Struggles with estimating square roots

My #1 struggle in teaching my kids is failing to anticipate the topics that are going to give them extra difficulty. This week produced a shining example of that struggle.

I did two short projects with my younger son involving estimate square roots. He seemed comfortable with basic estimation questions – find the nearest integer to \sqrt{35} for example, but the slightly more advanced problems – say finding the nearest integer to 2*\sqrt{5} – gave him a tremendous amount of difficulty.

Here’s our first time through with 4*\sqrt{5}. What I didn’t appreciate is that he would want to focus on the value of \sqrt{5} as a starting point. While I was able to place \sqrt{5} in between two integers, the later multiplication by 4 caused some problems.


We discussed this type of problem a little more and I thought that we’d had some really productive discussions. However, reviewing a pretty similar problem led to difficulties that were quite similar to what we’d encountered the first time through:


Following this second struggle, we talked a little more about this type of approximating and I think this second round of discussions has helped him understand these approximations a little better. I wish that I would have understood ahead of time how difficult the transition to these more difficult problems was going to be.

Using Numberphile’s “blob Pythagorean theorem” video in a lesson

Ran across an interesting problem in our Introduction to Geometry book today that reminded me of an old Numberphile video. My son didn’t make the connection right away, but he did have some interesting observations about the problem so I thought it would make a fun little project.

I wanted to do a quick review of the problem first, but an accidental mistake sent us down the wrong path at the start. We sort of start over about 2 minutes in to the video below. Solving problems isn’t always a straight line . . .


Now that we were on the right path, we finished up the calculation and talked about the geometric situation. He recognized that the Pythagorean theorem was hanging around somewhere, and that there might be an analogous theorem for half circles:


After we finished up I had him watch Numberphile’s excellent “blob Pythagorean theorem” video featuring Harvard’s Barry Mazur:


So, a fun morning project (even with the little false start). Nice to see some of the ideas in Numberphile’s videos coming up when we study geometry.

Square roots day 2 – approximating the square root of 2

One of the examples in my son’s Prealgebra book today was prove that \sqrt{2} is less than 2. We were having a pretty good discussion about the ideas in this example, so I thought it would be fun to see if we could go a little deeper. Since we just talked about continued fractions last weekend, I was hoping that end up being able to find something to say that was much more accurate than just “less than 2.”

Our initial discussion of the problem is here:


Next up was the beginning of looking at \sqrt{2} as a continued fraction. We’ve spent very little time on this subject, so it is still new to him and we had to go slowly through the process. Luckily the continued fraction starts to repeat fairly quickly.


We finished up by figuring out some of the fractions that approximate \sqrt{2}. This exercise was why I wanted to go down the path of calculating the continued fraction. First off, we’ll see some of the fractions that we saw already in part 1. Second, we’ll find a couple better approximations, which is neat. Third, we’ll get to see directly that these fractions are nearly equal to 2 when you square them. AND, we get lots of good fraction practice in the process. Yes!


A fun geometry problem with frisbees – or technically two Discraft discs for the purists :)

Nearly three years ago I ran across this problem:

Two circles of the same size are tangent to each other in a plane. One of the circles stays fixed and the other circle rolls around the first circle one time. How many times does the rolling circle turn around its center?

A super fun, and easy to state problem. Here’s our first run through it (FamilyMath 23 – sheesh – we are over 250 now!!):


Well . . . today I ran across the same problem in our Introduction to Geometry book. Fun to see the ideas the the boys have 3 years later:


A first look at square roots

I started in on a new section with my younger son yesterday – square roots.

We have, of course, talked about square roots informally through the various projects that we do, but we have not discussed properties of square roots in detail until now. It is always really interesting to me to see the initial reactions that the boys have to these new subjects.

The first discussion today was about some basic properties that he already knew, and then I threw in a few twists. The idea that a square root has to be non-negative caused a little confusion. We ended up with a short discussion of i.


