practices". Though not all states have adopted and/or are currently using the Common Core Standards, with its incorporation at the beginning of the text I initially thought that the Common Core standards would be revisited consistently throughout the text.
Though the "Think Pair Share" sections are great additions for discussion to the book they do not include common misconceptions or tips for instructors to use to help guide these discussion prompts. The focus on just one type of discussion "Think Pair Share" also does not give future teachers a broader experience with different cooperative learning strategies in the classroom. There are many strategies in addition to “Think Pair Share” that are also great and seeing the same strategy over and over did not provide variation or keep me engaged as I read through the text.
There are a few key concepts that are not included in the text including Measurement & Data and Statistics & Probability.
The text also does not include an effective index and/or glossary. I have found that students do use the index and/or glossary that is typically in the back of the book to help them find information in the text quickly.
Content Accuracy rating: 3
The content is error-free however some of the images included on the PDF version are blurry and hard to read. There does not seem to be consistency between the different readable versions of the text.
There also seems to be a bias for the dots and boxes strategy throughout the text and the content lacks current practices of teaching concepts.
Just like any text, this textbook needs to be updated to match current best practices and research in math education. Since this text is Attribution-ShareAlike which allows “others to remix, adapt, and build upon your work even for commercial purposes, as long as they credit you and license their new creations under the identical terms” it does seem that updates and instructor/course-specific content will be relatively easy to implement as needed.
Clarity rating: 3
This text is written in a way unique way that makes it easier for students to read through and follow. It is very student friendly however might not be as useful as an instructor text since the instructor needs to fill-in-the-blanks on their own.
Consistency rating: 3
The text is written with consistent terminology however the framework for each chapter is not consistent. Some chapters include Explorations and additional sections while others end consistently with a problem bank.
Modularity rating: 3
The text is divided into smaller reading sections however the titles of each section are not easily recognized by students. Though I imagine the titles were meant to be creative for each section, having something more straight forward to make it easier for students to navigate is more important than creativity especially for future teachers who might be teaching these concepts for the first time.
Organization/Structure/Flow rating: 3
It would be good to organize the material consistently throughout the text (e.g.each section should end with a problem bank). The variation in the different sections can be confusing to both the instructor and student when trying to find something in the text.
I also noticed that the online version does not include page numbers while the PDF version does. This is not helpful when referring students to particular sections of the book. The PDF version also has many completely blank pages. I am not sure if this was meant to be on purpose (for printing purposes) but these pages can be very distracting to the reader.
Interface rating: 2
Navigation throughout the text is fine however, there are noticeable differences between the online and PDF versions of the text. The images in the PDF versions are noticeably blurry and lower quality than those in the online version. In some instances, it seems as though images were screenshot and copied and pasted which could account for the image quality.
Some images, in particular, should not have been included at all and are unreadable, for example, the Hokulea on page 441.
I did not notice grammatical errors.
The connection to the Hawaiian culture was a nice touch.
I would use this text as a reference but would not adopt this book as the main text for my class.
Reviewed by Thomas Starmack, Professor, Bloomsburg University of Pennsylvania on 3/26/20
The book is somewhat dated and does not include current research based best practices like concrete, representational, then abstract. Like most authors, they make assumptions that students have the ability to understand abstract and start the... read more
The book is somewhat dated and does not include current research based best practices like concrete, representational, then abstract. Like most authors, they make assumptions that students have the ability to understand abstract and start the lesson there, which is contradictory to how the brain works and what current research says about effective math instruction and learning.
I agree the content is accurate, but in many areas the learner must have a very strong understanding of mathematical concepts, structures, and applications. There lacks current best practice and current NCTM recommendations to approaching the teaching of mathematical content.
Relevance/Longevity rating: 1
Although mathematical concepts at the elementary level remain the same, the approach to engaging students in learning and the methods of instruction have evolved greatly. The book lacks many of the newer approaches and is outdated. The arrangement of the concepts is okay. I would recommend that the big ideas of teaching math are in the beginning and providing an overview of what is mathematics and best approaches to teaching/learning mathematics. Then scaffold the specific concepts. Fractions is one of the most complex and abstract, and this book starts there as a first topic.
Once again, the book is okay in terms of math learning but dated on best practice approaches. The book does not use jargon per say, but does not provide the best approaches for students to learn how to effectively teach mathematics.
Consistency rating: 4
Yes the book is consistent throughout.
The text is divisible, just not relevant to today nor provides current approaches. The order of the content is not in line with a methods of teaching course I would follow.
Organization/Structure/Flow rating: 2
I think the topics are clear but dated and not in the order as described above.
The text provides a variety of interfaces, none of which are confusing for the student who has a very strong math background. The text does mislead students to think starting with abstract is how to instruct elementary students, which is contradictory to brain research and current best practices.
Grammatical Errors rating: 4
I did not notice any grammar errors.
Cultural Relevance rating: 3
I think the text is culturally appropriate. Not certain about the final chapter as it focuses on one population. Having a chapter or theme woven throughout the text that provides students with a stronger understanding that although mathematics is a universal language, there are cultural differences to teaching and learning as evidenced in the 1999 TIMSS report.
The text is outdated. The text is an okay resource but I would not be able to use as the main guide for learning in a college level methods of teaching elementary mathematics course.
Reviewed by Jamie Price, Assistant Professor, East Tennessee State University on 3/20/20
This book introduces the reader to the standards for mathematical practice (SMP) from the Common Core standards in the introduction. I appreciated this as these standards cover all grades and are a unifying theme of the Common Core standards, yet... read more
This book introduces the reader to the standards for mathematical practice (SMP) from the Common Core standards in the introduction. I appreciated this as these standards cover all grades and are a unifying theme of the Common Core standards, yet many times overlooked. In addition, many states, including mine, that are not following Common Core directly have adopted the SMPs. The book does not cover two of the mathematical strands, namely measurement and statistics/data. Among the strands that are covered, however, the author does a thorough job of explaining the content, using a unified theme throughout, such as dots and boxes introduced in place value that appear again in number operations. I particularly liked the final chapter of the book and its connection to Hawaiian culture. The author could easily incorporate ideas related to teaching and learning measurement into this chapter in order to make the book more comprehensive.
The content was very accurate. I did not come across any mathematical errors or biases. The author did a good job of incorporating "think, pair, share" elements throughout each chapter as a model for future teachers. To further guide future teachers, I would have liked to see the author include information in each chapter about common misconceptions students have when learning the related material and ideas on how to address those misconceptions. In my experience, I find that pre-service teachers are unaware of these misconceptions and it is helpful to make them aware of them so that they can anticipate them in their own classrooms.
The content presented in this book is up-to-date and will remain relevant for a long time. Due to the fact that this book focuses more on content rather than methods, I do not foresee a need for many updates moving forward.
The book is written in a very clear and concise way that is approachable to future and current elementary teachers. The author presents key words in bold throughout the book to draw attention to them. I liked the way that the author included videos as well as written explanations of ideas, such as in the Number and Operations chapter, section titled Addition: Dots and Boxes. The author explains, in words, how to use this method to add multi-digit numbers and follows the written example with a video explanation. This helps to reach a variety of learners and learning styles. The author also addresses common "jargon" associated with particular mathematical concepts, such as proper and improper fractions (section titled What is a Fraction?), and discusses how this jargon can be misleading for students.
Each chapter in the book includes an introduction, multiple opportunities for think-pair-share discussions, and several problem sets to practice. I appreciated the consistency in the Dots and Boxes method introduced in the Place Value chapter and then carried into the Number and Operations chapter.
The book uses a modular approach to present the material. Each module contains numerous sections that help to break up the content into smaller chunks so that the content does not seem overwhelming. The modules are set up in an order that makes sense for the mathematics, but a reader could begin reading at any module and still make sense of the content.
The organization of the topics makes sense according to the mathematics presented and is logical.
I did not find anything distracting or confusing in relation to the interface of the text. The book was easy to navigate, with a clearly defined table of contents. I was able to easily click through the various modules and sections within each module. The book uses figures well to provide engagement to the reader as well as to further clarify content. The use of videos embedded within the modules helps to strengthen understanding of the content. It did take me a minute to find the navigation link that allowed me to move to the next section in a module (right arrow at bottom right corner of the page), but once I found it I was able to navigate seamlessly to each subsequent section.
I did not find any grammatical errors in the text.
In my opinion, this was one of the biggest strengths of this text. The author did a nice job of incorporating Hawaiian culture into the text. For example, the author includes an image in the Place Value chapter (Number Systems section) that references the use of tally marks on a sign at Hanakapiai Beach. In addition, a full chapter was devoted to Voyaging on Hōkūle`a. I particularly liked how the author connected this idea to beginning teaching of elementary mathematics and encouraged future teachers to think about ways to see mathematics outside of traditional mathematical settings.
I am glad that I came across this resource. I primarily teach math methods courses for elementary pre-service teachers, but I found many aspects of this text that I can incorporate into my classes to help students think more deeply about the mathematics that they will teach. I appreciated the author's attempt to challenge students in their thinking about elementary mathematics. Initially, I was surprised to find that there was no "answer key" provided for the many problem sets that were included throughout the text. After reading the quote presented on the introductory page to the Problem Solving chapter, I realized that this may have been an intentional decision made by the author to encourage readers to go beyond "a trail someone else has laid." I find that many pre-service elementary teachers want to "just know the answer" when it comes to mathematics; a no answer key approach will encourage discussion and justification, two elements important to ensuring equity in the teaching and learning of mathematics.
