How to Use Tape Diagrams in Math to Solve Word Problems

Tape diagrams are visual tools that can assist students in solving math word problems. They provide a simple approach to understanding complex mathematical concepts.

In this post, I will discuss a tape diagram and how it can be used in the elementary classroom. I will also provide examples of tape diagrams for addition, subtraction, multiplication, and division word problems.

What is a Tape Diagram?

A tape diagram is a simple, yet powerful tool used in math to help students understand, interpret, and solve word problems. Tape diagrams can also be called strip diagrams, bar models, fraction strips, or length models.

This tool uses varying lengths of rectangles, or ‘tapes’, to depict mathematical relationships and reveal parts of an equation. Each part or segment of the tape represents a problem component.

The entire tape represents the whole or total, and each segment represents a part. The sizes of the rectangles proportionally match the quantities they represent, allowing for a clear visual comparison and comprehension of the equation.

Prerequisite Skills Necessary to Use a Tape Diagram

To use a tape diagram, students need a conceptual understanding of quantities of numbers. This is a key number sense skill that transfers to solving operations. Developing this conceptual understanding looks different for each grade level.

Students generally move from concrete representations, like physical objects, to pictorial representations, to visual representations, like drawing dots, to abstract representations, like a box. A tape diagram is abstract, but students can use visual representations to understand quantities in tape diagrams.

In kindergarten and first grade, students work with concrete objects and then move to representational drawings. In each process step, place a circle or a box around the objects to signify that they are a group. Be sure to add a digit that represents that number.

In second and third grade, students are introduced to multiplication and division. Their understanding of quantities is now in the form of equal groups. Students will revert to drawing discrete objects to deepen their understanding of equal groups.

A similar situation will occur when students are introduced to fractions. They will need to return to using objects to represent the fractions.

Our goal as math teachers is to move students toward efficiency. Tape diagrams are one way to help students represent their mathematical thinking and move toward more efficient models.

How do You use a Tape Diagram in Math?

To use a tape diagram, start by identifying the components of the problem. Each component will be represented by a rectangle, or ‘tape’ on the diagram. Each ‘tape’ length is then adjusted to represent its relative value in the problem.

When tackling problems with tape diagrams, students must start by asking themselves, “Do I know the entire amount?”

If it’s given in the problem, go ahead and fill it in. If not, simply use a question mark at the bottom of the diagram. Then, carefully examine the other information in the word problem and fill in those details.

How do you draw a tape diagram?

While there are conventions for drawing tape diagrams, there is some variability in how to draw one. Here are some key components:

• Rectangles will represent quantities.
• Rectangles will be labeled with a number, although they may also contain a visual representation like dots.
• The quantities will most likely have labels corresponding with the word problem’s context. Note: I do not label the quantities below.
• A question mark generally represents unknowns.
• The total or the whole can be represented as a rectangle or a bracket.
• Brackets are also used to show groups that are counted.

Questions to Ask Students When Working with Tape Diagrams

To develop students’ mathematical thinking when they work with word problems and tape diagrams, consider helping them internalize some questions about the problem. I have quite a few blog posts about how to solve word problems by problem type and go through a process of understanding the context of word problems.

To help students start to draw what they understand in the word problem, consider asking these questions;

• Can I draw something?
• What can I label?
• What do I see?
• How are these numbers related?

Examples of Tape Diagrams in Elementary Math

Let’s look at a few examples of how tape diagrams can be used for different mathematical operations.

Using Tape Diagrams for Addition

Consider this joining result unknown word problem: Jack has 5 apples, and he buys 3 more. How many apples does he have now?

The tape diagram will have two segments; one representing Jack’s initial 5 apples and the other segment representing the 3 apples he added.

The total length of the tape represents the total number of apples Jack has, which is 8.

Another Addition Example of a Tape Diagram

Most young students can represent 7 + 7 by drawing dots. To emphasize the part-part-whole relationship, line up the dots and draw a rectangle around 7 dots to represent each addend. Indicate the sum with a question mark. This moves students from representational draws of dots to labeling parts with a number, reinforcing the part-part-whole relationship of quantities in this visual model.

As students move into multiplication, they may revert to representational drawings, dots, and boxes because they lack a conceptual understanding of multiplication and division. Once students understand multiplication and division more deeply, they can move back to using numbers in their tape diagram. See below for examples of multiplication and division.

Using Tape Diagrams for Subtraction

For subtraction, imagine Sarah has 10 candies, and she gives 4 of them to her friend. How many candies does Sarah have left?

The tape diagram will start with a segment representing Sarah’s initial 10 candies. We will then “remove” or “subtract” a segment representing the 4 candies she gave away.

What remains in our diagram is 6, which is the number of candies Sarah has left.

Using Tape Diagrams for Multiplication

Here is an example of using a tape diagram for an elementary multiplication problem.

Suppose a flower pot holds 3 flowers, and there are 4 pots. How many flowers are there in total?

We would create a tape diagram with 4 equal segments (for each pot), each segment representing 3 flowers.

The total length of the tape represents the total number of flowers, which is 12.

Using Tape Diagrams for Division

For division, imagine there are 20 cookies that 5 children want to share equally. How many cookies does each child receive?

We would start with a tape representing 20 cookies and then divide this into 5 equal parts. Students can then distribute the cookies to each child. Since this is about distributing the cookies into equal groups, many students will start by drawing one or two cookies into each group, then go back to the beginning and draw one or two more until they have distributed all of the cookies.

Each segment (or child) gets 4 cookies.

Using Tape Diagrams for Fractions

Fractions: In the case of 3/4, you would divide a tape into 4 equal parts, then shade in 3 of those parts to represent the fraction.

To represent 3/4 of a whole number, students would draw the whole number, divide it into groups of 4 then count 3 of those groups.

Videos and More Resources about Using Tape Diagrams

If you’re looking for more resources about how to use tape diagrams in elementary school to develop deeper mathematical thinking about the relationship of numbers in word problems, check out these resources

A few other resources include:

• Khan Academy – Comparing Fractions
• PBS Learning – Modeling with Tape Diagrams

Tape diagrams can be an effective tool in visually representing and understanding various mathematical concepts. Through the use of tape diagrams, mathematical problems become a less abstract and more concrete process for students. This allows them to visualize the problem at hand, making it easier for them to solve even as the problems become more complex.

How to Solve Word Problems By Problem Type

If you’re for more information about how to help students solve word problems, take a look at these resources:

• 5 Tips for Teaching Word Problems by Problem Type
• The Problem with Using Keywords to Solve Word Problems

I also have a course all about teaching word problems by problem type. Click here to check it out.