Next up was today’s lesson about finding square roots of large integers. I enjoyed his approach to the specific problem below, and think that the time that we spent in the first half of this school year studying number theory has helped him become comfortable with this type of problem.

I found his explanation of why you take half of a power when you take a square root to be really fascinating.


So, a nice start to our new chapter on square roots, and nice to see a little bit of the work we did in the Introduction to Number Theory book paying off.

A fun and challenging geometry problem from twitter

Saw this problem posted on Twitter earlier today (via a John Golden retweet) –

Not the easiest problem in the world, but since my son and I are studying a new section about circles in our Introduction to Geometry book, I thought I’d give it a try.

If you are interested in watching the thoughts of a kid as he struggles through a tough problem, today is your lucky day 🙂

First, an introduction to the problem and maybe 5 minutes of his initial thoughts. He’s walking towards the solution the whole way – slowly to be sure, but steadily.


So, I just turned the camera on and off to break the last video at approximately 5 minutes. In this second video he continues working towards the solution. Eventually he sees that the circle is the circumcircle of the triangle he’s drawn. That plus the area formula:

Area = A * B * C / 4R, where A, B, and C are the side lengths and R is the radius of the circumscribed circle gets him to the finish line.

Finding the approximate value of (5/2) * \pi confused him a little at the end, but he eventually was able to conclude that this expression was less than 8.


So, a fun exercise for me watching my son work through this problem, and a pretty challenging problem for him. Made for a good night. Thanks, as always, Twitter!

The grazing goat on a rope problem and a nearly special triangle

Our Introduction to Geometry book had the classic “goat on a rope” as an example problem. Rather than working through the example, I gave it to my son as a challenge problem and it really gave him fits.

I decided to talk through it with him using our Zometool set and during the talk we discovered that a 3 – 4 – 6 triangle very nearly has a 120 degree angle. Fun little exercise with a surprising geometric result!

First – the problem:


Next – I asked my son to share a few of his initial thoughts about the problem and talk about why it gave him a little trouble. During the conversation here we stumble on the 3-4-6 triangle and think that it might have a 120 degree angle.


Next we went to the table to try to work out the calculation with pencil and paper. My son was a little tired and got tripped up a little by one bit of the calculation, but we did get to the end.

At the end of this piece, we started looking at the 3-4-6 triangle, but decided to go to Wolfram Alpha:


Last up was a quick look at the 3-4-6 triangle on Wolfram Alpha. I probably could have found a way to do this part with the geometry that we already know, but we were both a little tired and investigating the triangle this way seemed just fine.


So, our Zometool set gave us a fun way to look at a pretty standard geometry problem. It is always really fun for me to watch my kids see a problem like this for the first time – I never know where it will go. The extra surprise that came from this particular problem set up was nice, too – a 3 – 4 – 6 triangle nearly has a 120 degree angle!

A fun Zometool story 

Last night we had some neighbors over for dinner at our new house. There were 5 rowdy boys watching a movie in the living room and the adults + one girl were in the kitchen talking and eating pie.

The girl seemed bored of the adult conversation but I saw this Zometool shape catch her eye:

I told her it was from one of our old math projects and brought out the set for her to play around with. She’s in 6th grade, though I don’t know what math classes she’s had, but what she built was awesome.

This was her first shape:

So many great questions to ask with this shape – in fact, I’m going to use it for a project with my kids next weekend. Two of the questions that jumped to my mind were:

(1) What is the area inclosed by the outer hexagon in terms of the area of the inner hexagon?

(2) What is the area of the region between the large and small hexagons in terms of the area of the small hexagon?

I think there’s also a fun opportunity to use this shape to talk about how areas of similar objects scale when the sides scale.

The next shape she made was this:

Here you’ve got an opportunity to talk about how to calculate the area of a trapezoid since her shape shows exactly how two trapezoids can be arranged into a parallelogram. Sort of reminiscent of this recent post from Ben Orlin:

The final shape she built was absolutely awesome – the trapezoids from the previous shape can be rearranged into a pentagon!

