how can you solve the problem using your paper strip

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Using Strip Diagrams as a Problem-Solving Strategy

There’s a lot to be learned from the data that comes from state testing if we use it correctly . Oftentimes, we’ll take a problem that students, as a whole, perform poorly on and we create a bunch of problems just like that one and “teach” them how to work that type of problem. In the end, that’s not an effective strategy, because they’ll probably never see a problem exactly like that one again. A more effective way to use the data is to analyze the wrong answers to determine underlying misconceptions that resulted in the wrong answers and provide students with strategies and tools to improve their overall mathematical reasoning. As far as strategies go, drawing strip diagrams is one of the most powerful strategies students can have in their toolbox. And I have some super clear examples to persuade you!

This post contains affiliate links, which simply means that when you use my link and purchase a product, I receive a small commission. There is no additional cost to you, and I only link to books and products that I personally use and recommend.

The test items in this post come from the 2021 State of Texas Assessments of Academic Readiness (STAAR) test. A wonderful organization called lead4ward analyzes the test each year and provides error analysis statistics.

Analyzing errors

Let’s get started! As you can see from this test item, only roughly half of the students in the state got the correct answer. Look at the most common wrong answer, choice H. Can you see the error that the students who chose that answer made? Take a minute to figure it out before you scroll down.

Look closely at the order of the numbers in the problem. The smaller number comes first in the problem. Students who chose H realized it was a subtraction problem, but took the numbers in order from the problem and subtracted them. So they did 379 – 514. Nine minus 4 is 5. Seven minus 1 is 6. They couldn’t do 3 minus 5, so they did 5 minus 3. The answer they got was 265. Your first thought might be, I need to make sure my students always know to subtract the smaller number from the bigger number. Except that’s not true. If the temperature is 18 degrees and the temperature drops 29 degrees…  We need to be extremely careful to not teach “rules” that expire .

The students who chose H lacked an understanding of the meaning of the numbers in the problem. The way we help them be more successful with problems like this is to give them tools to improve their comprehension of word problems and the numbers they contain. Enter strip diagrams.

Using drawings to describe problems

Drawing strip diagrams is a process that actually begins in Kindergarten and 1st grade when students should be drawing pictures to represent word problems. Students should understand that the numbers in a problem represent something—pizzas, soccer players, apples, money saved—and their drawings should include labels identifying what the numbers represent. At this point, the drawings don’t need to resemble strip diagrams. What’s important is the labeling. It could look something like what you see here.

Notice a couple of things. First, the boys and girls are represented by circles. Easy to draw and count. Students need to understand that these are math pictures, not art pictures. Next, and of critical importance, are the labels.

Beginning in 2nd grade, students can begin to draw more formal strip diagrams. Strip diagrams, also called tape diagrams, are often associated with Singapore Math. Char Forsten’s Step-by-Step Model Drawing is the book I learned model drawing from. Another great resource is Math Playground’s Thinking Blocks .

Modeling how to draw a strip diagram

Now let’s get back to that released test item I started this post with and see what the model-drawing process might look like.

This problem is a comparison subtraction problem. We always want students to draw the model with labels first. they will add the numbers in the next step.

Teacher: [Reads problem out loud]  What is this story about? (lions) How many lions? (2)  What does the problem tell us about the lions? (their weight)  Do we know their weights? (yes) Which lion weighs more? (the older lion)  What is the problem asking us to find? (the difference in their weights)  Huh, what does that mean?  (The older lion weighs more than the younger lion. The problem is asking how much more.)

NOTE: Notice that we didn’t talk about the numbers at all! The point of this discussion is to help students make sense of the numbers in the problem and verbalize what the problem is asking them to find.

Teacher: Drawing a model really helps me understand what math I need to do to solve a problem. Let’s draw a model to represent this problem. We know that the older lion weights more, so his bar should be longer, right? [draws and labels the older lion’s bar] That means the younger lion’s bar should be shorter. [draws and labels younger lion’s bar] And you guys told me the problem is asking for the difference.  [adds the difference with a question mark]

Now we can plug in the numbers from the problem.

Notice that what I’ve described is very scripted. I want students to hear my mathematical thinking, and I’m teaching them the mechanics of drawing the model. But it’s important to let students use the tool to solve problems. Think how the models would look for these variations of the problem:

There are two lions at the zoo. The weight of the younger lion is 379 pounds. That’s 135 less than the weight of the older lion. How much does the older lion weigh?

There are two lions at the zoo. The weight of the older lion is 514. That’s 135 more than the weight of the younger lion. How much does the younger lion weigh?

Remember, our goal is for students to be able to use strip diagrams to solve new types of problems, so once they understand and can use the model, we have to give them new types of problems to solve without scripted instruction.

More examples of strip diagrams

Let’s take a look at a few more problems from the same test. Each of these problems had pretty dismal results.

This first problem is what we in Texas call a  gridable . That means it’s not multiple choice—students have to write and bubble in their answers. A lot of times students will miss gridables due to calculation errors. But I’m pretty sure that’s not the case here. I doubt they miscalculated 4 x 5. What that means is that 38% of the 3rd-grade students in Texas did not recognize this as a multiplication problem. They likely added 4 + 5. Teaching keywords could be the culprit. Students see the word total in the problem, and they’ve been taught that  total means addition. Teaching keywords basically gives students permission to  not read and understand the problem—just find the keyword and plug the numbers into the operation. Not a sound problem-solving strategy. Instead, we see how a strip diagram could be drawn to represent the problem.

Here’s another multiplication problem. You can see from the error analysis that only 52% of the 3rd-graders correctly answered this problem. Answers B and D are calculation errors. Can you figure out the error these students made? Doing so can help you prevent these types of errors by addressing them in your instruction. Answer choice C results from adding the two numbers, not multiplying. Again, we see the keyword total in the question. Drawing a model would not only help the students visualize the problem as multiplication, but it might also prevent calculation errors. Students who are not confident with the standard algorithm could solve the problem with repeated addition.

Last one and it’s a doozy! Look at that error distribution. When it’s spread out like that, it usually means the kids just didn’t have a clue and guessed. There’s a lot going on here. How could we help students tackle a problem like this?

First, of course, is drawing a model. We see that this is a part/whole problem with three parts, one of which is missing. If you looked carefully at the wrong answers, answer choice F was adding all three numbers. Pretty hard to look at this model and think you’re supposed to add all three numbers.

Aside from model drawing, however, students should learn to write equations to match their models. That’s really the other thing that was hard about this problem. They weren’t asked to solve the problem, just find the correct way to solve it.

Final thoughts

Strip diagrams have to be presented to students as a problem-solving tool and they have to be used consistently. Yes, it takes longer for students to draw strip diagrams to represent their problems, but it should be an expectation. That means we probably need to assign fewer problems to allow students the time to draw their strip diagrams.

The labels are a must! If students can’t label their numbers, it’s a huge red flag. Work with those students in small groups to help them develop comprehension skills.

Students who say they know the answer without drawing a strip diagram should be reminded that we draw models when the problems are easy so we can use them as a problem-solving tool when the problems get harder. And if a large percentage of students can really solve the problems without drawing models are we challenging them enough?

So there you have it. Have I sold you on having students draw strip diagrams? Do you have tips of your own to share? I hope you’ll sound off in the comments.

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13 Comments

Strip/bar/tape diagrams are becoming one of my favorite strategies. The labels are crucial and I need to be better at expecting that. I am nearly to the part of 3rd grade where I will be teaching 2 step problems again and I think continuing the strip/tape/bar diagram representation is going to be a wonderful tool. Here’s hoping!

Thanks for your comment, Jessica! They are one of my favorites as well. So very powerful! Come back and let us know how it goes.

Any way we can show visuals for problems are going to support student understanding! Thank you for adding another tool to my teaching tool box! I teach 1st grade and I want to build my students’ confidence and understanding of math!

That’s wonderful, Suzanne! It’s so important that our students develop a positive math identity early.

Hi Donna, I am increasingly interested in the connections between literacy and math. Your post about how kids start by representing their mathematical thinking/problem solving with pictures, then labels, then drawing a model and later abstract number sentences. seems so similar to how young children write stories first with pictures, then we ask them to label and later on write sentences with words, punctuation etc. Do you know of any research linking the two? Thanks! Jennifer

You are so very correct! There are tons of professional books connecting the two. Solving word problems requires comprehension of the problem. Just like reading teachers tell their students to “make a movie in their head” when they read, I tell my math students the very same thing!

Thank you so much for breaking down strip diagrams. This is a tool I try to encourage my students to use, but I feel like I need to do more modeling for them after reading this. I also like how you pointed out we should analyze the best wrong answers for misconceptions.

I have been using strip diagrams to teach my 2nd graders for a few years. They always seem to struggle with understanding where to put the numbers when the story is comparing. How many more toys does Grant have than Amanda? So, I always pointed out that it’s a comparison, and that seemed to help. I am now a K-5 math interventionist and some upper grades teachers think it will be confusing to call that a comparison when they are not using >, <, = symbols. How can I help them understand those problem types, and is there harm in using the word comparison? I think they are still comparing – "how many more".

A strip diagram for comparison looks different. It’s one bar on top of the other. The difference is where the longer bar is shorter than the shorter bar. That might help with the confusion. Absolutely use the word comparison!

Hi Donna. I’m wondering how strip diagrams are any different than the part part whole organizer. They seem so similar to me. I’ve done a lot of work getting my students to use and understand the PPW as a tool. I like the idea of the strip diagram but fear I’ll confuse them if I introduce a new tool to use in place of a PPW.

They are very similar! Often students transition to strip diagrams in the intermediate elementary grades. No need to do both though.

Hi Donna! I started using tape diagrams with my students when my district adopted the Engage NY/Eureka Math Curriculum. I found that students have had great success in using this strategy. Thank you for this clear explanation on using tape diagrams during problem solving.

It’s such a powerful strategy! I’m glad to hear your students found success using it!

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Because differences are our greatest strength

Dividing fractions using fraction strips: An evidence-based math strategy

By Brendan R. Hodnett, MAT

Expert reviewed by Sarah R. Powell, PhD

What you’ll learn

Download: printable fraction strips, read: how to use this strategy, understand: why fraction strips work, connect: link school to home, research behind this strategy.

Dividing fractions can be really tough for many students. It’s hard to visualize splitting a fraction up into groups of other fractions. To divide fractions, many students memorize the “keep-change-flip” algorithm without knowing why it works.  

Without a conceptual understanding of dividing fractions, students tend to get stuck when they come across problems (especially word problems) where they have to divide by non-unit fractions like 2/3 or 3/4, or problems where the divisor is larger than the dividend.

By sixth grade, when students need to divide mixed numbers, they often rely heavily on the multi-step “keep-change-flip” method that is hard to memorize and understand.  

You can help your students understand how to divide fractions by using fraction strips that they can move around. Manipulatives and visual representations are evidence-based strategies that support learning new math concepts. Fraction strips can help students not only understand the concept of dividing fractions, but literally see how to solve these problems without having to do any calculations. 

Fraction strips PDF - 32.9 KB

Objective: Students will use fractions strips to show and solve the division of two fractions. 

Grade levels (with standards): 

5 (Common Core 5.NF.B.7 : Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions)

5 (Common Core 5.NF.B.7.B : Interpret division of a whole number by a unit fraction, and compute such quotients)

6 (Common Core 6.NS.A.1 : Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem)

5–6 (Common Core Math Practice MP1: Make sense of problems and persevere in solving them; Common Core Math Practice MP5: Use appropriate tools strategically)

Best used for instruction with: 

Whole class

Small groups

Individuals

How to prepare:

Gather and explore materials. Give each student or pair of students a set of fraction strips. Have students cut each strip into unit fractions (with a numerator of 1). For students who struggle with fine motor skills, consider having a few sets already cut out. You could also make a few laminated or card stock sets for students. 

