Creative thinking in subtraction

Case 2 – Ann

Grade 6, October

When do I borrow? Is this where I cross out something? What if there is a zero in the column I want to take from? When do I stop borrowing?

1. In Ann’s case 2, examine the methods presented by Jason, Bert, Holly, and Joe. Apply their methods to this problem: 83-56.

Jason’s thinking: 

  • 83 – 56= ?
  • First subtract 50 from 83.
  • 83 – 56 = 33
  • Then subtract 3 (which is part of the 56)
  • 33 – 3= 30
  • Finally subtract the remaining 3 from the 30
  • 30- 3= 27
  1. Jason started on the left with the tens column.
  2. Then tried to make the number “friendlier”, by making the ones column a zero.
  3. For Jason, going from 83- 3= 80 and 33-3= 30 was easier than 33- 6= 27

Bert’s thinking:

  • 83 – 56= ?
  • 83 – 10 = 73
  • 73 – 10 = 63
  • 63 – 10 = 53
  • 53 – 10= 43
  • 43 – 10= 33
  • 33 – 3 = 30
  • 30 – 3= 27
  1. Bert subtracts 5 groups of 10, then subtracts 30 to bring his ones place to zero, then subtracts 3 more to get 27

Holly’s thinking:

  • 83 – 56 = ?
  • Take away 3 from both to make 80- 53
  • 80 – 10= 70
  • 70 – 10 = 60
  • 60 – 10= 50
  • 50 – 10= 40
  • 40 – 10= 30
  • 30- 3= 27
  1. Holly’s thinking is similar to Bert’s although she starts off with his last step.
3. What is it that children need to be able to understand about number in order to solve math problems in these ways?
  • If there wasn’t regrouping [as in 83-56] then “when the number is bigger” you don’t switch, and you add the answer to the final number instead of subtracting it.
4. In case 4, Emily writes that one of her second-graders, Ivan, invented a solution to a subtraction problem that was both new to her and confusing to figure out.

What Emily observed:

  • 52 – 28 = ?
  • Ivan’s first answer was 36
  • His friend got 24 and explains why Ivan is wrong
  • His friend: “20 away from 50 is 30, but you can’t take the 2 away from the 8; you have to take the 8 away from the 2 because the 8 is on the bottom.”
  • Ivan: “but you can’t take the 8 aways from the 2 because the 8 is too big.”

Ivan’s Thinking:

  • You take the 20 away from the 50 and get 30.
  • Then you take the 8 aways from the 2 which is -6.
  • Then you take the -6 away from the 30 and you get 24.
5. Does Ivan’s method resemble any of those presented in Ann’s case 2?

 

6. Emily declares this to be a time when she was learning mathematics by interacting with her students. What is your reaction to that?

Everyone you will ever meet knows something you don’t – Bill Nye

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