3 Chapter 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 base-ten multiplication and division. Mental multiplication and division skills and estimation techniques.

Properties of Exponents Definition of an If a, the base and n, the exponent, are whole numbers and n  0, then

Properties of Exponents Theorem 3-9 For any whole number a and natural numbers m and n: Theorem 3-10

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

Example Write each of the following with only one exponent. a. b. a. b.

Multiplication Algorithms

Multiplication Algorithms Multiplication by 10n 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 Algorithms

Multiplication Algorithms Multiplication using expanded addition

Multiplication Algorithms Multiplication with Two-Digit Factors

Multiplication Algorithms Lattice Multiplication

Division Algorithms or Using Repeated Subtraction to Develop the Standard Division Algorithm or

Division Algorithms or

Division Algorithms Using Base-Ten Blocks to Develop the Standard Division Algorithm 1. Represent 726 with base-ten blocks.

2. Determine how many sets of 6 flats (hundreds) there are 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.

3. Convert the one leftover flat to 10 longs.

4. Determine how many sets of 6 longs there are in the 12 longs 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.

5. Now determine how many sets of 6 units there are 5. Now determine how many sets of 6 units there are. There is 1 set of 6 units with no units left over.

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.

Division by a Two-Digit Divisor Four step method: 1. Estimate 2. Multiply 3. Subtract 4. Compare (check)

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.

3. Find the number of units in the quotient. 4. Check. 32 · 81 + 26 = 2618

Multiplication and Division in Different Bases Base-Five Multiplication Table

Multiplication and Division in Different Bases Multiply 21five · 3five. Fives Ones 2 1  3 → → →

Multiplication and Division in Different Bases

Example Multiply:

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

Mental Mathematics: Division 1. Breaking up the dividend Break up the dividend into parts. Divide both parts by 8. Add the answers.

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.

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.

Estimation: Multiplication and Division 1. Front-end estimation 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 474  9