1. MTH 231 Section 2.4 Multiplication and Division of Whole Numbers

2. Multiplication Some of the conceptual models mentioned in the section: 1.Multiplication as repeated addition 2.Array model 3.Rectangular area model 4.Skip-count model

3. Repeated Addition

4. Array

5. Rectangular Area

6. Skip-Count

7. Multiple Models

8. Properties of Whole-Number Multiplication Like addition, multiplication is: 1.Closed 2.Associative 3.Commutative However, there are three new properties we need to discuss.

9. 4. Multiplicative Identity Property There is a “special” element in the whole numbers. This element has the property that any whole number multiplied by it gives back the number you started with: a x 1 = a and 1 x a = a for all whole numbers a

10. 5. Multiplication-by-Zero Property Any whole number multiplied by 0 gives a result of 0 b x 0 = 0 and 0 x b = 0 for all whole numbers b

11. 6. Distributive Property If a, b, and c are any three whole numbers: a x (b + c) = (a x b) + (a x c) and (a + b) x c = (a x c) + (b x c) The official title of the property, “distributive property of multiplication over addition”, is reflected in the fact that both operations are present.

12. Images

13. More Images

14. Division of Whole Numbers Division is inherently more difficult to model than multiplication, yet there are fewer models: 1.Repeated-subtraction 2.Partition 3.Missing-factor

15. Repeated-Subtraction In this model, elements in a set are subtracted away in groups of a specified size. This model is also called division by grouping.

16. Partition In this model, elements in a set are separated into groups of a specified size.

17. Missing Factor In this model, division is recognized as the inverse of multiplication.

18. Division By Zero Consider the following questions: 1.John has 12 pieces of candy. He wants to give each of his friends 0 pieces. How many friends will receive 0 pieces of candy? (repeated-subtraction) 2.John has 12 pieces of candy. He wants to divide them in groups of 0 pieces. How many groups of 0 pieces can John make? (partition) 3.Find a whole number c such that 0 x c = 12. (missing-factor)

19. Division With Remainders Sticking with the missing-factor model, we now consider those situations where a whole number c cannot be found: Find a whole number c such that 5 x c = 7. The other models further support the idea that, in some cases, a remainder is needed to extend the division operation.

20. The Division Algorithm Let a and b be whole numbers with b not equal to zero (Why?). Then there exist whole numbers q and r such that a = q x b + r, with 0 < r < b. a is called the dividend. b is called the divisor. q is called the quotient. r is called the remainder.

21. 7 Divided By 5, 3 Ways