What Is a Tape Diagram?

A tape diagram is a drawing of rectangles, like strips of tape, whose lengths stand for the numbers in a word problem. It is also called a bar model in Singapore Math and a strip diagram in the Texas standards: three names for one drawing. A child draws how the numbers are related, which is the whole and which are the parts, or which amount is bigger and by how much, before deciding whether to add or subtract.

For “8 red cars and some blue cars make 13”, the child draws one bar for 13, splits it in two, and labels the parts 8 and ?. Children start tape diagrams in first grade, for adding, subtracting, and comparing, and use them through third grade to multiply and divide. A free blank tape diagram template and word problems with an answer key are at the end of this guide.

What is a tape diagram?

A tape diagram is a drawing of one or more bars, in which each bar, or each piece of a bar, stands for an amount in a word problem. The Common Core glossary defines it as “a drawing that looks like a segment of tape, used to illustrate number relationships”, which is where the name comes from. Four marks make one:

• Bars. One bar, or a few, drawn as plain rectangles.
• Lengths. A bigger amount gets a longer piece. The lengths are roughly right, never measured.
• Labels. Each number goes inside or beside its piece, with what it counts: “8 red”, not just “8”.
• A bracket and a ? A bracket over the bar marks the total, and a question mark marks the amount the problem asks for.

A tape diagram draws the relationship, not the objects. A child does not draw 13 cars. The bar shows which number is the whole and which are the parts, and that is enough to choose the operation.

Tape diagram, bar model, strip diagram: what is the difference?

There is no difference: tape diagram, bar model, and strip diagram are three names for one drawing. The Common Core glossary says so in its definition: a tape diagram is “also known as a strip diagram, bar model, fraction strip, or length model.” The name on your child’s homework depends on the program:

Name	Where you will see it
Tape diagram	Common Core materials, and curricula such as Eureka Math
Bar model	Singapore Math and the programs built on it, which also call it “the model method”
Strip diagram	The Texas standards (TEKS), from third grade

The bar model is taught in Singapore’s elementary math curriculum. Ng Swee Fong and Kerry Lee, writing in the Journal for Research in Mathematics Education in 2009, describe the model method as a heuristic Singapore teaches for word problems, one that “involves drawing a diagram to represent key information in the problem.”

What grade are tape diagrams introduced?

Tape diagrams are introduced in first grade, for adding, subtracting, and comparing within 20. Children use them through third grade, when the bars are cut into equal parts for multiplying and dividing. The Common Core’s progressions document for operations, written by the standards’ authors, says first graders compare amounts by matching objects and drawings, and “later in Grade 1, a tape diagram can be used.” Great Minds says its Eureka Math curriculum “introduces tape diagrams as early as Grade 1 with addition and subtraction.”

Grade	What children draw	Common Core	Texas (TEKS)
Kindergarten	Objects and simple drawings for adding and taking away, not yet bars	K.OA.A.2	K.3(B)
1st grade	One bar for parts and a whole, then two bars for comparing, within 20	1.OA.A.1, 1.OA.A.2	1.3(B), 1.5(D)
2nd grade	The same bars within 100, and two-step problems	2.OA.A.1	2.4(C), 2.7(C)
3rd grade	Bars cut into equal parts for multiplying and dividing within 100, and two-step problems	3.OA.A.3, 3.OA.D.8	3.5(A), 3.5(B)

The Common Core standards for grades 1–3 never use the words “tape diagram”. First grade’s 1.OA.A.1 asks for word problems solved “by using objects, drawings, and equations”, second grade’s 2.OA.A.1 and third grade’s 3.OA.A.3 ask for “drawings and equations”. The tape diagram is the drawing the progressions document recommends. The standards first name it in sixth grade, for ratios (6.RP.A.3).

Texas names the strip diagram first in third grade. Its standard 3.5(B) asks children to solve multiplication and division problems “using arrays, strip diagrams, and equations” (TEKS grade 3), and 3.3(A) uses strip diagrams for fractions. Texas first graders solve joining, separating, and comparing problems with “objects and pictorial models” (1.3(B), TEKS grade 1), so a bar can appear on a Texas first-grade worksheet before the standards call it a strip diagram.

Fourth grade takes the bars further: two bars show one amount as so many times another, which 4.OA.A.2 calls “multiplicative comparison”, and later grades use the same bars for fractions and ratios. At the other end, a kindergartner who draws dots or circles for a word problem, not bars, is on time, since K.OA.A.2 asks for “objects or drawings”.

