problem solving math lesson plan

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

FIND IT NOW!

Check me out on tpt.

CHECK THESE OUT

Three Types of Rocks and Minerals with Rock Cycle Circle Book

Partitioning Shapes Equal Share Fractions Halves, Thirds, Fourths Math Puzzles

Want to save time?

COPYRIGHT © 2016-2024. The Owl Teacher | Privacy page | Disclosure Page | Shipping | Returns/Refunds

BOGO on EVERYTHING!

• Try for free

Math Lesson Plans for Teachers

• Most Popular
• Most Recent

High Impact Tutoring Built By Math Experts

Personalized standards-aligned one-on-one math tutoring for schools and districts

Free ready-to-use math resources

Hundreds of free math resources created by experienced math teachers to save time, build engagement and accelerate growth

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

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 of 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.

Related articles

Ultimate guide to metacognition [free].

Looking for a summary on metacognition in relation to math teaching and learning?

Check out this guide featuring practical examples, tips and strategies to successfully embed metacognition across your school to accelerate math growth.

Privacy Overview

• Visit Two Rivers Public Charter School to see the school that inspired the Two Rivers Learning Institute.
• Course Login

Problem-Based Tasks in Math

Want to learn more about our work?

Dive deeper into our Instructional Practices and methodology by taking one of our online courses.

Providing students with opportunities to grapple with math has led to amazing things happening in my class. Students are totally excited and are driven to figure out not just how to solve a problem but why it works.

– Jessica Proffitt, Fifth-Grade Teacher at Two Rivers

Watch two rivers’s teachers and students at work on problem-based tasks in math.

Problem-Based Tasks Require Students to Apply Their Knowledge in New Contexts

Problem-based tasks are math lessons built around a single, compelling problem. The problems are truly “problematic” for students — that is, they do not offer an immediate solution.

The problems provide an opportunity for students to build conceptual understanding. Problem-based tasks require students to apply their current understanding and skills to new contexts that highlight core math concepts. For example, when students solve a problem that could be solved with multiplication before they have formally been taught what multiplication is and how it works, they build an understanding that multiplication is repeated addition.

Well-designed problem-based tasks provide multiple entry points for students to engage in problem solving, ensuring that all students have access to the same concepts. When students solve the problems in different ways—including drawing pictures, acting out the problem, writing algorithms, and using manipulatives—they make connections between the variety of models that all accurately illustrate the underlying mathematics.

Problem-Based Tasks in Math Resources

• Daily Games
• Strategy and Puzzles
• Vocabulary Games
• Junior Edition Games
• All problems
• High School Math
• MAML Problems
• Calculus Problems
• Loony Physics
• Pro Problems
• Getting Started
• Pro Control Panel
• Virtual Classroom
• Play My Favorites
• Select My Favorites

Problem Solving

Featured lesson plans.

Full Directory Listing

Blogs on this site.

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

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

You may also like...

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!

Leave a Reply Cancel reply

Your email address will not be published. Required fields are marked *

This site uses Akismet to reduce spam. Learn how your comment data is processed .

©2024 The Routty Math Teacher.   All Rights Reserved.  Designed by Ashley Hughes.

Privacy overview.

Solving Word Problems (Grades 1-2)

Our Solving Word Problems lesson plan provides students with strategies to help them solve word problems, such as using illustrations or drawings. Students practice solving example word problems using the given strategies.

Description

Additional information.

Our Solving Word Problems lesson plan develops math problem-solving strategies for young students. This interactive lesson equips students to identify and define keywords and use pictures or diagrams for math problem solving (addition and subtraction). Students are asked to work collaboratively, in pairs, to compose word problems that incorporate pictures or diagrams and exchange problems with other groups to solve. Students are also asked to individually complete practice problems in order to demonstrate their understanding of the lesson.

At the end of the lesson, students will be able to identify and define keywords and use pictures or diagrams for math problem solving (addition and subtraction).

State Educational Standards: LB.Math.Content.1.OA.A.2, LB.Math.Content.2.OA.A.1

Thank you for submitting a review!

Your input is very much appreciated. Share it with your friends so they can enjoy it too!

Great product

Love that it gives me a short lesson plan and the work to be done with it.

Solving Word Problems

This concept is hard for students. Thanks for explicit lesson on solving word problems.

We recently began homeschooling my son, after learning that he has not been being taught many subjects due to being in self-contained unit at school because of his behavioral issues. After explaining a few things to him, the material really helped to guide him on to the next part in our efforts to get him caught up. Thank you.

Related products

Printing Cursive Letters

All about Butterflies

Reading and Writing with Illustrations

Daylight Saving Time 1

Make your life easier with our lesson plans, stay up-to-date with new lessons.

• Lesson Plans
• For Teachers

© 2024 Learn Bright. All rights reserved. Terms and Conditions. Privacy Policy.

• Sign Up for Free

Want a daily email of lesson plans that span all subjects and age groups?

Subjects all subjects all subjects the arts all the arts visual arts performing arts value of the arts back business & economics all business & economics global economics macroeconomics microeconomics personal finance business back design, engineering & technology all design, engineering & technology design engineering technology back health all health growth & development medical conditions consumer health public health nutrition physical fitness emotional health sex education back literature & language all literature & language literature linguistics writing/composition speaking back mathematics all mathematics algebra data analysis & probability geometry measurement numbers & operations back philosophy & religion all philosophy & religion philosophy religion back psychology all psychology history, approaches and methods biological bases of behavior consciousness, sensation and perception cognition and learning motivation and emotion developmental psychology personality psychological disorders and treatment social psychology back science & technology all science & technology earth and space science life sciences physical science environmental science nature of science back social studies all social studies anthropology area studies civics geography history media and journalism sociology back teaching & education all teaching & education education leadership education policy structure and function of schools teaching strategies back thinking & learning all thinking & learning attention and engagement memory critical thinking problem solving creativity collaboration information literacy organization and time management back, filter by none.

• Elementary/Primary
• Middle School/Lower Secondary
• High School/Upper Secondary
• College/University
• TED-Ed Animations
• TED Talk Lessons
• TED-Ed Best of Web
• Under 3 minutes
• Under 6 minutes
• Under 9 minutes
• Under 12 minutes
• Under 18 minutes
• Over 18 minutes
• Algerian Arabic
• Azerbaijani
• Cantonese (Hong Kong)
• Chinese (Hong Kong)
• Chinese (Singapore)
• Chinese (Taiwan)
• Chinese Simplified
• Chinese Traditional
• Chinese Traditional (Taiwan)
• Dutch (Belgium)
• Dutch (Netherlands)
• French (Canada)
• French (France)
• French (Switzerland)
• Kurdish (Central)
• Luxembourgish
• Persian (Afghanistan)
• Persian (Iran)
• Portuguese (Brazil)
• Portuguese (Portugal)
• Spanish (Argentina)
• Spanish (Latin America)
• Spanish (Mexico)
• Spanish (Spain)
• Spanish (United States)
• Western Frisian

sort by none

• Longest video
• Shortest video
• Most video views
• Least video views
• Most questions answered
• Least questions answered

This piece of paper could revolutionize human waste

Lesson duration 05:35

2,433,132 Views

Can you solve the magical maze riddle?

Lesson duration 04:51

326,286 Views

How to clear icy roads, with science

Lesson duration 06:13

191,585 Views

How to make smart decisions more easily

Lesson duration 05:16

1,056,826 Views

Can you solve a mystery before Sherlock Holmes?

Lesson duration 05:17

464,186 Views

Can you solve the secret assassin society riddle?

Lesson duration 05:01

673,024 Views

How to overcome your mistakes

Lesson duration 04:52

887,531 Views

What the fossil fuel industry doesn't want you to know - Al Gore

Lesson duration 25:45

737,310 Views

Can you solve the cursed dice riddle?

Lesson duration 04:31

666,255 Views

How the water you flush becomes the water you drink

Lesson duration 05:23

385,879 Views

The growing megafire crisis — and how to contain it - George T. Whitesides

Lesson duration 10:42

56,594 Views

Can you solve the time traveling car riddle?

Lesson duration 05:18

619,115 Views

4 epidemics that almost happened (but didn't)

Lesson duration 06:26

393,561 Views

The return of Mongolia's "wild" horses

Lesson duration 04:53

204,703 Views

Whatever happened to the hole in the ozone layer?

Lesson duration 05:13

517,262 Views

The most important century in human history

Lesson duration 05:20

337,414 Views

This one weird trick will get you infinite gold

Lesson duration 05:08

989,034 Views

How to quit your job — without ruining your career - Gala Jackson

104,099 Views

How to design climate-resilient buildings - Alyssa-Amor Gibbons

Lesson duration 14:12

42,917 Views

The case for free, universal basic services - Aaron Bastani

Lesson duration 19:09

80,078 Views

Can you steal the most powerful wand in the wizarding world?

755,110 Views

How college loans exploit students for profit - Sajay Samuel

Lesson duration 11:49

229,159 Views

What’s the smartest age?

1,551,048 Views

The 4 greatest threats to the survival of humanity

Lesson duration 05:24

484,996 Views

Math Interventions

Introduction.