Reviewed by Shay Kidd, Assistant Professor- Mathematics Education, University of Montana - Western on 12/30/19
The content that elementary teachers need to have that is not covered in this book is graphing, probability, statistics, exponents, visual displays of data. The coverage of operations is very specific in the examples and does not cover the wide... read more
The content that elementary teachers need to have that is not covered in this book is graphing, probability, statistics, exponents, visual displays of data. The coverage of operations is very specific in the examples and does not cover the wide range that should be presented in this type of text.
Content Accuracy rating: 4
While the core topics presented are correct, the number of problems that are provided without any solutions is alarming. The majority of problems that are provided are meant for the reader to perform but do not provide any type of answer key for checking the work. In this way, the book seems to assume the reader to have a solid knowledge of the topics already and this book discusses a few different approaches to these topics.
The specific content presented is up-to-date and usable.
The book's prose seem to be more of a teaching guide than a textbook. This is nice for the conversational aspect that a reader may want in their learning, but should be explained more or possibly a change of title for the book. Something more like "Exploring the concepts of Elementary Mathematics" would provide a more reading friendly approach the book offers.
The author has a consistent voice of teaching and presenting the material.
Modularity rating: 2
The break-up of the text with boxes is difficult to follow the purpose of each box. While some of the box styles are clear, such as the think, pair, share or problem boxes, others seem to break up the line of discussion. A problem box may be discussed more directly immediately following the box and the presentation of the problem. Most of the problem boxes are not discussed again in the main text. This cased issues for wanting to read with a specific purpose. When the reader wants to understand a problem more, there is generally not more discussion, but unclear about when that would be provided or not. Other times boxes were used without any "box type" provided and these were just to break up the flow of the text.
Place value was a major topic to start the book and had good coverage, then operations and fractions were discussed, then a return to place value with decimals. It would seem that a connection of place value and decimals would work better to follow the other place value discussion.
Interface rating: 3
There are several pages that have large blank parts or are totally blank. This may be due to the PDF version that I chose. When I did use the internet-connected version, there seems to be a dependence on youtube to help do some of the teaching.
There are a few minor issues that would be resolved with a good proofread.
The book does seem to be written with the Hawaiian culture in mind. This may be difficult for other cultures to connect to or understand but does not present any insensitivities.
The book's title suggests a full discussion of the topics that elementary education pre-service teachers would need to know and teach, but this book is very lacking in the topics required for this. I selected this book to review because I teach classes that would use the textbook, but I would not use this textbook as is. There are a few topics that I plan to add to my own instruction, but the book as a whole needs additional help to be able to stand alone. This really appears to be a teaching guide based on the constant think-pair-share setup. This also is a specific teaching and method that seems to require the students to already have much of the content mastered. It does not teach all the content that is required to the level of the discussion had.
Reviewed by Ryan Nivens, Associate Professor, East Tennessee State University on 10/25/19
The book covers all the expected mathematical strands except for measurement and data/stats. There are some obvious connections to the strands of mathematical practice from the Common Core standards. While the abstract specifically lists MP1, MP2,... read more
The book covers all the expected mathematical strands except for measurement and data/stats. There are some obvious connections to the strands of mathematical practice from the Common Core standards. While the abstract specifically lists MP1, MP2, and MP3, the introduction clearly lists all 8. The chapter "Voyaging on Hōkūle`a" contains activities that will require use of measurement and units, but there is no explanation on how measurement topics should be taught or approached. However, this chapter does provide a good project-based learning set of materials, and is an exceptional resource for navigation. The book also includes a chapter on Problem Solving, which is important for those students who must complete the EdTPA and address the 3rd subject specific emphasis area. All embedded links to Youtube videos or Vimeo videos are working and play within the textbook pages.
I find the mathematics to be entirely accurate. There are many teaching strategies, such as "think pair share" that are found throughout the chapters. This is particularly helpful for future teachers.
This book should last a very long time in terms of relevance.
This book is very clear, with mathematical words in bold and proper definitions provided. The text also addresses common math classroom jargon. For an excellent example of this, see the heading "What is a Fraction" in the chapter on Fractions. Toward the bottom is a sub-heading "Jargon: Improper Fractions" that has students consider the usefulness of proper and improper fractions.
This book is consistently laid out, with multiple examples, problems to try, and diagrams to support the transfer of information.
This book is entirely modular. You can pick it up, and easily start in any chapter and not be lost. The heading, subheading, use of italics and boldface make it easy to locate information. As a mathematics education book, this is quite nice.
A mathematician wrote this, the layout is logical without question.
The book is extremely easy to navigate, with a logical structure to the table of contents that you can easily click through. A drop down menu in the upper left corner allows you to view the outline of the book while still viewing a page, and you can collapse/expand chapters within the menu.
The many figures that are present throughout the textbook are perfectly displayed and fit the reading material.
There is nothing I find distracting in the layout and interface.
I could not find any errors.
An entire chapter is dedicated to Voyaging on Hōkūle`a, with exceptional videos and diagrams to illustrate the cultural practices of the early Polynesians.
I was excited to find this book in the Open Educational Resources library. As a professor who frequently teaches methods courses in mathematics for elementary teachers, I feel that this book may be a terrific book to use to replace previous texts that I've adopted. I would like to see a chapter on Measurement to make the Voyaging on Hōkūle`a chapter more useful. It is obvious from the first page you open to that this book was well planned and thought out. I'm impressed.
Reviewed by Monica Rose Gilmore, Graduate Student, CU Boulder on 7/1/19
This textbook goes into depth about different mathematical concepts that are important for elementary school teachers to understand in teaching mathematics. However, the text is missing a focus on statistics and probability, which are key areas of... read more
Comprehensiveness rating: 3 see less
This textbook goes into depth about different mathematical concepts that are important for elementary school teachers to understand in teaching mathematics. However, the text is missing a focus on statistics and probability, which are key areas of focus in elementary math classrooms. The text is also missing an index or glossary but does define new terms as they are introduced.
The content, mathematical diagrams and depictions are accurate and error-free. Each chapter also accurately shows various ways to understand mathematical concepts. However, the diagrams are geared towards an audience that already has some understanding of advanced mathematics.
The content is organized in a way that necessary updates would be straightforward to implement. More specifically, much of the content reflects current mathematical practices and activities endorsed by up-to-date research in mathematics education.
The text is written in accessible prose and provides context for jargon and technical terminology. Additionally, the text clearly separates different terms for different strategies and concepts. For example, in the Problem Solving Strategy section, the interface is divided into different strategies for the reader to explore. This is helpful in keeping new concepts and strategies organized for the reader.
The text is written with consistent terminology. More specifically, the text consistently gives examples of what concepts are called by mathematicians and teachers. This is helpful for pre-service teachers that might be teaching mathematical concepts and strategies for the first time.
The text is easily divided into smaller reading sections. These sections include not only explanations of mathematical concepts, but also theorems, activities and diagrams which can be referenced by the teacher at any point. Also, the text gives teachers ideas for activities and additional problems to try with students.
Though the topics in the texts are presented in a logical, clear fashion, it might be beneficial for pre-service or elementary teachers to see how to specifically scaffold the different concepts within those topics for elementary students at different grade levels. Additionally, the text could also demonstrate how students typically confuse topics so teachers and pre-service teachers are prepared to navigate new concepts for the class.
The interface is easy to navigate since the content clearly outlines chapters and the topics within them. Sections such as notation and vocabulary, think pair shares and theorems are clearly outlined, organized and conceptually scaffolded. However, it might be helpful to have an index so the reader does not have to click within each topic to find the concept they are exploring.
This text is free from grammatical errors.
This text is not culturally insensitive or offensive and includes examples from the Hawaiian culture. Though the text is mainly made up of mathematical explanations, there are a variety of people's names in different problems that could be attributed to a variety of cultures. Additionally, the text reflects Polya's advice (1945) to try adapt the problem until it makes sense. Though the text includes mainly mathematical explanations, it does call for adapting problems which could potentially be applied to a variety of students of different backgrounds.
Reviewed by Glenna Gustafson, Professor, Radford University on 5/22/19
This is book is fairly comprehensive and I feel could be used by most foundational courses in elementary mathematics. The structure and writing provide a good foundation for students learning the "why" behind the mathematics and becoming... read more
This is book is fairly comprehensive and I feel could be used by most foundational courses in elementary mathematics. The structure and writing provide a good foundation for students learning the "why" behind the mathematics and becoming mathematical thinkers. There were some areas that could possibly use more development. In geometry for example there was no discussion of perimeter, area, and volume. Estimation, measurement of weight, time, and probability also appears to be missing. The text is well organized and written so that the chapters do not have to be completed in the order in which they are presented. While there is not index or glossary, the author uses colored text boxes to explain specific content or terms.
The content of the text is accurate and represented in a variety of formats to support learning, Not only does it provide solutions to problems, but also the mathematical thinking behind those solutions.
The text is very relevant for K-6 elementary pre-service teachers. It would be beneficial to know the specific grade levels that the author considers as "elementary" since this does vary by location. The content is "standard" for most elementary math courses and would not need to be updated often and the consistent layout and formation would make changes easy to make.
The text is written in a conversational tone. The simplicity and straight-forwardness of the text should appeal to those students that have sometimes been overwhelmed by writing in more traditional math texts.
