Today’s submitter asked the following question:

Here are a selection of student responses:

- Slope is how far apart the two points are. You can use the idea of rise over run. You rise a certain amount and then run right or left depending ona negative or positive.

- Slope is how you measure two or more points or equations on a graph.

- It’s the y-intercept divided by the x intercept and intercept is the coordinates.

- Slope is the change in the dependent (which can be anything) over the independent (which can be anything).

- Slope is one point of a line on a graph minus another point on the same line in that graph.

- The slope is the difference of the dependent variable (y) over the independent variable (x).

- Slope is the rate of change between the dependent and the independent. If the dependent went up by 2 every time x, the independent, went up by 1, then the slope would be y over x which in this case is 2.

- Slope is when you take one point on a graph and a second one and see the difference between them and the numbers whatever y go over the numbers of the difference of x and that’s your answer.

- Slope is the rate of change in an equation, graph or story. For example, 2x -4 = y, take two possible solutions the rate of change is 2.

- Slope is the unit that you “go up by”. Let’s say you buy 4 apples for $4. The slope is 1 because it always goes up by one, 6 apples would be $5.

- Slope is the rate of change in an equation.

- The rate of change, between an independent and depending things.

- You have two points so all you do is see how far apart they are. We see that it goes from (4, 2) to (2,1) so it’s 2/1.

- Slope shows how a number increases or decreases and by how much.

- A slope is the change of rate in a problem for example, a man gets paid $5 an hour How much does he get after working for 3 hours? He gets $15, so 5 is the slope.

- Slope is when you have 2 numbers and find out what the answer will be.

There’s a lot to work on here. In the comments, pick a student and dig into their understanding of slope. And also, can you infer what the approach of the unit was from the students’ responses? How could the unit be improved?

## 4 replies on “Explanations of Slope from 9th Graders”

A quick thought: One of the comments that recurs throughout this list is that slope has to do with measuring or comparing two points. That’s not right, because slope measures the line connecting two points. That’s subtle, so instruction has to show that slope has to do with lines, not pairs of points.I think that guessing reasonable slopes of lines without the coordinate grid could help this. That activity isolates the lines from any points, and still associates slope with them.

The students seem very definition focused, as if they are trying to explain how to calculate slope instead of explaining what slope means. Last year I introduced slope with pictures of lines for an entire day before we worked on calculating it. They made up their own ways to describe which slopes were larger and smaller. I was hoping they would understand that it was the steepness of that line and then see that you pick two points and do a calculation to more accurately describe this change. I wonder if my students would still attempt to give such technical descriptions. I have found that slope is one of the most difficult things for them, AFTER you put the numbers in.

I used legos and rulers and match box cars to illustrate slope. The team of students draw a ratio of bricks to clicks. make a ramp with the iteration of so many bricks high to so many clicks ( little round circles on top of lego) over. Many time students are surprised that 2/1 and 1/2 look very different. We race cars down the ramp of the ruler taped to the tower. Before we do this we predict who will win the race down the length of the table with out falling off, or to the end. Most pick the steepest slope 4/1 but the car just usually crashes. Then we do 100 days posters or something similar, what does 100 days mean to you, 100 showers, 200 teeth brushing, etc. Then we bring in numbers. what increases/decreases in each event. ie negative slope for toothpaste left in tube. What day will it be empty etc.

What a great question, and a scary thing for a teacher to ask.

Just a clarification about the point of our participating in this exercise — it’s to interpret the students’ demonstrated understanding, right? Not offer how to do lessons differently to sidestep misunderstandings?

Curious if anyone thinks any of these students really “got it”?

The ones that describe carrying out the slope formula, or describe slope as the distance between points, I worry about them, even though the procedural descriptions were largely correct (making references to subtracting or differences, also dividing dependent by independent.) For them, I’d want the opportunity to clarify the prompt “but the 11 year old wants to know what it means, not how to calculate a value…can you say more?”

The ones that use accurate definitional words “rate of change” — I’d want to probe their understanding further. It’s impossible to tell if they understand or if they are parroting words.

I have the best feeling about the ones that use phrases like “go up by” and “how a number increases or decreases.”