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.

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**