## pattern talk 9

*Kids didn't see the color version of this pattern, but in black and white, there was enough shading to make the distinctions.*

**Student 1**

- I see 2 squares. The top right square is n by n. The bottom one is (n + 1) by (n + 1).
- But I have to subtract 1 because they share this circle. My equation:
**C = n^2 + (n + 1)^2 – 1**.

**Student 2**

- I see the top square like Student 1 did. Then I see this rectangle of (n) by (n + 1), and the left over part is always n.
- Equation is
**C = n^2 + n(n + 1) + n**.

**Student 3**

- I see the whole thing as one big square of 2n by 2n.
- Then the two empty spaces are rectangles of n by (n – 1). My equation is
**C = (2n)^2 – 2[n(n - 1)]**.

**Student 4**

- The shadings helped me see my pattern. My equation is
**C = n^2 + (n – 1)^2 + 2n + 2(n – 1) + 1**.

## pattern talk 10

**Student 1**

- I see 2 rectangles of n by (n + 1), then the middle leftover is n. My equation:
**S = 2[n(n + 1)] + n**.

**Student 2**

- I see the whole rectangle, and a piece missing. My equation:
**S = (n + 1)(2n + 1) – 1**.

**Student 3**

- I see kinda the same as Ss2, but with legs. Each leg is n wide. My equation:
**S = n(2n + 1) + 2n**.

**Student 4**

- I see 2 bottom squares on n by n. The leftovers are 3 groups of n. My equation:
**S = 2n^2 + 3n**.

## pattern talk 11

**Student 1**

- Each step has these groups of 4 that matches the step number. But there are overlaps. The number of overlapped hexagons is always one less than the step number. My equation is
**H = 4n – (n – 1)**.

**Student 2**

- I see these groups of 3 plus 1 extra at the end. My equation is
**H = 3n + 1**.

**Student 3**

- Horizontally I always see there is 1 more hexagon than the step number (green). Then vertically there are groups of 2 that are the same as the step number. My equation is
**H = (n + 1) + 2n**.

**Student 4**

- I always see these 4 in every step (pink). Then there are these groups of 3. The number of groups is one less than the step number. The equation is
**H = 4 + 3(n -1)**.

## pattern talk 12

**Student 1**

- I don’t have the equation, but I know how it grows. Each step number adds the next cube. Step 4 has a 4x4x4 cube on the bottom.

**Student 2**

- It’s Gauss like. It’s kinda like adding consecutive numbers. Consecutive cube numbers. But I don’t have the equation.

**Student 3**

- I just squared the “Gauss equation” and it works!
**C = [(n^2 + n)/2]^2**.

*Before I saw Student 3’s equation, I asked him to give me the number of cubes in step 43, and he gave me the correct answer of 894,916. Then he shared how he got it.*