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Pattern talk 5

Picture
Pattern #6
Student 1
  • I don’t know the equation but I think I can build it. 
  • For step 43, you’d build 43 triangles on the bottom layer, then 42 on top of that, then so on and so forth until 1 at the top.

Student 2
  • I know the perimeter is always 3 times the step number. But I don’t know how to find the toothpicks inside the perimeter.

Student 3
  • I used a table of values and got (n + 1)(n/2) for the number of triangles, then multiply this by 3 because each triangle has 3 toothpicks.

Student 4
  • Oh, isn't what Student 3 shared the Gauss equation?

pattern talk 6

Picture
Pattern #115
Student 1
  • I see the top row as always (n + 1). The middle row is always (2n + 1). Then the next 2 rows are the same, each one is (2n – 1). So my equation is S = (n + 1) + (2n + 1) + 2(2n – 1).

Student 2
  • I always see these 4 squares on the sides. Then the top middle is always (n – 1). Then I see columns of 3 tall: an (n) column and an (n – 1) column. My equation is S = 4 + (n – 1) + 3n + 3(n – 1).

Student 3
  • I see the top row as (n + 1). Then there are always 2 leftover squares on the sides. The rest is a rectangle, always 3 high and (2n – 1) wide. My equation is S = (n + 1) + 2 + 3(2n – 1).

Mrs. Nguyen

I haven’t shared with kids how I see any of the patterns, but we had some extra time so I shared with them my way.

  • I see each step as a full rectangle, with pieces missing. The full rectangle is always 4 tall and (2n + 1) wide. But there are always 4 missing squares on both lower sides. And the top row is always missing (n) squares. My equation is S = 4(2n + 1) – 4 – n.

pattern talk 7

Picture
Pattern #102
Student 1
  • I see a square of (n) sides, then a piece missing in the corner. The equation is S = (n^2) – 1.

Student 2

  • I see a square of (n – 1) sides, then 2 groups of (n – 1) left over. My equation is S = [(n - 1)^2] + 2(n – 1).

Student 3

  • I see the top row as always (n – 1). The remaining pieces form a rectangle of height (n – 1) and width (n). The equation is S = (n – 1) + n(n -1).

pattern talk 8

Picture
Pattern #20
Student 1
  • I see that it’s always (n) tall and (2n + 1) wide. So my equation is H = n(2n + 1).

Student 2
  • I saw 2 sets of rectangles. One is always (n) tall and 2 wide — seen here in green, the other is always (n) tall and (2n – 1) wide. My equation is H = 2n + n(2n – 1).

Student 3
  • I see 2 squares of (n by n). Then there’s (n) leftover, H = 2n^2 + n.
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  • NT 1-4
    • PT 1-4
  • NT 5-8
    • PT 5-8
  • NT 9-12
    • PT 9-12
  • NT 13-16
    • PT 13-16
  • NT 17-20
    • PT 17-20
  • NT 21-24
    • PT 21-24
  • NT 25-28
    • PT 25-28
  • TEACHERS