I know. It sounds crazy. I thought so, too, when I first read Don Cohen’s brochure about teaching calculus to 7-year-olds. As it turns out, kids are perfect candidates for learning tough subjects, especially trigonometry and calculus because of they’re endless curiosity and wiggly-ness. They like to figure things out for themselves and they LOVE learning what the “big kids” are learning.
Ian is one of those kids. He came to us in 4th grade because his then tutor was taking a full time job. I will be forever grateful that she did. She had already been teaching him trigonometry because he likes it. I started teaching him infinite series and the unit circle. Over the next year and a half, Ian calculated the values for the sine, cosine and tangent functions, discovered some trigonometric identities, found the relationship of the sine and cosine and where they are positive and negative in the quadrants of the trig unit circle, and more. Some of his work is below.
This is Ian’s first attempt at a trig unit circle. He started with 0° angle on the left and increased clockwise instead of 0° angle on the right and increasing counterclockwise.
This is Ian’s corrected trig circle. The 0° angle is also the 360° angle.
These are Ian’s calculations for the sine and cosine functions. On his trig circle, he measured the horizontal distance from the center for the special angles 0° through 90° and divided each by the measure of the radius. He measured the vertical distance for the sine. Then Ian used a calculator to find the values of the cosine and sine, then took the different between those and his calculations to find the error. The difference was very small. Ian’s measurements were excellent.
Ian started to compare the values of the sine and cosine and found that, as the cosine goes down, the sine goes up. It took several sessions and lots of experimentation to figure out that the sin(n)=cos(90-n). Ian was excited and enjoyed it. His comment at the end was, “why didn’t I see that sooner.!” Funny kid.
Most students who take trig are juniors and seniors and never truly understand this identity.
Ian learned how to plot on the Cartesian coordinate system. After being told that he always had to do the horizontal motion first, he wrote the instructions for plotting the point using coordinate pairs.
Ian noticed that the sine and cosine was positive or negative in certain quadrants. He noticed that the sine is positive in the upper quadrants, I and II, and negative in the lower quadrants, III and IV. The cosine is positive in the right quadrants, I and IV, and negative in the left quadrants, II and III.
Ian did this work at home. You may notice that the top two quadrants weren’t correct but he figured out his mistake. He tried using two colors but then decided that the four-color circle was best. It’s a great visualization of the positive and negative values of the the sine and cosine.
Ian figured out that if he added a multiple of 360 to the angle 60°, that he got the same result for the cosine. Ian also found that it didn’t matter if the multiple was a positive or negative multiple of 360. Good work!
Ian’s excellent graph of the sine function. When he finished he wanted to know “how to make more waves.” See below.
These are Ian’s notes for the results of using a graphing calculator to find different was to make more or fewer waves, i.e. changing the frequency. Ian found that multiplying the angle by 2 double the waves and dividing by 2 halved that waves.
One of the last things Ian did with us was calculate the tangent function. He divided the horizontal measurement from center by the vertical measure of the special angles, as before.
Ian did work on some infinite series but his love is trigonometry and why I focused on it in this post. If you think Ian is gifted or a genius, guess again. I’m not saying than Ian is average because he isn’t. Ian LOVES math and is willing to work on it even when he can barely keep his eyes open. Ian IS advanced and had an advanced start that was fed by two parents who have a fearless love of learning. Ian was interested so they just saw what he could do.
The same work that you see here, we can teach to a kid who is flunking math. We do it all the time. In fact anyone who can count can learn what you’ve seen Ian do.
(UPDATE: Since then, Ian has skipped a grade and is now taking Algebra I in 7th grade, a class normally taken in 9th grade. So, technically, he’s three years ahead in math. YAY!)