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Addition and subtraction

Addition and subtraction. Math 123. Definitions. Addition of whole numbers: Let a and b be any two whole numbers. If A and B are disjoint sets with a = n ( A ) and b = n ( B ), then a + b = n ( A  B ).

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Addition and subtraction

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  1. Addition and subtraction Math 123

  2. Definitions • Addition of whole numbers: • Let a and b be any two whole numbers. If A and B are disjoint sets with a = n(A) and b = n(B), then a+b = n(A B). • This seems very complicated. But in reality, this is how children learn to count: if you have 3 apples, and I have 4 apples, to find out how much we have together, we will join your set of 3 and my set of 4 to see how many there are in the union of the two. (Like a JRU problem)

  3. Models • In addition to the set model that you have just seen, we can also use the number line model to represent addition.

  4. Subtraction of whole numbers: • Let a and b be any two whole numbers (a>b) and A and B be sets such that a = n(A) and b = n(B), and B  A. Then a-b = n(A- B). • Again, this looks complicated, but think about it. I have 5 apples, and you take 3 away from me. I had a set of 5 apples, and you took a subset of 3 from it. What is left is the number of apples I have left. (Like a SRU problem)

  5. Properties of addition • Closure: the sum of any two whole numbers is a whole number • Commutative property: the order in which numbers are added does not matter: a+b = b+a. • Associative property: numbers can be grouped differently: (a+b)+c = a+(b+c). • Identity property: a+0 = a = 0+a .

  6. Why are properties useful? Try to compute the following: • 34 + 29 + 76 + 66 + 24= Can you find an easier way to add the numbers? Which properties are you using?

  7. What about subtraction?

  8. The closure property does not hold. • The commutative property does not hold. • The associative property does not hold. • The identity property holds only somewhat: a – 0 = a, but 0 – a = -a.

  9. Different contexts for subtraction • Take away approach. • Missing addend approach. • Comparison approach Note that we can also use sets and number lines to represent subtraction.

  10. Standard algorithms Write this on paper – to turn in. Use blocks to do the following computations: • 169 + 357 = • 357 – 169 = After finishing, think about how what you have done is related to the standard “carrying” and “borrowing” algorithms. Then explain, using blocks, why the standard algorithms work and what is going on when we “carry” and “borrow.”

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