Teaching Trigonometry to a Fifth Grader

One of my students (I will name him Joy Boy) is in the fifth grade at an elementary school in Pleasant Valley. He and I have been working together for almost a year. His mother contacted Augustana College's math department to locate a tutor for him. Luckily for me, I happen to have a connection with one of the math professors there who knows about my penchant for working with kids doing enrichment for math.

Joy Boy sees patterns quickly. Last year he had a teacher who did not understand how to work with an advanced learner. (I am avoiding the term gifted, but it's probably accurate.) His parents wanted him to move ahead as he was bored, but no acceptable alternative was given to them by the school. This became my great fortune as I have the pleasure of watching Joy Boy's mind at work.

Joy Boy and I have done many things together over the past year including using Calculus By and For Young People by Don Cohen and working through worksheets from MathCounts. We discovered that Joy Boy needed to reinforce his skills when working with ratios and proportions. Ed Zaccaro's Challenge Math has a chapter on this subject, and Joy Boy already had a copy of the book. Guess which chapter follows Ratios and Proportions? Trigonometry.

Last Saturday Joy Boy and his mom met me at our usual time and place. He was not happy about trigonometry. He told me it was his "worst favorite" subject. His mom and I shared a quick glance and a shrug. Given that trig. is based on ratios, his second worst favorite subject, I told him it was no wonder.

Turns out that he didn't mind the ratios. He was just upset that instead of explaining how to figure out the trig. values for yourself, he was told to use a table or his calculator. He felt that he was cheating by doing so. He wanted to know how to figure everything out on his own. Oh, boy!

My first thought was to ask him why he would want to spend the time working out something that other people had already figured out, but I knew better than to go there. I asked him about pi instead.

Pi is a ratio (which is oxymoronic because pi is not a rational number, but that is a whole other discussion). It is the circumference of any circle divided by the diameter of the same circle. This ratio has a wonderful and rich history. I mentioned that pi crops up in the Bible (1 Kings 7:23). This led to some discussion as I had read that some people have used gematria to come up with a more accurate approximation of pi than the 3:1 ratio in the text. I showed Joy Boy how to convert the alphabet to numeric values and showed him how to find the numeric value of his name. We used digit sums and that brought up how we use those in our divisibility rules. I then told him that we were now entering the land of numerology, and I wanted to get back to math. Joy Boy was willing to accept that C/D = pi, which we approximate to 3.14. On to the next step.

Special Case Right Triangles to the rescue! I drew a picture of a 45-45-90 triangle and showed him how to figure out the sides using Pythagorean's Theorem (which he already knew). Then I showed him how to do the same for a 30-60-90. (I would like to point out that when I have helped to teach this concept to 8th graders, it usually sends them into a tailspin.) So far, he was with me. Then I showed him the mnemonic device to remember the trig. ratios: SOH, CAH, TOA. When we put it all together, I realized that neither of us had brought a calculator with trig. functions. Ooops.

Fortunately, his mom remembered that the book had a trig. table at the end of that chapter. Nice job, Mr. Zaccaro. The book is geared for elementary and middle school students who might not have an advanced calculator at their disposal (although one can be bought for just over ten dollars, even at a drugstore). After showing Joy Boy how to derive some of the values, he decided that he could use the table or a calculator to do the work. He just wanted to know that he could figure it out for himself and that he understood where the values were coming from.

Big sigh of relief from both his mom and me. He happily cranked out the answers to the rest of the problems with no signs of distress. Joy Boy, indeed!