1. 3 Chapter Numeration Systems and Whole Number Operations
Copyright © 2016, 2013, and 2010, Pearson Education, Inc.

2. 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.

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

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

5. 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

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

7. Multiplication Algorithms

8. 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.

9. Multiplication Algorithms

10. Multiplication Algorithms
Multiplication using expanded addition

11. Multiplication Algorithms
Multiplication with Two-Digit Factors

12. Multiplication Algorithms
Lattice Multiplication

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

14. Division Algorithms or

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

16. 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.

17. 3. Convert the one leftover flat to 10 longs.

18. 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.

19. 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.

20. 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.

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

22. 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.

23. 3. Find the number of units in the quotient.
4. Check. 32 · = 2618

24. Multiplication and Division in Different Bases
Base-Five Multiplication Table

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

26. Multiplication and Division in Different Bases

27. Example
Multiply:

28. Mental Mathematics: Multiplication
1. Front-end multiplying
2. Using compatible
numbers
3. Thinking money
78 → 70  4 = 280
 4 → 8  4 = 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

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

30. 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.

31. 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.

32. 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: = 4230
2. Compatible numbers
474
 9