After cutting each strip into unit fractions, have students reassemble each of the strips so they have a complete set that matches what’s in the printable. Give students a copy of the printable, or project an image of the printable so they have a visual model to refer to.     

Once students have all the strips arranged, reintroduce the concept of a whole. Remind students that the visual representation of 1 and the word “whole” are often interchangeable when we talk about fractions. Say, “At the top of our set we have 1 whole piece.” Then, ask students what they notice about the strips beneath the whole piece. Model an example. You could say, “I notice that each row of fractions strips is the same size.” Have students share what they notice with a partner. Then ask a few students to share with the entire class. Remind students of previous lessons where they worked on dividing whole numbers by fractions. 

How to teach:

1. Review dividing a whole number by a fraction. Ask students to place the 1 whole strip at the top of their desk. Beneath that strip, have students place as many 1/4 strips as needed to match the same size as 1 whole. Write the equation 1 ÷ 1/4 = 4 on the board and ask the students how they know this is true.  Students should refer to the strips in front of them to explain the answer.

Review all the common ways students could explain their answers by providing the following examples both visually and verbally:  

“1/4 fits into 1 whole 4 times” 

“There are 4 groups of 1/4 in 1 whole” 

“1 divided into groups of 1/4 is 4” 

“4 times 1/4 equals 1”

2. Explain how to use fraction strips to divide a fraction by a fraction. Use the “I Do, We Do, You Do” model (also known as a gradual release model of instruction) to guide students through the process of using the strips.  

I Do: Explain and model using the strips. Say, “Now we can use a similar strategy to solve division problems with two fractions in them. Let’s take a look at 1/2 ÷ 1/6. I’ll start by placing a 1/2 strip at the top like this. Next, underneath it, I’ll put as many 1/6 strips as I can to match the 1/2 strip. We can see that it takes one, two, three 1/6 strips to match the 1/2 strip. So I can conclude that 1/2 ÷ 1/6 = 3, or 1/6 fits into 1/2 three times.”

We Do: Guide students through trying it with you. Say, “Now let’s try this together. Start with 1/2 again. I will place 1/2 at the top. You do the same.” Model placing the 1/2 strip at the top. “This time divide 1/2 by 1/8. Let’s place as many 1/8 strips as we can to match the 1/2 strip.” Model it and then walk around to help students who may need support. For students who are able to set it up correctly, prompt them to write out the division problem with the solution. 

Once everyone has finished, discuss the answer as a class. Write the division number sentence for students who may not have gotten the correct answer. Describe the solution in multiple ways.

You Do: Choose three division problems using unit fractions for both the divisor and the dividend. Tell students they’ll be doing them on their own. Say, ”Try the next several problems by yourself or with your partner. Be sure to write the division sentence with your solution once you have set it up using your fractions strips.” Provide guidance as needed. When checking in, ask students to explain their solutions using the language discussed at the beginning of class, such as “____ groups of ____ fit into ____.”

Teaching tip: Many students, including English language learners (ELLs) and students who struggle with expressive language, benefit from having sentence frames at their desks. Print out a set of frames and put them in dry-erase pockets so students can write out their answers each time. 

3. Move on to practicing more challenging problems. Follow the “I Do, We Do, You Do” model again. 

I Do: Say, “Let’s try some problems that are a little more challenging. This time begin with 2/3.” Model for them. Students should follow by placing two 1/3 strips at the top of their desk.  “I want you to divide 2/3 by 1/6.” Demonstrate how to line up 1/6 strips below 2/3 until you have matched the whole. Count the number of strips you used out loud while pointing them out. “So, 2/3 ÷ 1/6 = 4.”  

We Do: “Now, let’s do this next one together. Let’s try 3/4 ÷ 1/8. I’m going to show 3/4 at the top using three 1/4 strips. You do the same.” Model placing the 3/4 at the top. “This time, divide 3/4 by 1/8. Let’s place as many 1/8 strips as we can to match the 3/4 strip.” Model it and then walk around to help students who may need support. For students who are able to set it up correctly, prompt them to write out the division problem with the solution. 

You Do: Provide several practice problems for students to try independently. Check in to see how students are using the fraction strips. Some students by this point may have figured out how to solve the problems without using the strips. Talk with these students about what procedure they are using. If you feel the class is ready at the end of the lesson, have these students demonstrate the keep-change-flip algorithm. This will help begin the transition to using the algorithm over the fraction strips. 

4. Continue to practice. Some students may begin to solve without the fraction strips. Others will continue to use them. As everyone practices, challenge the students who are using the strips to draw models for each division problem instead. They can refer back to the strips as a reference, but drawing their own versions of the strips will help transition from the concrete of strips to the abstract of numbers . 

Fraction strips (or fraction bars or tiles) help students see that the same “whole” can be broken up into different equal-size parts. When students move the strips and put them side by side, they can visualize the fractional amounts. They can also make comparisons between fractions and do addition, subtraction, multiplication, and division with fractions.    

Students who experience difficulty with math, particularly with fractions, often have trouble connecting meaning to the numbers and symbols in an algorithm. Fraction strips give students a conceptual representation of the problem. They can use the strips to create both the dividend and the divisor so that they can see both as fractional parts of a whole. Students can also count the number of groups (the quotient) made by dividing. 

With this strategy, you use explicit instruction in the “I Do, We Do, You Do” model. By showing students how to use the fraction strips and then allowing them to practice with both teacher and peer support, this strategy provides multiple learning opportunities. 

Using this strategy before moving on to divisors that are not unit fractions will help students have a better understanding of why the algorithm works, not just how it works.  

Once students are familiar with how to use the fraction strips, send a new fraction strip page home or a set that has already been cut out. Families can help reinforce using fraction strips at home when students are working on their homework.

If you find families are confused by the homework, consider sending this resource to help parents and caregivers understand why math is taught differently now than when they were in school. 

“The concrete-representational-abstract approach for students with learning disabilities: An evidence-based practice synthesis,” from Remedial and Special Education

“Explicit mathematics instruction: What teachers can do for teaching students with mathematics difficulties,” from Intervention in School and Clinic

Explore related topics

Strip Diagrams: Visualizing Math

What is a strip diagrams in math.

A strip diagram, sometimes known as a bar model, length model or fraction strip, is a kind of visual model used when teaching math in order to demonstrate numerical relationships and help students solve various numbers and problems a little more easily. 

A strip diagram is essentially a rectangular bar or box that can be divided up in different ways to represent known and unknown quantities. 

The overall idea is that with a drawing or picture students can better visualize a problem and then decide what operation they should use to get the answer.

How Are They Used In Math Lessons?

Strip diagrams are actually pretty flexible visual tools that can be used when learning different topics and when confronted by different types of word problems. 

Typically, strip diagrams are used as a tool to help students visualize problems and what operations they’re being asked to do (addition, subtraction, multiplication and division), something that can be particularly helpful with word problems. 

The overarching principle is that, by drawing out a model, students learn to pick out the different variables given in a word problem, what they need to find and can then figure out what operation and methods they need to use to solve the problem in question.

Different types of strip diagram strategies

In general, there are a few different ways that parents and teachers use strip diagrams to solve different math problems. 

Part-Part Whole/Equal parts of a whole

Generally speaking, the first strip diagram most students will learn is the part-part whole model. 

In essence, a large rectangle is placed at the top, representing a whole amount. 

Below it, a same-sized rectangle is divided up into two or more pieces, representing the different parts that can make up that whole. 

Depending on the problem, these can be uneven in size (representing two different numbers – such as in an addition or subtraction) or equal in size. 

A bar might even be broken down into several, separate parts, as might be found in multiplication and division.

Students are given some information but quickly realize they are going to find and fill in the missing pieces.

Comparison strip diagram

Compariston strip diagrams are, much as the name would imply, an intuitive and visual way of approaching problems where there is a difference in amounts and when there is a known/unknown component. 

They can be ideal, for example, when approaching problems that ask the student things like: 

• How much more than
• How many more
• How many fewer
• Larger than
• Smaller than 

Much like Part-Part Whole models, comparison strip diagrams have two bars arranged one on top of the other so that students can immediately grasp the difference in length between the two, with brackets pointing out relevant numbers and totals.

The idea here is that these very visual bars can be an easy starting point for understanding differences in amounts. 

More than that, however, we find that they can make understanding and breaking down comparison word problems a little easier for students, especially given the wide variation in vocabulary and terms that these problems can use (taller than, wider than, bigger than, as many as, as few as, fewer than, greater than, more expensive than, etc.). 

Using strip diagrams for more complex problems

One common misconception is that strip diagrams are only used for basic operations such as addition, subtraction, multiplication and division. 

The reality is they are quite flexible and can be used well into more advanced math and for more complex and tricky word problems that often throw students. 

For example, strip diagrams can be used to illustrate and work out ratio problems, such as below.

How To Set up A Strip Diagram Effectively

It’s important to keep in mind that strip diagrams are a tool and that a good homeschooling parent can’t be too rigid when it comes to its precise use and application. 

It goes without saying that one needs to keep a student’s unique personality, understanding, abilities and skills in mind when doing any kind of problem solving. 

That said, there are some steps that most programs will involve when teaching students to use a strip diagram. These tend to be:

• Read the question carefully and to completion.
• Highlight or circle any important information provided in the question, paying special attention to numbers and important words or phrases (more, together, as many). 

• Who is the problem about/ what is the problem about
• What is known
• What information is missing
• Rereading the key phrase of the question (and paying special attention to any particular phrases – how many, what is the total, who/what is bigger, how much, etc ), decide what operation needs to be used to solve the problem.
• Draw and label the bars using the information collected

• Review the problem to make sure the bars make sense given the information presented
• Complete the calculation using the operation in order to work out the answer

Why Some Curricula Use Strip Diagrams

A systematic approach to confusing word problems.

Younger students often struggle with word problems for a variety of reasons.

For one thing, word problems (unlike computational exercises) involve much more than math.

Rather than being presented with a straight forward math problem, students are suddenly faced with a paragraph of words and numbers. 

From this jumble of words, they have to keep track of a lot of different information, figure out what’s important, determine what the question actually is (since it’s usually cloaked in specific math vocabulary) and, finally, they have to figure out what to do.

Strip diagrams can help by providing students with a ready and systematic method of approaching these problems. 

In order to build a strip diagram, students need to parse out the important information, collecting information on:

• What the problem is about
• What information is given to me

As a student gets more practice in building strip diagrams, they can become more adept at highlighting key information, ignoring irrelevant information, and identifying key math terms as hints for what the operation is supposed to be about. 

In addition, as they get more experience at drawing out these diagrams, they can get better at organizing and presenting that information in a meaningful and more organized way, something that’s helpful for students who have a hard time showing their work or communicating their thoughts succinctly or mathematically. 

Visual approach to help younger students with abstract concepts

Another challenge word problems present is that they often challenge students to reason or think through problems abstractly. 

Typically, below the age of 12 or so , students are far better at seeing the world in concrete terms than they are at understanding abstractions.  

They think logically, sure, and they can understand symbols for the most part, but the way in which they tend to understand things best is by being able to directly observe it. 

A word problem involving two trains speeding along the track between two cities, or age differences between several kids, or different sizes of different items, or different prices can all be very hard for kids to get their heads around. 

Learning to build a strip diagram can help younger students by letting them see and work with a visual (and concrete) representation of the question’s information rather than just relying on the written word. 

A gentler introduction to algebraic thinking

Yet, it’s important to note that despite being visual, strip diagrams still remain somewhat symbolic and standardized, abstracted to a degree from the various topics and characters that pop up in a word problem. 

