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**AHA! MOMENT**

In a recent blog post we took a look at how tape diagrams help students move from a model to more efficient methods of multiplying and dividing with fractions. Let’s take this a step further and see how tape diagrams can be used to solve algebraic equations where students need to write or solve equations.

Students start to use tape diagrams as early as first grade. When it comes to sixth grade algebra and beyond, a tape diagram is a well-known tool that students may use with confidence.

Check out the tape diagram in action with this seventh grade problem from Module 3, Lesson 9:

**Every day Heather practices soccer and piano. Each day she practices piano for 2 hours. If after 5 days she practiced both piano and soccer for a total of 20 hours, how many hours, h, per day did Heather practice soccer?
**

Students are expected to write an equation to represent this situation, which can be a difficult task. However, the tape diagram can help provide a visual of the problem first.

Start by representing the amount of time practicing both piano and soccer in one day. We know that Heather is practicing piano for 2 hours and soccer for *h *hours each day.

Now, we can show that Heather spends this same amount of time practicing on each of the five days. And we know that the total for all five days is 20 hours.

At this point, students may be able to see the equation emerge. There are 5 copies of (*h* + 2) and the sum of all the pieces together is 20.

Or they could even add each piece together.

Having students write the equation in different ways leads to a beautiful discussion on whether or not these equations really are the same, building even more mathematical understanding. This is, in fact, the RDW (Read, Draw, Write) process at work: students are reading the word problem, drawing a model that represents the information given, and then writing an equation from that drawing.

Students often struggle going from word problem to equation. This model really helps students move from reading a problem and feeling lost to having a plan of action that will build the equation.

Using tape diagrams early on in algebra will also help build the skills necessary for solving much more difficult problems later on. Consider this problem from Grade 9, Module 1, Lesson 25:

**16 years from now, Pia’s age will be twice her age 12 years ago. Find her present age.
**

This problem can be quite difficult to set up at first. However, building a model can help students go from word problem to equation as they did in earlier grades.

Start by representing Pia’s age right now.

We can add to the diagram to show Pia’s age in 16 years.

Now what about Pia’s age 12 years ago? We know that this piece will be smaller than P. The problem also states that in 16 years, Pia will be twice as old as she was 12 years ago. So we can divide the P + 16 piece in half to represent her age at P - 12. Showing that 2 copies of P - 12 is the same as P + 16.

Now students can use the model as a visual to write the equation that is needed.

We can see that P + 16 is equal to 2(P - 12), leading to the equation, P + 16 = 2(P - 12). Students now have an equation and can use their algebra skills to solve for Pia’s current age.

Creating models for every problem can be very tedious. The true goal is to guide students to the most efficient method of solving, but students must understand how these equations are being developed before they will be able to write them on their own. The tape diagram is a great tool for building that understanding. Over time, students will have a better sense of how to form and solve equations from the word problems without having to draw models, but they will always have it as a tool to fall back on for more challenging problems.

*This post is by Debby Grawn, a Eureka Math writer for A Story of Ratios (Grades 6–8). She taught seventh and eighth grade math for 10 years and currently teaches college-level math.*