Multiplication and Division Algorithm and Properties

1. Multiplication and Division Algorithm and Properties THursday, July 13

2. Objectives and agenda • Understand the properties of number • Use properties to justify steps in multiplication and division problems • Warm Up – Order of Operations • Properties Instruction • Commutative Property • Associative Property • Distributive Property • Practice • Connect Properties to Multiplication and Division Algorithms • Exit Slip

3. Warm up • Directions: Make the largest (or smallest) expression by using the whole numbers 0-9 in the boxes below. http://www.openmiddle.com/order-of-operations-2/

4. Order of operations • Parentheses or Grouping Symbols • Exponents • Multiplication or Division whichever comes first from left to right • Addition or Subtraction whichever comes first from left to right • More practice on the website – July 13 Order of Operations Practice or continue on the Open Middle practice

5. Properties of number • Commutative Property • Associative Property • Distributive Property • What are they? When do you use them?

6. Commutative property • What is it? • a + b = b + c or a • b = b • c • Why is it important? • Makes calculations easier --- 8 x 2 vs. 2 x 8 • 8 x 2 = 8 + 8 • 2 x 8 = 2 + 2 + 2 + 2 + 2 + 2 + 2 • Makes half of the multiplication tables easier to learn

7. Associative Property • What is it? • 7 • (6 • 5) = (7 • 6) • 5 • Volume

8. Power of commutative and associative properties • 4 • 7 = (2 • 2) • 7 • 8 • 17 • 25 Associative property alone does not help • With more than 3 factors 5 x 8 x 6 x 4

9. Distributive property • What is it? • a(b + c) or a(b – c) • Use in multiplication facts • 6 • 7 = (3 + 3)7 • Decomposing numbers to multiply • How would you use the distributive property to multiply… • 23 x 102 • 14 x 201

10. Using properties to justify algorithms 30 + 40 (3 x 10) + (4 x 10) (3 + 4) x 10 7 x 10 70 Given Place Value Distributive Property Order of Operations Multiplication

11. Connecting the properties to the algorithms Use partial products to find 4 x 23 How are the methods similar? Directions: Provide an explanation for each step of 4 x 23 4 x (20 + 3) (4 x 20) + (4 x 3) 80 + 12 80 + (10 + 2) (80 + 10) + 2 90 + 2 92

12. Multiplication Algorithm • Why is the 2 written in the answer line? What does it mean? • What does the “carried” 1 mean? Why is it placed where it is? What other term(s) besides carry could be used to develop a more meaningful understanding? • Why is 9 placed where it is? What does it represent? • How would your answers to questions 1 – 3 help provide a mathematical justification for a traditional, standard algorithm? How would they contribute to a deeper understanding of the traditional algorithm?

13. Justify each step in 24 x 35 • 24 x (30 + 5) • (24 x 30) + (24 x 5) • [(20 + 4) x 30)] + [20 + 4) x 5] • (20 x 30) + (4 x 30) + (20 x 5) + (4 x 5) • 600 + 120 + 100 + 20 • 840

14. Solve 24 x 35 below using partial products and the standard algorithm. • What does the “carried” 2 mean? Why is it written where it is? Can you think of another term besides carry that might help to develop a more meaningful understanding? • Why is 120 written below the first horizontal (total) line? What does it represent? • Why is a 1 “carried”? What does it represent? Why is it written where it is? Again, what term might be a better choice than carry? • What does the 72 represent? Why? • How are these numbers connected to the numbers in the area representation and to the partial product algorithm?

15. Exit Slip • How will you use what you learned today in your classroom?