1. 3 Chapter Numeration Systems and Whole Number Operations
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2. 3-3 Multiplication and Division of Whole Numbers
Meanings of multiplication and division by examining various models.
The inverse relationship between multiplication and division.
Properties of multiplication and division and how to use them to develop computational strategies.
The special cases of multiplication and division by 0 and 1.

3. Multiplication of Whole Numbers
Multiplication of whole numbers can be modeled in several different ways:
Repeated-Addition Model – multiplication can be thought of as repeatedly adding the multiplicand, multiplier times.
Array and Area Model – multiplication can be visualized by constructing an array of crossed sticks or a grid, and counting intersection points or grid components, respectively.

4. Multiplication of Whole Numbers
Multiplication of whole numbers can be modeled in several different ways:
Cartesian-Product Model – Multiplication can be thought of in terms of Cartesian products.

5. Multiplication of Whole Numbers
Repeated Addition Model
If we have 4 groups of 3 brushes, we can use addition to put the groups together. When we put equal-sized groups together we can use multiplication. We can think of this as combining 4 sets of 3 objects into a single set.
= 12
four 3’s

6. Multiplication of Whole Numbers
Repeated Addition Model

7. Multiplication of Whole Numbers
The Array and Area Models
4 • 5 = 20

8. Definition Multiplication of Whole Numbers
For any whole numbers a and natural number n,

9. Multiplication of Whole Numbers
Cartesian-Product Model
Suppose you plan to take two Web-based courses at your local college. You can take one of the following – World History, Ancient History. For your second course, you can take one of the following – Latin, French, or German. To show the number of different schedules you could have, you can use a tree diagram.

10. Multiplication of Whole Numbers
Cartesian-Product Model
The ways of designing your schedule are a result of selecting a history course from the set H = {World History, Ancient History} and a foreign language from L = {Latin, French, German}.

11. Multiplication of Whole Numbers
Cartesian-Product Model
Design the Cartesian product H × L using a tree.
Latin
German
French
World History, Latin
World History
World History, German
World History, French
Schedule
Latin
German
French
Ancient History, Latin
Ancient History
Ancient History, German
Ancient History, French

12. Multiplication of Whole Numbers
Cartesian-Product Model
The Fundamental Counting Principle tells us that the number of ordered pairs in H × L is 2 • 3 = 6.

13. Definition Multiplication of Whole Numbers
For finite sets A and B, if n(A) = a and n(B) = b, then
In this alternate definition, sets A and B do not have to be disjoint. The expression a · b, or simply ab, is the product of a and b, and a and b are factors.
Note that A  B indicates the Cartesian product. We multiply numbers, not sets.

14. Properties of Whole-Number Multiplication
Closure Property of Multiplication of Whole Numbers
If a and b are whole numbers, then a · b is a unique whole number.
Commutative Property of Multiplication of Whole Numbers
If a and b are any whole numbers, then a · b = b · a.

15. Properties of Whole-Number Multiplication
Associative Property of Multiplication of Whole Numbers
If a, b, and c are whole numbers, then (a · b) · c = a · (b · c).
Identity Property of Multiplication of Whole Numbers
There is a unique whole number 1 such that for any whole number a, a · 1 = a = 1 · a.

16. Properties of Whole-Number Multiplication
Zero Multiplication Property of Whole Numbers
For any whole number a, a · 0 = 0 = 0 · a.

17. Commutative Property of Multiplication

18. Associative Property of Multiplication

19. Distributive Property of Multiplication Over Addition
The area of the large rectangle equals the sum of the areas of the two smaller rectangles.

20. Distributive Property of Multiplication Over Addition for Whole Numbers
For any whole numbers a, b, and c,

21. Distributive Property of Multiplication Over Addition for Whole Numbers
For any whole numbers a, b, and c with b > c,

22. Distributive Property
Because the commutative property of multiplication of whole numbers holds, the distributive property of multiplication over addition can be rewritten as
(b + c)a = ba + ca.
The distributive property can be generalized to any finite number of terms. For example,
a(b + c + d) = ab + ac + ad.

23. Distributive Property
When the distributive property is written as
ab + ac = a(b + c),
this is called factoring.
The factors of ab + ac are a and (b + c).

24. Example Use an area model to show that
(x + y)(z + w) = xz + xw + yz + yw.

25. Example (continued)
Use the distributive property of multiplication over addition to show that
(x + y)(z + w) = xz + xw + yz + yw.
(x + y)(z + w) = (x + y)z + (x + y)w The distributive property of
multiplication over addition
= xz + yz + xw + yw The distributive property of
= xz + yz + xw + yw The commutative property of addition

26. Division of Whole Numbers
Three models for division
Set (Partition) Model – the set of elements representing the dividend is partitioned into divisor-number of subsets.
Missing-Factor Model – a divided by b is a unique number c, provided that b • c = a.
Repeated Subtraction Model – the divisor is continually subtracted from the dividend until only the remainder is left.

27. Division of Whole Numbers
Set (Partition) Model
Suppose we have 18 cookies and want to give an equal number of cookies to each of three friends: Bob, Dean, and Charlie. How many should each person receive?

28. Division of Whole Numbers
Missing Factor Model
If each friend receives c cookies, then the three friends will receive 3c, or 18 cookies. Therefore, 3c = 18. Since 3 • 6 = 18, we have answered the division computation by using multiplication.

29. Definition Division of Whole Numbers
For any whole numbers a and b, with b ≠ 0, a ÷ b = c, if and only if, c is the unique whole number such that b · c = a.
The number a is the dividend, b is the divisor, and c is the quotient.
a ÷ b can also be written as

30. Division of Whole Numbers
Repeated-Subtraction Model
Suppose we have 18 cookies and want to package them in cookie boxes that hold 6 cookies each. How many boxes are needed?
If one box is filled, then there are 18 − 6 = 12 cookies left.
If one more box is filled, then there are 12 − 6 = 6 cookies left.
The last 6 cookies will fill another box.

31. Division Algorithm
Given any whole numbers a and b with b ≠ 0, there exist unique whole numbers q (quotient) and r (remainder) such that
When a is “divided” by b and the remainder is 0, we say that a is divisible by b or that b is a divisor of a or that b divides a.

32. Example
If 123 is divided by a number and the remainder is 13, what are the possible divisors?
Solution
If 123 is divided by b, then from the division algorithm
123 = bq + 13 and b > 13.
From the definition of subtraction
bq = 123 − 13, so bq = 110.

33. Example (continued)
Now we are looking for two numbers whose product is 110, where one number is greater than 13. 1
110 2 55 5 22 10 11
The table shows the pairs of whole numbers whose product is 110.
We see that 110, 55, and 22 are the only possible values for b because each is greater than 13.

34. Relating Multiplication and Division as Inverse Operations

35. Division by 0 and 1 Let n be any nonzero whole number. Then,
n ÷ 0 is undefined.
0 ÷ n = 0.
0 ÷ 0 is undefined.

36. Order of Operations
When no parentheses are present, multiplications and divisions are performed before additions and subtractions.
The multiplications and divisions are performed in the order they occur, and then the additions and subtractions are performed in the order they occur.