week 15

What is 3.75 x 32?

Student 1

  1. 32 x 3 = 96
  2. I think of 0.75 as 3/4, and 1/4 of 32 is 8, so 3/8 is 24
  3. 96 + 24 = 120

Student 2

  1. I rounded 3.75 to 4, and did 32 x 4 = 128
  2. Since it’s 1/4 less, and 1/4 of 32 is 8, I subtracted 8 from 128 to get 120

Student 3

I changed the problem by multiplying 3.75 by 4 and divide 32 by 4.

  1. 3.75 x 4 = 3(4) + 0.75(2) + 0.75(2) = 12 + 1.5 + 1.5 = 15
  2. 32 ÷ 4 = 8
  3. 15 x 8 = 120

Student 4

  1. 3 x 32 = 96
  2. To multiply 0.75 and 32, I kept doubling 0.75 until it’s easy. So, 0.75 doubled is 1.5, and add another pair of 0.75 would give me 3
  3. Since it took 4 sets of 0.75 to make 3, then there are 8 of these sets in 32
  4. 3 x 8 = 24
  5. Add this 24 to 96, I get 120

Mrs. Nguyen

(I shared this strategy with them as no one had mentioned it.)

  1. 3/8 = 0.375
  2. So 3.75 is 3/8 of 10
  3. 32 x 10 = 320
  4. 3/8 of 320 = 120

Another strategy, similar to Student 3′s:

  1. 3.75 x 2 = 7.5 and 32 ÷ 2 = 16
  2. 7.5 x 2 = 15 and 16 ÷ 2 = 8
  3. 15 x 2 = 30 and 8 ÷ 2 = 4
  4. 30 x 4 = 120

Pattern #136

136

Student 1

I see 2 squares and 1 rectangle: S = 2(n^2) + n(n+1).

136a

Student 2

I see a full rectangle, then subtract the empty space: S = 2n(2n+1) – n(n+1).

136b

Student 3

I see 2 rectangles: S = n(2n) + n(n+1).

136cStudent 4

I move the whole left portion down to bottom of the right portion to have 1 rectangle: S = n(3n+1).

136d


What is 37.5 x 16?

Student 1

  1. 37.5 x 10 = 375
  2. 37.5 + 37.5 = 75
  3. 75 x 3 = 225
  4. 375 + 225 = 600

Student 2

  1. 37.5 x 2 = 75
  2. 16 ÷ 2 = 8
  3. 75 x 8 = 600

Student 3

  1. 40 x 16 = 640 (I did 4 times 16, then add the 0)
  2. I went over by 2.5, so 2.5 x 16 = 2(16) + 0.5(16) = 32 + 8 = 40
  3. 640 – 40 = 600

Student 4

  1. 37.5 x 2 = 75
  2. 16 ÷ 2 = 8
  3. 75 x 2 = 150
  4. 8 ÷ 2 = 4
  5. 150 x 4 = 600

Pattern #134

134

Student 1

I see the middle rectangle separately from the 2 sides. Then I move the left side pieces to join the right side pieces to form 1 rectangle: S = n + (n^2) + n(n+1).

134aStudent 2

I move the left pieces over to the right to form one large rectangle: S = 2n(n+1).

134b

Student 3

I see a Gauss equation. I add the top and bottom row, which is (n+3n). The number of pairs of with this 4n sum is the height of the pattern divided by 2: S = 4n[(n+1)/2].

134c

week 14

You need to cook a 24-pound turkey. The cookbook says it takes 15 minutes to cook every pound of turkey weight. How long will take to cook the turkey?

I stole this question from David Coffey. Actually, I just asked the kids what 24 x 15 is.

Student 1

I broke the 15 into 3 chunks of 5.

  1. 24 x 5 = 120
  2. 3 x 120 = 360

Student 2

I changed the problem to make the numbers nicer.

  1. 24 ÷ 4 = 6
  2. 15 x 4 = 60
  3. 6 x 60 = 360

When some of my 6th graders weren’t sure why Student 2′s strategy would work, I showed them this area model.

Student 3

I just saw 15 on the clock face as a quarter. It takes 4 of these to make 1 whole. So I did 24 divided by 4 to get 6. Then 6 times 60 (full clock) to get 360.

Student 4

I changed the problem.

