One of the latest assessments I’ve given was a test over linear functions. Some of the questions from the test are posted below. The questions attached are specifically about writing an equation from a scenario in y=mx+b format and interpreting what the x- and y- intercepts would mean in this context.
Below are student samples from questions 3 and 4 on this test.
Based off these answers and the work I saw (done separately), the student answering question 3 understood the quantities for the slope and the y-intercept, which are needed for the equation format, y=mx+b. The student even correctly understood to replace the variable x with variable t. How I originally thought to grade this was to give one point for correctly identifying the y-intercept and one point for correctly identifying the slope. It would be 2 points in total. However, I decided to give this student 1.5 points out of 2 because the quantities this student wrote down were correct, yet the slope was positive instead of negative, and she added another variable. The feedback I gave this student therefore was to think about the rate of change in this problem. Was the amount of money Malik’s card going up or down? Was Malik gaining or spending money every month? Those are the questions I proposed to her, and my hope is she will be able to see this and understand that the slope should be negative because Malik is actually having less and less money as time goes on. Additionally, I addressed the extra variable by mentioning that in the linear equation y=mx+b, only one of the quantities has a variable attached to it. Is it the y-intercept or the slope that has a variable attached? My hope is that she’ll be able to look at the slope-intercept equation and recognize that only the slope should have a variable attached to it.
I wanted to talk about another question where I saw some confusion on the assessment, question 4 posted above. Originally, I was going to make this 2 points in total again, one point for correctly explaining the x-intercept and one point for correctly explaining the y-intercept. The two students answering question 4 had different misconceptions. The first student saw the two intercepts as actually only one point on the graph, so she really only addressed the x-intercept, the point signifying Malik is out of money on his card. She demonstrated a good understanding of what would happen there, but she just failed to distinguish this intercept from the y-intercept, leaving me uncertain of what her understanding for the y-intercept would be. I therefore let her know that the x- and y-intercepts are two different points on a graph. I asked her to think about which one she just described. Then, what would the other intercept represent in this context? To scaffold a little more, I reminded her that the x axis represents the amount of months passing and the y axis represents the amount of money on the card. When both x=0 and y=0, those are two separate points. What could each point signify in this problem? Since she gave a strong understanding for the x-intercept, I gave her 1 point out of 2 for this question.
The second student confused the intercepts for the axes. I let her know that she described the axes well and what quantities are attached to those quantities. Those are important to know to understand what kind of function is being described, which is a relationship between quantities. However, I left as feedback that the x- and y- intercepts are distinct points on those axes. I therefore left a question, what does it mean specifically at those specific points? What does it mean when x=0, which is the y-intercept, and what does it mean when y=0, which is the x-intercept, in this context? The hope is for her to use the knowledge she knows about the axes and then give a response that if x=0, that means no months have taken place, meaning that is where Malik starts with $300 that was given on the card. Additionally, when y=0, that is the point where Malik’s card has run out of money because the y represents the amount of money, and that will take place after a certain amount of months. Since she did not describe either intercept here but did mention the quantities involved, I gave her 0.5 points out of 2.