Approaching a challenging AMC 10 problem with our Zometool set

Last weekend we did a fun little project on Fibonacci numbers and pentagons that we found in Zome Geometry.

Fibonacci Spirals and Pentagons

I got a happy surprise this week when I saw that problem #22 on the 2015 was related to this project. Today I thought it would be fun to revisit the project and see if we could use our Zometool set to solve the problem. Although we didn’t get to the solution in a perfectly straight line, I think this is a great project for kids. If for no other reason, it shows how you can make progress on a really difficult problem using some basic ideas in math.

So, we started by looking at the problem on Art of Problem Solving’s website:

Problem 22 from the 2015 AMC 10b

 

The first step in solving the problem was building the shape out of our Zometool set:

 

Now that we had all of the pieces of the puzzle, we went to the living room floor to see if we could figure out any ways to approach the problem. My younger son came up with the idea of chopping up the long segments into segments that matched the one with length 1.

Right away we see that the length is 3 plus an extra medium blue. Then my older son has an interesting idea about how to measure the length of a medium blue.

 

Next we tested my older son’s idea that a medium blue was about 3/5 of a long blue. The great thing about this part of the project is that we ended up having a great talk involving fractions.

 

Having found one approximate answer in the previous video – the length is 3 3/13 – I thought it would be fun to try to find a different way of thinking about the problem. My younger son came up with the idea of measuring everything using short blues.

 

To save a little time, I had them build the models out of smalls off camera. Again we had a nice discussion about fractions.

 

Now we returned to the original problem to see if our two estimates for the answer would help us identify the right answer. The great part is the discussion of decimals (and numbers in general) we had here.

 

Finally, the last thing that I wanted to do was to try to connect this problem with the golden ratio. This is probably a bit of a stretch for younger kids, but it did spark an interesting thought from my older son – what if the lengths of the medium blue and long blue zome struts were related by the golden ratio?

 

The last part was a little unexpected, and perhaps not the most exciting way to end the project, but with my son speculating that the long and medium blues were related by the golden ratio, I wanted to show him what that would mean.

One interesting fact about the golden ratio is that it is easy to square. In fact, squaring the golden ratio is the same as adding one to it! That algebraic relationship helps is see that a relationship that we saw in the last video is exactly the same as multiplying by twice the golden ratio. Surprise!

 

So, a super fun project solving one of the most challenging AMC 10 problems. I think it is nice for kids to see that some simple math ideas can be used to solve some of these difficult problems. Along the way we got to have great conversations about fractions and decimals – so that was a nice bonus in this project. What a lucky break to have one of these contest problems almost match one of our prior projects!

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When we accidentally derived the law of cosines

We’ve been working through some of the challenging proofs in chapter 9 of Art of Problem Solving’s Introduction to Geometry book. As I’ve written, my son’s been finding these proofs to be pretty difficult:

 

BUT – he’s staying with them and seeming to get a little more out of each one that we go through.

Today’s challenge was prove that in an obtuse triangle, if we call the sides A, B, and C, with C opposite the obtuse angle, then C^2 > A^2 + B^2.

We spent about 30 minutes working through this proof and then I wanted to go through it again on camera. To my surprise my son took the proof in a completely different direction in the video and we ended up essentially proving the law of cosines by accident. Ha!

The first 5 minutes was introducing the problem, drawing a little picture, and starting down the path toward the proof:

 

In the second half, we looked at how our picture helped is sort through some complicated looking equations. It doesn’t seem to be progressing too quickly, then at 2:23 – “oooooooooohhh”

I wrap up at the end talking very briefly about the interesting expression that we stumbled on – the law of cosines!

 

Looking at some slightly more difficult proofs

We started a new section in our Geometry book this week and it starts with some properties of triangles that require slightly more difficult proofs that we’ve seen so far. My son is struggling with the difficulty and I have a lot of sympathy – learning techniques of proof was absolutely one of the most difficult things for me.

Today we were exploring non-right triangles and, in particular, when a triangle’s sides might satisfy a^2 + b^2 > c^2.

It took about 30 minutes to explore the idea this morning. Even with that long exploration, reviewing the idea again in our movie took a long time. Initially we just talked through about the problem and the initial ideas in the proof.

I broke this video after we write down three algebraic relationships in the triangle.

 

Once we had those relationships, the idea was to see how we could apply them to our problem. I think there are two parts of learning math that are particularly hard here. The first is understanding the connection between the geometry and the algebra. The second is understanding the relationships between the numbers / sides implied by the algebra.

 

Again, I have a lot of sympathy for my son as learning more difficult proofs really is hard. A line from an old Indigo Girls song comes to mind – “the hardest to learn was the least complicated.”

A simple to state math problem from finance

I asked my colleagues at work as well as my Facebook wall the following question today. It led to quite an interesting discussion in both places!

Consider the following transaction:

(1) You hand me $100 today.
(2) One year from now I give you $100.
(3) One year after that I give you an additional $10.

That’s it – no further cash flows of any kind happen.

Question – no calculators or anything like that – what is your gut feeling about the annual return you’d be receiving in this transaction?

When things don’t go so well.

I think I bit off more than I could chew this morning . . . .

