Question

Dear Mitch,

I teach two classes of middle school math and one class of tenth grade math. There is one question in particular that comes up in all three classes as we cover geometry and approach the subject of triangles. It involves the concept that no side of a triangle can ever be equal to or greater than the combined length of the other two sides joined together. Do you happen to know of some hands-on method that would help the students see for themselves that this is always true and actually demonstrates exactly* why* it is so?

Thank you for taking the time to read this question and (hopefully) posting it on your wonderful website.

Respectfully,

Mrs. Hamilton

Washington State

Answer

Dear Mrs. Hamilton,

As a matter of fact, I do happen to have a 'hands-on' method that I've used for a long time -- and continue to use -- to enable the students to discover the exact nature of the concept and the real-world reason behind its truth. Here is what I do:

1. I write about twenty different sets of numbers on the chalkboard, with each set comprised of three separate numbers. For example, I might include sets such as these: (2,3,5), (7,15,5), and (4,5,6), among many, many others.

2. I break the class up into groups of two-to-three students, and pass around a large container of fresh (i.e., unused) Q-tips. That container, incidentally, is clearly labeled "I.Q. Tips". (A little joke/pun, there... obviously, a VERY little pun).

3. Next, I tell them that each Q-tip represents one line segment. (And they certainly do resemble the typical chalkboard drawing of a line segment, with one big endpoint of cotton on either side and one straight line connecting them, with each segment of virtually identical size.

4. I circle some of the sets of numbers on the board, and make the following offer: If anyone can make a triangle out of these numbers, using one of the 'hands-on' representative segments for each unit of the number, such as four Q-tips to create a line four units long, I will reward each member of that group with a gift of $100.00 per member. The RULES are simple and easy to follow: The segments that comprise any particular side of a potential triangle must, of course, be touching end to end (in other words, no gaps between the Q-tips,

5. Each line must be reasonably straight, which means the students must check each leg of their triangle for straightness aganst the side of a yardstick,

6. There is to be no overlapping (in other words, no line segment can be made of sets of parallel Q-tips.)

7. All of a circled group of lengths for a particular triangle's legs must be the same length indicated by the number of units depicted on the chalkboard, and

8. No side of a triangle can end up extending past any of that triangle's three 'corners' (or vertices).

The sets of numbers I have circled for them to use if they wish to 'go for the money' are, of course, the sets that have one of the three numbers equal to -- or greater than --- the combined lengths of the other two sides.

Everyone eagerly begins to work, almost feverishly, as there is a lot of money at stake and each possible set of numbers seems as likely as any other to be workable.

Then, within five to ten minutes, the enthusiasm becomes hushed as the students find they have to keep changing the angles of their triangles to accommodate the required lengths. And, almost always, within another five to ten minutes, the hushed enthusiasm diminishes to the silence of perplexed learners as, one-by-one, the students realize the problem. Then, frustration turns to jokes and nervous giggles as they sit back and realize that the lengths must have a certain relationship to each other for there to be any space remaining inside. In short, rather than triangles, they realize they are creating a set of seemingly parallel lines, with some groups of students creating two lines of equal length, and other groups of students staring at one short line and one long one.

Try it. I have had such success with this particular 'hands-on' experience (as have other teachers I've guided through it), that I have promised myself that one day I will film the lesson, making sure to zoom in on the faces of the students toward the end, as they discover for themselves something that one cannot help but think the students will never forget.

Hope this helps,

Mitch