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Multiply and Divide, Big and Small
- 1. Multiply and Divide, Big and Small NCCTM 2014 Amanda Northrup anorthrup@haywood.k12.nc.us www.linkyy.com/teachandlearn @msnorthrup
- 2. The Need… 2 Ma & Pa Kettle Math
- 3. CRA – A Sequence of Instruction 3 C = Concrete. Use materials to focus on the development of conceptual understanding, while starting to make connections to procedures. During this stage, students might work with base ten blocks, fraction bars, red and yellow chips, tiles, cubes, etc...
- 4. CRA – A Sequence of Instruction 4 R = Representational. Connect the previous work with concrete materials to other representations, especially drawings. Students think more deeply about both concepts and procedures. They might use circles, tallies, rectangles, drawings, etc...
- 5. CRA – A Sequence of Instruction 5 A = Abstract. Use previous work with materials and drawings to make sense of procedures with numbers and symbols.
- 7. CRA for Multiplying Large Whole Numbers 7 Concrete… 36 x 4 36 x 24
- 8. CRA for Multiplying Large Whole Numbers 8 How would you use representational? 36 x 4 36 x 24
- 9. CRA for Multiplying Large Whole Numbers 9 Now for abstract 36 x 4 Any abstract method should MAKE SENSE based on what we learned from the concrete and representational strategies. If we can’t explain the connection for the strategy, we shouldn’t be using it.
- 10. CRA for Multiplying Large Whole Numbers 10 Abstract strategies for 36 x 4 36 x 24 436 x 24 • Partial products • Array • Repeated addition • Lattice • Standard algorithm (5th grade)
- 11. CRA for Division with Whole Numbers 11 Concrete – Use 15 chips to show 15 divided by 5
- 12. CRA for Division with Whole Numbers 12 Concrete 78 3
- 13. CRA for Division with Whole Numbers 13 Concrete: Use base ten blocks to show 78 3
- 14. CRA for Division with Whole Numbers 14 Representational: Show 78 3 using a drawing Circle division: 672 5
- 15. CRA for Division with Whole Numbers 15
- 16. CRA for Division with Whole Numbers 16 Abstract: 672 5 Try it with more than one algorithm
- 17. DECIMALS
- 18. Strategies for Multiplying Decimals 18 How could you model 3 x 0.4 using concrete materials?
- 19. Strategies for Multiplying Decimals 19 How could you model 3 x 0.4 using a picture? 0.4 + 0.4 + 0.4 = 1.2
- 20. Strategies for Multiplying Decimals 20 0.1 x 0.1 Things to think about: • What is 1 x 0.1? (Justify your answer) • Will our answer be more or less? • What is ½ of 0.1 • Will our answer be more or less? • What could the context be?
- 21. Strategies for Multiplying Decimals 21 How could you model 0.1 x 0.1 using concrete materials?
- 22. Strategies for Multiplying Decimals 22 How could you model 0.1 x 0.1 using a picture?
- 23. Strategies for Multiplying Decimals 23 What about the standard algorithm? Solve these using only repeated addition and/or decimal grids: 3 x 7 3 x 0.7 7 x 0.03 What do you notice stays the same? What differences do you notice?
- 24. Strategies for Multiplying Decimals 24 What do you notice stays the same? What differences do you notice? 34 x 78 = 2652 3.4 x 0.78 = 2.652 34 x 7.8 = 265.2
- 25. Strategies for Multiplying Decimals 25 Using reasoning to place the decimal – Define a context for 7.9 x 4.6 Let’s find the digits. 7.9 x 4.6 = 3 6 3 4 0.3634 3.634 36.34 363.4
- 26. Strategies for Multiplying Decimals 26 What reasoning can you use to place the decimal? 7.9 x 4.6 = 3 6 3 4 8 x 5 = 40 7 x 5 = 35 7 x 4 = 28 8 x 4 = 32
- 27. Strategies for Multiplying Decimals 27 What reasoning can you use to place the decimal? 5 1 4 4 64.3 x 0.8 = 64 x 1 = 64 60 x 1 = 60 64 x ½ = 32 60 x ½ = 30
- 28. Strategies for Dividing Decimals 28 What’s the context? 0.9 3 What’s the concrete model?
- 29. Strategies for Dividing Decimals 29 What’s the context? 0.9 3 How could you draw a pictorial model?
- 30. Strategies for Dividing Decimals 30 What’s the context? 2 0.5 What’s the concrete model? 1 2 3 4
- 31. Strategies for Dividing Decimals 31 2 0.5 How could you draw a pictorial model?
