## pattern talk 21

**Student 1**

- I see the diagonal, then 3 groups of n on the outside, then the bottom group. My equation is
**D = (n+2) + 3n + (n+1)**.

**Student 2**

- Left and right, each is n. Top and bottom, each is (n+2). The leftover dot is 1 less than n. My equation:
**D = 2n + 2(n+2) + (n-1)**.

**Student 3**

- I saw these 4 in the right corner in every step. My equation is
**D = 4 + 2n + 2(n-1) + (n+1)**.

## pattern talk 22

**Student 1**

- I see top and bottom as 2 groups of (n-1). The middle has (n+1) by (n). My equation is
**R = 2(n-1) + n(n+1)**.

**Student 2**

- I move one rod from the bottom row to the top to make it a complete “rectangle” of (n+2) by (n). The leftover rods on the bottom row is (n-2). My equation is
**R = n(n+2) + (n-2)**.

## pattern talk 23

**Student 1**

- I always see 2 groups of (n+1). The leftover dots are (n-1). My equation is
**D = 2(n+1) + (n-1)**.

**Student 2**

- I see the bottom dot separately. The rest are n groups of 3. So, it’s
**D = 3n + 1**.

**Student 3**

- I saw step 1 in every step, so a constant of 4. Then I see (n-1) groups of 3. My equation is
**D = 4 + 3(n-1)**.

**Student 4**

- I saw (n+1) groups of 3 dots. But there are always 2 [red] dots missing. So,
**D = 3(n+1) – 2**.

**Student 5**

- I see 4 groups of n. But there’ll be overlaps to subtract. That overlap is (n-1). My equation is
**D = 4n – (n-1)**.

**Student 6**

- I always see two groups of 2 on the outside. The middle dots are (n-1) groups of 3. My equation is
**D = 2(2) + 3(n-1)**.

## pattern talk 24

*Didn't realize we repeated a pattern. We did this already in Pattern Talk 15.*

**Student 1**

- I see a square of side n. The leftover is 2 groups of n with overlap of 1. My equation is
**A = n^2 + 2n – 1**.

**Student 2**

- I see 2 overlapping squares. The overlapped region is also a square. My equation is
**A = 2n^2 – (n-1)^2**.

**Student 3**

- I see a large square that’s always missing 2 pieces:
**A = (n+1)^2 – 2**.

**Student 4**

- I see 2 groups of n on top and bottom. The middle is a rectangle of dimensions (n-1) and (n+1). My equation is
**A = 2n + (n-1)(n+1)**.