problem solving strategies grade 3

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Third Grade (Grade 3) Problem Solving Strategies Questions

You can create printable tests and worksheets from these Grade 3 Problem Solving Strategies questions! Select one or more questions using the checkboxes above each question. Then click the add selected questions to a test button before moving to another page.

• The bells have red bows tied to them.
• A box holds 4 rows of bells.
• Each row has 5 bells.
• How many bells are in the box?

• 8 snowmen have red mittens.
• 5 snowmen have green mittens.
• 7 snowmen have blue mittens.
• How many more snowmen have red mittens than green mittens?

• A toy store has 4 shelves of Santa toys.
• Each of the Santa toys is 5 inches tall.
• There are a total of 28 Santa toys.
• How many Santa toys are on each shelf?

• Taylor's family has 6 stockings hanging by the chimney.
• Cedric's family has 4 stockings hanging by the chimney.
• All of the stockings are red and green.
• How many total stockings are hanging by the chimneys?

• Julia hangs 12 candy canes on the tree.
• Clare hangs 8 snowmen on the tree.
• Alex hangs 9 candy canes on the tree.
• How many candy canes were hung on the tree?
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20 Effective Math Strategies To Approach Problem-Solving 

Katie Keeton

Math strategies for problem-solving help students use a range of approaches to solve many different types of problems. It involves identifying the problem and carrying out a plan of action to find the answer to mathematical problems.  

Problem-solving skills are essential to math in the general classroom and real-life. They require logical reasoning and critical thinking skills. Students must be equipped with strategies to help them find solutions to problems.

This article explores mathematical problem solving strategies, logical reasoning and critical thinking skills to help learners with solving math word problems independently in real-life situations. 

What are problem-solving strategies?

Problem-solving strategies in math are methods students can use to figure out solutions to math problems. Some problem-solving strategies: 

• Draw a model
• Use different approaches
• Check the inverse to make sure the answer is correct

Students need to have a toolkit of math problem-solving strategies at their disposal to provide different ways to approach math problems. This makes it easier to find solutions and understand math better. 

Strategies can help guide students to the solution when it is difficult ot know when to start.

The ultimate guide to problem solving techniques

Download these ready-to-go problem solving techniques that every student should know. Includes printable tasks for students including challenges, short explanations for teachers with questioning prompts.

20 Math Strategies For Problem-Solving

Different problem-solving math strategies are required for different parts of the problem. It is unlikely that students will use the same strategy to understand and solve the problem. 

Here are 20 strategies to help students develop their problem-solving skills. 

Strategies to understand the problem

Strategies that help students understand the problem before solving it helps ensure they understand: 

• The context
• What the key information is
• How to form a plan to solve it

Following these steps leads students to the correct solution and makes the math word problem easier .

Here are five strategies to help students understand the content of the problem and identify key information. 

1. Read the problem aloud

Read a word problem aloud to help understand it. Hearing the words engages auditory processing. This can make it easier to process and comprehend the context of the situation.

2. Highlight keywords 

When keywords are highlighted in a word problem, it helps the student focus on the essential information needed to solve it. Some important keywords help determine which operation is needed.  For example, if the word problem asks how many are left, the problem likely requires subtraction.  Ensure students highlight the keywords carefully and do not highlight every number or keyword. There is likely irrelevant information in the word problem.

3. Summarize the information

Read the problem aloud, highlight the key information and then summarize the information. Students can do this in their heads or write down a quick summary.  Summaries should include only the important information and be in simple terms that help contextualize the problem.

4. Determine the unknown

A common problem that students have when solving a word problem is misunderstanding what they are solving. Determine what the unknown information is before finding the answer.  Often, a word problem contains a question where you can find the unknown information you need to solve. For example, in the question ‘How many apples are left?’ students need to find the number of apples left over.

5. Make a plan

Once students understand the context of the word problem, have dentified the important information and determined the unknown, they can make a plan to solve it.  The plan will depend on the type of problem. Some problems involve more than one step to solve them as some require more than one answer.  Encourage students to make a list of each step they need to take to solve the problem before getting started.

Strategies for solving the problem 

1. draw a model or diagram.

Students may find it useful to draw a model, picture, diagram, or other visual aid to help with the problem solving process.  It can help to visualize the problem to understand the relationships between the numbers in the problem. In turn, this helps students see the solution.

Similarly, you could draw a model to represent the objects in the problem:

2. Act it out

This particular strategy is applicable at any grade level but is especially helpful in math investigation in elementary school . It involves a physical demonstration or students acting out the problem using movements, concrete resources and math manipulatives .  When students act out a problem, they can visualize and contectualize the word problem in another way and secure an understanding of the math concepts.  The examples below show how 1st-grade students could “act out” an addition and subtraction problem:

3. Work backwards

Working backwards is a popular problem-solving strategy. It involves starting with a possible solution and deciding what steps to take to arrive at that solution.  This strategy can be particularly helpful when students solve math word problems involving multiple steps. They can start at the end and think carefully about each step taken as opposed to jumping to the end of the problem and missing steps in between.

For example,

To solve this problem working backwards, start with the final condition, which is Sam’s grandmother’s age (71) and work backwards to find Sam’s age. Subtract 20 from the grandmother’s age, which is 71.  Then, divide the result by 3 to get Sam’s age. 71 – 20 = 51 51 ÷ 3 = 17 Sam is 17 years old.

4. Write a number sentence

When faced with a word problem, encourage students to write a number sentence based on the information. This helps translate the information in the word problem into a math equation or expression, which is more easily solved.  It is important to fully understand the context of the word problem and what students need to solve before writing an equation to represent it.

5. Use a formula

Specific formulas help solve many math problems. For example, if a problem asks students to find the area of a rug, they would use the area formula (area = length × width) to solve.   Make sure students know the important mathematical formulas they will need in tests and real-life. It can help to display these around the classroom or, for those who need more support, on students’ desks.

Strategies for checking the solution 

Once the problem is solved using an appropriate strategy, it is equally important to check the solution to ensure it is correct and makes sense. 

There are many strategies to check the solution. The strategy for a specific problem is dependent on the problem type and math content involved.

Here are five strategies to help students check their solutions. 

1. Use the Inverse Operation

For simpler problems, a quick and easy problem solving strategy is to use the inverse operation. For example, if the operation to solve a word problem is 56 ÷ 8 = 7 students can check the answer is correct by multiplying 8 × 7. As good practice, encourage students to use the inverse operation routinely to check their work. 

2. Estimate to check for reasonableness

Once students reach an answer, they can use estimation or rounding to see if the answer is reasonable.  Round each number in the equation to a number that’s close and easy to work with, usually a multiple of ten.  For example, if the question was 216 ÷ 18 and the quotient was 12, students might round 216 to 200 and round 18 to 20. Then use mental math to solve 200 ÷ 20, which is 10.  When the estimate is clear the two numbers are close. This means your answer is reasonable. 

3. Plug-In Method

This method is particularly useful for algebraic equations. Specifically when working with variables.  To use the plug-in method, students solve the problem as asked and arrive at an answer. They can then plug the answer into the original equation to see if it works. If it does, the answer is correct.

