problem solving maths lesson

Teaching Problem Solving in Math

  • Freebies , Math , Planning

Problem solving tends to REALLY throw students for a loop when they're first introduced to it. Up until this point, math has been numbers, but now, math is numbers and words. I discuss four important steps I take in teaching problem solving, and I provide you with examples as I go. You can also check out my math workshop problem solving unit for third grade!

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.

Problem solving tends to REALLY throw students for a loop when they're first introduced to it. Up until this point, math has been numbers, but now, math is numbers and words. I discuss four important steps I take in teaching problem solving, and I provide you with examples as I go. You can also check out my math workshop problem solving unit for third grade!

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.

Problem solving tends to REALLY throw students for a loop when they're first introduced to it. Up until this point, math has been numbers, but now, math is numbers and words. I discuss four important steps I take in teaching problem solving, and I provide you with examples as I go. You can also check out my math workshop problem solving unit for third grade!

There’s also this visuals strategy wheel practice.

Problem solving tends to REALLY throw students for a loop when they're first introduced to it. Up until this point, math has been numbers, but now, math is numbers and words. I discuss four important steps I take in teaching problem solving, and I provide you with examples as I go. You can also check out my math workshop problem solving unit for third grade!

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.

Problem solving tends to REALLY throw students for a loop when they're first introduced to it. Up until this point, math has been numbers, but now, math is numbers and words. I discuss four important steps I take in teaching problem solving, and I provide you with examples as I go. You can also check out my math workshop problem solving unit for third grade!

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!

Problem solving tends to REALLY throw students for a loop when they're first introduced to it. Up until this point, math has been numbers, but now, math is numbers and words. I discuss four important steps I take in teaching problem solving, and I provide you with examples as I go. You can also check out my math workshop problem solving unit for third grade!

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.

Problem solving tends to REALLY throw students for a loop when they're first introduced to it. Up until this point, math has been numbers, but now, math is numbers and words. I discuss four important steps I take in teaching problem solving, and I provide you with examples as I go. You can also check out my math workshop problem solving unit for third grade!

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.

Problem solving tends to REALLY throw students for a loop when they're first introduced to it. Up until this point, math has been numbers, but now, math is numbers and words. I discuss four important steps I take in teaching problem solving, and I provide you with examples as I go. You can also check out my math workshop problem solving unit for third grade!

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.

Problem solving tends to REALLY throw students for a loop when they're first introduced to it. Up until this point, math has been numbers, but now, math is numbers and words. I discuss four important steps I take in teaching problem solving, and I provide you with examples as I go. You can also check out my math workshop problem solving unit for third grade!

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.

Problem solving tends to REALLY throw students for a loop when they're first introduced to it. Up until this point, math has been numbers, but now, math is numbers and words. I discuss four important steps I take in teaching problem solving, and I provide you with examples as I go. You can also check out my math workshop problem solving unit for third grade!

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.

Problem solving tends to REALLY throw students for a loop when they're first introduced to it. Up until this point, math has been numbers, but now, math is numbers and words. I discuss four important steps I take in teaching problem solving, and I provide you with examples as I go. You can also check out my math workshop problem solving unit for third grade!

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.

Problem solving tends to REALLY throw students for a loop when they're first introduced to it. Up until this point, math has been numbers, but now, math is numbers and words. I discuss four important steps I take in teaching problem solving, and I provide you with examples as I go. You can also check out my math workshop problem solving unit for third grade!

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.

Problem solving tends to REALLY throw students for a loop when they're first introduced to it. Up until this point, math has been numbers, but now, math is numbers and words. I discuss four important steps I take in teaching problem solving, and I provide you with examples as I go. You can also check out my math workshop problem solving unit for third grade!

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!

Problem solving tends to REALLY throw students for a loop when they're first introduced to it. Up until this point, math has been numbers, but now, math is numbers and words. I discuss four important steps I take in teaching problem solving, and I provide you with examples as I go. You can also check out my math workshop problem solving unit for third grade!

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.

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The resources on this page will hopefully help you teach AO2 and AO3 of the new GCSE specification - problem solving and reasoning.

This brief lesson is designed to lead students into thinking about how to solve mathematical problems. It features ideas of strategies to use, clear steps to follow and plenty of opportunities for discussion.

problem solving maths lesson

The PixiMaths problem solving booklets are aimed at "crossover" marks (questions that will be on both higher and foundation) so will be accessed by most students. The booklets are collated Edexcel exam questions; you may well recognise them from elsewhere. Each booklet has 70 marks worth of questions and will probably last two lessons, including time to go through answers with your students. There is one for each area of the new GCSE specification and they are designed to complement the PixiMaths year 11 SOL.

These problem solving starter packs are great to support students with problem solving skills. I've used them this year for two out of four lessons each week, then used Numeracy Ninjas as starters for the other two lessons.  When I first introduced the booklets, I encouraged my students to use scaffolds like those mentioned here , then gradually weaned them off the scaffolds. I give students some time to work independently, then time to discuss with their peers, then we go through it as a class. The levels correspond very roughly to the new GCSE grades.

Some of my favourite websites have plenty of other excellent resources to support you and your students in these assessment objectives.

@TessMaths has written some great stuff for BBC Bitesize.

There are some intersting though-provoking problems at Open Middle.

I'm sure you've seen it before, but if not, check it out now! Nrich is where it's at if your want to provide enrichment and problem solving in your lessons.

MathsBot  by @StudyMaths has everything, and if you scroll to the bottom of the homepage you'll find puzzles and problem solving too.

I may be a little biased because I love Edexcel, but these question packs are really useful.

The UKMT has a mentoring scheme that provides fantastic problem solving resources , all complete with answers.

I have only recently been shown Maths Problem Solving and it is awesome - there are links to problem solving resources for all areas of maths, as well as plenty of general problem solving too. Definitely worth exploring!

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5 Teaching Mathematics Through Problem Solving

Janet Stramel

Problem Solving

In his book “How to Solve It,” George Pólya (1945) said, “One of the most important tasks of the teacher is to help his students. This task is not quite easy; it demands time, practice, devotion, and sound principles. The student should acquire as much experience of independent work as possible. But if he is left alone with his problem without any help, he may make no progress at all. If the teacher helps too much, nothing is left to the student. The teacher should help, but not too much and not too little, so that the student shall have a reasonable share of the work.” (page 1)

What is a problem  in mathematics? A problem is “any task or activity for which the students have no prescribed or memorized rules or methods, nor is there a perception by students that there is a specific ‘correct’ solution method” (Hiebert, et. al., 1997). Problem solving in mathematics is one of the most important topics to teach; learning to problem solve helps students develop a sense of solving real-life problems and apply mathematics to real world situations. It is also used for a deeper understanding of mathematical concepts. Learning “math facts” is not enough; students must also learn how to use these facts to develop their thinking skills.

According to NCTM (2010), the term “problem solving” refers to mathematical tasks that have the potential to provide intellectual challenges for enhancing students’ mathematical understanding and development. When you first hear “problem solving,” what do you think about? Story problems or word problems? Story problems may be limited to and not “problematic” enough. For example, you may ask students to find the area of a rectangle, given the length and width. This type of problem is an exercise in computation and can be completed mindlessly without understanding the concept of area. Worthwhile problems  includes problems that are truly problematic and have the potential to provide contexts for students’ mathematical development.

There are three ways to solve problems: teaching for problem solving, teaching about problem solving, and teaching through problem solving.

Teaching for problem solving begins with learning a skill. For example, students are learning how to multiply a two-digit number by a one-digit number, and the story problems you select are multiplication problems. Be sure when you are teaching for problem solving, you select or develop tasks that can promote the development of mathematical understanding.

Teaching about problem solving begins with suggested strategies to solve a problem. For example, “draw a picture,” “make a table,” etc. You may see posters in teachers’ classrooms of the “Problem Solving Method” such as: 1) Read the problem, 2) Devise a plan, 3) Solve the problem, and 4) Check your work. There is little or no evidence that students’ problem-solving abilities are improved when teaching about problem solving. Students will see a word problem as a separate endeavor and focus on the steps to follow rather than the mathematics. In addition, students will tend to use trial and error instead of focusing on sense making.

Teaching through problem solving  focuses students’ attention on ideas and sense making and develops mathematical practices. Teaching through problem solving also develops a student’s confidence and builds on their strengths. It allows for collaboration among students and engages students in their own learning.

Consider the following worthwhile-problem criteria developed by Lappan and Phillips (1998):

  • The problem has important, useful mathematics embedded in it.
  • The problem requires high-level thinking and problem solving.
  • The problem contributes to the conceptual development of students.
  • The problem creates an opportunity for the teacher to assess what his or her students are learning and where they are experiencing difficulty.
  • The problem can be approached by students in multiple ways using different solution strategies.
  • The problem has various solutions or allows different decisions or positions to be taken and defended.
  • The problem encourages student engagement and discourse.
  • The problem connects to other important mathematical ideas.
  • The problem promotes the skillful use of mathematics.
  • The problem provides an opportunity to practice important skills.

Of course, not every problem will include all of the above. Sometimes, you will choose a problem because your students need an opportunity to practice a certain skill.

Key features of a good mathematics problem includes:

  • It must begin where the students are mathematically.
  • The feature of the problem must be the mathematics that students are to learn.
  • It must require justifications and explanations for both answers and methods of solving.

Needlepoint of cats

Problem solving is not a  neat and orderly process. Think about needlework. On the front side, it is neat and perfect and pretty.

Back of a needlepoint

But look at the b ack.

It is messy and full of knots and loops. Problem solving in mathematics is also like this and we need to help our students be “messy” with problem solving; they need to go through those knots and loops and learn how to solve problems with the teacher’s guidance.

When you teach through problem solving , your students are focused on ideas and sense-making and they develop confidence in mathematics!

Mathematics Tasks and Activities that Promote Teaching through Problem Solving

Teacher teaching a math lesson

Choosing the Right Task

Selecting activities and/or tasks is the most significant decision teachers make that will affect students’ learning. Consider the following questions:

  • Teachers must do the activity first. What is problematic about the activity? What will you need to do BEFORE the activity and AFTER the activity? Additionally, think how your students would do the activity.
  • What mathematical ideas will the activity develop? Are there connections to other related mathematics topics, or other content areas?
  • Can the activity accomplish your learning objective/goals?

problem solving maths lesson

Low Floor High Ceiling Tasks

By definition, a “ low floor/high ceiling task ” is a mathematical activity where everyone in the group can begin and then work on at their own level of engagement. Low Floor High Ceiling Tasks are activities that everyone can begin and work on based on their own level, and have many possibilities for students to do more challenging mathematics. One gauge of knowing whether an activity is a Low Floor High Ceiling Task is when the work on the problems becomes more important than the answer itself, and leads to rich mathematical discourse [Hover: ways of representing, thinking, talking, agreeing, and disagreeing; the way ideas are exchanged and what the ideas entail; and as being shaped by the tasks in which students engage as well as by the nature of the learning environment].

