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Kindergarten, basic geometry, pre-algebra, algebra basics, high school geometry.

## Statistics and probability

- 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
- Measurement
- 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
- Quadrilaterals
- 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
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- Properties of numbers
- Variables & expressions
- Equations & inequalities introduction
- Data and statistics
- Negative numbers: addition and subtraction
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- Rates & proportional relationships
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- Numbers and operations
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- Linear equations and functions
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- 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
- Foundations
- Algebraic expressions
- Linear equations and inequalities
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- 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
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- Complex numbers
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- Advanced regression (inference and transforming)
- Analysis of variance (ANOVA)
- Scatterplots
- Data distributions
- Two-way tables
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- More on regression
- Prepare for the 2020 AP®︎ Statistics Exam
- AP®︎ Statistics Standards mappings
- Polynomials
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- Probability and combinatorics
- Limits and continuity
- Derivatives: definition and basic rules
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- AP Calculus AB solved free response questions from past exams
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## 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!

- Teaching Tools
- Subtraction
- Multiplication
- Mixed operations
- Ordering and number sense
- Comparing and sequencing
- Physical measurement
- Ratios and percentages
- Probability and data relationships

A jolt of creativity would help. But it doesn’t come.

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

Best for: 1st grade, 2nd grade

## Subtraction word problems

Best for: 1st grade, second grade

## Practice math word problems with Prodigy Math

## Multiplication word problems

Best for: 2nd grade, 3rd grade

## Division word problems

Best for: 3rd grade, 4th grade, 5th grade

## Mixed operations word problems

## Ordering and number sense word problems

33. Composing Numbers: What number is 6 tens and 10 ones?

## Fractions word problems

Best for: 3rd grade, 4th grade, 5th grade, 6th grade

## Decimals word problems

Best for: 4th grade, 5th grade

## 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?

## Time word problems

## Money word problems

Best for: 1st grade, 2nd grade, 3rd grade, 4th grade, 5th grade

## Physical measurement word problems

Best for: 1st grade, 2nd grade, 3rd grade, 4th grade

## Ratios and percentages word problems

Best for: 4th grade, 5th grade, 6th grade

## Probability and data relationships word problems

Best for: 4th grade, 5th grade, 6th grade, 7th grade

## Geometry word problems

Best for: 4th grade, 5th grade, 6th grade, 7th grade, 8th grade

99. Understanding 2D Shapes: Kevin draws a shape with 4 equal sides. What shape did he draw?

102. Understanding 3D Shapes: Martha draws a shape that has 6 square faces. What shape did she draw?

## Variables word problems

Best for: 6th grade, 7th grade, 8th grade

## How to easily make your own math word problems & word problems worksheets

- 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.

## Final thoughts about math word problems

WOULD YOU LIKE ACCESS TO ALL THE FREEBIES FOR ELEMENTARY TEACHERS? ➔

## Get a collection of FREE MATH RESOURCES for your grade level!

Open ended math questions and problems for elementary students.

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?

## What are the Differences Between Open-Ended and Closed-Ended Problems in Math?

## What are the Advantages and Disadvantages of Open Ended Math Problems?

Advantages of open ended math problems.

- 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

- 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

- 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?

## Get Our Open-Ended Math Prompts

Math resources for 1st-5th grade teachers.

## Try a Collection of our Math Resources for Free!

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

## Word Problems

Solve a word problem and explore related facts.

## Solve a word problem:

## Story Problems

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

What is the key word in 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.

Therefore, our answer is 11 toy trucks altogether.

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?

Now, let’s try a couple harder problems.

This problem is a multiplication problem. Now, let’s get it set up. How will this problem look?

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.

This problem is a division problem. Now, let’s get it set up. What would our equation be?

Now, perform the division. What is $27.89 / 3? (Round to the nearest cent)

$5.95 + $5.93 + $3.50 = $15.38

Thus, our final answer is that Carly will spend $15.38 on fabric for her dress.

a) How long will it take them to get to Florida? (in hours)

b) How much money should they leave for gasoline (going one way)?

- 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

## Previously Covered:

## Approaches to Problem Solving

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.

Recall that the clues’ relevance were identified and prioritized as follows:

Let’s solve this problem by representing it in a visual way , in this case, a diagram:

h = the height of triangle EFA and the height above the ground at which the ropes intersect

Given a pickle jar filled with marbles, about how many marbles does the jar contain?

How many more faces does a cube have than a square pyramid?

- 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.

## Mathematical Mistakes

## Circular Arguments

- 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.

## Assuming the Truth of the Converse

## Assuming the Truth of the Inverse

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.

## Faulty Generalizations

## Faulty Analogies

False (or tenuous) analogies are often used in persuasive arguments.

An arrow is used to indicate that q is derived from p, like this:

A direct proof will attempt to lay out the shortest number of steps between p and q.

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

Example: For all primes x ≥ 3, x is odd.