• Subitizing Interventions
• Counting Interventions: Whole Numbers Less Than 30
• Counting Interventions: Whole Numbers Greater Than 30 (Place Value)
• Counting Interventions: Fractions
• Counting Interventions: Decimals
• Composing and Decomposing Numbers Interventions
• Rounding Interventions
• Number Sense Lesson Plans
• Addition and Subtraction Facts
• Multiplication and Division Facts
• Computational Fluency Lesson Plans
• Understanding the Problem Interventions
• Planning and Executing a Solution Interventions
• Monitoring Progress & Reflecting on a Solution Interventions
• Problem-Solving Process Interventions
• Problem-Solving Lesson Plans
• Identifying Essential Variables Interventions
• Direct Models Interventions
• Counting On/Back Interventions
• Deriving Interventions
• Interpreting the Results Interventions
• Mathematical Modeling Lesson Plans
• Math Rules and Concepts Interventions
• Math Rules and Concepts Lesson Plans

You can use either  Explicit Instruction  or Self-Regulated Strategy Development when you intervene to support your student's problem solving skills. The following lesson plan targets a specific problem-solving skill using explicit instruction. As you read this plan, consider: 

How does this plan support objective mastery?

Problem Solving Intervention Plan

Art, E. (2017). Problem solving intervention packet. New York, NY: Relay Graduate School of Education.

This lesson plan supports objective mastery because the teacher employs Principles of Specialized instruction to help the student visualize Part/Part/Whole (Part Unknown) problems to identify what she is supposed to figure out. In this lesson plan, she isolates the skill (identifying what the problem is), and uses explicit instruction to teach the student how to identify two what is happening in the story and what she is trying to figure out. After she explains the process she'll take, she uses metacognition and shows the student how she asks these two questions as she is reading. Finally, she builds in multiple at-bats so that the student has the opportunity to practice this strategy over and over so that she can reach her objective.

• << Previous: Problem-Solving Process Interventions
• Next: Identifying Essential Variables Interventions >>
• Last Updated: Feb 14, 2024 6:46 PM
• URL: https://relay.libguides.com/math-interventions

Chapter 1, Lesson 1: A Plan for Problem Solving

• Extra Examples
• Group Activity Cards
• Personal Tutor
• Self-Check Quizzes

The resource you requested requires you to enter a username and password below:

Please read our Terms of Use and Privacy Notice before you explore our Web site. To report a technical problem with this Web site, please contact the site producer .

• Benefits to Participating Communities
• Participating School Districts
• Evaluations and Results
• Recognition Accorded
• National Advisory Committee
• Establishing New Institutes
• Topical Index of Curriculum Units
• View Topical Index of Curriculum Units
• Search Curricular Resources
• View Volumes of Curriculum Units from National Seminars
• Find Curriculum Units Written in Seminars Led by Yale Faculty
• Find Curriculum Units Written by Teachers in National Seminars
• Browse Curriculum Units Developed in Teachers Institutes
• On Common Ground
• Reports and Evaluations
• Articles and Essays
• Documentation
• Video Programs

Have a suggestion to improve this page?

To leave a general comment about our Web site, please click here

Share this page with your network.

Real World Problem Solving in Second Grade Mathematics

Introduction.

Edgewood Magnet School in New Haven, Connecticut is an arts magnet school, integrating the arts across the curriculum. Students in this environment are encouraged to use the strategies of observation, interpretation, and analysis to increase their thinking skills in every subject. With that mission, both teachers and students use unique and exciting approaches to “the basics” and work together to ensure that all learners are included.

For most second graders, the beginning of the year is a time for refreshing knowledge and skills from first grade. The summer away from direct instruction and opportunities for practice and guidance sometimes means a loss of solid understanding of learned concepts in mathematics. This three- to four-week unit is designed to review and build new understanding of one-step word problem solving using addition and subtraction as students develop skills and strategies they will use all year. The students, through a series of mathematical scenarios, will use the problem types identified in Table 1 of the Common Core Mathematics Glossary which covers addition and subtraction. 1

The Common Core concentrates on a clear set of math skills and concepts. Students learn concepts in an organized way during the school year as well as across grades. The standards encourage students to solve real-world problems. 2

The Common Core calls for greater focus in mathematics. Rather than racing to cover many topics in a mile-wide, inch-deep curriculum, the standards ask math teachers to significantly narrow and deepen the way time and energy are spent in the classroom. This means focusing sharply on the major work of each grade, which for grades Kindergarten through second grade includes concepts, skills, and problem solving related to addition and subtraction.

The New Haven Public School district uses the Math in Focus Singapore Approach, a Common Core-based curriculum for students from Kindergarten to Fifth Grade. The student books and workbooks follow an instructional pathway that includes learning concepts and skills through visual lessons and teacher instruction for understanding the how and why; consolidating concepts and skills through practice, activities and math journals for deep math understanding, hands-on work in pairs and in small groups; and, applying concepts and skills through extensive problem solving practice and challenges to build real world problem solvers. 3

This approach embeds problem solving throughout each lesson and encourages frequent practice in both computation and problem solving. The word problems appear throughout each chapter and progress from 1-step to 2-step to multi-step. Each chapter concludes with a challenging problem or set of problems that require students to solve some non-routine questions. To solve these problems, the students need to draw on their deep prior knowledge as well as recently acquired concepts and skills, combining problem solving strategies with critical thinking skills, including classifying, comparing, sequencing, identifying parts and whole, identifying patterns and relationships, induction and deduction and spatial visualization.

The second grade text begins with numbers to 1000. Students begin by expressing numbers in standard form (231), expanded form (200 + 30 + 1), and word form (two hundred thirty-one). This is accompanied by concrete representations via base ten blocks, and, for two digit numbers and a few three digit numbers, representation by trains of rods, of lengths 1, 10 and 100. This initial chapter also includes sequencing numbers and comparing using greater than and less than terminology, and then moving right into addition and subtraction of two- and three-digits numbers. Here the take-away should be, if you have more hundreds, the tens and ones don’t make any/much difference; and if you have the same number of hundreds, but more tens, then the ones don’t make any/much difference. Most of my students (if not all) struggle from the start! They do not seem to have a solid foundation of understanding numbers to 100 or the concept of place value in general. This unit is designed to get ahead of the frustration that the students feel when pushed too quickly before they have a firm understanding of principles of place value and the properties of operations.

This unit launches the school year with 1-step addition and subtraction problems of all types using numbers to 10. The goal is to spend time practicing basic computations with numbers that student can work comfortably with before jumping right into the district curriculum. Once there is a level of understanding with these problem sets (numbers to 10), students will move on to solving 1-step problems using teen numbers and then onto numbers to 100. Most of the curriculum problems at the start of the year require addition and subtraction of 3-digit numbers. Some students will move quickly through the problem sets with numbers to 100 and will be ready to work with the regular curriculum.

For the duration of the unit, the focus will be steadily on solving and later constructing a collection of word problems that provide robust and balanced practice. Problems sets will be based on a scenario which will provide the substance of the story. Each scenario will allow us to extract several problems, changing the numbers and ensuring each set of numbers makes a reasonable problem. This idea looks like the following: John has 8 crayons in his box. He shares 3 with Sam. How many crayons does John have left in the box? John has some crayons in his box. He shares 3 with Sam. John has 5 crayons left in his box. How many crayons did John start with? John has 5 crayons. Sam has 2 fewer than John. How many crayons does Sam have? John and Sam are sharing crayons. John has 5 and Sam has 3. How many crayons do the friends have together? The two students participate in several crayon-sharing stories that use the same set of numbers but in slightly different situations. Some situations are more obvious and direct while others take more thinking. It is important to provide opportunities for students to work with and solve the different problem types that can be created from one set of numbers. 4

Background: Problem Types

The taxonomy of addition and subtraction problem types as identified in the Common Core State Standards of Mathematics Glossary is a framework that sorts one-step problems into three broad classes: change , comparison , and part-part-whole . Each of the three classes is then separated further into a total of 14 problem types sorted out as follows: change , in which some quantity is either added to or taken away from another quantity over time; comparison , in which one amount is described as more or less than another amount; and part-part-whole , in which an amount is made up of two parts. 5

Within the group of change problems, there are two subgroups: change-increase , in which a quantity is added to an initial amount and change-decrease , in which a quantity is taken from an initial amount. We might recognize these subgroups more familiarly as “add to” or “take from.” Additionally within each of these subgroups, there are three possible unknown quantities. One scenario to show change-increase : 2 kittens were playing with some yarn. 3 more kittens join them. Now there are 5 kittens playing with the yarn. Using these quantities, the unknown might be the result (2 + 3 = ?), an unknown quantity of change (2 + ? = 5) or an unknown initial amount (? + 3 = 5). In the change-decrease subgroup, there are again three possible unknowns. A scenario for this example: 5 birds are sitting on the branch. 2 fly away. Now there are 3 birds sitting on the branch. Here again the students might solve for the final amount (5 – 2 = ?), the amount of change (5 - ? = 3), or the initial amount (? – 2 = 3). This gives in all six types of change problems.

Similarly comparison problems can also be categorized into two subgroups: comparison-more , in which one quantity is described as more or greater than another, and comparison-less, in which one quantity is described as less or fewer than another. Here again, each of these two subgroups has three possible unknowns, for a total of 6 types. Sam has 6 marbles. James has 8 marbles. James has 2 more marbles than Sam. The unknown quantity may be the lesser amount (? + 2 = 8), an unknown greater amount

(6 + 2 = ?), or the unknown difference (8 – 6 = ?) one quantity that is more and one that is less. Using this same scenario for a set of comparison-less problems, the language need to change from “more than” to “less than.” Here is a way to present this set with the language adjustment: Sam has 6 marbles. James has 8 marbles. Sam has 2 fewer marbles than James.