Students are working with rectangles, after all, and aren’t being asked to literally draw out every problem. 

As a result, they can still act to help kids get used to algebraic thinking, that is the ability to represent, generalize or reason through a problem using abstract math symbols and relationships. 

There is strong evidence for their usefulness

Being something of a different way to tackle word problems than many educators (and parents) are used to, some have questioned whether using strip diagrams in approaching math problems has any real effect on success.

In recent years, studies have been run measuring how well strip diagrams can help students understand and solve math operations and word problems.

The findings from these have been quite positive. Studies have found, for example, using strip diagrams as a tool can be effective in helping elementary students answer word problems correctly, and there is some evidence that it can be an effective tool for remediation.

Pros and Cons of Using Strip Diagrams in your Homeschool Learning

It is important to remember that strip diagrams are just one of many tools that parents and students can use to solve word problems.

It is also important to note that, as a tool, strip diagrams can have their pros and cons and integrating their use into teaching isn’t always the best fit for every student.

Advantages of Strip Diagrams

Easy to learn for both parents and students.

Learning to use strip diagrams and weaving them into a problems solving approach isn’t really all that hard once you get the hang of it. 

By and large, strip diagrams are really just a particular way of sketching out the key details of a question in a few different ways and can be pretty intuitive for both parents and students to learn and use. 

Easy to understand, visual representation of word problem information

As mentioned previously, creating a strip diagram provides students with a pictorial representation that they can use to better understand what they’re being asked to do. 

Seeing the size difference between two or more squares can be a lot easier to deal with for students than a paragraph involving trains, planes, clothing, chocolates, people, heights, sizes and more. 

Encourages students to break apart and analyze word problems systematically

In order to set up a strip diagram properly, students have to learn to read a sentence, identify key elements, identify what is missing and understand certain important key phrases. 

As a result, practicing strip diagrams can help students learn to approach word problems (and lay out their results) in a more organized and logical manner.  

Easy to integrate with manipulatives and hands-on learning objects

Strip diagrams lend themselves quite easily and naturally to hands-on math learning approaches.

Strips can be constructed using blocks, pennies, sticks or whatever else might be handy, turning learning into a multisensory activity that can better suit tactile learners. 

Disadvantages

Not every student needs to use strip diagrams (or wants it).

Although a useful tool, strip diagrams aren’t a tool that every student needs or prefers to use.  

Some students “get” word problems very easily, being able to almost intuitively pick out key information and have no problem thinking about and representing them using equations and symbols. 

Other students are more procedural in nature and don’t much care for the use of visuals and other models. Such students might prefer a more straightforward, by the numbers approach to solving word problems (i.e. memorizing  a series of steps to follow) or may simply respond well to more practice and drill. 

In either case, such students may find being forced to use strip diagrams frustrating. 

In the case of more gifted math students they may even see the process as superfluous and something of a burden on them. 

It can slow things down when solving problems

Strip diagrams do involve identifying and collecting information from a word problem, organizing it and drawing a model that makes some kind of sense in terms of scale. 

Consequently, setting up and working with strip diagrams can take time, especially when students are still getting the hang of the process, which in turn can increase the length of a math lesson or practice session. 

Not applicable to all word problems in math

While they tend to work well for most word problems involving operations and even algebra, creating strip diagrams doesn’t always make sense (or even really work) with all types of math problems. 

Statistics and geometry, for example, can have interesting and challenging word problems associated with them but don’t necessarily lend themselves as subjects to strip diagrams and require their own models and algorithms to follow. 

What Curricula Use Strip Diagrams And Other Easier Math Tools?

Strip diagrams, in particular, are most notably (although not exclusively) a characteristic of programs based on the Singapore approach to math and these programs tend to do a pretty good job at guiding parents and students through their use in a step-by-step manner.  

The Singapore approach to math (so-called as it was originally developed as the main math curriculum for elementary school students in Singapore) is a conceptual math method that focuses on teaching students the why behind math and why it works the way it does, as well as to think critically and to solve math problems without relying excessively on memorization and drill.   

In doing so, and by teaching for mastery, these programs tend to give students a deeper understanding of number relationships, strong math fluency and problem solving skills compared to more traditional programs. 

The Singapore method or approach believes very strongly in what they call the Concrete-Pictorial-Abstract stages of learning, i.e. bringing a student gently from the concrete stage of understanding through a visual modeling stage and finally to a point where the student is comfortable using abstract symbols and equations. 

Alongside number bonds, strip diagrams (which are more commonly referred to as bar models in these programs), tend to fit comfortably in the visual/pictorial stage of learning. 

If this type of math program interests you or you have a student who you think would benefit from approaching math problems more visually, such as with strip diagrams, then we highly recommend the following programs.

Singapore Math 

One of the most famous and widely used examples in the homeschool community of the Singapore approach, Singapore Math offers a thorough and rigorous math curriculum for students in K-8 that emphasizes hands-on learning and visual representation of math problems. 

Offering a couple different editions (Primary Mathematics and Dimensions), Singapore Math’s visually-rich books encourage a deeper, critical thinking approach to math that has produced very strong and capable math students over the years 

Read our in-depth review of Singapore Math 

Read our in-depth review of their latest curriculum Primary Mathematics 2022 Edition

Math in Focus

Released in 2009, produced by Marshall Cavendish and published by Houghton Mifflin Harcourt, Math in Focus is also based on the Singapore approach, teaches math similarly and also offers a rigorous and thorough math curriculum. 

Singapore Math and Math in Focus are similar and the difference largely lies in the fact that Math in Focus is designed a bit more as a traditional school book, touches on a little more content and tends to have a little more built-in differentiation help. 

Read our review of Math in Focus

Bottom Line: 

A popular math model often associated with the Singapore approach, a strip diagram can be an excellent tool to help students more effectively deal with word problems. 

While not always for every student, stripe diagrams can help students systematically analayze a complex word problem, better visualize what they are being asked to do and communicate their results in an organized manner. 

And, for some students at least, that can make all the difference.

About the Author

David Belenky is a freelance writer, former science and math tutor and a tech enthusiast. When he’s not writing about educational tech, he likes to chill out with his family and dog at home.  

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What is a Strip Diagram in Math? A Powerful Tool for Problem Solving

When it comes to solving math problems, having a clear visual representation can make all the difference. That’s where strip diagrams come in. In this

Feni Rutcher

February 25, 2024

When it comes to solving math problems, having a clear visual representation can make all the difference. That’s where strip diagrams come in. In this article, we will explore what strip diagrams are and how they can help students better understand and solve math problems. Whether you’re a student struggling with word problems or a teacher looking for a new teaching tool, strip diagrams could be the key to unlocking mathematical success.

Strip diagrams, also known as bar models or tape diagrams, are visual tools used in math to represent and solve problems. They provide a pictorial representation of the relationships between quantities in a problem, making it easier for students to comprehend complex mathematical concepts. By breaking down problems into smaller, more manageable parts, strip diagrams help students identify the necessary information and develop strategies to solve them.

Table of Contents

Understanding the Basics of Strip Diagrams

Strip diagrams are a valuable tool for students to develop a deeper understanding of math concepts. In this section, we will delve into the fundamental components of strip diagrams. We will explore how to create and interpret strip diagrams, including their different formats and uses.

Creating a Strip Diagram

Creating a strip diagram involves identifying the relevant quantities and representing them visually. Start by determining the main quantities involved in the problem. For example, if the problem involves two quantities, such as the number of apples and the number of oranges, draw two separate strips or bars to represent each quantity.

Next, label each strip with the corresponding quantity and any additional information that may be provided in the problem. For instance, if the problem states that there are 5 apples and 3 oranges, label one strip as “apples” and the other as “oranges.”

Finally, divide each strip into smaller segments to represent the specific values of each quantity. For example, if there are 5 apples, divide the “apples” strip into five equal segments. Similarly, if there are 3 oranges, divide the “oranges” strip into three equal segments.

Interpreting a Strip Diagram

Once you have created a strip diagram, it’s important to understand how to interpret it. Each segment of the strip diagram represents a specific quantity or value. By visually comparing the lengths of the segments, you can determine the relationships between the quantities in the problem.

For example, if the “apples” strip has three segments filled and the “oranges” strip has two segments filled, you can immediately see that there are more apples than oranges. This visual representation allows you to comprehend the relative quantities without relying solely on numerical values.

Applying Strip Diagrams to Addition and Subtraction Problems

Addition and subtraction problems can sometimes be challenging for students, especially when dealing with multiple quantities. In this section, we will showcase how strip diagrams can simplify these problems by visually representing the relationships between numbers. We will provide step-by-step examples to illustrate how to use strip diagrams effectively in solving addition and subtraction problems.

Using Strip Diagrams for Addition

Strip diagrams are particularly helpful when solving addition problems that involve two or more quantities. By visually representing the quantities involved, students can better understand the relationships between the numbers and the overall problem.

Let’s consider an example: John has 4 red marbles and 3 blue marbles. To find the total number of marbles, you can create a strip diagram with two sections, one for red marbles and one for blue marbles. Each section is divided into smaller segments to represent the specific quantities.

In this case, the “red marbles” section would have four filled segments, while the “blue marbles” section would have three filled segments. By counting the total number of filled segments, students can determine that John has a total of seven marbles.

Utilizing Strip Diagrams for Subtraction

Strip diagrams are equally useful when solving subtraction problems. They enable students to visualize the quantities involved and easily identify the relationships between numbers.

Let’s take an example: Sarah has 8 cookies, and she gives away 3 cookies to her friend. To determine how many cookies Sarah has left, you can create a strip diagram with two sections, one for the initial number of cookies and one for the cookies she gives away.

The “initial number of cookies” section would have eight filled segments, while the “cookies given away” section would have three filled segments. By subtracting the number of filled segments representing the cookies given away from the total number of filled segments, students can determine that Sarah has five cookies remaining.

Solving Multiplication and Division Problems with Strip Diagrams

Strip diagrams are not limited to addition and subtraction; they are equally powerful when applied to multiplication and division problems. This section will demonstrate how strip diagrams can help students better understand the concepts of multiplication and division. Through practical examples, we will guide you on how to create strip diagrams for these types of problems.

Using Strip Diagrams for Multiplication

Multiplication problems often involve finding the product of two or more quantities. Strip diagrams can provide a visual representation of the relationships between these quantities, making it easier for students to grasp the concept.

Let’s consider an example: Mary has 4 bags, and each bag contains 5 apples. To find the total number of apples, you can create a strip diagram with four sections, each representing a bag, and divide each section into smaller segments to represent the apples.

In this case, each “bag” section would have four filled segments, and each “apples” segment within each bag section would have five filled segments. By counting the total number of filled segments, students can determine that there are 20 apples in total.

Utilizing Strip Diagrams for Division

Division problems involve dividing a quantity into equal parts. Strip diagrams can help students visualize the division process and understand the relationships between the quantities involved.

Let’s take an example: There are 15 candies, and they need to be distributed equally among 3 friends. To represent this situation using a strip diagram, create a section for the total number of candies and divide it into smaller segments to represent the equal parts.

In this case, the “candies” section would have 15 filled segments, and by dividing these segments equally among the three friends, each friend would have five filled segments. By visually comparing the lengths of the segments, students can determine the equal distribution of candies.

Advanced Problem-Solving Techniques with Strip Diagrams

Once students have mastered the basics, strip diagrams can be taken to the next level to solve more complex math problems. In this section, we will explore advanced problem-solving techniques, such as ratio and proportion, using strip diagrams. By the end, you will have a comprehensive understanding of how strip diagrams can be applied to a wide range of mathematical scenarios.

