Wednesday, July 23, 2008

Everything's a Square! Motivating Students to Use Deductive Reasoning

Figures Not Drawn To Scale

Hi y'all! Enjoying a restful peaceful summer? Just a few thoughts...

1. Only one correct submission to this month's math anagram"


Please follow the rules here and email me by the end of the month.

2. Ok, so what's going on with the 3 "squares" in the above diagram? In a recent post, we looked at using "Figure Not Drawn To Scale" as an effective way to encourage student reasoning and to become more cautious about making assumptions. Making a variety of quadrilaterals all "appear" to be squares as in the above diagrams is consistent with this approach.

Even before teaching the formal theorems and definitions regarding quadrilaterals in "Chapter 5" of the text, why not begin with a preview activity? If you prefer to wait until students have the necessary definitions, theorems and postulates, then one can use this as an application. Your choice...

Suggested Questions:
Does the given information in each diagram guarantee that each is a square?
If you don't think so, your mission is to draw a quadrilateral with the given information but clearly does NOT look like a square.
Alright, think about the first one. After a minute share your thoughts, diagrams with your partner. Go!

If using this to review the standard definitions and theorems on quadrilaterals, I would still encourage the drawing of diagrams to illustrate that the first two figures do not have to be squares.
Your thoughts...


Alex said...

You can get a counter-example for the third one by drawing it on a balloon...

Nice idea, by the way. Might use it.

Dave Marain said...

Thanks, Alex!
This activity played out nicely with "live" students. I like your spherical application -- nothing like asking students to show how a "triangle" can have THREE right angles!

Eric Jablow said...


That leads to a fascinating episode in the history of mathematics. Giovanni Girolamo Saccheri [1667–1733] was a Jesuit priest and mathematician who tried to prove Euclid's parallel postulate from the others. He eventually analyzed them by what became known as Saccheri quadrilaterals: figures ABCD where ∠A = ∠B = 90° and AC = BD. He showed that ∠C = ∠D, and worked out what would happen were these angles obtuse, right, or acute.

Saccheri showed that the obtuse hypothesis led to a contradiction. He showed that if the angles were right angles, all Saccheri quadrilaterals had right angles, and in fact that the parallel postulate holds. He finally worked out the implications of the acute case; he found so many odd and bizarre results that he assumed the results impossible, and claimed that this proved the parallel postulate.

Basically, he lost his nerve. However, people in 1733 simply didn't have the tools to think of such things. Had he kept on his course, he would have discovered Bolyai-Lobachevski-Gauss non-Euclidean geometry a century early.

YzW731 said...

For Fig I and II, I predict many students would realize that the horizontal sides of the 'square' can be simply extended to form a rectangle.

Though I doubt many would think of the triangle on a sphere...that's pretty tricky.