Friday, November 22, 2019
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Algebra 1 PARCC question: graph Ay + Bx = C

The following interactive graphing 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:

On the coordinate plane provided, graph the line with equation

5y - 3x = -15

by selecting the x– and y-intercepts. A correct response must have the points placed at the intercepts.

Answer and references

Correct answers: A line with points at (0, –3) and (5, 0).

PARCC evidence statement(s) tested: F-IF.7a-1:

Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

a) Graph linear functions and show intercepts.

The evidence statement above references Math Practice 1, Math Practice 5, and Math Practice 6 in the Common Core:

[MP.1] Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. … They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.

The question tests students’ understanding of the eighth-grade Common Core standard 8.F.A.3, found under Grade 8 (functions), which states that they should be able to “interpret the equation y = mx + b as defining a linear function, whose graph is a straight line.” To get the equation in this form is a straightforward algebraic manipulation.

Example of a solution strategy (there are others)

Find and plot the intercepts from the general equation for the line.

The equation is not in slope-intercept form but in that’s OK. To find the x intercept, set y to 0 in the equation and solve for x:

5(0) - 3x = -15
-3x = -15
x = 5

That means the x-intercept is at (5, 0). The y-intercept is the point on the line where x is equal to 0. So, just set x = 0 and find y:

5y - 3(0) = -15
5y = -15
y = -3

The y-intercept is therefore at (0, –3). Here’s the line:

Resources for further study

Purple Math, developed by Elizabeth Stapel, a math teacher from the St Louis area, has a few pages on different forms for the equation of a line. The pages start here.

The Khan Academy, developed by Sal Khan, an engineer who has created a library of thousands of video lessons, has a video dealing strictly with intercepts for linear functions, including an entire video about finding the intercepts given an equation in a form like the one used in this PARCC problem, here. The video is part of an entire section of the site devoted to linear equations in two variables, here.

Chapter 3 of Paul A Foerster’s book Algebra and Trigonometry deals with linear functions, and Section 3-2 specifically deals with graphing them. He says, “The y-intercept of a function is the value of y when x = 0″ and “the x-intercept of a function is the value of x when y = 0.” He points out that the form of the equation shown here is just an equivalent way of writing the equation in slope-intercept form, y = mx + b.

  • Slope-intercept form: y = mx + b
  • Point-slope form: y - y_1 = m(x - x_1)
  • General form: Ax + By = C

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 graph a line by plotting the x– and y-intercepts. 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. Students online may experience difficulties using the graphing tool in that they have to click on two points, which causes a line to be drawn through those points, and then pick up the points by selecting a region in a shaded halo around them and dragging them to the intercept points to receive credit for the question.

If you get a graphing question like this, make sure you drag the two intercept points to the intercepts on the axes. You could have exactly the right line, which goes through the intercepts, but if the two big points aren’t at those intercepts, the question will be marked wrong.

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

Challenge

In point-slope form, the coordinate (x1, y1) is on the graph and the slope is m. Quickly plot the function

y - 4 = 2(x-3)

Note that the point-slope form, the slope-intercept form, and the Ax+By=C form used in this PARCC problem are all equivalent ways of writing an equation for the same function.

Did you know there’s a lot more to equivalence in life than there is in just an algebra class? John Troutman McCrann, a high school math teacher and a National Board-Certified Teacher, writes in Education Week. “Any number, measure, numerical expression, algebraic expression, or equation can be represented in an infinite number of ways that have the same value,” he writes. How many ways are there to represent the term “good leader” or “good student”?

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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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