- 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

## 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.

## How and When Can I Use Them?

Solve and Explain Tasks Cards are very versatile. You can use them for:

## Shop Recommended Resources

## You might also like...

## Reflect and Reset: Tips for Becoming a Better Math Teacher

## Student Math Reflection Activities That Deepen Understanding

## 5 Math Mini-Lesson Ideas that Keep Students Engaged

## A Rigorous Elementary Math Curriculum for Busy Teachers

## What We Offer:

- 3.1 Use a Problem-Solving Strategy
- Introduction
- 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

## Learning Objectives

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

Use a Problem-Solving Strategy for Word Problems

## 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.

## Example 3.1

If this were a homework exercise, our work might look like this:

Let’s try this approach with another example.

## Example 3.2

## Example 3.3

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.

## Example 3.4

The sum of twice a number and seven is 15. Find the number.

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.

## Example 3.5

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.

## Example 3.6

## Try It 3.11

## Try It 3.12

The sum of two numbers is −18 . −18 . One number is 40 more than the other. Find the numbers.

## Example 3.7

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.

## Example 3.8

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.

## Example 3.9

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 .

## Example 3.10

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 .

## Example 3.11

## Try It 3.21

## Try It 3.22

## Section 3.1 Exercises

Use the Approach Word Problems with a Positive Attitude

In the following exercises, prepare the lists described.

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 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 .

## Everyday Math

Buying in Bulk Alicia bought a package of eight peaches for $3.20. Find the cost of each peach.

## Writing Exercises

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.

ⓑ If most of your checks were:

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Access for free at https://openstax.org/books/elementary-algebra/pages/1-introduction

- Authors: Lynn Marecek, MaryAnne Anthony-Smith
- Publisher/website: OpenStax
- Book title: Elementary Algebra
- Publication date: Feb 22, 2017
- Location: Houston, Texas
- Book URL: https://openstax.org/books/elementary-algebra/pages/1-introduction
- Section URL: https://openstax.org/books/elementary-algebra/pages/3-1-use-a-problem-solving-strategy

## 5 Problem-Solving Activities for Elementary Classrooms

## Teach the problems

## No. 1 – Create a visual image

## No. 2 – Use manipulatives

## No. 3 – Make a guess

## No. 4 – Patterns

## No. 5 – Making a list

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## Teaching Problem Solving in Math

## The Problem Solving Strategies

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.

## The Problem Solving Steps

- 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.

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

- 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!

- 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

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.

You can grab these problem-solving bookmarks for FREE by clicking here .

## FIND IT NOW!

## CHECK THESE OUT

## Force and Motion Science Mystery

## Natural Disasters Vocabulary Interactive Booklet

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## Mathematics for Elementary Teachers

Publisher: University of Hawaii Manoa

## Formats Available

Reviewed by Kevin Voogt, Assistant Professor, Grace College on 4/20/23

Comprehensiveness rating: 4 see less

I did not find mathematical errors in the text during my review.

The wording was quite clear and had nice explanations throughout.

It seems consistent throughout - with recurrent use of the same technical terms as needed.

Organization/Structure/Flow rating: 4

I did not see any issues with the interface. It was pretty user-friendly.

I did not notice any errors during my review.

I did not see anything insensitive or offensive in the text.

Reviewed by Sandra Zirkes, Teaching Professor, Bowling Green State University on 4/14/23

All information in the text is mathematically accurate and the writing and diagrams are error-free.

Organization/Structure/Flow rating: 5

The text is well written with no grammatical errors.

There is no apparent cultural insensitivity in the text.

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 use understandable and consistent terms.

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.

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 topics in this text are organized from basic to complex concepts in a logical, clear fashion.

I did not spot any grammatical errors in this text.

Reviewed by April Slack, Math Instructor, Aiken Technical College on 5/13/21

The text is free from grammatical errors.

Reviewed by Kane Jessen, Math Instructor, Community College of Aurora on 8/13/20

Content was found to be accurate, error-free and unbiased.

Reviewed by Reina Ojiri, Assistant Professor, Leeward Community College on 7/27/20

Comprehensiveness rating: 2 see less

Organization/Structure/Flow rating: 3

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

Yes the book is consistent throughout.

Organization/Structure/Flow rating: 2

I think the topics are clear but dated and not in the order as described above.

I did not notice any grammar errors.

Reviewed by Jamie Price, Assistant Professor, East Tennessee State University on 3/20/20

The organization of the topics makes sense according to the mathematics presented and is logical.

I did not find any grammatical errors in the text.

The specific content presented is up-to-date and usable.

The author has a consistent voice of teaching and presenting the material.

There are a few minor issues that would be resolved with a good proofread.

Reviewed by Ryan Nivens, Associate Professor, East Tennessee State University on 10/25/19

This book should last a very long time in terms of relevance.

A mathematician wrote this, the layout is logical without question.

