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10 Strategies for Problem Solving in Math
Jessica Kaminski
8 minutes read
June 19, 2022
Kids often get stuck when it comes to problem solving. They become confused when you offer them word problems or include an unknown variable like x in their math question. In such cases, teachers have to guide kids through this problem-solving maze, which is why this article covers the strategies for problem solving in math and the ways your students can leverage them.
What Are Problem Solving Strategies in Math?
To solve an issue, one must have a reliable strategy. Strategies for problem solving in math refer to methods of approaching math questions to ensure accurate results and increased efficiency. Such strategies simplify math for kids with no experience in problem solving and those already familiar with it.
There are various ways to implement problem solving strategies in math, and each method is different. While none is foolproof, they can improve your student’s problem-solving skills, especially with exercises and examples. The keyword here is practice — the more problems students solve, the more strategies and methods they pick up.
Strategies for Problem Solving in Math
Even if a student is not a math whiz, appropriate strategies for problem-solving in math can help them find solutions. Students may solve math issues in many ways, but here are ten math strategies for problem solving with high success rates. Depending on usage and preference, the strategies give kids renewed confidence as they work through difficulties.
Understand the Problem
Before solving a math problem, kids need to know and understand their nature. They should identify if the question is a fraction problem , a word problem, a quadratic equation, etc. An excellent way to boost their understanding is to look for keywords in the problem, revisit other similar questions, or check online. This step keeps the student on track.
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Guess and Check
The guess and check approach is one of the time-intensive strategies for problem solving in math. Students are to keep guessing until they find the proper answer.
After assuming a solution, kids need to put it back into the math problem to determine its accuracy. The procedure may seem laborious, but it often uncovers patterns in a child’s thought process.
Work It Out
When kids are working on a math problem, please encourage them to write down every step. This strategy is a self-monitoring method for math students since it demands that they first understand the problem. If they immediately start solving the problem, they risk making mistakes.
Using this strategy, students will keep track of their ideas and correct mistakes before arriving at a final answer. Even after working out their math problems in the supplementary sheet, a child may still ask you to explain the processes. This confirmation stage etches the steps they took to solve the problem in their minds.
Work Backwards
There are times when math problems may be best solved by looking at them differently. Kids need to understand that recreating math problems will be handy for project management and engineering careers.
Using the “Work Backwards” strategy, students anticipate challenges in real-world situations and prepare for them. They can start with the final result and reverse engineer it to arrive at the initial problem.
A math problem that may seem confusing to kids can generally become simpler once you represent it visually. Having kids visualize and act out the math problem are some of the most effective math strategies for problem solving.
Drawing a picture or making tally marks on a sheet of working-out paper is a visualization option. You could also model the process on the whiteboard and give students a marker to doodle before writing down the solution.
Find a Pattern
Pattern recognition strategies help kids understand math fundamentals and remember formulas. The best way to uncover patterns in a math problem is to teach pupils to extract and list relevant details. They can use the strategy when learning shapes and repetitive concepts, which makes the approach one of the most effective elementary math strategies for problem solving.
Using this method, students will recognize similar information and find the missing details. Over time, this approach will help students solve math problems faster.
One of the best problem solving strategies for math word problems is asking oneself, “what are some possible solutions to this issue?” It helps you consider the question more carefully, think outside the box, and avoid tunnel vision when facing challenges. So, encourage kids to muse over math problems and not settle for the first answer that enters their minds.
Draw a Picture or Diagram
Like visualization, creation of a diagram of a math problem will help kids figure out the best ways to approach it. Use shapes or numbers to represent the forms to keep things basic. Depending on the situation, patterns and graphs may also be valuable, and you can encourage kids to use dots or letters to represent the items.
Diagrams are even beneficial in many non-geometrical situations. After studying, students can create sketches of the concepts they read about for later revision. The approach will help kids determine what kind of math problem they are dealing with and the steps needed whenever they encounter a similar idea.
Trial and error method
Trial and error approach may be one of the most common strategies for solving math problems. However, the efficiency of this strategy depends on its application. If students blindly try solving math questions without specific formulas or directions, the chances of success will be low.
On the other hand, if they start by making a list of possible solutions based on preset guidelines and then attempting each one, they increase their odds of finding the correct answer. So, don’t be quick to discourage kids from using the trial and error strategy.
Review answers with peers
Strategies for problem solving in math that involve reviewing solutions with peers are enjoyable. If students come up with different answers to the same question, encourage them to share their thought processes with the rest of the class.
You could also have a session with the class to compare children’s working techniques. This way, students can discover loopholes in their ideas and make the necessary adjustments.
Check out the Printable Math Worksheets for Your Kids!
Many strategies for problem solving in math influence students’ speed and efficiency in tests. That is why they need to learn the most reliable approaches. By following the problem solving strategies for math listed in this article, students will have better experiences dealing with math problems.
Jessica is a a seasoned math tutor with over a decade of experience in the field. With a BSc and Master's degree in Mathematics, she enjoys nurturing math geniuses, regardless of their age, grade, and skills. Apart from tutoring, Jessica blogs at Brighterly. She also has experience in child psychology, homeschooling and curriculum consultation for schools and EdTech websites.
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Math Problem Solving Strategies That Make Students Say “I Get It!”
Even students who are quick with math facts can get stuck when it comes to problem solving.
As soon as a concept is translated to a word problem, or a simple mathematical sentence contains an unknown, they’re stumped.
That’s because problem solving requires us to consciously choose the strategies most appropriate for the problem at hand . And not all students have this metacognitive ability.
But you can teach these strategies for problem solving. You just need to know what they are.
We’ve compiled them here divided into four categories:
Strategies for understanding a problem
Strategies for solving the problem, strategies for working out, strategies for checking the solution.
Get to know these strategies and then model them explicitly to your students. Next time they dive into a rich problem, they’ll be filling up their working out paper faster than ever!
Before students can solve a problem, they need to know what it’s asking them. This is often the first hurdle with word problems that don’t specify a particular mathematical operation.
Encourage your students to:
Read and reread the question
They say they’ve read it, but have they really ? Sometimes students will skip ahead as soon as they’ve noticed one familiar piece of information or give up trying to understand it if the problem doesn’t make sense at first glance.
Teach students to interpret a question by using self-monitoring strategies such as:
- Rereading a question more slowly if it doesn’t make sense the first time
- Asking for help
- Highlighting or underlining important pieces of information.
Identify important and extraneous information
John is collecting money for his friend Ari’s birthday. He starts with $5 of his own, then Marcus gives him another $5. How much does he have now?
As adults looking at the above problem, we can instantly look past the names and the birthday scenario to see a simple addition problem. Students, however, can struggle to determine what’s relevant in the information that’s been given to them.
Teach students to sort and sift the information in a problem to find what’s relevant. A good way to do this is to have them swap out pieces of information to see if the solution changes. If changing names, items or scenarios has no impact on the end result, they’ll realize that it doesn’t need to be a point of focus while solving the problem.
Schema approach
This is a math intervention strategy that can make problem solving easier for all students, regardless of ability.
Compare different word problems of the same type and construct a formula, or mathematical sentence stem, that applies to them all. For example, a simple subtraction problems could be expressed as:
[Number/Quantity A] with [Number/Quantity B] removed becomes [end result].
This is the underlying procedure or schema students are being asked to use. Once they have a list of schema for different mathematical operations (addition, multiplication and so on), they can take turns to apply them to an unfamiliar word problem and see which one fits.
Struggling students often believe math is something you either do automatically or don’t do at all. But that’s not true. Help your students understand that they have a choice of problem-solving strategies to use, and if one doesn’t work, they can try another.
Here are four common strategies students can use for problem solving.
Visualizing
Visualizing an abstract problem often makes it easier to solve. Students could draw a picture or simply draw tally marks on a piece of working out paper.
Encourage visualization by modeling it on the whiteboard and providing graphic organizers that have space for students to draw before they write down the final number.
Guess and check
Show students how to make an educated guess and then plug this answer back into the original problem. If it doesn’t work, they can adjust their initial guess higher or lower accordingly.
Find a pattern
To find patterns, show students how to extract and list all the relevant facts in a problem so they can be easily compared. If they find a pattern, they’ll be able to locate the missing piece of information.
Work backward
Working backward is useful if students are tasked with finding an unknown number in a problem or mathematical sentence. For example, if the problem is 8 + x = 12, students can find x by:
- Starting with 12
- Taking the 8 from the 12
- Being left with 4
- Checking that 4 works when used instead of x
Now students have understood the problem and formulated a strategy, it’s time to put it into practice. But if they just launch in and do it, they might make it harder for themselves. Show them how to work through a problem effectively by:
Documenting working out
Model the process of writing down every step you take to complete a math problem and provide working out paper when students are solving a problem. This will allow students to keep track of their thoughts and pick up errors before they reach a final solution.
Check along the way
Checking work as you go is another crucial self-monitoring strategy for math learners. Model it to them with think aloud questions such as:
- Does that last step look right?
- Does this follow on from the step I took before?
- Have I done any ‘smaller’ sums within the bigger problem that need checking?
Students often make the mistake of thinking that speed is everything in math — so they’ll rush to get an answer down and move on without checking.
But checking is important too. It allows them to pinpoint areas of difficulty as they come up, and it enables them to tackle more complex problems that require multiple checks before arriving at a final answer.
Here are some checking strategies you can promote:
Check with a partner
Comparing answers with a peer leads is a more reflective process than just receiving a tick from the teacher. If students have two different answers, encourage them to talk about how they arrived at them and compare working out methods. They’ll figure out exactly where they went wrong, and what they got right.
Reread the problem with your solution
Most of the time, students will be able to tell whether or not their answer is correct by putting it back into the initial problem. If it doesn’t work or it just ‘looks wrong’, it’s time to go back and fix it up.
Fixing mistakes
Show students how to backtrack through their working out to find the exact point where they made a mistake. Emphasize that they can’t do this if they haven’t written down everything in the first place — so a single answer with no working out isn’t as impressive as they might think!
Need more help developing problem solving skills?
Read up on how to set a problem solving and reasoning activity or explore Mathseeds and Mathletics, our award winning online math programs. They’ve got over 900 teacher tested problem solving activities between them!
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Strategies for Math Problem Solving
How do you teach strategies for solving math word problems?
Is there a step by step problem solving method that my students can use?
Do your students struggle to solve math word problems? Students often find it difficult to understand what to solve, how to start and find out the unknown. Solving math word problems doesn’t have to be hard. Teaching students how to solve math word problems is important. There are strategies for math problem solving that they can use today!