The strengths of using Low Floor High Ceiling Tasks:

  • Allows students to show what they can do, not what they can’t.
  • Provides differentiation to all students.
  • Promotes a positive classroom environment.
  • Advances a growth mindset in students
  • Aligns with the Standards for Mathematical Practice

Examples of some Low Floor High Ceiling Tasks can be found at the following sites:

  • YouCubed – under grades choose Low Floor High Ceiling
  • NRICH Creating a Low Threshold High Ceiling Classroom
  • Inside Mathematics Problems of the Month

Math in 3-Acts

Math in 3-Acts was developed by Dan Meyer to spark an interest in and engage students in thought-provoking mathematical inquiry. Math in 3-Acts is a whole-group mathematics task consisting of three distinct parts:

Act One is about noticing and wondering. The teacher shares with students an image, video, or other situation that is engaging and perplexing. Students then generate questions about the situation.

In Act Two , the teacher offers some information for the students to use as they find the solutions to the problem.

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

“Math in 3 Acts” is a fun way to engage your students, there is a low entry point that gives students confidence, there are multiple paths to a solution, and it encourages students to work in groups to solve the problem. Some examples of Math in 3-Acts can be found at the following websites:

  • Dan Meyer’s Three-Act Math Tasks
  • Graham Fletcher3-Act Tasks ]
  • Math in 3-Acts: Real World Math Problems to Make Math Contextual, Visual and Concrete

Number Talks

Number talks are brief, 5-15 minute discussions that focus on student solutions for a mental math computation problem. Students share their different mental math processes aloud while the teacher records their thinking visually on a chart or board. In addition, students learn from each other’s strategies as they question, critique, or build on the strategies that are shared.. To use a “number talk,” you would include the following steps:

  • The teacher presents a problem for students to solve mentally.
  • Provide adequate “ wait time .”
  • The teacher calls on a students and asks, “What were you thinking?” and “Explain your thinking.”
  • For each student who volunteers to share their strategy, write their thinking on the board. Make sure to accurately record their thinking; do not correct their responses.
  • Invite students to question each other about their strategies, compare and contrast the strategies, and ask for clarification about strategies that are confusing.

“Number Talks” can be used as an introduction, a warm up to a lesson, or an extension. Some examples of Number Talks can be found at the following websites:

  • Inside Mathematics Number Talks
  • Number Talks Build Numerical Reasoning

Light bulb

Saying “This is Easy”

“This is easy.” Three little words that can have a big impact on students. What may be “easy” for one person, may be more “difficult” for someone else. And saying “this is easy” defeats the purpose of a growth mindset classroom, where students are comfortable making mistakes.

When the teacher says, “this is easy,” students may think,

  • “Everyone else understands and I don’t. I can’t do this!”
  • Students may just give up and surrender the mathematics to their classmates.
  • Students may shut down.

Instead, you and your students could say the following:

  • “I think I can do this.”
  • “I have an idea I want to try.”
  • “I’ve seen this kind of problem before.”

Tracy Zager wrote a short article, “This is easy”: The Little Phrase That Causes Big Problems” that can give you more information. Read Tracy Zager’s article here.

Using “Worksheets”

Do you want your students to memorize concepts, or do you want them to understand and apply the mathematics for different situations?

What is a “worksheet” in mathematics? It is a paper and pencil assignment when no other materials are used. A worksheet does not allow your students to use hands-on materials/manipulatives [Hover: physical objects that are used as teaching tools to engage students in the hands-on learning of mathematics]; and worksheets are many times “naked number” with no context. And a worksheet should not be used to enhance a hands-on activity.

Students need time to explore and manipulate materials in order to learn the mathematics concept. Worksheets are just a test of rote memory. Students need to develop those higher-order thinking skills, and worksheets will not allow them to do that.

One productive belief from the NCTM publication, Principles to Action (2014), states, “Students at all grade levels can benefit from the use of physical and virtual manipulative materials to provide visual models of a range of mathematical ideas.”

You may need an “activity sheet,” a “graphic organizer,” etc. as you plan your mathematics activities/lessons, but be sure to include hands-on manipulatives. Using manipulatives can

  • Provide your students a bridge between the concrete and abstract
  • Serve as models that support students’ thinking
  • Provide another representation
  • Support student engagement
  • Give students ownership of their own learning.

Adapted from “ The Top 5 Reasons for Using Manipulatives in the Classroom ”.

any task or activity for which the students have no prescribed or memorized rules or methods, nor is there a perception by students that there is a specific ‘correct’ solution method

should be intriguing and contain a level of challenge that invites speculation and hard work, and directs students to investigate important mathematical ideas and ways of thinking toward the learning

involves teaching a skill so that a student can later solve a story problem

when we teach students how to problem solve

teaching mathematics content through real contexts, problems, situations, and models

a mathematical activity where everyone in the group can begin and then work on at their own level of engagement

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

Mathematics Methods for Early Childhood Copyright © 2021 by Janet Stramel is licensed under a Creative Commons Attribution 4.0 International License , except where otherwise noted.

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problem solving maths lesson

Problem Solving Activities: 7 Strategies

  • Critical Thinking

problem solving maths lesson

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. 

This is an example of seasonal problem solving activities.

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.

This is an example of a math starter.

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.

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Shametria Routt Banks

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

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

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Solving equations

Here you will learn about solving equations, including linear and quadratic algebraic equations, and how to solve them.

Students will first learn about solving equations in grade 8 as a part of expressions and equations, and again in high school as a part of reasoning with equations and inequalities.

What is solving an equation?

Solving equations is a step-by-step process to find the value of the variable. A variable is the unknown part of an equation, either on the left or right side of the equals sign. Sometimes, you need to solve multi-step equations which contain algebraic expressions.

To do this, you must use the order of operations, which is a systematic approach to equation solving. When you use the order of operations, you first solve any part of an equation located within parentheses. An equation is a mathematical expression that contains an equals sign.

For example,

\begin{aligned}y+6&=11\\\\ 3(x-3)&=12\\\\ \cfrac{2x+2}{4}&=\cfrac{x-3}{3}\\\\ 2x^{2}+3&x-2=0\end{aligned}

There are two sides to an equation, with the left side being equal to the right side. Equations will often involve algebra and contain unknowns, or variables, which you often represent with letters such as x or y.

You can solve simple equations and more complicated equations to work out the value of these unknowns. They could involve fractions, decimals or integers.

What is solving an equation?

Common Core State Standards

How does this relate to 8 th grade and high school math?

  • Grade 8 – Expressions and Equations (8.EE.C.7) Solve linear equations in one variable.
  • High school – Reasoning with Equations and Inequalities (HSA.REI.B.3) Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

[FREE] Math Equations Check for Understanding Quiz (Grade 6 to 8)

[FREE] Math Equations Check for Understanding Quiz (Grade 6 to 8)

Use this quiz to check your grade 6 to 8 students’ understanding of math equations. 10+ questions with answers covering a range of 6th, 7th and 8th grade math equations topics to identify areas of strength and support!

How to solve equations

In order to solve equations, you need to work out the value of the unknown variable by adding, subtracting, multiplying or dividing both sides of the equation by the same value.

  • Combine like terms .
  • Simplify the equation by using the opposite operation to both sides.
  • Isolate the variable on one side of the equation.

Solving equations examples

Example 1: solve equations involving like terms.

Solve for x.

Combine like terms.

Combine the q terms on the left side of the equation. To do this, subtract 4q from both sides.

The goal is to simplify the equation by combining like terms. Subtracting 4q from both sides helps achieve this.

After you combine like terms, you are left with q=9-4q.

2 Simplify the equation by using the opposite operation on both sides.

Add 4q to both sides to isolate q to one side of the equation.

The objective is to have all the q terms on one side. Adding 4q to both sides accomplishes this.

After you move the variable to one side of the equation, you are left with 5q=9.

3 Isolate the variable on one side of the equation.

Divide both sides of the equation by 5 to solve for q.

Dividing by 5 allows you to isolate q to one side of the equation in order to find the solution. After dividing both sides of the equation by 5, you are left with q=1 \cfrac{4}{5} \, .

Example 2: solve equations with variables on both sides

Combine the v terms on the same side of the equation. To do this, add 8v to both sides.

7v+8v=8-8v+8v

After combining like terms, you are left with the equation 15v=8.

Simplify the equation by using the opposite operation on both sides and isolate the variable to one side.

Divide both sides of the equation by 15 to solve for v. This step will isolate v to one side of the equation and allow you to solve.

15v \div 15=8 \div 15

The final solution to the equation 7v=8-8v is \cfrac{8}{15} \, .

Example 3: solve equations with the distributive property

Combine like terms by using the distributive property.

The 3 outside the parentheses needs to be multiplied by both terms inside the parentheses. This is called the distributive property.

\begin{aligned}& 3 \times c=3 c \\\\ & 3 \times(-5)=-15 \\\\ &3 c-15-4=2\end{aligned}

Once the 3 is distributed on the left side, rewrite the equation and combine like terms. In this case, the like terms are the constants on the left, –15 and –4. Subtract –4 from –15 to get –19.

Simplify the equation by using the opposite operation on both sides.

The goal is to isolate the variable, c, on one side of the equation. By adding 19 to both sides, you move the constant term to the other side.

\begin{aligned}& 3 c-19+19=2+19 \\\\ & 3 c=21\end{aligned}

Isolate the variable to one side of the equation.

To solve for c, you want to get c by itself.

Dividing both sides by 3 accomplishes this.

On the left side, \cfrac{3c}{3} simplifies to c, and on the right, \cfrac{21}{3} simplifies to 7.

The final solution is c=7.

As an additional step, you can plug 7 back into the original equation to check your work.

Example 4: solve linear equations

Combine like terms by simplifying.

Using steps to solve, you know that the goal is to isolate x to one side of the equation. In order to do this, you must begin by subtracting from both sides of the equation.

\begin{aligned} & 2x+5=15 \\\\ & 2x+5-5=15-5 \\\\ & 2x=10 \end{aligned}

Continue to simplify the equation by using the opposite operation on both sides.

Continuing with steps to solve, you must divide both sides of the equation by 2 to isolate x to one side.

\begin{aligned} & 2x \div 2=10 \div 2 \\\\ & x= 5 \end{aligned}

Isolate the variable to one side of the equation and check your work.

Plugging in 5 for x in the original equation and making sure both sides are equal is an easy way to check your work. If the equation is not equal, you must check your steps.

\begin{aligned}& 2(5)+5=15 \\\\ & 10+5=15 \\\\ & 15=15\end{aligned}

Example 5: solve equations by factoring

Solve the following equation by factoring.

Combine like terms by factoring the equation by grouping.

Multiply the coefficient of the quadratic term by the constant term.

2 x (-20) = -40

Look for two numbers that multiply to give you –40 and add up to the coefficient of 3. In this case, the numbers are 8 and –5 because 8 x -5=–40, and 8+–5=3.

Split the middle term using those two numbers, 8 and –5. Rewrite the middle term using the numbers 8 and –5.

2x^2+8x-5x-20=0

Group the terms in pairs and factor out the common factors.

2x^2+8x-5x-20=2x(x + 4)-5(x+4)=0

Now, you’ve factored the equation and are left with the following simpler equations 2x-5 and x+4.

This step relies on understanding the zero product property, which states that if two numbers multiply to give zero, then at least one of those numbers must equal zero.

