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Multiplication and Division Errors

Multiplication and Division Errors. 11/8/06. Various Models of Multiplication. Multiplication as Cartesian or Cross-Product 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

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Multiplication and Division Errors

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  1. Multiplication and Division Errors 11/8/06

  2. Various Models of Multiplication • Multiplication as Cartesian or Cross-Product • 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 X X Friend 1 X X Friend 2 X X Friend 3 X X Friend 4 X X

  3. Various Models of Multiplication • Multiplication as Repeated Addition • Number Line Models • 4 jumps of 5 0 5 10 15 20

  4. Various Models of Multiplication • Area Models of Multiplication

  5. Representational Errors • Misapplication of action language • Steve puts 5 cookies in 4 plastic bags. How many cookies are in all of the bags? • 5 bags of 4 or 4 bags of 5? • In both cases the answer is 20. • So, why does it matter?

  6. Concrete and Representational Multiplication Error Types

  7. Concrete and Representational Multiplication Error Types

  8. Representational Errors Your students work with base-10 blocks to model 32 x 5. One student shows: How would the representations below have helped?

  9. Symbolic Errors • Let’s play detective!

  10. 7*4= 28 + 3 = 36 • 9*2 = 18 + 5 = 21 • 8*6= 48 + 4= 51

  11. Division Errors • 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

  12. 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?

  13. 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.

  14. 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)

  15. Division Errors- Concrete • Assimilating Measurement and Partition Division into a Generalized Concept • Billy has 12 marbles. He wishes 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? • 12 / 3 = ? Alter the problem

  16. Division Errors- Concrete • Assimilation of Remainders to the Concept of Whole Numbers Added to a Quotient • 17 dogs have to be walked on two days • Probing reveals: “the remainder wasn’t counted”

  17. Division Errors- Representational

  18. Symbolic Errors • Let’s look at some common errors

  19. Division Symbolic Errors • Error 1: Basic Facts (#4) • Memorization • Actively use concrete objects to develop an understanding of basic facts • Error 2: Errors associated with other operations in the algorithm • Incorrect subtraction (#2) • Incomplete subtraction (#5) • Multiplication error (#11)

  20. Division Symbolic Errors • Error 3: Errors related to Base-10 misunderstandings • Forgetting the zero is a place holder in the quotient (#3) • Failing to account for complete divisor when it ends in a zero (#10)

  21. Division Symbolic Errors • Error 4: Errors within the division algorithm • Treating first partial quotient as the solution (#1) • Failing to multiply by the final partial quotient (#7) • Ignoring remainders equal to or greater than the divisor (#8) • Omitting the remainder altogether (#9) • Dividing the divisor by the first factor of the dividend (#6)

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