  1. 24 ÷ 2 = 12
  2. 15 x 2 = 30
  3. 12 x 30 = 360

Student 5

I broke up 24 into 20 and 4.

  1. 20 x 15 = 300
  2. 4 x 15 = 60
  3. 300 + 60 = 360

Student 6

I broke up the numbers.

  1. 24 x 10 = 240
  2. 20 x 5 = 100
  3. 4 x 5 = 20
  4. 240 + 100 + 20 = 360

Student 7

  1. 20 x 10 = 200
  2. 20 x 5 = 100
  3. 4 x 10 = 40
  4. 4 x 5 = 20
  5. Add them all up to get 360.

Student 8

I divided both numbers by 3 to get easier numbers.

  1. 24 ÷ 3 = 8
  2. 15 ÷ 3 = 5
  3. 8 x 5 = 40

Because I divided both numbers by 3, I have to multiply the 40 by 3, and then by 3 again. So 40 times 3 is 120, then 120 times 3 is 360.

 


Pattern #44

Student 1

I see the shape as 2 halves, top and bottom. For each half, I see n groups of 3. But the groups overlap, and the overlap is always (n-1). My equation is H = 2(3n) – 2(n-1).

Student 2

I see the two middle rows of hexagons of (n+1) each. Then there are leftover groups of n on top and bottom: H = 2(n+1) + 2n.

Student 3

I always see 2 hexagons at the front. Then I see n groups of 5, but that includes the white inside hexagons, so I have to subtract them — and there are always n white ones. My equation is H = 2 + 5n – n.

Student 4

I see n groups of 6. But there are (n-1) overlaps of 2 each. My equation is H = 6n – 2(n-1).


Today is March 17. Write down as many equations as you can, each equaling to 17.

102 – 83 = 17

sqrt(289) = 17

(15 + 4) – 2 = 17

42 + 1 = 17

sqrt(9) + 14 = 17

abs(-16 – 1) = 17

25 – 15 = 17

52 – 8 = 17

 


Pattern #52

Student 1

I moved the front pieces to the left side and the back pieces to the right side, so it’s one flat rectangle. The equation is C = (n+1)(2n+1).

Student 2

I see the center column as (n+1). Then there are 4 identical groups around the center, each one is a Gauss addition. My equation is C = (n+1) + 4[(n2+n)/2].

Student 3

I see the center column as (n+1). Then I rearranged the front and back pieces together to form a rectangle. I do the same with the left and right pieces. My equation is C = (n+1) + n(n+1) + n(n+1).

(Similar to how Student 1 sees it, but she creates 2 separate rectangles instead of 1.)

Student 4

I take the left and right pieces (including the middle column and flatten them out into a rectangle like everyone else has done, but I see that I can move the pieces to form a square of (n+1) sides. So I have 2 of these because I do the same to the front and back pieces. Because I count the middle column of (n+1) for both squares, so I have to subtract it from my equation: C = 2(n+1)2 – (n+1).


Today is March 21. Write down as many equations as you can, each equaling to 21.

620 ÷ 2 – 300 + 11 = 21

52 – 4 = 21

85 – 64 = 21

sqrt(36) × 3 + 3 = 21

sqrt(400) + 1 = 21

42 × 2 – 11 = 21

4! – 3 = 21

week 13

A pair of jeans costs $60. The sales tax is 8.25%. What is the total cost for the pair of jeans?

Student 1

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.

Student 2

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.

Student 3

  1. 8% of $60 is $4.80
  2. .25% of $60 is 15 cents
  3. Add above to get $4.95

Student 4

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.


Pattern #130

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).


How would you find the area of the shaded?

This 4-leaf clover was created by drawing 4 semicircles of radius equal to 1/2 of side of square

Student 1

  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

Student 2

  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

Student 3

(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


Pattern #106

Student 1

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.

Student 2

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

week 12

What fraction of this square region is shaded? Stripes are equal in width, and the figure is drawn to scale.

Student 1

I don’t have the exact answer, but I know it’s more than 1/2. I visually drew in grid lines.

Student 2

I mentally drew in grid lines like Ss1 did. It’s a 6×6 square, so 36 square units. Starting in the left corner, I saw 1 white square. Diagonally from there, I notice these are consecutive odd numbers. I just added the white parts to get 15. Subtract 15 from 36 gives 21. So, 21/36 is the answer.

