Section 4.1 Mental Math

1. Section 4.1 Mental Math The availability and widespread use of calculators and computers have permanently changed the way we compute. Consequently, there is an increasing need to develop students’ skill in estimating answers when checking the reasonableness of results obtained electronically. Computational estimation, in turn, requires a good working knowledge of mental math. Thus this section begins with several techniques for doing mental calculations. NTCM Standard Grade 4 and 5: Students select appropriate methods and apply them accurately to estimate products and calculate them mentally, depending on the context and numbers involved.

2. Compatible Numbers (friendly numbers) are numbers whose sums, differences, products, or quotients are easy to calculate mentally. • Examples • 3 and 7 are compatible under addition because 3 + 7 = 10 • 86 and 14 are compatible under addition because 86 + 14 = 100 • 25 and 4 are compatible under multiplication because 25 × 4 = 100 • 480 and 24 are compatible under division because 480 ÷ 24 = 20

3. Compensation is a process that reformulates a sum, difference, product, or quotient to another one that is easier to compute. Additive compensation 98 + 56 = (98 +2) + (56 –2) = 100 + 54 = 154 Subtractive compensation (equal additions method) 47 – 29 = (47 +1) – (29 +1) = 48 – 30 = 18 Multiplicative compensation 16 × 35 = (16 ÷2) × (2× 35) = 8 × 70 = 560 Division compensation 180 ÷ 15 = (180 ×2) ÷ (15 ×2) = 360 ÷ 30 = 12 270 ÷18 = (270 ÷9) ÷ (18 ÷9) = 30 ÷ 2 = 15

4. Left-to-Right Methods 478 – 263 342 + 136 2 1 5 4 7 8

5. Multiplying Powers of 10 Examples • 45 × 1000 = 45000 i.e. we simply attach 3 zeros to the end of 45. • 20 × 300 = 6000 i.e. we multiply 2 by 3 and then attach 3 zeros to the end of our answer. Remark: It is not correct to say “adding 3 zeros to our answer”, because adding 0’s is the same as adding nothing. The most precise way to say is “concatenating” 3 zeros to the end of our answer.

6. Multiplying by Special Factors Numbers such as 5, 25, 11, and 99 are considered as special factors. • 628 × 5 = (628 × 10) ÷ 2 = 6280 ÷ 2 = 3140 • 47 × 5 = 470 ÷ 2 = 235 • 46 × 99 = 46 × (100 – 1) = 4600 – 46 = 4554 • 231 × 11 = 231 × 10 + 231 = 2310+ 231 = 2541

7. Special squares The squares of 15, 25, 35, … , 95 can be computed mentally, such as 152 = 225, 452 = 2025, … Secret The squares of these numbers always end in 25, and to compute the first 2 digits, we do the following 852 = 7225 where the 72 comes from 8×8+8

8. Special Products (I) 43 × 47 = 2021 can be performed mentally by 1. 4×4 + 4 = 20 2. 3×7 = 21 3. Linking 20 to 21 and we get 2021 • Remark: • This method works only for multiplying two-digit numbers, and with the extra conditions that • The 1st digits must be equal, • The 2nd digits add up to 10. More examples: 24×26, 62×68, 73×77, 41×49.

9. Special Products (II) 63 × 43 = 2709 can be performed mentally by 1. 6×4 + 3 = 27 2. 3×3 = 9 3. Linking 27 to 09 and we get 2709 • Remark: • This method works only for multiplying two-digit numbers, and with the extra conditions that • The 2nd digits must be equal, • The 1st digits add up to 10. More examples: 24×84, 96×16, 37×77, 54×54.

