Happy Spring! Addition and subtraction (last month) Multiplication and division (today )

1. Happy Spring! • Addition and subtraction (last month) • Multiplication and division (today) • Measuring length, working with time and money (Apr. 17)

2. Agenda • Your experiences in the last few weeks • Establishing fluency with addition and subtraction • The different ways that multiplication is represented and learned • A sequential approach • Number talks and other fluency approaches • Internet resources for practice and drill • Connection of multiplication to area of rectangles • Extensions to division

3. 2nd grade critical areas (1) extend understanding of base-ten notation; (2) build fluency with addition and subtraction; (3) use standard units of measure; and (4) describe and analyze shapes.

4. 3rd grade critical areas (1) develop understanding of multiplication and division and strategies for multiplication and division within 100; (2) develop understanding of fractions, especially unit fractions (fractions with numerator 1); (3) develop understanding of the structure of rectangular arrays and of area; (4) describe and analyze two-dimensional shapes.

5. How do we get to fluency?

6. Addition and Subtraction fluency 2.OA.2 Fluently add and subtract within 20 using mental strategies. By end of Grade 2, know from memory all sums of two one-digit numbers. Teaching strategy: 1. Lots of real-world problems to make sure students know the concepts 2. Focus on combinations that aren’t known. 3. Practice in various ways to develop useful and efficient strategies 4. Drill in ways that reward speed See wiki and fluency packet

7. 1. Real-world problems Joining, separating, comparing, part-whole • Lucy has 8 fish. She wants to buy 5 more fish. How many fish would Lucy have then? • TJ had 13 chocolate chip cookies. At lunch she ate 5 of those cookies. How many cookies did TJ have left? • Mark has 3 mice. Joy has 7 mice. Joy has how many more mice than Mark? • 10 children were playing soccer. 6 were boys and the rest were girls. How many girls were playing soccer?

8. 2. Focus on specific combinations

9. 3. Develop useful and efficient strategies

10. 3. Practice Sum Search Math Squares • Always follow the small group work with a whole class discussion where students explain their methods. Number Talks 8+6

11. Fluency – Practice and Drill From Van de Walle, Elementary and Middle School Mathematics: Teaching Developmentally “Practice” refers to lessons that are problem-based and that encourage students to develop flexible and useful strategies that are personally meaningful. “Drill” is repetitive non-problem-based activity to help children become facile with strategies they know already in order to internalize (remember) the fact combinations.

12. 4. Drill http://www.fun4thebrain.com/addition.html Print triangle flash cards

13. Addition and Subtraction fluency 2.NBT.5 Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction. Teaching strategy: 1. C 2. R 3. A 4. Practice with corrective feedback (I do it, we do it, you do it – Explain your thinking)

14. “…using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.” What’s a problem for which this solution strategy would be an easy approach?

15. Johanna has 12 fish in her aquarium. Over the summer, she gets some more fish from the pet store. Now she has 32 fish. How many more fish did she get over the summer? • 12 + ___ = 32

16. Multiplication and Division • A silent activity: Write all the things you know about multiplication. • Discussion: Go around the table and discuss what you wrote, one item at a time.

17. 2nd Grade 2.OA.4 Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends. 2.G.2 Partition a rectangle into rows and columns of same-size squares and count to find the total number of them.

18. 3rd Grade 3.OA.1 Interpret products of whole numbers, e.g., interpret 5 × 7 as the total number of objects in 5 groups of 7 objects each. For example, describe a context in which a total number of objects can be expressed as 5 × 7. 3.OA.3 Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.

19. Multiplication – Equal Groups • Skip counting

20. Acquisition-Fluency-Generalization Acquisition • modeling word problems • skip counting in arrays • using visual representations • developing the connection to area

22. Whole-class activities Developing Number Concepts Book 3: Place Value, Multiplication and Division by Kathy Richardson Dale Seymour Publications, 1999 $34.95 through Math Perspectives – Teacher Development Center

23. Act. 2-23 Lots of Rectangles The child selects a card, fills the rectangular shape with cubes or tiles, and writes the multiplication equation that describes the array that is formed.

24. Looking for equal groups in the real world I see six panes in the window by my desk. Are there any other windows in our room that have six panes? I see two more. Yes, we have three windows, each with six panes. Six panes and six panes and six panes.

25. Acting out multiplication stories 1: Using real objects • 4 chairs around 3 tables, 2 pencils for 4 children, 4 stacks of 3 books, 5 boxes of crayons with 8 in each box

26. Acting out multiplication stories 2: Using counters with word problems There are 4 houses on Letitia’s street. The family in each house has 2 cars. How many cars is that in all? • Use your counters to show only the cars. How many cars are in front of the first house? • How many cars are in the front of the second house? Show this with counters. • Have you shown the cars for all the houses? Show me the cars in front of the next house. • You made four groups of two cars. How many cars is that in all?