Student 3

I got 7/12. I split it up into smaller squares too and got 36 because it’s sides are 6 each. There are 21 shaded squares, 21/36 simplifies to 7/12.


Pattern #124

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

I have the exact same equation as Ss1, but I saw it differently, like this:

Student 3

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 4

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


Can you figure out the rule: 2, 4, 8, …?

I had fun watching this video from Veritasium, so I posed the question to the kids. The quickest answer came from a 6th grader, and he even used the word “ascending” in his rule!


Pattern #126

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).


Three corn dogs cost $5.25. How much does one corn dog cost?

I asked this question because Justin Lanier shares how he and his friend had done this math problem mentally.

Student 1

I just divided like I normally would on paper. Yea, I just set it up in my head because it’s not too bad.

Student 2

I knew that if the price had been $6, then each corn dog would cost $2. But it costs 75 cents less than that, so 75 divided by 3 is 25. I subtracted 25 cents from $2 to get $1.75 for each.

Student 3

I knew each must cost at least $1. Then I split up the leftover $2.25 into thirds: three 25-cents are not enough, three 50-cents are still not enough, so I tried three 75-cents, and that added up to $2.25.

Student 4

I used the divisibility rule to see if $5.25 is divisible by 3: 5+2+5 = 12, and 12 is divisible by 3, so I knew the amount is divisible by 3. I just tried some numbers and $1.75 works.

Student 5

I changed $5.25 to 21 quarters. 21 divided by 3 is 7 — 7 quarters is $1.75.

Student 6

I tried $1.50 first, that got me $4.50 for 3 corn dogs. I tried $2, and that’s $6 total. So I tried the middle number of $1.75, and it worked.

When I shared with the students how Mr. Lanier had done the problem, they said they would never have guessed that! I said to them, “Math Munch people are just cool like that.” :)

week 11

Mike Lawler did this week’s math talks with his two young sons and shared his post. Thank you, Mike!

What is 0.48 × 650?

Student 1

  1. 650/2 = 325
  2. 0.01(650) = 6.5
  3. 6.5 + 6.5 = 13
  4. 325 – 13 = 312

Student 2

I only did 650 divided by 2 to get 325. But I knew the real answer would be less than 325.

Student 3

  1. 600(.08) = 48
  2. 50(.08) = 4
  3. 48 + 4 = 52
  4. 600(0.4) = 240
  5. 240 + 52 = 292
  6. 50(0.4) = 20
  7. 292 + 20 = 312

Student 4

  1. 650/2 = 325
  2. .02(100) = 2
  3. 2(6) = 12 (there are 6 one-hundreds in 600)
  4. 50 [from the 650 given] is 1/2 of 100, therefore I added 1 to the 12 to get 13
  5. 325 – 13 = 312

Pattern #6

I tossed out this pattern for 2 reasons:

  1. They’d already seen this pattern toward the beginning of our math talks — it was during week 3 — and I was curious to learn of students’ growth in pattern reasoning.
  2. Zack Patterson‘s students saw them as triangles [instead of toothpicks in the answer key] – and there were at least 3 ways to see the triangles — thus I was grateful he’d pointed this out.

Student 1

I always saw 3 groups of consecutive numbers: along the horizontals and diagonals. For example, on step 3, I saw 3 groups of 1 + 2 + 3. My equation is T = 3(n + 1)(n/2).

Student 2

I saw the base as always the same as step number, so base is n. Then I see the 2 sides also as n each. I did the table of values for the middle toothpicks. Step 1 had none, step 2 had 3, step 3 had 9, step 4 had 18, step 5 had 30 — but I ran out of time.

Student 3

I just saw the pattern as stacks of triangles. So, step 4 would add 4 more triangles at the bottom. Step whatever would have that many triangles at the bottom, and on and on to the top with one triangle. Each triangle has 3 toothpicks, so it’s really Gauss equation times 3. My equation is T = 3([(n^2)+n]/2).


There are 2,790 miles of flight distance from Los Angeles to New York City. This distance is equal to how many inches?