10. Section 4.2 Written Algorithms for whole number operations

11. Intermediate Algorithms for Addition 134 + 325 = ?

12. Intermediate Algorithms for Addition 37 + 46 = ?

13. Chip Abacus Model for base five Compute 123five + 34five 123five 34five Now we have more than four cubes, so we need to join them end to end and form a long. (click to see)

14. Chip Abacus Model for base five Compute 123five + 34five 123five 34five

15. Chip Abacus Model Chip Abacus Model for base five Compute 123five + 34five 123five 34five

16. Chip Abacus Model Chip Abacus Model for base five Compute 123five + 34five 123five 34five

17. Chip Abacus Model Chip Abacus Model for base five Compute 123five + 34five 123five 34five Now we have more than four longs, so we need to glue them together and form a flat. (click to see)

18. Chip Abacus Model Chip Abacus Model for base five Compute 123five + 34five 123five 34five

19. Chip Abacus Model Chip Abacus Model for base five Compute 123five + 34five 123five 34five

20. Chip Abacus Model Chip Abacus Model for base five Compute 123five + 34five 123five 34five

21. Chip Abacus Model Chip Abacus Model for base five Compute 123five + 34five 123five 34five The answer is therefore 212five.

22. Lattice method for Addition

23. Intermediate Algorithms for Subtraction 357 – 123 = ?

24. Subtraction in base five Compute 243five – 112five 243five take away 112five Do we have enough chip to take away in each column? YES

25. Subtraction in base five Compute 243five – 112five 243five take away 112five

26. Subtraction in base five Compute 243five – 112five take away 112five

27. Subtraction in base five Compute 243five – 112five take away 112five

28. Subtraction in base five Compute 243five – 112five take away 112five

29. Subtraction in base five Compute 243five – 112five take away 112five

30. Subtraction in base five Compute 243five – 112five take away 112five 131five The answer is therefore:

31. Intermediate Algorithms for Subtraction 423 – 157 = ?

33. Subtraction in base five Compute 31five – 12five 31five take away 12five No Do we have enough chips to take away in each column?

34. Subtraction again Compute 31five – 12five 31five take away 12five What can we do? Trade one long for five small cubes.

35. Subtraction again Compute 31five – 12five 31five take away 12five

36. Subtraction again Compute 31five – 12five 31five take away 12five

37. Subtraction again Compute 31five – 12five 31five take away 12five Now we can start to take chips away.

38. Subtraction again Compute 31five – 12five 31five take away 12five Now we can start to take chips away.

39. Subtraction again Compute 31five – 12five take away 12five Now we can start to take chips away.

40. Subtraction again Compute 31five – 12five take away 12five 14five The answer is therefore:

41. One more subtraction problem Compute 102five – 23five 102five take away 23five Do we have enough chips to take away? NO

42. One more subtraction problem Compute 102five – 23five 102five take away 23five What can we do? Trade a flat for five longs etc.

43. One more subtraction problem Compute 102five – 23five 102five take away 23five

44. One more subtraction problem Compute 102five – 23five 102five take away 23five Next trade a long for five small cubes.

45. One more subtraction problem Compute 102five – 23five 102five take away 23five

46. One more subtraction problem Compute 102five – 23five 102five take away 23five

47. One more subtraction problem Compute 102five – 23five take away 23five Now we can beginning taking chips away

48. One more subtraction problem Compute 102five – 23five take away 23five The answer is therefore: 24five

49. Equal Addend Subtraction Algorithm The term “borrow” was dropped from elementary math books about fifteen years ago because we are not really borrowing, we regroup or trade. The object that we took out for trading will never be returned. Hence the new terminology “regrouping” is introduced. However, this term “borrow”, no matter how incorrect it is, is stuck to everyone’s mind. Even if the teachers are not using it, the parents will still do. It is therefore desirable to create an algorithm that allows us to borrow and return. This is called the Equal Addend (Subtraction) Algorithm

50. The standard algorithm for subtraction can become quite complicated when there are several zeros in the minuend (the 1st number in the subtraction problem). For instance, in the problem 9 9 3 10 10 16 4 0 0 6 • 1 3 2 8 ¯¯¯¯¯¯¯¯ 2 6 7 8 we don’t have anything in the tens column hence we need to go to the thousands column to regroup and bring something back to the ones column. (click to see animation) Most students will not be able to memorize the long process correctly.