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1. Multiplication and Division Errors

2. There are 57 teams in the Charlotte Area Basketball League (CABL). If there are 13 players on each team, how many players are in the league? • Solve the task in two different ways.

3. CABL • Solve 3 x 7 • Solve 3 x 50 • Solve 50 x 10 • Solve 7 x 10 • How do they relate to 57 * 13?

4. CABL • Solve 10 x 13 = 130 • Take the product and multiply it by 5, 130x5 = 650 • Solve 7 x 10= 70 • Solve 7 x 3= 21 • How do they relate to 57 * 13? • 5x(10x13) 7x(10+3), 50x13, 7x13 • 57x13 = (50x13)*(7x13)

5. CABL • Solve 10 x 13 = 130 • Take the product and multiply it by 6, 130x6 = 780 • Solve 3 x 13 = 39 • How do they relate to 57 * 13? • 6x(10x13), 3x13 • 60x13, 3x13– what’s the relationship to 57x13? • 60-3 = 57. 57x13= (60x13) – (3x13)

6. There are 57 teams in the Charlotte Area Basketball League (CABL). If there are 13 players on each team, how many players are in the league? But what if we added 3 more teams to the league? How many more players are needed?

7. There are 57 teams in the Charlotte Area Basketball League (CABL). If there are 13 players on each team, how many players are in the league? But what if we added 3 more teams to the league? How many more players are needed? 57x13- now Add 3 teams- add 3x13= 39

8. Various Models of Multiplication • Multiplication as Cartesian (Groups) Use concrete objects to represent the various items and various locations. begin w/ concrete object or representation progress to just the number Team 1 Team 2 Team 3 Team 55 Team 56 Team 57 13 13 13 13 13 13

9. Various Models of Multiplication • Multiplication as Cartesian (Groups) • Tomas and his four friends each have two pencils. How many total pencils do they have? Use concrete objects to represent the various items and various locations. Tomas 2 Friend 1 2 Friend 2 X X Friend 3 X X Friend 4 X X

10. Various Models of Multiplication • Multiplication as Repeated Addition • Number Line Models • 57 jumps of 13…. 13 26 39 52 65

11. Various Models of Multiplication • Multiplication Arrays 50 7 10 50 * 10 =500 7 * 10 =70 3 50 * 3 =150 7 * 3 =21 500 + 150 + 70 + 21 = 741

12. Errors in Multiplication

13. Representational Errors • Misapplication of action language • Steve puts 5 cookies in 4 plastic bags. How many cookies are in all of the bags?, 5x4=20 • 5 bags of 4 or 4 bags of 5? • In both cases the answer is 20. • So, why does it matter? • Extension: Steve’s mom adds two cookies to each bag. How many cookies are there now? • 4 bags w/ 7 in each, 4x7=28, error- 6x5 = 30

14. Wordy Word Problems There is a difference between donuts and muffins. There are five donuts in a pack, while there are six muffins in a pack. If I had five packs of each how many more muffins would I have than donuts total?

15. Wordy Word Problems A total of 39 people come to a reception. Altogether there are 12 tables that fit 4 people each. How many people can still come to the reception?

16. Representational Errors Your students work with base-10 blocks to model 32 x 5. One student shows: How is the representation below different?

17. Factors: An intro to division • Make multiple lists of the first 6 multiples for: • 2s, 3s, 4s, 5s, 6s • What do you notice about the multiples in each list?

18. 2, 4, 6, 8, 10, 16 • Ends in an even number • 3, 6, 9, 12, 15, 18 • Sum of the digits • 12- 1+2 = 3, 1+5=6, 1+8 = 9 • If sum of digits is a multiple of 3, the number is divisible by 3

19. 4, 8, 12, 16, 20, 24, 28, 32 • 4 is double of 2 • If half a number is divisible by 2 then it’s divisible by 4, 24- half is 12. • 458- if the last two digits together are divisible by 4 then then number is • 444, 448, 1,048, 5,112; 1,000,516 • 100 is a multiple of 4…last 2 digits determine divisibility

20. 5, 10, 15, 20, 25, 30 • Ends in 5s or 0s • 6, 12, 18, 24, 30, 36, 42 • If sum of digits is a multiple of 3, the number is divisible by 3 • Every multiple of 6 is a multiple of 3 • If half the number is divisible by 3, the number is divisible by 6 • 1,212- 606, 6+6= 12. 1,212- divisible by 3.

21. Types of Division Problems • I have 15 cookies. If I divide them equally and place them into three bags, how many cookies will be in each bag? • I have 15 cookies. If I put 5 in each bag, how many bags will I fill? • Similarities? • Differences? Measurement model Partition model

22. Misapplication of Action Language Teacher: Teresa, would you please read the number sentence? Teresa: Eight divided by two. Teacher: Can you use the materials here (Cuisenaire Rods) to solve this problem? Teresa: I think so, let me try. (Teresa then solved the problem 8 x 2 = 16 as follows.) Teacher: Teresa, that is good work, but is that the solution to the problem you read from the number sentence? Teresa: I think so. Teacher: Let me read it for you and then you see if you might be able to solve it a different way? A set of eight grouped into sets of two, how many times?

23. Misapplication of Action Language Teresa: Oh, that's different, let me try that one. Teresa then solves the problem as follows. Teacher: Teresa, that is excellent. You solved the problem "Eight grouped into sets of two" rather than the problem "Eight sets of two." One of these sentences is division action. Can you tell me which one? Teresa: The first one you said. Teacher: Do you mean, "eight grouped into sets of two?" Teresa: Yes, that's the one.

24. Misapplication of Action Language Teacher: That's right, now see if you can read this problem? 6  3 = ? Teresa: Six grouped into sets of three, how many times? Teacher: That's great, can you solve it using the rods? Teresa: Sure. (Teresa's solution)

25. Division Errors- Concrete • Assimilating Measurement and Partition Division into a Generalized Concept • Billy has 12 marbles. He wants to put them into three bags. How many marbles should Billy place in each bag so that there are the same number in all bags?

26. Division Errors- Concrete • Assimilating Measurement and Partition Division into a Generalized Concept • Billy has 14 marbles. He wants to put them into three bags so that each bag has the same number of marbles. How many are in each bag? What do you notice?

27. Division Errors- Concrete • Dealing with Remainders • There are 53 slices of bread in a bag. While making sandwiches for the homeless, you use two slices of bread per sandwich. If your class uses 5 bags of bread, how many sandwiches can you make? • 5 bags of 53= 265. 265/2 = 132 R 1 • 53 slices divided by 2= 26 sand. 1 slice leftover • 26x5 = 130

28. Sold out UGA Game • There are 92,352 at the sold out University of Georgia football game. If the stadium is divided into 8 equal sections how many people would be in each section?

29. Long division • Carrying the divisor down • Fraction as a model for division • Kenyan model

30. Symbolic Errors • Let’s play detective!