Wednesday, September 30, 2020

# Algebra 1 PARCC question: linear functions

#### The following multiple-select 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:

Which of these represent a linear function?

• (3, 6), (0, 2), (3, 5)
• For each square whose sides have length s, the perimeter is 4s.
• $y = |x|$

Explanation and solution(s)

PARCC evidence statement(s) tested: F-IF.1, subclaim A:

Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).

The evidence statement references Math Practice 2 in the Common Core: Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.

The question tests students’ understanding of the eighth-grade Common Core math standard 8.F.A.1, which states that they should “understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.” It also touches 8.F.B.5, which says they should be able to “describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.”

Although the question begins to tap into HSF.LE.A.1, a high school math standard in “Linear, Quadratic, and Exponential Models,” most of the standards assessed by this question are in eighth grade within the Common Core and, I suspect, most states’ math standards.

The first option, a set of ordered pairs, is incorrect. Two of the ordered pairs have an x value of 3 but have different y values. As stated above, each element of the domain in a linear function must map to exactly one element of the range. If the same x value gives two different y values, the set of ordered pairs does not represent a linear function (or any type of function).

The graph is correct. As long as any line plotted on a set of coordinate axes isn’t straight up-and-down vertical, the graph represents a function. The fact that the graph is a “line” means it’s “linear.” Therefore, this is a perfectly good way to represent a linear function.

The table is incorrect. As with the set of ordered pairs, the same x value yields more than one y value. This table, therefore, doesn’t represent a function, linear or otherwise.

The description of how to find the perimeter of a square is correct. The word problem translates to an equation, which can be plotted as a non-vertical line and therefore represents a linear function, since it meets all the requirements for a function—each valid input has exactly one output—and for being linear. If s = 4, for example, then the perimeter is 16. It can’t be anything but 16, so only one output results from the input of 4. Expressing that in the format y = mx + b would give us something like p = 4s + 0.

The absolute value function is incorrect. It’s not linear, because it looks like a “V” (see below), but it is a function. For each input, whether it’s a positive or negative number, there is exactly one output. On the graph of y = |x|, no points are directly on top of another point, although some points are side-to-side.

## Resources for further study

Purple Math, developed by Elizabeth Stapel, a math teacher from the St Louis area, has a series of four webpages that deal with linear functions and how they are represented, here. She describes how they can be represented in tables, graphs, equations, and word problems, as shown here.

The Khan Academy, developed by Sal Khan, an engineer who has created a library of thousands of video lessons, has a series of tutorial videos, here, that explain how to recognize linear functions and distinguish them from nonlinear functions.

Chapter 3 of Paul A Foerster’s book Algebra and Trigonometry deals with linear functions. He comes right out and says: A linear function is a function specified by an equation of the form

$y = m x + b$

Complete reference: Foerster, Paul A. Algebra and Trigonometry: Functions and Applications, revised edition. Addison-Wesley, 1980, 1984. The book is used in several algebra classes taught in middle and high schools in both Illinois and Maryland.

## 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 analyze different situations by stripping away the context, whether it be a word problem, a graph, a table of values, or whatever, in order to get at the underlying mathematics involving linear functions. 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. The question is best tested online, since erasures of multiple answers may be interpreted by document scanners as marks rather than erasures if the student erases them incompletely.

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

## Challenge

Helmut Landsberg developed a report he entitled “Weather and Health” using data from soldiers during World War II. Many documents and papers he produced can be found in the archives of the University of Maryland, here. He showed that people burn about 30 calories per day more for every drop of 1°C in the air temperature.

If you normally burn about 2,000 calories a day at room temperature (22°C), how hot would it have to be for you to burn, according to his predictions, zero calories? Does the relationship between air temperature and calories burned in a day represent a linear function? Do you think it’ll ever get that hot?

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

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