Using Strip Diagrams for Ratios

Ratios compare the quantities of two or more objects or values. Strip diagrams can provide a visual representation of these ratios, making it easier for students to understand and solve problems involving ratios.

Consider an example: A recipe calls for a ratio of 2 cups of flour to 3 cups of sugar. To represent this ratio using a strip diagram, create two sections, one for flour and one for sugar. Divide each section into smaller segments to represent the cups.

In this case, the “flour” section would have two filled segments, and the “sugar” section would have three filled segments. By visually comparing the lengths of the segments, students can see the ratio of 2:3 between flour and sugar.

Utilizing Strip Diagrams for Proportions

Proportions involve comparing two ratios and determining if they are equivalent. Strip diagrams can help students visualize these proportions and better understand the relationships between the quantities involved.

Let’s take an example: A recipe calls for a ratio of 3 cups of flour to 4 cups of milk. Another recipe calls for a ratio of 6 cups of flour to 8 cups of milk. To determine if these ratios are equivalent, create strip diagrams for each ratio, with sections for flour and milk divided into smaller segments to represent the cups.

In the first strip diagram, the “flour” section would have three filled segments, and the “milk” section would have four filled segments. In the second strip diagram, the “flour” section would have six filled segments, and the “milk” section would have eight filled segments. By visually comparing the lengths of the segments in both strip diagrams, students can determine if the ratios are equivalent.

Strip Diagrams in Real-Life Situations

In this section, we will showcase how strip diagrams can be used to solve real-life problems. From calculating expenses to determining the best deal, strip diagrams provide a practical approach to applying math concepts in everyday situations. By incorporating real-life examples, we aim to demonstrate the versatility and relevance of strip diagrams beyond the classroom.

Using Strip Diagrams for Shopping

Strip diagrams can be helpful when comparing prices or determining the best deal while shopping. Let’s consider an example: You are at the supermarket and want to buy a box of cereal. There are two brands available, one priced at $4 for 250grams and another priced at $6 for 400 grams. To determine which brand offers a better value, create strip diagrams for each brand, with sections representing the price and the quantity of cereal.

In the strip diagram for the first brand, the “price” section would have four filled segments, and the “quantity” section would have 250 filled segments. In the strip diagram for the second brand, the “price” section would have six filled segments, and the “quantity” section would have 400 filled segments.

By visually comparing the lengths of the segments, students can determine that the second brand offers a better value for money, as the “quantity” section is longer for the same price compared to the first brand. Strip diagrams allow individuals to make informed decisions based on visual representations of quantities and prices.

Using Strip Diagrams for Budgeting

Strip diagrams can also be applied to budgeting situations. For example, let’s say you have a monthly income of $200 and need to allocate it among different expenses. Create a strip diagram with sections representing different expenses, such as rent, groceries, transportation, and entertainment.

Divide each section into smaller segments to represent the amount of money allocated to each expense. For instance, if you allocate $80 for rent, create four filled segments in the “rent” section. Repeat this process for the other expenses, allocating an appropriate amount of money to each category.

By visually comparing the lengths of the segments, you can easily see how your income is distributed among different expenses. Strip diagrams provide a clear representation of budget allocations, making it easier to manage finances and make adjustments if necessary.

Strip Diagrams as a Teaching Tool

Strip diagrams not only benefit students but also serve as a valuable teaching tool for educators. This section will explore how teachers can incorporate strip diagrams into their lesson plans to enhance students’ understanding of math. We will provide tips and strategies for effectively incorporating strip diagrams into the classroom, ensuring an engaging and interactive learning experience for students.

Incorporating Strip Diagrams into Math Lessons

Teachers can introduce strip diagrams to students as a problem-solving strategy for various math concepts. For instance, when teaching addition and subtraction, teachers can demonstrate how strip diagrams can help students visualize the relationships between quantities and solve word problems more effectively.

When introducing strip diagrams, teachers can start with simple examples and gradually increase the complexity as students become more comfortable with the concept. They can provide guided practice exercises and encourage students to work collaboratively, discussing their strategies and findings.

Teachers can also incorporate technology, such as interactive whiteboards or online tools, to create and manipulate strip diagrams. This can enhance student engagement and provide opportunities for interactive learning experiences.

Using Strip Diagrams for Cross-Curricular Connections

Strip diagrams can be used beyond the realm of mathematics and integrated into other subjects as well. For example, in science class, students can use strip diagrams to represent the relationships between different variables in experiments or data analysis.

In language arts, strip diagrams can be utilized to analyze and interpret information in graphs, charts, and other visual representations of data. Students can create strip diagrams to better understand the relationships between different elements in a story or a piece of literature.

By incorporating strip diagrams into various subjects, teachers can reinforce the concept and provide students with opportunities to apply their problem-solving skills in different contexts, fostering a holistic understanding of the tool.

Advantages and Limitations of Strip Diagrams

While strip diagrams offer numerous benefits, they also have their limitations. In this section, we will discuss the advantages and limitations of strip diagrams. By understanding their strengths and weaknesses, educators and students can make informed decisions about when and how to best utilize strip diagrams in their math problem-solving endeavors.

Advantages of Strip Diagrams

Strip diagrams offer several advantages in math problem-solving. They provide a visual representation that helps students grasp complex concepts and relationships between quantities. Strip diagrams break down problems into smaller, more manageable parts, making it easier for students to identify the necessary information and develop problem-solving strategies.

Additionally, strip diagrams can be used across different math topics and grade levels, making them a versatile tool for educators. They promote critical thinking and logical reasoning skills, as students need to analyze and interpret the visual representation of the problem.

Furthermore, strip diagrams can be particularly beneficial for students who struggle with language barriers or have difficulty understanding word problems. The visual representation allows these students to focus on the mathematical relationships rather than getting overwhelmed by the language used in the problem.

Limitations of Strip Diagrams

While strip diagrams are a valuable tool, they do have some limitations. One limitation is that strip diagrams may not be suitable for every math problem. They are most effective for problems that involve quantities or relationships that can be visually represented.

Strip diagrams may also be limited in their ability to represent more abstract math concepts or problems that require complex mathematical reasoning. In such cases, alternative problem-solving strategies may be more appropriate.

Another limitation is that strip diagrams may not provide a complete solution to a math problem. They are a valuable tool in the problem-solving process, but students must also be able to translate the visual representation into mathematical equations or calculations to arrive at a final answer.

Strip Diagrams and Common Core State Standards

Strip diagrams have gained popularity due to their alignment with the Common Core State Standards in math education. This section will explore how strip diagrams support and reinforce the concepts and skills outlined in the Common Core State Standards. By incorporating strip diagrams into instruction, students can meet the rigorous expectations set by these standards.

Alignment with Mathematical Practice Standards

Strip diagrams align with several Mathematical Practice Standards outlined in the Common Core State Standards. For example, strip diagrams promote the standard of “Model with Mathematics” by providing a visual representation that helps students make sense of problems and persevere in solving them.

Strip diagrams also support the standard of “Reason Abstractly and Quantitatively” by helping students understand the relationships between quantities and translate them into mathematical reasoning. They encourage students to use precise mathematical language when discussing and interpreting the strip diagram.

Furthermore, strip diagrams align with the standard of “Attend to Precision” by requiring students to carefully consider the lengths and proportions of the segments in the diagram. Students must attend to the details and accurately represent the given information in the strip diagram.

Reinforcing Content Standards

Strip diagrams reinforce several content standards in mathematics education. For example, they align with the standards of Operations and Algebraic Thinking, as they help students visualize and solve problems involving addition, subtraction, multiplication, and division.

Strip diagrams also support the standards of Ratios and Proportional Relationships by providing a visual representation of ratios and proportions. They help students understand the relationships between quantities and solve problems involving ratios, rates, and percentages.

Additionally, strip diagrams align with the standards of Measurement and Data, as they assist students in visualizing and interpreting data representations, such as graphs and charts. They help students analyze and compare different quantities in a meaningful way.

Enhancing Math Skills with Strip Diagrams: Practice Exercises

Practice makes perfect, and this section offers a collection of practice exercises to enhance math skills using strip diagrams. These interactive exercises will provide students with an opportunity to apply their knowledge and problem-solving techniques to various scenarios. By practicing with strip diagrams, students can build confidence and improve their ability to solve math problems effectively.

Practice Exercise 1: Addition and Subtraction

In this exercise, students will be presented with word problems that require them to use strip diagrams to solve addition and subtraction problems. They will create strip diagrams and use them to determine the total or difference between quantities.

Example: John has 5 baseball cards. He gives away 2 cards to his friend. How many cards does John have now?

Students can create a strip diagram with segments to represent the initial number of cards and the cards given away. By visually comparing the lengths of the segments, students can determine the answer.

Practice Exercise 2: Multiplication and Division

This exercise focuses on multiplication and division problems that can be solved using strip diagrams. Students will create strip diagrams to represent the quantities involved and use them to solve the problems.

Example: A box contains 6 bags of candies. Each bag has 8 candies. How many candies are there in total?

Students can create a strip diagram with sections for the bags and the candies. By visually comparing the lengths of the segments, students can determine the answer.

Practice Exercise 3: Ratios and Proportions

In this exercise, students will practice solving problems involving ratios and proportions using strip diagrams. They will create strip diagrams to represent the given ratios and use them to determine equivalent ratios or solve proportion problems.

Example: The ratio of boys to girls in a classroom is 3:5. If there are 24 boys in the classroom, how many girls are there?

Students can create a strip diagram with sections for boys and girls. By visually comparing the lengths of the segments and using the given information, students can determine the answer.

Strip diagrams are a powerful tool in math education, providing students with a visual representation thatenhances their problem-solving skills. By breaking down complex problems into manageable parts, strip diagrams enable students to see the relationships between quantities and develop effective strategies for solving them. Whether used as a teaching tool or a problem-solving technique, strip diagrams offer a versatile approach to tackling math problems.

Throughout this article, we have explored the basics of strip diagrams, including how to create and interpret them. We have seen how strip diagrams can be applied to addition and subtraction problems, as well as multiplication and division problems. Strip diagrams have proven to be effective in simplifying word problems and helping students visualize the relationships between numbers.

Furthermore, we have delved into advanced problem-solving techniques with strip diagrams, such as ratios and proportions. Strip diagrams provide a visual representation that aids in understanding these concepts and solving problems that involve them. By incorporating strip diagrams into real-life situations, such as shopping or budgeting, students can apply their math skills to practical scenarios.

Strip diagrams also serve as a valuable teaching tool for educators. By incorporating strip diagrams into math lessons, teachers can enhance students’ understanding of mathematical concepts and promote critical thinking skills. Strip diagrams can be used in various subjects, making cross-curricular connections and reinforcing the Common Core State Standards.

While strip diagrams offer numerous advantages, it is important to acknowledge their limitations. Strip diagrams may not be suitable for every math problem and may not provide a complete solution on their own. Students must be able to translate the visual representation into mathematical equations or calculations to arrive at a final answer.

In conclusion, strip diagrams are a powerful tool in math education that can greatly enhance students’ problem-solving skills. By providing a visual representation of quantities and relationships, strip diagrams enable students to comprehend complex math concepts and develop effective strategies for solving problems. Whether used as a teaching tool or a problem-solving technique, strip diagrams offer a versatile approach to tackling math problems and promoting a deeper understanding of mathematical concepts. By incorporating strip diagrams into instruction and practice exercises, students can build confidence and excel in their math skills. So, embrace the power of strip diagrams and unlock the potential of math problem-solving.

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Tape Diagrams in Math: A Problem Solving Strategy

What is your go-to strategy for teaching word problems? I have always encouraged my students to model, model, model when solving word problems. Modeling is a way for students to make sense of a problem and catch mistakes before they happen. Sometimes, what seems like a good answer just doesn't work out once it's drawn as a model.

