Mental Math in Math Essentials 11

1. Mental Math in Math Essentials 11 Implementation Workshop November 30, 2006 David McKillop, Presenter

2. Mental Math Outcomes • B1 Know the multiplication and division facts • B2 Extend multiplication and division facts to products of tens, hundreds, and thousands by single-digit factors • B3 Estimate sums and differences • B4 Estimate products and quotients

3. Mental Math Outcomes • B5 Mentally calculate 25%, 33⅓%, and 66⅔% of quantities compatible with these percents • B6 Estimate percents of quantities

4. Why should students learn number facts? • They are the basis of all mental math strategies, and mental math is the most widely used form of computation in everyday life • Knowing facts is empowering • Facilitates the development of other math concepts

5. How is fact learning different from when I learned facts? 1. Facts are clustered in groups that can be retrieved by the same strategy. • Students can remember 6 to 8 strategies rather than 100 discrete facts. 3. Students achieve mastery of a group of facts employing one strategy before moving on to another group.

6. General Approach • Introduce a strategy using association, patterning, contexts, concrete materials, pictures – whatever it takes so students understand the logic of the strategy • Practice the facts that relate to this strategy, reducing wait time until a time of 3 seconds, or less, is achieved. Constantly discuss answers and strategies. • Integrate these facts with others learned by other strategies. • IT WILL TAKE TIME!

7. Facts with 2s:2 x ? and ? X 2 • Strategy: Connect to Doubles in Addition (Math Essentials 10) • Start with 2 x ? • Relate ? X 2 to 2 x ?

8. 17 facts

9. Webs Dice games Card games Flash cards Practice the Facts

10. Nifty Nines Strategy: Two Patterns -Decade of answer is one less than the number of 9s and the two digits of the answer sum to 9 9 x 9 = 81 8 x 9 = 72 7 x 9 = 63 6 x 9 = 54 5 x 9 = 45 4 x 9 = 36 3 x 9 = 27 Facts with 9s:? X 9 and 9 x ?

11. 13 facts 30 Total

12. Calculator Practice the Facts

13. To 10s, 100s, 1000s 4 x 90 9 x 60 5 x 900 9 x 700 6 x 9 000 9 x 3 000 To estimating 6.9 x $9 9 x $4.97 3.1 x $8.92 7 x $91.25 9 x $199 4 x $889 8.9 x $898.50 Extend Nifty Nines

14. To division: 36 ÷ 9 54 ÷ 9 63 ÷ 9 27 ÷ 3 81 ÷ 9 45 ÷ 5 Extend Nifty Nines

15. The Clock Strategy: The number of 5s is like the minute hand on the clock – it points to the answer. For example, for 4 x 5, the minute hand on 4 means 20 minutes; therefore, 4 x 5 = 20. Facts with 5s

16. 13 new facts 43 Total

17. Which facts can use The Clock Strategy? Which facts can use the Nifty Nines Strategy? Which facts can use the Doubles Strategy? 3 x 5 5 x 9 8 x 2 9 x 7 9 x 2 2 x 5 7 x 5 6 x 9 Practice Strategy Selection

18. To 10s, 100s, 1000s 5 x 80 7 x 50 5 x 400 6 x 500 9 x 5 000 5 x 3 000 To estimating 4.9 x $5 3 x $4.97 3.89 x $50 5 x $61.25 7 x $499 5 x $399 4.9 x $702.50 Extend Clock Facts

19. To division: 25 ÷ 5 45 ÷ 5 30 ÷ 5 20 ÷ 4 15 ÷ 3 35 ÷ 5 Extend Clock Facts

20. The Tricky Zeros: All facts with a zero factor have a zero product. (Often confused with addition facts with 0s) If you have 6 plates with 0 cookies on each plate, how many cookies do you have? Facts with 0s

21. 19 facts 62 Total

22. The No Change Facts: Facts with 1 as a factor have a product equal to the other factor. If you have 3 plates with 1 cookie on each plate OR 1 plate with 3 cookies on it, you have 3 cookies. Facts with 1s

23. 13 new facts 75 Total

24. The Double and One More Set Strategy. For example, for 3 x 6, think: 2 x 6 is 12 plus one more 6 is 18. Facts with 3s

25. 9 new facts 84 Total

26. To 10s, 100s, 1000s 5 x 80 7 x 50 5 x 400 6 x 500 9 x 5 000 5 x 3 000 To estimating 4.9 x $5 3 x $4.97 3.89 x $50 5 x $61.25 7 x $499 5 x $399 4.9 x $702.50 Extend Threes Facts