Let’s relate this back to the factored equation (2x-5)(x+4)=0

Because of this property, either (2x-5)=0 or (x+4)=0

Isolate the variable for each equation and solve.

When solving these simpler equations, remember that you must apply each step to both sides of the equation to maintain balance.

\begin{aligned}& 2 x-5=0 \\\\ & 2 x-5+5=0+5 \\\\ & 2 x=5 \\\\ & 2 x \div 2=5 \div 2 \\\\ & x=\cfrac{5}{2} \end{aligned}

\begin{aligned}& x+4=0 \\\\ & x+4-4=0-4 \\\\ & x=-4\end{aligned}

The solution to this equation is x=\cfrac{5}{2} and x=-4.

Example 6: solve quadratic equations

Solve the following quadratic equation.

Combine like terms by factoring the quadratic equation when terms are isolated to one side.

To factorize a quadratic expression like this, you need to find two numbers that multiply to give -5 (the constant term) and add to give +2 (the coefficient of the x term).

The two numbers that satisfy this are -1 and +5.

So you can split the middle term 2x into -1x+5x: x^2-1x+5x-5-1x+5x

Now you can take out common factors x(x-1)+5(x-1).

And since you have a common factor of (x-1), you can simplify to (x+5)(x-1).

The numbers -1 and 5 allow you to split the middle term into two terms that give you common factors, allowing you to simplify into the form (x+5)(x-1).

Let’s relate this back to the factored equation (x+5)(x-1)=0.

Because of this property, either (x+5)=0 or (x-1)=0.

Now, you can solve the simple equations resulting from the zero product property.

\begin{aligned}& x+5=0 \\\\ & x+5-5=0-5 \\\\ & x=-5 \\\\\\ & x-1=0 \\\\ & x-1+1=0+1 \\\\ & x=1\end{aligned}

The solutions to this quadratic equation are x=1 and x=-5.

Teaching tips for solving equations

  • Use physical manipulatives like balance scales as a visual aid. Show how you need to keep both sides of the equation balanced, like a scale. Add or subtract the same thing from both sides to keep it balanced when solving. Use this method to practice various types of equations.
  • Emphasize the importance of undoing steps to isolate the variable. If you are solving for x and 3 is added to x, subtracting 3 undoes that step and isolates the variable x.
  • Relate equations to real-world, relevant examples for students. For example, word problems about tickets for sports games, cell phone plans, pizza parties, etc. can make the concepts click better.
  • Allow time for peer teaching and collaborative problem solving. Having students explain concepts to each other, work through examples on whiteboards, etc. reinforces the process and allows peers to ask clarifying questions. This type of scaffolding would be beneficial for all students, especially English-Language Learners. Provide supervision and feedback during the peer interactions.

Easy mistakes to make

  • Forgetting to distribute or combine like terms One common mistake is neglecting to distribute a number across parentheses or combine like terms before isolating the variable. This error can lead to an incorrect simplified form of the equation.
  • Misapplying the distributive property Incorrectly distributing a number across terms inside parentheses can result in errors. Students may forget to multiply each term within the parentheses by the distributing number, leading to an inaccurate equation.
  • Failing to perform the same operation on both sides It’s crucial to perform the same operation on both sides of the equation to maintain balance. Forgetting this can result in an imbalanced equation and incorrect solutions.
  • Making calculation errors Simple arithmetic mistakes, such as addition, subtraction, multiplication, or division errors, can occur during the solution process. Checking calculations is essential to avoid errors that may propagate through the steps.
  • Ignoring fractions or misapplying operations When fractions are involved, students may forget to multiply or divide by the common denominator to eliminate them. Misapplying operations on fractions can lead to incorrect solutions or complications in the final answer.

Related math equations lessons

  • Math equations
  • Rearranging equations
  • How to find the equation of a line
  • Solve equations with fractions
  • Linear equations
  • Writing linear equations
  • Substitution
  • Identity math
  • One step equation

Practice solving equations questions

1. Solve 4x-2=14.

GCSE Quiz False

Add 2 to both sides.

Divide both sides by 4.

2. Solve 3x-8=x+6.

Add 8 to both sides.

Subtract x from both sides.

Divide both sides by 2.

3. Solve 3(x+3)=2(x-2).

Expanding the parentheses.

Subtract 9 from both sides.

Subtract 2x from both sides.

4. Solve \cfrac{2 x+2}{3}=\cfrac{x-3}{2}.

Multiply by 6 (the lowest common denominator) and simplify.

Expand the parentheses.

Subtract 4 from both sides.

Subtract 3x from both sides.

5. Solve \cfrac{3 x^{2}}{2}=24.

Multiply both sides by 2.

Divide both sides by 3.

Square root both sides.

6. Solve by factoring:

Use factoring to find simpler equations.

Set each set of parentheses equal to zero and solve.

x=3 or x=10

Solving equations FAQs

The first step in solving a simple linear equation is to simplify both sides by combining like terms. This involves adding or subtracting terms to isolate the variable on one side of the equation.

Performing the same operation on both sides of the equation maintains the equality. This ensures that any change made to one side is also made to the other, keeping the equation balanced and preserving the solutions.

To handle variables on both sides of the equation, start by combining like terms on each side. Then, move all terms involving the variable to one side by adding or subtracting, and simplify to isolate the variable. Finally, perform any necessary operations to solve for the variable.

To deal with fractions in an equation, aim to eliminate them by multiplying both sides of the equation by the least common denominator. This helps simplify the equation and make it easier to isolate the variable. Afterward, proceed with the regular steps of solving the equation.

The next lessons are

  • Inequalities
  • Types of graph
  • Coordinate plane

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10 Fun Math Problem Solving Activities

I love using fun games and activities to get my kids excited about math. That’s why I was so excited to discover Beast Academy Playground! The site includes a wide variety of math problem solving activities, games, puzzles, and ideas that can be used with your current homeschool curriculum. If you’re looking for some fun ways to get your child interested in math again or some new ideas for teaching math at home, this is a great place to start.

Fun first grade math activities for active kids

I received compensation in exchange for my honest review, but all the opinions in this post are my own.

What is Beast Academy Playground?

As a homeschool parent, I know that fun math problem solving activities are hard to come by. Practicing math can be frustrating and boring for kids. And, it’s not enough to just teach math facts – children need to learn how to solve complex problems too!

Beast Academy Playground is the perfect solution. This site was created by Beast Academy, a homeschool curriculum for kids ages 8-13 that’s written as a comic book. Beast Academy Playground is a website that includes a growing library of fun activity ideas for kids ages 4-11.

The site includes both tabletop games that can be played with paper and pencil and more active games that can be played outside. Parents can sort the activities by concept, age, number of players, and more to find the perfect activity to complement any math lesson. New activities, games, and crafts are added to the site weekly.

Key Features of Beast Academy Playground

I love that every game on the site includes variations. These are different ways to adapt the game to fit the number of kids in your family or the age and grade level of your child.

In addition, most of the games can be played alone or in small groups. This makes Beast Academy Playground so easy for the whole family to use together!

Beast Academy Playground was developed by the experts at Art of Problem Solving, who are global leaders in K-12 math education. Each activity is intentionally designed to help kids learn new math concepts.

I was excited to see that each game also includes learning notes. This section helps parents understand what their kids will be learning when they play the game. I felt like I was prepared to be a better math teacher after I read the tips in this section!

Check out Beast Academy Playground and learn more now!

Math Problem Solving Activities with Beast Academy Playground

How to use Math Problem Solving Activities in Your Homeschool

Introduce a new concept.

One way to use problem-solving activities in your math lessons is to help introduce a new concept. For example, when we were learning about even and odd numbers, we started our math lesson by playing the Odds vs. Evens game from Beast Academy Playground. This simple math problem solving activity is a variation on the game Rock Paper Scissors. This was a fun way to help us review addition facts while introducing the concept of odd and even numbers.

Practice Problem-Solving Skills

Another great way to use Beast Academy Playground is to help kids practice their problem-solving skills. One fun problem solving game that my kids love is the Fox and Hare game. In this outdoor game, the fox needs to use strategy to try to catch the hare on a grid, while the hare tries to avoid capture. You can change the size of the grid to create a variety of problems for kids to solve.

Practice Math Facts

If your kids need extra practice with their math facts, games and math puzzles can be a fun way to practice these important skills. Beast Academy Playground has games for addition and subtraction, as well as concepts like skip counting that will help kids learn their multiplication facts.

One favorite that we enjoyed was Troll Hole . In this game, we took turns writing numbers on a special game board. In the end, we had to add up all the numbers to see who was the winner. My kids had so much fun with the theme of this game and loved getting to draw the troll in the hole!

Independent learning

As homeschool parents, we always need ideas for independent learning activities. Beast Academy Playground includes some great math problem solving games that are perfect for self-directed math learning. The many different activities on the site include several fun activities for one player that kids can work through independently to reinforce their math skills.

For extra review

Hands-on activities are also helpful when kids need a little extra review on a topic. For example, we played Kanga Ruler to help review skip counting. The kids loved this game because it was active and fun!

Beast Academy Playground

Top 10 Math Problem Solving Activities from Beast Academy Playground

Here are 10 of our favorite games that help kids develop a deeper understanding of mathematical concepts, all while having fun at the same time.

Trashketball

Trashkeball Math Problem Solving Activities

In this game, kids aim and try to score a basket into a trash can. Then, they add up the points they score. You can modify this game to add extra intellectual challenges for older kids or adapt the math so that younger kids can play too!

Trashketball was my kids’ favorite Beast Academy game, hands down. They loved trying to score as many baskets as they could. They had so much fun that they decided to keep playing even after our math lesson was finished!

Learn how to play Trashketball here.

In this math game for young children, kids race to stack towers of number cards. This is a great way to review numbers and counting. My kids really enjoyed trying to build the tallest tower that they could!

Learn how to play Towers here.

Bumper Cars

Although it took us a few tries to understand the strategy behind this game, my kids loved the concept of trying to figure out new ways to move the cars on the road. This was another great mathematical practice for strategy and solving difficult math problems.

Learn how to play Bumper Cars here.

Hungry Monster

This was a great way to practice inequalities and comparing numbers, and my kids loved feeding the correct answer in each inequality to the hungry monster!

Learn how to play Hungry Monster here.

Fruit Flies

Fruit flies Math Problem Solving Activities

In this math game for older problem solvers, kids try to claim as many grapes as they can for themselves, while blocking their opponent from getting any. This game is like an interactive logic puzzle, and it’s a good way for kids to learn critical thinking skills.

Learn how to play Fruit Flies here.

Blind Heist

In this game, Battleship meets addition as kids try to build the highest towers on their own secret side of the board. There are many different solutions and strategies to be successful, and my kids loved trying different solutions to this open-ended problem.

Learn how to play Blind Heist here.

Möbius Madness

This is a classic brain teaser for a reason- my kids were fascinated by the magic of a piece of paper with only one side. My kids were able to follow the directions easily and afterwards, they were excited to show their magical piece of paper to everyone who would watch.

Learn how to play Möbius Madness here.

This fast-paced card game was the perfect way to help my kids practice their addition facts.

Learn how to play Fifteen here.

