Illustrative Mathematics

Task

A company uses two different-sized trucks to deliver concrete blocks. The first truck can transport $x$ blocks per trip, and the second can transport $y$ blocks per trip. The first truck makes $S$ trips to a job site, while the second makes $T$ trips. What do the following expressions represent in the context of the problem? Use the drop-down menus to construct a two-part description of the expression.

$S+T$
$xS+yT$
$\displaystyle\frac{xS+yT}{S+T}$

IM Commentary

This task is part of a joint project between Student Achievement Partners and Illustrative Mathematics to develop prototype machine-scorable assessment items that test a range of mathematical knowledge and skills described in the CCSSM and begin to signal the focus and coherence of the standards.

Task Purpose

The primary purpose of this task is to assess students' knowledge of certain aspects of the mathematics described in the A.SSE.1:

A-SSE.1: Interpret expressions that represent a quantity in terms of its context.
a. Interpret parts of an expression, such as terms, factors, and coefficients
b. Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret $P(1+r)^n$ as the product of $P$ and a factor not depending on $P$.

The task has students look for structure in algebraic expressions related to a context, and asks them to relate that structure to the context.

While the task is not a direct assessment of N-Q.1:

N-Q.1: Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.

it is certainly the case that mastery of working with units will help students construct correct phrases.

Cognitive Complexity

Mathematical Content

For students who view algebra simply as the mechanics of algebraic expressions and their manipulations, this will be a difficult problem. The key conceptual prerequisite being required of students is the ability to process a context with several named variable quantities, and the direct skill being tested is the ability to identify the relationships between these quantities and their manifestation in algebraic form. Parts (a) and (b) of the task directly illustrate A-SSE.1a, and parts (c) and (d) are similar, but are intricate enough expressions that students are likely to make progress only by parsing the expression into simpler parts and interpreting them one at a time.

Mathematical Practices

The task has strong connections to several of the standards for mathematical practice. For one, MP2 requires students to contextualize and and decontextualize as is required by this task.

The language of MP6 on precision includes language on appropriate use of units. While the task does not explicitly ask students to reason about units, the requirement is present nonetheless -- for example, students must interpret $xS$ as a measurement of blocks per trip times a number of trips, resulting in a quantity whose units are simply blocks. This insight is sufficient to correctly select the first clause of the requested interpretation.

MP7 has obvious ties to the content standard being illustrated here.

Linguistic Demand

The language of the task is reasonably direct, and also with a minimal amount of mathematical jargon. The linguistic demand is raised by the fact that the response mode involves the construction of a valid English phrase. There are some choices of a first and second clause which lead to solutions which are not semantically (and possibly not syntactically) valid, so fluent English speakers can easily eliminate these choices.

Stimulus Material

Students are presented with a block of text to read and process, and an interactive display for submitting an answer.

Response Mode

Students piece together two components of a sentence whose union forms an interpretation of an algebraic expression from a context. Likely the simplest response mode for this selection is two drop-down menus, though fancier options present themselves readily (e.g., dragging and dropping the clauses into a text box to form the sentence).

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

Solution

$S$ is the number of trips the first truck makes to a job site, and $T$ is the number of trips the second truck makes to a job site. It follows that

$S + T$ is the total number of trips that both trucks make to a job site.
We can think of $xS + yT$ in separate terms. The first term, $xS$, multiplies $x$, the number of blocks the first truck can transport, by $S$, the number of trips the first truck makes to a job site. This means that

$xS$ is the total number of blocks being delivered to a job site by the first truck.

In the second term, $y$, the number of blocks the second truck can transport, is being multiplied by $T$, the number of trips the second truck makes. This means that

$yT$ is the total number of blocks being delivered to a job site by the second truck.

We then have that

$xS + yT$ is the total number of blocks that both trucks deliver to the job site together.
From part (c), we know that $xS + yT$ is the total number of blocks being delivered to a job site. We also know from part (a) that $S + T$ is the number of total trips being made to a job site. By dividing $xS + yT$ by $S + T$, we are averaging out the number of blocks being transported over the total number of trips. So,

$\frac{xS + yT}{S+T}$ is the average number of blocks that are being transported per trip.

Delivery Trucks, assessment variation