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Track Practice


Alignments to Content Standards: 7.RP.A 7.RP.A.1

Task

Angel and Jayden were at track practice. The track is $\frac25$ kilometers around.

  • Angel ran 1 lap in 2 minutes.
  • Jayden ran 3 laps in 5 minutes.

  1. How many minutes does it take Angel to run one kilometer? What about Jayden?
  2. How far does Angel run in one minute? What about Jayden?
  3. Who is running faster? Explain your reasoning.

IM Commentary

Parts (a) and (b) of the task ask students to find the unit rates that one can compute in this context. Part (b) does not specify whether the units should be laps or km, so answers can be expressed using either one.

The purpose of part (c) is to give students an opportunity to make use of the unit rates that they found in parts (a) and (b). While it is possible for students to solve part (c) in other ways, the solution shown represents the kind of reasoning with unit rates that 7th graders should be able to do. It is important to note that the answer can be determined using different unit rates as long as the reasoning behind it is correct.

Solution

  1. We can create a table that shows how far each person runs for a certain number of laps:

    Number of lapsNumber of km
    1 $\frac25$
    2 $\frac45$
    3 $\frac65$

    We can see from the table that 1 km is exactly half way between 2 and 3 laps. So it will take 2.5 laps to run 1 km.

    Since it takes Angel 2 minutes to run 1 lap, she will take $$\frac{2.5 \text{ laps}}{1 \text{ km}} \cdot \frac{2 \text{ minutes}}{1 \text{ lap}} = \frac{5 \text{ minutes}}{1 \text{ km}}.$$ So it takes Angel 5 minutes to run 1 km.

    Since it takes Jayden 5 minutes to runs 3 laps, she runs 1 lap in $\frac53$ minutes. Thus, it takes Jayden $$\frac{2.5 \text{ laps}}{1 \text{ km}} \cdot \frac{5 \text{ minutes}}{3 \text{ laps}} = \frac52 \cdot \frac53 \text{ minutes/km} = \frac{25}{6} \text{ minutes/km}= 4 \frac16 \text{ minutes/km}.$$ So it takes Jayden $4 \frac16$ minutes to run 1 km.

  2. Angel runs 1 lap in 2 minutes so she runs $\frac12$ lap in 1 minute. Since 1 lap is $\frac25$ km, $\frac12$ lap is $\frac15$ km. So she also runs $\frac15$ km in one minute.

    Since Jayden runs 1 lap in $\frac53$ minutes, she will run $\frac35$ laps in 1 minute. Since Jayden runs 1 km in $\frac{25}{6}$ minutes, she will run $\frac{6}{25}$ km in 1 minute.

  3. Jayden runs the same distance in less time than Angel (alternatively, Jayden runs farther in the same time than Angel), so Jayden is running faster than Angel.

Jeff Meyer says:

almost 6 years

(a) the answer here seems to involve some unnecessary steps and disassociates the fractions from their physical units, which is important. For example, encourage the students to recognize that the written expression already implies a fraction - e..g. Angel runs 1 lap "per" 2 minutes is the same as:

1 lap

2 minutes

And, from the other information:

1 lap

2.5 km

The answer we're looking for is:

? minutes

1 km

so, the student should look for ways of cancelling the units....

1 lap 2 minutes 2 minutes 2 / (2/5) min 5 min
----- * --------- = ---------- = ------------ = ---------- = 5 min/km 2/5 km 1 lap 2/5 km 2/5 / 2/5 km 1 km

Not sure if this is too advanced for a 7th grade student, but it seems more straightforward and keeps the focus on the real world application of measurement.

(b) The answer here is very prose oriented, as opposed to mathematical.

(c) can be solved a variety of ways, such that it could be easily answered without any use of the domain or standard skills. For example, Angel runs 3 laps in 6 minutes, therefore Jayden is faster.

A question possibly better aligned with the standards might be, what is the ratio of the speed of the faster runner to the slower runner?

Kristin says:

almost 6 years

Thanks for your comments, Jeff.

Re: (a) A lot of people like a unit analysis approach, but students are not actually required to manipulate units in that way until high school (see N-Q.1). Whether or not the approach you suggest is too advanced for a seventh grade student probably depends on the student--the point here is that students have other ways they can tackle the problem and the way the RP standards are written, students are expected to be able to do it in some of these other ways.

Having said that, it would be nice to see other solution approaches than those shown here. What I would like to see is some real student work so that the solutions could more closely mirror the reasoning the seventh graders are likely to employ.

Re: (b) It's interesting that you describe this solution as "prose" which you put in opposition to "mathematical." In my mind the mathematics is the reasoning part, and you can represent that reasoning with words or symbols. I do agree that it would be good to have a symbolic representation of the argument above as well as the wordy one.

Re: (c) Agreed that there are other ways to do this that may not reflect the competencies described in 7.RP.1, but it is given here to show how one might actually use the unit rates found in the earlier parts of the problem. So part (c) is really an extension beyond the particulars of 7.RP.1. Based on your comments, I added to the commentary and aligned this task to the cluster as well. Let me know what you think of the revised commentary.