- 32. Strategies for Dividing Decimals 32 What do you notice? What is tricky here? Why do these results make sense? 0.2 4 = 0.05 4 0.2 = 20
- 33. Strategies for Dividing Decimals 33 Using reasoning to place the decimal – 33.8 13 Let’s find the digits. 33.8 13 = 2 6 0.26 2.6 26 260
- 34. Strategies for Dividing Decimals 34 What reasoning can you use to place the decimal? 33.8 13 = 2 6 30 10 = 3 30 15 = 2 36 12 = 3
- 35. Strategies for Dividing Decimals 35 Using reasoning to place the decimal – 97.5 6.5 Let’s find the digits. 97.5 6.5 = 1 5 150 15 1.5 0.15
- 36. Strategies for Dividing Decimals 36 What reasoning can you use to place the decimal? 97.5 6.5 = 1 5 90 10 = 9 100 10 = 10 100 5 = 20 90 5 = 18
- 37. FRACTIONS
- 38. A Context to Consider: 38 Half of Jimmy’s garden is roses. Of the roses, two-thirds are red. What fraction of Jimmy’s whole garden is red roses? Model with CONCRETE materials: Fraction bars; Paper strip 2 1 2 of 3
- 39. Multiplying Fractions 39 How can you use a drawing to represent this problem? 2 1 2 3
- 40. Multiplying Fractions 40 What’s the context? 2 1 3 4 How can you use concrete objects to model this problem?
- 41. Multiplying Fractions 41 How can you use a drawing to represent this problem? 2 1 3 4 When and how do you introduce the standard algorithm for multiplying fractions?
- 42. Multiplying Fractions 42 What’s the context? 2 1 1 2 2 3 How can you use concrete objects to model this problem?
- 43. Multiplying Fractions 43 How can you use a drawing to represent this problem? 2 1 1 2 2 3 What algorithms can you use for multiplying with mixed numbers?
- 44. Multiplying Fractions 44 2 1 1 2 2 3 What algorithms can you use for multiplying with mixed numbers?
- 45. Multiplying Fractions 45 Tasks like this are important for developing reasoning: Use words and numbers to explain whether the product is larger or smaller than the underlined factor. 1 1 2 18 x 1 16 x 36 x ½ 2 72 x ½ x 2 2 3 1 2 3
- 46. Division with Fractions 46 It’s all about reasoning! Use only pictures and words to solve these problems. 1. Choose a problem 2. Solve and discuss with your partner 3. Flag any problems you would like to debrief with the group 4. Move to any other problem
- 47. Division with Fractions 47 How do these problems differ? Write a context for each. Draw a representation for each. ½ 5 5 ½
- 48. Multiply and Divide, Big and Small NCCTM 2014 Amanda Northrup anorthrup@haywood.k12.nc.us www.linkyy.com/teachandlearn @msnorthrup
Notas del editor
- Take pics of representations – 5 groups of 3, 3 groups of 5, a 3x5 array How is division related to multiplication? Do all of these representations have the same meaning? Emphasize that a context can help us understand the meaning of division (and create a matching representation)
- Do and discuss Show video of my students solving this problem
- Do and discuss connection between the representational strategies and the previous work with blocks. If time, demonstrate this problem on iPad app “Dare to Share” as well Using Explain Everything demo circle division for 78 divided by 3, then challenge group with 672 / 5
- Allow participants time to solve and discuss, then present their strategies.
- Handouts – decimal grids, number lines
- Ample discussion here to work out the meaning of multiplication in this context.
- Ample discussion before clicking to reveal images Note – model with the blocks how these are done concrete
- The digits are the same. The only uncertainty is – where does the decimal go? How can we use reasoning to determine where to place the decimal?
- Step through each.
- Solicit ideas. Common possibilities included after click
- Solicit ideas. Common possibilities included after click Before moving on, note that the next lesson would have the students solve several problems using reasoning – then analyze them to look for patterns. Students DISCOVER the algorithm of counting decimal places this way
- .3 in each of 3 groups
- Circles Decimal grids Number line Bar model
- Notice that we are now dividing a whole number by a decimal instead of dividing a decimal by a whole number. How does that change things in the problem?
- Number line Bar model Why won’t circle division work here?
- Number lines can help understand why one quotient is so small, and one is so large
- Step through each one
- Solicit ideas. Common possibilities included after click
- Step through each one
- Solicit ideas. Common possibilities included after click This one is especially challenging because the decimal isn’t needed for the quotient. Point out the common misconception that the algorithm of counting decimal places doesn’t work here, as in multiplication. Invite discussion on this point. Goal end time – 3:30
- Note both area model and bar model if no one else mentions these
- Note both area model and bar model if no one else mentions these
- Note and demonstrate the use of area model algorithm if no one else does A good time to demonstrate Think Blocks app – Fractions Version
- Note and demonstrate the use of area model algorithm if no one else does A good time to demonstrate Think Blocks app – Fractions Version
- Talk through each example End goal time (2:30 + break)
- 10 problems are posted around the room for groups to solve (20 mins) Debrief with an emphasis on pictures for problems 1, 2, 4, and 5 End goal time is 4:05
- Note the tendency of students to look for a rule/short cut which causes problems when ½ div by 5 is much different than 5 div by ½ Allow time to work independently, share in group, and discuss