Cookie Cutter

This game helps kids practice both spatial reasoning skills and fine motor skills at the same time. And, the result is a fun picture that they can color!

Learn how to play Cookie Cutter here.

Odd Knights

odd knights Math Problem Solving Activities

This was a fun way to practice even and odd (and it even led to a history lesson about the Knights of the Round Table!)

Learn how to play Odd Knights here.

Beast Academy Playground

What math problem solving activities will you use?

Whether it’s for extra practice or math review , Beast Academy Playground has something for every math learner. This site is a great resource to find exciting games that help kids develop number sense, problem solving, and logical thinking skills. If you want fun and engaging math activities that don’t require a textbook, this is the place to go. For more math problem solving activities and fun math games, check out Beast Academy Playground !

Find hands on activities to teach spelling and reading here!

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problem solving maths lesson

The Lesson Study Group

at Mills College

Teaching Through Problem-solving

An elementary-age male student points while his female teacher stands beside him and observes

  • TTP in Action

What is Teaching Through Problem-Solving?

In Teaching Through Problem-solving (TTP), students learn new mathematics by solving problems. Students grapple with a novel problem, present and discuss solution strategies, and together build the next concept or procedure in the mathematics curriculum.

Teaching Through Problem-solving is widespread in Japan, where students solve problems before a solution method or procedure is taught. In contrast, U.S. students spend most of their time doing exercises– completing problems for which a solution method has already been taught.

Why Teaching Through Problem-Solving?

As students build their mathematical knowledge, they also:

  • Learn to reason mathematically, using prior knowledge to build new ideas
  • See the power of their explanations and carefully written work to spark insights for themselves and their classmates
  • Expect mathematics to make sense
  • Enjoy solving unfamiliar problems
  • Experience mathematical discoveries that naturally deepen their perseverance

Phases of a TTP Lesson

Teaching Through Problem-solving flows through four phases as students 1. Grasp the problem, 2. Try to solve the problem independently, 3. Present and discuss their work (selected strategies), and 4. Summarize and reflect.

Click on the arrows below to find out what students and teachers do during each phase and to see video examples.

  • 1. Grasp the Problem
  • 2. Try to Solve
  • 3. Present & Discuss
  • 4. Summarize & Reflect
  • New Learning

WHAT STUDENTS DO

  • Understand the problem and develop interest in solving it.
  • Consider what they know that might help them solve the problem.

WHAT TEACHERS DO

  • Show several student journal reflections from the prior lesson.
  • Pose a problem that students do not yet know how to solve.
  • Interest students in the problem and in thinking about their own related knowledge.
  • Independently try to solve the problem.
  • Do not simply following the teacher’s solution example.
  • Allow classmates to provide input after some independent thinking time.
  • Circulate, using seating chart to note each student’s solution approach.
  • Identify work to be presented and discussed at board.
  • Ask individual questions to spark more thinking if some students finish quickly or don’t get started.
  • Present and explain solution ideas at the board, are questioned by classmates and teacher. (2-3 students per lesson)
  • Actively make sense of the presented work and draw out key mathematical points. (All students)
  • Strategically select and sequence student presentations of work at the board, to build the new mathematics. (Incorrect approaches may be included.)
  • Monitor student discussion: Are all students noticing the important mathematical ideas?
  • Add teacher moves (questions, turn-and-talk, votes) as needed to build important mathematics.
  • Consider what they learned and share their thoughts with class, to help formulate class summary of learning. Copy summary into journal.
  • Write journal reflection on their own learning from the lesson.
  • Write on the board a brief summary of what the class learned during the lesson, using student ideas and words where possible.
  • Ask students to write in their journals about what they learned during the lesson.

How Do Teachers Support Problem-solving?

Although students do much of the talking and questioning in a TTP lesson, teachers play a crucial role. The widely-known 5 Practices for Orchestrating Mathematical Discussions were based in part on TTP . Teachers study the curriculum, anticipate student thinking, and select and sequence the student presentations that allow the class to build the new mathematics. Classroom routines for presentation and discussion of student work, board organization, and reflective mathematics journals work together to allow students to do the mathematical heavy lifting. To learn more about journals, board work, and discussion in TTP, as well as see other TTP resources and examples of TTP in action, click on the respective tabs near the top of this page.

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Problem Solving

 A selection of resources containing a wide range of open-ended tasks, practical tasks, investigations and real life problems, to support investigative work and problem solving in primary mathematics.

Problem Solving in Primary Maths - the Session

Quality Assured Category: Mathematics Publisher: Teachers TV

In this programme shows a group of four upper Key Stage Two children working on a challenging problem; looking at the interior and exterior angles of polygons and how they relate to the number of sides. The problem requires the children to listen to each other and to work together co-operatively. The two boys and two girls are closely observed as they consider how to tackle the problem, make mistakes, get stuck and arrive at the "eureka" moment. They organise the data they collect and are then able to spot patterns and relate them to the original problem to find a formula to work out the exterior angle of any polygon. At the end of the session the children report back to Mark, explaining how they arrived at the solution, an important part of the problem solving process.

In a  second video  two maths experts discuss some of the challenges of teaching problem solving. This includes how and at what stage to introduce problem solving strategies and the appropriate moment to intervene when children find tasks difficult. They also discuss how problem solving in the curriculum also helps to develop life skills.

Cards for Cubes: Problem Solving Activities for Young Children

Quality Assured Category: Mathematics Publisher: Claire Publications

This book provides a series of problem solving activities involving cubes. The tasks start simply and progress to more complicated activities so could be used for different ages within Key Stages One and Two depending on ability. The first task is a challenge to create a camel with 50 cubes that doesn't fall over. Different characters are introduced throughout the book and challenges set to create various animals, monsters and structures using different numbers of cubes. Problems are set to incorporate different areas of mathematical problem solving they are: using maths, number, algebra and measure.

problem solving maths lesson

Problem solving with EYFS, Key Stage One and Key Stage Two children

Quality Assured Category: Computing Publisher: Department for Education

These three resources, from the National Strategies, focus on solving problems.

  Logic problems and puzzles  identifies the strategies children may use and the learning approaches teachers can plan to teach problem solving. There are two lessons for each age group.

Finding all possibilities focuses on one particular strategy, finding all possibilities. Other resources that would enhance the problem solving process are listed, these include practical apparatus, the use of ICT and in particular Interactive Teaching Programs .

Finding rules and describing patterns focuses on problems that fall into the category 'patterns and relationships'. There are seven activities across the year groups. Each activity includes objectives, learning outcomes, resources, vocabulary and prior knowledge required. Each lesson is structured with a main teaching activity, drawing together and a plenary, including probing questions.

problem solving maths lesson

Primary mathematics classroom resources

Quality Assured Collection Category: Mathematics Publisher: Association of Teachers of Mathematics

This selection of 5 resources is a mixture of problem-solving tasks, open-ended tasks, games and puzzles designed to develop students' understanding and application of mathematics.

Thinking for Ourselves: These activities, from the Association of Teachers of Mathematics (ATM) publication 'Thinking for Ourselves’, provide a variety of contexts in which students are encouraged to think for themselves. Activity 1: In the bag – More or less requires students to record how many more or less cubes in total...

8 Days a Week: The resource consists of eight questions, one for each day of the week and one extra. The questions explore odd numbers, sequences, prime numbers, fractions, multiplication and division.

Number Picnic: The problems make ideal starter activities

Matchstick Problems: Contains two activities concentrating upon the process of counting and spotting patterns. Uses id eas about the properties of number and the use of knowledge and reasoning to work out the rules.

Colours: Use logic, thinking skills and organisational skills to decide which information is useful and which is irrelevant in order to find the solution.

problem solving maths lesson

GAIM Activities: Practical Problems

Quality Assured Category: Mathematics Publisher: Nelson Thornes

Designed for secondary learners, but could also be used to enrich the learning of upper primary children, looking for a challenge. These are open-ended tasks encourage children to apply and develop mathematical knowledge, skills and understanding and to integrate these in order to make decisions and draw conclusions.

Examples include:

*Every Second Counts - Using transport timetables, maps and knowledge of speeds to plan a route leading as far away from school as possible in one hour.

*Beach Guest House - Booking guests into appropriate rooms in a hotel.

*Cemetery Maths - Collecting relevant data from a visit to a local graveyard or a cemetery for testing a hypothesis.

*Design a Table - Involving diagrams, measurements, scale.

problem solving maths lesson

Go Further with Investigations

Quality Assured Category: Mathematics Publisher: Collins Educational

A collection of 40 investigations designed for use with the whole class or smaller groups. It is aimed at upper KS2 but some activities may be adapted for use with more able children in lower KS2. It covers different curriculum areas of mathematics.

problem solving maths lesson

Starting Investigations

The forty student investigations in this book are non-sequential and focus mainly on the mathematical topics of addition, subtraction, number, shape and colour patterns, and money.

The apparatus required for each investigation is given on the student sheets and generally include items such as dice, counters, number cards and rods. The sheets are written using as few words as possible in order to enable students to begin working with the minimum of reading.

NRICH Primary Activities

Explore the NRICH primary tasks which aim to enrich the mathematical experiences of all learners. Lots of whole class open ended investigations and problem solving tasks. These tasks really get children thinking!

Mathematical reasoning: activities for developing thinking skills

Quality Assured Category: Mathematics Publisher: SMILE

problem solving maths lesson

Problem Solving 2

Reasoning about numbers, with challenges and simplifications.

Quality Assured Category: Mathematics Publisher: Department for Education

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Maths Problem Solving Booklets

Maths Problem Solving Booklets

Subject: Mathematics

Age range: 11-14

Resource type: Worksheet/Activity

Paul Tyler

Last updated

23 August 2022

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pdf, 424.8 KB

Maths problem solving booklets covering a wide range of mathematical problems designed to improve problem solving strategies as well as numeracy and mathematical ability.

Designed to be printed as A5 booklets.

Disclaimer: These are free because the problems are from a wide variety of sources, most of which I have forgotten. I am a maths problem magpie and collect maths problems wherever I find them. All I have done is bundled these ones up to make it convienent.

If you find a problem that is yours and you don’t want it used please comment and I’ll remove it - cheers.

Creative Commons "Sharealike"

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ardglassie1

Very impresses. Ta.

Empty reply does not make any sense for the end user

Excellent resource

Great resource to use at KS3

mummygoth23

Just what I have been looking for. Thanks so much!

Good resources. Thanks you!

Report this resource to let us know if it violates our terms and conditions. Our customer service team will review your report and will be in touch.

Not quite what you were looking for? Search by keyword to find the right resource:

Robert Kaplinsky

  • Open Middle Problems
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Problem-Based Lesson Search Engine

problem solving maths lesson

This search engine searches all of the sites below to quickly help you find a problem-based lesson (also called 3-Act Task, mathematical modeling, or application problem):

The links below are the pages that are being searched by the search engine:

  • 101 Questions
  • Andrew Gael
  • Andrew Stadel
  • Catherine Castillo
  • Dane Ehlert
  • Emergent Math’s Problem Based Curriculum Maps
  • Geoff Krall
  • Graham Fletcher
  • Kendra Lomax
  • Kristen M. Acosta
  • Kyle Pearce
  • Jennifer Barker
  • John Scammell
  • Matt Vaudrey
  • Michael Van Etten
  • Mike Wiernicki
  • Robert Kaplinsky
  • Timon Piccini

There must be many great sources of lessons that I am missing.  Please leave me a comment to let me know which websites I need to add to the search engine.

