This is a follow up to a previous post, Intercepts, The Zero Product Property, and Other Such Mathematical Disasters in which I wrote about some common misconceptions I was seeing in my students’ work while I was grading a mid-unit quiz. I wanted to dive deep into what the root of the problem was, and determine ways in which I can better address the issue as we move forward into the unit.
In this post, I will break down the mathematics behind each of the scenarios presented in the student work below.
Scenario #1 Breakdown
The student correctly factors each equation (even though the approach to question 1a is still massively triggering), identifies the values of x that make each equation true, and then incorrectly states their solution as an ordered pair (x, y). If we look at question 1 carefully, we see that each equation only has a single variable – that’s x. No y’s! I failed in getting students to recognize WHEN stating an ordered pair (x, y) response was appropriate.
Here’s what I think happened…
Last year many students struggled with understanding the difference between an x-intercept and a y-intercept. Visually, x-intercepts are where the graph crosses the x-axis (horizontal axis), and the y-intercept is where the graph crosses the y-axis (vertical axis).
Easy enough, right?
What I noticed in the math classroom was that students were having difficulties transferring their visual understanding of what x and y-intercepts are to the numerical.
Here are two key facts to understand:
- Any point on the x-axis has a y-value of zero
- Any point in the y-axis has an x-value of zero
If we think about the xy-coordinate plane as representing horizontal and vertical movements, this should make sense. To move left and right, we are moving along the horizontal, or x-axis. To move up and down, we need to shift along the vertical, or y-axis. So, if a point exists on the y-axis only, it means there is no horizontal movement. Thus the x-value anywhere along the y-axis is zero.
Now, let’s say we were given the equation instead of the graph. This is the equation of a line:
y = 3x + 12
In order to find the x-intercept(s), this means we want to know what the horizontal position is when there is no vertical movement. Numerically, this means we set y = 0 and solve for x.
0 = 3x + 12
After some basic solving, we arrive at x = 4.
Therefore, the x-intercept is located at (4, 0), since the original question was asking about a specific location on the xy-coordinate plane.
For the y-intercept, we want to know what the horizontal position is when there is no horizontal movement. Numerically, this means we set x = 0 and solve for y.
y = 3(0) + 12
We get y = 12. This means our y-intercept is located at (0, 12) on the xy-coordinate plane.
I wanted to make it clear to students that when we are looking at features of a graph (such as the x-intercepts, y-intercept, vertex, etc.), oftentimes we are a specific location in 2-dimensional space. In order to specify a point in space we need both an x-coordinate, and a y-coordinate (just like longitude and latitude). I really emphasized this and DRILLED it into my students minds that what we saw on the graph, has a logical and equivalent counterpart when dealing with the numeric.
Yet, students got so used to writing ordered pair answers that they did so even when it made no sense in the context of the problem. So, in order to counter this type of behaviour I need to try and implement a new habit: once students are finished writing their solution to a problem, go back through the steps and ask themselves, line-by-line, why did I write this?
The method the student takes to solve the problem here will lead to a correct answer; when done correctly, use of the quadratic formula to solve a quadratic equation will always lead to correct results. The fact that I was seeing this approach from several students on this specific question, however, shows they they are lacking fundamental understanding of a major conceptual idea. The answer to an equation presented in this way should be obtainable in a matter of seconds.
To understand how to solve this question easily, it’s helpful to think about what it means when two numbers multiply to make zero. We call this the zero product property.
The zero product property
The zero product property states that when two or more things (called factors) are multiplied and result in zero, it means that one or more of the factors must equal zero.
ab = 0
So, either a = 0 or b = 0, or both!
If I had the following equation
Instead of a and b, my factors are (x-4) and (x+2). One of those expressions must equal zero in order for the entire equation to equal zero. By that logic then, if
x – 4 = 0 or x + 2 = 0,
it must mean that either x = 4 or x = -2 (or both).
I can check to see if my answers are correct by plugging in the values I’ve obtained for x and seeing if that makes the original equation true.
Check if x = 4, then my original equation
(x – 4) (x + 2) = 0 becomes
( 4-4)(4+2) = 0
(0)(6) = 0
0 = 0
if x = -2, then my original equation
(x – 4) (x + 2) = 0 becomes
( -2-4)(-2+2) = 0
(-6)(0) = 0
0 = 0
Test your understanding
Does the zero product property apply in each of these cases?
- (x-1) + 3 = 0
- (x-5)y =0
- No. We do not have two things that multiply to make zero.
- Yes, the two factors are (x-5) and y. Both multiply to make zero.
- No. The two factors do not multiply to make zero.