The text is organized consistently from chapter to chapter. The table of contents and chunking of content in the chapters is logical and clear, Each chapter includes graphics as well as sections for: Think-Pair-Share; Definitions; Theorems (when appropriate); and, Problems. This consistent structure makes navigation easy.
The table of contents and chunking of content in the chapters is logical and clear. This also makes it easy to not necessary to move sequentially through the text, but to have the option of reviewing or using only needed topics. Subtitles and graphic captioning are appropriate for the content.
The text is easy to read and the organization within each chapters makes navigation easy.
This text is easy to navigate. The inclusion of graphics, charts, photos, and videos support learning. There are several pages where graphics in the Geometry chapter are skewed in the PDF version, but this does not seem to be a problem in the online version, Not all of the video links work within the PDF version.
There were no obvious grammatical errors. Several of the errors that were found were typos and/or word omissions.
The text is culturally inclusive. One thing that should be noted is that it seems male names are over-represented in the Problem sections. A reference to Hawaiian culture and life is evident. The Hōkūle`a voyage found in the last chapter is a good example of problem based learning and the integration of math with other subject areas.
This would be a wonderful text to use as a supplement or compliment to an elementary math methods course. It is not as overwhelming as other math texts, and would provide pre-service teachers with a good foundational review of math concepts, including vocabulary and some pedagogy.
Reviewed by Karise Mace, Mathematics Instructor, Kuztown University on 5/16/19, updated 11/9/20
This book is fairly comprehensive for a one-semester course, although it does not include much detail about several topics. The section on number systems barely touches on Roman numerals and only mentions Mayan and Babylonian counting systems.... read more
This book is fairly comprehensive for a one-semester course, although it does not include much detail about several topics. The section on number systems barely touches on Roman numerals and only mentions Mayan and Babylonian counting systems. The sections on addition, subtraction, and division would be more robust if the author included other algorithms for these operations. The chapter on Geometry does not address perimeter, area, surface area and volume. The book does not include an index or glossary.
While the book is not error free, it is unbiased. Most of the errors seem to be typographical and/or related to web links or LaTeX. In the section on number systems, the author incorrectly explains how one million would be represented using Roman numerals and incorrectly claims that the Mayans did not use a symbol for zero. Further, the Mayan number system was not a true vigesimal system, as the text indicates.
This text uses a constructivist approach to help students build their understanding of the mathematics included in the book. It is well organized and written so that the chapters do not have to be completed in the order in which they are presented. Because of this, the text should be easy to update. When concepts that are presented earlier in the text are used in later chapters, the author includes a brief but thorough review that would allow students to understand the later chapter even if they had not read and completed the problems in the earlier chapter. The "dots and boxes" approach is timely, as it uses the idea of the "exploding dots" that are part of the Global Math Project (https://www.globalmathproject.org).
The textbook is clearly written and enjoyable to read...even for the math-phobic student. The tone is conversational and is even funny at times. The author defines important mathematical terminology in a way that is both mathematically accurate and accessible to students. The chapter on problem solving is fantastic and really gives students insight into how to think and problem solve like a mathematician. The pies per child model for fractions is not the most effective model for helping students understand fractions and this part of the text would be improved if the author replaced this type of modeling with pattern block modeling.
Overall, the text is consistent in its chapter structure and terminology use. However, there is inconsistent notation when using "dots and boxes."
The text is well-organized but can be reorganized in order to suit an instructor's preference. However, it would be best to complete the chapter on problem solving first, as it sets the stage for the rest of the book. Most of the chapters are structured more like an activity book with lots of great problems and thought provoking questions that will help students think deeply about the mathematical concepts being presented. With the exception of the chapter on problem solving, there is not a whole lot of text for students to read.
Although the topics presented could be reorganized to meet student needs, the order in which they are presented is logical and clear.
With only a few exceptions, the images it the text are clear. In the section titled "Careful Use of Language in Mathematics: =" some of the scale images need to be modified so that the items on the scale appear to actually sit on the pans. The same issue occurs in the section titled "Structural and Procedural Algebra." Some of the images in the sections titled "Platonic Solids" and "Symmetry" spill off of the page. The image that appears on page 89 and then again on page 144 would be more clear if a different font was used to label the line segments.
No grammatical errors were noted. However, there were a few typographical errors that could cause confusion for students as on page 219.
The text was culturally sensitive and nothing offensive was noted. As the focus of the text is purely mathematical, there are not many cultural references at all, unless they are references to historical cultures. The author does use names for hypothetical students that are diverse and represent a variety of ethnicities. The last chapter is an integrated unit that focuses on the Hawaiian culture. Unfortunately, the links and web addresses in this chapter do not work and/or are no longer active.
The book includes three sections at the end of the problem solving chapter in which the author articulately explains the language that mathematicians use to succinctly and precisely explain their problem solving and solutions. These sections will help students who may not think of themselves as mathematicians learn to think like mathematicians. So many mathematics textbooks are full of exercises but no true problems. On the other hand, this text is full of wonderful problem solving and critical thinking problems that are embedded in the sections as well as in the problem banks. The author also includes many "Think/Pair/Share" exercises and questions that will facilitate mathematical thinking and conversation among students. The constructivist approach used by the author will help students build deep understanding about the mathematics covered in the text. While there is some room for revision and improvement, this is a very good text to use with elementary education majors, and I definitely plan to use this book the next time I teach them.
Reviewed by Desley Plaisance, Associate Professor, Nicholls State University on 4/29/19
This textbook seems to be appropriate for the first course typically taught for elementary teachers which usually includes topics of problem solving, place value, number and operations. Most books are able to be used for a second course which... read more
This textbook seems to be appropriate for the first course typically taught for elementary teachers which usually includes topics of problem solving, place value, number and operations. Most books are able to be used for a second course which focuses on geometry. This book could not be used for the second course.
Content seems to be accurate.
Topics are somewhat static for a course like this, so the textbook will not become obsolete within a short period of time.
Appears to be clear.
The flow from topic to topic is consistent in presentation.
Divided into clear sections.
Topics are presented logically and in a similar order to most books of this type.
Easy to navigate with clear images and other items such as tables.
Book is written in simple language and appears to be free of grammatical errors.
Appears to be culturally diverse.
This book could definitely be used for a first course of elementary math for teachers with the teacher providing resources. As with many open books, the print and layout is very simple without cluttering pages with unnecessary items.
Reviewed by Lisa Cooper, Assistant Professor, LSUS on 4/26/19
This text covers many concepts appropriately; however, a few concepts are missing, such as; data analysis and statistics. For more than ten years, data-driven instruction has been a major focus in education along with many other uses. This text... read more
This text covers many concepts appropriately; however, a few concepts are missing, such as; data analysis and statistics. For more than ten years, data-driven instruction has been a major focus in education along with many other uses. This text has a table of contents but not an index and/or glossary; however, does define words in chapters when needed.
The content is well organized and accurate. Multiple representations and diverse examples are provided throughout the text which supports an unbiased approach to those entering elementary education.
The text is quite relevant to the classroom today, incorporating such resources as YouTube, varied strategies to promote differentiated instruction, scaffolding between concepts, and problem-solving opportunities. Some states may find issues with Common Core standards being addressed; however, mathematical practices could be interchanged with the "standards."
The text is written free from educational jargon; it is straightforward and easy to understand.
The text is consistent in its structure; color is not distracting, problems, strategies, diagrams, charts, and definitions are provided throughout.
The text is appealing with the page layout; it's not too busy or distracting. Colors are attractive and text is broken down into appropriate amounts.
The text has a well-organized flow with the layout of each topic/chapter.
The text has charts, pictures, diagrams, real-world examples throughout; several different versions of the text are offered too.
No grammatical errors were observed in my review of the text.
The text provides a variety of backgrounds, races, and ethnicities while providing learning experiences and pedagogical approaches to support student engagement and learning.
Reviewed by Demetrice Smith-Mutegi, Instructor/Coordinator, Marian University on 3/6/19
This text covers place value, numbers and operations, fractions, patterns, algebraic thinking, decimals, and geometry. However, elementary teachers are expected to also know and understand statistics and probability. This text does not address... read more
This text covers place value, numbers and operations, fractions, patterns, algebraic thinking, decimals, and geometry. However, elementary teachers are expected to also know and understand statistics and probability. This text does not address this mathematical concept.
The text makes non-traditional, yet, accurate representations of mathematical concepts. In some sections, different solutions are presented and explained. This eliminates bias and provides a diverse representation of ideas when solving math problems.
The text is representative of common core problem-solving standards, however, it does require mathematical knowledge beyond elementary school. The problem-solving nature of the text is very relevant to elementary pre-service and in-service teachers (the audience for the book).
The text language is clear and accessible. There is a section on terminology, which is very helpful. Additional diagrams would help to improve the clarity in some cases.
I was expecting to see videos embedded throughout, after seeing them in the first section. It would be great to have a consistent format throughout the text, however, I understand that it is not always feasible to do so. There were other obvious and clear patterns presented, color-coded sections (think/pair/share), problems, examples.
The chapters and subchapters can be easily accessed, breaking the material into smaller sections.
The topics were presented in a logical, clear fashion, however, not all of the chapters would end with a problem bank. In some cases, there were additional sections after the problem bank. It would be great if each section included key objectives or goals of the section.
The text comes in pdf, XML, and an online web version. The search feature on the online version was a valuable addition.
I did not observe any obvious grammatical errors.