This question was inspired by this tweet:

Student 1

I knew the math was 2790 times 5280 because that’s how many feet are in a mile, then times 12 for number of inches, but I just blanked out, didn’t know where to even begin. I’m tired. Sorry.

Student 2

I rounded all the numbers.

  1. I multiplied 2800 by 5000. I did 28 times 5, which is 140, then I added 5 zeros, so that’s 14 million. Right?
  2. Anyway, 14 million times 12… I just did 14 million times 10, so that’s 140 million.

Student 3

I wasn’t sure if there were 5280 feet in a mile or 528 feet in a mile. I went with the wrong one, thinking there was about 500 feet in a miles, so I multiplied 3000 — I bumped 2790 to 3000 — by 500. That’s 15, then add 5 zeros. I got lost a little bit. I think I got 1,500,000. Then I ran out of time.

Student 4

I rounded 2790 up to 3000, then I rounded 5280 down to 5000. I figured if I rounded one number up and one number down, then my answer would be good. It’d be pretty close.

  1. 3000(5000) = 15,000,000
  2. To get inches, I separated the 12 into 10 and 2.
  3. 15 million times 10 is 150 million
  4. 15 million times 2 is 30 million
  5. 150 + 30 = 180… 180 million

Pattern #75

These are all from my 6th graders!! I’m so proud of them. This pattern is too easy apparently for my 8th graders :)

Student 1

I always see this top one alone. Then I see 2 groups, one on each side, each group has twice the n number. But they share 1, so minus 1. My equation is D = 1 + 2(2n) – 1.

Student 2

I see these 2 in the middle. The I see the outer wings as n each, the inner wings as (n – 1) each. My equation is D = 2 + 2n + 2(n – 1).

Student 3

I have D = 2(n + 1) + 2(n – 1).

Student 4

I see D = 4n. Hehe!

(So damn cute how she took joy in her simple equation!)

Student 5

(This one blew my socks off!)

I see D = 4n too, but I rearranged the dots into a parallelogram: its base is always 4, its height always n.

week 10

Which is greater, 65 x 47 or 67 x 45?

These answers are from 6th graders. My 8th graders had quizzes today.

Student 1

I think they are equal because 67 minus 65 is 2, and 47 minus 25 is also 2.

Student 2

We had something like this before. They both have 65 x 45 in them, so there are still 2 x 65 in the first one and 2 x 45 in the second one. It’s obvious 2 x 65 is bigger, so I know 65 x 47 is bigger.

Student 3

I rounded the first one to 65 x 50, and I can do that in my head to get 3250. I ignored the zero then put it back later. Then I rounded the second one to 70 x 45, which is 3150. So the first one is bigger. I’m 99.8% confident.

When I asked why her confidence was not 100%, she replied, “Okay, it’s 100%.”

Student 4

I have this picture in my mind [she came up and drew this sketch]. You told us that a multiplication problem is like the area of a rectangle, so I imagined these two overlapping rectangles. Kinda like what Ss2 shared — the overlap is 45 times 65. So you can see the leftover areas: the 2 by 65 is bigger than the 2 by 45.


Pattern # 77

Holy cow. Students had more ways to see this pattern than any other so far. I had to extend the talk over two days because we’d already used up 20 minutes of class on the first day! I’ve been calling on students (first 3 at random) to share the equation if they have it, then I ask the class to try to see where each term in the equation comes from.

Student 1

I see the perimeter as 4 groups of (n-1), plus the 4 corners. Then I see these diagonals inside [brown] — there are always 3 groups of (n-1). The rest [green] makes a rectangle if I flip them and put them together; the dimensions of the rectangle are (n-2) and (n-1). My equation is T = 4(n-1) + 4 + 3(n-1) + (n-2)(n-1).

Student 2

(I love how this kid saw this!!)

I see a square, or the toothpicks make up a square of (n+1) sides. Then there are always 2 left over groups of (n-1), so T = [(n+1)^2] + 2(n-1).

Student 3

I see the perimeter as 4n. The inside diagonals [orange] are 3 groups of (n-1). The these two groups [purple] are Gauss, with (n-2) as the highest term. My equation is T = 4n + 3(n-1) + 2[1 + (n-2)][(n-2)/2].

Student 4

I see this middle [green] rectangle with sides (n-1) and (n+2). Top and bottom are (n-1), then this extra [light blue] row is (n-1), plus the 4 corners. My equation is T = (n-1)(n+2) + 2(n-1) + (n-1) + 4.