Part-part-whole problems are a set of two quantities, the parts that, when put together, make up a whole quantity.  This problem type seems very like to change category but in this problem type there is no change over time. The two parts play equivalent roles, which allow for only two possible unknown categories: either a part is unknown or the whole is the unknown. There are 4 large dogs and 3 small dogs. There are 7 dogs in all. One of the parts may be unknown (4 + ? = 7 or ? + 3 = 7) or the unknown may be the size of the whole (4 + 3 = ?). Since the parts are interchangeable, there are only 2 types in this class of problems.

The following chart sort these classes and categories into the framework. Located in Appendix A of this unit is a set of example problems illustrating each of these 14 types.

The Scenarios of the Problems

For second graders, life at school is a large part of their world. Most of my students arrived at Edgewood for their Kindergarten year and stayed through First Grade making the year in second grade essentially their third year at the same school. They are comfortable in the building and know many of the other students. They will become the active players in the math stories that I, and we together, will construct. Activities that occur in the classroom, in the cafeteria, on the playground, and on the bus seem to be recognizable situations that will help with the basic understanding of context.

Additionally, there are opportunities for students to incorporate the topics and learning that occur in the other subjects, such as science, social studies, literacy, art, music and, in our school, dance and drama. One example might be to create set of story problems centered on the life cycle of the butterfly, a unit of study each year in second grade. With the common knowledge the students will be obtaining, this content could become the scenarios for word problems. An example might be: Seven caterpillars climbed up the branch and formed their chrysalises. Later that day, three more caterpillars climbed up the branch and formed their chrysalises. How many chrysalises are hanging from the branch? Similarly, using the characters in a book read together as a class could provide the characters in a new set of problems. Curious George had a bunch of bananas. He ate 4 of them. Now he has 3. How many bananas did Curious George begin with? The use of common or thematic content will not only connect all the thinking and practicing, it will provide tangible and real situations. With an established scenario, students will work with a set of numbers, determining the unknown within each of problems types.

Creating the Problems

A question that is frequently answered with a guess is “What should we do to answer the question to solve the word problem?” The fundamental understanding of what is being asked is not apparent to the students, making the solution inaccessible. Most first graders entering second grade have a basic understanding when the story (problem) is categorized as final unknown or whole unknown , but most other components of the taxonomy are unfamiliar to them or just too difficult to decode. To begin to help them with their thinking, they will use concrete models, such as themselves (2 children are sitting at the reading table, 4 more join them) acting out scenarios. Many basic materials in the classroom – pencils, notebooks, folders, crayons – can be used to create and design scenarios, with each type of problem represented.

Solving the Problems

Following the overall plan of the Singapore Math program, the students will solve problems using the concrete, pictorial and abstract approach. Because this is a standard approach in our district mathematics instruction throughout the year, the students will begin with this set of strategies to solve problem sets.

Word problems are written as stories and scenarios making language a consideration in crafting the problems for the beginning second graders. Word problems are as much about language and reading as they are about math. If the story is not understandable, how can students begin the know what to do with the numbers they’ve been given and the question they’ve been asked? Thus, words and vocabulary need to be appropriate and useful for the variety of reading levels of the incoming students.  The structure of word problems should be understandable and clear, accessible in language as well as numbers. Also, the language, especially the words that express the relationship between the quantities involved, should be discussed to ensure that it is familiar to all students.

This is a clear integration of Language Arts and Mathematics and a method in which students can connect math to the real world, in this case, through the activities they engage in at school. Reading skills and computation skills come together with even the simplest of word problems.

Structure of Problem Collection

The content introduction over the duration of this unit includes a certain sequencing and scaffolding to guide students through the 14 problem types. To begin the unit, students will only be working with numbers to 10. This is an important starting place to ensure that understanding is occurring. Most of my second graders are capable with addition and subtraction to 10, but are not so comfortable with the word problem language. So first, students will be challenged more by the language than the arithmetic. Students will practice figuring out what exactly the problems are asking with problems that they are familiar with before moving on to a new step. Practicing all the problem types will improve and increase strategies and confidence!

With addition and subtraction within 10 mastered, the next phase of the unit moves to numbers to 20. The key is to continue with scenarios that are obvious and repeated as new numbers are introduced. An example of this transition would be these parallel problems:

6 students got on the bus at the first stop. 3 students got on the bus at the second   stop. After the second stop, how many students are on the bus? ( change-increase, final unknown)

Some students got on the bus at the first stop. 3 students got on the bus at the second stop. Now there are 9 students on the bus. How many students got on at the first stop? ( change-increase, initial unknown)

These now become:

11 students got on the bus at the first stop. 7 students got on the bus at the second stop. After the second stop, how many students are on the bus? ( change-increase, final unknown)

Some students got on the bus at the first stop. 7 students got on the bus at the second stop. Now there are 18 students on the bus. How many students got on at   the first stop? ( change-increase, initial unknown)

When working with numbers to 20, it is essential that students understand that the “teen” numbers (11-19) are really 10 and some ones. Students should work with numbers within 20, creating equations using their knowledge and skill of making a ten first. In the case of 7 + 6, making a new ten looks like this:

7 + 6 = 7 + 3 + 3 = 10 + 3 = 13

Because 7 needs a 3 to make ten, and 6 is composed of 3 + 3, this equation shows the progression of making 10 and some more. Practicing this method using two ten frames demonstrates the process concretely. In the example above, students use the ten-frames to show 7 and 6 separately. To make the new 10, students will move 3 from the 6, which now shows 10 and 3 more or 13.

As mentioned earlier, it is obvious that the most accessible problem types for students entering the second grade are the change-increase or change-decrease, result unknown and part-part-whole, whole unknown. The general go-to strategy for solving a word problem seems to be to just take the two numbers you see and add them together, or maybe subtract, but often the students are just unsure. It seems that these are the most practiced problem types, which leaves students without balanced experience with all 14 types and ultimately without some strategies to employ as they problem-solve. Students need to see a broad range of problems to gain a strong understanding of how addition and subtraction are used and how they are related to each other. The notion of example sufficiency means students should be exposed to a wide array of examples to provide well-rounded practice with the concept. 6

Teaching Strategies

The approaches for this curriculum unit vary to reflect the learning styles of all students.

The general format is based on the workshop model. The concepts and skills are taught through a series of mini-lessons focused on the objective with the following methods used throughout:

Experiential Learning: Most young students need to begin with hands-on learning. Using concrete models to work out math stories allows students to see the problem and manipulate the pieces as the story progresses. This type of learning is an important first step.

Differentiated Instruction: Lessons and activities will be targeted to maximize learning. The students will use a variety of approaches, working sometimes individually and sometimes in small groups, determined by the complexity of the work. Some students will move more quickly as they master skills and some will need more opportunities for practice.

Cooperative Learning: The students will be given opportunities to work as cooperative groups to create math stories to present to the class. This strategy will allow students to work collaboratively taking on various roles necessary to complete the work, with a focus on success for all.

Classroom Activities

Activity 1: sequenced problem types – problems to 10.

The introduction (and review) portion of the unit covers all problem types but in a sequenced manner. The objective is for students to read and interpret a word problem with guided instruction followed by independent practice. Because of the many problem types, this part will take several days of review and practice before students are comfortable beginning to write their own sets of problems. Based on student need and pace of understanding, I expect this section to be a four- to six-day set of lessons, more if needed.

The sequence is as follows: part-part-whole ; change-increase and change-decrease ; and finally, compare-greater and compare-fewer . The following introductory sessions are designed as a whole group activity, with students either at their desks or gathered on the rug close to the board or easel. The whole group portion should be 20 minutes at most. At the close of each session, I will give students between 5 and 10 similar problems to solve. More capable students can begin to generate their own problems during the independent work time.

Beginning with the fundamentals provides a good opportunity to get to know students’ skills which is helpful in preparing differentiated work and creating groups,

In this lesson, students will interpret real world problems and with the help of manipulatives and pictures, solve part-part-whole stories using addition and subtraction.

6 girls are playing

3 boys are playing with them.

How many children are playing in all?

Begin the story with the whole unknown as in this example. This type of story is perfect for students to act out right in the classroom. Write the story on the board or chart paper and have students volunteer as actors. Once the students have solved the problem, write the math sentence to show what happened: 6 + 3 = 9 students. Explain that the 2 parts (boys and girls) have made a whole (children). With the students still in acting position, present a new approach to this scenario:

9 students are playing.

6 of them are girls.

How many boys are playing?

With this visual example, students should see right away how many. The important concept to demonstrate is that the parts can be determined when the whole and one part are known, in this case 9 is known as the whole and 6 as one part. Again, write the math sentence to show this calculation: 6 + ? = 9 and include the strategy of starting with the whole to determine the missing part as a subtraction sentence 9 – 6 = 3. Practicing both approaches to the solution will help students connect addition and subtraction and recognize how they are used together.

