Visually exploring negative exponents and fractions

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What if the characters from The Lion's Share took 2/3rds of the cake rather than half? We modeled this with math tiles.

Math tiles arranged to show repeatedly taking two thirds of what is left.

 

Last week I explained some of the ways we were studying exponents here at my house.  This week we’ve been doing even more exploration. We started with the book The Lion’s Share which I’ve mentioned previously, and I asked my seven year old to make me a chart based on the book. The first column was to show what fraction of the cake each animal ate and how we could show that fraction with exponents. The second column was to show what number of cakes each animal promised to bake and how we could show that number with exponents. Easy exponent practice.

When the animals are taking 1/2 of what comes to them the amount taken and the amount left behind can both be described by the same number, an exponent of two. But what if each animal takes 2/3rds of the cake? Now the amount taken and the remaining cake care not equal. The amount remaining will be 3-n and the amount taken will be twice that 2(3-n).

 We modeled this using square inch tiles and a cake of 81 tiles. After the fourth animal takes his piece there is only a 1/81 piece of the cake, also known as 3-4. We discussed how in order to keep the smallest piece of cake a whole math tile we would need a larger cake, and quite a few more math tiles.

Using paper to model negative exponents

Using math tiles is limiting in some ways so I started ripping up paper. One task was to have my seven year old fold a sheet of paper, rip a third off, then fold that over, rip that off, and so on while we wrote out the exponent that he had created. Rip once and you have 3-1 and 2(3-1). Discard the larger piece and fold the smaller into three parts, ripping off one of them. That piece you ripped of makes 3-2 of the original sheet. Or in other words, 1/9th. Repeat. Repeat. Repeat while talking about what we’re doing as being repeated division. We talked also about how 30 describes the sheet before we divided it by three.

Modeling taking away 2/3rds of the paper, but using ripping the paper a different direction each time.

Then it was time to gather up the pieces of paper and make some art!

In picture number two each piece of paper is (approximately) two thirds the size of the paper it is on top of. The papers in picture number three have the same relationship (each are 2/3rds the size of the paper it is on top of) but I switched which directions I folded and ripped the thirds.

By ripping the thirds different directions we could explore the way the area is preserved but the perimeter changed.

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