This shape has almost endless opportunities and I couldn’t resist showing her a recent project that we did with our Zometool set involving pentagons and Fibonacci numbers since (I think) a similar Fibonacci pattern will emerge from the pentagon she made:

Fibonacci Spirals and Pentagons

So, a super fun evening watching a middle school kid play around with a Zometool set for the first time. It really is incredible what comes out of just playing around with these amazing geometry tools.

Stuart Price and Joshua Bowman’s PIth roots of unity exercise

Saw this amazing post about the \pi^{th} roots of unity yesterday and wanted to use it for a quick extension of our \pi day activity:

Unfortunately, kids up at 11:00 pm, dog up at 1:00 am, cat up at 4:00 am and then everyone up at 6:00 am let to “one of those nights” . . . . So, instead of making use of a really great exercise, I sort of totally butchered it – but it is the idea that counts, right 🙂 Despite stumbling through our project this morning, I can’t recommend Price’s post and Bowman’s Desmos program enough.

For the kids to understand the project a little better, I wanted to do a quick introduction to the complex plan and how the roots of unity show up on the unit circle. We’ve talked a little bit about i before, so the ideas here aren’t totally new to the kids, but a quick re-introduction seemed appropriate:


Next up was a reminder of some of the rational approximate to \pi that we found yesterday in our activity inspired by Evelyn Lamb:

The fractions that we found that approximate \pi are 22/7, 333/106, and 355/113. We reviewed these fractions and also what the similar approximations to 2\pi would be.


With the background out of the way, we moved on to Bowman’s Desmos activity. First I just like the kids play around with it and see if they could find a situation in which we nearly had a regular polygon using powers of the \pi^{th} roots of unity. This was a fun “what do you notice” exercise.

Also, sorry for the extra blue screen – don’t know what happened to the camera here. Double also, ignore all of my talking for the first minute, please . . . . I was tired, confused, and incoherent.


Finally, having found the number 44 as a case where the dots where nearly equally spaced and having seen that this approximation was the same number we saw in the numerator of our 44/7 approximate for 2\pi, we looked to see if we’d see something interesting at 666 and 710. Right around 2:00 is the “wow” moment.


So, a fun project showing a geometric representation of some continued fraction approximations for \pi. Definitely one I’d like to have a 2nd, non-exhausted chance at, but oh well. Hopefully the awesome work of Stuart Price and Joshua Bowman shines through over my several stumbles in this project.

Celebrating Pi day with Evelyn Lamb’s idea

Last week Evelyn Lamb wrote a nice piece about \pi and continued fractions.

Since we’ve talked a little bit about continued fractions in the past, this seemed like a great way to celebrate \pi day. We started with a quick reminder about continued fractions:


After the quick introduction, we used my high school teacher’s fun continued fraction technique – Split, Flip, and Rat – to calculate the continued fraction for \sqrt{2}. This exercise gives you a great opportunity to talk with kids about fractions and decimals.


Next up was today’s activity – the continued fraction for \pi! Unfortunately, for this continued fraction split, flip, and rat doesn’t work so well. Nonetheless, we do get to have a good discussion about decimals while calculating the first two pieces of the continued fraction for \pi.


To calculate a few more parts of the continued fraction we went to Wolfram Alpha. Turned out to be a pretty neat way (and obviously a much quicker way) to see the next few numbers in the continued fraction. Again, we got to have a great discussion about decimals and reciprocals.


Now, having found a few terms in the continued fraction, we went and looked at what fractions other than 22/7 were good approximations to \pi. Happy 333/106 day everyone 🙂


Finaly (and sorry for the camera screw up on this one), I wanted to show a different continued fraction for \pi. In a previous video my younger son thought that we’d find a pattern in the continued fraction for \pi. We didn’t in the first one that we looked at, but there are indeed continued fractions for \pi that do have amazingly simple patterns.


So, a fun little project for pi day. A great opportunity to review lots of arithmetic in the context of learning about continued fractions and \pi.