Since this lesson requires students to read story problems, I will pair fluent readers with those who are less fluent, provide counters for those who want them, and allow partners to work together to solve and problems and share the strategies that they used.

I will use two more examples, like the ones below, to demonstrate, remembering to write the word problem on the board as well as the math sentence. I will also reword the problems to have the part as the unknown.

Hannah has 5 red markers.

She has 3 blue markers

How many markers does Hannah have in all?

7 students are drawing with crayons.

2 students are drawing with colored pencils.

How many students are drawing?

Continuing with this same idea, the next set of problem types includes change-increase and change-decrease . Although part-part-whole is language that students can adopt and use while discussing their work, the change-increase and change decrease language is a bit trickier. The use of the word change is more appropriate for students to demonstrate that some amount has been either added or subtracted from an initial amount.

Introduce the word problem below which is an example of the unknown result in the change-increase category.

Jason had 8 “caught being good” stickers on his chart at the beginning of the day.

During the school day, he earned 2 more stickers.

How many stickers does Jason have on his chart at the end of the day?

Student can solve the problem as written and, using the same scenario, challenge them to create the change-unknown and initial unknown story. One example might be:

Jason had some “caught being good” stickers on his chart at the

beginning of the day.

At the end of the day, he has 10 stickers.

How many stickers did Jason have at the beginning of the day?

This is an oral activity, with me writing the adjusted version across the board, placing the math sentence underneath. It is important to allow students to work on composing the problem so they can begin the see the relationship between the problems and what the problems are asking.

The goal is for students to understand and not just solve. I can informally assess during the discussion of rewriting the text of the word problem, with more formal assessment later in the unit.

The next category to introduce is the change-decrease problem types. Following the same format as before, I will introduce the result unknown, change unknown and then initial unknown.

Crystal collected 7 leaves for her project.

2 leaves blew away in the breeze.

How many leaves does Crystal have left for her project?

Again, the goal is for students to understand and not just solve.

The third broad class, compare, is more difficult for my 2 nd grade students. This requires the text of the word problems to be very straight-forward. Students should not get tangled up when they are learning to take the data from the problem. Remember that using the exact terminology is not the goal, but rather understanding what the problem is asking. Here are three ways I will present a scenario that shows the problem types comparison-more , and three ways to show comparison-less. Students need to be exposed to and have opportunities to practice all types. Of course, not all of these examples should be used at one time. As I write the problems out on chart paper and post them in the classroom, the students can begin to see and do their own comparing and contrasting as one scenario is explained in different ways. The use of the words “more” and “fewer” should be highlighted and explained as the problem set is introduced and worked on. My role here is to let the students begin to notice the subtle differences in the wording and how it changes the thinking. Simpler is better to start with!

Throughout these introductory sessions, the students and I will brainstorm scenarios that can eventually be used in own word problems. Ideas should generate from school activities and materials, guiding students to think of what students can actually use for manipulatives or, as in the first scenario, be able to act out to solve. By keeping a list of ideas on chart paper as reference material, students won’t struggle with vocabulary or appropriate scenarios; they will be on to the task of crafting their problems. This list will prepare the students for the second part of the unit.

Activity 2: Classroom / school scenarios

As stated earlier, words and vocabulary should be accessible to students and not a challenge or hurdle. The goal is to get to the thinking of the stories and plugging in the information that was gathered during the brainstorming session. To begin this portion, review the charts and add more if students have new ideas. It may be helpful for the purposes of composing word problems to have the information in categories, such as these:

Materials We Use

Classes We Attend

Activities at School

Classmate’s Names

I will create groups of two or three students to have them write problems of their own to share with the class.  Since this lesson requires students to read and write story problems, again, fluent readers and writers with those who are less fluent, provide counters for those who want them, and allow partners to work together to solve and problems and share the strategies that they used.

The goal during this period of time is to challenge students to write the same problem but try it another way, choose a different type as they tell the story. The timing for the student groups to work together will be during arrival time as morning work and during the math workshop portion of math instruction time. This will allow students to work as much as 30 minutes per day with their partners to create some math stories.

I will stress that it is important to keep their collection together as much of their work will become part of the workbook they will create at the end of this unit. Folders and math journals can be helpful, or my collecting the work-in-progress daily is another option.

Activity 3: Science Scenarios

The first unit in 2 nd grade science is investigation and research on the life cycle of the butterfly. Students receive caterpillars at the start of the semester and observe and record the changes the caterpillar’s life. The work that the students do during their science lessons can become the information and scenarios they can use for crafting word problems.

Using all different problem types, we will write several together as a class.  This is an additional opportunity to integrate math very specifically into our science research and work. It is important for students to recognize that, although their learning has been compartmentalized into subject areas, it is essentially impossible to separate it all out into categories. So this portion of the unit uses math, science and reading to help students learn about the life cycle of a butterfly (and other animals as well).

Students will create problem sets that use their daily experiences tracking their caterpillars. Each student has 2-3 caterpillars to observe and record information on, which can become the start of word problems. Examples to start: If Table 3 has 8 caterpillars and 2 caterpillars join that group, how many caterpillars are being observed at Table 3? Here are 28 students in the class. Each student need one cup of caterpillar food.  There are 30 cups of caterpillar food. How many more cups of food are there than students?

There are often students who have great interest in other areas of science. This is an area to encourage if students are excited about sharing their knowledge. Some students will be more inclined to use the unit of study going on in class, but throughout the literacy portion of the day, students are exposed to a great deal of non-fiction, or informational text, that could certainly enrich our science word problems.

Throughout the duration of the science unit, students will continue to write word problems of various types to eventually include in our final project, the workbook. These problems can be written during the morning work session, during math workshop, and at the end of science class. By the end of the unit, each student should have two problems to add to the Science chapter of the workbook.

Activity 4: Creation of Workbook / Publishing Celebration

The goal of this portion of the unit is to sort the word problems into “chapters” and create a workbook to share at the Publication Celebration. Chapters will be titled by subject or category, depending on student choice and teacher suggestion. Ideas include Beginning Stories, Classroom Activities, Playground Fun, Science & Math, and Social Studies Connections. Let students be creative with titles!

Students will submit their work which will include at least one word problem for each chapter. They must also submit the solutions to their problems so that they can be included in an answer key. Each chapter will have at least 25 problems, with examples of all types and with varying levels of difficulty. Word problems can either be typed or hand-written for the final workbook, depending on what the students decide as a class. One workbook per student will need to be copied and bound in some manner for the Celebration.

Two weeks before the Publication Celebration, students will create an invitation to give to their family and friends, inviting them to come for a “Celebration of Problem Solving.” Parents and other VIP guests will spend some time working on word problems, moving around the room, visiting many students. The students will share their own specific work with the guests (the word problems they themselves created) and “help” their visitors figure out the answers.

Each student will have a “Comments” sheet for guests to sign and leave comments on their experience working with the student. I will encourage visitors to stop to talk with each student or as many as they can during their visit.

Additionally, this is an opportunity to have some students work as editors and publishers. Creating the workbook will require review and assembly time and these tasks can be delegated and shared by the students who are interested.

Common Core State Standards for Mathematics,

http://www.corestandards.org/the-standards/mathematics.

Fong, Ho Kheong. Math in Focus Singapore Math by Marshall Cavendish . Final ed. Singapore: Marshall Cavendish Education, 2015.

Fuson, Karen C. Math Expressions . 2011 ed. Orlando, FL: Houghton Mifflin Harcourt ;, 2011.

Howe, Roger. Three Pillars of First Grade Mathematics and Beyond

Howe, Roger, The Most Important Thing for Your Child to Learn about Arithmetic

Howe, Roger and Harold Reiter, The Five Stages of Place Value

Ma, L. Knowing and Teaching Elementary Mathematics , Erlbaum Associates, Mahwah, NJ, 1999.

The Moscow Puzzles: 359 Mathematical Recreations . New York: Dover Publications, 1992.

Polya, George. How to Solve It: A New Aspect of Mathematical Method . New Princeton Science Library ed., 2014.

Appendix A: Problem Set

Change Increase / Result Unknown

7 students are in the classroom. 2 more students join them. How many students are in the classroom now?

Change Increase / Change Unknown

7 students are in the classroom. Some more students join them. Then there were 9 students. How many students join the first 7?

Change Increase / Initial Unknown

Some students are in the classroom. 2 more students joined them. Then there were 9 students. How many students were in the classroom at the beginning?

Change Decrease / Result Unknown

9 students were in the classroom. 2 students went home. How many students are in the classroom now?

Change Decrease / Change Unknown

9 students were in the classroom. Some students went home. Now there are 7 students. How many students went home?

Change Decrease / Initial Unknown

Some students were in the classroom. 2 students went home. Then there were 7 students in the classroom. How many students were in the classroom in the beginning?

Compare More / Difference Unknown

Sam has 10 French fries. Emily has 6 French fries. How many more does Sam have than Emily?

Compare More / Greater Unknown

Sam has 4 more French fries than Emily. Emily has 6 French fries. How many French fries does Sam have?

Compare More / Smaller Unknown

Sam has 4 more French fries than Emily. Sam has 10 French fries. How many French fries does Emily have?

Compare Fewer / Difference Unknown

Sam has 10 French fries. Emily has 6 French fries. How many fewer does Emily have than Sam?