If students use the equation 20m+80=300 to solve this problem and find that m = 11, they can plug that value back into the equation to see if it is correct. 20m + 80 = 300 20 (11) + 80 = 300 220 + 80 = 300 300 = 300 ✓

4. Peer Review

Peer review is a great tool to use at any grade level as it promotes critical thinking and collaboration between students. The reviewers can look at the problem from a different view as they check to see if the problem was solved correctly.   Problem solvers receive immediate feedback and the opportunity to discuss their thinking with their peers. This strategy is effective with mixed-ability partners or similar-ability partners. In mixed-ability groups, the partner with stronger skills provides guidance and support to the partner with weaker skills, while reinforcing their own understanding of the content and communication skills.  If partners have comparable ability levels and problem-solving skills, they may find that they approach problems differently or have unique insights to offer each other about the problem-solving process.

5. Use a Calculator

A calculator can be introduced at any grade level but may be best for older students who already have a foundational understanding of basic math operations. Provide students with a calculator to allow them to check their solutions independently, accurately, and quickly. Since calculators are so readily available on smartphones and tablets, they allow students to develop practical skills that apply to real-world situations.  

Step-by-step problem-solving processes for your classroom

In his book, How to Solve It , published in 1945, mathematician George Polya introduced a 4-step process to solve problems. 

Polya’s 4 steps include:

• Understand the problem
• Devise a plan
• Carry out the plan

Today, in the style of George Polya, many problem-solving strategies use various acronyms and steps to help students recall. 

Many teachers create posters and anchor charts of their chosen process to display in their classrooms. They can be implemented in any elementary, middle school or high school classroom. 

Here are 5 problem-solving strategies to introduce to students and use in the classroom.

How Third Space Learning improves problem-solving 

Resources .

Third Space Learning offers a free resource library is filled with hundreds of high-quality resources. A team of experienced math experts carefully created each resource to develop students mental arithmetic, problem solving and critical thinking. 

Explore the range of problem solving resources for 2nd to 8th grade students. 

One-on-one tutoring 

Third Space Learning offers one-on-one math tutoring to help students improve their math skills. Highly qualified tutors deliver high-quality lessons aligned to state standards. 

Former teachers and math experts write all of Third Space Learning’s tutoring lessons. Expertly designed lessons follow a “my turn, follow me, your turn” pedagogy to help students move from guided instruction and problem-solving to independent practice. 

Throughout each lesson, tutors ask higher-level thinking questions to promote critical thinking and ensure students are developing a deep understanding of the content and problem-solving skills.

Problem-solving

Educators can use many different strategies to teach problem-solving and help students develop and carry out a plan when solving math problems. Incorporate these math strategies into any math program and use them with a variety of math concepts, from whole numbers and fractions to algebra. 

Teaching students how to choose and implement problem-solving strategies helps them develop mathematical reasoning skills and critical thinking they can apply to real-life problem-solving.

READ MORE :

• 8 Common Core math examples
• Tier 3 Interventions: A School Leaders Guide
• Tier 2 Interventions: A School Leaders Guide
• Tier 1 Interventions: A School Leaders Guide

There are many different strategies for problem-solving; Here are 5 problem-solving strategies: • draw a model  • act it out  • work backwards  • write a number sentence • use a formula

Here are 10 strategies for problem-solving: • Read the problem aloud • Highlight keywords • Summarize the information • Determine the unknown • Make a plan • Draw a model  • Act it out  • Work backwards  • Write a number sentence • Use a formula

1. Understand the problem 2. Devise a plan 3. Carry out the plan 4. Look back

Some strategies you can use to solve challenging math problems are: breaking the problem into smaller parts, using diagrams or models, applying logical reasoning, and trying different approaches.

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Problem-Solving Strategies

October 16, 2019

There are many different ways to solve a math problem, and equipping students with problem-solving strategies is just as important as teaching computation and algorithms. Problem-solving strategies help students visualize the problem or present the given information in a way that can lead them to the solution. Solving word problems using strategies works great as a number talks activity and helps to revise many skills.

Problem-solving strategies

1. create a diagram/picture, 2. guess and check., 3. make a table or a list., 4. logical reasoning., 5. find a pattern, 6. work backward, 1. create a diagram/draw a picture.

Creating a diagram helps students visualize the problem and reach the solution. A diagram can be a picture with labels, or a representation of the problem with objects that can be manipulated. Role-playing and acting out the problem like a story can help get to the solution.

Alice spent 3/4 of her babysitting money on comic books. She is left with $6. How much money did she make from babysitting?

2. Guess and check

Teach students the same strategy research mathematicians use.

With this strategy, students solve problems by making a reasonable guess depending on the information given. Then they check to see if the answer is correct and they improve it accordingly.  By repeating this process, a student can arrive at a correct answer that has been checked. It is recommended that the students keep a record of their guesses by making a chart, a table or a list. This is a flexible strategy that works for many types of problems. When students are stuck, guessing and checking helps them start and explore the problem. However, there is a trap. Exactly because it is such a simple strategy to use, some students find it difficult to consider other strategies. As problems get more complicated, other strategies become more important and more effective.

Find two numbers that have sum 11 and product 24.

Try/guess  5 and 6  the product is 30 too high

  adjust  to 4 and 7 with product 28 still high

  adjust  again 3 and 8 product 24

3. Make a table or a list

Carefully organize the information on a table or list according to the problem information. It might be a table of numbers, a table with ticks and crosses to solve a logic problem or a list of possible answers. Seeing the given information sorted out on a table or a list will help find patterns and lead to the correct solution.

To make sure you are listing all the information correctly read the problem carefully.

Find the common factors of 24, 30 and 18

Logical reasoning is the process of using logical, systemic steps to arrive at a conclusion based on given facts and mathematic principles. Read and understand the problem. Then find the information that helps you start solving the problem. Continue with each piece of information and write possible answers.

Thomas, Helen, Bill, and Mary have cats that are black, brown, white, or gray. The cats’ names are Buddy, Lucky, Fifi, and Moo. Buddy is brown. Thoma’s cat, Lucky, is not gray. Helen’s cat is white but is not named Moo. The gray cat belongs to Bill. Which cat belongs to each student, and what is its color?

A table or list is useful in solving logic problems.

Since Lucky is not gray it can be black or brown. However, Buddy is brown so Lucky has to be black.

Buddy is brown so it cannot be Helen’s cat. Helen’s cat cannot be Moo, Buddy or Lucky, so it is Fifi.

Therefore, Moo is Bill’s cat and Buddy is Mary’s cat.

5. Find a pattern.

Finding a pattern is a strategy in which students look for patterns in the given information in order to solve the problem. When the problem consists of data like numbers or events that are repeated then it can be solved using the “find a pattern” problem-solving strategy. Data can be organized in a table or a list to reveal the pattern and help discover the “rule” of the pattern.

 The “rule” can then be used to find the answer to the question and complete the table/list.

Shannon’s Pizzeria made 5 pizzas on Sunday, 10 pizzas on Monday, 20 pizzas on Tuesday, and 40 pizzas on Wednesday. If this pattern continues, how many pizzas will the pizzeria make on Saturday?

6. Working backward

Problems that can be solved with this strategy are the ones that  list a series of events or a sequence of steps .

In this strategy, the students must start with the solution and work back to the beginning. Each operation must be reversed to get back to the beginning. So if working forwards requires addition, when students work backward they will need to subtract. And if they multiply working forwards, they must divide when working backward.