(California’s area is 163,695 square miles, about 1.7 times larger.)

Student 1

Twice as big.

I rounded up to 100000 square miles. That times 2 to get 200000. I subtracted 1400 from 200000 because I’d rounded up, so about 198600.

Student 2

I think California is about 1.5 times bigger.

I took half of 98380 to get 49190. Then I added this to 98380 to get 147570.

Student 3

I think it’s 1.7 times bigger.

I rounded up to 100000 square miles, so around 170000 is my estimate.


Pattern #78

Student 1

I see n columns of 3. There are always 2 horizontal [green] groups of (n+1). Top and bottom [yellow] are each (n-1). The left over on the sides is always 6. My equation is T = 3n + 2(n+1) + 2(n-1) + 6.

Student 2

I see the pattern growing from the middle out. So the outside has 5 toothpicks on each side, that’s always 10. Vertically, there are n groups of 3. Horizontally, there are 4 groups of (n-1). So, T = 10 + 3n + 4(n-1).

Student 3

The middle here [red] are two groups of (n+1). Then these vertical pieces [green] are 3 groups of (n+2). Top and bottom make up the rest, each is (n-1). My equation is T = 2(n+1) + 3(n+2) + 2(n-1).


source: Would You Rather

Student 1

I’d choose 50 acre at 42%.

  1. I changed 58% to 66% because 66% is 2/3, and 2/3 of 30 [acre] is 20.
  2. I did 50% of 42 is 21.
  3. I bumped up step 1 to 66% and it’s still lower than the 2nd step (20 versus 21), so I know the 2nd option is higher.

Student 2

I chose the 30 acre at 58% because…

  1. 60% of 30 is 18.
  2. 40% of 50 is 20.
  3. Even though 20 is higher than 18, I’m wasting 30 acres of land [50 minus 20], and with the other one [18], I’m wasting only 12 acres [30 minus 18].

Student 3

I agree with and my math is the same as Ss2′s. A lot of water is wasted on the larger 50-acre plot because of the lower survival percentage.

Student 4

I compared the two percentages with each other and the two acres with each other.

  1. There’s a 2/5 difference between 30 and 50 acres. And 2/5 is 40%. So, I’m gaining 40% in land.
  2. 58% minus 42% is 16%, so I’m losing only 16% of the survival rate.
  3. This is why I’d chose the 50-acre plot.

week 9

What is the area of this rectangle?

Student 1

  1. I multiplied 5.25 by 2, but I actually just added 5.25 to itself to get 10.5.
  2. To multiply 5.25 by .5, I just divided 5.25 by 2. Half of 5 is 2.5, and half of .25 is .125, so that’s 2.625.
  3. Then time was up before I could add the answers together.

Student 2

I split the rectangle into two, one of them being a square of 2.5 by 2.5 because I already know that’s 6.25. The other is a 2.75 by 2.5 rectangle, but then I couldn’t do that in my head.

Student 3

  1. 2.5 = 2  1/2 = 5/2
  2. 5.25 = 5 1/4 = 21/4
  3. 5/2 x 21/4 = 105/8

Student 4

I just realized that my answer is wrong.

I doubled the dimensions just because those numbers would be easier to do, then I multiplied them to get area. Then I divided this by 2 because of my earlier doubling, but that’s not right when I look at my answer.

(We then had a discussion about why Student 4”s answer was incorrect. We talked about linear and area measurements, that doubling the linear dimensions which then get multiplied means the 2-dimensional area will be 4 times what it should be.)


Pattern #106

Student 1

I see a full square then 2 corner pieces missing. My equation is S = (n + 1)^2 – 2.

Student 2

Top and bottom rows are n each. The middle rectangle is (n-1) by (n+1). My equation then is S = 2n + (n-1)(n+1).

Student 3

Mine is similar to Ss2′s, but I thought of the middle as a square with (n-1) sides and 2 groups of (n-1): S = 2n + (n-1)^2 + 2(n-1).

Student 4

I see two overlapping squares of n sides. Then just subtract the overlap: S = 2n^2 – (n-1)^2.


How do these two containers compare?

