3 Chapter Whole Numbers and Their Operations

3 Chapter Whole Numbers and Their Operations
Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

3-3 Multiplication and Division of Whole Numbers
Multiplication of Whole Numbers Repeated-Addition Model Properties of Whole-Number Multiplication Division of Whole Numbers The Division Algorithm Relating Multiplication and Division as Inverse Operations Division by 0 and 1 Order of Operations Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Multiplication and Division of Whole Numbers
Students understand the meanings of multiplication and division of whole numbers through the use of representations (e.g., equal-sized groups, arrays, area models, and equal “jumps” on number lines for multiplication, and successive subtraction, partitioning, and sharing for division). NCTM grade 3 Focal Points, p. 15 Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

They use properties of addition and multiplication (e.g., commutativity, associativity, and the distributive property) to multiply whole numbers and apply increasingly sophisticated strategies based on these properties to solve multiplication and division problems involving basic facts. By comparing a variety of solution strategies, students relate multiplication and division as inverse operations. NCTM grade 3 Focal Points, p. 15 Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Understanding properties of multiplication and the relationship between multiplication and division is a part of algebra readiness that develops at grade 3. The creation and analysis of patterns and relationships involving multiplication and division should occur at this grade level. NCTM grade 3 Focal Points, p. 15 Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Students build a foundation for later understanding of functional relationships by describing relationships in context with such statements as, “The number of legs is 4 times the number of chairs.” NCTM grade 3 Focal Points, p. 15 Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

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. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Multiplication of whole numbers can be modeled in several different ways: Cartesian-Product Model – Multiplication can be thought of in terms of Cartesian products. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

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 Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

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. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

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}. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

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 Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

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. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

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. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

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. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Commutative Property of Multiplication

Associative Property of Multiplication

Distributive Property of Multiplication Over Addition
The area of the large rectangle equals the sum of the areas of the two smaller rectangles. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

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. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

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). Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Example 3-2 Use an area model to show that
(x + y)(z + w) = xz + xw + yz + yw. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

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 Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

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. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

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? Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

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. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

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 Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

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. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

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. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Example 3-3 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. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

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. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Relating Multiplication and Division as Inverse Operations

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. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

3 Chapter Whole Numbers and Their Operations

Similar presentations

Presentation on theme: "3 Chapter Whole Numbers and Their Operations"— Presentation transcript:

Similar presentations

About project

Feedback

Log in

Auth with social network:

3 Chapter Whole Numbers and Their Operations

Similar presentations

Presentation on theme: "3 Chapter Whole Numbers and Their Operations"— Presentation transcript:

Similar presentations

About project

Feedback