Finding the slope seems to trip students up every year, even though we work step-by-step through the different variations. Students seem to master each individual piece (*EXCEPT maybe the slope formula… hence, this blog post! We’ll get there…). We always have great conversations about types of slopes, making connections between all the different representations:
- special slopes (positive, negative, zero, undefined)
- slope from a graph
- slope from 2 points
- slope from a table
- slope from an equation (slope-intercept, point-slope, or standard form). For standard form, students must isolate the y first.
But then, when students take their end-of-unit assessment that pulls everything together, they fall miserably short. They forget what an undefined slope looks like. They forget that their slope should have been negative because the line goes down on the right. They forget that 3/6 does not equal 2, and that y goes on top! My honest opinion is that it’s too much for them to juggle all at once, regardless of how easy each individual step is.
Here are a few tricks I’ve used to help students remember some key information. Then we’ll get to why you’re *really* here:
(1) SLOPE DUDE! This may be (definitely) my favorite Algebra video of all time. I have probably single-handedly contributed 200,000 views. The kids think it’s silly–stupid even (My apologies, Teacher Tube Math). But they LOVE it. And when they don’t love it and forget that positive slopes go up on the right, I quote: “puff puff positive!” “Niiice, negative…” for the negative slopes. “This is zero fun,” for the zero slopes. My all-time favorite, though, is “THE WORST CURSE WORD IN ALL OF ALGEBRA: UNDEFIIIIINED!” Watch it. It’s great.
(2) Slope Man: A quick pictorial representation of positive, negative, undefined, and zero slopes using a face. The kids can sketch it themselves. I like the version with the negative sign on the left because it makes slope man look angry – students are always forgetting their slopes! More importantly, subliminally it makes sense because the positive x-axis is on the right side of the graph.
(3) VUX, HOY: Our 3-letter words we use to remember our special lines:
Vertical lines have Undefined slopes, and their equations are in the form X = #.
Horizontal lines have 0 Slopes, and their equations are in the form Y = #. HOY also happens to be the Spanish word for “today.”
Check out the VUX HOY Foldable I created for Teachers Pay Teachers
(4) Slope from a graph: When students count slope on a graph, I always tell them to pick 2 exact points where the graphed line hits the “city block corners” of the graph paper. Then we draw and label the triangle that results from the “rise” and the “run.” I prefer that students always start on the left and think of the “rise or fall” as the negative number–that way they always Run Right. I also emphasize the fact that they have to rise up out of bed before they can run to school.
(5) Slopes with Names: Have a student write his name on the graphed line. If his name goes up, then the slope is positive. If his name goes down, the slope of the line is negative.
(6) Dividing with zero: I have unabashedly adopted a song from CarFax (a fellow teacher) that goes to the tune of If You’re Happy and You Know It, Clap Your Hands. Some day I’ll put it out there on Youtube for the world to see… But in the meantime I am content with singing it “at” students who forget their 0 division rules:
“Iiiiiif the zero’s on the bottom, undefined!” (x 3)
“If the zero’s on the top, then zero’s what you got.
“If the zero’s on the bottom, undefined!”
Slope Between Two Points
I find that the slope formula works really well for honors students. They understand that (x1, y1) and (x2, y2) must be stacked vertically, with the y-values in the numerator. They understand that they must subtract the second ordered pair, which means a double negative turns into a positive.
However, general and special education Algebra 1 students often confuse the formula at the high school level. While we have conversations about the slope formula that they may have seen the year prior, I just about abandon the formula altogether and focus on a table approach. The best part is, students then don’t have to “learn how to find slope from a table!” Let’s be honest–we as *math* teachers and successful mathematicians understand the connections (that it’s the difference in 2 y-values, divided by the difference in 2 x-values), but my struggling students often think of the table as *just another thing* to learn. By using the table approach, we’re taking one more “thing” off their mental plates.
(1) Create an x-y table. I usually find that students need the reminder to label their x- and y-values, or they put the numbers in the wrong location. By the way, POINTS ARE ALPHABETICAL. x comma y. Think about it. Share it with them!
(2) Subtract the bottom row. We usually turn this step into “change the signs of the bottom row” after we discuss the fact that they’re subtracting y minus y and x minus x. I always show my sign changes in a different color to make the change clear to students.
(3) Write the fraction. y goes on top! Label the fraction m, to sear into their brains the fact that m = slope FOR-EV-ER! FOR-EV-ER!
(4) Simplify. Focus on the difference between 1/3 and 3. I started talking this year about how eating 4 slices out of 12 in a pizza is wayyyy different from eating 3 whole pizzas. And if kids need to use the (MATH) >> (FRAC) option in their graphing calculators, I let them. For now. We keep chipping away at number sense throughout the year…
*The only difference between finding the slope between 2 points in this manner and finding the slope from a table is that students have to choose only 2 points from the table! And the table is already set up for them! I actually have kids cross out the other rows so as to avoid confusion.*