There are five strategies for math problem solving to word problems that you can teach your students in thirty minutes class. Before introducing these skills make sure you have reviewed how to read word problems first . The second step in the problem solving process is to teach strategies that will help your students become better problem solvers. Try one or all of them today!
1. Drawing a Picture or Diagram.
This is a great strategy to use with visual learners. Students who are visual learners process information that they can see better than information that they hear. Drawing a picture helps them see the problem.
Here’s an example of using the strategy of a picture. What’s the problem tell us? There are four apple juice boxes in the cooler and those apple juice boxes are 1/3 of the juice boxes in the cooler. (Also Step 1) Draw the Problem. Draw 4 apple juice boxes. Say these are 1/3 of the juice boxes. Draw one circle around the 4 apple juice boxes, and then draw 2 empty circles. Question what would go in the other circles and how to get to the correct answer. This is great for math chats about the possibilities.
2. Find a Pattern.
Students should list the information already given in the problem. This list should reveal some very critical information about the problem. Examine the list of information for a pattern. What looks alike in the numbers? Does it repeat? Does it double? After finding the pattern, students should be able to identify the answer to the word problem.
3. Guess and Check.
The strategy is exactly like the name. Students guess the answer and then check their guess to fit the conditions of the problem. It’s a simple strategy, but very powerful to get students thinking.
4. Make a List.
This strategy is one of the most powerful ones. Students decide what information goes on the list from the word problem given. Organize the list by categories and make sure all the pieces of the problem are on the list. Lastly have students review the information that they organized on a list. Does it make sense? Can you reach a conclusion to solve the problem?
5. Use Reasoning.
To use reasoning students first need to organize the information given into a chart. Examine the relationships between the numbers. Think about the data and form a logical conclusion. Students may have to eliminate information to find the answer. Reasoning is not always easy to teach. Here are some questions to help guide students through using reasoning.
- Does the information make sense?
- What do these numbers have in common?
- Is there a pattern or relationship between the numbers?
- What can you conclude about the information?
- Does this word problem ask you to find something?
The most important thing you can do when teaching strategies for math problem solving is share as many as possible. You are teaching your students how to become problem solvers. The more strategies they know, the more independent and confident in problem solving they will become. As students become fluent problem solvers, they will be able to solve any word problem.
Try one or all the strategies and download the problem solving guide today!
Hi I’m Kelly!
Hello! I'm Kelly McCown, the Teacher Author and Math Consultant behind this website. Thank you for taking the time to learn more about me and why I share classroom resources with fellow teachers. I started as a 5th grade teacher over 16 years ago. I loved getting to teach in a K-6 setting and be an advisor to the drama club. I moved to middle school and taught 6th, 7th, 8th grades and Algebra 1 Honors. I was a Middle School Math Club Coach for 3 years. I've had teaching certificates in elementary {K-6}, middle {English 5-9 and Mathematics 5-9}, and high school {Mathematics 6-12}. In 2013 I became a teacher author and started creating math curriculum for other teachers. I love teaching math to Elementary and Middle school students. Helping students conquer Math is something I take pride in. 100% of my Algebra I Honors students passed the state end of course … read more
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Module 1: Problem Solving Strategies
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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 1985.1
1. Image of Pólya by Thane Plambeck from Palo Alto, California (Flickr) [CC BY
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!
Problem Solving Strategy 1 (Guess and Test)
Make a guess and test to see if it satisfies the demands of the problem. If it doesn't, alter the guess appropriately and check again. Keep doing this until you find a solution.
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
We are given in the problem that there are 25 chickens and cows.
All together there are 76 feet.
Chickens have 2 feet and cows have 4 feet.
We are trying to determine how many cows and how many chickens Mr. Jones has on his farm.
Step 2: Devise a plan
Going to use Guess and test along with making a tab
Many times the strategy below is used with guess and test.
Make a table and look for a pattern:
Procedure: Make a table reflecting the data in the problem. If done in an orderly way, such a table will often reveal patterns and relationships that suggest how the problem can be solved.
Step 3: Carry out the plan:
Notice we are going in the wrong direction! The total number of feet is decreasing!
Better! The total number of feet are increasing!
Step 4: Looking back:
Check: 12 + 13 = 25 heads
24 + 52 = 76 feet.
We have found the solution to this problem. I could use this strategy when there are a limited number of possible answers and when two items are the same but they have one characteristic that is different.
Videos to watch:
1. Click on this link to see an example of “Guess and Test”
http://www.mathstories.com/strategies.htm
2. Click on this link to see another example of Guess and Test.
http://www.mathinaction.org/problem-solving-strategies.html
Check in question 1:
Place the digits 8, 10, 11, 12, and 13 in the circles to make the sums across and vertically equal 31. (5 points)
Check in question 2:
Old McDonald has 250 chickens and goats in the barnyard. Altogether there are 760 feet . How many of each animal does he have? Make sure you use Polya’s 4 problem solving steps. (12 points)
Problem Solving Strategy 2 (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 thinking visually can help!
Videos to watch demonstrating how to use "Draw a Picture".
1. Click on this link to see an example of “Draw a Picture”
2. Click on this link to see another example of Draw a Picture.
Problem Solving Strategy 3 ( Using a variable to find the sum of a sequence.)
Gauss's strategy for sequences.
last term = fixed number ( n -1) + first term
The fix number is the the amount each term is increasing or decreasing by. "n" is the number of terms you have. You can use this formula to find the last term in the sequence or the number of terms you have in a sequence.
Ex: 2, 5, 8, ... Find the 200th term.
Last term = 3(200-1) +2
Last term is 599.
To find the sum of a sequence: sum = [(first term + last term) (number of terms)]/ 2
Sum = (2 + 599) (200) then divide by 2
Sum = 60,100
Check in question 3: (10 points)
Find the 320 th term of 7, 10, 13, 16 …
Then find the sum of the first 320 terms.
Problem Solving Strategy 4 (Working Backwards)
This is considered a strategy in many schools. If you are given an answer, and the steps that were taken to arrive at that answer, you should be able to determine the starting point.
Videos to watch demonstrating of “Working Backwards”
https://www.youtube.com/watch?v=5FFWTsMEeJw
Karen is thinking of a number. If you double it, and subtract 7, you obtain 11. What is Karen’s number?
1. We start with 11 and work backwards.
2. The opposite of subtraction is addition. We will add 7 to 11. We are now at 18.
3. The opposite of doubling something is dividing by 2. 18/2 = 9
4. This should be our answer. Looking back:
9 x 2 = 18 -7 = 11
5. We have the right answer.
Check in question 4:
Christina is thinking of a number.
If you multiply her number by 93, add 6, and divide by 3, you obtain 436. What is her number? Solve this problem by working backwards. (5 points)
Problem Solving Strategy 5 (Looking for a Pattern)
Definition: A sequence is a pattern involving an ordered arrangement of numbers.
We first need to find a pattern.
Ask yourself as you search for a pattern – are the numbers growing steadily larger? Steadily smaller? How is each number related?
Example 1: 1, 4, 7, 10, 13…
Find the next 2 numbers. The pattern is each number is increasing by 3. The next two numbers would be 16 and 19.
Example 2: 1, 4, 9, 16 … find the next 2 numbers. It looks like each successive number is increase by the next odd number. 1 + 3 = 4.
So the next number would be
25 + 11 = 36
Example 3: 10, 7, 4, 1, -2… find the next 2 numbers.
In this sequence, the numbers are decreasing by 3. So the next 2 numbers would be -2 -3 = -5
-5 – 3 = -8
Example 4: 1, 2, 4, 8 … find the next two numbers.
This example is a little bit harder. The numbers are increasing but not by a constant. Maybe a factor?
So each number is being multiplied by 2.
16 x 2 = 32
1. Click on this link to see an example of “Looking for a Pattern”
2. Click on this link to see another example of Looking for a Pattern.
Problem Solving Strategy 6 (Make a List)
Example 1 : Can perfect squares end in a 2 or a 3?
List all the squares of the numbers 1 to 20.
1 4 9 16 25 36 49 64 81 100 121 144 169 196 225 256 289 324 361 400.
Now look at the number in the ones digits. Notice they are 0, 1, 4, 5, 6, or 9. Notice none of the perfect squares end in 2, 3, 7, or 8. This list suggests that perfect squares cannot end in a 2, 3, 7 or 8.
How many different amounts of money can you have in your pocket if you have only three coins including only dimes and quarters?
Quarter’s dimes
0 3 30 cents
1 2 45 cents
2 1 60 cents
3 0 75 cents
Videos demonstrating "Make a List"
Check in question 5:
How many ways can you make change for 23 cents using only pennies, nickels, and dimes? (10 points)
Problem Solving Strategy 7 (Solve a Simpler Problem)
Geometric Sequences:
How would we find the nth term?
Solve a simpler problem:
1, 3, 9, 27.
1. To get from 1 to 3 what did we do?
2. To get from 3 to 9 what did we do?
Let’s set up a table:
Term Number what did we do
Looking back: How would you find the nth term?
Find the 10 th term of the above sequence.
Let L = the tenth term
Problem Solving Strategy 8 (Process of Elimination)
This strategy can be used when there is only one possible solution.
I’m thinking of a number.
The number is odd.
It is more than 1 but less than 100.
It is greater than 20.
It is less than 5 times 7.
The sum of the digits is 7.
It is evenly divisible by 5.
a. We know it is an odd number between 1 and 100.
b. It is greater than 20 but less than 35
21, 23, 25, 27, 29, 31, 33, 35. These are the possibilities.
c. The sum of the digits is 7
21 (2+1=3) No 23 (2+3 = 5) No 25 (2 + 5= 7) Yes Using the same process we see there are no other numbers that meet this criteria. Also we notice 25 is divisible by 5. By using the strategy elimination, we have found our answer.
Check in question 6: (8 points)
Jose is thinking of a number.
The number is not odd.
The sum of the digits is divisible by 2.
The number is a multiple of 11.
It is greater than 5 times 4.
It is a multiple of 6
It is less than 7 times 8 +23
What is the number?