Cultural awareness was very obvious in this text. While it was more relevant to Hawaiian culture, it also included cultural awareness of other cultures and backgrounds.
Overall, this text assumes that the student has successfully completed mathematics through basic calculus. There should be more support in this area, as some elementary math students are not prepared to complete problems with this focus.
This a great "discussion" text.
Reviewed by Kandy Noles Stevens, Assistant Professor of Education, Southwest Minnesota State University on 12/28/18
This text covers the areas applicable to elementary mathematics extremely well (with the exception of omitting probability and data analysis) and provides graphically visual boxes within the text to define terms and instructional strategies. ... read more
This text covers the areas applicable to elementary mathematics extremely well (with the exception of omitting probability and data analysis) and provides graphically visual boxes within the text to define terms and instructional strategies. Additionally, the text provides thinking routines that support understanding more than just the concept, but also, the how's and why's of conceptual understanding.
The content is accurate and organized in a way to supports student learning for those training to become elementary teachers of mathematics.
The content of the book is relevant to today's elementary classroom in that it provides future elementary educators with the content knowledge, but also pedagogical approaches that would support student learning. Additionally, the text is organized in a way that is consistent and provides scaffolding support for those who might struggle with any one of the concepts. For Minnesota standards the only item of note is that there is not a section devoted to probability or data analysis, but the latter is touched upon in other chapters. There are three mentions of the Common Core standards in the text. Minnesota is not an adopter of the CC mathematics standards, but the references to the CCSS are in regards to the practices of mathematics and not on standards specifically.
While a great text for training future math teachers, this book does not read as a "typical" mathematics textbook. Students who have struggled in the past with mathematics might find the authors' writing style to be approachable and accessible for all levels of mathematics competence and confidence.
The text is consistent in its terminology and the structure of the framework is uniform throughout, relying on supporting student learning through exercises, think-pair-share activities, and continuous dialog and reflection.
A majority of the chapters begin with a section that introduces the strand of elementary mathematics covered. Not all chapters have this introduction which may pose challenging to interrupt the mathematical progression of some established courses.
The text is very well organized and has an easy-to-read format and flow.
The text is graphically rich with succinct advanced organizers, diagrams, and photos to support learning.
The text is written with professional level writing and is free of grammatical errors.
A variety of races, ethnicities, and backgrounds are present in the exercises used to support student learning throughout. The end of the text involves a Hōkūle`a voyage as a part of a problem-based learning (integrated curriculum) experience. This was something that really made this text stand out in that it gave future elementary teachers an example of using mathematical concepts in authentic (and exciting) learning experiences. This Polynesian voyage would provide many students with an introduction to life culturally different from their own.
I have been a STEM educator for more than two decades and I come from a long line of mathematics educators. While wrapping up my reading of this text, I happened to have my father (a 46 year veteran mathematics educator) here visiting. I shared the text with him and several times I heard him utter, "I like the way this problem is set up". We both found the book to be very knowledgeable for mathematical conceptual understandings, but even more so for introducing ideas for instructional strategies and classroom discourse to help future teachers become equipped with speaking the "language of mathematics" to guide their future students.
Table of Contents
I. Problem Solving
- Problem or Exercise?
- Problem Solving Strategies
- Beware of Patterns!
- Problem Bank
- Careful Use of Language in Mathematics
- Explaning Your Work
- The Last Step
II. Place Value
- Dots and Boxes
- Other Rules
- Binary Numbers
- Other Bases
- Number Systems
- Even Numbers
III. Number and Operations
- Addition: Dots and Boxes
- Subtration: Dots and Boxes
- Multiplication: Dots and Boxes
- Division: Dots and Boxes
- Number Line Model
- Area Model for Multiplication
- Properties of Operations
- Division Explorations
- What is a Fraction?
- The Key Fraction Rule
- Adding and Subtracting Fractions
- What is a Fraction? Revisited
- Multiplying Fractions
- Dividing Fractions: Meaning
- Dividing Fractions: Invert and Multiply
- Dividing Fractions: Problems
- Fractions involving zero
- Egyptian Fractions
- Algebra Connections
- What is a Fraction? Part 3
V. Patterns and Algebraic Thinking
- Borders on a Square
- Careful Use of Language in Mathematics: =
- Growing Patterns
- Matching Game
- Structural and Procedural Algebra
VI. Place Value and Decimals
- Review of Dots & Boxes Model
- Division and Decimals
- More x -mals
- Terminating or Repeating?
- Operations on Decimals
- Orders of Magnitude
- Triangles and Quadrilaterals
- Platonic Solids
- Painted Cubes
- Geometry in Art and Science
VIII. Voyaging on Hokule?a
- Worldwide Voyage
- Submit ancillary resource
About the Book
This book will help you to understand elementary mathematics more deeply, gain facility with creating and using mathematical notation, develop a habit of looking for reasons and creating mathematical explanations, and become more comfortable exploring unfamiliar mathematical situations.The primary goal of this book is to help you learn to think like a mathematician in some very specific ways. You will:• Make sense of problems and persevere in solving them. You will develop and demonstrate this skill by working on difficult problems, making incremental progress, and revising solutions to problems as you learn more.• Reason abstractly and quantitatively. You will demonstrate this skill by learning to represent situations using mathematical notation (abstraction) as well as creating and testing examples (making situations more concrete).• Construct viable arguments and critique the reasoning of others. You will be expected to create both written and verbal explanations for your solutions to problems. The most important questions in this class are “Why?” and “How do you know you're right?” Practice asking these questions of yourself, of your professor, and of your fellow students.Throughout the book, you will learn how to learn mathematics on you own by reading, working on problems, and making sense of new ideas on your own and in collaboration with other students in the class.
About the Contributors
Michelle Manes, Associate Professor, Department of Mathematics, University of Hawaii
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How to Use Real-World Problems to Teach Elementary School Math: 6 Tips
- Share article
When you think back on elementary school math, do you have fond memories of the countless worksheets you completed on adding fractions or solving division problems? Probably not.
Researchers and educators have been pushing for years for schools to move away from teaching math through a set of equations with no context around them, and towards an approach that pushes kids to use numerical reasoning to solve real problems, mirroring the way that they’ll encounter the use of math as adults.
The strategy is largely about setting kids up for success in the professional world, and educators can lay the groundwork decades earlier, even in kindergarten .
Here are some tips for using a real world problem-solving approach to teaching math to elementary school students.
1. There’s more than one right answer and more than one right method
A “real world task” can be as simple as asking students to think of equations that will get them to a particular “target” number, say, 14. Students could say 7 plus 7 is 14 or they could say 25 minus 11 is 14. Neither answer is better than the other, and that lesson teaches kids that there are multiple ways to use math to solve problems.
2. Give kids a chance to explain their thinking
The process you use to solve a real world math problem can be just as important as arriving at the correct answer, said Robbi Berry, who teaches 5th grade in Las Cruces, N.M. Her students have learned not to ask her if a particular answer is correct, she said, because she’ll turn the question back on them, asking them to explain how they know that it is right. She also gives her students a chance to explain to one another how they arrived at a particular solution, “We always share our strategies so that the kids can see the different ways” to arrive at an answer, she said. Students get excited, she said, when one of their classmates comes up with an approach they never would have thought of. “Math is creative,” Berry said. “It’s not just learning and memorizing.”
3. Be willing to deal with some off-the-wall answers
Problem solving does not necessarily mean going to the word problems in your textbook, said Latrenda Knighten, a mathematics instructional coach in Baton Rouge, La. For little kids, it can be as simple as showing a group of geometric shapes and asking what they have in common. Students may go off track a bit by talking about things like color, she said, but teachers can steer them towards thinking about things like how a rectangle differs from a triangle.
4. Let your students push themselves
Tackling these richer, real-world problems can be tougher than solving equations on a worksheet. And that is a good thing, said Jo Boaler, a professor at Stanford University and an expert on math education. “It’s really good for your brain to struggle,” she said. “We don’t want kids getting right answers all the time because that’s not giving their brains a really good workout.” These types of problems require collaboration, a skill that many don’t associate with math, but that is key to how math reasoning works beyond the classroom. The complexity and difficulty of the tasks means that students “have to talk to each other and really figure out what to do, what’s a good method?”
5. Celebrate ‘favorite mistakes’ to encourage intellectual risk taking
Wrong answers should be viewed as learning opportunities, Berry said. When one of her students makes an error, she asks if she can share it with the class as a “favorite mistake.” Most of the time, students are comfortable with that, and the class will work together to figure how the misstep happened.
6. Remember there’s no such thing as a being born with a ‘math brain’
Some teachers believe that certain students are just naturally good at math, and others are not, Boaler said. But that’s not true. “Brains are constantly shaping, changing, developing, connecting, and there is no fixed anything,” said Boaler, who often works alongside neuroscientists. What’s more, many elementary school teachers lack confidence in their own math abilities, she said. “They think they can’t do [math],” Boaler said. “And they often pass those ideas on” to their students.
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Problem Solving Activities: 7 Strategies
- Critical Thinking
Problem solving can be a daunting aspect of effective mathematics teaching, but it does not have to be! In this post, I share seven strategic ways to integrate problem solving into your everyday math program.
In the middle of our problem solving lesson, my district math coordinator stopped by for a surprise walkthrough.
I was so excited!
We were in the middle of what I thought was the most brilliant math lesson– teaching my students how to solve problem solving tasks using specific problem solving strategies.