Student 1

  1. I ignored the 6.2 because they both have that dimension, then I multiplied the other two.
  2. But I didn’t fully do 3.4 x 2.8. I only did enough to check. I did 4 times 8 (the tenths place digits), that’s 32, so the answer ends in a 2. I carry the 3. Then I did 3 times 2 (the ones digits), which is 6, 6 plus the carried 3 is 9. So I know the answer starts with 9 and ends with 2.
  3. I did the same with 5.6 times 1.7 and learned that the answer also starts with 9 and ends with 2.
  4. I’m confident the 2 containers have the same volume.

Student 2

  1. I rounded all the numbers.
  2. 6 x 3 x 3 = 54
  3. 6 x 2 x 6 = 72
  4. So I think the one on the right is bigger.

Student 3

  1. I noticed the 6.2 is the same on both containers. Then I noticed that 1.7 is half of 3.4 and 5.6 is twice 2.8. I then tried with an easier set of numbers to see if the products would be the same.
  2. So one container might have dimensions of 3, 4, and 5.
  3. The other container would then be 3, 2, and 10.
  4. Both of these sets, when multiplied together, give the same answer of 60. So they both have the same volume.

Pattern #68

Student 1

I see a full thing then 2 corner pieces missing. My equation is P = (n + 1)^2 – 2.

Student 2

You really don’t see it, but what I did was I moved these 3 pieces on the bottom to the white space, and that forms a square. Then the right column is always n+1. My equation is P = (n^2) + (n+1).

Student 3

I see (n+1) groups of n, and the corner square. My equation is P = n(n+1) + 1.

week 8

Student 1

I chose the feet in mile because…

  1. 365 x 10 = 3650
  2. 365 x 4 = 300(4) + 60(4) + 5(4) = 1200 + 240 + 20 = 1200 + 260 = 1420 (realized when he was telling us that he’d added incorrectly)
  3. 3650 + 1420 = 5070

Student 2

I knew I had to multiply 13 by 365 but I got stuck. I can’t think today.

Student 3

I also chose the feet in mile.

  1. 365 x 10 = 3650
  2. 365 x 5 = 1825 (I really just divided 3650 by 2.)
  3. 3650 + 1825 = 5475 (so this is if he were 15 years old)
  4. 5475 – 365 = 5110 (but I subtracted a year to get 14 years old)

Student 4

Me too.

  1. 365 x 10 = 3650
  2. 365 x 3 = 360(3) + 5(3) = 1080 + 15
  3. 3650 + 1080 = 4730
  4. 4730 + 15 = 4745
  5. My birthday is May 22, so I figure that’s about 150 days.
  6. And 150 days plus 4745 is still less than 5280 feet.

Pattern #70

The equation for the blue parallelograms was straightforward enough as B = n – 1, so we focused our attention on discussing just the orange squares.

No one came up with the equation for whatever pattern they saw for the orange squares. Most common thinking was was 4, 1, 0, 1, 4, 1, 0, 1, …. One student thought it might be 4, 1, 0, 1, 4, 2, 0, 2, 4, 3, 0, 3, ….; others felt it was okay to see it that way too.

In the most common pattern seen, they figured that in every even-numbered step, the number of orange squares will always be 1. To figure out step 43, you’d take 43 divide by 4 because it the pattern repeats after every 4, leaving a remainder of 3 and that puts us at 0 orange squares.

Because no one came up with an equation, I told them I had an equation that would work and that I saw 9 orange squares in step 6. Not enough information for them yet, so I gave them 16 orange squares for step 7, and a few guessed that the next steps must be 25, 36, 49, 64, etc. Then someone said, it’s n minus 3, then square that.


 

source: Would You Rather

Student 1

I chose to drive 1.2 miles because it’s only 0.9 miles more, and 18 cents per gallon cheaper seems to make the extra driving worth it.

Student 2

Depends on the MPG of the car.

Student 3

Not enough information. I’d need to know how many gallons are in the tank.

Student 4

My family’s car can take 12 gallons, so I’d go 1.2 miles because 18 cents per gallon times 12 gallons is worth it.

Student 5

What if I just wanted to add 1 gallon to the tank? So I don’t mind the extra 18 cents. (When he shared this with the class, someone responded with, “Really?? What idiot would pull up to just get one gallon?” Rough crowd.)