Compare Fewer / Smaller Unknown

Emily has 4 fewer French fries than Sam. Sam has 10 French fries. How many French fries doe Emily have?

Compare Fewer / Greater Unknown

Emily has 4 fewer French fries than Sam. Emily has 6 French fries. How many French fries does Sam have?

Part-Part-Whole / Whole Unknown

Sam has 4 cookies for lunch. He has 2 more for dinner. How many cookies does Sam have?

Sam ate 6 cookies today. He had 4 of them for lunch. How many did he have for dinner?

Appendix B – Implementing Common Core Standards

This unit integrates, quite naturally, literacy and math. Both reading and writing are essential parts of the students’ ability to solve word problems involving both addition and subtraction.

Students will work most specifically toward the Common Core State Standard in Mathematics, 2.OA.A.1 which states that second graders should, by the end of the year, be able to “represent and solve problems involving addition and subtraction within 100 to solve one- step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions. During this unit students will begin solving and crafting word problems with numbers to 10, advance to numbers to 20 and continue on to the 100’s and up to 1000 as they master place value concepts.

This unit also addresses the Language Arts Common Core State Standards of Reading Informational Text, RI.2.1, in which students work on locating key ideas and details by asking and answering such questions as who, what, where, when, why, and how to demonstrate understanding of key details in a text. Throughout this unit on word problems, students will be working asking these questions as they determine what information the math story is providing. As they begin to write their own word problems, they will need to consider these questions to craft a meaningful story for the text of their problem.

• Common Core State Standards for Mathematics, http://www.corestandards.org/the-standards/mathematics.
• Common Core State Standards for Mathematics
• Ho Kheong Fong, Math in Focus Singapore Math by Marshall Cavendish , 8.
• Roger Howe, Three Pillars of First Grade Mathematics and Beyond, 2 .
• Roger Howe, Three Pillars of First Grade Mathematics and Beyond, 1 . Common Core State Standards for Mathematics
• Roger Howe, Three Pillars of First Grade Mathematics and Beyond, 2

Comments (0)

Be the first person to comment

• Career & Technical Education (CTE)
• Lesson Plans & Activities
• Professional Development
• Student Engagement
• Student Achievement
• School Funding
• Special Education

By: Margo Ensz on April 30th, 2014

Print/Save as PDF

Math Problem Solving Strategies (Primary) - Lesson Plan & Templates

Grade Level: Primary (PK-2) | Featured Topics: Lessons and Activities | Academic Subject: Mathematics | Featured Topics: Templates for FREE download

This lesson plan is part of our partnership with two super creative educators from  The Curriculum Corner , bringing you free Common Core aligned lesson ideas, teacher resources, and templates you can easily print with your VariQuest Perfecta Poster Design System - and be sure you have the latest version of the VariQuest Software ! 

Lesson:  introduction to problem solving strategies (two-day lesson) , level:  primary, objectives & ccss alignment:  students will increase independence in using various strategies to solve different types of word problems., *note: ccss has problem solving strategies spread throughout the domains for math.  this lesson focuses on giving them a review of each of the various strategies.  it is suggested that teachers follow up this review with a more in-depth lesson in each of the strategies as they fit naturally into the curriculum or as there is student need. , materials & visuals: , anchor chart – problem solving strategies, my math notes for problem solving.

Click on each image to download the PDF and print to a poster!

Preparation:

Think through a sample problem for each of the strategies listed on the anchor chart.  these will be shown to the students as examples., prepare “problem solving” anchor chart for mini-lesson., copy the “my math notes for problem solving” page for each child in the classroom.  you may choose to three-hole-punch this page for use in a math notes binder/folder to be kept by students and referred to throughout the year. , mini-lesson: , ask students what a “strategy” is.  accept several answers and then lead the class to the conclusion that a strategy is a plan to reach a goal.  in math a strategy would be a plan to reach a final answer., display the anchor chart titled “problem solving strategies” for the students.  tell them that these are some ways that they can use to solve all kinds of different math problems., pass out “my math notes for problem solving” to each student and explain that this page is a place for them to write down some notes that will help them to remember what each strategy is and how to use it.  they will keep this in their math binder/folder and can pull it out as a reminder when needed., *note: the boxes of this resource have been left blank so that the teacher can decide the level of support needed to fill it out.  students may need to draw or write exactly what the teacher shares for each strategy or they may be allowed to draw or write whatever they feel will help them to remember the strategy., briefly explain a few of the eight strategies and guide students as to how you would like for them to take their notes (copying the teacher notes or creating their own depending on grade or skill level).  introduce however many strategies that time allows for in this first lesson, and then complete the note-taking for the final strategies the following day., have students place their math notes into an organized binder or folder that will be kept and used throughout the year..

Did you like this lesson? Leave a comment and let us know what topics you would like us to cover in future posts and we will work on delivering free resources to help you prepare for your lessons!

3 Ways to Strengthen Math Instruction

• Share article

Students’ math scores have plummeted, national assessments show , and educators are working hard to turn math outcomes around.

But it’s a challenge, made harder by factors like math anxiety , students’ feelings of deep ambivalence about how math is taught, and learning gaps that were exacerbated by the pandemic’s disruption of schools.

This week, three educators offered solutions on how districts can turn around poor math scores in a conversation moderated by Peter DeWitt, an opinion blogger for Education Week.

Here are three takeaways from the discussion. For more, watch the recording on demand .

1. Intervention is key

Research shows that early math skills are a key predictor of later academic success.

“Children who know more do better, and math is cumulative—so if you don’t grasp some of the earlier concepts, math gets increasingly harder,” said Nancy Jordan, a professor of education at the University of Delaware.

For example, many students struggle with the concept of fractions, she said. Her research has found that by 6th grade, some students still don’t really understand what a fraction is, which makes it harder for them to master more advanced concepts, like adding or subtracting fractions with unlike denominators.

At that point, though, teachers don’t always have the time in class to re-teach those basic or fundamental concepts, she said, which is why targeted intervention is so important.

Still, Jordan’s research revealed that in some middle schools, intervention time is not a priority: “If there’s an assembly, or if there is a special event or whatever, it takes place during intervention time,” she said. “Or ... the children might sit on computers, and they’re not getting any really explicit instruction.”

2. ‘Gamify’ math class

Students today need new modes of instruction that meet them where they are, said Gerilyn Williams, a math teacher at Pinelands Regional Junior High School in Little Egg Harbor Township, N.J.

“Most of them learn through things like TikTok or YouTube videos,” she said. “They like to play games, they like to interact. So how can I bring those same attributes into my lesson?”

Part of her solution is gamifying instruction. Williams avoids worksheets. Instead, she provides opportunities for students to practice skills that incorporate elements of game design.

That includes digital tools, which provide students with the instant feedback they crave, she said.

But not all the games are digital. Williams’ students sometimes play “trashketball,” a game in which they work in teams to answer math questions. If they get the question right, they can crumble the piece of paper and throw it into a trash can from across the room.

“The kids love this,” she said.

Williams also incorporates game-based vocabulary into her instruction, drawing on terms from video games.

For example, “instead of calling them quizzes and tests, I call them boss battles,” she said. “It’s less frightening. It reduces that math anxiety, and it makes them more engaging.

“We normalize things like failure, because when they play video games, think about what they’re doing,” Williams continued. “They fail—they try again and again and again and again until they achieve success.”

3. Strengthen teacher expertise

To turn around math outcomes, districts need to invest in teacher professional development and curriculum support, said Chaunté Garrett, the CEO of ELLE Education, which partners with schools and districts to support student learning.

“You’re not going to be able to replace the value of a well-supported and well-equipped mathematics teacher,” she said. “We also want to make sure that that teacher has a math curriculum that’s grounded in the standards and conceptually based.”

Students will develop more critical thinking skills and better understand math concepts if teachers are able to relate instruction to real life, Garrett said—so that “kids have relationships that they can pull on, and math has some type of meaning and context to them outside of just numbers and procedures.”

It’s important for math curriculum to be both culturally responsive and relevant, she added. And teachers might need training on how to offer opportunities for students to analyze and solve real-world problems.

“So often, [in math problems], we want to go back to soccer and basketball and all of those things that we lived through, and it’s not that [current students] don’t enjoy those, but our students live social media—they literally live it,” Garrett said. “Those are the things that have to live out in classrooms right now, and if we’re not doing those things, we are doing a disservice.”

Sign Up for EdWeek Update

Edweek top school jobs.

Sign Up & Sign In

Word Problems Lesson Plans

Explore our word problems lesson plans for teachers! Find curriculum-aligned lessons focused on solving word problems using arithmetic operations, tackling fraction word problems, and solving mixed word problems. These plans offer fun learning activities, targeted goals, and useful teaching strategies. ... Read more Our word problem lesson plans cover everything from comprehending problem statements to analyzing how to solve them. Let's help kids connect math concepts to real life. Start now for free!

