Chapter 4 - ASN Explain

Many teachers ask students what we call "True/False" in worksheets and on tests and quizzes. A "true or false question" might look like this: " The diagonals of a parallelogram bisect each other." The correct answer is "True". The student has a 50 per cent chance of guessing and coming up with the right answer.

A question about the properties of a parallelogram could, instead, be asked in this way: "You are given the following statement, and are to decide if the statement is always true, sometimes true, or never true: "The diagonals of a rhombus are congruent." The answer would be "The diagonals of a parallelogram are sometimes congruent", or simply "Sometimes". Phrasing the question in this way gives the students a one in three chance of guessing the answer, and therefore is a better test of his knowlege.

But a more interesting question would be as follows: "Given the sentence "The diagonals of a parallelogram are congruent" decide if the sentence is always true (true for any parallelogram), sometimes true, or never true, and explain why." This not only eliminates guessing, it causes the student to think carefully about not only what the answer is, but why. It also gives the student practice in an informal type of proof. Of course, this all depends on the definition that your text book uses; in this case we are defining a parallelogram as a polygon with exactly two pair of parallel sides.

A student might answer this question as follows: The definition of a paralellogram is a quadrilateral with both pairs of opposite sides parallel. But this does not mean that one pair must be congruent to the other pair, as you can see in the diagram below:

If the parallelogram happens to have all angles right angles, which makes it a rectangle, and in that case the diagonals are congruent as shown:

Another example of an "ASN Explain" question might be the following:

If all sides of a polygon are congruent than all of the interior angles are congruent. Is this statement always, sometimes, or never true?

A student might answer as follows:

"If all sides of a polygon are congruent, it is only sometimes true that all of the interior angles are congruent. (If a triangle is equilateral then it is equiangular.) But, let's look at a hexagon for example, as in the drawing below: All of its sides are congruent, and all the interior angles appear to be congruent also.

But in the hexagon below, all the sides are congruent, but that the angles are clearly not all congruent to each other:"

The student might even choose to take this explanation further: "This is not only true for hexagons, but even for quadrilaterals every other polygon with the single exception of the triangle." (If a triangle is equilateral then it is equiangular.)

Asking the question in this way gives students opportunities to make conjectures and then test them, and to explore the issue by sketching various figures, making judgments and conclusions on his or her own rather than referring to a memorized list of rules.

In their reflections on this process, students expressed their thoughts about doing projects that were "outside the box", and not just problems directly from the textbook. Lina said this about the problem above: "When you read a theorem in the book you just believe it, because the book is always right. And it's easy to think that if a theorem works for one figure it will be true for another. This assignment really made me think again about that. I guess we have to keep our brains working, and not just make assumptions. What is true for a triangle is not necessarily true for anything else!"

And Jordan had this to say: "When you think about it, there are so many ways to look at math. And you really have to pay attention because what happens in one situation isn't what will happen in another. Hmmm. That's pretty true with life in general, isn't it!"

A somewhat related exercise is the one we did on "Proving Quadrilaterals Congruent". The idea again was to compare what is true about triangles with what is true about other geometric figures. As with the hexagons in the project above, the information one needs to be sure two quadrilaterals (or any other pair of polygons) congruent is different for each type of polygon.

The central question was this: "What is it about triangles that allows us to use these 3-letter "shortcuts" to prove them congruent? Why do we NOT have to prove all 6 pairs of sides and angles? Could you use these same shortcuts to prove two quadrilaterals congruent? If not, why not? What would it take to prove two quadrilaterals congruent? Always, If 3 sides of one quadrilateral is congruent to 3 sides of another quadrilateral are the quadrilaterals Always, Sometimes or Never congruent"? The students worked in groups of three or four to experiment and come to some conclusions.

Adam, Leah, Sara and Stacy began with these thoughts and conclusions: "In a triangle, there are a total of 6 sides and angles. In the three letter SAS, ASA etc. "shortcuts", you only have to prove 3 pairs of parts congruent to prove the triangles congruent. So you really only need to be given 3 pieces of information, like 3 sides, or 2 sides and the angle between. Once you have half (3 out of 6) pieces of information, you can prove the triangles congruent. You can think of a quadrilateral as being made up of 2 triangles. Therefore it would seem that you would need six pieces of information (sides and/or angles) to prove a pair of quads congruent.

Actually, we found out that you only need 5 pieces of information (sides and/or angles). The reason for this is that all quadrilaterals are made up of 2 triangles, if you just draw any diagonal."

In explaining how they did this, one group (Adam, Leah, Sara and Stacie) said: "If you can prove the two triangles of the one quadrilaterals congruent to the two triangles of the other quadrilateral, then you will have proved the two quadrilaterals congruent. Since it only takes 3 pieces of information (2 sides and an angle, for example) to prove a pair of triangles congruent, and since in each quadrilateral there a pair of triangles, there are 6 pieces of information except that the diagonal is shared, so there are only 5 pieces of information needed to prove the 2 quadrilaterals congruent. And, if you said "SAS" for triangles, we could use the same type of lettering for the quads and say: that the following 5-letter "shortcuts" would do it: SASAA, SASSS, SASAS, and ASASA".

The students proved each of these separate cases. You can see the proof of SASSS below:

In her reflection on this project, Nella said "This assignment really pushed my mind to its fullest. We had to try to decide if it was even possible to prove something, and if so, how? This challenged us beyond what we were taught; we really were "out there" in a new world!"

"No human investigation can be called real science if it cannot be demonstrated mathematically."

Leonardo da Vinci