Until recently, I really didn't care what kind of model my kids used. Drawings, tally marks, symbols: anything was fair game. But as we've worked our way through multiplication and into division, I've found myself returning over and over again to the tape diagram. It's a nearly fail-proof way to work through a story problem and my students have become problem solving masters!

So, what exactly is a tape diagram?

A tape diagram is a drawing that looks like a segment of tape, used to illustrate number relationships. Also known as a strip diagram, bar model, fraction strip, or length model."

In practice, a tape diagram in math looks like this...

Tape diagrams have become my go-to model when teaching math. They work for any type of word problem, but I especially love them for solving multiplication and division.

how do you solve a tape diagram?

To solve problems using tape diagrams, students must first ask, "Do I know the whole amount?" If it's supplied by the problem, fill it in. If not, put a question mark at the bottom of the diagram. Then look for other information and fill that in. The question mark always represents whatever piece of information is missing.

Here is an example of how I use it to teach division problems:

Students will quickly begin to see a pattern... the bottom number is always the product of the top two numbers.

Therein lies the beauty of this model:

It is self-correcting. If the top two numbers, when multiplied, do not equal the bottom, you've done something wrong.

When teaching with this model, it's a good idea to have students write all four possible equations. So in the above example, we would also write ? x 4 = 32 and 32 ÷ ? = 4. This is important because it reinforces the concept of inverse operations and fact families. It also gives students a tool to use when the encounter problems with a missing factor or divisor.

Once my students are able to interpret story problems and solve using tape diagrams, I teach them to analyze a tape diagram and write their own story problems to go with it. This requires high-order thinking and really develops their mathematical minds.

how to use tape diagrams

To use tape diagrams with your students, I suggest following this instructional sequence...

Teacher supplies the story problem, draws the tape diagram, and models how to solve.

Teacher supplies the story problem, draws the tape diagram, and students help solve.

Teacher supplies the story problem, students help draw the diagram, and students solve on their own.

Teacher supplies the story problem, students draw and solve alone.

Teacher draws a tape diagram and students create a story problem to go with it.

Along the way, you'll want to move from including one equation that represents the problem to showing all four possible equations. This process, from introduction to proficiency, might take several weeks depending on the skills of your students. But it will pay off in the long run when your students become experts at solving word problems !

If you're interested in teaching your students to use this strategy, you may want to check out this resource:

Problem Solving with Tape Diagrams: A Model for Multiplication and Division

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Strip Diagram Questions

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20 word problems that can be solved using strip diagrams.

• 20 mixed topic word problems with a strip diagram template.
• Word problems include questions on number and operations in base ten, measurement and data and operations and algebraic thinking.
• Answer key includes completed strip diagrams.

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Comparison Problems with Tape Diagrams/Strip Diagrams

Comparison problems: using tape diagrams to represent math thinking.

Word problems are a key part of rigorous standards everywhere, but teachers and students alike often dread tackling them.  Even textbooks sometimes save them for the bottom few problems on a page or a separate lesson toward the end of a chapter.  The thing is–word problems (at least good ones!) are the “real world” part of math.  One type of word problems,  comparison problems, can be particularly challenging for students.  Let’s look at some teaching tips that might make them more accessible!

Critical Reading of Math Problems

As educators, we should always be striving to help our students understand that the skills we are teaching are them are FOREVER…not just to complete a math page or worksheet.  One skill that we really want to make sure our students understand is the need to critically read math problems to figure out what is being asked, what information is given, and to make a plan for solving.  So often we do the thinking and hard work for our students.

For example, just consider how many of our math books are organized.  A lesson entitled “Solving Addition Stories” doesn’t leave much room for student thinking, does it?  It seems pretty clear what operation students will need to choose! Providing students with a constantly spiraling variety of problems forces them to think for themselves, learn to look for key information in problems, and make solution decisions accordingly.

One idea?  Use highlighters to find and notate important information.  Underline or circle the question.  One thing I do NOT recommend? Looking for key words like “fewer” or “total”.  These words may seem like a quick fix for students…but they can lead them down the wrong path.  How?  How about this problem…

“Larry has 14 baseball cards.  This is 25 fewer than his sister Kara has.  How many cards does Kara have?” 

Visualization and Modeling with Comparison Problems

One strategy that can really help students make sense of problems is to be able to visualize and draw models of different problem types.  Comparison problems–sometimes represented with “tape” or “strip” diagrams are a GREAT way to help students visualize the math!  I thought I’d share a few ways that these can be super helpful for students–whether used as whole class lessons or for intervention groups.

Using a blank “template” or outline of a tape diagram with manipulatives can really help students see the comparison that appears in a problem.  The photo below showcases a visual representation of the following problem:

Next Steps with Comparison Problems

Another strategy to get students really visualizing is to take a blank diagram and create different stories to go along with them.  This is a fantastic way to help students get flexible with their thinking.  I start with modeling some problems and then have students progress to writing and sharing their own.  It truly helps them understand the problems at a deeper level.

Hope you found this information valuable!  These problems don’t HAVE to be as challenging if we help students bridge the gap between the problem and a meaningful strategy to solve them!

Looking to try some of these on your own?

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Using Strip Models to Explain Percent

A Strip Model is a rectangular model used to show numerical relationships. It is also used to display fractions or solve equations involving operations.

To explain percent, it is better to write the percentage as a fraction. Then, simplify and then signify it using a strip model.

Step-by-Step Guide Using Strip Models to Explain Percent

A strip model is a visual tool that can be used to represent and explain percent. Here’s a step-by-step guide to using strip models to explain percent:

• Understand the problem: Read the problem carefully and identify the information provided and the information that needs to be found.
• Draw a strip model: Draw a strip divided into equal parts, with each part representing a certain percentage. The strip can be divided into 100 equal parts, with each part representing 1%.
• Identify the whole: Identify the total amount or quantity that the percent represents.
• Shade in the appropriate number of parts: Shade in the number of parts that represent the percent. For example, if the problem states that an item is 20% off, shade in 20 parts of the strip.
• Label the shaded parts: Label the shaded parts with the percent or fraction that they represent.
• Use the strip model to find the part or whole: Use the strip model to find the part or whole by counting the number of shaded parts or the number of unshaded parts and multiplying by the whole.
• Write the final answer in a complete sentence: Use the information from the strip model to express the final answer.

Using Strip Models to Explain Percent – Examples 1

Problem: A shirt is on sale for 30% off. The original price of the shirt is $60. What is the sale price of the shirt?

• We are given that the shirt is on sale for 30% off and the original price is $60.
• Draw a strip divided into 100 equal parts.
• Identify the whole: The original price is $60.
• Shade in 30 parts of the strip
• Label the shaded parts: 30%
• Use the strip model to find the part or whole: 30% of $60 is $18
• The sale price of the shirt is $60 – $18 = $42.

It’s worth noting that this is just one way to use strip models to explain percent, and you can use different models depending on the information provided in the problem.

Using Strip Models to Explain Percent – Examples 2

Using a strip model to show 20%. Solution: Step 1: Write 20% as a fraction. It is \(\frac{20}{100}\). It means 20 parts out of 100. Step 2: Simplify the fraction. \(\frac{20}{100}=\frac{1}{5}\) Step 3: Now represent 1 part out of 5 total parts.

Using Strip Models to Explain Percent – Examples 3

Using a strip model to show 40%. Solution: Step 1: Write 40% as a fraction. It is \(\frac{40}{100}\). It means 40 parts out of 100. Step 2: Simplify the fraction. \(\frac{40}{100}=\frac{2}{5}\) Step 3: Now represent 2 parts out of 5 total parts.

by: Effortless Math Team about 1 year ago (category: Articles )

Effortless Math Team

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Science News

An enduring möbius strip mystery has finally been solved.

A mathematician’s journey to prove how short the loops can be had some twists and turns

A Möbius strip (one shown) is a loop of paper with a half-twist in it. A mathematician has now proven the shortest possible Möbius strip for a given width.

MirageC/Moment/Getty Images Plus

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By Emily Conover

October 10, 2023 at 8:00 am

Any attempt to better understand Möbius strips is bound to run into some kinks.

The twisted loops are so strange that mathematicians have struggled to answer some basic questions about them. For example: “What’s the shortest Möbius strip you can make for a paper band of a given width?” 

The question hooked mathematician Richard Evan Schwartz. A mistake in a computer program almost prevented him from finding the answer. Simply messing around with strips of paper finally helped him solve the mystery.

A Möbius strip is a mathematical oddity that anyone can make. Cut a strip of paper, twist one end halfway around, and tape the two ends together to form a loop with a twist in it. The result is a one-sided surface. The strips have inspired mathematicians, artists and scientists in a variety of fields ( SN: 5/27/22 ).

A long, skinny Möbius strip is easier to make than a stumpy one. With a very short strip, the paper has to contort so much that it flattens into an equilateral triangle ( SN: 7/24/07 ). (You can see this shape form if you slowly pull one end of an untaped Möbius strip to shorten it.) The triangular Möbius strip is made from a piece of paper that has a length that’s √3, or about 1.73, times its width.

In 1977, mathematicians hypothesized that the triangular Möbius strip was as short as you can go. Specifically, it’s the limit for an idealized mathematical version of paper that is infinitely thin, smooth and nonstretchy, and which, like real-world paper, can’t pass through itself. But in the nearly 50 years that followed, no one had been able to prove it. Mathematicians could show only that the ratio between a Möbius strip’s length and width must be greater than π/2, or about 1.57.

The stumper piqued Schwartz’s interest. He is fond of simple problems that befuddle mathematicians. “I like it when no one has any idea what to do,” he says. A bonus: “If I bomb out on this, there’s no shame in it. I’m just like everybody else.”

Schwartz focused on a key property of Möbius strips: While the paper curves this way and that to form the loop, at every point on the band there’s a direction in which the paper follows a straight line from edge to edge, with no curvature at all. (That’s not true of all surfaces. Think of a bowl: There are no straight lines to be found.) He realized that, in any Möbius strip, there must always be two such lines that are perpendicular and in the same plane, as in the letter T.

Based on how the paper contorts to form this T shape, Schwartz found a new minimum length-to-width ratio. To his disappointment, it was not √3 but a number achingly close to it , about 1.69, he reported in Geometriae Dedicata in 2021.

Schwartz moved on to other topics but couldn’t stop thinking about the problem. One day, on a whim, he began playing with strips of paper. In a head-smacking jolt, he realized he’d made an error.

Schwartz had assumed that slicing open a Möbius strip along a diagonal and flattening it forms a parallelogram. But when Schwartz cut open one of his paper Möbius strips, he saw in front of him not a parallelogram, but a trapezoid. “I instantly said, ‘uh oh,’” he says.

It was a simple mistake. But Schwartz had been investigating Möbius strips primarily on the computer. He’d flubbed the setup of his computer program, which led to the parallelogram whoopsie. “Once I’d made the mistake,” he says, “it’s like it got locked into my brain.”

Schwartz says he hardly ever used paper Möbius strips in his research. But that’s what it took to jolt him out of his stagnant thought pattern. It’s a bit curious that Schwartz didn’t turn to paper earlier. He fiddles with paper as a hobby, designing elaborate masks of dangling paper.

Once Schwartz redid the calculation with the trapezoid fix, √3 popped out. He’d finally proved , that the length of a Möbius strip must be greater than √3 times its width, Schwartz reported August 24 at arXiv.org. The triangular Möbius strip is truly the limit for paper Möbius strips.

Now, Schwartz is interested in taking this work further. What, he wonders, is the minimum length for a loop with two twists, or three twists, instead of one? This time, perhaps, he’ll spend more time playing with paper.

