Symmetry and counting in geometry – a fun project for kids

[had to publish without editing, sorry – a little time constrained today]

A seemingly innocent problem in Art of Problem Solving’s Introduction to Geometry sparked a fun conversation about counting and symmetry this morning. It was such an interesting topic, and little background in geometry was required, so I asked my younger son to join in, too.

The original problem was this – If a regular polygon has 27 diagonals, how many sides does it have?

We started by talking about connecting points in a plane. Sorry for the blurry start at the beginning of this video, I wasn’t paying attention to the camera – only lasts about 10 seconds 🙂

It was interesting to me to hear how the boys counted the inside and outside line segments in the picture we drew here.

In the second step we looked at the number of line segments that you would need to connect “n” dots. Other than an odd comment about a factorial, it seemed like the boys caught on to the counting process. I tried to connect the formula we got here to adding up 1 + 2 + 3 + . . . + (n-1), but the kids didn’t see it, so I moved back to the geometry.

In the last part we returned to the original problem – how many diagonals are there? My younger son picked up on the (n-3) line segments that were diagonal, which was nice.

I also liked hearing their approach to solving (n)(n-3) = 54 at the end.

So, a nice little project coming out of a review problem in our geometry book. I like opportunities like this to show the kids a little more math than just the problem they were working on.

An old James Tanton-inspired 3d printing project pays off today!

Last summer James Tanton posted this problem on Twitter:

That problem led to a super fun 3D printing project with the boys:

James Tanton’s Geometry Problem and 3D Printing

To my great (and wonderful) surprise, today my older son and I came across a similar problem from an old AMC 10:

Problem #17 from the 2008 AMC 10A hosted at Art of Problem Solving

Here’s the problem:

“An equilateral triangle has side length 6. What is the area of the region containing all points that are outside the triangle but not more than 3 units from a point of the triangle?”

Of course we had to dig out our old 3d printed objects to start the problem, and then my son drew in the analogous 2 dimensional figures:

 

Next we moved to a second sheet of paper and my son solved the problem. I like his “inclusion / exclusion” way of counting the area here, though I understand that some people would probably prefer a more geometric approach.

 

How fun to have one of our old 3d printing projects help us solve a challenging AMC 10 problem!

Zonohedron part 2

After staring at the screen for 20 minutes, it appears that I’m to wiped out from shoveling another foot of snow to write. But, since we did squeeze in another fun Zonohedron project this morning in between shoveling sessions, here’s a brief description.

First, this is a follow up to yesterday’s project from Unit 14 of Zome Geometry:

A 242-sided Zonohedron

That project started at the end of the unit (literally the last paragraph). Today we went back to the beginning to talk about symmetry. We built some rhombohedra with 3 and 5 fold symmetry following the ideas in part 4 of section 14.1. Here are my sons describing the shapes that they built (~1 min each):

Older son, 3-fold symmetry:

 

older son, 5-fold symmetry:

 

younger son, 3-fold symmetry:

 

younger son, 5 fold symmetry:

 

After that, we moved on to section 14.2 to talk about an activity that the book calls “around the world.” The ideas here involve parallel edges of the polyhedra and symmetry. We first looked at the “around the world” activity with the 3 and 5-fold symmetric objects that we’d just built:

 

Finally, doing the “around the world” activity with the giant 242 sided object that we built yesterday. I think the activity here shows the enthusiasm that kids have for the zome projects better than any other activity that we’ve ever done:

 

A 242-Sided Zonohedron

Was flipping through Zome Geometry looking for a project for today and found a really cool picture at the end of Unit 14. The shape – a 242-Sided Zonohedron – was described as “the largest convex polyhedron you can build with red, blue, and yellow struts.”

This was a doubly nice description since our green struts haven’t made the move up to Boston yet!

Interestingly there was really no further description, and no instructions on how to build it. Seemed like a nice challenge, though, and perhaps also a lesson in perseverance. So, off we went:

 

Building this object was a struggle right from the start. Luckily my younger son noticed that there were 10 sided shapes and 12 sided shapes. We know how to build decagons out of blues, so here was the beginning of our first build:

 

But we got stuck after a while – as the shape curved up, it wasn’t forming the same shapes that we saw in the picture. Quite a puzzle:

 

Our next try felt right. The only surprise is that the shape is absolutely huge. I was proud of the kids for figuring out that we could use other zome pieces for support struts. If you look carefully at the remaining videos, the support struts became more and more eloborate:

 

The next video comes when we are about half way through. It is fun to hear the kids talk about the shape – especially the “almost squares.” It seems as though we are building the right shape now, so that’s good. We broke for lunch and a little shoveling after this video since we’d been building for about an hour already.