Click on this link for a quick review of the problem solving strategies.
https://garyhall.org.uk/maths-problem-solving-strategies.html
3 Reads Strategy for Successful Problem Solving in Math
Word Problems are often the hardest part of our math instruction. They can visually overwhelm students. They often contain extraneous information or multiple steps for completion. Students often struggle to persevere through complex problems. But, ultimately, it is through these complex problems that we are able to truly see our students’ understanding of math concepts and standards. Our students are expected to persevere through solving them, to demonstrate understanding, and to use a variety of strategies. I detail my experience with difficulties with story problems in my post Why Your Students Struggle with Word Problems . I use a modified 3 Reads Strategy in my classroom to help students make sense of complex word problems during our Word Problem of the Day . I connect it to the Close Reading we do during E/LA.
We have to read the problem closely to truly understand what is being asked of us as mathematicians.
The 3 Reads Strategy is a series of steps that helps students make sense of word problems. It’s focused on understanding the context. There are a variety of interpretations of the protocol. I have found my students have been increasingly successful following the 3 Read protocol daily. We do it during our Word Problem of the Day routine so we practice nearly every single day. At the beginning of the year, I walk my students through the 3 reads and we talk about the steps with each read. As the weeks go on, my scaffolding decreases as I expect students to apply the same steps independently. I often, especially with more complex word problems, do the first reading orally to provide access for all students. Here are the steps we take during our 3 Reads Routine.
3 Reads Strategy for Word Problems
1st read: read for gist.
The purpose of the first read is to get the gist of the word problem. Students should be able to answer what the problem is about; the context . Students should be able to retell, in their own words, what is happening in the word problem.
2nd Read: Read for the Unknown
The second read is focused on the unknown ; what is being solved for. Identifying the unknown helps students identify important and necessary information for solving during the third read. This helps them parse extraneous information out. It also helps students ensure they’re solving for what is actually being asked. I have my students underline important information in the question and also write a sentence frame for the solution.
3rd Read: Read for Quantities
In this read, students identify the quantities and relevant units. During this read, I have students circle the numbers and underline the key words (most often the units) for solving. It’s important to note that I do not mean keywords that are typically words relating to operations such as more. In this read, we focus on what is known ; the information given. With the unknown already being identified. Students then write an equation or expression to solve. They may also draw a picture if it’s helpful understanding the steps needed for solving.
Make a Plan
The last step in the 3 read protocol is to make a plan for solving. Now that students have identified what is being asked, the information that’s given to them, and what they are solving for, the last step is to actually solve. That may include modeling the problem with base ten blocks. It may also include using the standard algorithm to solve. Whatever strategy students feel they need, they do.
After students have worked through the problem, we share solutions and strategies. The focus is on so much more than a correct solution! I have students show their work and explain their thinking. Through our conversation we may critique someone else’s work to identify their mistake. We may share a variety of strategies for solving the equation. We may compare equations or expressions that were written for the problem. Because I’m walking around while students are working independently, I’m able to give on the spot support to some kids, while also identifying things I want to highlight for the group. This routine, and our steps after, go through so many of the Standards for Mathematical Practice!
It’s great to have a 3 Reads anchor chart or poster for student reference. A co-created anchor chart constructed with students while solving a complex problem would be great! I also have free black & white 3 Read Strategy posters that are perfect for printing on colored Astrobrights paper and made into a bulletin board. There’s also a 1-page 3 Reads Math Routine Poster that’s designed for student use.
The posters will be sent to you when you fill out the form below.
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We use our 3 Reads Strategy during our Word Problem of the Day routine. You can read more about it in the blog post linked below. If you want to take a closer look at my Word Problem of the Day Bundles for 1st, 2nd, and 3rd grades, I have them in my TpT store. Each bundle includes a free Back to School version that gives you a great look at the format of the problems.
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I am one of the teachers that are teaching Word Problems wrong! I’ve read through all you said and it makes sense. Will use your strategy as our term starts tomorrow and this term the focus is on Word problems. Thanks for sharing. Do appreciate it.
I hope it helps you and your kiddos!
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I’ve spent the last 15 years teaching in 1st, 2nd, and 3rd grades, and working beside elementary classrooms as an instructional coach and resource support. I’m passionate about math , literacy , and finding ways to make teachers’ days easier . I share from my experiences both in and out of the elementary classroom. Read more About Me .
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Math Problem-Solving: Combining Cognitive & Metacognitive Strategies
- Reading the problem. The student reads the problem carefully, noting and attempting to clear up any areas of uncertainly or confusion (e.g., unknown vocabulary terms).
- Paraphrasing the problem. The student restates the problem in his or her own words.
- ‘Drawing’ the problem. The student creates a drawing of the problem, creating a visual representation of the word problem.
- Creating a plan to solve the problem. The student decides on the best way to solve the problem and develops a plan to do so.
- Predicting/Estimating the answer. The student estimates or predicts what the answer to the problem will be. The student may compute a quick approximation of the answer, using rounding or other shortcuts.
- C omputing the answer. The student follows the plan developed earlier to compute the answer to the problem.
- Checking the answer. The student methodically checks the calculations for each step of the problem. The student also compares the actual answer to the estimated answer calculated in a previous step to ensure that there is general agreement between the two values.
- The student first self-instructs by stating, or ‘saying’, the purpose of the step (‘ Say ’).
- The student next self-questions by ‘asking’ what he or she intends to do to complete the step (‘ Ask ’).
- The student concludes the step by self-monitoring, or ‘checking’, the successful completion of the step (‘ Check ’).
- Verifying that the student has the necessary foundation skills to solve math word problems
- Using explicit instruction techniques to teach the cognitive and metacognitive strategies
- Ensuring that all instructional tasks allow the student to experience an adequate rate of success
- Providing regular opportunities for the student to be engaged in active accurate academic responding
- Offering frequent performance feedback to motivate the student and shape his or her learning.
Attachments
- Say-Ask-Check Student Self-Coaching (Metacognitive) Prompts
- Burns, M. K., VanDerHeyden, A. M., & Boice, C. H. (2008). Best practices in intensive academic interventions. In A. Thomas & J. Grimes (Eds.), Best practices in school psychology V (pp.1151-1162). Bethesda, MD: National Association of School Psychologists.
- Montague, M. (1992). The effects of cognitive and metacognitive strategy instruction on the mathematical problem solving of middle school students with learning disabilities. Journal of Learning Disabilities, 25, 230-248.
- Montague, M., & Dietz, S. (2009). Evaluating the evidence base for cognitive strategy instruction and mathematical problem solving. Exceptional Children, 75, 285-302.
Crocodile Maths – A Problem-Solving Strategy
by Romero Esposito | Mar 7, 2023 | Crocodiles
Crocodile maths is a problem-solving strategy that can be used to solve mathematical problems . It is a strategy that is based on the principles of trial and error and is suitable for solving problems that are not too complex. The crocodile maths strategy can be used to solve addition, subtraction, multiplication, and division problems.
Can you solve the crocodile math problem that stumped Scottish students? There was an uproar on social media because of the difficulty, and Scottish students cried. Check out my video for a quick explanation of the problem and how it was solved. = T(8) = 98 if (*)1/2(36 x2), then x = 8 if (*)1/2(36 x2), then T(19) = 8 if (1). What is the shortest path a crocodile can walk? Physical principles can be used in the real world. Light travels in a given direction at a given speed.
The phenomenon of light alternating between different medium is similar to that of light alternating between different medium. The physical properties of light bends and angles are described in terms of Snell’s Law. My blog Mind Your Decisions was established in 2007 to share a wealth of knowledge about mathematics, personal finance, and personal thoughts. Using math to get an edge over your competitors is demonstrated in The Joy of Game Theory. Irrationality Illusions: How to Make Smart Decisions and Overcome Bias explain the various ways in which we are biased when it comes to making decisions. Mind Your Puzzles is a collection of three math puzzle books: Book 1, Book 2, and Book 3. Mathematical concepts such as geometry, probability, logic, and game theory are covered in the puzzles. Since education can have such a significant impact on our society, I try to make it as widely available as possible at the lowest possible price.
When there is a large number, the alligator mouth is opened towards it. If you were told to show which number was greater than or less than than 5 and an alligator was given the number 8, the alligator would open his mouth. In this sense, the number of 5 is less than 8.
What Is The Crocodile Symbol Called In Math?
Another way to remember which symbol is greater than sign is to imagine it as a crocodile. The crocodile is hungry and wants to consume a large number of people. In other words, its mouth has a 3-degree angle.
What Does ≥ Mean?
The symbol * indicates that there is more than or equal to something.
The Power Of Symbols In Text Messaging
The majority of messages in text messages contain symbols that convey the meaning. A (*) asterisk indicates that a word is censored, for example. A swastika, an asterisk, or another expression can be used to indicate that a text message is not a joke. Symbols can be useful for communicating. It is simple to understand why they are useful in assisting people in expressing themselves.
What Is This Maths Symbol Called?
Math symbols can be used to represent basic math concepts . *asteriskmultiplication*times signmultiplication28 more rowsplus – minus both plus and minus operationsplusboth plus and minus operations, plusboth minus and minus operations, plusboth minus and plus operations, plusboth minus
The Use Of ∈ In Math And Linguistics
In mathematics, the symbol * denotes the unit of membership and is an element. The statement x*A means x is an element of the set A, which means that x is one of the many objects in the set A that are of the same element. In linguistics, the symbol * is sometimes used to represent the zero symbol, as Bourbaki suggests, and is sometimes used to represent the empty set as well.
I am Romero Esposito, and I am passionate about reptiles. I have been keeping reptiles as pets for over 20 years, and I have also worked with reptiles in zoos and nature centers.
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Math Problem Solving Strategies
How many times have you been teaching a concept that students are feeling confident in, only for them to completely shut down when faced with a word problem? For me, the answer is too many to count. Word problems require problem solving strategies. And more than anything, word problems require decoding, eliminating extra information, and opportunities for students to solve for something that the question is not asking for. There are so many places for students to make errors! Let’s talk about some problem solving strategies that can help guide and encourage students!
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Update for 2020: Scroll down to the bottom to read how we address showing your work during distance learning.
1. C.U.B.E.S.
C.U.B.E.S stands for circle the important numbers, underline the question, box the words that are keywords, eliminate extra information, and solve by showing work.
- Why I like it: Gives students a very specific ‘what to do.’
- Why I don’t like it: With all of the annotating of the problem, I’m not sure that students are actually reading the problem. None of the steps emphasize reading the problem but maybe that is a given.
2. R.U.N.S.
R.U.N.S. stands for read the problem, underline the question, name the problem type, and write a strategy sentence.
- Why I like it: Students are forced to think about what type of problem it is (factoring, division, etc) and then come up with a plan to solve it using a strategy sentence. This is a great strategy to teach when you are tackling various types of problems.
