Grade 8 Unit 2 Lessons 10-12
When I say “equations of lines” what do you think of? Go ahead, write down the first 5 or so things that come to your mind .
Here is a collection of words and ideas I got from a few of my coworkers:
If I asked your students, what would their list look like?
I dug into the 7th grade curriculum to get an idea. A search of the course guide for the phrase equation of a line got zero hits. When I searched for just line, I got “vertical line” – not exactly what I was looking for. When I search for equations I got 29 hits. Many of them were about solving equations, but at least an equal number were about writing equations to describe a relationship. The focus for most of these equations was making sense of patterns specific to a context, but they also wrote equations to express a proportional relationship between numbers in a ratio table. Students also learned that graphing a proportional relationship created a line going through (0,0), but they never use the term equation of a line. That is a new idea for us to develop this year.
So my advice: Set aside all your preconceived notions about equations of lines. Because we will not be starting where our thinking about them begins. We are starting at where your students are currently at. And we will get to all those goodies eventually, but not during Unit 2. Be patient.
So in Unit 2 lesson 9, the IM curriculum introduces students to the idea that “the quotient of a pair of side lengths in a triangle is equal to the quotient of the corresponding side lengths in a similar triangle.” So in the figure below we could say:
In lesson 10, they focus our attention on two specific sides of a right triangle drawn along a given line, the vertical side and the horizontal side. See what a tiny step in thinking it is to make those the two sides we are comparing? And for every right triangle we draw along the same line, we find the ratio of the vertical side/ horizontal side are equivalent.
And finally in lesson 11 we talk about writing an equation to show the relationship of all the points that are on that line. Notice we are connecting to their 7th grade work that revolved around using equations to express relationships. So any two points you chose on that line can be used to draw a slope triangle. All of those triangles will have the exact same ratio of vertical side to horizontal side. So in unit 2, THIS is what we mean when we talk about equations of lines:
Ratio of the sides of one triangle = ratio of the corresponding sides of a similar triangle
More specific to slope:
Ratio of the vertical side to the horizontal side on a slope triangle drawn along a line = ratio of the vertical side to the horizontal side on another slope triangle drawn along the same line
And more general to make an equation that shows the relationship for any points on the line
Using the general point (x,y) as one of the points creating the slope triangle, the ratio of the vertical side to the horizontal side on that slope triangle drawn along the line = ratio of the vertical side to the horizontal side on another slope triangle drawn along the same line.
So shake your head and try hard not to see a messy version of y=mx+b. The equation we are writing is two equivalent ratios.
Lesson 11 takes on the work of saying how long the vertical and horizontal sides are if one of the points is going to be (x,y).
Let one point be at (3,1). “Slide” your (x,y) around and keep asking for the vertical distance.
What if y was at 4?
What if y was up at 7?
What if it was down at 2?
Instead of just writing
on those vertical sides, write
4 -1 = 3
7 – 1 = 6
2 – 1 = 1.
Then slide (x,y) way up past the edge of the grid, so they cannot tell the y value. Label it (x, y) and encourage students to use a variable expression to name that vertical length
y – 1
Do a rerun of the same logic with x.
The big idea for these equations of lines is that EVERY point that is on that line will make this proportional relationship true. Because no matter what point on the line you pick to draw your slope triangle, the slope ratio will be the same.
Here is an extra practice problem for the ideas from this section. Notice in the final part we use the equation to check and see if a point is one the line.