71 Comments

Wow! This is amazing! Thank you so much for doing this. One other great site I’ve found is HungryTeacher.com It has some very cool project ideas, similar to Mathalicious but not as detailed. Keep up the great work!

@EdCamposJR

Thank you Ed. I actually added HungryTeacher.com to the search engine but forgot to add it to the list of sites on this page so thanks for the heads up.

This is going be amazing for me.

This is so cool and helpful. Thank you so much for this. http://www.gogeometry.com has some challenging geometry problems.

Hi Teresa. Glad it is useful. I searched around gogeometry.com but couldn’t find anything like a problem-based lesson. Could you give me a specific link to a lesson to check out?

Thank you for this search engine. It is very helpful.

What about adding this site? http://ell.stanford.edu/teaching_resources/math

There are only a few tasks here but they focus in on our ELs.

Hi Jennifer. There are some cool lessons there but I don’t know if they would fall under the category of problem based lessons I am aspiring to capture here. Thanks for your idea!

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nice work, could you share how you set this up? I like the idea of embedding a search to only look through certain sites.

Also I suggest Dan Meyer’s Graphing Stories, Fawn Nguyen’s blog and the math lab: http://www.themathlab.com/Algebra/linear%20functions%20regressions%20slope/regression%20lessons/barbie%20bungee/barbbungee.htm

Hi Shaun. Check out Google’s Custom Search at https://www.google.com/cse/ . That is where you can create your own search engine. It is rather robust but not perfect. For example, it is very challenging to include a site but exclude the blog posts.

Regarding your suggestions, those are all great sites but Graphing Stories and Fawn’s awesome blog are not really within the problem-based learning domain so that is why I haven’t included them. That being said, both of them have been links on the right side of my blog for quite some time so I definitely value them both.

I am having trouble getting the Barbie Bungee jump to load but I will try again another time to check it out.

This is an old post, but I am trying to develop my own 3Acts and have a resource page of them as well.

Hi Bryan. All of your lessons are included since they are posted on 101qs.com. Thanks!

Amazing resource!!! Thank You for creating this search engine!!! 🙂

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Have you checked out the website: http://davidwees.com/

He has some pretty useful articles, especially on strategies for formative assessment.

Full disclosure: He’s a colleague of mine so I am biased, but I do get a lot out of what he shares and apply to my own work.

Thanks Blue. David is great but I don’t believe he is creating any problem-based lessons. If he is, please list a link to where I can find them.

http://reasonandwonder.com

Thanks Kevin. Can’t believe I forgot Michael. Problem fixed.

Rob, have you come across Tuva yet? Focused on data-based problems for math, science, and many other subjects. You can check it out at tuvalabs.com.

Hi, thanks for the resource!

Would you consider these to be examples of problem-based learning?

https://aofradkin.wordpress.com/2015/03/18/facing-the-impossible/ https://aofradkin.wordpress.com/2014/11/11/stick-figures/ https://aofradkin.wordpress.com/2014/10/10/fun-with-tetraminos/

Hello. To me the answer is both yes and no. Clearly these are problems that you can base a lesson around. However they are not of the style that we are including in this search engine. If you are interested in examples of elementary tasks of this style, check out Graham Fletcher’s problems here: http://gfletchy.com/3-act-lessons/ .

I was so excited to find this resource that I immediately emailed all of the teachers in my department a link to this post. Then, I realized that I had a double block of Algebra I approaching in 15 minutes and I could use this search tool to find a resource that might raise the engagement level for my students. Sure enough…I typed “graphing quadratic equations” into the search box, and numerous resources appeared. I selected the second hit on “Angry Birds”,and it fit seamlessly into my lesson plan. The students asked such great questions and were able to clearly answer questions about the vertex, x-intercepts, and orientation of the quadratic equations that modeled the paths of the angry bird. Of course, then I had to send out another email to my department ranting and raving about my math rush.

Isn’t it cool when it works out! Glad to hear it was useful for you in a tight situation.

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What about http://www.openmiddle.com ? Thanks so much for this! You are a life saver.

Hi Alex. Obviously I love Open Middle, but it doesn’t seem to be a good fit here. This search engine is more for math modeling problems also known as 3-Act Tasks or problem-based lessons. What do you think?

Will. Change. My. Life. So excited to share this!!!!

Hi Robert, I will be sharing this with my colleagues at school that focus on the Math contents… Thanks for a great resource on behalf of all those teacher who need a little more help in the topics area.

That’s great Sonia. I hope they find it useful.

Thanks for this very useful resource.

Great search engine. Didn’t know that you could have a custom search engine.

Have you looked at nzmaths?

https://nzmaths.co.nz/problem-solving

Hi Ben. I haven’t heard of this but it seems like there are some useful problems there. I’ll add it to the list. Much appreciated!

I LOVE THIS! I’m so excited to have a search engine that will only search for mathematical tasks! I’m sharing it with everyone I know:). Thank you so much, Robert!!

Glad you like it Karen. Definitely a time saver.

Hi Robert. I love that you did this. It makes it easier to get teachers to try these out. I also have another site – tapintoteenminds.com. He has developed some tasks using the 3-Act format.

Hi Nadirah. I actually have that site on the list under Kyle Pearce, who created that site and those lessons.

Of course you did Robert! Silly me.

Thanks! This is amazing and such a great idea! Is there a way to embed this link into our own sites?

Can you explain more about what you mean to “embed this link”? You can put a link to this blog post on your own site, that’s no problem.

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Hey! I think this is a great list of resources and so thankful for the list. It has saved me in the last a lot. I am in my grad school class and researching on PBL. I am a little confused how this qualifies has PBL and not just projects or really good math lessons. My research suggests that with PBL the learning is happening through the project and the project is not a activity we tack on at the end. I also read, “While the learning context is common to all groups, the paths may differ considerably—all leading to distinct learning. In project-based learning all students engage in a common project with unclear processes but clearly identified expected outcomes.” Can you speak a little bit about your thoughts on PBL. I am trying to find some math teachers I look up to and follow to help me learn about PBL in a secondary classroom. Please respond or email me at [email protected]

Hi Virginia. Thanks for this question. I should be clear that this is a PROBLEM-based search engine and not a PROJECT-based search engine. While the two have similarities, it sounds like you are talking about project based lessons and not what I am focusing on.

I don’t do a lot with project based learning so unfortunately I’m not a good resource for you there. Sorry! If you want to learn more about problem based learning, you can read more here: http://robertkaplinsky.com/tag/problem-based-learning-2/ .

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Youcubed certainly has many options as well!

Thanks Merryl. I’m not aware of any problems that Youcubed has that would be considered problem-based lessons like 3-act tasks and the ones on my site. If you can point me in the right direction, I’d love to check them out.

This is awesome! Is there any way to take out Mathalicious since it is now requires a subscription?

Hi Danielle. I understand where you’re coming from, but that isn’t a problem for me. You often get what you pay for, and as much as it feels nice to get lessons for free, the Mathalicious team’s full time job is creating high quality lessons for educators. They deserve to be compensated.

Personally, I prefer to be aware of what’s out there so that I can choose what’s best from me. Sometimes it will be worth paying for.

Have you looked at illustrative mathematics or exemplars? Both have good problem based questions.

Thanks Sarah. I think this issue is stems from the word “problem” being so broad. Yes, I have looked at those problems, but they aren’t like the 3-act tasks and other problem-based lessons I am looking for with this search engine. I want problems that are more like these: https://robertkaplinsky.com/lessons/

Is it possible to add the Phillips Exeter sets? Could it search through their PDF files?

I’m not sure Dave. Do you have a link to an example?

What about Ilustrative Mathematics? https://www.illustrativemathematics.org/

Hi Brian. I think this issue is stems from the word “problem” being so broad. Yes, I have looked at those problems, but they aren’t like the 3-act tasks and other problem-based lessons I am looking for with this search engine. I want problems that are more like these: https://robertkaplinsky.com/lessons/

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So many comments – did anyone else suggested the Illustrative Mathematics site

Hi Angela. Yes, a few others have suggested it. It’s not a good fit for the kinds of problems I’m referring to (which are more like you’d find on my lessons page or 3-act tasks). Thanks!

Hi. Thanks for creating that.

I’m not sure if either of these qualify, but one you may want to consider is https://mathsolutions.com/free-resources/ because it has stuff from Marilyn Burns.

I also love Steve Wyborney’s site: https://stevewyborney.com/2018/11/esti-mysteries-estimation-meets-math-mysteries/

My students really enjoy the esti-mysteries.

Thanks Jeff. Those are all great resources. The kinds of problems we’re trying to include here are more real-world or three-act task style problems, so this is not a great fit for this particular search engine.

This is AWESOME! Thank you. Wide Open School ( https://wideopenschool.org/#grades-6-8/ ) is a great resource also.

Thanks Jenny. There’s lots of great stuff there. Unfortunately, the kinds of problems we’re trying to include here are more real-world or three-act task style problems, so this is not a great fit for this particular search engine.

What a wonderful one-stop shop for problem solving tasks. I would love to link them all to my site called CURIOSEDY. I wonder if you think Curiosedy could be added to the list as it has open-ended rich tasks as well as multi-step problem solving. Would love your thoughts. Thanks again for this.

Thanks for sharing this, Francois. I can see lots of potential in what you’re doing. The kinds of tasks in this search engine (more like 3-act tasks) are a bit different from your approach, but both have value. So it’s not a great fit for this search engine but keep it going as it’ll help other students.

Hi Robert, Mark Chubb has some great stuff, especially around spatial reasoning. He deserves a spot! Thanks.

Mark definitely has some great resources. Unfortunately they’re not the kinds of problems for this search engine. I’m thinking more like problems with real world context.

I believe the hyperlink for Beth Brandenburg is not working.

Thanks for catching that. I guess her site is no longer active. I just went through all of them and updated it accordingly.

Hi Robert, Thanks for sharing. Check out https://www.resolve.edu.au/teaching-resources There are many lessons on this that have problem solving and inquiry tasks. My favourites are the circumference and area of circles. Hope this fits the style of tasks you are looking for.

Thanks Lorna. This is a bit different style of problem than what I’m going for. Ideally, I’m looking for problems that are more contextual without necessarily being a word problem. These are definitely great problems, but a different kind than I’m using with this search engine. Thanks again.

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Math Lesson Plans for Teachers

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Teaching and Learning

Elevating math education through problem-based learning, by lisa matthews     feb 14, 2024.

Elevating Math Education Through Problem-Based Learning

Image Credit: rudall30 / Shutterstock

Imagine you are a mountaineer. Nothing excites you more than testing your skill, strength and resilience against some of the most extreme environments on the planet, and now you've decided to take on the greatest challenge of all: Everest, the tallest mountain in the world. You’ll be training for at least a year, slowly building up your endurance. Climbing Everest involves hiking for many hours per day, every day, for several weeks. How do you prepare for that?