Mom bought a box of candy. Mary took 5 of them, Nick took 4 of them and 31 were given out on Halloween night. The next morning they found 8 pieces of candy in the box. How many candy pieces were in the box when mom bought it.

For this problem, we know that the final number of candy was 8, so if we work backward to “put back” the candy that was taken from the box we can reach the number of candy pieces that were in the box, to begin with.

The candy was taken away so we will normally subtract them. However, to get back to the original number of candy we need to work backward and do the opposite, which is to add them.

8 candy pieces were left + the 31 given out + plus the ones Mary took + the ones Nick took

8+31+5+4= 48   Answer: The box came with 48 pieces of candy.

Selecting the best strategy for a problem comes with practice and often problems will require the use of more than one strategies.

Print and digital activities

I have created a collection of print and digital activity cards and worksheets with word problems (print and google slides) to solve using the strategies above. The collection includes 70 problems (5 challenge ones) and their solution s and explanations.

sample below

How to use the activity cards

Allow the students to use manipulatives to solve the problems. (counters, shapes, lego blocks, Cuisenaire blocks, base 10 blocks, clocks) They can use manipulatives to create a picture and visualize the problem. They can use counters for the guess and check strategy. Discuss which strategy/strategies are better for solving each problem. Discuss the different ways. Use the activities as warm-ups, number talks, initiate discussions, group work, challenge, escape rooms, and more.

Ask your students to write their own problems using the problems in this resource, and more, as examples. Start with a simple type. Students learn a lot when trying to compose a problem. They can share the problem with their partner or the whole class. Make a collection of problems to share with another class.

For the google slides the students can use text boxes to explain their thinking with words, add shapes and lines to create diagrams, and add (insert) tables and diagrams.

Many of the problems can be solved faster by using algebraic expressions. However, since I created this resource for grades 4 and up I chose to show simple conceptual ways of solving the problems using the strategies above. You can suggest different ways of solving the problems based on the grade level.

Find the free and premium versions of the resource below. The premium version includes 70 problems (challenge problems included) and their solutions

There are 2 versions of the resource

70 google slides with explanations + 70 printable task cards

70 google slides with explanations + 11 worksheets

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CHALLENGE ZONE 3rd Grade Math Problems

Welcome to our 3rd Grade Math Problems. Here you will find our range of challenging math problem worksheets which are designed to give children the opportunity to apply their skills and knowledge to solve a range of longer problems.

These problems are also a great way of developing perseverance and getting children to try different approaches in their math.

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3rd Grade Math Problems

Here you will find a range of problem solving worksheets.

The 3rd grade math problems on the sheets are longer math problems designed to encourage children to use a range of math skills to solve them.

The skills the problems will help to develop include:

• systematic working
• logical thinking
• number fact knowledge
• fraction problems
• trial and improvement strategies
• working systematically
• searching for all possible answers.

At third grade, the problems are starting to become more advanced with children needing to become more systematic in their approach and experimenting using trial and improvement strategies.

3rd Grade Math Word Problems

• Bikes and Trikes

Bikes and Trikes is an activity to encourage children to work systematically and find all possibilities for the number of bikes and trikes at a show.

• PDF version
• Captain's Coins

Captain's Coins is a 3rd grade math problem activity involving sharing out a pile of coins according to certain rules. This activity is good to use for developing trial and improvement strategie.

• Broken Calculator Problem 2

The Broken Calculator problem is a number problem involving using an imaginary broken calculator with only the 3, 5, +, - and = buttons working to make different totals.

There are 2 versions of the problem sheet, one with a pre-prepared template for filling in, and a second blank version for children to show their own recording system.

• Blank version
• Color that Shape

Color that Shape is a coloring activity which uses logical thinking to work out which shape needs to be shaded which color.

• Dilly's Eggs #2

Dilly's Eggs is an activity involving counting in different size steps. The aim is to work out how many eggs Dilly had from two separate rules. It is a good activity to develop systematic working and the use of lists or tables to solve a problem.

• Frogs in Ponds

Frogs in Ponds is a 3rd grade math problem activity which involves using trial and improvement and logical thinking to get the correct number of frogs into each of the ponds.

Join Me Up is an logical puzzle where the aim is to place the numbers from 1 to 7 into the puzzle so that no consecutive numbers are next to each other.

Place It RIght #3

Place It Right is a place value activity designed to get children thinking hard about where to put the beads on a place value abacus to get numbers closest to a specific total.

• Place It Right #3
• Sally's Rectangular Puzzle #1

Sally's Rectangular Puzzle is an adding puzzle where the aim is to give each circle a value so that the sides of the rectangles add together to make a given total.

Sally's Fruit Punch Challenge #1

Sally's Fruit Punch is a scaling problem involving changing a recipe for fruit punch for one glass into a recipe for ten glasses. In the challenge, you have to firstly work out the ingredients you need, and then work out the cost of the ingredients. This challenge is great for developing real-life math solving problems involving money and measures.

• Sally's Fruit Punch Challenge 1
• Sally's Fruit Punch Challenge 1 (UK version)
• Share the Treasure #3

Share the Treasure is a fraction sharing activity where the aim is to share 24 bars between a group of pirates. It is a good activity for developing fraction problem solving.

• Spot the Digits

Spot the Digits is a logic activity where children have to find out the values of the letters a, b, c and d. The values can be determined by using the 3 clues.

The Rock Race Challenge 1

The Rock Race is a number activity which needs some perseverance to complete. The aim of the activity is to try different routes around the 6 rocks to determine which route is the shortest.

• The Rock Race #1

The Three Spiders Problem

The Three Spiders Problem is another problem which involves finding all the possibilites. The aim of this task is to find all triples of odd numbers that add up to 23. Good for odd and even practice as well as adding three small numbers together and strategic thinking.

• Three Spiders Problem

Looking for some easier math problems?

We have a range of easier word problems at our 2nd-grade-math-salamanders website.

The problems on this page are at a simpler level than those here.

Many of the problems, e.g. Place It Right, Pick the Cards and Share the Treasure have easier versions on this page.

Using the link below will open our 2nd grade site in a new tab.

• Second grade Math Problems

Looking for some harder word problems

We have a range of more challenging word problems on our 4th grade problem solving page.

The problems on this page are at a trickier level than those here.

Some of the problems, e.g. The Rock Race and Share the Treasure have harder versions on this page.

• 4th Grade Math Problems

Looking for some more third grade math problems?

Here are a range of problems solving sheets for 3rd graders. Most of the sheets contain 'real-life' problems which involve real data.

Using the sheets will help your child to:

• apply their addition, subtraction, and multiplication skills;
• develop their knowledge of fractions;
• apply their knowledge of rounding and place value;
• solve a range of 'real life' problems.

All the 3rd grade sheets in this section support Elementary math benchmarks.

• Math Word Problems for kids 3rd Grade
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3rd Grade Money Worksheets

On this webpage, you will find our 3rd grade Money Challenges.

These challenges involve solving money problems and are a great way for kids to develop their reasoning skills and practice their counting money at the same time.

• 3rd Grade Money Challenges
• 3rd Grade Math Puzzles

Here you will find a range of printable 3rd grade math puzzles for your child to enjoy.

The puzzles will help your child practice and apply their addition, subtraction and multiplication facts as well as developing their thinking and reasoning skills in a fun and engaging way.

