 This post is a follow-up post to my post on making a Golden Ratio Gauge. Watching some you-tube videos, making the gauge and searching around the house for golden ratios was just the start of a week or so of studying the golden ratio… here’s what else we did…

Making Stars

The first step is to create your own little golden ruler. I outlined this in the first four steps of making the golden ratio gauge. You draw a square and expand it to make a golden-rectangle, then expand the rectangle to make a square and expand that square to make another rectangle. At the edge of the page you should have marks showing you a large space, a smaller space and then another large space. (On my star picture the larger space is marked “length A” and the smaller “length B.”) Use your ruler to draw one line of the star, horizontally about two thirds of the way up the page. Mark off the golden ratios in it. Set a compass to allow you to draw an arc “length A” and set the tip at one of the edges. Draw the arch lightly and then set up the ruler to find the line length B that goes through the arc at the corner (marked “C”) and one other time. This tells you how to measure the angle for the second and third lines. Once those two are measured out the last one is just joining the tips.

Practicing with Decimals

At some point my son announced that the larger line in a golden rectangle is almost one and a half the smaller line. I pointed out that it was close, but not quite. The ratio is Phi, the irrational number that starts off with 1.61803. If it was one and a half, I said, it would be 1.5.

I wanted to check if he could really understand what the number Phi means. So I cut strips of paper ten centimeters long, marking off the centimeters as I went. I asked him to use the strips to make a golden rectangle using the number Phi and I gave him scissors so he could cut one of the strips apart to show 6/10ths. I asked him to make another rectangle with the smaller side 20 centimeters long, and he did that. Then I gave him strips of paper 10 inches long, without the inches marked out, and had him repeat the activity that way.

We talked about how his rectangles were based on 1.6, but Phi is a bit more than that. How would we get the rectangles closer to Phi? We could take one of the 1/10th pieces he hadn’t used and cut that into ten pieces, and add one of those to the longer side. Then we could take another of them and cut that into ten pieces and add eight of those. We could take one of the remaining of those pieces and cut it into ten. This time we wouldn’t add any of them, but we would take one of those and cut it into ten and add three of those. We didn’t really do all this, just talked about how we could do it, and by this time he was giggling because he could understand.

Phi and Fibonacci Numbers

I had drawn out squares using Fibonacci numbers on graph paper. The first square was one unit squared, the second one unit squared, the third two units squared, and so on. The children had learned about this spiral from Vi Hart’s video. We used our golden ratio gauges to see that the ratios were close to a golden ratio.

Then I had my oldest write out the Fibonacci sequence and I showed him with a calculator that if we divide a Fibonacci number by the Fibonacci number that comes before it the answer is reasonably close to Phi, and the larger the two numbers are the closer it comes to Phi.

Making a Poster

After we had done several explorations of Phi and the Golden Ratio we made a poster hanging some samples of our drawings and calculations. The poster is hanging in the boys room as a reminder of what they learned, though at some point I’ll probably pull everything off it to allow them to make another poster about a different number…. 🙂

I know that the children don’t really need to know anything about Phi or golden ratios. Yet exploring them gave us a chance to play with math. We could practice using a straight-edge and compass. We could practice using a calculator and dividing.  We could review decimals and ratios.

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