1. 3 Chapter Numeration Systems and Whole Number Operations
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2. 3-2 Addition and Subtraction of Whole Numbers
Number relationships including comparing and ordering.
The meaning of addition and subtraction by studying various models and in turn learn addition and subtraction facts.
Properties of addition and subtraction and how to use them to develop computational strategies.
The inverse relationship between addition and subtraction.

3. Addition of Whole Numbers
Counting On
Counting on is an addition strategy where
addition is performed by counting on from one of the numbers, for example, can be computed
by starting at 5 and counting 6, 7, 8.

4. Addition of Whole Numbers
Set Model
Suppose Jane has 4 blocks in one pile and 3 in another. If she combines the two groups, how many objects are there in the combined group?
Note that the sets must be disjoint (have no elements in common) or an incorrect conclusion can be drawn.

5. Definition Addition of Whole Numbers
Let A and B be two disjoint finite sets. If n(A) = a and n(B) = b, then a + b = n(A U B).

6. Number-Line Model
Josh has 4 feet of red ribbon and 3 feet of white ribbon. How many feet of ribbon does he have altogether?
One day, Gail drank 4 ounces of orange juice in the morning and 3 ounces at lunchtime. If she drank no other orange juice that day, how many ounces of orange juice did she drink for the entire day?

7. Number-Line Model
Students need to understand that the sum represented by any two directed arrows can be found by placing the endpoint of the first directed arrow at 0 and then joining to it the directed arrow for the second number with no gaps or overlaps.
The sum of the numbers can then be read.

8. Definition
Less Than: For any whole numbers a and b, a is less than b, written a < b, if, and only if, there exists a natural number k such that a + k = b.
a ≤ b means a < b or a = b.
a > b is the same as b < a.

9. Whole Number Addition Properties
Closure Property of Addition of Whole Numbers
If a and b are whole numbers, then a + b is a whole number.
The closure property implies that the sum of two whole numbers exists and that the sum is a unique whole number.
For example, is a unique whole number, 7.

10. Whole Number Addition Properties
Commutative Property of
Addition of Whole Numbers
If a and b are any whole numbers, then a + b = b + a.

11. Whole Number Addition Properties
Associative Property of
Addition of Whole Numbers
If a, b, and c are any whole numbers, then (a + b) + c = a + (b + c).

12. Whole Number Addition Properties
Identity Property of
Addition of Whole Numbers
There is a unique whole number, 0, the additive identity, such that for any whole number a, a + 0 = a = 0 + a.

13. Example Which properties are illustrated in each of the following?
b is a unique whole number.
Commutative property of addition
Closure property of addition

14. Example (cont)
Which properties are illustrated in each of the following?
c. (3 + 5) + 7 = (5 + 3) + 7
d. (8 + 5) + 2 = 2 + (8 + 5) = (2 + 8) + 5
Commutative property of addition
Commutative and associative properties of addition

15. Mastering Basic Addition Facts
Counting on: Start with the greater addend then count on the smaller addend. For example: 4 + 2, start with 4, then count on another two, 5, 6.
Doubles: After students master doubles (such as 3 + 3), doubles + 1 and doubles plus 2 can be learned easily. For example, if a student knows = 12, then is (6 + 6) + 1 = = 13.

16. Mastering Basic Addition Facts
Making 10: Regroup to form a group of 10 and a leftover. For example: can be added as follows:

17. Mastering Basic Addition Facts
Counting back: Usually used when one number is 1 or 2 less than 10. For example, because 9 is 1 less than 10, then is 1 less than or 16.

18. Subtraction of Whole Numbers
Subtraction of whole numbers can be modeled in several different ways:
Take-Away Model – views subtraction as a second set of objects being taken away from the original set
Missing Addend Model – an algebraic-type of reasoning is used where students compute a difference by determining the value of an “unknown” addend.

19. Subtraction of Whole Numbers
Comparison Model – students determine “how many more” of one quantity exists than another.
Number-Line Model – subtraction is represented by moving left on the number line a given number of units.

20. Subtraction of Whole Numbers
Take-Away Model

21. Subtraction of Whole Numbers
Missing-Addend Model
8 − 3 =
This can be thought of as the number of blocks that must be added to 3 in order to get 8.
The number 8 – 3 is the missing addend in the equation
= 8 5

22. Subtraction of Whole Numbers
Missing-Addend Model

23. Definition Subtraction of Whole Numbers:
For any whole numbers a and b, such that a ≥ b, a − b is the unique whole number c such that b + c = a.

24. Subtraction of Whole Numbers
Comparison Model
Juan has 8 blocks and Susan has 3 blocks. How many more blocks does Juan have than Susan?
8 − 3 = 5

25. Subtraction of Whole Numbers
Number-Line (Measurement) Model
5 − 3 = 2

26. Properties of Subtraction
It can be shown that if a < b, then a − b is not meaningful in the set of whole numbers. Therefore, subtraction is not closed on the set of whole numbers.

27. Introductory Algebra Using Whole-Number Addition and Subtraction
Sentences such as = ☼ and 12 − ◊ = 4 can be true or false depending on the values of ☼ and ◊.
For example, if ☼ = 10, then = ☼ is false. If ◊ = 8, then 12 − ◊ = 4 is true.
If the value that is used makes the equation true, it is a solution to the equation.