We started a new section in our geometry book today and we were looking at an interesting question – prove that the side opposite the obtuse angle in an obtuse triangle is the longest side. My son did a nice job working through the problem. Even looking at the proof in two slightly different ways which was great:

 

But then for some reason I wanted to take a deeper dive, and oh my goodness did this deep dive not go well.

My idea was to get him to think about why some lines he drew in the prior proof were either inside or outside of the triangle. I just did a terrible job of helping him see the ideas when he was stuck and the whole discussion wasn’t that productive, unfortunately

Compounding the problem was that we had to get out the door for an appointment and we needed to stop the film. Ugh. I need to study this a little bit and figure out how to talk through the ideas here a little lot better next time.

 

Interleaved Practiced and math contest problems

Last week I read this interesting article posted by Pam Wilson on Twitter:

I don’t claim to know the ins and outs of interleaved practice, but I do something that sounds similar with my kids. Today seemed like a good day to write about it.

Each morning I have my kids spend a little time on some old math contest problems. Currently my younger son works on old MOEMS tests and my older son works on old AMC 10 problems. The goal is not speed or getting good scores, rather the goal is just to get exposure to math topics that we don’t happen to be covering just now. When problems given them a little extra trouble I spend a little time in the evening talking through them.

Today my older son struggled with an absolute value problem and my younger son struggled with a problem about averages. Here are those talks.

The AMC 10 problem with my older son is here:

Problem 13 from the 2010 AMC 10b

This is a tough problem for kids and requires a pretty good understanding of the absolute value function. After a rocky start by me (!) my son is able to talk through the basic ideas that are required to solve the problem:

 

After we have the four equations set up, we work through the solution for each 4. I went through this part in detail to emphasize that going back to the beginning to check the answers is important. Sometimes in equation solving you introduce solutions that are not actually solutions. We have one here that my son hasn’t noticed yet, but he does notice it later:

 

For my younger son, the MOEMS problems are wonderful. The two books we have with problems from this contest can be purchased here (oooh, and I didn’t even know until getting the link that there’s now a 3rd book out!!):

The MOEMS books at Art of Problem Solving

The problem that gave my son trouble today is about averages. We work through it two different ways and it seemed that the solutions made sense to him.

 

So, I like the concept of interleaved practice, and, I think, have been doing it unintentionally. I like using the old math contest problems for a couple of reasons. First, good problems for kids are hard to write and the problem from these contests are an easy source of well-written problems. Second, I like the variety – it has been quite a while since my older son and I talked about absolute value problems. This problem today gave me a nice opportunity to review that concept with him. With my younger son, I’m not sure that we’ve ever talked about averages formally. This problem served as a nice, informal way to talk through this concept.

I think there are lots of fun and important math ideas that kids can take away from these old contest problems.

Counting geometric properties in 4 and 6 dimensions

During yesterday’s project, my younger son asked me about tesseracts a couple of times. I’m not sure what caused his interest in 4-dimesional cubes yesterday, but I decided to explore tesseracts starting with Cifford Pickover’s Math Book as a way of catering to that interest. Pickover’s section on tesseracts has an interesting chart that served as a nice way to get the project going.

To start off I asked the boys about what patterns they saw in Pickover’s chart:

 

Next we moved to the living room to explore the beginning of the chart with our Zometool set. The goal was for the boys to see a connection between the corners and line segments as you moved up in dimension. It took a little extra time for them to see this connection, but eventually they were able to see the pattern.

 

Next we made a cube and counted up the corners, edges, and faces. Here the boys saw that there was a relationship between the number of faces in dimension n and the faces and edges in dimension (n-1). In fact, the relationship is exactly the same as the relationship we saw in the last video!

At the end of this video, my younger son describes how he thinks we can make a Zometool hypercube.

 

We made our hypercube off camera. I asked the boys to describe the new shape – it was interesting to hear their descriptions since I did not expect them to describe this one in terms of sliding. Their descriptions involved scaling rather than sliding.

After their descriptions, we tried to count the shapes that were in the chart in the Pickover book. They had a tough time seeing the 3-dimensional shapes at first, but eventually were able to count the 8 cubes (or “pyramids” as they called them).

The real trick was counting the 2-dimensional faces – this part was hard!

 

I took a break in the last video after about 5 minutes. The problem of counting the faces of the hyper cube was giving the boys a tough time. They knew from the chart that there were 24 faces, but their initial count was 36, so they knew they were over counting. But what and by how much were they over counting?

We worked through two different ideas in the counting process – one where you count each face twice, and one where you only count the “pyramid” sides. This two part process seemed to help them see how to count the faces.

Interestingly, counting the edges and corners was easy.

 

To wrap up the project, we returned to the kitchen table and the Pickover chart. We used the patterns that we saw from our Zometool set to fill in the numbers for a 6 dimensional cube. It was neat to hear them talking about the various geometric pieces of a 6 dimensional cube. Showing how to count all of these shapes without being able to visualize the complete 6-dimensional shape is a nice example of the power of math!

 

So, this project was a little longer than normal. Counting the properties of the hypercube was hard, but I’m glad we were able to work all the way through it. The neat thing is that the patterns themselves are not too difficult – multiple one thing by 2 and add another thing –