- Why I don’t like it: Though I love the opportunity for students to write in math, writing a strategy statement for every problem can eat up a lot of time.
3. U.P.S. CHECK
U.P.S. Check stands for understand, plan, solve, and check.
- Why I like it: I love that there is a check step in this problem solving strategy. Students having to defend the reasonableness of their answer is essential for students’ number sense.
- Why I don’t like it: It can be a little vague and doesn’t give concrete ‘what to dos.’ Checking that students completed the ‘understand’ step can be hard to see.
4. Maneuvering the Middle Strategy AKA K.N.O.W.S.
Here is the strategy that I adopted a few years ago. It doesn’t have a name yet nor an acronym, (so can it even be considered a strategy…?)
UPDATE: IT DOES HAVE A NAME! Thanks to our lovely readers, Wendi and Natalie!
- Know: This will help students find the important information.
- Need to Know: This will force students to reread the question and write down what they are trying to solve for.
- Organize: I think this would be a great place for teachers to emphasize drawing a model or picture.
- Work: Students show their calculations here.
- Solution: This is where students will ask themselves if the answer is reasonable and whether it answered the question.
Here is where I typically struggle with problem solving strategies: 1) modeling the strategy in my own teaching weeks after I have taught students to use the strategy and 2) enforcing students to do it. So… basically everything. This might be why I haven’t been able to stick with a strategy from year to year.
5. Digital Learning Struggle
Many teachers are facing how to have students show their work or their problem solving strategy when tasked with submitting work online. Platforms like Kami make this possible. Go Formative has a feature where students can use their mouse to “draw” their work. If your students don’t have access to a touchscreen, then ha ving them submit images of their work might be your best bet. To simplify this process, I would recommend asking students to submit an image for all of their work — not individual problems. We do not want to create additional barriers for students.
If you want to spend your energy teaching student problem solving instead of writing and finding math problems, look no further than our All Access membership . Click the button to learn more.
Students who plan succeed at a higher rate than students who do not plan. Do you have a go to problem solving strategy that you teach your students?
Editor’s Note: Maneuvering the Middle has been publishing blog posts for nearly 6 years! This post was originally published in September of 2017. It has been revamped for relevancy and accuracy.
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Reader Interactions
18 comments.
October 4, 2017 at 7:55 pm
As a reading specialist, I love your strategy. It’s flexible, “portable” for any problem, and DOES get kids to read and understand the problem by 1) summarizing what they know and 2) asking a question for what they don’t yet know — two key comprehension strategies! How about: “Make a Plan for the Problem”? That’s the core of your rationale for using it, and I bet you’re already saying this all the time in class. Kids will get it even more because it’s a statement, not an acronym to remember. This is coming to my reading class tomorrow with word problems — thank you!
October 4, 2017 at 8:59 pm
Hi Nora! I have never thought about this as a reading strategy, genius! Please let me know how it goes. I would love to hear more!
December 15, 2017 at 7:57 am
Hi! I am a middle school teacher in New York state and my district is “gung ho” on CUBES. I completely agree with you that kids are not really reading the problem when using CUBES and only circling and boxing stuff then “doing something” with it without regard for whether or not they are doing the right thing (just a shot in the dark!). I have adopted what I call a “no fear word problems” procedure because several of my students told me they are scared of word problems and I thought, “let’s take the scary out of it then by figuring out how to dissect it and attack it! Our class strategy is nearly identical to your strategy:
1. Pre-Read the problem (do so at your normal reading speed just so you basically know what it says) 2. Active Read: Make a short list of: DK (what I Definitely Know), TK (what I Think I Know and should do), and WK (what I Want to Know– what is the question?) 3. Draw and Solve 4. State the answer in a complete sentence.
This procedure keep kids for “surfacely” reading and just trying something that doesn’t make sense with the context and implications of the word problem. I adapted some of it from Harvey Silver strategies (from Strategic Teacher) and incorporated the “Read-Draw-Write” component of the Eureka Math program. One thing that Harvey Silver says is, “Unlike other problems in math, word problems combine quantitative problem solving with inferential reading, and this combination can bring out the impulsive side in students.” (The Strategic Teacher, page 90, Silver, et al.; 2007). I found that CUBES perpetuates the impulsive side of middle school students, especially when the math seems particularly difficult. Math word problems are packed full of words and every word means something to about the intent and the mathematics in the problem, especially in middle school and high school. Reading has to be done both at the literal and inferential levels to actually correctly determine what needs to be done and execute the proper mathematics. So far this method is going really well with my students and they are experiencing higher levels of confidence and greater success in solving.
October 5, 2017 at 6:27 am
Hi! Another teacher and I came up with a strategy we call RUBY a few years ago. We modeled this very closely after close reading strategies that are language arts department was using, but tailored it to math. R-Read the problem (I tell kids to do this without a pencil in hand otherwise they are tempted to start underlining and circling before they read) U-Underline key words and circle important numbers B-Box the questions (I always have student’s box their answer so we figured this was a way for them to relate the question and answer) Y-You ask yourself: Did you answer the question? Does your answer make sense (mathematically)
I have anchor charts that we have made for classrooms and interactive notebooks if you would like them let me me know….
October 5, 2017 at 9:46 am
Great idea! Thanks so much for sharing with our readers!
October 8, 2017 at 6:51 pm
LOVE this idea! Will definitely use it this year! Thank you!
December 18, 2019 at 7:48 am
I would love an anchor chart for RUBY
October 15, 2017 at 11:05 am
I will definitely use this concept in my Pre-Algebra classes this year; I especially like the graphic organizer to help students organize their thought process in solving the problems too.
April 20, 2018 at 7:36 am
I love the process you’ve come up with, and think it definitely balances the benefits of simplicity and thoroughness. At the risk of sounding nitpicky, I want to point out that the examples you provide are all ‘processes’ rather than strategies. For the most part, they are all based on the Polya’s, the Hungarian mathematician, 4-step approach to problem solving (Understand/Plan/Solve/Reflect). It’s a process because it defines the steps we take to approach any word problem without getting into the specific mathematical ‘strategy’ we will use to solve it. Step 2 of the process is where they choose the best strategy (guess and check, draw a picture, make a table, etc) for the given problem. We should start by teaching the strategies one at a time by choosing problems that fit that strategy. Eventually, once they have added multiple strategies to their toolkit, we can present them with problems and let them choose the right strategy.
June 22, 2018 at 12:19 pm
That’s brilliant! Thank you for sharing!
May 31, 2018 at 12:15 pm
Mrs. Brack is setting up her second Christmas tree. Her tree consists of 30% red and 70% gold ornaments. If there are 40 red ornaments, then how many ornaments are on the tree? What is the answer to this question?
June 22, 2018 at 10:46 am
Whoops! I guess the answer would not result in a whole number (133.333…) Thanks for catching that error.
July 28, 2018 at 6:53 pm
I used to teach elementary math and now I run my own learning center, and we teach a lot of middle school math. The strategy you outlined sounds a little like the strategy I use, called KFCS (like the fast-food restaurant). K stands for “What do I know,” F stands for “What do I need to Find,” C stands for “Come up with a plan” [which includes 2 parts: the operation (+, -, x, and /) and the problem-solving strategy], and lastly, the S stands for “solve the problem” (which includes all the work that is involved in solving the problem and the answer statement). I find the same struggles with being consistent with modeling clearly all of the parts of the strategy as well, but I’ve found that the more the student practices the strategy, the more intrinsic it becomes for them; of course, it takes a lot more for those students who struggle with understanding word problems. I did create a worksheet to make it easier for the students to follow the steps as well. If you’d like a copy, please let me know, and I will be glad to send it.
February 3, 2019 at 3:56 pm
This is a supportive and encouraging site. Several of the comments and post are spot on! Especially, the “What I like/don’t like” comparisons.
March 7, 2019 at 6:59 am
Have you named your unnamed strategy yet? I’ve been using this strategy for years. I think you should call it K.N.O.W.S. K – Know N – Need OW – (Organise) Plan and Work S – Solution
September 2, 2019 at 11:18 am
Going off of your idea, Natalie, how about the following?
K now N eed to find out O rganize (a plan – may involve a picture, a graphic organizer…) W ork S ee if you’re right (does it make sense, is the math done correctly…)
I love the K & N steps…so much more tangible than just “Read” or even “Understand,” as I’ve been seeing is most common in the processes I’ve been researching. I like separating the “Work” and “See” steps. I feel like just “Solve” May lead to forgetting the checking step.
March 16, 2020 at 4:44 pm
I’m doing this one. Love it. Thank you!!
September 17, 2019 at 7:14 am
Hi, I wanted to tell you how amazing and kind you are to share with all of us. I especially like your word problem graphic organizer that you created yourself! I am adopting it this week. We have a meeting with all administrators to discuss algebra. I am going to share with all the people at the meeting.
I had filled out the paperwork for the number line. Is it supposed to go to my email address? Thank you again. I am going to read everything you ahve given to us. Have a wonderful Tuesday!
Math problem solving strategies with examples
Math problem solving strategies with examples is a software program that supports students solve math problems.
Problem Solving Strategies
Strategies for Problem Solving in Math Understand the Problem Guess and Check Work It Out Work Backwards Visualize Find a Pattern Think.
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17 Maths Problem Solving Strategies Boost your Learning
Look for a pattern. Example: Solution: Make an organized list. Example: Find the median of the following test scores: 73, 65, 82, 78, and 93. Guess and check
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Math app always seems to get the job done in an understandable fashion. Amazing app that can truly help you on your math needs, it always know how to answer my math equation either if I have to evaluate or simplify it. Its help A Lot, meet your match MATH.
It is very very helpful for my studies. Also, if you've already written the answer, it can tell younif you're right. This should be any Student’s companion for all calculations, it also helps by showing to steps it took to get an answer so you can get an idea of how to do the problem or other problems related to it.
Math Problem Solving Strategies That Make Students Say I
In these lessons, we will learn some math problem solving strategies for example, Verbal Model (or Logical Reasoning), Algebraic Model, Block Model (or
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5 Strategies to Learn to Solve Math Word Problems
5 Strategies to Learn to Solve Math Word Problems Cross out Extra Information. Along with highlighting important keywords students should also
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Problem-Solving Strategies and Obstacles
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From deciding what to eat for dinner to considering whether it's the right time to buy a house, problem-solving is a large part of our daily lives. Learn some of the problem-solving strategies that exist and how to use them in real life, along with ways to overcome obstacles that are making it harder to resolve the issues you face.