CONTENT TYPE

• Lesson Plans
• Number Sense (24)
• Counting (9)
• Compare and Order Numbers (13)
• Place Value (9)
• Addition (42)
• Subtraction (28)
• Multiplication (29)
• Times Tables (8)
• Division (11)
• Fractions (25)
• Fractions Operations (17)
• Decimals (9)
• Geometry (5)
• Measurement (8)
• Counting Money (5)
• Algebra (6)
• Word Problems (36)
• Reading (160)
• Phonics (44)
• Books and Genres (4)
• Reading Skills (101)
• Communication Skills (13)
• Writing (10)
• Creative Writing (9)
• Grammar (93)
• Adverbs and Adjectives (15)
• Nouns and Pronouns (21)
• Prepositions and Conjunctions (12)
• Punctuation (15)
• Sentences (10)
• Verbs and Tenses (20)
• Vocabulary (22)
• Affixes (5)
• Making Connections in Reading (5)

Jungle Quest: Master Add & Subtract Lesson Plan

Develop fluency in addition and subtraction within 5 with this interactive jungle adventure lesson plan.

Word Problem Wizardry Adventure Lesson Plan

Become a Word Problem Wizard with this interactive math lesson!

Bar Models: Mastering Comparison Word Problems Lesson Plan

Master comparison word problems with sets and bar models!

Master Subtraction Word Problems Lesson Plan

Master subtraction word problems with this fun-filled math adventure!

Tackle Real-World Addition Word Problems & Win Lesson Plan

Solve addition word problems with ease!

Addition & Subtraction Problems Magic Lesson Plan

Master addition and subtraction word problems with our interactive math lesson. Join the Word Problem Wizards on a magical journey of learning!

Addition Word Problems Adventure Lesson Plan

Unleash your problem-solving skills with this interactive math lesson on addition word problems. Solve mysteries while mastering the count on strategy!

3-Number Summation: An Epic Math Adventure Lesson Plan

Master addition skills with this interactive math lesson. Learn how to add three numbers using various strategies and solve word problems along the way!

Solve Subtraction Word Problems Lesson Plan

Solve subtraction word problems with confidence!

Math Mysteries: Crack Addition Word Problems Lesson Plan

Solve addition word problems step by step in this interactive math adventure.

Detective Math: Subtraction Word Problem Fun Lesson Plan

Solve subtraction word problems like a detective!

Quantitative Quest: Solving Comparison Problems Lesson Plan

Master the skill of comparing quantities with this engaging math lesson. Explore bar models, solve comparison problems, and develop critical thinking skills!

Embrace the Art of Quantitative Comparisons Lesson Plan

Master the skill of comparing quantities with this interactive math lesson. Explore bar models, solve comparison problems, and develop critical thinking skills!

Solve & Conquer: Comparison Word Problems Await Lesson Plan

Unleash your problem-solving skills with this interactive math lesson on comparison word problems.

Add & Subtract: Word Problems Made Fun Lesson Plan

Master addition and subtraction word problems with this engaging math lesson. Develop problem-solving skills and learn strategies to tackle real-world scenarios.

Solve 2-Step Word Problems: Math Adventure Time Lesson Plan

Master two-step addition and subtraction word problems using the part-part-whole model!

Two-Step Word Problem Solving Lesson Plan

Develop problem-solving skills with this interactive math lesson. Solve two-step word problems using strategies like counting on, counting forward/backward, and the part-part-whole model.

Word Problem Mastery: Model, Solve & Understand Lesson Plan

Master word problem comprehension with interactive activities. Understand addition, subtraction, and multiplication scenarios through models and equations.

Division Mastery: Facts & Problem-Solving Fun Lesson Plan

Master division facts and problem-solving skills with this interactive math lesson.

Multiplication Word Problem Dive Lesson Plan

Analyze and solve multiplication word problems with this interactive math lesson!

Dive into Multiplication Word Problems Lesson Plan

Enhance students' multiplication skills through engaging word problems. Develop problem-solving strategies and apply them to real-life situations.

Multi-Step Word Problems Mastery Lesson Plan

Master multi-step word problems with this engaging math lesson!

Multiplication Word Problems Dive Lesson Plan

Unleash your problem-solving skills with this interactive math lesson on solving multiplication word problems. Explore different strategies and learn how to apply them to real-life situations.

Dive into Division: Comparing Expressions Made Easy Lesson Plan

Master the skill of comparing division expressions with this engaging math lesson. Explore various scenarios and learn how to analyze and compare different division situations.

Tackle Fraction Word Problems Lesson Plan

Master fraction word problems with this interactive math lesson!

Whole Number Addition & Subtraction Mastery Awaits Lesson Plan

Master addition and subtraction of whole numbers with this interactive math lesson. Explore column method/standard algorithm, estimation, and word problems!

Arrays & Equal Groups Multiplication Lesson Plan

Master multiplication using arrays and equal groups in this engaging math lesson. Explore different models, solve word problems, and enhance your mathematical reasoning skills.

Magical Multi-Step Word Problems Lesson Plan

Unleash your problem-solving skills in this magical math adventure!

Multiplication & Division Relationship Lesson Plan

Explore the connection between multiplication and division in this engaging math lesson. Discover how these two operations are related and learn strategies for solving problems using their connection.

Multiplication Journey: 3 & 4 Digits Lesson Plan

Master the art of multiplying large numbers with this engaging lesson plan. Explore strategies for multiplying three- and four-digit numbers by one-digit numbers, both with and without regrouping. Develop a deep understanding of the concepts through hands-on activities and problem-solving tasks.

Multiply 2-Digits by 1-Digit Lesson Plan

Mastering multiplication with two-digit and one-digit numbers in an interactive lesson.

Estimate Fraction Sums: A Math Dive Lesson Plan

Join us on a fun-filled math adventure as we learn how to estimate the sums of fractions using benchmark fractions. Engage in hands-on activities and problem-solving tasks to develop your estimation skills!

Multiplying Fractions: Deep Dive Lesson Plan

Master fraction multiplication through engaging word problems!

Fraction Word Problems Mastery Lesson Plan

Conquer word problems with fractions using key strategies in this interactive math lesson.

Decimals Word Problems Mastery Lesson Plan

Develop problem-solving skills with this interactive math lesson. Learn strategies to solve word problems involving decimals and apply them to real-world scenarios.

Mastering Multi-Step Problems Lesson Plan

Master multi-step word problems with this engaging math lesson. Explore real-life scenarios and apply mathematical operations to find solutions.

Browse Our Comprehensive Collection of Word Problems Lesson Plans for Teachers!

Discover SplashLearn's engaging word problems lesson plans for effective teaching and joyful learning! Aligned with the Common Core Standards, these plans empower teachers to deliver interactive lessons on solving word problems. Students will explore real-world scenarios, learn to identify and comprehend the key information, apply mathematical operations, and solve word problems across various contexts.

Word problems help kids connect math with everyday life. Introducing kids to word problems is essential for boosting their problem-solving abilities. Our interactive online lesson plans make learning word problems enjoyable. Each lesson features engaging activities and clear instructions, allowing teachers to focus on teaching without the hassle of extensive preparation.

With our lesson plans, teachers can effectively teach word problems by providing clear explanations, breaking down problems into manageable steps, encouraging students to identify key information, and guiding them at every step of finding the solution.

Join us to help children master word problems in an enjoyable and playful way. Our online word problems lesson plans will enable teachers to improve their teaching preparation effortlessly. Happy teaching! 

Topics Covered in Word Problems Lesson Plans across Different Grades

Let’s take a grade-wise overview of concepts covered under word problems lesson plans.  

Benefits of Word Problems Lesson Plans for Teachers

• Free Word Problems Lesson Plans for Teachers: Teachers can access free lesson plans aligned with the curriculum, facilitating streamlined instruction and highly effective lesson preparation.
• Discover Effective Teaching Strategies: Explore engaging strategies to teach word problems in an enjoyable way. Help students grasp the question statements and analyze the necessary math operations to solve them.
• Use Fun Teaching Activities: Engage students in interactive activities within each lesson to reinforce problem-solving skills. These lessons use real-life examples and visual aids to boost learning.
• I mprove Classroom Time Management: Our plans provide a detailed class overview, outlining learning objectives, activities, and allocated time, enabling teachers to manage classroom time effectively.
• Additional Practice and Assessment: Each lesson offers extra practice exercises to help teachers identify areas where students may need additional support and reinforcement in their problem-solving skills.

What Are the Best 5 Word Problems Lesson Plans for Teachers?

Here are 5 interactive lessons focused on solving different word problems:

• Mastering Comparison Word Problems
• Solving Two-Step Word Problems
• Fraction Word Problems Mastery
• Decimals Word Problems Mastery
• Mastering Multi-Step Word Problems

Your one stop solution for all grade learning needs.

• Share full article

For more audio journalism and storytelling, download New York Times Audio , a new iOS app available for news subscribers.

A.I.’s Original Sin

A times investigation found that tech giants altered their own rules to train their newest artificial intelligence systems..

This transcript was created using speech recognition software. While it has been reviewed by human transcribers, it may contain errors. Please review the episode audio before quoting from this transcript and email [email protected] with any questions.

From “The New York Times,” I’m Michael Barbaro. This is “The Daily.”

[MUSIC PLAYING]

Today, a “Times” investigation shows how as the country’s biggest technology companies race to build powerful new artificial intelligence systems, they bent and broke the rules from the start.

My colleague Cade Metz on what he uncovered.

It’s Tuesday, April 16th.

Cade, when we think about all the artificial intelligence products released over the past couple of years, including, of course, these chatbots we’ve talked a lot about on the show, we so frequently talk about their future their future capabilities, their influence on society, jobs, our lives. But you recently decided to go back in time to AI’s past, to its origins to understand the decisions that were made, basically, at the birth of this technology. So why did you decide to do that?