The answer, as in many situations, lies in math. Climbers maximize their training by measuring their heart rate. When they train, they aim for a heart rate between 60 and 80 percent of their maximum. More than that, and they risk burning out. A heart rate below 60 percent means the training is too easy — they’ve got to push themselves harder. By combining this strategy with other types of training, overall fitness will increase over time, and eventually, climbers will be ready, in theory, for Everest.

problem solving maths lesson

Knowledge Through Experience

The influence of constructivist theories has been instrumental in shaping PBL, from Jean Piaget's theory of cognitive development, which argues that knowledge is constructed through experiences and interactions , to Leslie P. Steffe’s work on the importance of students constructing their own mathematical understanding rather than passively receiving information .

You don't become a skilled mountain climber by just reading or watching others climb. You become proficient by hitting the mountains, climbing, facing challenges and getting right back up when you stumble. And that's how people learn math.

problem solving maths lesson

So what makes PBL different? The key to making it work is introducing the right level of problem. Remember Vygotsky’s Zone of Proximal Development? It is essentially the space where learning and development occur most effectively – where the task is not so easy that it is boring but not so hard that it is discouraging. As with a mountaineer in training, that zone where the level of challenge is just right is where engagement really happens.

I’ve seen PBL build the confidence of students who thought they weren’t math people. It makes them feel capable and that their insights are valuable. They develop the most creative strategies; kids have said things that just blow my mind. All of a sudden, they are math people.

problem solving maths lesson

Skills and Understanding

Despite the challenges, the trend toward PBL in math education has been growing , driven by evidence of its benefits in developing critical thinking, problem-solving skills and a deeper understanding of mathematical concepts, as well as building more positive math identities. The incorporation of PBL aligns well with the contemporary broader shift toward more student-centered, interactive and meaningful learning experiences. It has become an increasingly important component of effective math education, equipping students with the skills and understanding necessary for success in the 21st century.

At the heart of Imagine IM lies a commitment to providing students with opportunities for deep, active mathematics practice through problem-based learning. Imagine IM builds upon the problem-based pedagogy and instructional design of the renowned Illustrative Mathematics curriculum, adding a number of exclusive videos, digital interactives, design-enhanced print and hands-on tools.

The value of imagine im's enhancements is evident in the beautifully produced inspire math videos, from which the mountaineer scenario stems. inspire math videos showcase the math for each imagine im unit in a relevant and often unexpected real-world context to help spark curiosity. the videos use contexts from all around the world to make cross-curricular connections and increase engagement..

This article was sponsored by Imagine Learning and produced by the Solutions Studio team.

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10 Helpful Worksheet Ideas for Primary School Math Lessons

M athematics is a fundamental subject that shapes the way children think and analyze the world. At the primary school level, laying a strong foundation is crucial. While hands-on activities, digital tools, and interactive discussions play significant roles in learning, worksheets remain an essential tool for reinforcing concepts, practicing skills, and assessing understanding. Here’s a look at some helpful worksheets for primary school math lessons.

Comparison Chart Worksheets

Comparison charts provide a visual means for primary school students to grasp relationships between numbers or concepts. They are easy to make at www.storyboardthat.com/create/comparison-chart-template , and here is how they can be used:

  • Quantity Comparison: Charts might display two sets, like apples vs. bananas, prompting students to determine which set is larger.
  • Attribute Comparison: These compare attributes, such as different shapes detailing their number of sides and characteristics.
  • Number Line Comparisons: These help students understand number magnitude by placing numbers on a line to visualize their relative sizes.
  • Venn Diagrams: Introduced in later primary grades, these diagrams help students compare and contrast two sets of items or concepts.
  • Weather Charts: By comparing weather on different days, students can learn about temperature fluctuations and patterns.

Number Recognition and Counting Worksheets

For young learners, recognizing numbers and counting is the first step into the world of mathematics. Worksheets can offer:

  • Number Tracing: Allows students to familiarize themselves with how each number is formed.
  • Count and Circle: Images are presented, and students have to count and circle the correct number.
  • Missing Numbers: Sequences with missing numbers that students must fill in to practice counting forward and backward.

Basic Arithmetic Worksheets

Once students are familiar with numbers, they can start simple arithmetic. 

  • Addition and Subtraction within 10 or 20: Using visual aids like number lines, counters, or pictures can be beneficial.
  • Word Problems: Simple real-life scenarios can help students relate math to their daily lives.
  • Skip Counting: Worksheets focused on counting by 2s, 5s, or 10s.

Geometry and Shape Worksheets

Geometry offers a wonderful opportunity to relate math to the tangible world.

  • Shape Identification: Recognizing and naming basic shapes such as squares, circles, triangles, etc.
  • Comparing Shapes: Worksheets that help students identify differences and similarities between shapes.
  • Pattern Recognition: Repeating shapes in patterns and asking students to determine the next shape in the sequence.

Measurement Worksheets

Measurement is another area where real-life application and math converge.

  • Length and Height: Comparing two or more objects and determining which is longer or shorter.
  • Weight: Lighter vs. heavier worksheets using balancing scales as visuals.
  • Time: Reading clocks, days of the week, and understanding the calendar.

Data Handling Worksheets

Even at a primary level, students can start to understand basic data representation.

  • Tally Marks: Using tally marks to represent data and counting them.
  • Simple Bar Graphs: Interpreting and drawing bar graphs based on given data.
  • Pictographs: Using pictures to represent data, which can be both fun and informative.

Place Value Worksheets

Understanding the value of each digit in a number is fundamental in primary math.

  • Identifying Place Values: Recognizing units, tens, hundreds, etc., in a given number.
  • Expanding Numbers: Breaking down numbers into their place value components, such as understanding 243 as 200 + 40 + 3.
  • Comparing Numbers: Using greater than, less than, or equal to symbols to compare two numbers based on their place values.

Fraction Worksheets

Simple fraction concepts can be introduced at the primary level.

  • Identifying Fractions: Recognizing half, quarter, third, etc., of shapes or sets.
  • Comparing Fractions: Using visual aids like pie charts or shaded drawings to compare fractions.
  • Simple Fraction Addition: Adding fractions with the same denominator using visual aids.

Money and Real-Life Application Worksheets

Understanding money is both practical and a great way to apply arithmetic.

  • Identifying Coins and Notes: Recognizing different denominations.
  • Simple Transactions: Calculating change, adding up costs, or determining if there’s enough money to buy certain items.
  • Word Problems with Money: Real-life scenarios involving buying, selling, and saving.

Logic and Problem-Solving Worksheets

Even young students can hone their problem-solving skills with appropriate challenges.

  • Sequences and Patterns: Predicting the next item in a sequence or recognizing a pattern.
  • Logical Reasoning: Simple puzzles or riddles that require students to think critically.
  • Story Problems: Reading a short story and solving a math-related problem based on the context.

Worksheets allow students to practice at their own pace, offer teachers a tool for assessment, and provide parents with a glimpse into their child’s learning progression. While digital tools and interactive activities are gaining prominence in education, the significance of worksheets remains undiminished. They are versatile and accessible and, when designed creatively, can make math engaging and fun for young learners.

The post 10 Helpful Worksheet Ideas for Primary School Math Lessons appeared first on Mom and More .

Mathematics is a fundamental subject that shapes the way children think and analyze the world. At the primary school level, laying a strong foundation is crucial. While hands-on activities, digital tools, and interactive discussions play significant roles in learning, worksheets remain an essential tool for reinforcing concepts, practicing skills, and assessing understanding. Here’s a look […]

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The Booming Business of Cutting Babies’ Tongues

One family’s story of “tongue-tie release” surgery on their newborn..

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 Sabrina Tavernise, and this is “The Daily.”

A “Times” investigation has found that doctors are increasingly performing unnecessary medical procedures that generate huge profits while often harming patients.

Today, my colleague Katie Thomas — on the forces driving this emerging and troubling trend in American health care and the story of one family caught in the middle of it. It’s Monday, February 19.

So Katie, tell me about this investigation.

So I am a health care reporter who writes about the kind of intersection of health care and money. And I was working with two other colleagues, Sarah Kliff and Jessica Silver-Greenberg. And together, the three of us had long been interested in, are the medical procedures and the tests and other things that we get when we go to the doctor or into a hospital — are they always necessary?

But what we were really interested in exploring was not just are these procedures and are these tests, et cetera — are they necessary, but in some situations, could they actually be harmful to patients? And so that’s what we decided to try and take a look at. And so we had gotten started in our reporting when we got a tip. And it was from a mom in Boise, Idaho. And her name was Lauren Lavelle.

Nice to meet you.

Hi, how are you?

And my colleague Jessica Silver-Greenberg and I went to her house to meet with her.

And where does her story start?

I am a mom of two. I live in Boise. My daughter, June, is four, and I have a 17-month-old, Flora.

Her story starts when Lauren gets pregnant with her daughter, June.

So by the time we got pregnant with June, November of 2018, about eight months after we had the miscarriage, I think I was just more hesitant and nervous than anything.

Lauren and her husband had trouble conceiving, and so they were so happy when they learned that they were going to have June. And like most first-time parents, they were also a little bit nervous.

But being type-A and super prepared, I did all my homework. We hired a doula. I wanted an epidural. Having a natural childbirth absolutely was not for me.

And Lauren is very organized. She’s always on top of everything, and she makes all sorts of plans. And she gets a lot of different providers lined up ahead of time —

I didn’t know anything about breastfeeding, like zero things.

— including one that she has hired to help her with breastfeeding.

Where did you find out about her?

So I asked our doula for a list of recommendations, and she gave me a very short list. At the time, there were very few lactation consultants in the Valley. And Melanie was one of them.

She ended up deciding to work with Melanie Henstrom, who was a local lactation consultant in Boise.

She sold this package at the time. I don’t know if she still did, but it was like prenatal visit breastfeeding class. And then, she’ll come to the hospital and help you latch, and then she’ll come to the house a couple of times after. And I thought, well, this sounds perfect. Great. You know, I’m covered there.

So one week after her due date, she gives birth. And it was a difficult labor. It took 24 hours. Lauren was completely exhausted. But once June arrived, the family was very, very excited to have her.

And I remember June coming out and that surreal feeling have when you see your first baby for the first time, like oh, my God, there’s a baby in the room.

And June was a healthy baby, but she was having trouble breastfeeding.

She would not latch. Like, she wouldn’t even attempt. She would scream. It was the only time she ever cried — if you tried to make her to breastfeed.

And so as her pediatrician was making the rounds, they noticed that June was having trouble and said that June’s tongue is really tight.

We can clip it if you’d like.

And that they could clip it.

What does that mean exactly, Katie — clipping her tongue?

What it means is that there’s a small percentage of babies whose tongue is very tightly tethered to the bottom of their mouth. And for a very small percentage of babies, their tongue is almost tied so tightly down that they can’t nurse well.

So it makes breastfeeding very difficult if a baby has a tongue like this.

Exactly. If you bottle-feed your baby, the baby can basically adjust and make do. But if you want to breastfeed, some babies have trouble, basically, latching on to their mother when they don’t have that tongue motion. And so some version of clipping these tongue ties has been done for centuries. Midwives have been doing it. Pediatricians do it.

