Addition and subtraction Math 123 Addition of whole
Addition and subtraction Math 123
Addition of whole numbers: definition • 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)
What the book says • “When quantities are combined additively, they are joined together, so the appropriate arithmetic operation on their values is usually addition. This operation is called an additive combination. The result of an additive combination is a sum of the values combined. ”
Different contexts • Addition comes up in two different types of problems: ▫ Joining ▫ Part-part-whole • For us, these problems are similar, but for children they are not because the second type does not imply addition like the first one does.
Word of caution about keywords • Consider this example: Josie needs to make 15 tacos for lunch. She has made 7 already. How many more tacos does she have to make? • What is the keyword in this example? • What operation does the keyword imply?
Models • There are two models for representing addition: ▫ Set model. This corresponds to, for example using counters or blocks to combine quantities ▫ Number line model. Number lines have a prominent role in the CCSSM, and are used throughout elementary and even middle school
Subtraction of whole numbers: definition • 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)
Terminology • You might not know that the words for the terms in a subtraction problem are subtrahend – minuend = difference
Contexts • Subtraction comes up in three different ways ▫ Take-away (the traditional one) ▫ Missing addend ▫ Comparison • We have seen these before. Can you connect them to the CGI problem types? • Give examples of each.
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.
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?
What about subtraction?
• • 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.
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