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

3-4Algorithms for Whole-Number Multiplication and Division  Multiplication Algorithms  Division Algorithms  Division by a Two-Digit Divisor  Multiplication and Division in Different Bases Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Algorithms for Whole-Number Multiplication and Division They select appropriate methods and apply them accurately to estimate products or calculate them mentally, depending on the context and numbers involved. NCTM grade 4 Focal Points, p. 16 They develop fluency with efficient procedures, including the standard algorithm, for multiplying whole numbers, understand why the procedures work (on the basis of place value and properties of operations), and use them to solve problems. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Properties of Exponents Definition of a n for whole numbers If a, the base and n, the exponent, are whole numbers and n  0, then Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Properties of Exponents Theorem 3-9 For any whole number a and natural numbers m and n: Theorem 3-10 For any whole number a and natural numbers m and n: Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Properties of Exponents Theorem 3-11 For any whole number a and natural numbers m and n: Theorem 3-12 If a, m, and n are natural numbers with m > n, then Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Example 3-4 Write each of the following with only one exponent. a.b. a. b. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

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

Multiplication Algorithms To multiply by 10, replace each piece with a base- ten piece that represents the next higher power of 10. Replace each unit with a long, and replace each long with a flat. Multiplication by 10 n Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

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

Multiplication Algorithms Multiplication using expanded addition Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Multiplication Algorithms Multiplication with Two-Digit Factors Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Multiplication Algorithms Lattice Multiplication Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Division Algorithms Using Repeated Subtraction to Develop the Standard Division Algorithm or Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Division Algorithms or Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Division Algorithms Using Base-Ten Blocks to Develop the Standard Division Algorithm 1. Represent 726 with base-ten blocks. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

2.Determine how many sets of 6 flats (hundreds) there are. There is one set of 6 flats with 1 flat, 2 longs, and 6 units left over. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

3. Convert the one leftover flat to 10 longs. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

4. Determine how many sets of 6 longs there are in the 12 longs. There are 2 set of 6 longs and 6 units left over. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

5. Now determine how many sets of 6 units there are. There is 1 set of 6 units with no units left over. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Division Algorithms Short Division Decide where to start. Divide the hundreds. Write the remainder by the tens. Divide the tens. Write the remainder by the ones. Divide the ones. Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Division by a Two-Digit Divisor Four step method: 1.Estimate 2.Multiply 3.Subtract 4.Compare (check) Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Divide 1.Estimate the quotient. The quotient is between 10 and Find the number of tens in the quotient. Division by a Two-Digit Divisor Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

3.Find the number of units in the quotient. 4.Check. 32 · = 2618 Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Multiplication and Division in Different Bases Base-Five Multiplication Table Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Multiplication and Division in Different Bases Multiply 21 five · 3 five. FivesOnes 21  3 → →→ Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Multiplication and Division in Different Bases Copyright © 2013, 2010, and 2007, Pearson Education, Inc.

Example 3-4a Multiply: Copyright © 2013, 2010, and 2007, Pearson Education, Inc.