5/9 Square Roots

Today I returned to a topic, square roots, I’ve done before with younger kids : 53-square-roots.  My thinking was that I had a cool video I wanted to show that mentioned approximating the numeric value of a square root and doing it ourselves would motivate that part of the video and provide some embedded practice calculating with decimals.

But before we could start in we needed to go over the old problem of the week.

“In an abandoned chemistry lab Gerome found a two-pan balance scale and three 1-gram weights, three 5-gram weights, and three 50-gram weights. By placing one pile of chemicals and as many weights as necessary on the pans of the scale, Gerome can measure out various amounts of the chemicals in the pile. Find the number of different positive weights of chemicals that Gerome could measure”

This one lent itself really well to whiteboard demos and I had kids present two different approaches. One was combinatoric, the other just brute force listed the cases. Since there is a max of 168 total in weight and only 9 weights either way is very approachable. However, none of the kids thought about putting the weights on both sides which presented me with a dilemma. I decided to approach it as follows “All the ideas you’ve mentioned are correct with some assumptions, lets make those explicit and double check if we can relax any of them.” That way I could honor the thinking already done and point out the further avenue to explore.

Next we started on the square root investigation. I framed the problem as follows:

“How does the square root key work on the calculator? What do you think its doing? I’m going to put up some simple square roots $\sqrt{2}, \sqrt{5}, \sqrt{7}$ can you find an approach to calculate value of these to a few decimal places (without a calculator)?”

What followed was a fairly useful exercise on two levels. There was a ton of calculation practice and several variants on the bound and search for a better fit algorithm emerged. At the end I also raised the question “For what other calculator keys do you wonder about how they work and are there any themes in the implementations?”  Logarithms were probably the best candidate for a future session. One possibility is stressing the use of series and iterative algorithms.

From there I tried to fit in an old problem on the same theme I’ve been saving:

Find the cube root of $x^6  - 9x^5 + 33x^4 - 63x^3 + 66x^2 - 36x + 8 $.   This we ended up doing as a group discussion. The kids eventually found the first and last coefficients of the root Ax^2 + Bx + C  but were stuck on the middle one B.  I really wanted to carefully go through the distributive law work and finish the solution but I was short on time and had to call this to a close. Moral: This is more than a 10 minute problem (and maybe more problems requiring polynomial mult. are called for)

In the midst of all of this I had audio issues with the projector and had to rush around to find another room we could use with a working system. So we filed over next door and spent the last 25 minutes watching this really fascinating 3Blue1Brown video which I alluded to originally:

Right at the beginning the square root approximation problem is discussed and hopefully it had extra resonance after trying it ourselves.

P.O.T.W.

A fun triangle inequality / golden ratio problem from @mpershan:

https://drive.google.com/open?id=1y-LAjQmdOuyTYd7NlNXfayi-AX73IF4CzkujMZta3PE

I’ve now seen an excellent numberphile video on Phi that I could easily integrate with this. So maybe we’ll do a golden ratio day. Although I also want to fit in a 3-D weaving art project as well in the next few weeks.