Addition & Subtraction: Closure

1. Addition & Subtraction: Closure Use the fact family 3, 6, 9 to write one story problem for each row.

2. Multiplication & Division 2012

3. Draw pictures for the following problems. • Jacob has four baskets. He has three apples in each basket. How many apples does Jacob have? • Juanita’s family took a vacation to Mexico for a week. Juanita and her three sisters spent $9 each day. How much money did Juanita and her sisters spend in one week? • Mario’s Boy Scout troop has three cars to take them to the park. If there are 14 Boy Scouts and 3 adult leaders who will drive the cars, how many children will ride in each car? • Noah has 20 toy cars. He has 4 times as many toy cars as Sam. How many cars does Sam have?

4. Get To 1000! Rules to Play: Take turns rolling die. Player decides to multiply by 1, 10, 100. Each player records own score. Go until you have 10 products Find the sum The person the closest to 1000 wins!

5. Favorite Activity from the book?

6. How do you define multiplication? division?

7. "Application" of knowledge of x and ÷ in real life…http://www.youtube.com/watch?v=ieKTU94-BgI

8. The meanings of multiplication • Write a story problem that can be solved with: 2 x 4 = 8. • Draw a picture to represent your problem. • Compare your problem and picture with those of the person sitting next to you. In what ways are they similar or different?

9. Examples of story problems that can be solved with 2 x 4 = 8 • Rob has 2 sets of counters with 4 counters in each set. How many counters does he have altogether? • Last year a plant was 2 cm high. Now it is 4 times higher. How high is the plant? • The dimensions of a rectangle are 2 cm and 4 cm. What is its area? • John has 2 pairs of trousers of different colours and 4 shirts of different colours. How many different outfits can he make? union of equivalent sets scale factor: “so many times bigger/higher/…” area of a rectangle “combination” problems: mapping between the members of two different sets

10. Examples of story problems that can be solved with 2 x 4 = 8 ڭ • Rob has 2 sets of counters with 4 counters in each set. How many counters does he have altogether? • Last year a plant was 2 cm high. Now it is 4 times higher. How high is the plant? • The dimensions of a rectangle are 2 cm and 4 cm. What is its area? • John has 2 pairs of trousers of different colours and 4 shirts of different colours. How many different outfits can he make? Repeated addition union of equivalent sets scale factor: “so many times bigger/higher/…” Scaling area of a rectangle Area “combination” problems: mapping between the members of two different sets Cartesian product

11. Multiplication as repeated addition • Concerns the union of equivalent sets – “so many sets of” • Relates to the aggregation meaning of addition e.g.: 2 x 4 = 4 + 4. Rob has 2 sets of counters with 4 counters in each set. How many counters does he have altogether? A garden has 2 rows of trees with 4 trees in each row. How many trees are there altogether? Helen walked 4 miles every day, 2 days in a row. How many miles did she walk in total?

12. Problems from the over-emphasis on multiplication as repeated addition • Task:Calculate 1.20 x 0.3 = ? and see whether you can write a “repeated addition” story problem that can be solved with this calculation. • Compare: • Misconception:“multiplication always makes bigger” • Difficulties: • with situations that involve multiplication of two decimals or fractions • to recognise multiplicative structures related to percentages, ratio, and proportion 1.20 x 0.3 = 0.36 1.20 x 5 = 1.20 + 1.20 + 1.20 + + 1.20 + 1.20 = 6 1.20  0.3 = 0.36

13. Multiplication as scaling • Relates to the augmentation meaning of addition, but in this case a quantity changes multiplicatively, by a scale factor e.g., 2 scaled by a factor of 4 becomes 2 x 4. • Central to situations that involve percentages, ratio, and proportion • Last year a plant was 2 cm high. Now it is 4 times higher. How high is the plant? • 2. The price of a shirt was £37.60. Its new price during sales is 75% the old price. How much does it cost now? 2 cm 8 cm scaled by a factor of 4 2 x 4 37.60 x 0.75 = 28.20

14. Multiplication as area • The dimensions of a rectangle are 2 cm and 4 cm. What is its area? • Ronaldo built a rectangular pen for his small rabbit that was 1.40 m by 0.90 m. What was its area? • The product a x b is the area of a rectangle with dimensions a and b. • Exemplifies well the commutative property of multiplication: a x b = b x a. 2 x 4 = 4 x 2 2 x 4 = 4 x 2 2 2 4 2 x 4 = 4 x 2 4 4 2 1.40 m x 0.90 m = 1.26 m2

15. Multiplication as Cartesian product • Relates to “combination” problems • Cartesian product is the mapping between the members of two sets. • John has 2 pairs of trousers of different colours and 4 shirts of different colours. How many different outfits can he make? • Lara bought 2 kinds of ice cream and 4 kinds of topping. If she serves one kind of ice cream and one kind of topping for each desert, how many different deserts can she offer her guests?