And traditionally, what it’s been is a very quick snip right underneath the tongue just to loosen up the tongue. And traditionally, that procedure is extremely straightforward. There’s little to no follow-up care. And basically, the baby naturally heals as it learns to breastfeed.

And so we said, OK. They explained that it was completely painless. They’d do it with scissors. She wouldn’t even feel it. And all of that was true. They clipped it. I don’t even think she woke up.

But in June’s case, it didn’t seem to help much, and she and Lauren were still having problems breastfeeding afterwards. So while she’s still in the hospital, she calls up the lactation consultant that she had hired — Melanie Henstrom — just to let her know what was going on. And from talking to her on the phone, Melanie said that the situation was actually much worse than Lauren had thought and that Lauren’s baby needed another tongue-tie procedure — a deeper cut under the tongue.

How did she make this diagnosis, Katie? Was it over the phone? How did she know this?

Yes, Lauren told us that it was from a phone conversation. And in addition to that, she also warned her that, basically, Lauren and her husband should really take this seriously and consider getting it done, because if she doesn’t get it fixed, it could lead to a whole host of problems beyond just problems breastfeeding.

She’ll have scoliosis, and she’ll suffer from migraines, and she’ll never eat, and she’ll have a speech impediment, and she won’t sleep — I mean, just, like, the long list of things over the phone.

And Lauren starts panicking.

I mean, first of all, I felt — I’ve never felt more terrible in my life than that first day or so after giving birth. Like, the comedown from the hormones, the drugs — all of it — the sleep deprivation. And then, here was this baby we’d wanted, we were told we probably would never have after one miscarriage. And she’s so perfect, like, the most beautiful baby I’d ever seen. And you think that she has some deformity that’s going to ruin her.

But Melanie says it’s OK. She has a solution. And she tells Lauren that there’s a dentist in town who can handle cases that are as severe as June’s.

A dentist? Why a dentist?

Well, there’s a procedure that’s done in a dentist’s office that’s a laser surgery. And dentists use this high-powered laser machine that can quickly cut the flesh that connects the lips and the cheeks to the gums. So according to Lauren, Melanie tells her that by chance, this dentist has an opening, because she said a family coming in from Oregon had just canceled their Saturday appointment.

So I thought, OK, wow, people are coming in from Oregon to see him. So we talked about it. We both felt unsure. But we said, well, let’s at least take the appointment, and then we can at least meet with the dentist, and also, someone can look at her mouth and assess.

And so Lauren agrees to go in and meet the dentist.

Like, I think some people, when they hear this story, think, like, why would you believe that? It just sounds so scammy. But to me, there is a lot of things that you hear in the hospital that sound insane. Like, it’s no different than someone saying, like, your baby’s orange because their bilirubin levels are too high, so we got to go put them under these lights. Like, that sounds insane. That sounds more insane than, your baby’s having a hard time eating because their tongue is too tight and it needs to be cut. Like, that seems rational, actually.

And all of this seemed really weird to Lauren at the time. But in the context of the hospital and having a baby, lots of things about health care are weird.

So one day after they got back home from the hospital, Lauren and her husband pack up the car and go to the office early in the morning.

You know, I was wearing my hospital diaper and an ice pack, took the elevator up to his office, and —

And what happens?

So Melanie greets them at the door. They sign some paperwork, and pretty soon, the dentist, Dr. Samuel Zink, arrives.

And then, he, like, very briefly — very briefly — looks in her mouth and is like, yeah, she’s got whatever — however he classified it — grade 4 or whatever he says — class 4 — and she has a lip tie, which — that had never been mentioned to us before, so it’s very much on the spot, this new piece of information.

You know, pretty quickly, the dentist diagnosed June as having a couple of ties. He confirmed that she had a tongue tie, and he said it was severe. He also said that she had tightness under her top lip, called a lip tie. And so the baby actually needed to get two cuts. And again, Lauren said that the dentist and the consultant told her how important it was for her to do this for her baby.

One of us says, like, what happens if we don’t do the procedure? , Like what are our alternatives? And it was basically like, there’s no alternative. Like, you have to do this. Otherwise, again, long —

So Lauren and her husband decided to do it. But before the procedure starts, Melanie actually stopped Lauren from coming into the room.

Melanie turned around and put a hand on my shoulder and said, oh, no. And I said, oh, am I not going with you? She goes, well, we can’t tell you no, but if you hear her cry, it’ll impact your milk supply, like, adversely.

What do I know? So I said, oh, OK. And she pulled out the white-noise machine and said, what do you want to listen to? And I had no idea what she was talking about. I had no idea what it was. And so then she just turned it on — white noise — and left.

What happens next is, Melanie turns on a white-noise machine in the room.

And that was the moment that I was like, get your baby and get out of here. And I didn’t listen to it. It was like all of my mom intuition firing, being like this isn’t right, you know. It’s like, I don’t know how to describe it, but your full body — you have to get your baby and get out of here. And I just ignored it.

She said her maternal instincts really kicked in, and she just had this instinctive fear about the procedure and whether June would be OK. But the procedure itself was very quick. Within just a couple of minutes, Melanie returns with June.

And she was screaming. Like, screaming, and so worked up. This was, like, hysterical, inconsolable. And she was also choking on something, like, gagging.

And June was so worked up. Lauren had only had her for a couple of days, but she said that this was on a different level than any other way she had ever seen June crying. And June just wouldn’t stop crying.

And she looked over to Melanie, and Lauren said that she remembered Melanie saying this was very typical. And so they pay the dentist. They pay $600 for the procedure, and then they go home.

Over the next several days, June did not get better as Melanie had assured them. You know, she was basically inconsolable, Lauren said — just crying hysterically. And Lauren and her husband — they don’t know how to comfort her. They’re new parents. They’ve only had a baby a couple of days. And they’re almost beside themselves.

There was nothing we could do. And I remember finally, I said, like, this is not normal. We’re going to an emergency room.

And they decided to go to the emergency room, where a doctor looks inside June’s mouth and finds a large sore in her mouth that he says is probably causing her so much pain.

And so he said, you know, it breaks my heart to see a sore that big in a baby this small. It was like the floodgates opened, and there was nothing but guilt and shame. Like, unmanageable guilt and shame.

Like, what have we done? Who are these people? What have I done to my baby? Will she ever be the same? Like, what did I do?

So at this point, Lauren is really understanding that her intuition about this surgery was probably right and that she and her husband may have really made a mistake with this. What does June’s recovery look like?

So June never did end up breastfeeding successfully, which was the main reason why Lauren and her husband had decided to do this procedure.

That was the whole point, right?

That was the whole point. Right. And over the next couple of years, June had a number of issues that there’s no official medical diagnosis for, but Lauren has attributed a lot of her behaviors to what had happened to her when she was just a few days old.

I mean, you couldn’t close a fridge door too loud, or else it would set her off. Or, we would attempt to take her for a stroller walk on the Greenbelt, which is the walking path, and she’d be asleep in her car seat, you know, stroller, and someone would try to pass us on their bike and ring their bell, and it would startle her, and it would just set her off. So she just was very, very, very fragile.

So Lauren just wanted to get answers, and she really wanted to hold Melanie and the dentist accountable. So she gathered all of the paperwork that she had — texts, emails, other correspondence — and she went to the Idaho Board of Dentistry, where she filed a complaint against the dentist. And then, she also went to a professional organization that certifies lactation consultants and filed a complaint with them as well.

And did she get anywhere with either of them?

At first, no. The Idaho dentistry board didn’t want to investigate, and Lauren appealed, and she lost her appeal. And she didn’t initially hear back at all from the lactation board.

No one wanted to take responsibility. That’s the thing. No one wanted to stick their neck out there. What’s the alternative? The story never gets told?

And that’s when she decided to reach out to us. And after our story came out, the lactation board finally told Lauren that they were investigating Melanie.

And Katie, you guys were reporting the story. I’m assuming you reached out to both the dentist and to Melanie. What did they say?

Beyond a very brief phone conversation that I had with Melanie in which she defended her work and she said that she had a number of very satisfied customers, she didn’t respond to detailed questions about Lauren’s story or the stories of her former clients. And Dr. Zink did not respond to our requests for comment, but he did tell the dentistry board that Lauren’s baby’s procedure was uneventful and that an extremely small percentage of patients do not respond well to the procedure.

And how big of an issue is this, Katie? I mean, how common is it for mothers to have an experience like Lauren’s?

So after we got the tip from Lauren and we dug deeper into her story, we found ourselves really surprised by how big this industry was for tongue-tie releases. And in part, it’s been driven by this movement for breastfeeding and the Breast is Best campaign and a growing number of parents who are choosing to breastfeed their children.

In turn, that has sparked this big boom in tongue-tie releases. One study that we found showed that these procedures have grown 800 percent in recent years.

Yeah. And also, as we started talking to other parents around the country, we learned that some of them had similar stories to what Lauren had told us. There’s plenty of instances where there’s no harm done to the baby at all when they get these procedures.

But we also found cases where babies were harmed, you know, where they developed oral aversions, which basically means that they don’t want to eat because they fear having anything put in their mouth, including a bottle. We found cases where babies became malnourished, had to be hospitalized. We found more than one instance in which babies had to be given a feeding tube just weeks after the procedure.

So these sounds so painful and awful for a newborn — these problems. But I guess there’s always a risk, Katie, in any medical procedure, right? I mean, how much of this is just the risk you sign up for when you agree that your baby should have a surgery?

Well, that’s true. I mean, there’s always a risk. But what you’re supposed to do is weigh the risks against what the potential benefits of a procedure are. And when we really started drilling down into what those benefits were and into the medical research, we found there just wasn’t a lot of potential benefit for these procedures, if at all, in many cases.

Really? So the procedures don’t have a medical reason to exist?

That’s right. We reviewed all of the best-quality medical research on this. And other than that old-fashioned snip under the tongue, which does show that in some cases, it can reduce pain for breastfeeding mothers, but otherwise, all of this growth and all of these other more invasive procedures — we found there just wasn’t good evidence that they helped babies. And the more we looked into tongue ties and started to connect it to the other reporting we were doing, we started to realize that it was driven by some really big forces in our health care system that really had the potential to harm patients.

We’ll be right back.

So Katie, we talked about this new surge in a procedure that surgically unties infants’ tongues from the bottom of their mouths, often needlessly, sometimes even harmfully. And you said your reporting found that this surgery was actually part of a broader trend. Tell me about this trend and what’s driving it.

So that’s what this investigation was really about — to find the procedures that are doing unnecessary harm to patients and to really understand why this is happening. You know, like, what’s driving the prevalence of these procedures? And there’s just a lot of unnecessary surgeries out there, but we decided to center our reporting on three particular surgeries that had the potential to harm patients, in addition to tongue ties. We focused on a particular hernia surgery, a bariatric surgery, which can be overdone and cause harm, and a vascular surgery done on patients’ legs to help us understand the forces that were at work that were driving all of this.

And what did you find when you dug deeper into those surgeries?

Well, it’s very complex, but we ultimately found three main drivers that were underlying all of these. First, there’s a financial incentive for the doctors to perform these surgeries. There’s also a real push from the medical device companies that make these surgeries possible. And last, there’s a huge information void for solid medical advice that a lot of these doctors and companies take advantage of in order to push the surgeries.