Using these puzzles will help your child to:

• learn and practice their addition facts to 100;
• practice adding multiples of 10;
• practice their subtraction facts to 100;
• practice multiplication and division facts;
• develop problem solving skills and reasoning.

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Unit 3: Addition, subtraction, and estimation

About this unit.

The world of math is full of twists and turns but luckily, this unit on addition, subtraction, and estimation is here to help you navigate. Whether it's mastering the art of rounding or learning how to regroup, Khan Academy is your secret weapon for success.

Rounding to nearest 10 or 100

• Rounding to the nearest 10 on the number line (Opens a modal)
• Rounding to the nearest 100 on the number line (Opens a modal)
• Rounding to nearest 10 and 100 (Opens a modal)
• Round to nearest 10 or 100 on the number line Get 5 of 7 questions to level up!
• Round to nearest 10 or 100 Get 3 of 4 questions to level up!
• Round to nearest 10 or 100 challenge Get 3 of 4 questions to level up!

Estimate to add multi-digit numbers

• Estimating when adding multi-digit numbers (Opens a modal)
• Estimate to add multi-digit whole numbers Get 3 of 4 questions to level up!

Strategies for adding two and three-digit numbers

• Breaking apart 3-digit addition problems (Opens a modal)
• Addition using groups of 10 and 100 (Opens a modal)
• Adding and subtracting on number line (Opens a modal)
• Break apart 3-digit addition problems Get 3 of 4 questions to level up!
• Add using groups of 10 and 100 Get 3 of 4 questions to level up!
• Add on a number line Get 3 of 4 questions to level up!
• Select strategies for adding within 1000 Get 3 of 4 questions to level up!

Adding with regrouping within 1000

• Using place value to add 3-digit numbers: part 1 (Opens a modal)
• Using place value to add 3-digit numbers: part 2 (Opens a modal)
• Adding 3-digit numbers (Opens a modal)
• Add within 1000 Get 3 of 4 questions to level up!

Estimate to subtract multi-digit numbers

• Estimating when subtracting large numbers (Opens a modal)
• Estimate to subtract multi-digit whole numbers Get 3 of 4 questions to level up!

Strategies for subtracting two and three-digit numbers

• Subtraction by breaking apart (Opens a modal)
• Methods for subtracting 3-digit numbers (Opens a modal)
• Break apart 3-digit subtraction problems Get 3 of 4 questions to level up!
• Subtract on a number line Get 3 of 4 questions to level up!
• Select strategies for subtracting within 1000 Get 3 of 4 questions to level up!

Subtracting with regrouping within 1000

• Worked example: Subtracting 3-digit numbers (regrouping) (Opens a modal)
• Subtracting 3-digit numbers (regrouping) (Opens a modal)
• Worked example: Subtracting 3-digit numbers (regrouping twice) (Opens a modal)
• Worked example: Subtracting 3-digit numbers (regrouping from 0) (Opens a modal)
• Subtract within 1000 Get 3 of 4 questions to level up!

Addition and subtraction missing value problems

• Missing number for 3-digit addition within 1000 (Opens a modal)
• Missing number for 3-digit subtraction within 1000 (Opens a modal)
• Find the missing number (add and subtract within 1000) Get 3 of 4 questions to level up!

Addition and subtraction word problems

• Three digit addition word problems (Opens a modal)
• Three digit subtraction word problems (Opens a modal)
• Addition, subtraction, and estimation: FAQ (Opens a modal)
• Add and subtract within 1000 word problems Get 3 of 4 questions to level up!

Estimation Strategies - Grade 3

Examples, solutions, videos, worksheets, games and activities to help Grade 3 students learn estimation strategies for addition and subtraction.

In this lesson, we will learn how to apply estimation strategies to predict sums and differences of two 2-digit numerals in a problem solving context. Some of the estimation strategies are: Front-End Strategy and Closest-Ten Strategy (or rounding).

Estimating sums and differences is valuable because it helps predict an answer and check a calculation. When using estimation in a problem solving context, there are important things to keep in mind. What is best, an exact answer or an estimate? How important is it for the estimate to be close to the exact value? Is it better to have a low or high estimate?

Front-End Strategy

The front-end strategy is a method of estimating computations by keeping the first digit in each of the numbers and changing all the other digits to zeros. This strategy can be used to estimate sums and differences. Note that the front-end strategy always gives an underestimate for sums.

Example: When estimating 77 - 24 Think: 77 -> 70 and 24 -> 20. 70 − 20 = 50. The estimate is about 50. Example: Karen is taking piano lessons and her piano teacher asked her approximately how much time she practiced on Saturday and Sunday. Karen knew she practiced 43 minutes on Saturday and 56 minutes on Sunday. To find an estimate for 43 + 56, Think: 43 -> 40 and 56 -> 50. 40 + 50 = 90. Karen could say she practiced about 90 minutes. This video shows how to estimate addition using front-end estimation.

Front End Estimation with Adjustment Strategy for solving addition word problems.

Closest Ten Strategy (Rounding)

Write each number as an approximation by rounding the number to the closest ten. Example: When estimating 77 − 24 77 is 3 away from 80, so we round to 80. 24 is 4 away from 20, so we round to 20. 80 − 20 = 60. The estimate is about 60.

Example: Erin has 83 colored beads to make necklaces for her friends. She uses 37 beads to make a necklace for Julia. About how many beads does Erin have left? 83 round to 80. 37 round to 40 The estimate is 80 − 40 = 40 beads

How to estimate sums by rounding and by compatible numbers

Rounding & estimation

Estimation strategies require the use of rounding techniques. If you need help to review rounding techniques then see the following videos. Description of Rounding and Estimation.

Use estimation and rounding to deal with problems.

Front-End Estimation, Clustering, and Compatible Numbers

Front-End Estimation 3,025 + 498 + 32 613 + 91 + 1,231

Clustering 51, 48, 53, 50, 47 93, 103, 99, 97, 101, 104

Compatible Numbers 73.8 + 7.9 22.6 + 3.04

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Teaching Problem Solving in Math

• Freebies , Math , Planning

Every year my students can be fantastic at math…until they start to see math with words. For some reason, once math gets translated into reading, even my best readers start to panic. There is just something about word problems, or problem-solving, that causes children to think they don’t know how to complete them.

Every year in math, I start off by teaching my students problem-solving skills and strategies. Every year they moan and groan that they know them. Every year – paragraph one above. It was a vicious cycle. I needed something new.

I put together a problem-solving unit that would focus a bit more on strategies and steps in hopes that that would create problem-solving stars.

The Problem Solving Strategies

First, I wanted to make sure my students all learned the different strategies to solve problems, such as guess-and-check, using visuals (draw a picture, act it out, and modeling it), working backward, and organizational methods (tables, charts, and lists). In the past, I had used worksheet pages that would introduce one and provide the students with plenty of problems practicing that one strategy. I did like that because students could focus more on practicing the strategy itself, but I also wanted students to know when to use it, too, so I made sure they had both to practice.

I provided students with plenty of practice of the strategies, such as in this guess-and-check game.

There’s also this visuals strategy wheel practice.

I also provided them with paper dolls and a variety of clothing to create an organized list to determine just how many outfits their “friend” would have.

Then, as I said above, we practiced in a variety of ways to make sure we knew exactly when to use them. I really wanted to make sure they had this down!

Anyway, after I knew they had down the various strategies and when to use them, then we went into the actual problem-solving steps.

