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:
This task is part of a set of three assessment tasks for [standard 7.RP.A.2].
7.RP.A.2 Art Class requires students to decide whether two quantities are in a proportional relationship by testing for equivalent ratios in a table, to find a unit rate for a ratio defined by non-whole numbers, and to represent a proportional relationship with an equation. Part (a) essentially asks students to partition a set of ratios displayed in a table into two sets of equivalent ratios. Part (b) asks students to identify all the ratios in the table that are equivalent to a given ratio. These two parts work together: the first question asks students to make a judgment about how many different proportional relationships are represented in the table, and the second asks students to specifically identify all of the ratios that go with one of those relationships. This task shows a shift in the standards that expand upon common approaches to "proportional reasoning” because it requires students to understand different aspects of proportional relationships, not just their ability to set up and solve a proportion.
Mathematical Content:
Students must work with ratios of whole numbers and common decimals between 0-5. Ratios involving only whole numbers were introduced in the prior grade; the 7th grade expectation is that students will work with ratios of non-whole numbers. Additionally, this task addresses the transition between working with ratios in isolation to thinking of ratios as defining proportional relationships.
Mathematical Practices:
The second task addresses several standards for mathematical practice. While it is possible that students have thought about what makes one paint mixture the same shade as another, it is unlikely they have thought about this from a mathematical perspective. Thus, students will need to make sense of the context and choose a mathematical approach to answer the questions given (there are multiple approaches). Most approaches require multiple steps, so students will need to make sense of the problem and persevere in solving it ([standard MP.1]). Students solving this task may look for structure ([standard MP.7]) by converting all five ratios into unit ratios and then grouping the ratios that have the same unit ratio. Students might also find equivalent ratios with the same amount of one kind of paint or the same total amount of paint. Any solution approach requires students to decontextualize and contextualize ([standard MP.2]). The complexity of the item could be lowered by asking Part (b), Part (c), and then Part (a) because it would suggest a solution approach to Part (a). Complexity could be increased by removing Part (c) which helps students choose a solution method for Part (d).
Solution:
- The students made $2$ different shades of paint.
- Mixtures D and E make the same shade as mixture A.
- A student should add $\frac{2}{3}$ cup of yellow paint to $1$ cup of blue paint to make the same shade as mixture A.
- Either of these equations would be correct:
- $b=\frac{3}{2}y$ (or $\frac{3}{2}y=b$ if this is a fill-in-the-blank)
- $y=\frac{2}{3}b$ (or $\frac{2}{3}b=y$ if this is a fill-in-the-blank)