It was a proud moment for me!
Each week, I presented a new problem solving strategy and the students completed problems that emphasized the strategy.
After observing my class, my district coordinator pulled me aside to chat. I was excited to talk to her about my brilliant plan, but she told me I should provide the tasks and let my students come up with ways to solve the problems. Then, as students shared their work, I could revoice the student’s strategies and give them an official name.
What a crushing blow! Just when I thought I did something special, I find out I did it all wrong.
I took some time to consider her advice. Once I acknowledged she was right, I was able to make BIG changes to the way I taught problem solving in the classroom.
When I Finally Saw the Light
To give my students an opportunity to engage in more authentic problem solving which would lead them to use a larger variety of problem solving strategies, I decided to vary the activities and the way I approached problem solving with my students.
Problem Solving Activities
Here are seven ways to strategically reinforce problem solving skills in your classroom.
Seasonal Problem Solving
Many teachers use word problems as problem solving tasks. Instead, try engaging your students with non-routine tasks that look like word problems but require more than the use of addition, subtraction, multiplication, and division to complete. Seasonal problem solving tasks and daily challenges are a perfect way to celebrate the season and have a little fun too!
Cooperative Problem Solving Tasks
Go cooperative! If you’ve got a few extra minutes, have students work on problem solving tasks in small groups. After working through the task, students create a poster to help explain their solution process and then post their poster around the classroom. Students then complete a gallery walk of the posters in the classroom and provide feedback via sticky notes or during a math talk session.
Notice and Wonder
Before beginning a problem solving task, such as a seasonal problem solving task, conduct a Notice and Wonder session. To do this, ask students what they notice about the problem. Then, ask them what they wonder about the problem. This will give students an opportunity to highlight the unique characteristics and conditions of the problem as they try to make sense of it.
Want a better experience? Remove the stimulus, or question, and allow students to wonder about the problem. Try it! You’ll gain some great insight into how your students think about a problem.
Start your math block with a math starter, critical thinking activities designed to get your students thinking about math and provide opportunities to “sneak” in grade-level content and skills in a fun and engaging way. These tasks are quick, designed to take no more than five minutes, and provide a great way to turn-on your students’ brains. Read more about math starters here !
Create your own puzzle box! The puzzle box is a set of puzzles and math challenges I use as fast finisher tasks for my students when they finish an assignment or need an extra challenge. The box can be a file box, file crate, or even a wall chart. It includes a variety of activities so all students can find a challenge that suits their interests and ability level.
Use calculators! For some reason, this tool is not one many students get to use frequently; however, it’s important students have a chance to practice using it in the classroom. After all, almost everyone has access to a calculator on their cell phones. There are also some standardized tests that allow students to use them, so it’s important for us to practice using calculators in the classroom. Plus, calculators can be fun learning tools all by themselves!
Three-Act Math Tasks
Use a three-act math task to engage students with a content-focused, real-world problem! These math tasks were created with math modeling in mind– students are presented with a scenario and then given clues and hints to help them solve the problem. There are several sites where you can find these awesome math tasks, including Dan Meyer’s Three-Act Math Tasks and Graham Fletcher’s 3-Acts Lessons .
Getting the Most from Each of the Problem Solving Activities
When students participate in problem solving activities, it is important to ask guiding, not leading, questions. This provides students with the support necessary to move forward in their thinking and it provides teachers with a more in-depth understanding of student thinking. Selecting an initial question and then analyzing a student’s response tells teachers where to go next.
Ready to jump in? Grab a free set of problem solving challenges like the ones pictured using the form below.
Which of the problem solving activities will you try first? Respond in the comments below.
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This is a very cool site. I hope it takes off and is well received by teachers. I work in mathematical problem solving and help prepare pre-service teachers in mathematics.
Thank you, Scott! Best wishes to you and your pre-service teachers this year!
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4 Ways to Build Student-Centered Math Lessons
To help more kids thrive in math class, keep student identities in mind during the lesson planning stage.
In traditional math classrooms, teachers present lessons, students work through problems individually or in groups, or perhaps they’re asked questions or take a quiz to demonstrate understanding, and then teachers correct or affirm answers.
On the surface, it’s a concise, logical teaching model where the teacher “starts with an instructional objective and then designs a lesson with the goal of students demonstrating proficiency,” write educators Sam Rhodes and Christopher R. Gareis for ASCD’s In Service . But it’s an approach they say tends to “relegate equity to an afterthought, inadvertently positioning many students as passive observers of mathematics.” Over time, this passive positioning impacts students’ math identity—especially kids from diverse backgrounds who might already struggle to connect classroom learning to their own life experiences—building a “sense of disinterest, inadequacy, and disenfranchisement,” note Rhodes and Gareis.
A more equitable model of math instruction, they believe, begins with student identities firmly in mind during the lesson planning phase, with the teacher thinking first about students’ beliefs around math—for example, considering their comfort level with language, or how they view themselves academically and as mathematicians. “We cannot leave considerations of student identities to the end,” write Rhodes and Gareis, an assistant professor of elementary math education at Georgia Southern University and a professor of educational leadership at William and Mary, respectively. “Rather we need to consider what dispositional outcomes we intend for students,” and then intentionally design curriculum backward , keeping “the aims of equity and sense of self in mind” so that more kids begin to see themselves as competent mathematical thinkers.
Here are four things to keep in mind when designing student-centered math lessons.
Develop a Clear Mission Statement
Consider creating a mission statement that articulates the intentions of your school’s math curriculum and communicates “what teaching mathematics should look and feel like in the school,” write Rhodes and Gareis. This is a simple way to “codify the beliefs and identities that [teachers] aim to foster in students.”
The mission statement might highlight the importance of creating a community of learners “who are seen as doers of mathematics,” for example, and set a goal of giving every student the chance to “develop and communicate deeper understandings of mathematics through flexible thinking, reasoning, and problem-solving.”
Connect to Students’ Experience
Children are naturally drawn to explore the math around them. “From young ages, we quantify, recognize patterns, and question the equivalence of things, even before we have those words for it,” say Rhodes and Gareis. “As we grow, these informal learning opportunities are intrinsically tied to home and cultural experiences and identities.” Drawing on these experiences in the classroom can be a powerful way for teachers to “create mathematical understandings that are inherently connected to the lives of their students.”
While clearly not everything in the math curriculum can directly relate to students’ life experiences, it’s important to plan for lessons that include more connection points for kids in your classroom—similar to how a thoughtfully assembled classroom library would include a rich variety of options that reflect students’ diverse tastes, cultural backgrounds, reading levels, and specific interests.
In his seventh-grade math classroom, Kwame Sarfo-Mensah plans a unit where students investigate an issue of interest . It’s an effort to help students “make sense of the world in which we live,” he says, and in the process, connect them more deeply to mathematics. He starts the unit with a survey to gauge students’ areas of interest. One year, the responses led to a three-week project examining the intersection between law enforcement and communities of color in Boston.
Sarfo-Mensah helped students come up with a focus question and brainstorm the different math-related data points needed to investigate it—statistics, graphical representations, geometric diagrams, and functional relationships—and he made sure to align the work with the appropriate academic standards. He gave students three options for their final product, providing “multiple access points for diverse learners,” he writes.
Allow for Multiple Solving Pathways
In vibrant math classrooms, teachers often “show different ways to solve the same problem and encourage students to come up with their own creative ways to solve them,” writes Matthew Beyranevand , a K–12 math and science department coordinator for Massachusetts Public Schools. “The more strategies and approaches that students are exposed to, the deeper their conceptual understanding of the topic becomes.”
After students solve a problem using a single method, encourage them to brainstorm alternative solving pathways, then discuss the various options as a class. It’s a subtle shift that puts emphasis on developing critical thinking and encourages students to embrace asking questions and sharing strategies as a way to make sense of complex material. “Whereas a focus on [right or wrong] answers results in judgments of correctness, a focus on thinking builds and refines understandings from what students know and understand,” write Rhodes and Gareis.
Encourage Productive Struggle
Problem-solving is an integral component of math, and allowing students to struggle productively as they attempt to solve complex problems “sends the message that the teacher believes students are capable of doing and creating mathematics,” write Rhodes and Gareis.
High school math teacher Solenne Abaziou, in an effort to build up her students’ problem-solving skills and stamina, assigns weekly open-ended math tasks called problem solvers—problems like Dice in a Corner and Snowmen Buttons . “Students often struggle with persistence—they’re uncomfortable with the idea of trying a solution if they’re not confident that it will yield the desired results, which leads them to refuse to take risks,” Abaziou writes . “Helping students get past this fear will give them a big advantage in math and in many other areas of daily life.”
A good problem solver “has a low floor and high ceiling,” Abaziou notes. “The skills needed to tackle the problem should be minimal, to allow weaker students to engage with it, but it should have several levels of complexity, to challenge high-flying students.” As students engage with the problem, they should be “confused at the beginning, which encourages them to struggle until they get on a path that will likely lead them to the solution.” It’s only by working through that initial frustration that students begin to build “problem-solving resilience,” she writes.
All students are capable of doing math, Rhodes and Gareis insist. “We believe that diversity of thought enhances understandings of mathematics for all students, and we believe that allowing student voices and experiences to shine in mathematics classrooms is a crucial step towards rehumanizing the subject,” they conclude.