Student 6

Modern cars have better MPG, so the 0.9 extra miles won’t be a big deal and I’d save in gas overall.


Pattern #28

Student 1

I see the floor and the 2 walls sharing a corner. My equation is C = (n-1)^2 + (2n^2) – n.

Student 2

I see a full cube of n sides. But it’s not a full cube, so missing is a smaller cube of (n – 1) sides: C = (n^3) – (n-1)^3.

Student 3

I see the entire floor as n^2, then the 2 walls, and the corner column. My equation is C = (n^2) + 2(n-1)^2 + (n-1).

Student 4

I see the exposed floor part. Then I see the left and right walls as one piece, so C = (n-1)^2 + n(2n – 1).

week 7

Which two are closer: 66.6%, 2/3, 0.67?

Student 1

2/3 and 0.67 are closer because I divided 2 by 3, and I got .66. And 66.6 is way more than a whole. Oh… wait! I never saw the percent sign! I thought it was just 66.6.

Student 2

I think 66.6% and .67 are closer. 2/3 is near 50% because 1.5/3 is 50%.

Student 3

66.6% and .67 are closer because I know 1/3 is 30, and 2/3 is 60.

Student 4

66.6% and 2/3 are closer because I changed them all to decimals with 3 decimal places.

  1. 66.6% = .666
  2. 2/3 = .66666
  3. 0.67 = .670

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).


Which two are closer: 3^2, 2^3, sqrt(72)?

Student 1

I think 3^2 and 2^3 are closer because I know these are 9 and 8, and I don’t know what sqrt(72) is.

Student 2

I know sqrt(72) is between 8 and 9 because 8^2 is 64 and 9^2 is 81. Well, 72 is 8 away from 64 and 9 away from 81. So sqrt(72) is closer to 8.

Student 3

(6th grader)

I know 8 times 9 is 72. So I think about what 8.5 times 8.5 is, and it’s 72.25. Therefore sqrt(72) is closer to 8, probably 8.4 something.

This 6th grader, after others had shared, tried a few more numbers and made a conjecture that when you multiply two consecutive numbers, like 5 and 6, then the square of the number between them, 5.5, is the same, plus 0.25 more. This works for 2 or 3-digit numbers too. For example, 20 x 21 = 420, and 20.5^2 = 420.25.

I tweeted this, and Dave @daveinstpaul returned with this visual representation of the student’s claim.

Then, Mike @mikeandallie kicked it up a notch and showed off his creations. :) Why I love Twitter!


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.

week 6

Which two are closer: 1/4, 1/2, 3/5?

Student 1

I changed them all to percents, 1/2 is 50%, 3/5 is 60%, so these two are closer because 1/4 is only 25%.

Student 2

I thought of the chocolate candy bar that my sister gave me.

Student 3

I imagined pizza, slices of pizza. I knew the half slice is twice the quarter slice, and the 3/5 slice is about like that.

Student 4

I looked at 1/4 and 1/2. I need to add another 1/4 to reach 1/2. When I compare 3/5 to 1/2, however, I need to subtract only to subtract a little bit. Well, in my head 3/5 is a little bit more than 1/2 because 2.5/5 is 1/2. So, I’d need to subtract .5/5 to get to 1/2, and that’s less than 1/4.


Pattern #85

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 Ss1 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. (Kids didn’t see the color version of this pattern, but in black and white, there was enough shading to make the distinctions.) My equation is C = n^2 + (n – 1)^2 + 2n + 2(n – 1) + 1.


Estimate 4953/68

Student 1

  1. I rounded the numbers to 5050 and 70.
  2. That’s 505 divided by 7. I got 72.

Student 2

  1. 4900/70 = 70
  2. Because I rounded 68 to 70, that’s 2 more 70s or 140.
  3. I added 140 to 53 to get 193.
  4. 193/68 = 2 something
  5. So, my answer is 72 something.

Student 3

  1. I rounded 4953 to 4956 because I knew that 49 and 56 are both divisible by 7.
  2. 4956/70 = 70.8
  3. I think the answer is bit lower because I rounded up.

Student 4

  1. I first did 4900/70 = 70.
  2. But there was time left, so I wanted to see if I could get closer, so I did 4970/70 = 71.

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.