Graphing High School: Modeling linear functions Linear, Quadratic, and Exponential Models*

Linear Modeling and Graphing

What’s the mistake? How would you help?

Thanks to John Weisenfeld for the submission.

Algebra 1 linear functions Linear, Quadratic, and Exponential Models* slope

Slippery Slope

The submitter of this mistake reports that the student is a “high flyer.”

  1. What is this student’s confusion?
  2. Why made this mistake so tempting? After all, if a student just doesn’t understand “slope” there are a million things that they could do wrong. Why this one?

Thanks to Jeff de Varona for the submission. Go check out his blog and follow him on twitter.

Graphing Linear, Quadratic, and Exponential Models* slope

Slopedy Slope Slope

Say something intelligent about this bit of student work.

Algebra 1 factoring horizontal and vertical lines linear functions

“Graph x = 3”


The mistake is pretty easy. But what’s the underlying misconception?

Algebra 1 linear functions slope

Explanations of Slope from 9th Graders

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?