Addition and subtraction Math 123 October 3-8, 2008

Sections 3.1 and 4.2 We will jump around the text a bit: we will simultaneously cover the properties of operations and corresponding algorithms. In addition, we will read cases about young students doing mathematics and discuss them in class.

Operations in base 5 Try to solve the following problems using base 5 blocks. When you finish these, you can make up your own problems. 1.In a class, there are 24 5 girls and 23 5 boys. How many students are there total? 2.The post office delivered mailings on Monday in one neighborhood. On Tuesday, the same homes received mailings. How many more mailings were delivered on Tuesday than on Monday?

Generalizing Which strategies did you use to add? Which strategies did you use to subtract? How did you “carry” and “borrow?”

Base 10: mental math Work on handouts (Exploration 3.2 and 3.4). Feel free to use base 10 blocks when you are finished, to understand what you did. How did your strategies compare to those in base 5? Which of the two assignments made it easier for you to understand what’s really going on when we add?

What do children do? Look at Part 1 of Exploration 3.3 and 3.5 (handouts provided). Do you understand the children’s strategies? Which one(s) do you think is/are easiest to use? How do these strategies use place value? Feel free to use base 10 blocks to understand the strategies.

What do children do wrong? Look at some examples of children’s strategies. What were the mistakes that the students were making? How many of the mistakes you encountered are related to place value?

Terminology What do you think about terms “carry” and “borrow?” Do they accurately portray what you are doing when you add and subtract? Are there terms you would rather use?

In math education a term more commonly used these days is “regroup.”

Alternative algorithms Work on Part 2 of Exploration 3.3 and 3.5 (handout provided). What do you think about these algorithms? Are they easier to use than the standard ones? Why do you think the one in use today has prevailed? I have seen the lattice algorithm used by school children.

Standard algorithm Try to justify the standard algorithms for addition and subtraction with the base 10 blocks. Work on the following problems. Feel free to try other problems when you finish these. – – (The standard algorithms are explained in detail on pages of the book.)

Washington standards athematics/RevisedStandards/k8- operations.pdfhttp:// athematics/RevisedStandards/k8- operations.pdf

Definitions This is where we will be needing sets 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.

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.

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.

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.

Different contexts for subtraction Take away approach. Missing addend approach. Try to model both using blocks.