Identifying and Interpreting the Intercepts
Lesson 7 of 10
Objective: SWBAT identify both the x- and y-intercepts of a linear function, and they will be able to interpret these intercepts in the context of a problem.
I love teaching the standard form of a linear equation because it's always easy to find a context that will first capture the attention of students, and then help them see that, in context, algebra is a reasonable, common sense, useful tool for helping us make sense of the world. As a case in point, here are the two problems I use to open today's lesson. The first asks students to consider all of the ways that $20 can be split among two people. In the second we count the number of ways that 30 points can be scored in basketball game.
When I start with an approachable context, students have the chance to flex their interpretation muscles, as they decide which solutions will make sense. In this narrative video, I express my excitement about the use of the word interpretation in the CCSS, before discussing this lesson a bit.
The Opening Problem
As today's class opens, students set to work on the "Xavier and Yadi" problem. It's a can't-miss punchline to ask the kids why a math teacher like me would give the young couple names like these. I give students a few minutes to recognize that there are many solutions to this problem, and to write a few in their notes. Some kids are really curious about how many answers there are to this problem. To answer that, I begin, "Xavier could have $0 and Yadi could have $20. Xavier could have $0.01 and Yadi could have $19.99. Xavier could have $0.02 and Yadi could have $19.98..." I continue for a few more, and it's important to note that I do this without any sarcasm, and I try hard not to be obnoxious here. I'm really trying to get some algebraic thinking going by listing these "points" in order. Kids do get it, and we come to the agreement that there are 2001 possible solutions.
Solutions in a Table and Points on a Graph
Next, I move to the second slide of today's notes, which will help us formalize things a bit. The first two questions help us name the intercepts - both x and y - then the third one just gets students to organize the thinking they've already done. Students then graph these points.
It's not too difficult to interpret what each point means, which makes it easy enough to talk about the domain and range of this function. After students spend a little while making their graphs, I start to draw mine on the board. The kids have graph paper, but on the board I do not. I model craftsmanship by using a ruler to carefully scale the axes. I say that I'm only drawing the first quadrant for my graph, and I ask if anyone knows why. We discuss why it doesn't make sense to have any negative numbers here.
Once I've recorded the suggestions of a few students on the board and on my graph, we talk about whether or not to connect the dots with a line. I remind students about y = -x + 20, I say that I have a different way to write this equation. I go back to the table, and once again use repetition to build algebraic reasoning: "What is 0 plus 20? What is 20 plus 0? What is 10 plus 10? What is 5 plus 15..." and finally, "In this table, what is x plus y?" Now we've got the equation x + y = 20, and many students are delighted by how reasonable this equation looks beside the previous one. I point to both forms and say that these each say the same thing, before showing in one step how to change the latter to the former. The notes look like this.
As I put up the second problem, I say, "This one has less than 2001 solutions, and I'd like you to find all of them." Once again, it's just a tiny bit of context that's enough to capture everyone's attention. When I choose these problems, I'm thinking about my kids. If there there were not a critical mass of basketball fans in my classes, I would do something else.
We begin to work through this problem in the same way as the first. Then there are two key notes I'll want to get to:
- Why can't we connect the dots here? This problem has the same domain and range limits as the previous problem, at least in terms of negative numbers, and then it goes a step further than that, because there is no such thing as a fraction of a field goal in basketball. Students point out that if we include free-throws, then there are more possible answers. (That's a great extension on which to send off curious students tonight.)
- How can we be certain that we've found all possible solutions? Here is where the table comes in handy. I circulate as students are working, and when they think they've found all possibilities, I ask that they prove it. When we come back together to discuss the problem, I emphasize what many of them have already discovered: listing the points in order really helps! In the table, we see how both variable columns make their own arithmetic sequences. Then we note that it's impossible to have less than 0 or more than 10 3-point baskets, and that the only numbers we've found so far are even. After writing the equation 2x + 3y = 30, we then see that, if we substitute an odd number for y, then x will no longer be an integer. There are few ways to see this, which leaves some room for improvisation.
My notes are similar to those of the previous problem. The last thing I do is show how to convert the equation in standard form to slope-intercept form. Returning to the table, I check to see if students can confirm that the slope should be -2/3.
The "Systematic Lists" chapter of Crossing the River With Dogs by Ken Johnson, Ted Herr, and Judy Kysh is a fantastic source of problems like this one, in which students must find ways to organize possible answers to a question with the goal of proving that their list is complete. More generally, if you've never seen this book, I promise that it's a great one to have lying around!
After investigating two contexts listing possibilities and creating linear equations in standard form, we revisit the same learning target as yesterday: "Yesterday, we looked at the slope and y-intercept of a linear function," I say. "Today, we've seen one more key feature of linear functions: the x-intercept."
We discuss the idea that two points is really enough to graph a line. Even though students have been plotting other points today, if the goal is to graph a line, the two intercepts are really enough.
Tonight's homework is a Kuta Software-generated worksheet with 12 equations in standard form. The instructions I've written ask students to do the following:
- Find both intercepts.
- Use these to roughly sketch a graph of the line.
- Rewrite the equation in slope-intercept form.
I use the first problem as an example, and then students have the last few minutes of class to try this for themselves.