What Is Problem-Solving?
In cognitive psychology , the term 'problem-solving' refers to the mental process that people go through to discover, analyze, and solve problems.
A problem exists when there is a goal that we want to achieve but the process by which we will achieve it is not obvious to us. Put another way, there is something that we want to occur in our life, yet we are not immediately certain how to make it happen.
Maybe you want a better relationship with your spouse or another family member but you're not sure how to improve it. Or you want to start a business but are unsure what steps to take. Problem-solving helps you figure out how to achieve these desires.
The problem-solving process involves:
- Discovery of the problem
- Deciding to tackle the issue
- Seeking to understand the problem more fully
- Researching available options or solutions
- Taking action to resolve the issue
Before problem-solving can occur, it is important to first understand the exact nature of the problem itself. If your understanding of the issue is faulty, your attempts to resolve it will also be incorrect or flawed.
Problem-Solving Mental Processes
Several mental processes are at work during problem-solving. Among them are:
- Perceptually recognizing the problem
- Representing the problem in memory
- Considering relevant information that applies to the problem
- Identifying different aspects of the problem
- Labeling and describing the problem
Problem-Solving Strategies
There are many ways to go about solving a problem. Some of these strategies might be used on their own, or you may decide to employ multiple approaches when working to figure out and fix a problem.
An algorithm is a step-by-step procedure that, by following certain "rules" produces a solution. Algorithms are commonly used in mathematics to solve division or multiplication problems. But they can be used in other fields as well.
In psychology, algorithms can be used to help identify individuals with a greater risk of mental health issues. For instance, research suggests that certain algorithms might help us recognize children with an elevated risk of suicide or self-harm.
One benefit of algorithms is that they guarantee an accurate answer. However, they aren't always the best approach to problem-solving, in part because detecting patterns can be incredibly time-consuming.
There are also concerns when machine learning is involved—also known as artificial intelligence (AI)—such as whether they can accurately predict human behaviors.
Heuristics are shortcut strategies that people can use to solve a problem at hand. These "rule of thumb" approaches allow you to simplify complex problems, reducing the total number of possible solutions to a more manageable set.
If you find yourself sitting in a traffic jam, for example, you may quickly consider other routes, taking one to get moving once again. When shopping for a new car, you might think back to a prior experience when negotiating got you a lower price, then employ the same tactics.
While heuristics may be helpful when facing smaller issues, major decisions shouldn't necessarily be made using a shortcut approach. Heuristics also don't guarantee an effective solution, such as when trying to drive around a traffic jam only to find yourself on an equally crowded route.
Trial and Error
A trial-and-error approach to problem-solving involves trying a number of potential solutions to a particular issue, then ruling out those that do not work. If you're not sure whether to buy a shirt in blue or green, for instance, you may try on each before deciding which one to purchase.
This can be a good strategy to use if you have a limited number of solutions available. But if there are many different choices available, narrowing down the possible options using another problem-solving technique can be helpful before attempting trial and error.
In some cases, the solution to a problem can appear as a sudden insight. You are facing an issue in a relationship or your career when, out of nowhere, the solution appears in your mind and you know exactly what to do.
Insight can occur when the problem in front of you is similar to an issue that you've dealt with in the past. Although, you may not recognize what is occurring since the underlying mental processes that lead to insight often happen outside of conscious awareness .
Research indicates that insight is most likely to occur during times when you are alone—such as when going on a walk by yourself, when you're in the shower, or when lying in bed after waking up.
How to Apply Problem-Solving Strategies in Real Life
If you're facing a problem, you can implement one or more of these strategies to find a potential solution. Here's how to use them in real life:
- Create a flow chart . If you have time, you can take advantage of the algorithm approach to problem-solving by sitting down and making a flow chart of each potential solution, its consequences, and what happens next.
- Recall your past experiences . When a problem needs to be solved fairly quickly, heuristics may be a better approach. Think back to when you faced a similar issue, then use your knowledge and experience to choose the best option possible.
- Start trying potential solutions . If your options are limited, start trying them one by one to see which solution is best for achieving your desired goal. If a particular solution doesn't work, move on to the next.
- Take some time alone . Since insight is often achieved when you're alone, carve out time to be by yourself for a while. The answer to your problem may come to you, seemingly out of the blue, if you spend some time away from others.
Obstacles to Problem-Solving
Problem-solving is not a flawless process as there are a number of obstacles that can interfere with our ability to solve a problem quickly and efficiently. These obstacles include:
- Assumptions: When dealing with a problem, people can make assumptions about the constraints and obstacles that prevent certain solutions. Thus, they may not even try some potential options.
- Functional fixedness : This term refers to the tendency to view problems only in their customary manner. Functional fixedness prevents people from fully seeing all of the different options that might be available to find a solution.
- Irrelevant or misleading information: When trying to solve a problem, it's important to distinguish between information that is relevant to the issue and irrelevant data that can lead to faulty solutions. The more complex the problem, the easier it is to focus on misleading or irrelevant information.
- Mental set: A mental set is a tendency to only use solutions that have worked in the past rather than looking for alternative ideas. A mental set can work as a heuristic, making it a useful problem-solving tool. However, mental sets can also lead to inflexibility, making it more difficult to find effective solutions.
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How to Improve Your Problem-Solving Skills
In the end, if your goal is to become a better problem-solver, it's helpful to remember that this is a process. Thus, if you want to improve your problem-solving skills, following these steps can help lead you to your solution:
- Recognize that a problem exists . If you are facing a problem, there are generally signs. For instance, if you have a mental illness , you may experience excessive fear or sadness, mood changes, and changes in sleeping or eating habits. Recognizing these signs can help you realize that an issue exists.
- Decide to solve the problem . Make a conscious decision to solve the issue at hand. Commit to yourself that you will go through the steps necessary to find a solution.
- Seek to fully understand the issue . Analyze the problem you face, looking at it from all sides. If your problem is relationship-related, for instance, ask yourself how the other person may be interpreting the issue. You might also consider how your actions might be contributing to the situation.
- Research potential options . Using the problem-solving strategies mentioned, research potential solutions. Make a list of options, then consider each one individually. What are some pros and cons of taking the available routes? What would you need to do to make them happen?
- Take action . Select the best solution possible and take action. Action is one of the steps required for change . So, go through the motions needed to resolve the issue.
- Try another option, if needed . If the solution you chose didn't work, don't give up. Either go through the problem-solving process again or simply try another option.
You can find a way to solve your problems as long as you keep working toward this goal—even if the best solution is simply to let go because no other good solution exists.
Sarathy V. Real world problem-solving . Front Hum Neurosci . 2018;12:261. doi:10.3389/fnhum.2018.00261
Dunbar K. Problem solving . A Companion to Cognitive Science . 2017. doi:10.1002/9781405164535.ch20
Stewart SL, Celebre A, Hirdes JP, Poss JW. Risk of suicide and self-harm in kids: The development of an algorithm to identify high-risk individuals within the children's mental health system . Child Psychiat Human Develop . 2020;51:913-924. doi:10.1007/s10578-020-00968-9
Rosenbusch H, Soldner F, Evans AM, Zeelenberg M. Supervised machine learning methods in psychology: A practical introduction with annotated R code . Soc Personal Psychol Compass . 2021;15(2):e12579. doi:10.1111/spc3.12579
Mishra S. Decision-making under risk: Integrating perspectives from biology, economics, and psychology . Personal Soc Psychol Rev . 2014;18(3):280-307. doi:10.1177/1088868314530517
Csikszentmihalyi M, Sawyer K. Creative insight: The social dimension of a solitary moment . In: The Systems Model of Creativity . 2015:73-98. doi:10.1007/978-94-017-9085-7_7
Chrysikou EG, Motyka K, Nigro C, Yang SI, Thompson-Schill SL. Functional fixedness in creative thinking tasks depends on stimulus modality . Psychol Aesthet Creat Arts . 2016;10(4):425‐435. doi:10.1037/aca0000050
Huang F, Tang S, Hu Z. Unconditional perseveration of the short-term mental set in chunk decomposition . Front Psychol . 2018;9:2568. doi:10.3389/fpsyg.2018.02568
National Alliance on Mental Illness. Warning signs and symptoms .
Mayer RE. Thinking, problem solving, cognition, 2nd ed .
Schooler JW, Ohlsson S, Brooks K. Thoughts beyond words: When language overshadows insight. J Experiment Psychol: General . 1993;122:166-183. doi:10.1037/0096-3445.2.166
By Kendra Cherry Kendra Cherry, MS, is an author and educational consultant focused on helping students learn about psychology.
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5 Teaching Mathematics Through Problem Solving
Janet Stramel
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
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.
Using “Worksheets”
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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4 Math Problem Solving Strategies
These useful Math problem-solving strategies will help you pass the GED Math test.
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The four problem-solving strategies discussed here come with examples as well and include:
- Working backward
- Making a table or list
- Solving a simpler problem
- Guess & check
Problem-Solving Strategy 1: Work Backward
Usually, in Math problems, you are given a set of facts or conditions after which you must find the end result. But there are also Math problems that begin with the end result and ask you to find something that occurred earlier.
To solve this type of Math problem, you can very well use the strategy of working backward. If you use this strategy, you begin with the result and then undo each of the steps.
Annie spent half of the money she had in the morning on lunch. Then, she gave her best friend one dollar. Now Annie has $1.50. Now, what was the amount of money that Annie had initially?
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So we start with our end result, $1.50. Then, we work our way back to find the amount of money Annie had initially. Annie now has an amount of $1.50. First, we undo the one dollar she gave her best friend. That gives us $2.50, right?
Annie spent half of her initial money on lunch. So we have to multiply what she has by two to undo her spending half of what she had initially. Well, that gives us $5.00 as Annie’s starting amount.
Let’s check this. Annie began with $5.00. If she spent half of this amount ($2.50) on lunch and then gave her friend $1.00, she would have exactly $1.50 left, right? Because this result matches what is stated in our given problem, this solution is correct! So we worked our way backward to solve this Math problem without having to use all sorts of Math formulas . Great, isn’t it?
Problem-Solving Strategy 2: Make A List Or A Table
Another strategy for solving Math problems is to make a list or a table. Lists or tables allow you to organize information or numbers in an easy-to-understand way. Let’s look at a few examples.
A fruit vending machine accepts dollars. Each piece of fruit will set you back 65 cents. The machine returns only quarters, dimes, and nickels. Now, which combinations of coins are options as change for one dollar?