How do you draw a tape diagram?

To draw a tape diagram, read the story, draw a bar for each amount, label what you know with a ? for what you do not, and only then choose whether to add or subtract. The six steps below work through one problem: Maya has 7 stickers. Leo has 12. How many more stickers does Leo have?

1. 1

   Read the story and name what is counted

   Read the problem aloud and ask what is being counted and whose it is. Here it is stickers, some of them Maya's and some of them Leo's.

2. 2

   Decide on one bar or two

   Use one bar when parts make up a whole. Use two bars, one above the other, when the story compares two amounts. This story compares Maya's stickers with Leo's, so it gets two bars.

3. 3

   Draw the bars roughly to size

   Draw Leo's bar longer than Maya's, with both starting at the same place on the left. The lengths only need to be roughly right. Nobody measures them.

4. 4

   Write what you know and a ? for what you do not

   Write 12 for Leo's whole bar and 7 on Maya's bar. The piece of Leo's bar that sticks out past Maya's is the difference, so it gets the ?.

5. 5

   Choose the operation from the picture

   The ? is a part of Leo's 12, and Maya's 7 is the other part, so subtract, 12 − 7, or count up from 7 to 12. Both are right.

6. 6

   Write the answer with its word and check it on the bars

   Write the answer as a sentence with the thing counted: Leo has 5 more stickers. Check it on the bars: 7 and 5 make Leo's 12.

Step 5 is the point of the whole drawing. Your child chose subtraction because the picture showed a missing part, not because the question said “more”.

Tape diagram examples: adding, subtracting, comparing, and equal parts

The tape diagrams in grades 1–3 come in three shapes: one bar split into parts, two bars compared, or one bar cut into equal parts. The story decides the shape, and the shape decides the operation.

Part-whole: one bar (first grade)

Sam has 8 red cars and 5 blue cars. How many cars does he have? The child draws one bar in two pieces, labels them 8 and 5, and puts a ? on the bracket over the whole: 8 + 5 = 13.

Now turn the problem around: Sam has 13 cars, and 8 are red. How many are blue? The bar is the same, with 13 on the bracket and the ? on a piece: 13 − 8 = 5, or count on from 8 to 13. One drawing serves the addition and the subtraction, so the reverse problem needs no new method, only a ? in a different place.

Compare: two bars (first and second grade)

Maya has 7 stickers and Leo has 12. Leo’s bar is longer, and the piece that sticks out past Maya’s is the difference.

“How many more does Leo have?” and “How many fewer does Maya have?” draw the same two bars and have the same answer, 5. Comparing which of two numbers is bigger comes first, with greater than and less than. The tape diagram adds how much bigger.

Equal parts: one bar cut into equal pieces (second and third grade)

There are 4 bags with 6 apples in each. How many apples are there? The child cuts one bar into 4 equal pieces, writes 6 in each, and puts a ? on the bracket: 4 × 6 = 24.

Sharing 24 apples equally into 4 bags uses the same bar with 24 on the bracket and a ? in one piece: 24 ÷ 4 = 6. An array, rows of equal groups drawn as dots, is the other third-grade picture of the same idea, explained in arrays in math.

Second graders also meet two-step problems, drawn as one bar in three parts or as two bars stacked, one step on each.

Tape diagrams vs. the old way: why schools teach them

The difference between a tape diagram and the old key-word method shows in one problem: Julie has 12 apples. She has 5 more apples than Lucy. How many apples does Lucy have?

	The key-word way	With a tape diagram
What the child notices	The word “more”	Julie has more, so Julie’s bar is the longer one
What the child draws	Nothing	Julie’s bar of 12, and Lucy’s bar 5 shorter
The calculation	12 + 5	12 − 5
The answer	17, which is wrong	7

The Common Core’s writing team names this kind of problem in its progression for operations: it is one where “the comparing sentence suggests an operation opposite to that needed for the solution.” Its answer is the compare drawing, and it calls such diagrams “a major step forward because the same diagrams can represent the adding and subtracting situations for all of the kinds of numbers students encounter in later grades”, from bigger numbers to fractions and algebra.