OK, so let’s start with the money, Katie. How exactly is that incentivizing doctors to perform a lot more of these procedures? Like, what are the mechanics of that?

So the reality of our health care industry today is that in many places, even in places like non-profit hospitals, the doctors who work there are not getting a salary, a straight salary that’s just kind of, you get paid for showing up to work that day. Instead, they’re actually getting paid based on the procedures that they’re doing, how complex those procedures are, or possibly how lucrative.

And it’s not every doctor. There are still doctors that get paid salaries. But it’s increasingly the case that doctors have — at least a part of their pay is tied to the procedures that they’re doing.

Interesting. So the procedure is growing in importance in terms of actual compensation for doctors.

Right. I mean, in part, it’s kind of baked into the health care system that we’ve always had. You can even think about it as the small-town doctor who operated his own independent practice or her own independent practice. It’s essentially a small business, and they would get paid based on the patients that they saw.

But increasingly, even in, for example, large hospital systems where you might think that a doctor is just getting paid a salary to work in a hospital, in fact, a chunk of their bonus, for example, can sometimes be tied to the procedures that they’re doing, and that is increasingly the case.

Interesting.

And so one particularly egregious example of this was at a hospital that’s in New York — Bellevue Hospital. And basically, what my colleagues found there was that they had basically turned their surgery department into an assembly line for bariatric surgery, which makes your stomach smaller and can lead to weight loss. But what we found was that they were greenlighting patients that, basically, didn’t meet the qualifications for the surgery, which is a serious surgery. And what they found was that there were several situations where people had very serious outcomes as a result of getting the bariatric surgery there.

OK, so this is an extreme case of a hospital turning to a particular surgery to drive profits. And it wasn’t uncommon in your reporting, it sounds like.

No, it wasn’t the only example, but it was the most striking. And when we reached out to Bellevue, they defended their work, and they said that their practices were helping patients who wouldn’t otherwise get care. But our reporting was pretty conclusive that the program was churning through a record number of surgeries.

So what else was driving this increase in harmful surgeries that you guys found?

So we found it wasn’t just the hospitals who were benefiting. The other major player that benefits are these companies that are making the tools and the products that doctors are using during the procedures. And in order for them to sell more of their products, a lot of time, what they end up doing is promoting the procedures themselves.

So like medical device makers, like the company that made the laser in June’s surgery.

Right. And they do this in a number of ways. They’re giving them loans to help them buy the equipment, and in some cases, they’re even lending them money to help set up those clinics where the procedures are performed.

So they’re really underwriting these doctors so that they can perform more surgeries and, ultimately, sell more machines.

Yes. And the other things that they do is — the laser companies, for example — they will host webinars where they will have dentists who frequently perform these procedures show other dentists how to do the procedures. We even discovered this conference that was created by one of the laser companies, and it had kind of a wild name. The name of the conference was Tongue Ties and Tequilas.

(CHUCKLING) Right. It brought in dentists to talk about how to make money off the procedures. You know, how to promote themselves on social media, how to actually perform the procedures, and of course, when they were all done, they got to celebrate with an open tequila bar.

OK, so a lot of this really amounts to these companies trying to popularize these procedures, basically, like, to get the word out, even if the procedures don’t really work or, in some cases, cause harm.

Right. But they also play a big role in the other factor that’s driving a lot of this, which is the information that they put out there about the surgeries. These companies often sponsor research, which doctors often rely on to guide their practices. And part of what we’ve found is that it can create this echo chamber where doctors feel more comfortable and justified in doing these procedures when they have this whole alternate universe that is telling them that it’s OK to do these procedures, and in fact, it’s beneficial to patients.

So tell me about this echo-chamber effect.

The best example of this we found was a doctor in Michigan named Dr. Jihad Mustapha. He calls himself “the Leg Saver.” And what we found was that he and several other doctors do these procedures called atherectomies, which is basically like inserting a tiny roto-rooter inside an artery to get the blood flowing.

And Dr. Mustapha in particular was not only a very prolific performer of these procedures, but he actually founded his own medical conference, and he even helped start a medical journal that was devoted to using these procedures. And you know, tongue ties — there’s really no good evidence that these are actually beneficial to patients. And in fact, despite his nickname as “the Leg Saver,” one insurance company told Michigan authorities that 45 people had lost their limbs after getting treated at Dr. Mustapha’s clinic over a four-year period.

45 people lost their limbs?

I mean, that is the ultimate version of harm, right?

Right. Now, he did speak to us, and he defended his work and said that he treats very sick people. And despite his best efforts, some of these patients are already so sick that they sometimes lose their limbs.

And how much did he receive for each procedure?

Doctors like him typically receive about $13,000 for each of these atherectomy procedures.

But we found that misinformation, or poor information, also applied when doctors were learning new types of surgeries.

Really? Like how?

So one of the areas we looked at was the area of hernia surgery that I mentioned. And there’s a particular type of surgery. It’s a very complex version of a hernia surgery, called component separation. And the expert surgeons that we spoke to said that it’s difficult to learn, and you have to practice it over and over and over again to get it right. But one recent survey of hernia surgeons said that one out of the four surgeons had taught themselves how to perform that operation.

Yeah, not by learning it from an experienced surgeon but by watching videos on Facebook and YouTube.

I mean, how unusual is that? I guess, to me, it strikes me as very unusual. I mean, I think of learning about how to take my kitchen faucet apart on YouTube, but I do not think of a doctor learning about how to perform a surgery on YouTube.

Right. And it has actually become increasingly popular in recent years, and there’s not good vetting of the quality of the instruction. We even found videos on a website run by a medical device company that was intended to be a how-to for how to do these surgeries, but the video contained serious mistakes.

Wow. And Katie, all of these videos — some of them with serious mistakes — I mean, is this something that would be subject to medical regulators? Like, is there any kind of rules of the road for this stuff?

You know, there’s less than you would expect. Sometimes hospitals have rules about what sort of education their doctors need before performing a surgery. But we were surprised that there was a lot less regulation than we thought there would be and much less vetting of these videos than we anticipated.

So essentially, what you found was this complex, oftentimes interconnected, group of forces — device companies pushing their products, hospitals bolstering their bottom line, and rampant misinformation that, as you said, all really trace back to the same motivating factor, which is money. But wouldn’t the fear of being sued for medical malpractice prevent a lot of this behavior?

You know, this kept popping up during the course of our reporting. I do think we have this idea that any time a doctor does anything wrong, they’re going to get sued. But it just wasn’t always the case in our reporting. There’s a lot of statutes of limitations, time limits on when somebody can file a lawsuit, and other ways that make it somewhat hard to really hold a doctor accountable.

One example is the regulatory organizations that oversee doctors. The one doctor that I mentioned earlier — Dr. Mustapha — state investigators had found that his overuse of procedures had led people to lose their legs. And yet, he ultimately settled with the state, and he was fined $25,000. That actually adds up to about two of these atherectomy procedures.

So it sounds like malpractice is not necessarily going to be the route to rectifying a lot of this. But I guess I’m wondering if the federal government could actually rein some of this in before the patients are harmed.

It’s possible. But this is just a very difficult issue. Some of the themes that we explored in this reporting are really just firmly embedded in our health care system in the way that it works. The fact is that we have a for-profit health care system, right? So everyone, from doctors to hospitals to the device companies, benefit when more procedures are done. All of the incentives are pointing in the same direction.

And so trying to find one or two simple solutions will probably not easily fix the issue, as much as we all hope that it could.

So is the lesson here, be much more discriminating and vigilant as a patient? I mean, to get a second opinion when you’re standing in front of a doctor — or a dentist — who’s telling you that you or your baby needs a procedure?

Yes. I think that is one of the takeaways. But look, we understood that even reporting on all of this was risky, because people could hear about these harmful surgeries and start wondering if everything that their doctors tells them is a scam. And of course, while some of these procedures are harmful, a lot of procedures are lifesaving. But ultimately, for now, patients are kind of left on their own to navigate what’s a pretty complex and opaque health care system. When you have somebody standing in front of you saying, you should do this, it can be very confusing.

And this is something that Lauren talked a lot about — just how confusing all of this was for her.

There’s a lot of information that you’re getting that is truly like someone is speaking a foreign language. And because they do it all day long, it’s not user-friendly. Like, it isn’t designed for the comfort or understanding of the person receiving the information.

There is so much blind trust and faith that you have in the system, in the providers who are giving you this information. You trust, like, this is what they do all day long. So there is no real reason to question. That is the system that we have in this country.

Katie, thank you.

Here’s what else you should know today. On Friday, the Russian authorities announced that opposition leader Alexei Navalny died in prison. He was 47.

Navalny, a charismatic anti-corruption activist, led the opposition to Vladimir Putin for more than a decade. His popularity was broad, extending far outside the realm of liberal Moscow. And that proved threatening to the Russian authorities, who attempted to poison him in 2020.

Navalny survived and later extracted a confession from his would-be assassin on tape. Navalny believed that Russia could be a free society, and he had the extraordinary ability, through sheer force of his personality, charisma, and confidence, to get others to believe it, too. Though he had been in prison since 2021, his death still came as a shock.

[SPEAKING RUSSIAN]

His wife, Yulia Navalnaya, made a surprise appearance at a security conference in Munich shortly after the Russian authorities announced her husband’s death.

She received an emotional standing ovation.

In Moscow, my colleague, Valerie Hopkins, spoke to Russians who were placing flowers in his honor —

— and expressing disbelief that he was gone.

Then I asked them if they believe in the beautiful Russia of the future that Navalny talked about. And they said, yes, but we don’t think we will survive to see it.

At least 400 people have been detained since his death, including a priest who had been scheduled to hold a memorial service in Saint Petersburg.

Today’s episode was produced by Asthaa Chaturvedi, Diana Nguyen, Will Reid, and Alex Stern, with help from Michael Simon Johnson. It was edited by Michael Benoist, with help from Brendan Klinkenberg, contains original music by Diane Wong and Dan Powell, and was engineered by Alyssa Moxley. Our theme music is by Jim Brunberg and Ben Landsverk of Wonderly.

That’s it for “The Daily.” I’m Sabrina Tavernise. See you tomorrow.

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Hosted by Sabrina Tavernise

Featuring Katie Thomas

Produced by Asthaa Chaturvedi ,  Diana Nguyen ,  Will Reid and Alex Stern

With Michael Simon Johnson

Edited by Michael Benoist and Brendan Klinkenberg

Original music by Diane Wong and Dan Powell

Engineered by Alyssa Moxley

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A Times investigation has found that dentists and lactation consultants around the country are pushing “tongue-tie releases” on new mothers struggling to breastfeed, generating huge profits while often harming patients.

Katie Thomas, an investigative health care reporter at The Times, discusses the forces driving this emerging trend in American health care and the story of one family in the middle of it.

On today’s episode

problem solving maths lesson

Katie Thomas , an investigative health care reporter at The New York Times.

A woman holding a toddler sits on a bed. The bed has white sheets and pink pillows.

Background reading

Inside the booming business of cutting babies’ tongues .

What parents should know about tongue-tie releases .

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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.

Katie Thomas is an investigative health care reporter at The Times. More about Katie Thomas

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