Strip Diagrams and Tape Diagrams: Definition, Method and Examples

What are strip and tape diagrams?

Strip diagrams and tape diagrams use a single continuous rectangle to represent a whole amount. This primary rectangle is then stacked with another rectangle of the exact same length that is partitioned into smaller segments.

The top rectangle typically displays the total sum or product. The partitioned sections directly underneath show the individual groups or parts that combine to make the total. Unknown values are often indicated by a variable, such as $x$, or a question mark.

Names for the same visual family

Depending on the curriculum, strip diagrams and tape diagrams are sometimes called bar models in math, length models, or fraction strips.

Despite the different names, they belong to the exact same visual family. They all use stacked rectangular bars of equal overall length to model mathematical equations and word problems.

Part-whole and comparison structures

Part-whole structures are used to model addition and subtraction. In these situations, the parts are usually different sizes, so the bottom sections of the strip diagram are drawn unevenly to reflect those different values.

When two parts are added together, they form the total. If the total and one part are known, learners use subtraction to find the missing part.

Equal-group structures

Equal-group structures model multiplication and division. Because multiplication represents equal-sized groups joining together, the partitioned sections of the bottom bar must be drawn to exactly the same size.

If the total is unknown but the number of groups and group size are given, multiply to find the top bar. If the total is known, divide by the number of equal boxes to find the value inside each box.

Use a strip diagram in a word problem

Visual models are extremely effective for decoding math word problems. Setting up a diagram reveals the relationships between the numbers before any calculation happens.

Follow these general steps:

1. Read the problem and identify the known amounts and the unknown amount.
2. Draw the top total strip. Place the total inside it if it is known, or a variable if it is unknown.
3. Draw the parts strip beneath it. Make the sections uneven for adding different values, or equally sized for equal groups.
4. Label every drawn section.
5. Use the diagram to write an equation and perform the calculation.

Worked examples

Example 1: Finding an unknown part

Question: Frank has $100$ marbles. $68$ of them are red and the rest are blue. How many marbles are blue?

Method:

1. Identify the quantities: The total is $100$. One part is $68$. The unknown part is the number of blue marbles, $b$.
2. Because this combines two different amounts, use a part-whole structure.
3. The diagram shows that the total minus the red part leaves the blue part. Calculate $100-68$.

Answer: Frank has $32$ blue marbles.

Check: $68+32=100$.

Example 2: Repeated equal parts

Question: Mason pays $34$ dollars every month for a membership. How much does his membership cost for $6$ months?

Method:

1. Identify the quantities: The total cost $t$ is unknown. There are $6$ equal monthly payments. Each payment is $34$ dollars.
2. Because there are equal groups, use an equal-group structure. The bottom bar is partitioned into $6$ equal boxes, each containing $34$.
3. To find the total, multiply the number of groups by the group size. This is one of the classic two-step word problems where finding the structure leads directly to the operation.
4. Calculate $34\times 6$.

Answer: The membership costs $204$ dollars.

Check: $204÷6=34$.

Example 3: Working through multiple steps

Question: A teacher bought $5$ boxes of pencils. Each box contained $24$ pencils. She divided all the pencils evenly among $8$ groups of students. How many pencils did each group receive?

Method:

1. This involves multi-step word problems and requires finding a hidden total first.
2. First, build an equal-group diagram for the $5$ boxes. The bottom bar has $5$ sections of $24$. The top bar represents the unknown total pencils.
3. Calculate the total: $5\times 24=120$ pencils.
4. Next, create a new diagram for the division phase. These two-step multiplication and division word problems require applying the new total to the next condition.
5. Draw a top bar labeled $120$. Draw a bottom bar with $8$ equal partitions for the groups of students. Let $p$ be the pencils per group.
6. Calculate $120÷8$.

Answer: Each group received $15$ pencils.

Check: $15\times 8=120$.

Common mistakes

A frequent error is drawing unequal sections in a model that requires equal groups. If a problem states there are $4$ equal jars, the strip diagram must feature $4$ identically sized boxes. Drawing them in different sizes makes the visual resemble an addition problem, confusing the learner.

Visual accuracy matters: Equal groups must look equal on the page.

Another common mistake is mixing up the total and a known part. If the problem asks for the total, the top rectangle should feature a variable. If the problem states the total amount up front, that number must go in the top rectangle.

Frequently asked questions

What is the difference between a strip diagram and a tape diagram?

There is no mathematical difference. They are synonyms for the same visual modeling tool. Some educational regions prefer one term over the other.

Can strip diagrams be used for fractions?

Yes. When representing fractions, these diagrams are often called fraction strips. They display equivalent fractions clearly by stacking bars of identical overall lengths partitioned into different numbers of pieces, such as halves directly above quarters.

Are tape diagrams only used for word problems?

While they are excellent for unpacking word problem scenarios, they are also useful for algebraic equations. A visual model can show why $3x=15$ means that three equal parts of $x$ combine to make $15$.