 

Now for the big finish!! My younger son is actually standing inside the shape in this video. It was fun to build and you can hear their enthusiasm as they talk about it. This video also shows why I love building shapes with our Zometool set – there are so many opportunities to hear kids talking about pretty advanced math concepts during (and after) the build.

For just a little throw away comment at the end of Unit 14 of Zome Geometry this was quite a project!

 

Powerball math

I heard some people in the office talking about Powerball today. I didn’t know how the prizes worked, but it turns out that some of the prizes have reasonably interesting math behind them. Thought it would be fun to talk them over with the boys.

First, the big prize – pick 5 number correctly from 1 to 59 plus get the Powerball correct:

 

Second, a small price ($4 it turns out) if you get the Powerball right, but no other number correct. This one is a little more tricky because you have to miss all of the numbers that were selected in the drawing (the non Powerball numbers, I mean):

 

As an aside, the FAQ on the Powerball site is hilarious. My favorite is this one:

The surprisingly awesome Powerball FAQ

Q: Your odds / probabilities are wrong.

A: Are not . . . .

ha ha.

What I learned working with my older son today

This problem – #15 from the 2006 AMC 10A – is a hard problem:

Problem 15 from the 2006 AMC 10A

“Odell and Kershaw run for 30 minutes on a circular track. Odell runs clockwise at 250 m/min and uses the inner lane with a radius of 50 meters. Kershaw runs counterclockwise at 300 m/min and uses the outer lane with a radius of 60 meters, starting on the same radial line as Odell. How many times after the start do they pass each other?”

I mean **hard**.

My number one struggle in teaching my kids is identifying ahead of time what problems are going to be relatively easy and which ones will be hard. I constantly miss on both sides, and I probably guessed incorrectly on this one by as much as I ever have.

But, spending 20 minutes talking through it helped me get a better feel for why this problem gave him trouble, and also a better feel for the how step by step process you need to go through to solve it. When I read the problem, that step by step process was essentially invisible to me, unfortunately.

A nice pentagon problem for kids learning geometry

Fun morning with my older son today as we slowly work our way through the section in our book about polygons. We stumbled on a neat problem about pentagons. It turned out to be a great problem for the white board and also a nice problem to look at with our Zometool set.

First on the white board:

 

Next with our Zometool set:

 

I was happy to run across a problem that we could approach two seemingly different ways. Happy also to see the Zometool set used not just for special project, but also for more day to day work in learning geometry.

Looking at the “Girls do better when their math exams are graded anonymously” paper

Last week the NYT had an article about girls and math that caused quite a stir. For example:

Girls Outscore Boys on Math Tests, Unless Teachers See Their Names

All of the mentions of the underlying paper by Victor Lavy and Edith Sand made me want to look a little deeper, so I bought a copy of the paper yesterday. It cost $5 if you are interested in getting a copy for yourself:

On The Origins of Gender Human Capital Gaps: Short and Long Term Consequences of Teachers’ Stereotypical Biases

After reading the paper, I think the headlines have gotten a little in front what the paper actually says. For example, this statement on page 11 of the paper might be surprising given the headlines:

“The distributions of this measure by subject are presented in Figure 1. English teachers in primary school over-assess girls (mean is -0.74) and the same pattern is seen for Hebrew teachers (mean is -0.41). Math teachers’ assessment in primary school, on the other hand, is on average gender neutral (0.01).”

The idea in the paper that seemed to get the most attention was the idea that girls did better when the graders didn’t know their names. The numbers backing up this claim are presented in Table 2 on page 32 of the paper.

With just over 4,000 boys and 4,000 girls tested, here were the results:

The “School” exam that was graded by teachers, with the overall exam average set to 0:

Boys: Mean score of 0.052 with a standard deviation of 0.985
Girls: Mean score of 0.003 with a standard deviation of 0.971

The “National” exam that was graded anonymously, with the overall exam average set to 0:

Boys: Mean score of -0.014 with a standard deviation of 1.034
Girls: Mean score of 0.014 with a standard deviation of 0.963

[post publication note – I had incorrectly typed “standard error” rather than “standard deviation” in the tables above when I published this. An accidental typo on my part. Table 2 in the paper clearly states they are presenting standard deviations not standard errors. Sorry about any confusion that caused.]

