The Shapes of Numbers — The Curve

Earlier in our Shapes of Numbers block, we explored triangle numbers, square numbers, and primes. We looked at the different ways they related to each other by exploring their sums and differences. Next up — curves — a particular curve — the parabola. I think I only mention the word once though, at the end. Really our work is with doubles with an introduction to exponents.   The beginning is a review of doubles that we’ve done before from my Addition and Subtraction curriculum.

We start out with the one white cube from our Cuisenaire rods, and just keep doubling. From here, we determine what each line column of rods were:

White – 1

Red – 1 x 2 = 2

Purple – 1 x 2 x 2 = 4

Brown – 1 x 2 x 2 x 2 = 8

Each time we were building the doubles, we are showing the multiplication with the red rods (not pictured,) and it was getting cumbersome, and we were running out of red rods. Enter Gattegno’s crosses for review. We talk about how we could use these crosses to represent the area model of multiplication, and each time we added another red rod, making a tower of red rods, it represented doubling the entire number.

Orange and dark green – 1 x 2 x 2 x 2 x 2 = 16

3 orange and a red – 1 x 2 x 2 x 2 x 2 x 2 = 32

Anyone else tired of writing ‘x 2’? Yea, me too. Let’s come up with a shortcut notation. This ‘2’ will be our base, and a superscript will represent how many times we are multiplying the 2. So how many 2’s have we multiplied now? 5, so 25. That works! We’ll call that superscript an ‘exponent.’ That simplifies things.

Oh, but wait, our towers keep falling. Geez! How many red rods can you stack and maintain stability?!? I know! Let’s come up with another system. What do you notice about our towers? The number of 2’s we are multiplying correspond with the height of a rod. Let’s make an L, with the red rod being the base, and the exponent being represented by how tall the second rod is (Gattegno). Whew! That’s easier.

So now we have  

26 = 64

27 = 128

28 = 256

29 = 512

210 = 1,024

Now let’s go back and compare a regular staircase (1, 2, 3, 4, 5….) to our doubles (1, 2, 4, 8, 16, 32.) What do you notice? What shape does the staircase have as it rises?

Place a dot at the upper left corner of each rod, move the rods over, and connect the dots. What do you have? A line! Now, place a dot on the upper left corner of each rod for the doubles sequence. Move the rods, and connect the dots. What shape do you have? A curve. This curve is called a parabola. 

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