27. To division: 18 ÷ 3 15 ÷ 3 12 ÷ 3 9 ÷ 3 21 ÷ 3 18 ÷ 6 Extend Threes Facts

28. The Double-Double Strategy. For example, for 4 x 6, think: double 6 is 12 and double 12 is 24. Facts with 4s

29. 7 facts 91 Total

30. To 10s, 100s, 1000s 4 x 40 7 x 40 8 x 400 4 x 600 8 x 4 000 4 x 6 000 To estimating 3.9 x $4 6 x $3.97 3.89 x $80 4 x $41.25 7 x $399 4 x $599 5.9 x $402.50 Extend Fours Facts

31. To division: 16 ÷ 4 28 ÷ 4 20 ÷ 4 32 ÷ 4 12 ÷ 4 28 ÷ 7 Extend Fours Facts

32. 6 x 6 6 x 7 and 7 x 6 6 x 8 and 8 x 6 7 x 7 7 x 8 and 8 x 7 8 x 8 Using helping facts: 6 x 6 = 5 x 6 + 6 7 x 6 = 5 x 6 + 2 x 6 6 x 8 = 5 x 8 + 8 7 x 8 = 5 x 8 + 2 x 8 8 x 8 = 4 x 8 x 2 Some know 8 x 8 is 64 because of a chess board What about 7 x 7? The Last Nine Facts

33. The 100 Facts

34. To 10s, 100s, 1000s 6 x 60 7 x 80 6 x 700 7 x 700 8 x 8 000 4 x 6 000 To estimating 6.8 x $7 6 x $5.97 7.89 x $80 7 x $61.25 6 x $799 8 x $699 5.9 x $702.50 Extend Last 9 Facts

35. To division: 36 ÷ 6 42 ÷ 7 64 ÷ 8 49 ÷ 7 56 ÷ 8 42 ÷ 6 Extend Last 9 Facts

36. Practice the Facts • Flash cards • Bingo • Dice Games • Card Games • Fact Bee • Calculators

37. B3 Estimate sums and differences Using a front-end estimation strategy prior to using a calculator would enable students to get a “ball-park” solutions so they can be alert to the reasonableness of the calculator solutions. Example: $42 678 + $35 987 would have a “ball-park” estimate of $40 000 + $30 000 or $70 000.

38. B3 Estimate sums and differences In other situations, especially where exact answers will not be found, rounding to the highest place value and combining those rounded values would produce a good estimate. Example: $42 678 + $35 987 would be rounded to $40 000 + $40 000 to get an estimate of $80 000.

39. About how many people live in the Maritime provinces? In the Atlantic provinces?About how many more people live in Nova than in New Brunswick?

40. B5 Mentally calculate 25%, 33 ⅓%, and 66 2/3% of quantities compatible with these percents B6 Estimate percents of quantities Percents

41. Find 3% of $800. Think: If $800 is distributed evenly in these 100 cells, each cell would have $8 – this is 1%. Therefore, there is 3 x $8 or $24 in 3 cells (3%). Visualization of Percent

42. Find 25% of $800. Think: If $800 is distributed evenly in these 4 quadrants, each quadrant would have $800 ÷ 4 or $200. Therefore, 25% of $800 is $200. Visualization of 25 Percent

43. Estimate: 25% of $35 25% of $597 26% of $48 24% of $439 26% of $118 25% of $4378 Estmating Percent

44. Find 33⅓% of $69. Think: $69 shared among three equal parts would be $69 ÷ 3 or $23. Therefore, 33⅓% of $69 is $23. Visualization of 33⅓% Percent

45. Find 33⅓% of: $96 $45 $120 $339 $930 $6309 Visualization of Percent

46. Estimate: 33⅓% of $67 33⅓% of $91 33% of $180 34% of $629 32% of $1199 33⅓% of $8999 Estimating Percent

47. Find 66⅔% of $36. Think: $36 divided by 3 is $12, so each one-third is $12, Therefore, 2-thirds is $24, so 66⅔% of $36 is $24. Visualization of 66⅔ Percent

48. Find 66⅔% of: $24 $60 $120 $360 $660 Visualization of 66⅔ Percent

49. Estimate: 67% of $27 65% of $90 68% of $116 65% of $326 67% of $894 Estimating Percent

50. Parting words… • It will take time. • Build on successes. • Always discuss strategies. • Use mental math/estimation during all classes whenever you can. • Model estimation before every calculation you make!