The Problem Solving Steps

I wanted students to understand that when they see a story problem, it isn’t scary. Really, it’s just the equation written out in words in a real-life situation. Then, I provided them with the “keys to success.”

S tep 1 – Understand the Problem.   To help students understand the problem, I provided them with sample problems, and together we did five important things:

• read the problem carefully
• restated the problem in our own words
• crossed out unimportant information
• circled any important information
• stated the goal or question to be solved

We did this over and over with example problems.

Once I felt the students had it down, we practiced it in a game of problem-solving relay. Students raced one another to see how quickly they could get down to the nitty-gritty of the word problems. We weren’t solving the problems – yet.

Then, we were on to Step 2 – Make a Plan . We talked about how this was where we were going to choose which strategy we were going to use. We also discussed how this was where we were going to figure out what operation to use. I taught the students Sheila Melton’s operation concept map.

We talked about how if you know the total and know if it is equal or not, that will determine what operation you are doing. So, we took an example problem, such as:

Sheldon wants to make a cupcake for each of his 28 classmates. He can make 7 cupcakes with one box of cupcake mix. How many boxes will he need to buy?

We started off by asking ourselves, “Do we know the total?” We know there are a total of 28 classmates. So, yes, we are separating. Then, we ask, “Is it equal?” Yes, he wants to make a cupcake for EACH of his classmates. So, we are dividing: 28 divided by 7 = 4. He will need to buy 4 boxes. (I actually went ahead and solved it here – which is the next step, too.)

Step 3 – Solving the problem . We talked about how solving the problem involves the following:

• taking our time
• working the problem out
• showing all our work
• estimating the answer
• using thinking strategies

We talked specifically about thinking strategies. Just like in reading, there are thinking strategies in math. I wanted students to be aware that sometimes when we are working on a problem, a particular strategy may not be working, and we may need to switch strategies. We also discussed that sometimes we may need to rethink the problem, to think of related content, or to even start over. We discussed these thinking strategies:

• switch strategies or try a different one
• rethink the problem
• think of related content
• decide if you need to make changes
• check your work
• but most important…don’t give up!

To make sure they were getting in practice utilizing these thinking strategies, I gave each group chart paper with a letter from a fellow “student” (not a real student), and they had to give advice on how to help them solve their problem using the thinking strategies above.

Finally, Step 4 – Check It.   This is the step that students often miss. I wanted to emphasize just how important it is! I went over it with them, discussing that when they check their problems, they should always look for these things:

• compare your answer to your estimate
• check for reasonableness
• check your calculations
• add the units
• restate the question in the answer
• explain how you solved the problem

Then, I gave students practice cards. I provided them with example cards of “students” who had completed their assignments already, and I wanted them to be the teacher. They needed to check the work and make sure it was completed correctly. If it wasn’t, then they needed to tell what they missed and correct it.

To demonstrate their understanding of the entire unit, we completed an adorable lap book (my first time ever putting together one or even creating one – I was surprised how well it turned out, actually). It was a great way to put everything we discussed in there.

Once we were all done, students were officially Problem Solving S.T.A.R.S. I just reminded students frequently of this acronym.

Stop – Don’t rush with any solution; just take your time and look everything over.

Think – Take your time to think about the problem and solution.

Act  – Act on a strategy and try it out.

Review – Look it over and see if you got all the parts.

Wow, you are a true trooper sticking it out in this lengthy post! To sum up the majority of what I have written here, I have some problem-solving bookmarks FREE to help you remember and to help your students!

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

You can do any of these ideas without having to purchase anything. However, if you are looking to save some time and energy, then they are all found in my Math Workshop Problem Solving Unit . The unit is for grade three, but it  may work for other grade levels. The practice problems are all for the early third-grade level.

• freebie , Math Workshop , Problem Solving

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No Series: Persistence in Problem Solving

Common core State Standards

• Math:  Math
• Practice:  Mathematical Practice Standards
• MP1:  Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, \"Does this make sense?\" They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.

Download Common Core State Standards (PDF 1.2 MB)

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Persistence in Problem Solving

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Discussion and Supporting Materials

• Supporting Materials

Thought starters

• How does the graphic organizer help scaffold problem solving for students?
• Why does Ms. Saul choose to have students work alone without help?
• How do "Heads Together Butts Up" and "Student-led Solutions" contribute to the class culture around problem solving?

92 Comments

Ana Harston Apr 17, 2021 8:16pm

• How does the graphic organizer help scaffold problem solving for students?  The graphic organizer help students problem solve independenly in as many ways while arriving at the same answer.
• Why does Ms. Saul choose to have students work alone without help? Student lead solution allows student to do work in their own way. and work with other students to see how thwy were able to resolve a solution to the answer.
• How do "Heads Together Butts Up" and "Student-led Solutions" contribute to the class culture around problem solving? Stud with the answer. They can w theowexplain. They come to the from and demonstrate without teacher telling them how to come up with the results.

NANCY PRONKO Mar 21, 2021 11:31pm

Persistence in Problem Solving is challenging!

Athena Morris Mar 15, 2020 2:14pm

The video I chose was Persistence in Problem Solving.  I think this video relates to the Math: Making Sense of Word Problems Mirco-Credential because it was a good example of a teacher giving a word problem that the students could relate to and then having them find 3 ways to solve the problem.  This allowed for the student to make sense of the problem in their own way, which would give them an opportunity to use different strategies.  The teacher allowed for individual time to solve (4 minutes) and then collaboration time, which would help to build their confidence and at the same time learn from another student.  After their group/partner discussion they came together as a class for 3 students to show their work and how they solved the problem. 

The demonstrated approach was really good.  I love the phrase  the teacher used, "heads together butts up."  I usually just say, "turn and talk with your face/shoulder partner."  I think I will incorporate that term into my room.  I usally have two groups/students come to the board/projector to display their work, but I like the 3 group/student approach better, so I will incorporate that too.  I will look online for the Find 3 Ways paper or create one myself.  I don't have any concerns with trying to integrate this approach into my classroom and I am looking forward to trying something new.  I think this will lead to improved learning outcomes because instead of just one way of solving a word problem they will have three ways.  Also, it will be something new for them to try and they love trying new things!  

carl sousa Mar 6, 2020 9:31am

What an exceptional strategy for building both independent and collaborative work. "Life skills that everybody needs." 

Kimberly Simmons Jan 27, 2020 10:44am

I submitted above.

• Choose Three Ways Worksheet
• Sara's thinking on Choose Three Ways
• Carlos's thinking on Choose Three Ways
• Transcript for Persistence in Problem Solving

Transcripts

Classroom Close Up: 3rd Grade Math: Persistence in Problem Solving with Jennifer Saul

[01:00:07;10] Jennifer: "Is it OK to make mistakes?

All: "YES!"

Jennifer: "So, you just fix them and learn from them. Mistakes are easy to fix."

Establishing a strong classroom culture is essential. You have to normalize error. They have to know that it's OK to try, and try, and try again. We want them to wrestle with a problem, and to stay with it. So, we came up with Find Three Ways.

Hopefully, this activity helps them take ownership for their own learning.

"OK I need some more help."

So, at the beginning, I present them with a problem, and really appeal to their sense of empathy.