27. Other story problems • Tim had three dogs. He gave each dog two bones. How many bones did he give all his dogs? • Five girls went to the library. They each checked out three books. How many books did they check out altogether? • There are five children in Dale’s family. Each child gets to carve one pumpkin for Halloween. How many jack-o’-lanterns will they have? • Robin’s mother went shopping for school clothes for her three children. She bought three shirts for each child. How many shirts did she buy?

28. Other types of multiplication problems Rate problems A baby elephant gains 4 pounds each day… Price problems How much would 5 pieces of bubble gum cost if…? Combination problems The Friendly Old Ice Cream Shop has 3 types of ice cream cones. They also have 4 flavors of ice cream.

29. Other types of multiplication problems Array and Area problems (symmetric problems) For the second grade play, the chairs have been put into 4 rows with 6 chairs in each row… A candy maker has a pan of fudge that measures 8 inches on one side and 9 inches on another side. If the fudge is cut into square pieces 1 inch on each side…?

30. Building models of multiplication problems • Towers or stacks • Rows • Groups or piles Build 2 rows of 3. How many altogether? Build 3 piles of 5. How many? Build 5 piles of 3. How many? Build 4 stacks of 2. How many? Build 2 stacks of 4. How many?

31. Modeling the recording of multiplication experiences Misty stacks books into two piles. She puts four in each pile. How many stacks is Misty making? Two. 2 stacks of How many books in each pile? Four. 2 stacks of 4 How many books altogether? Eight. 2 stacks of 4 = 8

32. Introducing the multiplication sign 2 stacks of 4 = 8 2 x 4 = 8 Read the sign as “stacks of” (not “times”) (or groups of, rows of, piles of) Interpreting symbols Write 4x3. Let’s make rows this time. How many stacks will you make? How many in each stack?

33. Acting out stories to go with multiplication problems What story can you tell for this problem?

34. Learning to write the multiplication sign • Copying equations • Writing equations and checking

35. Acquisition-Fluency-Generalization Fluency • independent and small group activities • games • developing strategies that build to fluency 3.OA.7 Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 × 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of one-digit numbers.

36. 3.OA.5 Apply properties of operations as strategies to multiply and divide. Examples: If 6 × 4 = 24 is known, then 4 × 6 = 24 is also known. (Commutative property of multiplication.) 3 × 5 × 2 can be found by 3 × 5 = 15 then 15 × 2 = 30, or by 5 × 2 = 10 then 3 × 10 = 30. (Associative property of multiplication.) Knowing that 8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive property.) (Students need not use formal terms for these properties.)

37. Doubling and Halving 1 x 16 2 x 8 4 x 4 8 x 2 16 x 1 Number Talks

38. Independent activities Developing Number Concepts Book 3: Place Value, Multiplication and Division by Kathy Richardson Dale Seymour Publications, 1999 $34.95 through Math Perspectives – Teacher Development Center

39. How many cups

40. How many groups?

41. How many rows?

42. How many towers?

43. Interpreting multiplication problems

44. Copying and solving problemsMaking up their own

45. Problems for partners

46. Roll and Multiply (Patterns)

47. Discovering Patterns: Cupfuls

48. Fluency – two steps 1.Practice – to learn strategies • The Product Game; Number Talks • Word problems clustered on 5’s (ORIGO) 2. Drill – to habituate the combinations • Some element of competition, where quick recall is important • Multiplication call-out; Computer games Number Talks Multiplication String: 7x7

49. Practice to develop strategies The numbers in the problems should be built up gradually and deliberately. While many teachers already focus on one set of “facts” at a time, this focused approach to story problems helps students see patterns and begin to develop strategies. Here is a set of “times 5” problems that would be used together over the course of several days: • There are bags of 5 apples for sale. If you buy 3 bags, how many apples will you have? • It takes 5 minutes to fill a wheelbarrow with soil. How long will it take to fill 6 wheelbarrows? • There are 5 rows of 8 chairs. How many people can be seated? • Nine cats each had 5 kittens. How many kittens is that altogether? • When Ben places 4 shoes end to end, they measure 1 yard. How many of these shoes would be lined up to measure 5 yards? • Jacob wants to plant 7 rows of 5 seeds. How many seeds is that?

50. Fluency – Practice and Drill From Van de Walle, Elementary and Middle School Mathematics: Teaching Developmentally “Practice” refers to lessons that are problem-based and that encourage students to develop flexible and useful strategies that are personally meaningful. “Drill” is repetitive non-problem-based activity to help children become facile with strategies they know already in order to internalize (remember) the fact combinations.