What we know is that a piece of fruit costs $0,65 and that the machine will return 35 cents in change. The combination is quarters, dimes, and nickels.
So let’s make a list of different possible combinations of quarters, dimes, and nickels that will total our 35 cents. Let’s organize our table by beginning with the combinations, including the most quarters.
We know that the total for each possible coin combination is 35 cents. If we list the combinations, we see that there are six (6) possible combinations:
quarters dimes nickels 1 1 0 1 0 2 0 3 1 0 2 3 0 1 5 0 0 7
This table gives us a clear overview of all possible combinations, right? We can use this strategy to list a number of possibilities. When making a list or table, we use a highly organized approach, so leaving out any important items should be avoided!
The Math problem is: Determine how many options or possibilities there are to receive change for one quarter when at least 1 coin is a dime.
So, let’s list the possible options. Let’s begin with the options that use the fewest coins. Option 1: dime-dime-nickel Option 2: dime-dime-5 pennies Option 3: dime-nickel-nickel-nickel Option 4: dime-nickel-nickel-5 pennies Option 5: dime-nickel-10 pennies Option 6: dime-15 pennies
So you see, there are in total 6 possibilities. Making a table gives us a clear picture of possible options. If you understand this sort of Math problem, chances are you’ll earn your GED fast.
Problem-Solving Strategy 3: Solve A Simpler Problem
A very useful strategy to solve Math problems is to first solve a far simpler problem. When we use this strategy, we first solve a more familiar or simpler case of a similar problem. Then, we can use the same relationships and concepts to solve the original Math problem.
The Math problem is: What is the sum of the integers 1 through 500? We begin by considering a much simpler problem, for example, finding the sum of the integers 1 through 10.
What you should notice is that we can group the addends in this list into partial sums. Check out this:
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 55
Notice the following: 1 + 10 = 11 2 + 9 = 11 3 + 8 = 11 4 + 7 = 11 5 + 6 = 11
In total, we have 5 (five) sums, right? This is half our total number of addends. Do you notice that all partial sums are 11? This is the sum of our first and last integers! So the sum of our list is 55 (5 x 11).
Now, we can use this same concept to come up with the sum of the integers 1 through 500. Use the same principles, and we’ll get:
1 + 2 + 3 … 499 + 500 = 250 x 501 = 125,250
So we multiply half the total number of addends (250) by the sum of our first and last integer (501), which gives us 125,250.
We can apply a similar problem-solving strategy by using subgoals. Let’s consider the following:
Two industry workers can produce two chairs in exactly two days. What’s the number of chairs that 8 workers can produce in 20 days when they work at exactly the same rate?
We begin by determining the number of chairs one worker can produce in two days. So we divide our two chairs by the two workers, which gives us 1 (2 divided by 2).
So now we know that one worker can produce one chair in 2 days. If we want to determine the number of chairs one worker can produce in 20 days, we have to divide 20 (days) by 2 (number of days needed to make one chair). Well, 20 divided by 2 gives us 10, right?
Now, when we have to determine the number of chairs that 8 workers can produce in 20 days, we have to multiply 8 by 10, which gives us 80 (8 x 10). So the correct answer is that 8 workers can produce 80 chairs over a period of 20 days.
Check here if you want to learn more about typical GED Division Vocabulary. You may need it to pass the GED Math test fast.
Problem-Solving Strategy 4: Guess & Check
Let’s take a look at one more useful strategy to solve Math problems. Sometimes, it is helpful to come up with a reasonable guess, after which you can check if it solves the problem.
Simply use your guess results to get closer to improving the problem’s solution. This strategy is what we call the ‘Guess & Check’ Strategy.
The problem asks us the following: Determine two even consecutive integers if the product of these integers is 1088. So what are these two integers?
Well, we know that the product of our integers is pretty close to 1000. So let’s make a guess. What happens if we use 24 and 26? The product of 24 and 26 is 624, so far too low.
We have to adjust our guess upward, so what about 30 and 32? The product of 30 and 32 is 960, still too low, but we’re getting closer, right?
Let’s adjust our guess upward one more time. Let’s see what happens if we use 34 and 36. Well, the product of 34 and 36 is 1224, which is too high.
So we’ll have to try even consecutive integers between 30 and 34, and we already tried 30 and 32! This leaves us with 32 and 34. And indeed, the product of 32 and 34 is 1088! So this is our correct answer. The integers we’re looking for are 32 and 34. So we guessed and checked until we came up with the correct solution.
Problem Solving in Mathematics
- Math Tutorials
- Pre Algebra & Algebra
- Exponential Decay
- Worksheets By Grade
The main reason for learning about math is to become a better problem solver in all aspects of life. Many problems are multistep and require some type of systematic approach. There are a couple of things you need to do when solving problems. Ask yourself exactly what type of information is being asked for: Is it one of addition, subtraction, multiplication , or division? Then determine all the information that is being given to you in the question.
Mathematician George Pólya’s book, “ How to Solve It: A New Aspect of Mathematical Method ,” written in 1957, is a great guide to have on hand. The ideas below, which provide you with general steps or strategies to solve math problems, are similar to those expressed in Pólya’s book and should help you untangle even the most complicated math problem.
Use Established Procedures
Learning how to solve problems in mathematics is knowing what to look for. Math problems often require established procedures and knowing what procedure to apply. To create procedures, you have to be familiar with the problem situation and be able to collect the appropriate information, identify a strategy or strategies, and use the strategy appropriately.
Problem-solving requires practice. When deciding on methods or procedures to use to solve problems, the first thing you will do is look for clues, which is one of the most important skills in solving problems in mathematics. If you begin to solve problems by looking for clue words, you will find that these words often indicate an operation.
Look for Clue Words
Think of yourself as a math detective. The first thing to do when you encounter a math problem is to look for clue words. This is one of the most important skills you can develop. If you begin to solve problems by looking for clue words, you will find that those words often indicate an operation.
Common clue words for addition problems:
Common clue words for subtraction problems:
- How much more
Common clue words for multiplication problems:
Common clue words for division problems:
Although clue words will vary a bit from problem to problem, you'll soon learn to recognize which words mean what in order to perform the correct operation.
Read the Problem Carefully
This, of course, means looking for clue words as outlined in the previous section. Once you’ve identified your clue words, highlight or underline them. This will let you know what kind of problem you’re dealing with. Then do the following:
- Ask yourself if you've seen a problem similar to this one. If so, what is similar about it?
- What did you need to do in that instance?
- What facts are you given about this problem?
- What facts do you still need to find out about this problem?
Develop a Plan and Review Your Work
Based on what you discovered by reading the problem carefully and identifying similar problems you’ve encountered before, you can then:
- Define your problem-solving strategy or strategies. This might mean identifying patterns, using known formulas, using sketches, and even guessing and checking.
- If your strategy doesn't work, it may lead you to an ah-ha moment and to a strategy that does work.
If it seems like you’ve solved the problem, ask yourself the following:
- Does your solution seem probable?
- Does it answer the initial question?
- Did you answer using the language in the question?
- Did you answer using the same units?
If you feel confident that the answer is “yes” to all questions, consider your problem solved.
Tips and Hints
Some key questions to consider as you approach the problem may be:
- What are the keywords in the problem?
- Do I need a data visual, such as a diagram, list, table, chart, or graph?
- Is there a formula or equation that I'll need? If so, which one?
- Will I need to use a calculator? Is there a pattern I can use or follow?
Read the problem carefully, and decide on a method to solve the problem. Once you've finished working the problem, check your work and ensure that your answer makes sense and that you've used the same terms and or units in your answer.
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Problem Solving
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. [1]
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?
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?
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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Problem Solving Strategies
What are problem solving strategies.
Strategies are things that Pólya would have us choose in his second stage of problem solving and use in his third stage ( What is Problem Solving? ). In actual fact he called them heuristics . They are a collection of general approaches that might work for a number of problems.
There are a number of common strategies that students of primary age can use to help them solve problems. We discuss below several that will be of value for problems on this website and in books on problem solving.
Common Problem Solving Strategies
- Guess (includes guess and check, guess and improve)
- Act It Out (act it out and use equipment)
- Draw (this includes drawing pictures and diagrams)
- Make a List (includes making a table)
- Think (includes using skills you know already)
We have provided a copymaster for these strategies so that you can make posters and display them in your classroom. It consists of a page per strategy with space provided to insert the name of any problem that you come across that uses that particular strategy (Act it out, Draw, Guess, Make a List). This kind of poster provides good revision for students.
An In-Depth Look At Strategies
We now look at each of the following strategies and discuss them in some depth. You will see that each strategy we have in our list includes two or more subcategories.
- Guess and check is one of the simplest strategies. Anyone can guess an answer. If they can also check that the guess fits the conditions of the problem, then they have mastered guess and check. This is a strategy that would certainly work on the Farmyard problem described below but it could take a lot of time and a lot of computation. Because it is so simple, you may have difficulty weaning some students away from guess and check. As problems get more difficult, other strategies become more important and more effective. However, sometimes when students are completely stuck, guessing and checking will provide a useful way to start to explore a problem. Hopefully that exploration will lead to a more efficient strategy and then to a solution.
- Guess and improve is slightly more sophisticated than guess and check. The idea is that you use your first incorrect guess to make an improved next guess. You can see it in action in the Farmyard problem. In relatively straightforward problems like that, it is often fairly easy to see how to improve the last guess. In some problems though, where there are more variables, it may not be clear at first which way to change the guessing.
- Young students especially, enjoy using Act it Out . Students themselves take the role of things in the problem. In the Farmyard problem, the students might take the role of the animals though it is unlikely that you would have 87 students in your class! But if there are not enough students you might be able to include a teddy or two. This is an effective strategy for demonstration purposes in front of the whole class. On the other hand, it can also be cumbersome when used by groups, especially if a largish number of students is involved. Sometimes the students acting out the problem may get less out of the exercise than the students watching. This is because the participants are so engrossed in the mechanics of what they are doing that they don’t see the underlying mathematics.
- Use Equipment is a strategy related to Act it Out. Generally speaking, any object that can be used in some way to represent the situation the students are trying to solve, is equipment. One of the difficulties with using equipment is keeping track of the solution. The students need to be encouraged to keep track of their working as they manipulate the equipment. Some students need to be encouraged and helped to use equipment. Many students seem to prefer to draw. This may be because it gives them a better representation of the problem in hand. Since there are problems where using equipment is a better strategy than drawing, you should encourage students' use of equipment by modelling its use yourself from time to time.