So, indeed the girls performed worse than the boys in the “school” exam and better than the boys in the “national” exam. But, the difference is miniscule – both groups essentially performed identically in both exams.

Maybe a little context helps understand the numbers, so I thought of the numbers this way:

Imagine that the 4,000 boys and 4,000 girls were flipping coins 100 times each in both exams. So, we’d expect to see “heads” 200,000 times from the boys and the girls in both exams.

In the first exam, the boys averaged (0.052) / (0.985) = 5.27% of a standard deviation higher. The standard deviation here is \sqrt{400,000}/2 = 316, so the boys got about 17 more “heads” than they were expecting to get. Similarly, the girls got 316*(0.003)/(0.971), or about 1 extra head.

So, in the “school” exam, instead of the 200,000 expected heads the boys got 200,017 heads and the girls got 200,001.

In the national exam the analogous numbers are: boys – 199,996 heads and girls – 200,005 heads.

Assuming that I’ve understood the presentation of numbers in the paper correctly (and I’d love for someone to double check my numbers here), I’m going to have a hard time attributing the difference between those two results to teacher bias, or anything else for that matter. You wouldn’t expect a measure of any group to be EXACTLY the same on two different tests. What we saw here was that the difference was next to nothing. Also, the difference in performance between the “anonymous” grading and “non-anonymous” grading for both the boys and for the girls seems to be, well, practically 0.

A nice surprise from my older son

Busy day today for me as I was shoveling snow from 5:00 am to 7:00 am and then drove 4 hours to a meeting. Yuck.

BUT, I did get to spend time doing a little math with the kids tonight, so yay!

In my tired state I thought I’d do a video about finding the area of a regular octagon with my older son. Unfortunately I gave no thought to the solution of the problem at all and figured we’d approach it the same way we approached the area of a hexagon.

Luckily my son surprised me with a much better solution:

 

To show him that his solution was much better than mine, I thought it would be instructive to show him the rabbit hole I’d planned to head down:

 

We’ve spent quite a bit of time talking about geometric solutions to problems rather than just algebraic ones. It is a nice little surprise to have my son come up with nice little geometric solution to a problem I didn’t really think all the way through! This might accidentally lead to a nice introduction to Area = (1/2) Apothem * Perimeter tomorrow.

Count like an Egyptian part 2 – Egyptian fractions

Last week I bought Count Like an Egyptian based on Evelyn Lamb’s recommendation, and we did our first project on multiplicaiton:

Going through Count Like an Egyptian with the Boys

Today we moved into the 2nd chapter of the book and looked at fractions. I’d actually heard the term “Egyptian fractions” before, but never really knew what it meant. The book gives a great explanation as well as several examples. With just two examples from me the boys were able to work through a couple of problems on their own.

I started with a quick introduction to the ideas behind Egyptian fractions:

 

With the introduction out of the way, I had the boys pick two numbers to form a fraction. They picked 8/13 which is quite a bit more complicated than the example in the first video, but also instructive. We work out that 8/13 = 1/2 + 1/10 + 1/70 + 1/910. There are also other ways to write 8/13 as a sum of reciprocals of integers. For example, 8/13 = 1/2 + 1/9 + 1/234. I wanted to stay with the “bread sharing” methods that the book used, though, and didn’t want to discuss multiple solutions just yet.

 

Now it was time for the kids to work through a problem on their own. My older son went first and worked through the fraction 7/10. He found it was equal to 1/2 + 1/6 + 1/30. Here I almost decided to show him how you could think of this as 1/2 + 1/5, but, again, I didn’t want to get into the multiple solutions today.

 

Finally my younger son gave it a shot and picked a big challenge – the fraction 11/17. He got a little confused by which number represented the loaves of bread and what number represented the people, but once we clarified that point he was able to work through to the end. He found that 11/17 = 1/2 + 1/8 + 1/48 + 1/816. Wow!

 

This was a fun project. It is definitely a neat historical lesson about fractions, but also provides a great way to review the way we think about fractions. Excited to try out more from this book with the boys next week.