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Unit 1: Algebra foundations
Unit 2: solving equations & inequalities, unit 3: working with units, unit 4: linear equations & graphs, unit 5: forms of linear equations, unit 6: systems of equations, unit 7: inequalities (systems & graphs), unit 8: functions, unit 9: sequences, unit 10: absolute value & piecewise functions, unit 11: exponents & radicals, unit 12: exponential growth & decay, unit 13: quadratics: multiplying & factoring, unit 14: quadratic functions & equations, unit 15: irrational numbers, unit 16: creativity in algebra, course challenge.
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Problem Solving Strategies
Think back to the first problem in this chapter, the ABC Problem . What did you do to solve it? Even if you did not figure it out completely by yourself, you probably worked towards a solution and figured out some things that did not work.
Unlike exercises, there is never a simple recipe for solving a problem. You can get better and better at solving problems, both by building up your background knowledge and by simply practicing. As you solve more problems (and learn how other people solved them), you learn strategies and techniques that can be useful. But no single strategy works every time.
Pólya’s How to Solve It
George Pólya was a great champion in the field of teaching effective problem solving skills. He was born in Hungary in 1887, received his Ph.D. at the University of Budapest, and was a professor at Stanford University (among other universities). He wrote many mathematical papers along with three books, most famously, “How to Solve it.” Pólya died at the age 98 in 1985. 
In 1945, Pólya published the short book How to Solve It , which gave a four-step method for solving mathematical problems:
- First, you have to understand the problem.
- After understanding, then make a plan.
- Carry out the plan.
- Look back on your work. How could it be better?
This is all well and good, but how do you actually do these steps?!?! Steps 1. and 2. are particularly mysterious! How do you “make a plan?” That is where you need some tools in your toolbox, and some experience to draw upon.
Much has been written since 1945 to explain these steps in more detail, but the truth is that they are more art than science. This is where math becomes a creative endeavor (and where it becomes so much fun). We will articulate some useful problem solving strategies, but no such list will ever be complete. This is really just a start to help you on your way. The best way to become a skilled problem solver is to learn the background material well, and then to solve a lot of problems!
We have already seen one problem solving strategy, which we call “Wishful Thinking.” Do not be afraid to change the problem! Ask yourself “what if” questions:
- What if the picture was different?
- What if the numbers were simpler?
- What if I just made up some numbers?
You need to be sure to go back to the original problem at the end, but wishful thinking can be a powerful strategy for getting started.
This brings us to the most important problem solving strategy of all:
Problem Solving Strategy 2 (Try Something!). If you are really trying to solve a problem, the whole point is that you do not know what to do right out of the starting gate. You need to just try something! Put pencil to paper (or stylus to screen or chalk to board or whatever!) and try something. This is often an important step in understanding the problem; just mess around with it a bit to understand the situation and figure out what is going on.
And equally important: If what you tried first does not work, try something else! Play around with the problem until you have a feel for what is going on.
Problem 2 (Payback)
Last week, Alex borrowed money from several of his friends. He finally got paid at work, so he brought cash to school to pay back his debts. First he saw Brianna, and he gave her 1/4 of the money he had brought to school. Then Alex saw Chris and gave him 1/3 of what he had left after paying Brianna. Finally, Alex saw David and gave him 1/2 of what he had remaining. Who got the most money from Alex?
After you have worked on the problem on your own for a while, talk through your ideas with a partner (even if you have not solved it). What did you try? What did you figure out about the problem?
This problem lends itself to two particular strategies. Did you try either of these as you worked on the problem? If not, read about the strategy and then try it out before watching the solution.
Problem Solving Strategy 3 (Draw a Picture). Some problems are obviously about a geometric situation, and it is clear you want to draw a picture and mark down all of the given information before you try to solve it. But even for a problem that is not geometric, like this one, thinking visually can help! Can you represent something in the situation by a picture?
Draw a square to represent all of Alex’s money. Then shade 1/4 of the square — that’s what he gave away to Brianna. How can the picture help you finish the problem?
After you have worked on the problem yourself using this strategy (or if you are completely stuck), you can watch someone else’s solution.
Problem Solving Strategy 4 (Make Up Numbers). Part of what makes this problem difficult is that it is about money, but there are no numbers given. That means the numbers must not be important. So just make them up!
You can work forwards: Assume Alex had some specific amount of money when he showed up at school, say $100. Then figure out how much he gives to each person. Or you can work backwards: suppose he has some specific amount left at the end, like $10. Since he gave Chris half of what he had left, that means he had $20 before running into Chris. Now, work backwards and figure out how much each person got.
Watch the solution only after you tried this strategy for yourself.
If you use the “Make Up Numbers” strategy, it is really important to remember what the original problem was asking! You do not want to answer something like “Everyone got $10.” That is not true in the original problem; that is an artifact of the numbers you made up. So after you work everything out, be sure to re-read the problem and answer what was asked!
Problem 3 (Squares on a Chess Board)
How many squares, of any possible size, are on a 8 × 8 chess board? (The answer is not 64… It’s a lot bigger!)
Remember Pólya’s first step is to understand the problem. If you are not sure what is being asked, or why the answer is not just 64, be sure to ask someone!
Think / Pair / Share
After you have worked on the problem on your own for a while, talk through your ideas with a partner (even if you have not solved it). What did you try? What did you figure out about the problem, even if you have not solved it completely?
It is clear that you want to draw a picture for this problem, but even with the picture it can be hard to know if you have found the correct answer. The numbers get big, and it can be hard to keep track of your work. Your goal at the end is to be absolutely positive that you found the right answer. You should never ask the teacher, “Is this right?” Instead, you should declare, “Here’s my answer, and here is why I know it is correct!”
Problem Solving Strategy 5 (Try a Simpler Problem). Pólya suggested this strategy: “If you can’t solve a problem, then there is an easier problem you can solve: find it.” He also said: “If you cannot solve the proposed problem, try to solve first some related problem. Could you imagine a more accessible related problem?” In this case, an 8 × 8 chess board is pretty big. Can you solve the problem for smaller boards? Like 1 × 1? 2 × 2? 3 × 3?
Of course the ultimate goal is to solve the original problem. But working with smaller boards might give you some insight and help you devise your plan (that is Pólya’s step (2)).
Problem Solving Strategy 6 (Work Systematically). If you are working on simpler problems, it is useful to keep track of what you have figured out and what changes as the problem gets more complicated.
For example, in this problem you might keep track of how many 1 × 1 squares are on each board, how many 2 × 2 squares on are each board, how many 3 × 3 squares are on each board, and so on. You could keep track of the information in a table:
Problem Solving Strategy 7 (Use Manipulatives to Help You Investigate). Sometimes even drawing a picture may not be enough to help you investigate a problem. Having actual materials that you move around can sometimes help a lot!
For example, in this problem it can be difficult to keep track of which squares you have already counted. You might want to cut out 1 × 1 squares, 2 × 2 squares, 3 × 3 squares, and so on. You can actually move the smaller squares across the chess board in a systematic way, making sure that you count everything once and do not count anything twice.
Problem Solving Strategy 8 (Look for and Explain Patterns). Sometimes the numbers in a problem are so big, there is no way you will actually count everything up by hand. For example, if the problem in this section were about a 100 × 100 chess board, you would not want to go through counting all the squares by hand! It would be much more appealing to find a pattern in the smaller boards and then extend that pattern to solve the problem for a 100 × 100 chess board just with a calculation.
If you have not done so already, extend the table above all the way to an 8 × 8 chess board, filling in all the rows and columns. Use your table to find the total number of squares in an 8 × 8 chess board. Then:
- Describe all of the patterns you see in the table.
- Can you explain and justify any of the patterns you see? How can you be sure they will continue?
- What calculation would you do to find the total number of squares on a 100 × 100 chess board?
(We will come back to this question soon. So if you are not sure right now how to explain and justify the patterns you found, that is OK.)
Problem 4 (Broken Clock)
This clock has been broken into three pieces. If you add the numbers in each piece, the sums are consecutive numbers. ( Consecutive numbers are whole numbers that appear one after the other, such as 1, 2, 3, 4 or 13, 14, 15.)
Can you break another clock into a different number of pieces so that the sums are consecutive numbers? Assume that each piece has at least two numbers and that no number is damaged (e.g. 12 isn’t split into two digits 1 and 2.)
Remember that your first step is to understand the problem. Work out what is going on here. What are the sums of the numbers on each piece? Are they consecutive?
After you have worked on the problem on your own for a while, talk through your ideas with a partner (even if you have not solved it). What did you try? What progress have you made?
Problem Solving Strategy 9 (Find the Math, Remove the Context). Sometimes the problem has a lot of details in it that are unimportant, or at least unimportant for getting started. The goal is to find the underlying math problem, then come back to the original question and see if you can solve it using the math.
In this case, worrying about the clock and exactly how the pieces break is less important than worrying about finding consecutive numbers that sum to the correct total. Ask yourself:
- What is the sum of all the numbers on the clock’s face?
- Can I find two consecutive numbers that give the correct sum? Or four consecutive numbers? Or some other amount?
- How do I know when I am done? When should I stop looking?
Of course, solving the question about consecutive numbers is not the same as solving the original problem. You have to go back and see if the clock can actually break apart so that each piece gives you one of those consecutive numbers. Maybe you can solve the math problem, but it does not translate into solving the clock problem.
Problem Solving Strategy 10 (Check Your Assumptions). When solving problems, it is easy to limit your thinking by adding extra assumptions that are not in the problem. Be sure you ask yourself: Am I constraining my thinking too much?
