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Running Laps


Alignments to Content Standards: 4.NF.A

Task

Cruz and Erica were both getting ready for soccer.

  • Cruz ran 1 lap around the school.

  • Erica ran 3 laps around the playground.

Erica said,

I ran more laps, so I ran farther.

Cruz said,

4 laps around the school is 1 mile, but it takes 12 laps around the playground to go 1 mile. My laps are much longer, so I ran farther.

Who is right? Draw a picture to help you explain your answer.

IM Commentary

The purpose of this task is for students to compare two fractions that arise in a context. Because the fractions are equal, students need to be able to explain how they know that. Some students might stop at the second-to-last picture and note that it looks like they ran the same distance, but the explanation is not yet complete at that point. Just because to pictures look the same doesn't mean they are. Think of a picture representing $\frac{49}{100}$ and $\frac{50}{99}$--these will look very similar, but they are not, in fact, equal. To explain why Erica and Cruz ran the same distance, we have to explain why $\frac14 = \frac{3}{12}$.

Note that this task illustrates the cluster "4.NF.A Extend understanding of fraction equivalence and ordering" because it addresses elements of both the standards within this cluster.

Solution

Let’s begin with a picture representing $1$ mile.

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With this picture in mind, we can see that one lap around the school is $\frac14$ mile:

8_8206d85bf6347412027c0e87daa624a1

and we can see that one lap around the playground is $\frac{1}{12}$ mile:

9_bc54e7e8059ffa25e8dfdff003dd6f31

Since Erica ran 3 laps around the playground, she ran $\frac{3}{12}$ mile. Since Cruz ran 1 lap around the school, he ran $\frac14$ mile:

10_68b0467b580da3b8d9979624dbfd9f6e

Now looking at the pictures above, it looks like they actually ran the same distance. We can show this is true if we divide each of the laps around the school into three equal pieces:

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When we subdivide each $\frac14$ mile lap into three equal pieces, our mile is divided into 12 equal pieces. The piece that represents Cruz's run is now divided into 3 pieces, and so $\frac14 = \frac{3}{12}$ he ran the same distance that Erica ran.

Jason says:

about 5 years

Currently, the solution says at one point:

 "Since Erica ran 3 laps around the playground, she ran 3/12 mile". 

No arithmetic operations are visible in this reasoning. However, the major advance in fractions in grade 4 is that fractions - first introduced in grade 3 as numbers but in a pre-operational way - are now being integrated into the system of arithmetic. Thus, in grade 4, we don't just "view fractions as built out of unit fractions" (grade 3 overview, p. 21); we "build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers."

It would be helpful to exhibit that progression of ideas here, and to move beyond the kind of pre-operational thinking portrayed in the grade 3 standards.

So the sentence in the solution could change to read:

"Erica ran 3 laps, or 3 x 1/12 mile, which equals 3/12 mile (4.NF.4c). Alternatively, Erica ran 3 laps, or 1/12 mile + 1/12 mile + 1/12 mile, which again equals 3/12 mile (4.NF.3d)."

This shows the operations of arithmetic being used as expected in grade 4.

Kristin says:

about 5 years

I completely agree with you about the transition from third to fourth grade, but this task is about equivalent fractions, where the key insight is about subdividing unit fractions that make up a fraction. Students haven't forgotten the meaning of a fraction, and using addition seems like overkill in a task like this. In fact, I think it distracts from the key reasoning that needs to happen here (in my experience, this is hard enough as it is).

We do have quite a few tasks that do what you ask, but they are illustrations for 4.NF.B.3. Here are two examples:

http://www.illustrativemathematics.org/illustrations/837 http://www.illustrativemathematics.org/illustrations/856