"I need your help figuring out for four hungry teachers, is my $20.00 enough to buy four burritos and, if it is enough, do I have any left over to maybe get sodas? You're going to try to find three different ways to solve the problem."

Once the problem's been recorded on the board, they are sent back to their seats, with their record sheet. One side has areas where they can do work, and ideas for strategies to use. And, the backside's purposefully left blank. So, they have a lot of freedom.

"Remember our initial tackling and wrestling with this problem is independent work."

They have four minutes to wrestle with the problem independently, and to try to solve it as many ways as they can think of.

"What did you do? Oh, from your table?"

If they can solve it three ways, and arrive at the same answer three times, that would help with their confidence level. They can assure themselves and don't have to wait for the teacher to come around, and say "Yep, you got it."

"OK...discuss with your table."

And then, we do what's called "Heads Together, Butts Up!" We want their heads together, showing each other what they were successful with, maybe others who need support, or didn't know where to go with it, can get an idea.

Sara: "How did you get this answer? I got it by adding this."

Lisette: "I got it by adding four and four, and adding twelve and twelve."

Sara: "OK."

Jennifer: They're allowed to copy each other's notes as long as they're having the discussion about why something was done in a particular way. It's a chance for them to practice using the language, and have discourse over what they're doing.

During that whole time, I try to rotate around the room, offer support. And, so I'll have in my mind, who I would like coming up to the front of the class, and sharing their work.

"Show me learning positions please."

They turn to the front of the room, and three students are called up, and I use my document camera and they can show and explain their work.

Carlos: "Then I added the ones, 1,2,3,4."

Establan: "Then I multiplied the four and the four, equals 16."

Sara: "First, I made a table. In the first two boxes, I put the teachers and amount."

Jennifer: Having a student-led solution enhances our culture of learning as a class. It's not me dictating how they should get to a particular result. It allows them to choose their own path, but it also opens up doors for consulting with each other, and collaboration, which are life skills that everybody needs.

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Problem Solving Activities: 7 Strategies

• Critical Thinking

Problem solving can be a daunting aspect of effective mathematics teaching, but it does not have to be! In this post, I share seven strategic ways to integrate problem solving into your everyday math program.

In the middle of our problem solving lesson, my district math coordinator stopped by for a surprise walkthrough. 

I was so excited!

We were in the middle of what I thought was the most brilliant math lesson– teaching my students how to solve problem solving tasks using specific problem solving strategies. 

It was a proud moment for me!

Each week, I presented a new problem solving strategy and the students completed problems that emphasized the strategy. 

Genius right? 

After observing my class, my district coordinator pulled me aside to chat. I was excited to talk to her about my brilliant plan, but she told me I should provide the tasks and let my students come up with ways to solve the problems. Then, as students shared their work, I could revoice the student’s strategies and give them an official name. 

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

I took some time to consider her advice. Once I acknowledged she was right, I was able to make BIG changes to the way I taught problem solving in the classroom. 

When I Finally Saw the Light

To give my students an opportunity to engage in more authentic problem solving which would lead them to use a larger variety of problem solving strategies, I decided to vary the activities and the way I approached problem solving with my students. 

Problem Solving Activities

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

Seasonal Problem Solving

Many teachers use word problems as problem solving tasks. Instead, try engaging your students with non-routine tasks that look like word problems but require more than the use of addition, subtraction, multiplication, and division to complete. Seasonal problem solving tasks and daily challenges are a perfect way to celebrate the season and have a little fun too!

Cooperative Problem Solving Tasks

Go cooperative! If you’ve got a few extra minutes, have students work on problem solving tasks in small groups. After working through the task, students create a poster to help explain their solution process and then post their poster around the classroom. Students then complete a gallery walk of the posters in the classroom and provide feedback via sticky notes or during a math talk session.

Notice and Wonder

Before beginning a problem solving task, such as a seasonal problem solving task, conduct a Notice and Wonder session. To do this, ask students what they notice about the problem. Then, ask them what they wonder about the problem. This will give students an opportunity to highlight the unique characteristics and conditions of the problem as they try to make sense of it. 

Want a better experience? Remove the stimulus, or question, and allow students to wonder about the problem. Try it! You’ll gain some great insight into how your students think about a problem.

Math Starters

Start your math block with a math starter, critical thinking activities designed to get your students thinking about math and provide opportunities to “sneak” in grade-level content and skills in a fun and engaging way. These tasks are quick, designed to take no more than five minutes, and provide a great way to turn-on your students’ brains. Read more about math starters here ! 

Create your own puzzle box! The puzzle box is a set of puzzles and math challenges I use as fast finisher tasks for my students when they finish an assignment or need an extra challenge. The box can be a file box, file crate, or even a wall chart. It includes a variety of activities so all students can find a challenge that suits their interests and ability level.

Calculators

Use calculators! For some reason, this tool is not one many students get to use frequently; however, it’s important students have a chance to practice using it in the classroom. After all, almost everyone has access to a calculator on their cell phones. There are also some standardized tests that allow students to use them, so it’s important for us to practice using calculators in the classroom. Plus, calculators can be fun learning tools all by themselves!

Three-Act Math Tasks

Use a three-act math task to engage students with a content-focused, real-world problem! These math tasks were created with math modeling in mind– students are presented with a scenario and then given clues and hints to help them solve the problem. There are several sites where you can find these awesome math tasks, including Dan Meyer’s Three-Act Math Tasks and Graham Fletcher’s 3-Acts Lessons . 

Getting the Most from Each of the Problem Solving Activities

When students participate in problem solving activities, it is important to ask guiding, not leading, questions. This provides students with the support necessary to move forward in their thinking and it provides teachers with a more in-depth understanding of student thinking. Selecting an initial question and then analyzing a student’s response tells teachers where to go next. 

Ready to jump in? Grab a free set of problem solving challenges like the ones pictured using the form below. 

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

Shametria Routt Banks

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2 Responses

This is a very cool site. I hope it takes off and is well received by teachers. I work in mathematical problem solving and help prepare pre-service teachers in mathematics.

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

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The 3 Reads Protocol for Solving Word Problems

Raise your hand if you wish your students were more confident and successful in solving word problems. Right, that’s what I thought. And the answer probably doesn’t change much based on your grade level. Face it, word problems are just plain hard!

What DOESN’T WORK

Over the years, very well-intentioned teachers have developed strategies designed to help students solve word problems . Two such strategies that are still quite prevalent are “problem-solving” models and the use of keywords. The idea is that if you follow these steps and look for these keywords, you will be able to solve any word problem. Unfortunately, it’s just not that simple, and despite their widespread use, these strategies are not very effective.

If you look at the CUBES problem-solving model, reading the problem is not even one of the steps! And if you’re thinking,  Well, of course students know to read the problem! you might want to watch this model in action. I have more often than not seen students just literally start circling numbers (and not even the labels that go with the numbers…) without ever having read the problem. And keywords are not reliable either. Some word problems have no keywords, and keywords in multi-step problems end up confusing students because of the mixed messages they send.

So can we just agree that something else is needed and put these “strategies” to rest? Students fail at solving word problems for one reason—they don’t understand what the problem is asking them to do. It’s a comprehension problem, so students need reading comprehension skills.

The 3 Reads Protocol

Let me first say that if you search the Internet for 3 Reads Protocol , you’ll find that there are slightly differing versions. What I’m about to describe is the version that I find to be particularly effective. Regardless of the version, we are reading the problem three different times and each reading has a different focus.