- It is fairly clear that a picture has to be used in the strategy Draw a Picture . But the picture need not be too elaborate. It should only contain enough detail to help solve the problem. Hence a rough circle with two marks is quite sufficient for chickens and a blob plus four marks will do a pig. All students should be encouraged to use this strategy at some point because it helps them ‘see’ the problem and it can develop into quite a sophisticated strategy later.
- It’s hard to know where Drawing a Picture ends and Drawing a Diagram begins. You might think of a diagram as anything that you can draw which isn’t a picture. But where do you draw the line between a picture and a diagram? As you can see with the chickens and pigs, discussed above, regular picture drawing develops into drawing a diagram. Venn diagrams and tree diagrams are particular types of diagrams that we use so often they have been given names in their own right.
- There are a number of ways of using Make a Table . These range from tables of numbers to help solve problems like the Farmyard, to the sort of tables with ticks and crosses that are often used in logic problems. Tables can also be an efficient way of finding number patterns.
- When an Organised List is being used, it should be arranged in such a way that there is some natural order implicit in its construction. For example, shopping lists are generally not organised. They usually grow haphazardly as you think of each item. A little thought might make them organised. Putting all the meat together, all the vegetables together, and all the drinks together, could do this for you. Even more organisation could be forced by putting all the meat items in alphabetical order, and so on. Someone we know lists the items on her list in the order that they appear on her route through the supermarket.
- Being systematic may mean making a table or an organised list but it can also mean keeping your working in some order so that it is easy to follow when you have to go back over it. It means that you should work logically as you go along and make sure you don’t miss any steps in an argument. And it also means following an idea for a while to see where it leads, rather than jumping about all over the place chasing lots of possible ideas.
- It is very important to keep track of your work. We have seen several groups of students acting out a problem and having trouble at the end simply because they had not kept track of what they were doing. So keeping track is particularly important with Act it Out and Using Equipment. But it is important in many other situations too. Students have to know where they have been and where they are going or they will get hopelessly muddled. This begins to be more significant as the problems get more difficult and involve more and more steps.
- In many ways looking for patterns is what mathematics is all about. We want to know how things are connected and how things work and this is made easier if we can find patterns. Patterns make things easier because they tell us how a group of objects acts in the same way. Once we see a pattern we have much more control over what we are doing.
- Using symmetry helps us to reduce the difficulty level of a problem. Playing Noughts and crosses, for instance, you will have realised that there are three and not nine ways to put the first symbol down. This immediately reduces the number of possibilities for the game and makes it easier to analyse. This sort of argument comes up all the time and should be grabbed with glee when you see it.
- Finally working backwards is a standard strategy that only seems to have restricted use. However, it’s a powerful tool when it can be used. In the kind of problems we will be using in this web-site, it will be most often of value when we are looking at games. It frequently turns out to be worth looking at what happens at the end of a game and then work backward to the beginning, in order to see what moves are best.
- Then we come to use known skills . This isn't usually listed in most lists of problem solving strategies but as we have gone through the problems in this web site, we have found it to be quite common. The trick here is to see which skills that you know can be applied to the problem in hand. One example of this type is Fertiliser (Measurement, level 4). In this problem, the problem solver has to know the formula for the area of a rectangle to be able to use the data of the problem. This strategy is related to the first step of problem solving when the problem solver thinks 'have I seen a problem like this before?' Being able to relate a word problem to some previously acquired skill is not easy but it is extremely important.
Uses of Strategies
Different strategies have different uses. We’ll illustrate this by means of a problem.
The Farmyard Problem : In the farmyard there are some pigs and some chickens. In fact there are 87 animals and 266 legs. How many pigs are there in the farmyard?
Some strategies help you to understand a problem. Let’s kick off with one of those. Guess and check . Let’s guess that there are 80 pigs. If there are they will account for 320 legs. Clearly we’ve over-guessed the number of pigs. So maybe there are only 60 pigs. Now 60 pigs would have 240 legs. That would leave us with 16 legs to be found from the chickens. It takes 8 chickens to produce 16 legs. But 60 pigs plus 8 chickens is only 68 animals so we have landed nearly 20 animals short.
Obviously we haven’t solved the problem yet but we have now come to grips with some of the important aspects of the problem. We know that there are 87 animals and so the number of pigs plus the number of chickens must add up to 87. We also know that we have to use the fact that pigs have four legs and chickens two, and that there have to be 266 legs altogether.
Some strategies are methods of solution in themselves. For instance, take Guess and Improve . Supposed we guessed 60 pigs for a total of 240 legs. Now 60 pigs imply 27 chickens, and that gives another 54 legs. Altogether then we’d have 294 legs at this point.
Unfortunately we know that there are only 266 legs. So we’ve guessed too high. As pigs have more legs than hens, we need to reduce the guess of 60 pigs. How about reducing the number of pigs to 50? That means 37 chickens and so 200 + 74 = 274 legs.
We’re still too high. Now 40 pigs and 47 hens gives 160 + 94 = 254 legs. We’ve now got too few legs so we need to guess more pigs.
You should be able to see now how to oscillate backwards and forwards until you hit on the right number of pigs. So guess and improve is a method of solution that you can use on a number of problems.
Some strategies can give you an idea of how you might tackle a problem. Making a Table illustrates this point. We’ll put a few values in and see what happens.
From the table we can see that every time we change the number of pigs by one, we change the number of legs by two. This means that in our last guess in the table, we are five pigs away from the right answer. Then there have to be 46 pigs.
Some strategies help us to see general patterns so that we can make conjectures. Some strategies help us to see how to justify conjectures. And some strategies do other jobs. We’ll develop these ideas on the uses of strategies as this web-site grows.
What Strategies Can Be Used At What Levels
In the work we have done over the last few years, it seems that students are able to tackle and use more strategies as they continue with problem solving. They are also able to use them to a deeper level. We have observed the following strategies being used in the stated Levels.
Levels 1 and 2
- Draw a Picture
- Use Equipment
- Guess and Check
Levels 3 and 4
- Draw a Diagram
- Guess and Improve
- Make a Table
- Make an Organised List
It is important to say here that the research has not been exhaustive. Possibly younger students can effectively use other strategies. However, we feel confident that most students at a given Curriculum Level can use the strategies listed at that Level above. As problem solving becomes more common in primary schools, we would expect some of the more difficult strategies to come into use at lower Levels.
Strategies can develop in at least two ways. First students' ability to use strategies develops with experience and practice. We mentioned that above. Second, strategies themselves can become more abstract and complex. It’s this development that we want to discuss here with a few examples.
Not all students may follow this development precisely. Some students may skip various stages. Further, when a completely novel problem presents itself, students may revert to an earlier stage of a strategy during the solution of the problem.
Draw: Earlier on we talked about drawing a picture and drawing a diagram. Students often start out by giving a very precise representation of the problem in hand. As they see that it is not necessary to add all the detail or colour, their pictures become more symbolic and only the essential features are retained. Hence we get a blob for a pig’s body and four short lines for its legs. Then students seem to realise that relationships between objects can be demonstrated by line drawings. The objects may be reduced to dots or letters. More precise diagrams may be required in geometrical problems but diagrams are useful in a great many problems with no geometrical content.
The simple "draw a picture" eventually develops into a wide variety of drawings that enable students, and adults, to solve a vast array of problems.
Guess: Moving from guess and check to guess and improve, is an obvious development of a simple strategy. Guess and check may work well in some problems but guess and improve is a simple development of guess and check.
But guess and check can develop into a sophisticated procedure that 5-year-old students couldn’t begin to recognise. At a higher level, but still in the primary school, students are able to guess patterns from data they have been given or they produce themselves. If they are to be sure that their guess is correct, then they have to justify the pattern in some way. This is just another way of checking.
All research mathematicians use guess and check. Their guesses are called "conjectures". Their checks are "proofs". A checked guess becomes a "theorem". Problem solving is very close to mathematical research. The way that research mathematicians work is precisely the Pólya four stage method ( What is Problem Solving? ). The only difference between problem solving and research is that in school, someone (the teacher) knows the solution to the problem. In research no one knows the solution, so checking solutions becomes more important.
So you see that a very simple strategy like guess and check can develop to a very deep level.
What are the strategies for problem solving
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5 Strategies for Problem Solving
What Are Problem Solving Strategies? Guess (includes guess and check, guess and improve) Act It Out (act it out and use equipment) Draw (this includes
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Teaching Math Problem Solving Strategies
Teaching math problem solving strategies in middle school.
There are many math problem solving strategies out there and some are very beneficial to upper elementary and middle school math students.
Before we jump into them, I’ll share a little bit of my experiences in teaching math problem solving strategies over the past 25+ years.
During my second year of teaching ( in the early 90s) , I was teaching 5th grade, and our state math testing began to include a greater focus on problem solving and writing in math.
Over the next several years, the other math teachers and I used standard sentence starters to help math students practice explaining their problem solving process. These were starters like:
- “In this problem, I need to….”
- “From the problem, I know….”
- “I already know…”
- “To solve the problem, I will…”
- “I know my answer is correct because…”
Benefits of Sentence Starters
By using these sentence starters, students ended up with several paragraphs (some short, some long) to explain how they approached and solved the math problem, AND how they knew they were correct .
Sometimes this process took quite a long time, but it was helpful, because:
- It made many students slow down and think a bit more about what they were doing mathematically.
- Students took a little more time to analyze the problem (rather than picking out the numbers and guessing at an operation!).
I was teaching 5th grade in elementary school at this time, and we had a full hour for math every day. So, fitting in problem solving practice a few times a week was pretty easy, after students understood the process.
I really liked spending the time on these types of math problems, because they often led to discussion of other math concepts, and they reinforced concepts already learned. I used math problems from a publication that focused on various strategies, like:
- Guess and Check
- Work Backwards
- Draw a Picture
- Use Logical Reasoning
- Create a Table
- Look for a Pattern
- Make an Organized List.
I LOVED these…I really did (do)! And the students I taught during those years became very good at reasoning and solving problems.
Teaching Math Problem Solving in Middle School
When I moved to 6th grade math in middle school, I tried to keep teaching these strategies, but our math periods are only 44 minutes.
- I tried to use the problem solving as warm-ups some days, but it would often take 30 minutes or more, especially if we got into a good discussion, leaving little time for a lesson.
- I found that spending too many class periods using the problem solving ended up putting me too far behind in the curriculum (although I’d argue that my students became better thinkers:-), so I had to make some alterations.