The What Works Clearinghouse practice guide Assisting Students Struggling with Mathematics (2021), written for intervention teachers, says the same about key words. Its panel advises teachers to “avoid teaching key words that link specific words to operations”, and its table of key words that fail includes “more” and “fewer”. The guide rates two related practices as having strong evidence: using drawings, with strip diagrams on its list, alongside objects (Recommendation 3), and teaching word problems with “diagrams, tables, or equations that directly link to the problem type” (Recommendation 5).

The tape diagram is how your child decides which calculation to do, and the calculation itself is the one you learned. The standard algorithm for adding and subtracting is still taught, in fourth grade, as the new math guide shows.

How can parents help with tape diagram homework?

Help with tape diagram homework by asking your child to explain the bars before anyone does any arithmetic. Six habits keep you in step with what the teacher asked for:

• Ask for the story of the bars first. “Whose bar is this? What does the ? stand for?” comes before any numbers.
• Let the drawing be rough. A bigger amount needs a longer bar, and that is all.
• Ask for the drawing even when your child knows the answer. The teacher asked for the drawing, and it still works when the numbers are too big to do in your head.
• Do not teach “more means add”. In “Julie has 12 apples, 5 more than Lucy”, the word “more” is there and Lucy’s apples are a subtraction, 12 − 5 = 7.
• Build bars from real things first. Two rows of crackers, LEGO bricks, or paper strips side by side show a compare problem before anyone draws it.
• Use a blank template so the drawing is not the hard part.

Common mistakes with tape diagrams, and how to fix them

Each of these seven tape diagram mistakes is something you can see in your child’s drawing, and each has a fix.

• Choosing the operation first. Your child decides to add, then draws bars to fit. Cover the numbers and ask who has more before anything is drawn.
• Drawing the bigger amount shorter. The bars then say the opposite of the story. Ask which amount is bigger and make that bar longer.
• One bar for a compare problem. Comparing two children’s stickers needs two bars, one each. Ask how many people or things are being compared.
• The total written inside a part. The total belongs on the bracket over the whole bar. Point to the bracket and ask what the whole bar stands for.
• Numbers with no labels. With a bare 7 and 12, a child loses track of whose is whose. Write the name or the thing beside each number.
• Measuring the bars. A child who measures every bar runs out of page and time. The bars only need to be roughly right.
• The ? in the wrong place. A ? on the whole when the story gives the whole means the story was misread. Read the question sentence again and ask what it wants to know.

When tape diagrams are hard, go back to objects in two rows, give your child bars already drawn so the only job is the labels, and use smaller numbers.

Tape diagrams vs. number lines (and number bonds)

Neither a tape diagram nor a number line is better: they answer different questions. A number line shows numbers as positions and jumps, which suits counting on and how far apart two numbers are. A tape diagram shows amounts as pieces of a whole, which suits finding the whole and comparing two amounts.

A number bond shows the same part-part-whole idea as a tape diagram, with circles for the whole and the parts in place of bars. The Common Core progressions document draws a tape diagram, a part-whole box, and a number bond side by side for one addition. Bars add something a bond does not have: two of them, one above the other, show a comparison.

Free tape diagram worksheets and blank templates

Our free tape diagram worksheets come as two PDFs: a blank tape diagram template, and tape diagram word problems with an answer key. Print the template first, for tonight’s homework. It has blank bars in two parts and in three parts, pairs of bars for comparing with the extra piece outlined in dashes, and bars already cut into equal parts, so your child can label any problem without drawing the bars.

Print the word problems once your child has drawn a few with you. They run in the order of the three shapes: parts and a whole within 20, then comparing, including a “more” problem that needs a subtraction, then problems within 100 where your child draws the bars, and last, equal parts. A first grader stops after the compare page. The problems are not numbered, and the key matches each answer to its problem by position. Both PDFs print in black and white, need no account or email, and end with a page of instructions for grown-ups.

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Free printable tape diagram template and word problems

Two free PDFs in black and white, with no account and no email, in teaching order: the blank tape diagram template, then tape diagram word problems from parts and a whole within 20 to equal parts. Start with the one your child can already do. The word problems have an answer key, and every PDF ends with a page of instructions for grown-ups on why, what for, and how to use it.

1. 1Blank Tape Diagram (PDF)19 tape diagrams · 5 pagesDownload
2. 2Tape Diagram Word Problems (PDF)15 problems · answer key included · 8 pagesDownload