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Updated:  31 May 2023

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What is a Strip Diagram? Tape Diagram? Bar Model?

Are your students struggling with problem-solving? Help them get back on track with the help of a strip diagram. The strip diagram is a handy visual model that can help your students solve problems with known and unknown quantities. These diagrams go by a few names, such as bar models, tape diagrams, fraction strips, length models, or part-part-whole models. But no matter what you call them, they’re a great tool to put in your students’ problem-solving toolboxes.

Printable Problem-Solving Anchor Charts

Use this resource when teaching your students how to problem solve using strips diagrams (a.k.a. bar models). This method is especially useful for those students who respond better to visual stimuli such as pictures, drawings, or  graphic organizers.

The pack illustrates the five basic variations and includes examples of how to apply them to different operations.

Print off a pack for each student to keep in their math folder to reference throughout the year. Or as an alternative, project each poster on the board as you teach each variation.

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September 12, 2023

Mathematician Solves 50-Year-Old Möbius Strip Puzzle

A trapezoid was key to discovering the elusive answer to a riddle about Möbius strips

By Rachel Crowell

Dmitri Otis/Getty Images

Möbius strips are curious mathematical objects. To construct one of these single-sided surfaces, take a strip of paper, twist it once and then tape the ends together. Making one of these beauties is so simple that even young children can do it, yet the shapes’ properties are complex enough to capture mathematicians’ enduring interest.  

The 1858 discovery of Möbius bands is credited to two German mathematicians—August Ferdinand Möbius and Johann Benedict Listing—though evidence suggests that mathematical giant Carl Friedrich Gauss was also aware of the shapes at this time, says Moira Chas, a mathematician at Stony Brook University. Regardless of who first thought about them, until recently, researchers were stumped by one seemingly easy question about Möbius bands : What is the shortest strip of paper needed to make one? Specifically, this problem was unsolved for smooth Möbius strips that are “embedded” instead of “immersed,” meaning they “don't interpenetrate themselves,” or self-intersect, says Richard Evan Schwartz, a mathematician at Brown University. Imagine that “the Möbius strip was actually a hologram, a kind of ghostly graphical projection into three-dimensional space,” Schwartz says. For an immersed Möbius band, “several sheets of the thing could overlap with each other, sort of like a ghost walking through a wall,” but for an embedded band, “there are no overlaps like this.”

In 1977 mathematicians Charles Sidney Weaver and Benjamin Rigler Halpern posed this question about the minimum size and noted that “their problem becomes easy if you allow the Möbius band you are making to have self-intersections,” says Dmitry Fuchs, a mathematician at the University of California, Davis. The remaining question, he adds, “was to determine, informally speaking, how much room you need to avoid self-intersections.” Halpern and Weaver proposed a minimum size, but they couldn’t prove this idea, called the Halpern-Weaver conjecture.

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Schwartz first learned about the problem about four years ago, when Sergei Tabachnikov, a mathematician at Pennsylvania State University, mentioned it to him, and Schwartz read a chapter on the subject in a book Tabachnikov and Fuchs had written . “I read the chapter, and I was hooked,” he says. Now his interest has paid off with a solution to the problem at last. In a preprint paper posted on arXiv.org on August 24, Schwartz proved the Halpern-Weaver conjecture. He showed that embedded Möbius strips made out of paper can only be constructed with an aspect ratio greater than √3, which is about 1.73. For instance, if the strip is one centimeter wide, it must be longer than √3 cm.

Solving the quandary required mathematical creativity. When one uses a standard approach to this type of problem, “it is always difficult to distinguish, by means of formulas, between self-intersecting and non-self-intersecting surfaces,” Fuchs says. “To overcome this difficulty, you need to have [Schwartz’s] geometric vision. But it is so rare!” 

In Schwartz’s proof, “Rich managed to dissect the problem into manageable pieces, each of which essentially necessitated only basic geometry to be solved,” says Max Wardetzky, a mathematician at the University of Göttingen in Germany. “This approach to proofs embodies one of the purest forms of elegance and beauty.” 

Before arriving at the successful strategy, however, Schwartz tried other tactics on and off again over a few years. He recently decided to revisit the problem because of a nagging sensation that the approach he had used in a 2021 paper should have worked.

In a way, his gut feeling was correct. When he resumed investigating the problem, he noticed a mistake in a “lemma”—an intermediate result—involving a “T-pattern” in his previous paper. By correcting the error, Schwartz quickly and easily proved the Halpern-Weaver conjecture. If not for that mistake, “I would have solved this thing three years ago!” Schwartz says. 

In Schwartz’s solution to the Halpern-Weaver conjecture, the T-pattern lemma is a critical component. The lemma begins with one basic idea: “Möbius bands, they have these straight lines on them. They’re [what are] called ‘ruled surfaces,’” he says. (Other paper objects share this property. “Whenever you have paper in space, even if it’s in some complicated position, still, at every point, there’s a straight line through it,” Schwartz notes.) You can imagine drawing these straight lines so that they cut across the Möbius band and hit the boundary at either end. 

In his earlier work, Schwartz identified two straight lines that are perpendicular to each other and also in the same plane, forming a T-pattern on every Möbius strip. “It is not at all obvious that these things exist,” Schwartz says. Showing that they do was the first part of proving the lemma, however.

The next step was to set up and solve an optimization problem that entailed slicing open a Möbius band at an angle (rather than perpendicular to the boundary) along a line segment that stretched across the width of the band and considering the resulting shape. For this step, in Schwartz’s 2021 paper, he incorrectly concluded that this shape was a parallelogram. It’s actually a trapezoid.

This summer, Schwartz decided to try a different tactic. He started experimenting with squishing paper Möbius bands flat. He thought, “Maybe if I can show that you can press them into the plane, I can simplify it to an easier problem where you’re just thinking of planar objects.” 

During those experiments, Schwartz cut open a Möbius band and realized, “Oh, my God, it’s not the parallelogram. It’s a trapezoid.” Discovering his mistake, Schwartz was first annoyed (“I hate making mistakes,” he says) but then driven to use the new information to rerun other calculations. “The corrected calculation gave me the number that was the conjecture,” he says. “I was gobsmacked.... I spent, like, the next three days hardly sleeping, just writing this thing up.” 

Finally, the 50-year-old question was answered. “It takes courage to try to solve a problem that remained open for a long time,” Tabachnikov says. “It is characteristic of Richard Schwartz’s approach to mathematics: He likes attacking problems that are relatively easy to state and that are known to be hard. And typically he sees new aspects of these problems that the previous researchers didn’t notice.” 

“I see math as a joint work of humanity,” Chas says. “I wish we could tell Möbius, Listing and Gauss, ‘You started, and now look at this....’ Maybe in some mathematical sky, they are there, looking at us and thinking, ‘Oh, gosh!’” 

As for related questions, mathematicians already know that there isn’t a limit on how long embedded Möbius strips can be (although physically constructing them would become cumbersome at some point). No one, however, knows how short a strip of paper can be if it’s going to be used to make a Möbius band with three twists in it instead of one, Schwartz notes. More generally, “one can ask about the optimal sizes of Möbius bands that make an odd number of twists,” Tabachnikov says. “I expect someone to solve this more general problem in the near future.”

Editor’s Note (9/14/23): This article was edited after posting to correct the descriptions of what a Möbius strip’s length is when it is wider than √3 centimeters and how the two lines that Richard Evan Schwartz identified form a T-pattern on every strip.

Mathematicians Solve A Key Möbius Strip Problem, After Almost 50 Years of Searching

For nearly fifty years, mathematicians have puzzled over a deceptively simple question: how small can you make a Möbius strip without it intersecting itself?

Now, Richard Schwartz, a mathematician at Brown University, has proposed an elegant solution to this problem, which was originally posed by mathematicians Charles Weaver and Benjamin Halpern in 1977.

In their paper, Halpern and Weaver pose a limit for Möbius strips based on the familiar geometry of folded bits of solid paper – that the ratio between the length and width of the paper must be greater than √3, or around 1.73.

For example, a Möbius strip one centimeter in length would need to be wider than √3 or 1.73 centimeters.

Schwartz says he became "hooked" on the Möbius strip problem after learning about it four years ago during a conversation with a colleague.

He had several stabs at solving it over the years and published a paper in 2021 with a promising approach that ultimately fell short.

Schwartz couldn't leave the problem alone and recently started experimenting with squashing paper Möbius strips in the hope that the 2D shape would be easier to tackle mathematically.

But when he cut open one of these loops at an angle (which was necessary to solve his optimization problem), he saw something he did not expect.

The 2D length of paper didn't look like a parallelogram, as he had reported in his first paper. Rather, it was a trapezoid – a shape with four, straight sides, where only two of the sides are parallel to one another .

"Embarrassingly, I discovered recently that I made an error in setting up the optimization problem," Schwartz writes .

Over three sleepless nights – and with some help from a few colleagues – Schwartz corrected his error and found "a really nice proof" for the intermediate step "that greatly simplified" the paper.

"I was amazed and delighted to discover that when I did the optimization problem correctly, I got … √3 right on the nose!" he writes .

Möbius strips have many strange properties, which has made them objects of fascination since they were described in 1858 by German mathematicians August Möbius and Johann Listing.

Möbius strips are unorientable . This means that an ant wandering around a Möbius strip is never truly on the 'inside' or 'outside' or 'top' or 'bottom' of the shape.

On their travels, the ants cover both sides of the ribbon in one continuous movement.

This ability to use both sides of a surface without the need to flip the ribbon has made Möbius strips useful for tape recorders, typewriters, conveyor belts, print cartridges and roller coasters.

Möbius strips are used in jewelry, the international symbol for recycling and in the Google drive logo because they are never-ending loops .

This paper was made available as a preprint through arXiv .

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Will Biden Withdraw?

Democratic worries about the president’s age have surged after thursday’s debate..

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 Natalie Kitroeff. This is “The Daily.”

[MUSIC PLAYING]

President Joe Biden’s disastrous debate performance last week set off a furious discussion among Democratic officials, donors, and strategists about whether and how to replace him as their party’s nominee. Today, chief White House correspondent Peter Baker takes us inside those discussions and Biden’s effort to shut that conversation down.

It’s Monday, July 1.

Peter, you’ve been reporting on what I think can be best described as the great Democratic freakout that started basically from the moment the debate began at 9:00 PM on Thursday night. Tell us about the aftermath.

Yeah, I’ve been covering politics for 38 years, and I’ve never seen a political panic like we saw after that debate. It was like a run on the bank. Everybody in the Democratic Party was suddenly confronted with what they didn’t want to admit up until then, which is that they have an 81-year-old candidate who would be 86 at the end of his second term. And it’s very possible that he was not capable of completing this campaign in a vigorous and competitive way against Donald Trump. That’s what really it comes down to for many Democrats. Can Joe Biden take the campaign to Donald Trump and stop what they think is an existential threat to the country?

I want to know more about who you were hearing from. Who are the people that are calling you? What are the big questions they’re asking? What are they struggling with?

Yeah, I don’t want to get into too many names. A lot of people don’t want to be out front. But you did see even publicly, people like Senator Claire McCaskill.

Joe Biden had one thing he had to do tonight and he didn’t do it.

The former Senator from Missouri, red state Democrat, was on MSNBC just minutes after the debate.

He had one thing he had to accomplish, and that was reassure America that he was up to the job at his age. And he failed at that tonight.

She talked about this was a crisis, that her phone was blowing up with a lot of Democrats. And she was very forthright about it. It was very striking that she said that.

I think there’s a lot of people who are going to want to see him consider taking a different course now,

People like van Jones, who was on CNN, he used to work in the Obama White House.