- highlight/underline the question in the problem
- shorten up the writing to bullet points
- highlight/underline the important information in the problem
Math Problem Solving Steps
Now, when I teach these problem solving strategies, our steps are: Find Out, Choose a Strategy, Solve, and Check Your Answer.
Find Out When they Find Out, students identify what they need to know to solve the problem.
- They underline the question the problem is asking them to answer and highlight the important information in the problem.
- They shouldn’t attempt to highlight anything until they’ve identified what question they are answering – only then can they decide what is important to that question.
- In this step, they also identify their own background knowledge about the concepts in that particular math problem.
Choose a Strategy This step requires students to think about what strategy will work well with the question they’ve been asked. Sometimes this is tough, so I give them some suggestions for when to use these particular strategies:
- Make an Organized List: when there are many possible answers/combinations; or when making a list may help identify a pattern.
- Guess and Check: when you can make an educated guess and then use an incorrect guess to help you decide if the next guess should be higher or lower. This is often used when you’re looking for 2 unknown numbers that meet certain requirements.
- Work Backwards: when you have the answer to a problem or situation, but the “starting” number is missing
- when data needs to be organized
- with ratios (ratio tables)
- when using the coordinate plane
- with directional questions
- with shape-related questions (area, perimeter, surface area, volume)
- or when it’s just hard to picture in your mind
- Find a Pattern: when numbers in a problem continue to increase, decrease or both
- when the missing number(s) can be expressed in terms of the same variable
- when the information can be used in a known formula (like area, perimeter, surface area, volume, percent)
- when a “yes” for one answer means “no” for another
- the process of elimination can be used
Solve Students use their chosen strategy to find the solution.
Check Your Answer I’ve found that many students think “check your answer” means to make sure they have an answer (especially when taking a test), so we practice several strategies for checking:
- Reread the question; make sure your solution answers the question.
- Redo the math problem and see if you get the same answer.
- Check with a different method, if possible.
- If you used an equation, substitute your answer into the equation.
- Ask – does your answer make sense/is it reasonable?
Teaching the Math Problem Solving Strategies
- Students keep reference sheets in their binders, so they can quickly refer to the steps and strategies. A few newer reference math wheels can be found in this blog post .
- For example, I often find that a ‘Guess and Check’ problem can be solved algebraically, so we’ll do the guessing and checking together first, and then we’ll talk about an algebraic equation – some students can follow the line of thinking well, and will try it on their own the next time; for others, the examples are exposure, and they’ll need to see several more examples before they give it a try.
Using Doodle Notes to Teach Problem Solving Strategies
This year, I’m trying something new – I created a set of Doodle Notes to use during our unit.
- The first page is a summary of the steps and possible strategies.
- There’s a separate page for each strategy, with a problem to work through AND an independent practice page for each
- There’s also a blank template, so I can create problem solving homework for students throughout the year, using the same format. I’m hoping that using the Doodle Notes format will make the problem solving strategies a little more fun, interesting, and easy to remember.
This was a long post about teaching math problem solving strategies! Thanks for sticking with me to the end!
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I’ve been creating resources for teachers since 2012 and have worked in the elearning industry for about five years as well!
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Problem Solving Strategies for Elementary-School Math Paperback – June 24, 2020
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The best way to learn math is by problem solving, but the challenge is that most elementary students don’t know how to start thinking about a math problem that they haven’t seen before. This book is especially designed to overcome this challenge by teaching seven basic problem solving strategies. The book contains more than 100 challenging problems that are suitable for elementary-school students, along with their step-by-step solution to help the reader master these strategies. This book will help you: - Learn seven useful problem solving strategies that can be used in many challenging math problems. - Ace your math tests in school, even the challenge problems that your teacher gives! - Get prepared for various math contests and education programs for gifted students, such as the GATE and Math Kangaroo. - Become an independent learner via the step-by-step instructions of this book. - Stay ahead of the curriculum when transitioning into higher grades and the middle school. - Become a creative thinker who can succeed in STEM fields. - Turn into a life-long math enthusiast who enjoys thinking and problem solving.
- Reading age 7 - 12 years
- Print length 124 pages
- Language English
- Grade level 2 - 6
- Dimensions 5.98 x 0.26 x 9.02 inches
- Publisher Now Publishers Inc
- Publication date June 24, 2020
- ISBN-10 1680839845
- ISBN-13 978-1680839845
- See all details
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Product details
- Publisher : Now Publishers Inc; Illustrated edition (June 24, 2020)
- Language : English
- Paperback : 124 pages
- ISBN-10 : 1680839845
- ISBN-13 : 978-1680839845
- Reading age : 7 - 12 years
- Grade level : 2 - 6
- Item Weight : 6.2 ounces
- Dimensions : 5.98 x 0.26 x 9.02 inches
- #3,720 in Children's Math Books (Books)
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This strategy is a self-monitoring method for math students since it demands that they first understand the problem. If they immediately start solving the problem, they risk making mistakes. Using this strategy, students will keep track of their ideas and correct mistakes before arriving at a final answer.
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.
This is a math intervention strategy that can make problem solving easier for all students, regardless of ability. Compare different word problems of the same type and construct a formula, or mathematical sentence stem, that applies to them all. For example, a simple subtraction problems could be expressed as:
There are five strategies for math problem solving to word problems that you can teach your students in thirty minutes class. Before introducing these skills make sure you have reviewed how to read word problems first . The second step in the problem solving process is to teach strategies that will help your students become better problem solvers.
Problem Solving Strategy 1 (Guess and Test) Make a guess and test to see if it satisfies the demands of the problem. If it doesn't, alter the guess appropriately and check again. Keep doing this until you find a solution. Example: Mr. Jones has a total of 25 chickens and cows on his farm.
3 Reads Strategy for Successful Problem Solving in Math Word Problems are often the hardest part of our math instruction. They can visually overwhelm students. They often contain extraneous information or multiple steps for completion. Students often struggle to persevere through complex problems.
First, the student is taught a 7-step process for attacking a math word problem (cognitive strategy). Second, the instructor trains the student to use a three-part self-coaching routine for each of the seven problem-solving steps (metacognitive strategy). In the cognitive part of this multi-strategy intervention, the student learns an explicit ...
Elementary Math Problem Solving -Acting it OutIn this video, we explore one of eight problem-solving strategies for the primary math student. Students are in...
The crocodile maths strategy can be used to solve addition, subtraction, multiplication, and division problems. Can you solve the crocodile math problem that stumped Scottish students? There was an uproar on social media because of the difficulty, and Scottish students cried. Check out my video for a quick explanation of the problem and how it ...
This strategy for selecting and teaching word problems guides students to develop their understanding of math concepts. By Tara Koehler, John Sammon. March 1, 2023. Johner Images / Alamy. Word problems in mathematics are a powerful tool for helping students make sense of and reason with mathematical concepts.
Math Problem Solving Strategies 1. C.U.B.E.S. C.U.B.E.S stands for circle the important numbers, underline the question, box the words that are keywords, eliminate extra information, and solve by showing work. Why I like it: Gives students a very specific 'what to do.'
Math Problem Solving Strategies That Make Students Say I In these lessons, we will learn some math problem solving strategies for example, Verbal Model (or Logical Reasoning), Algebraic Model, Block Model (or
Problem-solving helps you figure out how to achieve these desires. The problem-solving process involves: Discovery of the problem. Deciding to tackle the issue. Seeking to understand the problem more fully. Researching available options or solutions. Taking action to resolve the issue.
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.
There are 7 strategies that are normally covered in our math challenge program: DRAW A PICTURE/DIAGRAM/MODEL LOOK FOR PATTERNS WORK BACKWARD ACT IT OUT GUESS AND CHECK MAKE AN ORGANIZED LIST OR TABLE USE LOGICAL REASONING
The book illustrates various strategies for solving math problems; each chapter illustrates a specific strategy by providing completely worked out solutions for several problems (around 10 to 15 per chapter). The only thing I found missing was end-of-chapter exercises or even a set of miscellaneous problems at the end of the book for the ...
A very useful strategy to solve Math problems is to first solve a far simpler problem. When we use this strategy, we first solve a more familiar or simpler case of a similar problem. Then, we can use the same relationships and concepts to solve the original Math problem. Example 1: The Math problem is: What is the sum of the integers 1 through 500?
Define your problem-solving strategy or strategies. This might mean identifying patterns, using known formulas, using sketches, and even guessing and checking. If your strategy doesn't work, it may lead you to an ah-ha moment and to a strategy that does work. If it seems like you've solved the problem, ask yourself the following:
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.
Common Problem Solving Strategies Guess (includes guess and check, guess and improve) Act It Out (act it out and use equipment) Draw (this includes drawing pictures and diagrams) Make a List (includes making a table) Think (includes using skills you know already)
What are the strategies for problem solving - Discovery of the problem Deciding to tackle the issue Seeking to understand the problem more fully Researching ... Mathematicians work to clear up the misunderstandings and false beliefs that people have about mathematics. Determine math question Math is the study of numbers, space, and structure ...
Here are some strategies to solve a math problem. These strategies begin with Math Practice Standard 1: Make sense of problems and persevere in solving them. They all start with read the problem carefully to figure out what it asks. Read each sentence carefully to make sure you comprehend it. Decide what the problem includes that you need to ...
Using the guess and check problem solving strategy to help solve math word problems. Example: Jamie spent $40 for an outfit. She paid for the items using $10. Solving word questions. Word questions can be tricky, but there are some helpful tips you can follow to solve them.
Full set of problems available here. Great for differentiation, at math centers, or whole class. Practice with other problem solving techniques. You may also like: Problem Solving Strategies: Printables for the classroom and notebook Ratio: 3 ways to explore ratio (common core aligned) Rational vs. Irrational Numbers: 3 kinesthetic activities ...
Math Problem Solving Steps. Now, when I teach these problem solving strategies, our steps are: Find Out, Choose a Strategy, Solve, and Check Your Answer. Find Out. When they Find Out, students identify what they need to know to solve the problem. They underline the question the problem is asking them to answer and highlight the important ...
The book contains more than 100 challenging problems that are suitable for elementary-school students, along with their step-by-step solution to help the reader master these strategies. This book will help you: - Learn seven useful problem solving strategies that can be used in many challenging math problems.
Strategize it! The No. 1 issue math students struggle with is solving word problems. Math Problem Solving provides a solution. Each lesson teaches a key problem-solving strategy by breaking it down into manageable steps and then providing guided and independent practice to reinforce the learning. Plus-it aligns with your core math program and ...