Build the chair (part 2)

In class, we played with the NRICH Chairs and Tables activity (our outline here). We came up with an extension that we explored at home: making a sequence with smaller and larger chairs.

Kick-off

I knew that J2 had incorrectly counted the cubes in the NRICH sample chair, so I kicked-off by asking him to show me how he counted them. This was a surprising, and unintentional, kick-off. His method was to decompose the chair into three sections: seat, back, and legs. This upper left 2/3rd of this picture show how he determined the number of cubes in each section.

First Sequence

The previous picture also shows notes about how J2 thought of making the chair bigger or smaller. His idea was to keep the same size seat, but make the legs and back of the chair longer or shorter. You can see our drawing of back for the two next smaller chairs.

Along the way, we looked at the total number of cubes in each chair. A simple pattern jumped out to him: each step is a difference of 6 cubes. He quickly realized this was because each leg required one more cube (4) and there are two on the sides of the back.

I asked him how many cubes would be in the 10th chair. When I asked him to explain his thinking, he said, well, going backwards, the 0th chair should have 10 cubes, each step is +6, so I need 10 + 6x10.

From algebra back to geometry

His idea of the 0th chair really excited me as this was one of the ideas I had been hoping we would uncover. We talked about how this chair would look (a 3x3 seat with one cube in the middle of a side as the back). This was not something we would naturally have created when asked to build a chair.

What we had done is gone through a sequence translate from geometry to algebra, naturally extend the algebra to a new case, translate the new algebraic case back to geometry.

Second Sequence

One other delight in this activity was that J2's sequence was not one I had in mind when outlining the activity. Of course,  wanted to share my version, as well.

I followed his decomposition into seat, back, and legs. See if you can understand my notes and picture how my chairs are growing through the sequence. Chair D₃ is the starting example from NRICH.

I asked J1 to fill in the D₁ chair to see if he got the pattern.

Not linear

J2 noticed that, this time, the gaps between cube totals were not the same, but the second difference is constant.

To round up the discussion, we wrote down an equation for the nth chair and calculated how many cubes would be in the 10th chair. Finally, we tried our trick of extending to the 0th chair. 

This time, we realized that there could be something interesting in the 1st chair, too. See if you can build a version of our D₁ chair, using whatever your favorite building material might be.

2048 vs 2584

who: J1 and J2
Where: online
When: before and during violin lessons

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I don't have many games on my phone, but 2048 is there. It nicely fits J2's love of powers of 2, but J1 also really enjoys it. I will admit that I play a lot more than I think I should.

Assuming that you know the game, what do you make of this board:

Hmm, 2 and 8 are familiar, but 1, 3, 13, and 610?!

Searching recently for something related to the Fibonacci sequence, I found 2584, the Fibonacci sequence version of 2048. Thinking about it briefly, you will see why the sequence fits so nicely into this game structure. Of course, this was bound to be a favourite for the younger J's, too.

A bit of compare and contrast

J1 and J2 played back-to-back games, one round each, swapping in between. Then we talked about how the games compared. Because this was interspersed with other activities, J1 and I talked without J2 and then later got J2's opinions, but J1 waited to hear his thoughts before interjecting. Here were some snippets:

• J0: Which one is harder? 
• J1/J2: Fibonacci is harder. 
• J0; what does that mean, "harder"? Is it harder to play each step or harder to keep going in the game?
• J1/J2: Harder to play each step. For powers of 2, you just match up the number. For Fibonacci, you have to think about which numbers can combine together. 
• J0: which one do you think is harder to keep going?
• J1: the Fibonacci one is easier because each number can combine with two others. Like 3 can combine with 2 or 5, 5 can combine with 3 or 8, 8 can combine with 5 or 13, 13 can combine with 8 or 21.
• ---------------
• J0: are the games similar in anyway?
• J2: yes, both are on a 4 x 4 grid
• J1: yes, both have sliding number tiles that get added together
• ---------------
• J1: why do you win when you get 2584?
• J0: is it the closest Fibonacci number to 2048?
• J2: no, 1597 is closer
• J0: Let's see, what power of 2 is 2048?
• J2: 14
• J0: is that correct?
• J2: .... 11
• J0: is 2584 the 11th Fibonacci number?
• J2: let's count them
• <together>: 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584. That's 18 (or maybe we should call it 17 for purposes of the game?)
• J2: Oh, we knew it had to be more than 12 because 144 is the 12th (which is their favourite Fibonacci number right now because it is also 12 squared)