In the clock problem, because the first solution has the clock broken radially (all three pieces meet at the center, so it looks like slicing a pie), many people assume that is how the clock must break. But the problem does not require the clock to break radially. It might break into pieces like this:
Were you assuming the clock would break in a specific way? Try to solve the problem now, if you have not already.
- Image of Pólya by Thane Plambeck from Palo Alto, California (Flickr) [CC BY 2.0 (http://creativecommons.org/licenses/by/2.0)], via Wikimedia Commons ↵
Mathematics for Elementary Teachers by Michelle Manes is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License , except where otherwise noted.
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5 Teaching Mathematics Through Problem Solving
In his book “How to Solve It,” George Pólya (1945) said, “One of the most important tasks of the teacher is to help his students. This task is not quite easy; it demands time, practice, devotion, and sound principles. The student should acquire as much experience of independent work as possible. But if he is left alone with his problem without any help, he may make no progress at all. If the teacher helps too much, nothing is left to the student. The teacher should help, but not too much and not too little, so that the student shall have a reasonable share of the work.” (page 1)
What is a problem in mathematics? A problem is “any task or activity for which the students have no prescribed or memorized rules or methods, nor is there a perception by students that there is a specific ‘correct’ solution method” (Hiebert, et. al., 1997). Problem solving in mathematics is one of the most important topics to teach; learning to problem solve helps students develop a sense of solving real-life problems and apply mathematics to real world situations. It is also used for a deeper understanding of mathematical concepts. Learning “math facts” is not enough; students must also learn how to use these facts to develop their thinking skills.
According to NCTM (2010), the term “problem solving” refers to mathematical tasks that have the potential to provide intellectual challenges for enhancing students’ mathematical understanding and development. When you first hear “problem solving,” what do you think about? Story problems or word problems? Story problems may be limited to and not “problematic” enough. For example, you may ask students to find the area of a rectangle, given the length and width. This type of problem is an exercise in computation and can be completed mindlessly without understanding the concept of area. Worthwhile problems includes problems that are truly problematic and have the potential to provide contexts for students’ mathematical development.
There are three ways to solve problems: teaching for problem solving, teaching about problem solving, and teaching through problem solving.
Teaching for problem solving begins with learning a skill. For example, students are learning how to multiply a two-digit number by a one-digit number, and the story problems you select are multiplication problems. Be sure when you are teaching for problem solving, you select or develop tasks that can promote the development of mathematical understanding.
Teaching about problem solving begins with suggested strategies to solve a problem. For example, “draw a picture,” “make a table,” etc. You may see posters in teachers’ classrooms of the “Problem Solving Method” such as: 1) Read the problem, 2) Devise a plan, 3) Solve the problem, and 4) Check your work. There is little or no evidence that students’ problem-solving abilities are improved when teaching about problem solving. Students will see a word problem as a separate endeavor and focus on the steps to follow rather than the mathematics. In addition, students will tend to use trial and error instead of focusing on sense making.
Teaching through problem solving focuses students’ attention on ideas and sense making and develops mathematical practices. Teaching through problem solving also develops a student’s confidence and builds on their strengths. It allows for collaboration among students and engages students in their own learning.
Consider the following worthwhile-problem criteria developed by Lappan and Phillips (1998):
- The problem has important, useful mathematics embedded in it.
- The problem requires high-level thinking and problem solving.
- The problem contributes to the conceptual development of students.
- The problem creates an opportunity for the teacher to assess what his or her students are learning and where they are experiencing difficulty.
- The problem can be approached by students in multiple ways using different solution strategies.
- The problem has various solutions or allows different decisions or positions to be taken and defended.
- The problem encourages student engagement and discourse.
- The problem connects to other important mathematical ideas.
- The problem promotes the skillful use of mathematics.
- The problem provides an opportunity to practice important skills.
Of course, not every problem will include all of the above. Sometimes, you will choose a problem because your students need an opportunity to practice a certain skill.
Key features of a good mathematics problem includes:
- It must begin where the students are mathematically.
- The feature of the problem must be the mathematics that students are to learn.
- It must require justifications and explanations for both answers and methods of solving.
Problem solving is not a neat and orderly process. Think about needlework. On the front side, it is neat and perfect and pretty.
But look at the b ack.
It is messy and full of knots and loops. Problem solving in mathematics is also like this and we need to help our students be “messy” with problem solving; they need to go through those knots and loops and learn how to solve problems with the teacher’s guidance.
When you teach through problem solving , your students are focused on ideas and sense-making and they develop confidence in mathematics!
Mathematics Tasks and Activities that Promote Teaching through Problem Solving
Choosing the Right Task
Selecting activities and/or tasks is the most significant decision teachers make that will affect students’ learning. Consider the following questions:
- Teachers must do the activity first. What is problematic about the activity? What will you need to do BEFORE the activity and AFTER the activity? Additionally, think how your students would do the activity.
- What mathematical ideas will the activity develop? Are there connections to other related mathematics topics, or other content areas?
- Can the activity accomplish your learning objective/goals?
Low Floor High Ceiling Tasks
By definition, a “ low floor/high ceiling task ” is a mathematical activity where everyone in the group can begin and then work on at their own level of engagement. Low Floor High Ceiling Tasks are activities that everyone can begin and work on based on their own level, and have many possibilities for students to do more challenging mathematics. One gauge of knowing whether an activity is a Low Floor High Ceiling Task is when the work on the problems becomes more important than the answer itself, and leads to rich mathematical discourse [Hover: ways of representing, thinking, talking, agreeing, and disagreeing; the way ideas are exchanged and what the ideas entail; and as being shaped by the tasks in which students engage as well as by the nature of the learning environment].
The strengths of using Low Floor High Ceiling Tasks:
- Allows students to show what they can do, not what they can’t.
- Provides differentiation to all students.
- Promotes a positive classroom environment.
- Advances a growth mindset in students
- Aligns with the Standards for Mathematical Practice
Examples of some Low Floor High Ceiling Tasks can be found at the following sites:
- YouCubed – under grades choose Low Floor High Ceiling
- NRICH Creating a Low Threshold High Ceiling Classroom
- Inside Mathematics Problems of the Month
Math in 3-Acts
Math in 3-Acts was developed by Dan Meyer to spark an interest in and engage students in thought-provoking mathematical inquiry. Math in 3-Acts is a whole-group mathematics task consisting of three distinct parts:
Act One is about noticing and wondering. The teacher shares with students an image, video, or other situation that is engaging and perplexing. Students then generate questions about the situation.
In Act Two , the teacher offers some information for the students to use as they find the solutions to the problem.
Act Three is the “reveal.” Students share their thinking as well as their solutions.
“Math in 3 Acts” is a fun way to engage your students, there is a low entry point that gives students confidence, there are multiple paths to a solution, and it encourages students to work in groups to solve the problem. Some examples of Math in 3-Acts can be found at the following websites:
- Dan Meyer’s Three-Act Math Tasks
- Graham Fletcher3-Act Tasks ]
- Math in 3-Acts: Real World Math Problems to Make Math Contextual, Visual and Concrete
Number talks are brief, 5-15 minute discussions that focus on student solutions for a mental math computation problem. Students share their different mental math processes aloud while the teacher records their thinking visually on a chart or board. In addition, students learn from each other’s strategies as they question, critique, or build on the strategies that are shared.. To use a “number talk,” you would include the following steps:
- The teacher presents a problem for students to solve mentally.
- Provide adequate “ wait time .”
- The teacher calls on a students and asks, “What were you thinking?” and “Explain your thinking.”
- For each student who volunteers to share their strategy, write their thinking on the board. Make sure to accurately record their thinking; do not correct their responses.
- Invite students to question each other about their strategies, compare and contrast the strategies, and ask for clarification about strategies that are confusing.
“Number Talks” can be used as an introduction, a warm up to a lesson, or an extension. Some examples of Number Talks can be found at the following websites:
- Inside Mathematics Number Talks
- Number Talks Build Numerical Reasoning
Saying “This is Easy”
“This is easy.” Three little words that can have a big impact on students. What may be “easy” for one person, may be more “difficult” for someone else. And saying “this is easy” defeats the purpose of a growth mindset classroom, where students are comfortable making mistakes.
When the teacher says, “this is easy,” students may think,
- “Everyone else understands and I don’t. I can’t do this!”
- Students may just give up and surrender the mathematics to their classmates.
- Students may shut down.
Instead, you and your students could say the following:
- “I think I can do this.”
- “I have an idea I want to try.”
- “I’ve seen this kind of problem before.”
Tracy Zager wrote a short article, “This is easy”: The Little Phrase That Causes Big Problems” that can give you more information. Read Tracy Zager’s article here.
Do you want your students to memorize concepts, or do you want them to understand and apply the mathematics for different situations?
What is a “worksheet” in mathematics? It is a paper and pencil assignment when no other materials are used. A worksheet does not allow your students to use hands-on materials/manipulatives [Hover: physical objects that are used as teaching tools to engage students in the hands-on learning of mathematics]; and worksheets are many times “naked number” with no context. And a worksheet should not be used to enhance a hands-on activity.
Students need time to explore and manipulate materials in order to learn the mathematics concept. Worksheets are just a test of rote memory. Students need to develop those higher-order thinking skills, and worksheets will not allow them to do that.
