Chapter 2: Sets and Whole-Number Operations and Properties

2.1 Multiplication and Division of Whole Numbers

2.1.1. Using Models and Sets to Define Multiplication

2.1.1.1. Sets arranged in equal rows and columns are called rectangular arrays

2.1.1.2. To name an array we say the number of rows first, then the number of columns – RC cola – rows then columns

2.1.1.3. This is a 3 x 4 array of coins

2.1.1.4. This is a 4 x 3 array of coins

2.1.1.5. Area Model

2.1.1.5.1. The two numbers being multiplied represent the dimensions of a rectangle

2.1.1.5.2. The area of the rectangle is the result of the multiplication

To multiply 14 x 27

Draw a rectangle and label accordingly

Partition the rectangle into convenient areas

Finally, sum together the different areas to get the product or area of the rectangle: 14 x 27 = 100 + 100 + 70 + 40 + 40 + 28 = 378

2.1.1.6. Using repeated addition to define multiplication

2.1.1.6.1. 5 x 3 = 3 + 3 + 3 + 3 + 3 = 15

2.1.1.6.2. 3 x 5 = 5 + 5 + 5 = 15

2.1.1.6.3. multiplication is a shortcut for repeated addition

2.1.1.7. Using the language of sets to define multiplication

2.1.1.7.1. a x b is the number of elements in the union of a disjoint equivalent sets, each containing b elements

2.1.1.7.2. An ordered pair is a pair of elements where order is important: (a, b) is NOT the same as (b, a) as long as a ¹ b

2.1.1.7.3. a x b (read “a cross b”) is called a Cartesian product

2.1.1.7.4. Definition of Cartesian product: The Cartesian product of two sets A and B, A x B (read “A cross B”) is the set of all ordered pairs (x,y) such that x is an element of A and y is an element of B

2.1.1.7.4.1. Example: A = {1, 2, 3} and B = {a, b}, then

2.1.1.7.4.1.1. A x B = {(1, a), (2, a), (3, a), (1, b), (2, b), (3, b)}

2.1.1.7.4.1.2. B x A = {(a, 1), (a, 2), (a, 3), (b, 1), (b, 2), (b, 3)}

2.1.1.7.4.1.3. Note: A and B can also be equal sets – see ex. 2.11

2.1.1.8. Your turn p. 84: Do the practice and the reflect

2.1.1.9. Definition of multiplication of whole numbers: In the multiplication of whole numbers, if A and B are finite sets with a = n(A) and b = n(B), then a x b = n(A x B). In the equation a x b = n(A x B), a and b are called factors and n(A x B) is called the product

2.1.1.10. Your turn p. 86: Do the practice and the reflect

2.1.2. Properties of Multiplication

2.1.2.1. Closure property: For whole numbers a and b, a x b is a unique whole number

2.1.2.2. Identity property: There exists a unique whole number, 1, such that 1 x a = a x 1 = a for every whole number a. Thus 1 is the multiplicative identity element

2.1.2.3. Commutative property: For whole numbers a and b, a x b = b x a

2.1.2.4. Associative property: For whole numbers a, b, and c, (a x b) x c = a x (b x c)

2.1.2.5. Zero property: For each whole number a, a x 0 = 0 x a = 0

2.1.2.6. Distributive property of multiplication over addition: For whole numbers a, b, and c, a x (b + c) = (a x b) + (a x c)

2.1.3. Modeling Division

2.1.3.1. Finding how many in each subset model

2.1.3.1.1. How many elements are in a known amount of subsets?

2.1.3.1.2. A sharing or partitioning model of division

2.1.3.2. Finding how many subsets model

2.1.3.2.1. How many subsets of a known quantity can be removed from a set?

2.1.3.2.2. Repeated subtraction idea

2.1.3.2.3. This method also called the measurement model of division

2.1.3.2.4. see examples 89-90

2.1.3.3. Your turn p. 91: Do the practice and the reflect

2.1.4. Using Multiplication to Define Division

2.1.4.1. Division as the inverse of multiplication

2.1.4.1.1. Multiplication model: factor x factor = product

2.1.4.1.2. Division model: product ¸ factor = missing factor

2.1.4.1.3. Revised Division model: factor x missing factor = product

2.1.4.1.4. Definition of division: In the division of whole numbers a and b, b ¹ 0, a ¸ b = c if and only if c is a unique whole number such that c x b = a. In the equation, a ¸ b = c, a is the dividend, b is the divisor, and c is the quotient. The operation a ¸ b may also be written as , , or a:b = c

2.1.4.2. Division as finding the missing factor

2.1.4.2.1. takes advantage of the inverse relationship between multiplication and division

2.1.4.2.2. If I know my multiplication facts, I know my division facts

2.1.4.3. Why division by Zero is Undefined

2.1.4.3.1. viewing division as finding the missing factor helpful here

2.1.4.3.2. 8 ¸ 0 = ? translates to 0 x ? = 8 – answer no number times zero is 8

2.1.4.3.3. 0 ¸ 0 = ? translates to 0 x ? = 0 – answer anything and everything could be used as zero times anything is zero

2.1.4.3.4. Thus because either there is no solution or there are infinite solutions, we say DIVIDING BY ZERO IS UNDEFINED

2.1.5. Comparing Division to Multiplication

2.1.5.1. Your turn p. 92: Do the practice and the reflect

2.1.6. The Division Algorithm for Whole Numbers

2.1.6.1. The division algorithm: For any two numbers a and b, b ¹ 0, a division process for a ¸ b can be used to find unique whole numbers q (quotient) and r (remainder) such that a = bq + r and 0 £ r < b.

2.1.6.2. Query: According to the algorithm above 25 ¸ 4 = 6 remainder 1 and 31 ¸ 5 = 6 remainder 1. Does 25 ¸ 4 = 31 ¸ 5? Explain????

2.1.6.3. Be sure to help your students to avoid this pitfall… How?

2.1.7. Multiplication and Division Facts and Fact Families

2.1.7.1. Fact is a digit times a digit

2.1.7.1.1. How many multiplication facts are there?

2.1.7.1.2. If I know the zero property (0 x a = a x 0 = 0), how many more facts do I need to learn?

2.1.7.1.3. If I know the multiplicative identity (1 x a = a x 1 = a), how many more facts do I need to learn?

2.1.7.1.4. If I know zero and my ones, how many facts are left?

2.1.7.1.5. If I know zero, ones, and squares, how many facts are left?

2.1.7.1.6. If I know the commutative property of multiplication and squares, how many facts do I need to learn?

2.1.7.1.7. If I know the commutative property, squares, zero, and ones, how many facts are left for me to learn?

2.1.7.1.8. Which list is more intimidating - - 100 facts or the one we just figured out above? Remember to share this with your future students! It will empower them!! J

2.1.7.2. Division Facts

2.1.7.2.1. How many division facts are there?

2.1.7.2.2. Why are there less division facts than multiplication facts?

2.1.7.3. Fact Families

2.1.7.3.1. Another way of organizing and learning multiplication facts

2.1.7.3.2. Fact families deal with products and the facts that make up those products

0 Þ 0 x 0 0 x 1 0 x 2 0 x 3 … 3 x 0 2 x 0 1 x 0

1 Þ 1 x 1

2 Þ 2 x 1 1 x 2

3 Þ 3 x 1 1 x 3

4 Þ 4 x 1 2 x 2 1 x 4

5 Þ 5 x 1 1 x 5

6 Þ 6 x 1 3 x 2 2 x 3 1 x 6

etc.