Because if you’re thinking about the future of these chatbots, that is defined by their past. The thing you have to realize is that these chatbots learn their skills by analyzing enormous amounts of digital data.

So what my colleagues and I wanted to do with our investigation was really focus on that effort to gather more data. We wanted to look at the type of data these companies were collecting, how they were gathering it, and how they were feeding it into their systems.

And when you all undertake this line of reporting, what do you end up finding?

We found that three major players in this race OpenAI, Google, and Meta as they were locked into this competition to develop better and better artificial intelligence, they were willing to do almost anything to get their hands on this data, including ignoring, and in some cases, violating corporate rules and wading into a legal gray area as they gathered this data.

Basically, cutting corners.

Cutting corners left and right.

OK, let’s start with OpenAI, the flashiest player of all.

The most interesting thing we’ve found, is that in late 2021, as OpenAI, the startup in San Francisco that built ChatGPT, as they were pulling together the fundamental technology that would power that chatbot, they ran out of data, essentially.

They had used just about all the respectable English language text on the internet to build this system. And just let that sink in for a bit.

I mean, I’m trying to let that sink in. They basically, like a Pac-Man on a old game, just consumed almost all the English words on the internet, which is kind of unfathomable.

Wikipedia articles by the thousands, news articles, Reddit threads, digital books by the millions. We’re talking about hundreds of billions, even trillions of words.

So by the end of 2021, OpenAI had no more English language texts that they could feed into these systems, but their ambitions are such that they wanted even more.

So here, we should remember that if you’re gathering up all the English language text on the internet, a large portion of that is going to be copyrighted.

So if you’re one of these companies gathering data at that scale, you are absolutely gathering copyrighted data, as well.

Which suggests that, from the very beginning, these companies, a company like OpenAI with ChatGPT, is starting to break, bend the rules.

Yes, they are determined to build this technology thus they are willing to venture into what is a legal gray area.

So given that, what does OpenAI do once it, as you had said, runs out of English language words to mop up and feed into this system?

So they get together, and they say, all right, so what are other options here? And they say, well, what about all the audio and video on the internet? We could transcribe all the audio and video, turn it into text, and feed that into their systems.

Interesting.

So a small team at OpenAI, which included its president and co-founder Greg Brockman, built a speech-recognition technology called Whisper, which could transcribe audio files into text with high accuracy.

And then they gathered up all sorts of audio files, from across the internet, including audio books, podcasts —

— and most importantly, YouTube videos.

Hmm, of which there’s a seemingly endless supply, right? Fair to say maybe tens of millions of videos.

According to my reporting, we’re talking about at least 1,000,000 hours of YouTube videos were scraped off of that video sharing site, fed into this speech recognition system in order to produce new text for training OpenAI’s chatbot. And YouTube’s terms of service do not allow a company like OpenAI to do this. YouTube, which is owned by Google, explicitly says you are not allowed to, in internet parlance, scrape videos en masse from across YouTube and use those videos to build a new application.

That is exactly what OpenAI did. According to my reporting, employees at the company knew that it broke YouTube terms of service, but they resolved to do it anyway.

So, Cade, this makes me want to understand what’s going on over at Google, which as we have talked about in the past on the show, is itself, thinking about and developing its own artificial intelligence model and product.

Well, as OpenAI scrapes up all these YouTube videos and starts to use them to build their chatbot, according to my reporting, some employees at Google, at the very least, are aware that this is happening.

Yes, now when we went to the company about this, a Google spokesman said it did not know that OpenAI was scraping YouTube content and said the company takes legal action over this kind of thing when there’s a clear reason to do so. But according to my reporting, at least some Google employees turned a blind eye to OpenAI’s activities because Google was also using YouTube content to train its AI.

So if they raise a stink about what OpenAI is doing, they end up shining a spotlight on themselves. And they don’t want to do that.

I guess I want to understand what Google’s relationship is to YouTube. Because of course, Google owns YouTube. So what is it allowed or not allowed to do when it comes to feeding YouTube data into Google’s AI models?

It’s an important distinction. Because Google owns YouTube, it defines what can be done with that data. And Google argues that it has a right to that data, that its terms of service allow it to use that data. However, because of that copyright issue, because the copyright to those videos belong to you and I, lawyers who I’ve spoken to say, people could take Google to court and try to determine whether or not those terms of service really allow Google to do this. There’s another legal gray area here where, although Google argues that it’s OK, others may argue it’s not.

Of course, what makes this all so interesting is, you essentially have one tech company Google, keeping another tech company OpenAI’s dirty little secret about basically stealing from YouTube because it doesn’t want people to know that it too is taking from YouTube. And so these companies are essentially enabling each other as they simultaneously seem to be bending or breaking the rules.

What this shows is that there is this belief, and it has been there for years within these companies, among their researchers, that they have a right to this data because they’re on a larger mission to build a technology that they believe will transform the world.

And if you really want to understand this attitude, you can look at our reporting from inside Meta.

And so what does Meta end up doing, according to your reporting?

Well, like Google and other companies, Meta had to scramble to build artificial intelligence that could compete with OpenAI. Mark Zuckerberg is calling engineers and executives at all hours pushing them to acquire this data that is needed to improve the chatbot.

And at one point, my colleagues and I got hold of recordings of these Meta executives and engineers discussing this problem. How they could get their hands on more data where they should try to find it? And they explored all sorts of options.

They talked about licensing books, one by one, at $10 a pop and feeding those into the model.

They even discussed acquiring the book publisher Simon & Schuster and feeding its entire library into their AI model. But ultimately, they decided all that was just too cumbersome, too time consuming, and on the recordings of these meetings, you can hear executives talk about how they were willing to run roughshod over copyright law and ignore the legal concerns and go ahead and scrape the internet and feed this stuff into their models.

They acknowledged that they might be sued over this. But they talked about how OpenAI had done this before them. That they, Meta were just following what they saw as a market precedent.

Interesting, so they go from having conversations like, should we buy a publisher that has tons of copyrighted material suggesting that they’re very conscious of the kind of legal terrain and what’s right and what’s wrong. And instead say, nah, let’s just follow the OpenAI model, that blueprint and just do what we want to do, do what we think we have a right to do, which is to kind of just gobble up all this material across the internet.

It’s a snapshot of that Silicon Valley attitude that we talked about. Because they believe they are building this transformative technology, because they are in this intensely competitive situation where money and power is at stake, they are willing to go there.

But what that means is that there is, at the birth of this technology, a kind of original sin that can’t really be erased.

It can’t be erased, and people are beginning to notice. And they are beginning to sue these companies over it. These companies have to have this copyrighted data to build their systems. It is fundamental to their creation. If a lawsuit bars them from using that copyrighted data, that could bring down this technology.

We’ll be right back.

So Cade, walk us through these lawsuits that are being filed against these AI companies based on the decisions they made early on to use technology as they did and the chances that it could result in these companies not being able to get the data they so desperately say they need.

These suits are coming from a wide range of places. They’re coming from computer programmers who are concerned that their computer programs have been fed into these systems. They’re coming from book authors who have seen their books being used. They’re coming from publishing companies. They’re coming from news corporations like, “The New York Times,” incidentally, which has filed a lawsuit against OpenAI and Microsoft.

News organizations that are concerned over their news articles being used to build these systems.

And here, I think it’s important to say as a matter of transparency, Cade, that your reporting is separate from that lawsuit. That lawsuit was filed by the business side of “The New York Times” by people who are not involved in your reporting or in this “Daily” episode, just to get that out of the way.

I’m assuming that you have spoken to many lawyers about this, and I wonder if there’s some insight that you can shed on the basic legal terrain? I mean, do the companies seem to have a strong case that they have a right to this information, or do companies like the “Times,” who are suing them, seem to have a pretty strong case that, no, that decision violates their copyrighted materials.

Like so many legal questions, this is incredibly complicated. It comes down to what’s called fair use, which is a part of copyright law that determines whether companies can use copyrighted data to build new things. And there are many factors that go into this. There are good arguments on the OpenAI side. There are good arguments on “The New York Times” side.

Copyright law says that can’t take my work and reproduce it and sell it to someone. That’s not allowed. But what’s called fair use does allow companies and individuals to use copyrighted works in part. They can take snippets of it. They can take the copyrighted works and transform it into something new. That is what OpenAI and others are arguing they’re doing.

But there are other things to consider. Does that transformative work compete with the individuals and companies that supplied the data that owned the copyrights?

And here, the suit between “The New York Times” company and OpenAI is illustrative. If “The New York Times” creates articles that are then used to build a chatbot, does that chatbot end up competing with “The New York Times?” Do people end up going to that chatbot for their information, rather than going to the “Times” website and actually reading the article? That is one of the questions that will end up deciding this case and cases like it.

So what would it mean for these AI companies for some, or even all of these lawsuits to succeed?

Well, if these tech companies are required to license the copyrighted data that goes into their systems, if they’re required to pay for it, that becomes a problem for these companies. We’re talking about digital data the size of the entire internet.

Licensing all that copyrighted data is not necessarily feasible. We quote the venture capital firm Andreessen Horowitz in our story where one of their lawyers says that it does not work for these companies to license that data. It’s too expensive. It’s on too large a scale.

