Friday, April 23, 2021

Algebra 1 PARCC question: rate of change


The following multiple-choice question, explained here in hopes of helping algebra students in Maryland and Illinois prepare for the PARCC test near the end of this school year, appears on the released version of PARCC’s Spring 2015 test in algebra 1, here:

The graph shows the relationship between the weight, in pounds, of Vidalia onions and their cost.

Which best estimates the cost per pound?

  • A. $0.78
  • B. $1.29
  • C. $2.58
  • D. $10.32

Answer and references

Correct answer: B. $1.29

PARCC evidence statement(s) tested: F-IF.6-6a:

Estimate the rate of change from a graph of linear functions and quadratic functions.

Tasks have a real-world context.

The evidence statement above references Math Practice 1, Math Practice 4, Math Practice 5, and Math Practice 7.

The question tests students’ partial understanding of the eighth-grade Common Core math standard 8.F.B.4, which states that they should be able to “model a linear relationship between two quantities” and “determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.”

The introduction to the eighth-grade standards in math says, “Students recognize equations for proportions (y/x = m or y = mx) as special linear equations (y = mx + b), understanding that the constant of proportionality (m) is the slope, and the graphs are lines through the origin. They understand that the slope (m) of a line is a constant rate of change, so that if the input or x-coordinate changes by an amount A, the output or y-coordinate changes by the amount m×A. Students also use a linear equation to describe the association between two quantities in bivariate data (such as arm span vs height for students in a classroom). … Interpreting the model in the context of the data requires students to express a relationship between the two quantities in question and to interpret components of the relationship (such as slope and y-intercept) in terms of the situation.”

Solution strategy (there are others)

Determine an estimate for the unit rate, in dollars per pound, from the graph.

Since the x axis is in pounds, the cost per pound will be the slope of the graph. We could find it in any number of ways, so the way I chose to find it was by finding a point that looked like it was very close to the intersection of two grid lines:

(4 pounds, $5)

If I can buy 4 pounds for about $5, that means I can buy 1 pound for about

(¼)($5) = $1.25.

The only option that’s even close to that is (B) $1.29.

To check this, I could use another point, say (8 pounds, a little more than $10).

\frac{10}{8} = 1.25 \approx 1.29 \quad\textrm{check}

Resources for further study

Purple Math, developed by Elizabeth Stapel, a math teacher from the St Louis area, has a very nice description of what the slope and y intercept almost always mean in the context of a word problem like the one posed by this PARCC question. The page can be found here.

The Khan Academy, developed by Sal Khan, an engineer who has created a library of thousands of video lessons, has one that describes how Jordan can’t help but adopt more cats, which she does at a constant rate in the word problem he solves. What do the slope and y intercept mean in this context? He explains here.

Chapter 2, Section 2.2 of the book Algebra 2, Illinois edition by Ron Larson et al deals with slope and the rate of change. The authors use a Sequoia tree with a trunk that has a diameter of 137 inches in 1965 and 141 inches in 2005. The average rate of change, R, in inches per year, is given by the formula

R = \frac{d}{y}

where d is the change in diameter, and y is the number of years. To find the average rate of change, also known as the growth rate of the tree, in inches per year, we can use

R = \frac{(141 - 137)}{(2005 - 1965)} = \frac{4}{40} = 0.1

The trunk grows, on average, 0.1 inches in diameter every year.

Complete reference: Ron Larson, Laurie Boswell, Timothy D Kanold, Lee Stiff. Algebra 2, Illinois edition. Evanston, Ill.: McDougal Littell, a division of Houghton Mifflin Company, 2008. The book is used in several algebra classes taught in Illinois high schools.

Analysis of this question and online accessibility

The question measures knowledge of the Common Core standard it purports to measure and tests students’ ability to estimate the rate of change in a linear relationship given a graph. It is considered to have a low cognitive demand.

The question can be tested online and should yield results that are as valid and reliable as those obtained on paper.

No special accommodation challenges can be identified with this question, so the question is considered fair.


If the Sequoia tree in Mr Larson’s book continues to grow at this rate, what will the diameter of the trunk be in 2105?

When will the tree have a greater trunk diameter than another Sequoia tree in the same forest that grew from 149 inches in 1965 to 152 inches in 2005? Use graphs and tables to check your answer.

Purpose of this series of posts

Voxitatis is developing blog posts that address every algebra 1 question released to the public by the Partnership for Assessment of Readiness for College and Careers, or PARCC, in order to help students prepare to take the test this spring.

Our total release will run from February 27 through March 15, with one or two questions discussed per day. Then we’ll move to geometry at the end of March, algebra 2 during the first half of April, and eighth grade during the last half of April.

Paul Katula
Paul Katula is the executive editor of the Voxitatis Research Foundation, which publishes this blog. For more information, see the About page.

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