16. Multiplication as Cartesian product Cartesian product: Trousers x Shirts • Relates to “combination” problems • Cartesian product is the mapping between the members of two sets. • This mapping produces a set of different combinations T 8 combi-nations (= 2 x 4) • John has 2 pairs of trousers of different colours and 4 shirts of different colours. How many different outfits can he make? • Lara bought 2 kinds of ice cream and 4 kinds of topping. If she serves one kind of ice cream and one kind of topping for each desert, how many different deserts can she offer her guests? S

17. Multiplication as Cartesian product Cartesian product: Trousers x Shirts • Relates to “combination” problems • Cartesian product is the mapping between the members of two sets. • This mapping produces a set of different combinations T S • John has 2 pairs of trousers of different colours and 4 shirts of different colours. How many different outfits can he make? S 8 combinations (= 2 x 4)

18. The meanings of division • Write a story problem that can be solved with: 12 ÷ 2 = 6. • Draw a picture to represent your problem. • Compare your problem and picture with those of the person sitting next to you. In what ways are they similar or different?

19. divisor quotient dividend 12 ÷ 2 = 6 quotient 6 12 2 dividend divisor Mathematical terminology

20. 12 ÷ 2 = 6 Claire had 12 pencils to share equally among 2 children. How many pencils would each child receive? Sharing (partition) Claire had 12 pencils. She wanted to make packs of 2 pencils. How many packs could she make? Grouping (quotition or measurement)

21. divisor quotient dividend a ÷ b = c The two primary meanings of division • Sharing: • a ÷ b means a shared equally among b groups • the divisor tells you how many equal groups to make • the quotient tells you the size of each group • Grouping: • a ÷ b means a divided into groups of b • the divisor tells you the size of each group • the quotient tells you how many equal groups you can make of that size E.g.: Claire had 12 pencils to share equally among 2 children. How many pencils would each child receive? E.g.: Claire had 12 pencils. She wanted to make packs of 2 pencils. How many packs could she make?

22. 0 10 11 12 More on the grouping meaning: division as repeated subtraction 12 ÷ 2: 12 – 2 – 2 – 2 – 2 – 2 – 2 Claire had 12 pencils. She wanted to make packs of 2 pencils. How many packs could she make? Grouping Division as repeated subtraction Six twos 1 2 3 4 5 6 7 8 9

23. Problems from the over-emphasis on the sharing meaning of division • Task: Calculate 12÷ ¾ = ? and see whether you can write a “sharing” story problem that can be solved with this calculation. • A “grouping” story problem: How many ¾ litre bottles of milk can you make with 12 litres of milk? • Misconception:“division always makes smaller” • Difficulties: • with situations that involve division with a divisor that is a decimal or a fraction 12÷¾ = 16 12x¾ =

24. Summary and conclusion • Multiplication has four primary meanings of which repeated addition is over-emphasised • Division has two primary meanings of which sharing is over-emphasised • The emphasised meanings manipulate pupils’ problem-solving efforts and create difficulties for them when they try to solve problems where those meanings are not applicable • Students should encounter multiplication and division in different kinds of story problems so that they can develop a flexible and well-rounded understanding of their different meanings and domains of applicability

25. Models for Multiplication and Division • Set Models (Equal Groups) • Counters • Area/Array Models (Equal Groups) • Tiles • Grid Paper • Geoboards • Linear Models • Stacking Cubes • Number Line

26. On your note card • Using the number on your card, create your own story problem to match the structure of the problem from Table 2 (Common Core State Standards) [Handout]:

27. Jigsaw and Teach • Useful representations • Complete number strategies – Using Known Facts to find the Unknown • Partitioning Strategies • Compensation Strategies • Cluster Problems • Area Models • Lattice • Partial Products • Egyptian*

29. Resources for you… • http://www.lrsd.org/files/edservices/3xmMultiplicationandDivisionStrategies.pdf • http://www.aea267.k12.ia.us/paraeducators/files/ParaeducatorsSupportingMathManualupdateMarch2008.mw.pdf

30. Invented Strategies • How would you solve: 16 x 4 = ? 21 x 9 = ? 129 x 11 = ? 2005 x 13 = ? 98 ÷ 5 = ? 77 ÷ 25 = ? 488 ÷ 4 = ? 1287 ÷ 100 = ?

31. Conceptual development with the U.S. Standard Algorithm: Multiplication • How would you solve: 16 x 4 = ? 21 x 9 = ? 129 x 11 = ? 2005 x 13 = ?

32. Conceptual development with the U.S. Standard Algorithm: Division • How would you solve: 728 ÷ 7 = ? 77 ÷ 4 = ? 338 ÷ 6 = ? 1287 ÷ 9 = ?

33. Division Strategies • Invented Strategies p. 243-244 • Missing-Factor Strategies p.244 • Cluster p. 244 • Partial Quotients • Column Division 273 ÷ 5 =

34. Algebraic Thinking - Teaching Considerations • Emphasize appropriate vocabulary • Independent “input” and dependent variables “output” • Discrete and continuous • Domain and range • Multiple representations • Context • Table • Verbal Description • Symbols • Graphs • Connect representations • Algebraic thinking across the curriculum

35. Diagnostic Interviews • Video 6 (IMAP): Multiplication • Video 1 (IMAP): Adding • Video 15 (IMAP): Fractions • What do you notice about where the child is at developmentally? • What do you notice about the questions? The Interviewer?

36. DiagnosticInterviews • Review diagnostic interviews • Select one that you will use • Write your name on the post-it