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Miles to Kilometers


Alignments to Content Standards: 7.EE.A

Task

The students in Mr. Sanchez's class are converting distances measured in miles to kilometers. To estimate the number of kilometers, Abby takes the number of miles, doubles it, then subtracts 20% of the result. Renato first divides the number of miles by 5, then multiplies the result by 8.

  1. Write an algebraic expression for each method.

  2. Use your answer to part (a) to decide if the two methods give the same answer.

IM Commentary

In this task students are asked to write two expressions from verbal descriptions and determine if they are equivalent. The expressions involve both percent and fractions. This task is most appropriate for a classroom discussion since the statement of the problem has some ambiguity.

The Standards for Mathematical Practice focus on the nature of the learning experiences by attending to the thinking processes and habits of mind that students need to develop in order to attain a deep and flexible understanding of mathematics. Certain tasks lend themselves to the demonstration of specific practices by students. The practices that are observable during exploration of a task depend on how instruction unfolds in the classroom. While it is possible that tasks may be connected to several practices, the commentary will spotlight one practice connection in depth. Possible secondary practice connections may be discussed but not in the same degree of detail. 

This task helps illustrate Mathematical Practice Standard 2; where mathematically proficient students make sense of quantities and their relationships in problem situations, create coherent representations of the problem given and attend to the meaning of the quantities, not just how to compute them.  Students will translate the description of the situation into algebraic equations, decontextualizing.  Also, they will need to think about what quantities should be represented by variables and how those variables relate to each other.  This will require them to “make sense of quantities and their relationships in problem situations.”  The teacher might direct the students’ discussion by asking questions such as: “What do the numbers used in the task represent?”  “What operations will we need to use to solve this task?”  It is important that students just don’t know how to compute the numbers in the tasks, but are able to understand the meaning of the quantities and are flexible in the use of operations and their properties.

Adapted from Algebra: Form and Function, McCallum et al., Wiley 2010

Solution

  1. Abby's method starts by doubling $m$, giving $2m$. She then takes $20\%$ of the result, which we can write $0.2(2m)$. Finally she subtracts this from $2m$, giving $$2m -(0.2)2m$$

    Renato's method starts by dividing $m$ by $5$, giving $m\div5 = \frac{m}{5}$, and then multiplies the result by $8$, giving $$8\left( \frac{m}{5}\right)$$

  2. Abby's expression can be simplified as follows: $$ 2m - (0.2)2m = 2m - 0.4m = (2-0.4)m = 1.6m. $$ (The step where we rewrite $2m - 0.4m$ as $(2-0.4)$ uses the distributive property.)

    Renato's method gives $$ 8\cdot\frac{m}{5} = 8\cdot \frac{1}{5} \cdot m = \frac{8}{5} \cdot m = 1.6 m. $$ So the two methods give the same answer and the expressions are equivalent.

Bill says:

about 5 years

Caroline Sessions suggested changing "subtracts 20% from the result" to "subtracts 20% of the result". I've made this change, what do people think?

Cam says:

about 5 years

I think they're both pretty reasonable -- either is short-hand for "subtracts 20% of the result from itself." For that matter, I think just ", and then subtracts 20%" is also sufficiently clear.

Kristin says:

about 5 years

Both have some potential ambiguity. If it says, "subtracts 20% from the result," one could ask, "20%" of what? If it says "subtracts 20% of the result," one could ask, "subtracts from what?"

In either case, students need to think about what the reasonable interpretations might be and decide which one Abby most likely means. If we want it to be completely unambiguous, it should say, "Abby takes the number of miles, doubles it, then subtracts 20% of the result from the result." The problem with that statement is that Abby would never say that, only a task writer would. So if it is meant for classroom discussion, I think it would be better to leave the ambiguity in and discuss it (that's what I did when I used it with my students as a classroom discussion generator, and we had a very good discussion about both the mathematics of the task and the ambiguity). If somebody had the idea to use this as an assessment, it would need to be completely unambiguous. My guess is there are better assessment tasks for this standard.