1. Subtraction and computation Dr. Calvin J Irons

2. When/where do we need to be able to subtract? What types of situations lend themselves to subtraction?

5. I have $125. If I buy the $89 ticket, how much money will I have left? What pictures do students have of numbers/operations? What pictures do students need to calculate mentally with subtraction?

6. The mathematical structure of numbers include: Counting – a discrete representation: ungrouped grouped place value Measurement – a continuous representation: linear area

7. Counting is not an efficient strategy. The students do (and we stress) too much counting. Counting

8. $54 $38 How do we calculate if our picture of numbers is one that relates more to counting than some of the other representations? Find the difference

9. Place value is an abstract system that enables us to read and write numbers. Place value

10. $54 How do you calculate if the main picture of number is place value? $38 Find the difference

11. Remember, place value is just one of the structures. In some situations other representations may be more useful – particularly for mental strategies involving addition and subtraction (and of course for fractions and even decimals). Place value

12. The only model that can be used with ALL numbers. The best model for many strategies Not enough time (probably no time) is spent to develop this model. And then, the true linear aspects of the model are not stressed. Linear

13. 0102030405060708090100110120 How does a number line show a number? What is 38 on this number line? When asked, most people say a number such as three is at a particular ‘point.’ A number is a length (starting at 0).

14. When students are asked to work with a number such as thirty-eight, what ‘picture’ do they use? How would you work out the difference between the prices of these two items? $38 $53

15. Some important and less structured features of numbers include being able to : Partition a number in multiple ways: Double/halve with confidence:

16. Are their other pictures that we use to $190 $160 Are their other pictures that we use which are not based on a strict mathematical structure? Find the difference? How could you think?

17. How could we work out the difference in price of these two items? $3.75 $1.95 $1.75 $1.95

18. Addition Strategies Begin by extending a fact strategy Strategies Count-on 6 + 1 9 + 2 Use doubles 7 + 7 6 + 5 Bridge-ten 9 + 4 First extension Count-on 16 + 1 19 + 2 Use doubles 25 + 25 26 + 25 Bridge-ten 39 + 4 Further extensions Count-on 26 + 21 29 + 12 Use doubles 27 + 27 126 + 125 Bridge-ten 198 + 25 decimal extensions Count-on 3.6 + 2.1 2.9 + 1.2 Use doubles 2.5 + 2.5 1.26 + 1.25 Bridge-ten 1.98 + 0.6

19. Subtraction Strategies Begin by extending a fact strategy Strategies Take small 6 – 1 9 – 2 Use addition 6 – 5 9 – 7 12 – 6 15 – 7 14 – 9 First extension Take small 16 – 1 59 – 21 Use addition 26 – 25 19 – 17 120 – 60 30 – 15 23 – 9 Further extensions Use addition no bridging 67 – 53 bridging 85 – 59 126 – 98

21. What influences the teaching sequence for subtraction? The number combinations involved? The subtraction situations?

22. Joel has $75 for the day at Maze World. How much does he have left after paying the $24 admission charge? The type of problem and choice of numbers influence how you think. Change either, and you may want to use a different strategy. $29 take-away subtraction

23. What is the difference between the prices of these computer games? $75 $29 difference subtraction

24. How much more do you need? $35 missing addend subtraction

25. How much more do you need? $8.35 missing addend subtraction

26. Cube A: 8, 8, 8, 9, 9, 9 Cube B:11,12,13,14,15,16 2233 4555 6777 3 6 8 4 6 8 Bridging: Subtraction facts Use one board between 2 players. Roll both cubes. Write the numbers with the answer. Get three in a line.

27. Cube A: 8, 18, 28, 9, 19, 29 Cube B:31,32,33,34,35,36 Bridging: beyond subtraction facts Use one board between 2 players. Roll both cubes. Write the numbers with the answer. Get three in a line. 2345 12131415 22232425 6 16 26 7 17 27 8 18 28

28. 0.20.30.40.50.6 0.7 0.8 1.21.31.41.51.61.71.8 2.22.32.42.52.62.72.8 0.20.30.40.50.60.70.8 1.21.31.41.51.61.71.8 2.22.32.42.52.62.72.8 Cube A: 1.8, 2.8, 3.8, 1.9, 2.9, 3.9 Cube B: 4.1, 4.2, 4.3, 4.4, 4.5, 4.6 Bridging: subtracting decimals

29. Mental strategies for subtraction 1. Use addition $190 $160

30. Mental strategies for subtraction 2. Use place value (in some way) $190 $160

31. Mental strategies for subtraction 3. Use partitioning (other than place value) – and possibly use another strategy $190 $160

32. Where is subtraction used outside the realm of whole numbers and decimals?

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