We’re still far from our convention. And there is time for this party to figure out a different way forward, if you will allow us to do that.

He very candidly talked about how this was going to raise questions about whether the President should continue as the candidate.

Some Democrats are calling for Biden to step down. Andrew Yang —

Andrew Yang, who ran against Biden in 2020 for the Democratic nomination, popular with some younger voters, he said on social media it was time for Biden to step aside. Those are some of the public people. And obviously, in the hours and days that followed, more came out and said, well, this is something we need to think about.

But the people I was talking to were people behind the scenes, people who have run White Houses before, people who work for President Biden in this administration. I heard words like, “He can’t win.” “This is a disaster.”

“This is a nightmare.” And they were very, very concerned that he could not beat Donald Trump.

Right. And you saw these really prominent media figures, outlets, “The Times” as an actor in this situation calling for Biden to step aside. Our editorial board did this. We should say this is entirely separate from our newsroom from the show, but there was this real crescendo. And there was a sense that this was a turning point, right?

Absolutely. But it’s not just the media. I think what the Biden campaign would like it to be is about the media. It’s just that the media tends to be more out front and say things more openly than Democrats were saying. It really was rank and file Democrats. It really was high ranking Democrats, and they were absolutely flipped out.

Right. These doubts are coming from all over, from many corners. Take me through, Peter, the argument for why this poor performance meant that Biden should be replaced. How do they explain that thinking?

Well, look, a lot of people who defended President Biden will say is that incumbent presidents don’t do well in their first debate, and that is true. Historically, that’s been true. Ronald Reagan, George W. Bush, Barack Obama, Donald Trump all lost, arguably, their first debate when they were running for re-election.

But the difference is, if Obama doesn’t register a good performance against Mitt Romney, first of all, nobody thought that Obama wasn’t capable of being president as a result. And second of all, he had another debate about a week or two later in order to try to recover. Neither of those factors works here.

Biden’s problem from this debate is much more existential. It’s much more profound because it’s about whether he is able to perform the office of president, not just for the next few months, but for the next 4 and 1/2 years. And there’s not going to be another debate until September. So he doesn’t have another big audience opportunity to change people’s minds, to show that, in fact, he does still have it and can run the country. And that’s a real problem for him.

And there’s this broader context here, right. Voters have been telling pollsters for a year now that Biden’s age is a major concern for them. We’ve seen Biden’s age before our very eyes. We’ve seen him stumble in speeches, in public appearances. And, Peter, we had talked to you about this very issue a few months ago after a special counsel investigating Biden’s son, Hunter, issued this report focusing on Biden’s mental state, in part, saying that the president was, quote, a “well-meaning elderly man with a poor memory” and had, quote, “diminished faculties in advancing age.”

But at the time, the White House dismissed that report as a partisan hit job. So in a sense, this debate performance was the capstone of something that’s been in the air for a very long time. It’s just that this time, it was undeniable. There was no spinning it.

Well, I think that’s exactly right. There was no spinning it. One Democrat put it to me. He said, for a long time, the fear of Trump stifled Democratic criticism of Biden. People didn’t want to criticize him because they desperately want to beat Trump.

But now, that same fear, he said to me, now meant that they could no longer stand behind Biden, that they worried that he had been diminishing over a period of time and that his staff and the people around him had hidden that from the public. There’s a real anger out there among some Democrats. Now, what the Biden circle would tell you is, no, we didn’t hide anything from you.

Yes, he does have moments where he is not as lucid as you would want him to be, but that, broadly speaking, when they see him operate, when they’re sitting with him in the Oval Office or in the situation room, he is sharp. He asks good questions. He understands and grasps the issues that he is confronting.

And I mean, we all have good days and bad days. But when you’re 81, your good days and bad days may be more pronounced.

And if he has good days and bad days, well, Thursday night was a very bad night.

Peter, I want to ask you about that, about your view on all of this, because I do think all this has raised this fundamental question for a lot of Democrats, for a lot of journalists, for voters, which is what you’re getting at. Was what we saw on the debate stage the real Biden? And had the White House been hiding him from us? Or were the people around him just unable to recognize the perils of this themselves? Like, have they been gaslighting us all, or are they in denial?

Yeah, it’s a good question. That’s the question in some ways, right. I think that people who work closely with the president and like him, admire him, respect him want to see the best in him and want everybody else to see the best in him. And they have been unwilling to admit whether or not he has slipped in the last 3 and 1/2 years.

And part of it may be strategic. They recognize in their view that he is the president. They’ve got to build him u and make him as successful as possible. And they have shielded him as much as possible from public scrutiny.

He hasn’t give as many interviews or as many press conferences as any of his predecessors going back to Reagan. He’s never getting interview to “The New York Times,” or “The Washington Post,” or “The Wall Street Journal,” or the “LA Times,” or any other newspaper, which is, I think, the first president, certainly in my lifetime, who hasn’t done that. And that’s been part of a pattern of them trying to protect him. And I think there’s kind of a reckoning right now among other Democrats, wondering whether they went too far.

OK. So we have this huge reaction to this moment, the debate from the Democrats in the news media. Can you walk us through how the Biden team responds to the full-blown panic?

His campaign was thrown into full-blown damage control over the weekend. And the President himself set out to do two things. First, privately, he met with donors and assured them, yes, he’s still a viable candidate and that they should still support him.

And then publicly, he went on a campaign blitz, traveling to seven events in four states. And his first stop on Friday, in fact, was at a rally in Raleigh, North Carolina.

[CROWD CHEERING]

Hello, hello, hello.

This was already scheduled before the debate, but it gave him an opportunity to both show that he can do the job —

Thank you, North Carolina.

— to demonstrate vigor and vitality —

I don’t know what you did last night, but I spent 90 days and 90 minutes on the stage debating the guy who has the morals of an alley cat.

— and to address his own performance.

I know I’m not a young man. State the obvious. Well, I know.

And he says pretty candidly, he says, yeah, I’m not a young man.

I don’t walk as easy as I used to. I don’t speak as smoothly as I used to. I don’t debate as well as I used to.

But he goes on.

Well, I know what I do know. I know how to tell the truth.

And I know how to tell right from wrong.

And I know how to do this job. I know how to get things done. And I know, like millions of Americans know, when you get knocked down, you get back up.

And, you know, he comes across as pretty vigorous, pretty energetic. Of course, he’s reading from a teleprompter. Always a lot easier to read from a teleprompter. But I think more important than that was the body language and the spirit that he brought to the moment.

It sounds like teleprompter, not the Biden who appears at the rally is meaningfully different, at least in style from the President that we saw on the debate stage. I’m wondering if you think this has changed anything. Has the conversation changed?

No, not fundamentally. I think fundamentally that people still recognize that there’s an issue here. Now, there was pushback among Democrats saying, OK, take a breath. Get a hold of yourself. He’s not dropping out.

I don’t think you judge a person’s — the body of their work on one night. They don’t always go the way you want to. I have confidence in the President because he’s delivered.

And I understand that he had a raspy voice. But like I’ve told folks, who cares? We have a choice this November between someone that’s a good person, a good president with a real record of results, and someone that has brought shame on the presidency.

Don’t let 90 minutes define a career of a president who’s been in office for 3 and 1/2 years, been in politics for 50 years, and overshadow the important issues that he stands for. And so you heard that line of thinking in the spin room and on TV.

Look, I think Joe Biden had a bad debate night, but it doesn’t change the fact that Donald Trump was a bad president.

And by the way, Trump did terribly, too, which is a fair point. Trump may have been more lucid in the sense that he sounded stronger. But if you actually looked at what he said, listen to what he said, he said so many things that were just not true. And it helped Biden that former President Barack Obama put out a statement saying, hey, guys, I’ve seen bad debates. It’s fine. Don’t freak out, in effect, is what he said.

And Jim Clyburn —

And if he asked my opinion, I would give it, as I always do —

— who is his very close ally in Congress from South Carolina, the Congressman who helped get him the nomination in the first place, said, stay the course.

He should stay in this race. He should demonstrate going forward his capacity to lead the country.

So it was important to have those voices out there among prominent Democrats trying to calm the waters. But it only went so far because the waters are still churning underneath.

Peter, I’m curious how his donors are reacting to all this. I mean, you mentioned that part of his full court press is to reassure them that he’s got the mental acuity to run. How successful has he been at that?

Yeah, I think that there are certainly some donors who are resigned. They feel like there’s not much choice. But there are others who actually are considering jumping off the boat. Jumping on what Mika Brzezinski on “Morning Joe” called the hysteria train. And I think that it’s an open question.

But part of the thing is, of course, they’re waiting to see how the polls really shake out. The initial polls after any debate are often not really representative of how an event settles into the political narrative. And the polling and data so far have been kind of contradictory. On the one hand, it shows that Trump clearly beat Biden. Biden clearly lost, and that Biden has only reinforced the doubts that most voters have about his age and mental capacity. That’s absolutely true.

At the same time, there’s some polling showing that the overall horse race number, who are you going to vote for, hasn’t moved dramatically yet, if it does at all, and that it’s possible this is baked in that people who were going to vote against him are still going to vote against him. The people who would vote for him, holding their nose, may not be happy about it, may still be voting for him.

But there’s a tell. The tell was from the Biden campaign. When they put out a memo by Jen O’Malley Dillon, who was his top political person at the campaign, and she says if you see polls go down in the next few days or weeks, what’s telling is that she is, in fact, anticipating that polls would be bad for them and trying to set expectations for supporters and voters and donors saying, don’t let that panic you any further. That’s normal, and we’ll get past that just as we have other bumps in the road.

It seems like the Biden effort over the weekend has, in some sense, quieted some public doubts from key Democrats, right? There’s not a — we didn’t see a deluge of senior lawmakers going on Sunday talk shows and saying, Mr. President, step aside. But from what you’re saying and based on the reporting that we’ve seen from our colleagues, the effort has not, by any means, ended the discussion about replacing Biden. That is very much still happening under the surface.

Yeah. That discussion is very much alive among Democrats. Will Biden and should Biden remain as the candidate? And the question then becomes is if he doesn’t, what then?

We’ll be right back.

Peter, given that this discussion of Biden stepping aside is still, as you said, very much alive, what would it look like for someone to replace him on the presidential ticket at this point in the campaign just a few months before election day? It sounds like it would be pretty daunting.

Yeah. I mean, look, we have never had a situation like this, not certainly in modern times. No president has ever dropped out of the race so late in the cycle. And you have to remember a couple of things.

First of all, the Democratic National Convention, which would anoint a new nominee, is in late August. But they’re actually scheduled to take a roll call vote before the convention begins on August 7. So that means we have five weeks between now and when the roll call is scheduled to be held to decide a nominee.

If the president were to drop out, that would create this truncated, incredibly intense, incredibly wide open, incredibly volatile, short campaign to figure out who would be the nominee. And it’s complicated logistically. It’s complicated politically, it’s complicated in all sorts of ways. And we don’t really know what’s going to happen or how it would happen because we’ve never seen it before.

But it is conceivable. It is possible. The President has to decide that he’s not going to run. If that doesn’t happen, then there’s no contest. There’s no way anybody sees a forcing him off the ballot if he chooses to continue to run. That doesn’t seem to be any appetite for trying to find a way to undo his nomination other than with his consent.

He controls the 3,900 delegates that are going to be at the convention. They’re obligated to vote for him on the first ballot. So it has to be first, his decision on whether he continues to run. If he does, then that’s it. That’s the end of that question. But if he doesn’t, then it’s jump ball.

Would his replacement automatically be Kamala Harris as the vice president?

