Chapter 5 - Thoughts About Slope

This was a short project, but an interesting one. I asked the students a few questions concerning slope, and they wrote explanations and comments.

Here is the assignment: "What is slope? When say that a line has a slope of 2:3, what does that mean? What is the slope of a vertical line? And what is the slope of a horizontal line? What is the difference between 1/0 and 0/1? Explain using examples from algebra and examples. Explain your answers, and draw diagrams of each."

Mark explained what slope is in the following paragraph, by relating it to stairs:

"The simplest way to define slope is 'steepness', so if you are walking up a set of stairs, we call the vertical part of each stair is the 'rise' and the horizontal part (where you put your foot) is the 'run'. Most sets of stairs have the same rise and run as other stairs, so people don't have trouble walking up and down. When you climb a ladder, the slope of the ladder is usually much steeper than stairs are, so we would say that it the ladder is at a steep slope. A standard stair has a 7 inch rise and a 10 inch tread, which means that each goes up 7 inches vertically, and the flat part where you put your feet is 10 inches."

In geometry, slope is just like steepness, so if a line is at a steep angle, the slope of the line might be the ratio 7:3 and if the line is at a less steep angle, the slope might be 2:3. If the line is horizontal, we say it has a slope of zero (also called 'no slope') and if it is at a 45 degree angle we should say it's slope is 1 to 1 (or just call it 1). If it is vertical, we might want to say that the slope is 1:0 but that is called 'undefined' because 1:0 means 1 divided by zero and you can't divide by zero.

He included some diagrams with his explanation:

Meredith wrote some interesting comments about this assignment:

"This project was used to help us understand slope and parallelism. I enjoyed this assignment because it made me think about something that I had just always assumed to be true without thoroughly examining it. It took me a while to answer these questions because it was hard to explain in words why division by zero is undefined. It's easy to see why you can't divide something into zero groups but it's hard to explain in words why you can't do it. First I thought about it in everyday terms. I thought that if you have nothing, and try to divide that nothingness into 2 or 3 groups, you still have nothing. That's why 1/0 is 0. But try dividing something into zero groups! It's still there, so 1/0 is impossible! There's always something. Zero is just like infinity, in a way: they are both unimaginable to humans. It's kind of scary if you think about it too much. I thought I did a good job in my explantion, though. I tried to explain in a variety of ways so that someone reading it would understand and if they didn't get one explanation, the next might be clearer. The part which made it kind of challenging was trying to put my ideas into words so that other people would understand it. The way that I did this was by remembering what makes me understand. I guess that's what teacher and textbook writers have to do."

And Mark commented that he liked this assignment because it made him "think about something that I had just always assumed to be true without thoroughly examining it. It was a challenging assignment because it was hard to explain in words why division by zero is undefined. It's easy to see that you can't divide something into zero groups but it's hard to explain in words. However, I did my best and that's why I'm pretty proud of this paper. I kind of felt like I was the teacher, which was cool."

I am always happy to hear my students say that they are proud of their work. When students can feel that sense of accomplishment, they are inspired to do their best.


"To be a scholar of mathematics you must be born with talent, insight, concentration, taste, luck, drive and the ability to visualize and guess."

Paul R. Halmos

Go To Homepage         Go To Introduction

1) Constructions         2) Clock Problem         3) Test Corrections         4) ASN Explain         5) Thoughts About Slope         6) What is Proof?

7) Similar Triangles         8) Homework Corrections         9) Quads Midpoints         10) Quads Congruence         11) Polygons

12) Polygons Into Circles         13) Area and Perimeter         14) Writing About Grading         15) Locus         16) Extra Credit Projects

17) Homework Reflections         18) Students' Overall Reflections         19) Parents' Evaluate Method         20) In Conclusion