# Using Diagrams to Make Sense of Fraction Word Problems

I’ve been spending a lot of time in schools and classrooms these past two weeks using diagrams with students and teachers. As a result, I’ve been neglecting this blog but I’ve been learning so much. Hopefully, I’ll start making more time to write about what I’m doing.

Here’s the latest problem I’ve been tackling with students:

Each lap around the lake is 3 ¾ miles long. Nicora runs two-thirds of a lap and then stops.  How many miles did she run?

Some students have no idea how to get started. Others add the two fractions. Others convert the mixed number to an improper fraction and then multiply.  Of the ones who chose to multiply, many make mistakes–they convert the mixed number to an improper fraction incorrectly, they forget the rule for multiplying fractions or they forget how to convert the product to a mixed number.

Students with a strong understanding of fractions and good number sense could do the problem in their head. They thought about decomposing 3 ¾ into 3 parts and then taking two of them. The million dollar question is “why don’t the other students see the problem this way?”

In an effort to help them see the problem differently, this is how we worked on it together.

First, I asked them to tell me about the problem in their own words.  Was it about cupcakes?  Unicorns?  What is going on in the story?  I’ve found that many students need to be forced to think about what the problem is asking them to do because they are so used to circling numbers and randomly picking an operation to solve them.

I then asked them to draw a bar that shows how far a lap is.

Next, I asked them to show how far two-thirds is.

Finally, I asked them to figure out how many miles two-thirds of the lap was.  Some decomposed the three and then the three-fourths.  Others used guess and check.

After we did a series of similar problems, I asked students to solve the problem without actually drawing the diagram. When they had to picture the diagram in their head, they started to develop shortcuts. One student told me that she could just divide the whole number into three and divide the fraction into three and then add them together to get the amount for one-third.  To find two-thirds, she thought about doubling the amount for one-third.

I wanted to share this example with you all because I think it shows how representations can help make explicit what may be obvious to other students. I also think it shows a nice progression of how students can use a representation to build a strategy to solve the problem.  How is this different than telling them the procedure?  Because the students build a process to solve the problem that makes sense to them.  Furthermore, they can always go back to the diagram if they forget the procedure.  So often, I find that students don’t forgot things because they are lazy, but rather because the procedure never made sense to them to begin with. My goal is to help students build on what makes sense to them, not to show them an algorithm that makes sense to me, but may not make sense to them yet.

# Using diagrams to ease the transition to algebra

People sometimes feel compelled to tell me how much they hate math. When I ask when this started, many point to algebra (the rest say fractions, but we can talk about that another day).

We know that many students struggle when they get to algebra. Researchers have been studying how to ease this transition through “early algebra.” They found that simply introducing traditional algebra concepts at an early age isn’t the answer.

Instead, Blanton & Kaput (2005) talk about “algebrafying” the elementary school curriculum, in which elementary school concepts are developed in a way that allows generalizations about properties and relationships to become more explicit.  For example, students are asked to identify and generalize about patterns, relationships and structure in mathematics. They are given tasks that require them to reason about unknown quantities. At early ages, they develop the ability to identify, describe and analyze how quantities vary in relation to each other.

As you may have noticed, I’ve been obsessed with tape diagrams recently.  In fact, I’m doing a Global Math Department session about them tomorrow night if you want to hear more. One of the benefits I haven’t talked about yet is their potential for developing algebraic thinking.

Take a look at this problem.

Nicora wants to buy herself a new bicycle that costs \$240.  She has already saved \$32, but needs to make a plan so she can save the rest of the money she needs.  She decides to save the same amount of money each month for the next four months.    How much money must she save each month to meet her monthly goal of buying a bicycle?

Many of you probably solved it by setting up the equation \$240= \$32+ 4x and solving for x.

However, younger students could also solve it by setting up a tape diagram like the one below.

After creating the diagram and using it to solve the problem, students can talk about writing a number sentence to represent the situation. This allows the to give meaning to each part of the equation. Students can discuss what the \$240 represents, what \$32 represents and what the missing number, x, represents. They can also discuss how they solved for the missing number. In this way, the tape diagram can be used to build developing algebraic thinking.

Want to know more?

Blanton, M., & Kaput, J. (2005). Characterizing a classroom practice that promotes         algebraic reasoning. Journal for Research in Mathematics Education, 412-446.

# Using Tape Diagrams to Solve Division Problems

I’ve been working with K-5 teachers on how to use tape diagrams to solve problems. During one of these workshops, a first grader peeked in and asked, “Teachers have to go to school too?”  It got quite a laugh, but it’s true.  We need time to sit with our peers and try new ways to solve problems. I also think it reminds us of what it feels like to be a student and see things through a different lens.

One of the problems we worked on was:

Ms. Placa spends a total of \$42 for 3 sweaters.  Each sweater costs the same amount. How much does each sweater cost?

Most of the groups solved this by using long division.

But let’s say you have students who don’t remember the long division algorithm (most likely because it never it held any meaning to them).  If they have been working with a tape diagram or bar models, they can solve it using that.

First they can draw a bar and split it into three equal boxes to represent the 3 sweaters.

Then they can try guessing in order to determine what goes in the box. You want to work with them on using efficient strategies for guess and check. For example, 10’s are easy to count by, so they can start with 10s.

They can see that they still need 12 more.  At this point, some students may know that they can divide 12 by 3 and it will give them 4.  If not, they can count by 2’s and then by 2’s again.

When they add what is in each box, they get to \$14 for each sweater.

The tape diagrams may seem strange to us at first if we haven’t used them before.  We need to learn how to use them just like students need to learn how to use them in a meaningful way. However, once they do, the bar models begin to make sense to them. Later, the models can be used to build to the algorithm so that the algorithm will have meaning for them.

Research shows that students can do well when they use diagrams to solve problems. However, the diagram alone does not hold magical powers.  While simply putting a tape diagram alongside a word problem aided higher ability middle school students, it did not seem to aid lower level students who received no instruction on how to use the diagram. As with any model or representation, students need guidance and opportunities to foster their understanding of it.

Want to know more? Check out:

Beckmann, S. (2004). Solving algebra and other story problems with simple diagrams: A method demonstrated in grade 4–6 texts used in Singapore. The Mathematics Educator, 14(1), 42-46.

Booth, J. L., & Koedinger, K. R. (2012). Are diagrams always helpful tools? Developmental and individual differences in the effect of presentation format on student problem solving. British Journal of Educational Psychology, 82(3), 492-511.

Yancey, A. V., Thompson, C. S., & Yancey, J. S. (1989). Children must learn to draw diagrams. The Arithmetic Teacher, 36(7), 15-19.