There is nothing I find distracting in the layout and interface.

Reviewed by Monica Rose Gilmore, Graduate Student, CU Boulder on 7/1/19

Comprehensiveness rating: 3 see less

This text is free from grammatical errors.

Reviewed by Glenna Gustafson, Professor, Radford University on 5/22/19

The text is easy to read and the organization within each chapters makes navigation easy.

Reviewed by Karise Mace, Mathematics Instructor, Kuztown University on 5/16/19, updated 11/9/20

Reviewed by Desley Plaisance, Associate Professor, Nicholls State University on 4/29/19

The flow from topic to topic is consistent in presentation.

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.

Reviewed by Lisa Cooper, Assistant Professor, LSUS on 4/26/19

The text is written free from educational jargon; it is straightforward and easy to understand.

The text has a well-organized flow with the layout of each topic/chapter.

No grammatical errors were observed in my review of the text.

Reviewed by Demetrice Smith-Mutegi, Instructor/Coordinator, Marian University on 3/6/19

The chapters and subchapters can be easily accessed, breaking the material into smaller sections.

I did not observe any obvious grammatical errors.

This a great "discussion" text.

The text is very well organized and has an easy-to-read format and flow.

The text is written with professional level writing and is free of grammatical errors.

## Table of Contents

- Introduction
- Problem or Exercise?
- Problem Solving Strategies
- Beware of Patterns!
- Problem Bank
- Careful Use of Language in Mathematics
- Explaning Your Work
- The Last Step

- 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

- Review of Dots & Boxes Model
- Division and Decimals
- More x -mals
- Terminating or Repeating?
- Operations on Decimals
- Orders of Magnitude

## Ancillary Material

## About the Book

## About the Contributors

Michelle Manes, Associate Professor, Department of Mathematics, University of Hawaii

## Contribute to this Page

## How to Use Real-World Problems to Teach Elementary School Math: 6 Tips

## 1. There’s more than one right answer and more than one right method

## 2. Give kids a chance to explain their thinking

## 3. Be willing to deal with some off-the-wall answers

## 4. Let your students push themselves

## 5. Celebrate ‘favorite mistakes’ to encourage intellectual risk taking

## 6. Remember there’s no such thing as a being born with a ‘math brain’

## Sign Up for EdWeek Update

## Sign Up & Sign In

## Problem Solving Activities: 7 Strategies

What a crushing blow! Just when I thought I did something special, I find out I did it all wrong.

## When I Finally Saw the Light

## Problem Solving Activities

Here are seven ways to strategically reinforce problem solving skills in your classroom.

## Seasonal Problem Solving

## Cooperative Problem Solving Tasks

## Notice and Wonder

## Math Starters

## Calculators

## Three-Act Math Tasks

## Getting the Most from Each of the Problem Solving Activities

Which of the problem solving activities will you try first? Respond in the comments below.

## Shametria Routt Banks

- Assessment Tools
- Content and Standards
- Differentiation
- Math & Literature
- Math & Technology
- Math Routines
- Math Stations
- Virtual Learning
- Writing in Math

## You may also like...

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Thank you, Scott! Best wishes to you and your pre-service teachers this year!

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## Privacy Overview

## 4 Ways to Build Student-Centered Math Lessons

Here are four things to keep in mind when designing student-centered math lessons.

## Develop a Clear Mission Statement

## Connect to Students’ Experience

## Allow for Multiple Solving Pathways

## Encourage Productive Struggle

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## Unit 1: Algebra foundations

## Problem Solving Strategies

## Pólya’s How to Solve It

- 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?

- What if the picture was different?
- What if the numbers were simpler?
- What if I just made up some numbers?

This brings us to the most important problem solving strategy of all:

## Problem 2 (Payback)

## Think/Pair/Share

Watch the solution only after you tried this strategy for yourself.

## Problem 3 (Squares on a Chess Board)

## Think / Pair / Share

- 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?

## Problem 4 (Broken Clock)

- 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?

## 5 Teaching Mathematics Through Problem Solving

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.

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.

## Mathematics Tasks and Activities that Promote Teaching through Problem Solving

## Choosing the Right Task

- 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

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

Act Three is the “reveal.” Students share their thinking as well as their solutions.

- 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

- 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.

## Saying “This is Easy”

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:

## Using “Worksheets”

- 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 ”.

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

20 seconds to 2 minutes for students to make sense of questions

## Share This Book

## Problem Solvers: A Free Early Math Curriculum

A Free Early Math Curriculum for Children Aged 30 to 48 months

## New Winter Bonus Unit: “Chilly Patterns” now Available!!

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

## 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?

“It’s always been hard to think of different math activities so this curriculum gave us more ideas and resources!”

## Want more resources on early math?

Here are some of our favorite resources:

Early Math Counts : Explore resources for creating math-rich early childhood environments.

## IMAGES

## VIDEO

## COMMENTS

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.