pattern talk 9

Pattern #85

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.

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pattern talk 10

Pattern #45

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.

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pattern talk 11

Pattern #51

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).

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pattern talk 12

Pattern #100

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.