Analysis of Change: The Missing x-Intercept in “y=mx+b”

Contents

Theme
Common Core State Standards
Introduction
Symbolic Representation
Interviews with students
Mathematical Issues
Teaching Strategies
References
Your Experience

Theme

Interpreting graphs

Common Core State Standards

6.EE.9 8.F.1, 2, 3, and 4

Introduction

When working with linear functions, students frequently believe that the value of the x-intercept should be represented in the equation “y=mx+b.” This is not the case: “m” is the slope of the function and “b” is the y-intercept of the function. Many teachers might disregard this as a simple error, however this error might represent a complex misconception. This entry describes how the x-intercept value relates mathematically to the equation “y=mx+b” and offers suggestions on how to develop a correct understanding of the algebraic representation of the x-intercept.

Symbolic Representation

Problem Response
Write an equation for the following line:

Ac7.PNG

This line has a y-intercept of 1 and a slope of 2, so the correct solution is y=2x+1. However, common incorrect solutions placed the x-intercept value into the equation. For example:

A) y=2x+1/2, where the x-intercept value is used instead of the y-intercept value . B) y=(1/2)x+1, where the x-intercept value is used instead of the slope.

A group of 9th grade students took a test with problems like the one on the left. 72% of the students included the x-intercept at least once on the test and 55% used it two or more times.

Moschkovich, J. (1998). Resources for refining mathematical conceptions: Case studies in learning about linear function. The Journal of the Learning Sciences, 7(2), 209-237.

Interviews with students

PROBLEM #1

Graph the equation y=x+4. Jimmy said that this line would go through the axis at (4,0) because in the equation you add 4 to x. Do you think that Jimmy was right?

s: Yes, Jimmy is right. Because in the equation you add 4 to x.

(Graphs the equation on the calculator) Oh no! I forgot that it goes to the side.

 

Mathematical Issues

It is common for students to try to put the value of the x-intercept into equations of the form y=mx+b. Although the x-intercept is not typically explicit in equations of the form y=mx+b, there exists a relationship between the x-intercept and the equation of the line. For example, if m=1 then the x-intercept is -b, and if m=-1, then the x-intercept is b.

In the above graph, y=x+2 . Because the slope is equal to 1, the x-intercept, -2, has the negative value of the y-intercept. Thus the x-intercept is -b, where “b” is the value of the y-intercept. However in the previous example where y=2x+1, the x-intercept is not -b. The x-intercept is -b/m, where b=1 (the y-intercept) and m=2 (the slope). In general, the x-intercept is always -b/m. One way to see this is through algebra. By setting the y-value equal to zero in the equation, we can find the value of the x-intercept. Thus if y=mx+b, when y=0 we have 0=mx+b and x=-b/m. Visually, we can see that with a slope of 1, the x-intercept is always -b, but as the slope shifts, the x-intercept will shift as well. Larger slope values shift the x-intercept toward the origin, and slope values closer to 0 shift the x-intercept away from the origin. There is in fact an inverse proportional relationship between the x-intercept and the slope, precisely because the x-intercept value is -b/m.

Teaching Strategies

The idea that the value of the x-intercept should appear in the equation “y=mx+b” is persistent even with instructional intervention. One way to change this misconception is for students to understand the actual relationship between the x-intercept and the equation y=mx+b. Students should explore the case where m=1, and see the connection between how the changing value of “b” affects the x-intercept. Next, students should explore cases where the slope is not 1. As the value of the slope changes, students can see how the shift in “m” relates to the steepness of the slope of the graph, and a shift of the x-intercept toward the origin. This guides students away from seeing the x-intercept as a reflection of a change in “b” and seeing it as also depending on “m.”

References

Moschkovich, J. (1998). Resources for refining mathematical conceptions: Case studies in learning about linear function. The Journal of the Learning Sciences, 7(2), 209-237.

Moschkovich, J. (1999). “Students” use of the X-Intercept as an Instance of a Transitional Conception. Educational Studies in Mathematics, 37(2), 169.

Chiu, M. M., Kessel, C., Moschkovich, J., & Munoz-Nunez, A. (2001). Learning to graph linear functions: A case study of conceptual change. Cognition and Instruction, 19(2), 215-252.

Your Experience

  • What have you learned about your students’ thinking regarding this topic?
  • What have you learned that is an effective teaching strategy to help students understand this topic?