week 13

March 16, 2014 § Leave a comment

A pair of jeans costs $60. The sales tax is 8.25%. What is the total cost for the pair of jeans?
Ss1 — I rounded 8.25% to 10%, 10% of $60 is 6, so the total cost is $66. But I know the answer is slightly lower because I’d rounded up.
Ss2 — I knew 10% would be $6, so the tax is less than that. Then I just ignored the % sign and the 0 in 60, so that the math I did mentally was 825 times 6. I got 4950. Since I knew the answer was less than 6, I knew the number I wanted was $4.95. Add this to $60, that’s $64.95.
Ss3 — 
  1. 8% of $60 is $4.80
  2. .25% of $60 is 15 cents
  3. Add above to get $4.95
Ss4 — I set up the problem as 2 fractions being multiplied: (60/1) × (8/100) = 480/100 = 4.8. Add 60 to 4.8, that’s $64.8, but since I rounded the 8.25% to 8%, I think the answer is closer to $65.

What is the equation for the number of dots in this pattern?
Ss1 — I always see 2 groups of (n+1). The leftover dots are (n-1). My equation is D = 2(n+1) + (n-1).

Ss2 — I see the bottom dot separately. The rest are n groups of 3. So, it’s D = 3n + 1.

Ss3 — 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).

Ss4 — I saw (n+1) groups of 3 dots. But there are always 2 [red] dots missing. So, D = 3(n+1) – 2.

Ss5 — 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).

Ss6 — 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).


How would you find the area of the shaded?

4-leaf clover was created by drawing 4 semicircles of radius equal to 1/2 of side of square 
Ss1 — 
  1. Find area of square
  2. Find area of circle
  3. Subtract circle from square, this leaves area of the 2 white parts
  4. Subtract 2 white parts from circle gives only the shaded parts

Ss2 — 
  1. Find area of square
  2. Find area of 1 semicircle
  3. Multiply this by 4
  4. Subtract area of square from area of 4 semicircles gives you the shaded part only
Ss3 — (I drew in the 2 segments to hint students at this strategy.)
  1. Find area of 1/4 of circle
  2. Find area of right triangle within the quarter circle
  3. Subtract to get half the leaf part
  4. Multiply this by 8 


What is the equation for the area of this pattern?
Ss1 — I see a square of side n. The leftover is 2 groups of n with overlap of 1. My equation is A = n2 + 2n – 1.

Ss2 — I see 2 overlapping squares. The overlapped region is also a square. My equation is A = 2n2 – (n-1)2.

Ss3 — I see a large square that’s always missing 2 pieces: A = (n+1)2 – 2.

Ss4 — 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).

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