Chapter 3 Numeration Systems and Whole Number Operations
Chapter 3 Numeration Systems and Whole Number Operations Copyright © 2016, 2013, and 2010, Pearson Education, Inc.
3 -5 Multiplication and Division Algorithms, Mental Computation, and Estimation • • • Properties of exponents and how these can be used to develop multiplication and division algorithms. Models to develop algorithms for multiplication and division. Multiplication and division algorithms and how to use them to solve problems. Bases other than ten to provide insight into baseten multiplication and division. Mental multiplication and division skills and estimation techniques. Copyright © 2016, 2013, 2010 Pearson Education, Inc. Slide 2
Properties of Exponents Definition of an If a, the base and n, the exponent, are whole numbers and n 0, then Copyright © 2016, 2013, 2010 Pearson Education, Inc. Slide 3
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 © 2016, 2013, 2010 Pearson Education, Inc. Slide 4
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 © 2016, 2013, 2010 Pearson Education, Inc. Slide 5
Example Write each of the following with only one exponent. a. b. a. b. Copyright © 2016, 2013, 2010 Pearson Education, Inc. Slide 6
Multiplication Algorithms Copyright © 2016, 2013, 2010 Pearson Education, Inc. Slide 7
Multiplication Algorithms Multiplication by 10 n To multiply by 10, replace each piece with a baseten piece that represents the next higher power of 10. Replace each unit with a long, and replace each long with a flat. Copyright © 2016, 2013, 2010 Pearson Education, Inc. Slide 8
Multiplication Algorithms Copyright © 2016, 2013, 2010 Pearson Education, Inc. Slide 9
Multiplication Algorithms Multiplication using expanded addition Copyright © 2016, 2013, 2010 Pearson Education, Inc. Slide 10
Multiplication Algorithms Multiplication with Two-Digit Factors Copyright © 2016, 2013, 2010 Pearson Education, Inc. Slide 11
Multiplication Algorithms Lattice Multiplication Copyright © 2016, 2013, 2010 Pearson Education, Inc. Slide 12
Division Algorithms Using Repeated Subtraction to Develop the Standard Division Algorithm or Copyright © 2016, 2013, 2010 Pearson Education, Inc. Slide 13
Division Algorithms or Copyright © 2016, 2013, 2010 Pearson Education, Inc. Slide 14
Division Algorithms Using Base-Ten Blocks to Develop the Standard Division Algorithm 1. Represent 726 with base-ten blocks. Copyright © 2016, 2013, 2010 Pearson Education, Inc. Slide 15
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 © 2016, 2013, 2010 Pearson Education, Inc. Slide 16
3. Convert the one leftover flat to 10 longs. Copyright © 2016, 2013, 2010 Pearson Education, Inc. Slide 17
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 © 2016, 2013, 2010 Pearson Education, Inc. Slide 18
5. Now determine how many sets of 6 units there are. There is 1 set of 6 units with no units left over. Copyright © 2016, 2013, 2010 Pearson Education, Inc. Slide 19
Division Algorithms Short Division Decide where Divide the to start. hundreds. Write the remainder by the tens. Divide the tens. Write the remainder by the ones. Copyright © 2016, 2013, 2010 Pearson Education, Inc. Divide the ones. Slide 20
Division by a Two-Digit Divisor Four step method: 1. Estimate 2. Multiply 3. Subtract 4. Compare (check) Copyright © 2016, 2013, 2010 Pearson Education, Inc. Slide 21
Division by a Two-Digit Divisor Divide 1. Estimate the quotient. The quotient is between 10 and 100. 2. Find the number of tens in the quotient. Copyright © 2016, 2013, 2010 Pearson Education, Inc. Slide 22
3. Find the number of units in the quotient. 4. Check. 32 · 81 + 26 = 2618 Copyright © 2016, 2013, 2010 Pearson Education, Inc. Slide 23
Multiplication and Division in Different Bases Base-Five Multiplication Table Copyright © 2016, 2013, 2010 Pearson Education, Inc. Slide 24
Multiplication and Division in Different Bases Multiply 21 five · 3 five. Fives 2 � Ones 1 3 → → → Copyright © 2016, 2013, 2010 Pearson Education, Inc. Slide 25
Multiplication and Division in Different Bases Copyright © 2016, 2013, 2010 Pearson Education, Inc. Slide 26
Example Multiply: Copyright © 2016, 2013, 2010 Pearson Education, Inc. Slide 27
Mental Mathematics: Multiplication 1. Front-end multiplying 2. Using compatible numbers 3. Thinking money 78 → 70 4 = 280 4 → 8 4 = 32 280 + 32 = 312 2 8 5 40 5 = (2 5) (40 5) 8 = 10 200 8 = 16, 000 84 Think of the product 25 as 84 quarters. 21 dollars = 2100 cents Copyright © 2016, 2013, 2010 Pearson Education, Inc. Slide 28
Mental Mathematics: Division 1. Breaking up the dividend Break up the dividend into parts. Divide both parts by 8. Add the answers. Copyright © 2016, 2013, 2010 Pearson Education, Inc. Slide 29
Mental Mathematics: Division 2. Using compatible numbers Look for numbers that are divisible by 3 and whose sum is 105. Divide both parts by 3, then add the answers. Copyright © 2016, 2013, 2010 Pearson Education, Inc. Slide 30
Mental Mathematics: Division 2. Using compatible numbers (continued) Look for numbers that are divisible by 8 and whose difference is 232. Divide both parts by 8, then subtract the answers. Copyright © 2016, 2013, 2010 Pearson Education, Inc. Slide 31
Estimation: Multiplication and Division 1. Front-end estimation 474 9 § Start multiplying at the front: 400 9 = 3600 § Multiply the next digit: 70 9 = 630 § Add the two numbers: 3600 + 630 = 4230 2. Compatible numbers Copyright © 2016, 2013, 2010 Pearson Education, Inc. Slide 32
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