No, not at all. If it were after the convention and they were both nominated, and then he stepped aside at the last minute, then they probably would simply go to Kamala Harris because she had been ratified by the convention as the vice presidential candidate. That’s possible. But if we’re talking about a situation before the convention, it’s anybody’s guess. There’s about a dozen other prominent Democrats out there who are looking at jumping in if suddenly, the nomination is up for grabs.

But so who are we talking about? What are some of the most prominent names that have come up?

Well, other than Kamala Harris, you have a number of governors, particularly Gavin Newsom of California, Gretchen Whitmer of Michigan, J.B. Pritzker of Illinois. There are, of course, those who ran last time who might jump back in, in theory. Senator Cory Booker, Senator Amy Klobuchar, potentially even maybe Pete Buttigieg, who is currently the transportation Secretary.

But the ones who are going to have the best chance are those who have an apparatus already, who have a set of donors and fundraisers who can raise money instantly, and who have the ability to get on TV and get media attention without having to work as hard for it. And that does suggest, obviously, a sitting vice president or a sitting governor.

Basically hitting reset on the whole nomination process requires embracing a moment of genuine political chaos. I mean, that’s the theoretical downside of this. But of course, I mean, there’s also a potential upside, right? Massive media attention potentially for whoever the replacement Democrat is, the possibility that that replacement could energize a lot of Democratic voters and independents, and even potentially moderate Republicans out there who dislike Trump but just couldn’t get excited about Biden.

Yeah, absolutely. Whoever emerges will have a certain advantage of freshness, right. And that person will have a generational argument to make against Trump because whoever it would be would be younger than Trump. And suddenly, Trump is then the old candidate. He’s 78.

And that new Democratic candidate would be able to say, I’m the next generation. This guy is also too old to be president. And if you’re concerned about our guy was, can he make it through four years, then you should be picking me because the other guy can’t make it either through four years.

Now, the downside, of course, is these are people who are largely untested on a national stage, at least in this kind of an environment. And you don’t know how people will do once they actually jump in. They didn’t have the advantage of a year-long primary contest to prove themselves.

Before you jump into a race, you can look really attractive. Look at Ron DeSantis. Before he jumped in to the Republican primaries, on the Republican side, they all thought, wow, he’s really great. Didn’t turn out to be so great once he got on the campaign trail. The magic didn’t actually appear. So that’s the danger here, is we don’t know which of these Democrats, if any, would have the ability to shine when the big giant klieg lights are on them.

And we’re clearly in a delicate moment right now. But are any of these potential contenders trying to signal interest at this point? I mean, what does that look like? It sort of seems like the art of raising your hand for something but not wanting it to publicly.

Yeah, it’s a really good question because you obviously can’t do anything that seems disloyal to Biden, right. Nobody’s willing to take on Biden directly and say, I’m now running, and you should take him down. So the trick here is you have to be loyal, loyal, loyal right up to the minute that Biden says he’s not running, at which point then suddenly you’re off to the races.

And doing that from a standing start is not a good idea for any campaign. So they have to find ways of talking to their people, lining up donors, thinking about what kind of a campaign would look like, who might be their strategist, without letting anybody know that they’re doing that, or at least not let anybody in our business know that because it would obviously backlash on them. And that’s a very hard thing to do. I mean, I heard that there are people out there making phone calls who want to run, but nobody’s going to admit that outright because it would be damaging to them.

And in terms of logistics here — I know this is all very hypothetical — but if a new candidate were to become the nominee, do they get all the money Biden raised? Do they get his campaign team, or are they literally creating a presidential campaign from scratch with four months to go?

I mean, presumably, they would adopt a lot of Biden’s apparatus. As for the money, a lot of the money these days is in kind of superpacs and these sort of amorphous structures that can go immediately to a different candidate. And everybody who contributed to Biden can now contribute to the new candidate. And then Biden can still spend his money as he chooses in support of whoever the candidate is.

So there’s that advantage in a way it could actually increase some fundraising. But you’re right, they would be starting from scratch in a lot of ways, at least in terms of a national organization.

And obviously, for this to even happen, it relies on Biden stepping aside here, as you said. Everything you’ve laid out so far suggests that he is, for now at least, closed off to this suggestion. And I have to ask what you think from your reporting would change that.

Well, President Biden is a proud man. He’s a stubborn man. As a lot of people of any age are, he is reluctant to confront and face his own weaknesses, and he’s not going to be talked out of running by a bunch of media chattering class, pundits, and junior Democrats. I mean, think about it. He’s been running for president since 1987. And the idea that he is now in office and running the country and as he thinks it, running it pretty well, that he’s going to simply step aside because a bunch of people tell him he should, he reacts viscerally to that. Of course, he doesn’t want to do that.

In fact, when you talk to Democrats, they’re very conscious of not trying to push him because it could have the opposite reaction. It could trigger him to want to stay even more. The people who have influence with him, not that many people.

I mean, at this point, he’s been in politics since 1972 when he was elected to the Senate. And the people he considers his peers, most of them are gone. He’s not surrounded by people whose opinion he truly respects.

Obama and Clinton, the only two former presidents out there other than Jimmy Carter, who are Democrats, I don’t know that if they told him it was time to pull the plug, that he would listen to that. In fact, he might, again, do the opposite. He still resents Obama for discouraging from running in 2016.

Obviously, congressional leaders like Chuck Schumer, Nancy Pelosi, Hakeem Jeffries, Jim Clyburn, people like that, could have an influence if they were to go as a group to him in a way like the Republicans went to Nixon in 1974 and said that he wasn’t going to survive. Maybe that might influence him. But I don’t think they’re likely to do it. It doesn’t look like they’re likely to do that.

Who does have his ear? I mean, who does he really listen to? Who do we know he’s going to be listening to in this moment?

Well, the real people he listens to the most are his family. He’s a very family-oriented guy. Remember, he went home to Wilmington almost every night when he was a Senator. Even now as president, he flies home to Wilmington most every weekend.

And he’s at Camp David this weekend with his family. They’re there because they had already planned to be together for a photo shoot, ironically, with Annie Leibovitz. So the family was already gathering at Camp David —

— gives him an opportunity. Right. And it gives him an opportunity to have a heart-to-heart conversation with the people who mean the most to him, in particular, Jill Biden, of course, the first lady and his sister, Valerie, the kids, and the grandkids. We don’t know what that conversation looks like.

I mean, that’s as private as it comes. Someday they’ll be histories written and memoirs written. And I’m fascinated to read what’s happening this weekend at Camp David, because I think it’s kind of pivotal. But what we do know is that up until now, at least, Jill has been all in.

She’s been encouraging him to run. And it sounds like she’s been encouraging him to stay in. And a lot of people think her voice is the most important voice in this.

And just to pause on this for a second, Peter, it sounds like you’re saying something pretty remarkable, actually, which is that this very small handful of people have influence over what is going to be a massive decision that affects hundreds of millions of Americans and really the entire world.

Well, don’t underplay it. But yeah, that’s exactly right. And you’re right to point out the stakes here, because it’s not just about Joe Biden and his future. It is about this presidential race. And it is about whether you want Donald Trump back in office. That’s the way most Democrats look at it. And the difference between a Biden presidency and a Trump presidency is about as stark as you can imagine. And this will have consequences that will ripple out for years to come.

Peter, I kind of want to end where we began this conversation, which is with this question of Democrats’ mindset right now, post-debate, and the question of what’s motivating them in this moment. Privately, they’re talking about replacing Biden. They seem too scared to do that publicly for all the reasons we’ve discussed, including that the White House is basically telling them to stop talking about it and that Biden probably won’t step aside anyway. So all of this may just be useless hand-wringing.

And all of that raises a pretty essential question for me, which is, are Democrats about to recommit to a damaged nominee who had a disastrous debate, who may only get worse over time out of loyalty to President Biden? Or do you think on some level they really believe this is all just overblown, that it’s one debate, everybody has bad days, and Biden really can recover from all this?

I think there’s a lot of doubt in the party that he can really recover from. This is the argument obviously the Biden campaign is making. And it’s really the only argument they have available to them to say, look, don’t overreact here. It’s just the media hyperventilating. We will recover like we’ve recovered before.

And it is what they have to say because they don’t have anything else that they can say. But almost every Democrat outside of the inner circle that I talk to says this was a complete disaster. They are not sugarcoating it. They saw what they saw with their own eyes, and they think that it’s not tenable for the campaign to pretend otherwise.

And it’s particularly devastating because the weakness of the Biden campaign has always been concerns about his age. People have expressed that to pollsters and in interviews with reporters going back more than a year. And now they just saw it for themselves on TV, in their living rooms, more than 50 million of them. And to convince them not to believe what they saw with their own eyes is just a monumental task politically.

So for Biden’s team and for the Democrats, it’s a big roll of the dice. Do you stick with him and try to correct the damage in the four months you have? Or do you say, it’s time for somebody else, which is going to be a roll of the dice itself?

And assuming Biden continues, Peter, it sounds like we won’t know if that bet, that bet that he’s the only one that ever beat him. He’s the only guy that can get this done, if that was solid or diluted thinking until November 5.

On November 5, or whatever day of the election is resolved, we will know how history judges this. And maybe everybody looks back on this and says, well, that was close, but they were right to stick it out because they won. Or in fact, it was a disastrous decision, and they wasted an opportunity to fix campaign that was already having trouble. Hindsight will be perfect, but there’s no question that this weekend will be remembered as perhaps the most decisive of this campaign.

Peter, thanks for coming on.

Thanks for having me.

After we spoke with Peter, “The New York Times” reported that while at Camp David, Biden’s family urged the president to stay in the race, arguing that, yes, he could still show voters he’s capable of serving another four years.

Here’s what else you need to know today. Early projections show that France’s far-right national rally party won a decisive victory in the first round of voting for the country’s national assembly on Sunday. National rally, an anti-immigrant party long on the fringes of the French political scene, captured about a third of the vote, according to polls, and now appears poised to become the largest force in the lower house of parliament.

The results dealt a blow to French president Emmanuel Macron, who took a gamble by dissolving parliament last month and calling for snap elections. But his bet that the far right wouldn’t repeat its recent success in European parliament elections backfired. A run-off election between the leading parties will be held on July 7.

Today’s episode was produced by Asthaa Chaturvedi, Rachelle Bonja, Will Reid, and Rob Szypko, with help from Olivia Natt and Lynsea Garrison. It was edited by Lexie Diao, with help from Ben Calhoun, Paige Cowett, and Mike Benoist. And special thanks to Michael Barbaro.

Contains original music by Dan Powell, Marion Lozano, and was engineered by Chris Wood. Our theme music is by Jim Brunberg and Ben Landsverk of Wonderly.

That’s it for “The Daily.” I’m Natalie Kitroeff. See you tomorrow.

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Original music by Dan Powell and Marion Lozano

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President Biden’s disastrous debate performance last week set off a furious discussion among Democratic officials, donors and strategists about whether and how to replace him as the party’s nominee.

Peter Baker, who is the chief White House correspondent for The Times, takes us inside those discussions and Biden’s effort to shut them down.

On today’s episode

Peter Baker , the chief White House correspondent for The New York Times.

Background reading

President Biden’s allies can no longer wave away concerns about his capacity after his unsteady performance at Thursday’s debate.

Mr. Biden’s family is urging him to keep fighting .

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

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

Special thanks to Michael Barbaro .

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, 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, Maddy Masiello, Isabella Anderson, Nina Lassam and Nick Pitman.

Natalie Kitroeff is the Mexico City bureau chief for The Times, leading coverage of Mexico, Central America and the Caribbean. More about Natalie Kitroeff

Peter Baker is the chief White House correspondent for The Times. He has covered the last five presidents and sometimes writes analytical pieces that place presidents and their administrations in a larger context and historical framework. More about Peter Baker

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