The 3 Reads Protocol is a guided learning experience. Students are presented with the problem in stages, and with each read the teacher asks probing questions. Looking at an example is probably the easiest way to understand the protocol, so let’s dive in.

To begin the 3 Reads Protocol, the teacher presents the students with a problem, and the class reads the problem together. Probably the easiest way to do this is with a PowerPoint or Google Slides file. Notice that with the first read, there are no numbers and no question. We just want the students to understand what the story is about and make a mental picture. Without numbers, students have to focus on the meaning of the words! After reading the problem together, the teacher asks what the story is about and calls on students for responses. Don’t be surprised if the responses are very general at first ( girls, flowers,  etc.). Ask for additional details, if necessary. Ideally, for this problem, you’d like the students to offer the names of the girls and the types of flowers.

For the second read, the problem is again presented to the students, but this time it includes the numbers. Read the problem again whole class. The questions you will ask now are all related to the numbers in the story. Our goal is for the students to understand that it’s not just 10, it’s 10 daisies . Students might also offer relationships—e.g., Natassja picked more daisies than Ayriale.

Finally, with the third read, students are asked to generate questions that could be answered using the information in the problem. Even though the problem looks just like it did for the second read, don’t skip the reading part! Some problems won’t lend themselves to very many different questions. I like to use this problem as an example because many different questions can be generated. Why? Because there are lots of different numbers in the problem. Here’s a sampling of questions that could be asked. I’m sure you can think of many others.

• How many flowers did Ayriale pick?
• How many flowers did Ayriale and Natassja pick?
• Which girl picked more flowers? How many more?
• How many daisies did the girls pick?

That’s the protocol in a nutshell! Once a question or questions have been generated, you can have students go on to solve the problem.

Frequently asked questions

1. When students are solving word problems independently, do I ask them to ignore the numbers and the question?

No! That would be pretty much impossible for them to do. By routinely solving problems using the 3 Reads Protocol with either the whole class or in small groups, you are helping students develop good reading habits that will transfer to their independent work. When they are working independently, the idea is that they will automatically think about the context, identify what the numbers mean within that context, and better understand what the question is asking them to find.

2. Where do I find problems for the 3 Reads Protocol?

I’m sure you can find some that have already been prepared, but it’s super easy to make your own! Just set up a PowerPoint or Google Slides file and format it however you like. Maybe you want a colorful border or a particular font. Use problems that you already have from your resources—textbooks, supplemental books, etc. You’ll need two slides for each problem. On the first slide, type the problem from your resource, leaving out the numbers and the question. On the second slide, add in the numbers. Use a nice big font so students can easily read the problem when projected on your interactive whiteboard. That’s all there is to it!

If you have other questions, add them in the comments below, and I’ll add them to the FAQ. I’d also love to hear how the 3 Reads Protocol is working out in your classroom!

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16 Comments

fantastic thoughts! I LOVE this approach. I’ve used a resource from Lucky Little Learners that is called Numberless Word Problems. This 3 reads protocol will work perfectly with that resource!

Yes! It does include elements of numberless word problems, which I was first introduced to by Brian Bushart .

I appreciate your thoughts on problem solving however the use of CUBES has been totally effective for the past three years with my students who never attempt word problems. You stated that not circling the “labels”, I suppose you mean units connected to the problem. I use this strategy daily and we circle both numbers and units. I’ve made it my own. Let me also reiterate that this strategy works exceptionally well with students who have low reading abilities too. I spend 5 weeks on CUBES and we slowly work our way into the Three Read Protocol which also helps the “non-reader”

Thank you for sharing your experiences! Since you transition to Three Reads, I’m curious why you wouldn’t start with it. I’d also be interested to know what grade level you teach.

I teach both 7th and 8th grade, and many times they have not had any experience with either. CUBEs is embedded with 3-read. I read the question. Students read the questions, we use cubes to understand quantities and the operations ( C and B) used to bring these quantities together. Then we underline the question to provide clarity and to determine what type of answer we are seeking. In our data, we found that our students do not attempt word problems on any formative assessments so we had to start with decreasing student fear of problem-solving and reading. When I used CUBES, I found that many students could answer the question however they had no idea what to do (operations) with the numbers due to a lack of experience,(key operational words) not knowledge ( they always wanted to add). Immediately, students would attempt word problems because CUBES allowed them to pull out what they needed, producing increased math confidence. As we continued with problem-solving, the 3 read protocol was introduced when solving multi-step word problems, student growth was tremendous!

Excellent post!

Thank you! This strategy has so much potential for teaching students to approach word problems in a much more meaningful way!

Hello! This strategy looks great and reaffirms how I’ve been teaching my son to read the problems through as many times as he needs to understand. Now I have the right questions to ask with each reading! Thank you! My question, are there still some strategies to go along with this protocol that you have found to be helpful to use in conjunction i.e. drawing a picture, making a chart, etc. Thanks so much!

I’m so glad this provides you with the missing link! Drawing a picture or diagram is another powerful strategy that goes really well with 3 Reads because it helps students visualize what’s going on in the problem.

Thank you for your post. The Three Read strategy is a gentle reminder of the power of cross-curricular reading strategies; coupling this annotation and graphic organizers will help students develop ways to process and solve simple to more complex multi-step word problems.

I am a math interventionist at an elementary school. We see students K-5 and realize how important it is for us to have a common process/language when it comes to word problems. Looking forward to learning more about this and helping guide our school, thank you!

I absolutely agree about using common language and strategies campus-wide! It’s so much more effective than students having to adjust to their teacher’s language/strategies every new year.

Thank you! I am going to try Three Reads Protocol for the first time next week with my 4th grade students. I am excited. I have had students read word problems all the way through before trying to solve the problem, however, I love the 3 Reads approach and the questions that go with each read. Thank you for breaking it down.

It’s a very powerful strategy! Good luck.

Many of the problems already include the question or questions. When first teaching the strategy, I can take the questions away and then we can compare to see if one of our questions matched the questions posed, but how to do facilitate the strategy when the question is there and you can’t take it away (thinking of assessments).

The idea is that when done regularly, students will develop comprehension skills that will transfer to solving “normal” word problems. They’ll know, for example, that reading it through the first time and ignoring the numbers helps them focus on what’s happening in the problem. You certainly want to help them make that connection, though. You might even show them a normal problem and explicitly model how to use 3 read strategies when the numbers and question are present.

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Maneuvering the Middle

Student-Centered Math Lessons

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!

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.

Ideas for Promoting Showing Your Work

• White boards are a helpful resource that make (extra) writing engaging!
• Celebrating when students show their work. Create a bulletin board that says ***I showed my work*** with student exemplars.
• Take a picture that shows your expectation for how work should look and post it on the board like Marissa did here.

Show Work Digitally

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 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 8 years! This post was originally published in September of 2017. It has been revamped for relevancy and accuracy.

Problem Solving Posters (Represent It! Bulletin Board)

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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!

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6 Tips for Teaching Math Problem-Solving Skills

Solving word problems is tougher than computing with numbers, but elementary teachers can guide students to do the deep thinking involved.

A growing concern with students is the ability to problem-solve, especially with complex, multistep problems. Data shows that students struggle more when solving word problems than they do with computation , and so problem-solving should be considered separately from computation. Why?