Hmm, it would essentially make this technology economically impractical.

Exactly, so a jury or a judge or a law ruling against OpenAI, could fundamentally change the way this technology is built. The extreme case is these companies are no longer allowed to use copyrighted material in building these chatbots. And that means they have to start from scratch. They have to rebuild everything they’ve built. So this is something that, not only imperils what they have today, it imperils what they want to build in the future.

And conversely, what happens if the courts rule in favor of these companies and say, you know what, this is fair use. You were fine to have scraped this material and to keep borrowing this material into the future free of charge?

Well, one significant roadblock drops for these companies. And they can continue to gather up all that extra data, including images and sounds and videos and build increasingly powerful systems. But the thing is, even if they can access as much copyrighted material as they want, these companies may still run into a problem.

Pretty soon they’re going to run out of digital data on the internet.

That human-created data they rely on is going to dry up. They’re using up this data faster than humans create it. One research organization estimates that by 2026, these companies will run out of viable data on the internet.

Wow. Well, in that case, what would these tech companies do? I mean, where are they going to go if they’ve already scraped YouTube, if they’ve already scraped podcasts, if they’ve already gobbled up the internet and that altogether is not sufficient?

What many people inside these companies will tell you, including Sam Altman, the chief executive of OpenAI, they’ll tell you that what they will turn to is what’s called synthetic data.

And what is that?

That Is data generated by an AI model that is then used to build a better AI model. It’s AI helping to build better AI. That is the vision, ultimately, they have for the future that they won’t need all this human generated text. They’ll just have the AI build the text that will feed future versions of AI.

So they will feed the AI systems the material that the AI systems themselves create. But is that really a workable solid plan? Is that considered high-quality data? Is that good enough?

If you do this on a large scale, you quickly run into problems. As we all know, as we’ve discussed on this podcast, these systems make mistakes. They hallucinate . They make stuff up. They show biases that they’ve learned from internet data. And if you start using the data generated by the AI to build new AI, those mistakes start to reinforce themselves.

The systems start to get trapped in these cul-de-sacs where they end up not getting better but getting worse.

What you’re really saying is, these AI machines need the unique perfection of the human creative mind.

Well, as it stands today, that is absolutely the case. But these companies have grand visions for where this will go. And they feel, and they’re already starting to experiment with this, that if you have an AI system that is sufficiently powerful, if you make a copy of it, if you have two of these AI models, one can produce new data, and the other one can judge that data.

It can curate that data as a human would. It can provide the human judgment, So. To speak. So as one model produces the data, the other one can judge it, discard the bad data, and keep the good data. And that’s how they ultimately see these systems creating viable synthetic data. But that has not happened yet, and it’s unclear whether it will work.

It feels like the real lesson of your investigation is that if you have to allegedly steal data to feed your AI model and make it economically feasible, then maybe you have a pretty broken model. And that if you need to create fake data, as a result, which as you just said, kind of undermines AI’s goal of mimicking human thinking and language, then maybe you really have a broken model.

And so that makes me wonder if the folks you talk to, the companies that we’re focused on here, ever ask themselves the question, could we do this differently? Could we create an AI model that just needs a lot less data?

They have thought about other models for decades. The thing to realize here, is that is much easier said than done. We’re talking about creating systems that can mimic the human brain. That is an incredibly ambitious task. And after struggling with that for decades, these companies have finally stumbled on something that they feel works that is a path to that incredibly ambitious goal.

And they’re going to continue to push in that direction. Yes, they’re exploring other options, but those other options aren’t working.

What works is more data and more data and more data. And because they see a path there, they’re going to continue down that path. And if there are roadblocks there, and they think they can knock them down, they’re going to knock them down.

But what if the tech companies never get enough or make enough data to get where they think they want to go, even as they’re knocking down walls along the way? That does seem like a real possibility.

If these companies can’t get their hands on more data, then these technologies, as they’re built today, stop improving.

We will see their limitations. We will see how difficult it really is to build a system that can match, let alone surpass the human brain.

These companies will be forced to look for other options, technically. And we will see the limitations of these grandiose visions that they have for the future of artificial intelligence.

OK, thank you very much. We appreciate it.

Glad to be here.

Here’s what else you need to know today. Israeli leaders spent Monday debating whether and how to retaliate against Iran’s missile and drone attack over the weekend. Herzi Halevi, Israel’s Military Chief of Staff, declared that the attack will be responded to.

In Washington, a spokesman for the US State Department, Matthew Miller reiterated American calls for restraint —

^MATTHEW MILLER^ Of course, we continue to make clear to everyone that we talked to that we want to see de-escalation that we don’t want to see a wider regional war. That’s something that’s been —

— but emphasized that a final call about retaliation was up to Israel. ^MATTHEW MILLER^ Israel is a sovereign country. They have to make their own decisions about how best to defend themselves. What we always try to do —

And the first criminal trial of a former US President officially got underway on Monday in a Manhattan courtroom. Donald Trump, on trial for allegedly falsifying documents to cover up a sex scandal involving a porn star, watched as jury selection began.

The initial pool of 96 jurors quickly dwindled. More than half of them were dismissed after indicating that they did not believe that they could be impartial. The day ended without a single juror being chosen.

Today’s episode was produced by Stella Tan, Michael Simon Johnson, Muge Zaidi, and Rikki Novetsky. It was edited by Marc Georges and Liz O. Baylen, contains original music by Diane Wong, Dan Powell, and Pat McCusker, and was engineered by Chris Wood. Our theme music is by Jim Brunberg and Ben Landsverk of Wonderly.

That’s it for “The Daily.” I’m Michael Barbaro. See you tomorrow.

• April 24, 2024   •   32:18 Is $60 Billion Enough to Save Ukraine?
• April 23, 2024   •   30:30 A Salacious Conspiracy or Just 34 Pieces of Paper?
• April 22, 2024   •   24:30 The Evolving Danger of the New Bird Flu
• April 19, 2024   •   30:42 The Supreme Court Takes Up Homelessness
• April 18, 2024   •   30:07 The Opening Days of Trump’s First Criminal Trial
• April 17, 2024   •   24:52 Are ‘Forever Chemicals’ a Forever Problem?
• April 16, 2024   •   29:29 A.I.’s Original Sin
• April 15, 2024   •   24:07 Iran’s Unprecedented Attack on Israel
• April 14, 2024   •   46:17 The Sunday Read: ‘What I Saw Working at The National Enquirer During Donald Trump’s Rise’
• April 12, 2024   •   34:23 How One Family Lost $900,000 in a Timeshare Scam
• April 11, 2024   •   28:39 The Staggering Success of Trump’s Trial Delay Tactics
• April 10, 2024   •   22:49 Trump’s Abortion Dilemma

Hosted by Michael Barbaro

Featuring Cade Metz

Produced by Stella Tan ,  Michael Simon Johnson ,  Mooj Zadie and Rikki Novetsky

Edited by Marc Georges and Liz O. Baylen

Original music by Diane Wong ,  Dan Powell and Pat McCusker

Engineered by Chris Wood

Listen and follow The Daily Apple Podcasts | Spotify | Amazon Music

A Times investigation shows how the country’s biggest technology companies, as they raced to build powerful new artificial intelligence systems, bent and broke the rules from the start.

Cade Metz, a technology reporter for The Times, explains what he uncovered.

On today’s episode

Cade Metz , a technology reporter for The New York Times.

Background reading

How tech giants cut corners to harvest data for A.I.

What to know about tech companies using A.I. to teach their own A.I.

There are a lot of ways to listen to The Daily. Here’s how.

We aim to make transcripts available the next workday after an episode’s publication. You can find them at the top of the page.

The Daily is made by Rachel Quester, Lynsea Garrison, Clare Toeniskoetter, Paige Cowett, Michael Simon Johnson, Brad Fisher, Chris Wood, Jessica Cheung, Stella Tan, Alexandra Leigh Young, Lisa Chow, Eric Krupke, Marc Georges, Luke Vander Ploeg, M.J. Davis Lin, Dan Powell, Sydney Harper, Mike Benoist, Liz O. Baylen, Asthaa Chaturvedi, Rachelle Bonja, Diana Nguyen, Marion Lozano, Corey Schreppel, Rob Szypko, Elisheba Ittoop, Mooj Zadie, Patricia Willens, Rowan Niemisto, Jody Becker, Rikki Novetsky, John Ketchum, Nina Feldman, Will Reid, Carlos Prieto, Ben Calhoun, Susan Lee, Lexie Diao, Mary Wilson, Alex Stern, Dan Farrell, Sophia Lanman, Shannon Lin, Diane Wong, Devon Taylor, Alyssa Moxley, Summer Thomad, Olivia Natt, Daniel Ramirez and Brendan Klinkenberg.

Our theme music is by Jim Brunberg and Ben Landsverk of Wonderly. Special thanks to Sam Dolnick, Paula Szuchman, Lisa Tobin, Larissa Anderson, Julia Simon, Sofia Milan, Mahima Chablani, Elizabeth Davis-Moorer, Jeffrey Miranda, Renan Borelli, Maddy Masiello, Isabella Anderson and Nina Lassam.

Cade Metz writes about artificial intelligence, driverless cars, robotics, virtual reality and other emerging areas of technology. More about Cade Metz

Advertisement