Teaching Multiplication (and Division) Conceptually

1. Teaching Multiplication(and Division) Conceptually

2. Professional Learning Targets… • I can describe what it means and what it looks like to teach multiplication (and division) conceptually. • I can describe how standards progress across grade levels, giving details for the grade span in which I teach.

3. Agenda • Warming up with Multiplication and Division • Number Strings • Quick Images • Number Talks with Multiplication and Division • Big Ideas of Multiplication and Division • Types of Multiplication and Division Problems • Multiplication/Division Games

4. A warm–up mental number string • 100 x 13 • 2 x 13 • 102 x 13 • 99 x 13 • 14 x 99 • 199 x 34

5. A warm–up mental number string • 100 x 13 • 2 x 13 • 102 x 13 • 99 x 13 • 14 x 99 • 199 x 34 • What strategy does this string support? • What big ideas underlie this strategy?

6. Pictures for early multiplication

7. Small Group Discussions • What strategies would you expect to see? • How would you represent them?

8. How many apples? How many lemons?

9. How many tiles in each patio? • The furniture obscures some of the tiles possibly providing a constraint to counting by ones and supporting the development of the distributive property

11. Here’s one that I found.

12. Prior Understandings—Grades K-2 • Counting numbers in a set (K) • Counting by tens (K) • Understanding the numbers 10, 20, 30, 40, …, 90 refer to one, two, three, four, …, nine tens (1) • Counting by fives (2)

13. Prior Understandings • 2.G.2. Partition a rectangle into rows and columns of same-size squares and count to find the total number of them.

14. So What Else About Multiplication? X Thinking that multiplication should always be interpreted as repeated addition.

15. Multiplication can be interpreted in a variety of familiar ways. 3 x 5 = 15 3 + 3 + 3 + 3 + 3 3 3’ array area model repeated addition 5 + 5 + 5 5 5’

16. Array Cards

17. Scaling

18. “Gary’s flashlight shines three times farther than mine!” My Flashlight 3 times farther than 5 feet x 15 3 = 5 5 feet Gary’s Flashlight 15 feet

19. When can multiplication be interpreted as scaling? When it represents the relationship between the size of the product and the factors. 3x5 =15 15is 5 times > 3 15is 3 times > 5

20. So What About Division? How many of our students understand dividing a number by 3 is the same as multiplying the number by 1/3?

21. 169 ÷ 14 = To begin thinking about division, solve this problem using a strategy other than the conventional division algorithm. You may use symbols, diagrams, words, etc. Be prepared to show your strategy Hedges, Huinker and Steinmeyer. Unpacking Division to Build Teachers’ Mathematical Knowledge, Teaching Children Mathematics, November 2004, p. 4-8.

23. Forgiveness Method 21 12 252 - 120 10 132 - 120 10 12 - 12 1 0 21

24. Issic Leung, Departing from the Traditional Long Division Algorithm: An Experimental Study. Hong Kong Institute of Education, 2006.

26. Change it UP!!!! 1. Deal each player five cards. The remaining cards are placed face down on the center of the table. 2. Player one places a card face up on the table reads the division problem and provides the quotient. The next player must place a card with the same quotient on the first card. If the player cannot match, he/she may place a “Math Wizard” card on top and then a card with a different quotient. 3. If the player in unable to make either move, he/she must draw from the deck until a match is made. 4. The first player to use all of his/her cards is the winner.

27. Lies my teacher told me… • Division is about “fair sharing”. 35 ÷ 8 =

28. The Remainder • Can be discarded. • The remainder can “force the answer to the next highest whole number. • The answer is rounded to the nearest whole number for an approximate result.

29. Landon bought 80 piece bag of bubble gum to share with his five person soccer team. How many pieces did each player receive? • Brittany is making 7 foot jump ropes for the school team. She has a 25 foot piece of rope. How many can she make? • The ferry can hold 8 cars. How many trips will it need to make to carry 25 cars across the river?

30. Near Facts… Find the largest factor without going over the target number

31. Partial Quotients 18 R 25 26 493 - 260 10 233 - 130 5 103 - 783 25 18

32. The Remainder Game 1. To begin the game, both players place their token on START. 2. Player one spins the spinner and divides the number beneath his/her marker by the number on the spinner. If there is a remainder, he/she is allowed to move his/her token as many spaces as the remainder indicates. If the division does not result in a remainder, he/she must leave his/her marker where it is. 3. The play alternates between the two players (a new spin must occur each time) until some reaches HOME.

33. Lies my teacher told me… Any number divided by zero is zero! 6 ÷ 0 = How many times can 0 be subtracted from 6? How many 0 equal groups are there in six? What does six divided into equal groups of 0 look like? What number times 0 gives you 6?

34. Division Vocabulary Quotient Dividend Divisor

36. THE PARTITIVE PROBLEM

37. Partitive

38. Example 1: Write a word problem to represent this model of division?

39. THE MEASUREMENT PROBLEM

40. Measurement

41. Example 2: Write a word problem to represent this model of division?

42. Two basic types of problems in division Partitive (Sharing): You have a group of objects and you share them equally. How many will each get? Example: You have 15 lightning bugs to share equally in three jars. How many will you put in each jar?

43. Two basic types of problems in division Measurement: You have a group of objects and you remove subgroups of a certain size repeatedly. The basic question is—how many subgroups can you remove? Example: You have 15 lightning bugs and you put three in each jar. How many jars will you need?

44. Measurement Model

45. Ratio

46. Grade 3 Introduction • In Grade 3, instructional time should focus on four critical areas: (1) developing understanding of multiplication and division and strategies for multiplication and division within 100; … • Students develop an understanding of the meanings of multiplication and division of whole numbers through activities and problems involving equal-sized groups, arrays, and area models; • multiplication is finding an unknown product, and division is finding an unknown factor in these situations. • For equal-sized group situations, division can require finding the unknown number of groups or the unknown group size. • Students use properties of operations to calculate products of whole numbers, using increasingly sophisticated strategies based on these properties to solve multiplication and division problems involving single-digit factors. • By comparing a variety of solution strategies, students learn the relationship between multiplication and division.

47. Commutative Property • It is not intuitively obvious that 3 x 8 = 8 x 3. A picture of 3 sets of 8 objects cannot immediately be seen as 8 piles of 3 objects. Eight hops of 3 land at 24, but it is not clear that 3 hops of 8 will land at 24. • The array, however, can be quite powerful in illustrating the commutative property.

48. Distributive Property & Area Models 3 x 7 =__ 5 + 2 3 15 + 6 3 x 7 = 3 x (5 + 2) = (3 x 5) + (3 x 2)= 15 + 6 = 21

49. Grade 3 • 3.MD.7. Relate area to the operations of multiplication and addition. • Find the area of a rectangle with whole-number side lengths by tiling it, and show that the area is the same as would be found by multiplying the side lengths. • Multiply side lengths to find areas of rectangles with whole-number side lengths in the context of solving real world and mathematical problems, and represent whole-number products as rectangular areas in mathematical reasoning. • Use tiling to show in a concrete case that the area of a rectangle with whole-number side lengths a and b + c is the sum of a × b and a × c. Use area models to represent the distributive property in mathematical reasoning. • Recognize area as additive. Find areas of rectilinear figures by decomposing them into non-overlapping rectangles and adding the areas of the non-overlapping parts, applying this technique to solve real world problems.

50. Grade 3Represent and solve problems involving multiplication and division (Glossary-Table 2) Not until 4th Grade