Consider this. When we’re on the way to a new destination and we plug in our location to a map on our phone, it tells us what lane to be in and takes us around any detours or collisions, sometimes even buzzing our watch to remind us to turn. When I experience this as a driver, I don’t have to do the thinking. I can think about what I’m going to cook for dinner, not paying much attention to my surroundings other than to follow those directions. If I were to be asked to go there again, I wouldn’t be able to remember, and I would again seek help.

If we can switch to giving students strategies that require them to think instead of giving them too much support throughout the journey to the answer, we may be able to give them the ability to learn the skills to read a map and have several ways to get there.

Here are six ways we can start letting students do this thinking so that they can go through rigorous problem-solving again and again, paving their own way to the solution. 

1. Link problem-solving to reading

When we can remind students that they already have many comprehension skills and strategies they can easily use in math problem-solving, it can ease the anxiety surrounding the math problem. For example, providing them with strategies to practice, such as visualizing, acting out the problem with math tools like counters or base 10 blocks, drawing a quick sketch of the problem, retelling the story in their own words, etc., can really help them to utilize the skills they already have to make the task less daunting.

We can break these skills into specific short lessons so students have a bank of strategies to try on their own. Here's an example of an anchor chart that they can use for visualizing . Breaking up comprehension into specific skills can increase student independence and help teachers to be much more targeted in their problem-solving instruction. This allows students to build confidence and break down the barriers between reading and math to see they already have so many strengths that are transferable to all problems.

2. Avoid boxing students into choosing a specific operation

It can be so tempting to tell students to look for certain words that might mean a certain operation. This might even be thoroughly successful in kindergarten and first grade, but just like when our map tells us where to go, that limits students from becoming deep thinkers. It also expires once they get into the upper grades, where those words could be in a problem multiple times, creating more confusion when students are trying to follow a rule that may not exist in every problem.

We can encourage a variety of ways to solve problems instead of choosing the operation first. In first grade, a problem might say, “Joceline has 13 stuffed animals and Jordan has 17. How many more does Jordan have?” Some students might choose to subtract, but a lot of students might just count to find the amount in between. If we tell them that “how many more” means to subtract, we’re taking the thinking out of the problem altogether, allowing them to go on autopilot without truly solving the problem or using their comprehension skills to visualize it. 

3. Revisit ‘representation’

The word “representation” can be misleading. It seems like something to do after the process of solving. When students think they have to go straight to solving, they may not realize that they need a step in between to be able to support their understanding of what’s actually happening in the problem first.

Using an anchor chart like one of these ( lower grade , upper grade ) can help students to choose a representation that most closely matches what they’re visualizing in their mind. Once they sketch it out, it can give them a clearer picture of different ways they could solve the problem.

Think about this problem: “Varush went on a trip with his family to his grandmother’s house. It was 710 miles away. On the way there, three people took turns driving. His mom drove 214 miles. His dad drove 358 miles. His older sister drove the rest. How many miles did his sister drive?”

If we were to show this student the anchor chart, they would probably choose a number line or a strip diagram to help them understand what’s happening.

If we tell students they must always draw base 10 blocks in a place value chart, that doesn’t necessarily match the concept of this problem. When we ask students to match our way of thinking, we rob them of critical thinking practice and sometimes confuse them in the process. 

4. Give time to process

Sometimes as educators, we can feel rushed to get to everyone and everything that’s required. When solving a complex problem, students need time to just sit with a problem and wrestle with it, maybe even leaving it and coming back to it after a period of time.

This might mean we need to give them fewer problems but go deeper with those problems we give them. We can also speed up processing time when we allow for collaboration and talk time with peers on problem-solving tasks. 

5. Ask questions that let Students do the thinking

Questions or prompts during problem-solving should be very open-ended to promote thinking. Telling a student to reread the problem or to think about what tools or resources would help them solve it is a way to get them to try something new but not take over their thinking.

These skills are also transferable across content, and students will be reminded, “Good readers and mathematicians reread.” 

6. Spiral concepts so students frequently use problem-solving skills

When students don’t have to switch gears in between concepts, they’re not truly using deep problem-solving skills. They already kind of know what operation it might be or that it’s something they have at the forefront of their mind from recent learning. Being intentional within their learning stations and assessments about having a variety of rigorous problem-solving skills will refine their critical thinking abilities while building more and more resilience throughout the school year as they retain content learning in the process. 

Problem-solving skills are so abstract, and it can be tough to pinpoint exactly what students need. Sometimes we have to go slow to go fast. Slowing down and helping students have tools when they get stuck and enabling them to be critical thinkers will prepare them for life and allow them multiple ways to get to their own destination.

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3rd Grade Math Word Problems Worksheets

Word problems for third grade.

Math word problems help deepen a student's understanding of mathematical concepts by relating mathematics to everyday life. 

These worksheets are best attempted after a student has studied the underlying skill; for example, our 'addition in columns" word problem worksheets should not be attempted until students are comfortable with addition in columns. 

In many of our word problems we intentionally include superfluous data , so that students need to read and think about the questions carefully, rather than simply applying a computation pattern to solve the problems.

Addition word problems for third grade

Simple addition word problems

Column form addition word problems

Mixed add and subtract word problems

Subtraction word problems

Simple subtraction word problems

Subtraction in columns word problems

Multiplication word problems

Simple multiplication word problems

Multiples of 10

Multiplying in columns

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Simple division word problems

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Fraction word problems

Identifying and comparing fractions word problems

Adding and subtracting fractions word problems

Mixed 3rd grade word problems

The following worksheets contain a mix of grade 3 addition, subtraction, multiplication and division word problems .  Mixing math word problems tests the understanding mathematical concepts, as it forces students to analyze the situation rather than mechanically apply a solution.

Mixed word problems - mental math  

Mixed word problems - column math

Mixed word problems - simpler form   (shorter texts, no superfluous data)

Measurement word problems for grade 3

These word problems combine the 4 operations with real world units of length, time, volume and mass. There is no conversion of units.

Length word problems

Time word problems

Mass & weight word problems

Volume & capacity word problems.

Word problems with variables

These grade 3 word problems introduce students to using variables ("x, y, etc.") to represent unknowns. The problems are relatively simple, but emphasize the use of variables and the writing of equations.

Word problems with variables (variable is chosen for the student)

Writing variables to solve word problems (student chooses the variable)

Sample Grade 3 Math Word Problems Worksheet

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Problem Solving/Reading Strategies Workbook Grade 3 1st Edition

• ISBN-10 0153365242
• ISBN-13 978-0153365249
• Edition 1st
• Publisher HARCOURT SCHOOL PUBLISHERS
• Publication date April 1, 2002
• Language English
• Dimensions 8 x 0.25 x 10.5 inches
• Print length 168 pages
• See all details

Product details

• Publisher ‏ : ‎ HARCOURT SCHOOL PUBLISHERS; 1st edition (April 1, 2002)
• Language ‏ : ‎ English
• Paperback ‏ : ‎ 168 pages
• ISBN-10 ‏ : ‎ 0153365242
• ISBN-13 ‏ : ‎ 978-0153365249
• Reading age ‏ : ‎ 8 - 9 years
• Grade level ‏ : ‎ 3 and up
• Item Weight ‏ : ‎ 12.8 ounces
• Dimensions ‏ : ‎ 8 x 0.25 x 10.5 inches

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