One productive belief from the NCTM publication, Principles to Action (2014), states, “Students at all grade levels can benefit from the use of physical and virtual manipulative materials to provide visual models of a range of mathematical ideas.”
You may need an “activity sheet,” a “graphic organizer,” etc. as you plan your mathematics activities/lessons, but be sure to include hands-on manipulatives. Using manipulatives can
- Provide your students a bridge between the concrete and abstract
- Serve as models that support students’ thinking
- Provide another representation
- Support student engagement
- Give students ownership of their own learning.
Adapted from “ The Top 5 Reasons for Using Manipulatives in the Classroom ”.
any task or activity for which the students have no prescribed or memorized rules or methods, nor is there a perception by students that there is a specific ‘correct’ solution method
should be intriguing and contain a level of challenge that invites speculation and hard work, and directs students to investigate important mathematical ideas and ways of thinking toward the learning
involves teaching a skill so that a student can later solve a story problem
when we teach students how to problem solve
teaching mathematics content through real contexts, problems, situations, and models
a mathematical activity where everyone in the group can begin and then work on at their own level of engagement
20 seconds to 2 minutes for students to make sense of questions
Mathematics Methods for Early Childhood by Janet Stramel is licensed under a Creative Commons Attribution 4.0 International License , except where otherwise noted.
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Problem Solvers: A Free Early Math Curriculum
- August 31, 2022
A Free Early Math Curriculum for Children Aged 30 to 48 months
New Winter Bonus Unit: “Chilly Patterns” now Available!!
Research tells us that early math skills are a powerful predictor of overall school achievement. But what does “early math” look like for toddlers and young preschoolers? It looks like Problem Solvers.
Problem Solvers is a free, downloadable early math curriculum that includes:
- 22 play-based early math activities, spanning 7 domains of early math
- 22 specially-composed songs that support early math learning in each activity
- 22 book suggestions and extension activities that nurture early math language through read-alouds
- 22 parent resources (English/Spanish) that build bridges between school and home, and give parents ideas for engaging math play
- A Teacher’s Guide to help educators implement the curriculum
The development of Problem Solvers was made possible by the generous support of the Honda USA Foundation and the Dr. Seuss Foundation. We are deeply grateful to both foundations for allowing us to make this resource available to early education programs nationwide.
Get Problem Solvers today.
- Download a sample Problem Solvers activity .
- Download the entire Problem Solvers curriculum at no charge, including songs. (You’ll be asked to share your email.)
- Download the Problem Solvers Teacher’s Guide.
What are early educators saying about Problem Solvers?
“Problem Solvers helped us connect math to books, music, and free play. That was something new for me—fresh and different.”
“It’s always been hard to think of different math activities so this curriculum gave us more ideas and resources!”
“We have a full curriculum and we were still able to add Problem Solvers as an additional piece—and it wasn’t overwhelming. The kids wanted it and asked for it, even if the teachers missed it that day.”
“We saw the children focusing on this new math language and using it throughout the classroom – Is it longer? Is it more? Making lots of comparisons, doing lots of counting.”
Want more resources on early math?
Here are some of our favorite resources:
ZERO TO THREE’s Let’s Talk About Math video series: See how early math skills develop from birth to five. Includes free parent resources.
ZERO TO THREE’s Math4Littles resources: Explore this set of 36 activities for two- to three-year-olds that build math skills through parent-child play.
Erikson Early Math Collaborative : Browse a library of teacher-friendly instructional resources and fun activities for young children aligned to early math objectives.
Learning and Teaching With Learning Trajectories : Early Math Birth to Grade 3: Learn more about the math skills that children are developing in the early years.
Early Math Counts : Explore resources for creating math-rich early childhood environments.
Development and Research in Early Math Education ( DREME ): Check out math-focused children’s book suggestions with activity ideas.
Finding Math from the Institute for Learning & Brain Sciences, University of Washington: Access this suite of resources designed to help you discover the math in your everyday life.
Looking for training or professional development on early math?
Contact us to set up a professional development experience for your staff to build knowledge around early math instruction and support the implementation of Problem Solvers.
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Integral Calculus AP®︎/College Calculus AB AP®︎/College Calculus BC Calculus 1 Calculus 2 Multivariable calculus Differential equations Linear algebra Early math Counting Addition and subtraction
The list of examples is supplemented by tips to create engaging and challenging math word problems. 120 Math word problems, categorized by skill Addition word problems Best for: 1st grade, 2nd grade 1. Adding to 10: Ariel was playing basketball. 1 of her shots went in the hoop. 2 of her shots did not go in the hoop.
Example: Mr. Jones has a total of 25 chickens and cows on his farm. How many of each does he have if all together there are 76 feet? Step 1: Understanding the problem
This blog post will answer the following questions: What is an open ended math question? What are the differences between open-ended and closed-ended problems in math? Why should I implement open ended questions in my classroom? What are the disadvantages of using open-ended math problems?
She gives 9 to Sarah. How many apples does Rachel have now? Jack has 8 cats and 2 dogs. Jill has 7 cats and 4 dogs. How many dogs are there in all? if there are 40 cookies all together and A takes 10 and B takes 5 how many are left If Jane has 23 cats and I have 2 cats, and then Jane gives me 5 cats, how many more cats does Jane have than I?
are addition, subtraction, multiplication, or division. However, some story problems. have more than one step, involving more than one key word and/or operation. We'll. show you a few of these now. Carly is making a dress. She needs 1 yard of yellow fabric, 1.5 yards of purple. fabric, and .5 yards of green fabric.
Team Work Counts. After going through the process with the class, we decided to split the students into small groups of 3 and 4 to solve a math problem together. The groups were expected to use the same process that we used to solve the problem. It took a while but check out one of the final products below.
To help the students with their problem-solving "problem," let's look at some examples of mathematical problems and some general methods for solving problems: Problem Identify the following four-digit number when presented with the following information: One of the four digits is a 1.
Open-ended math problem solving tasks: promote multiple solution paths and/or multiple solutions. boost critical thinking and math reasoning skills. increase opportunities for developing perseverance. provide opportunities to justify answer choices. strengthen kids written and oral communication skills.
Brainstorming. Then students seek different solutions. As they read, they wonder, "Does it sound right? Does it make sense? How else could it sound to make more sense? What other sounds do those letters make?" Social problem-solving aspect: Students reflect on questions such as, "How can you solve the problem or make the situation better?
Use a Problem-Solving Strategy for Word Problems. We have reviewed translating English phrases into algebraic expressions, using some basic mathematical vocabulary and symbols. We have also translated English sentences into algebraic equations and solved some word problems. The word problems applied math to everyday situations.
No. 1 - Create a visual image One option is to teach children to create a visual image of the situation. Many times, this is an effective problem-solving skill. They are able to close their eyes and create a mind picture of the problem. For younger students, it may be helpful to draw out the problem they see on a piece of paper.
Step 1 - Understand the Problem. To help students understand the problem, I provided them with sample problems, and together we did five important things: read the problem carefully. restated the problem in our own words. crossed out unimportant information. circled any important information.
This text covers elementary mathematics strands including place value, numbers and operations, fractions, patterns, algebraic thinking, decimals, and geometry. Measurement and Data and Statistics strands are not included in this particular text. ... Reviewed by Kane Jessen, Math Instructor, Community College of Aurora on 8/13/20
1. There's more than one right answer and more than one right method A "real world task" can be as simple as asking students to think of equations that will get them to a particular "target"...
Part of solving a problem is understanding what is being asked, and knowing what a solution should look like. Problems often involve false starts, making mistakes, and lots of scratch paper! In an exercise, you are often practicing a skill. You may have seen a teacher demonstrate a technique, or you may have read a worked example in the book.
Problem solving can be a daunting aspect of effective mathematics teaching, but it does not have to be! In this post, I share seven strategic ways to integrate problem solving into your everyday math program. In the middle of our problem solving lesson, my district math coordinator stopped by for a surprise walkthrough. I was so excited!
Basic Elementary Math Problems with Solutions Basic Elementary Math Problems with Solutions In early elementary school, students learn to add, subtract, multiply and divide using whole numbers. Read on for tips and sample problems to help your child, where he's just beginning to add or learning the basics of multiplication.
Encourage Productive Struggle. Problem-solving is an integral component of math, and allowing students to struggle productively as they attempt to solve complex problems "sends the message that the teacher believes students are capable of doing and creating mathematics," write Rhodes and Gareis. High school math teacher Solenne Abaziou, in ...
The Algebra 1 course, often taught in the 9th grade, covers Linear equations, inequalities, functions, and graphs; Systems of equations and inequalities; Extension of the concept of a function; Exponential models; and Quadratic equations, functions, and graphs. Khan Academy's Algebra 1 course is built to deliver a comprehensive, illuminating, engaging, and Common Core aligned experience!
Problem Solving Strategy 9 (Find the Math, Remove the Context). Sometimes the problem has a lot of details in it that are unimportant, or at least unimportant for getting started. The goal is to find the underlying math problem, then come back to the original question and see if you can solve it using the math.
For example, students are learning how to multiply a two-digit number by a one-digit number, and the story problems you select are multiplication problems. Be sure when you are teaching for problem solving, you select or develop tasks that can promote the development of mathematical understanding. Teaching about problem solving
Problem Solvers is a free, downloadable early math curriculum that includes: 22 play-based early math activities, spanning 7 domains of early math. 22 specially-composed songs that support early math learning in each activity. 22 book suggestions and extension activities that nurture early math language through read-alouds.