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

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## Mental Math in Math Essentials 11

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**Mental Math in Math Essentials 11**Implementation Workshop November 30, 2006 David McKillop, Presenter**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**Mental Math Outcomes**• B5 Mentally calculate 25%, 33⅓%, and 66⅔% of quantities compatible with these percents • B6 Estimate percents of quantities**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**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.**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!**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 ?**Webs**Dice games Card games Flash cards Practice the Facts**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 ?**13 facts**30 Total**Calculator**Practice the Facts**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**To division:**36 ÷ 9 54 ÷ 9 63 ÷ 9 27 ÷ 3 81 ÷ 9 45 ÷ 5 Extend Nifty Nines**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**13 new facts**43 Total**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**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**To division:**25 ÷ 5 45 ÷ 5 30 ÷ 5 20 ÷ 4 15 ÷ 3 35 ÷ 5 Extend Clock Facts**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**19 facts**62 Total**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**13 new facts**75 Total**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**9 new facts**84 Total**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**To division:**18 ÷ 3 15 ÷ 3 12 ÷ 3 9 ÷ 3 21 ÷ 3 18 ÷ 6 Extend Threes Facts**The Double-Double Strategy.**For example, for 4 x 6, think: double 6 is 12 and double 12 is 24. Facts with 4s**7 facts**91 Total**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**To division:**16 ÷ 4 28 ÷ 4 20 ÷ 4 32 ÷ 4 12 ÷ 4 28 ÷ 7 Extend Fours Facts**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**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**To division:**36 ÷ 6 42 ÷ 7 64 ÷ 8 49 ÷ 7 56 ÷ 8 42 ÷ 6 Extend Last 9 Facts**Practice the Facts**• Flash cards • Bingo • Dice Games • Card Games • Fact Bee • Calculators**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.**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.**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?**B5 Mentally calculate 25%, 33 ⅓%, and 66 2/3% of**quantities compatible with these percents B6 Estimate percents of quantities Percents**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**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**Estimate:**25% of $35 25% of $597 26% of $48 24% of $439 26% of $118 25% of $4378 Estmating Percent**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**Find 33⅓% of:**$96 $45 $120 $339 $930 $6309 Visualization of Percent**Estimate:**33⅓% of $67 33⅓% of $91 33% of $180 34% of $629 32% of $1199 33⅓% of $8999 Estimating Percent**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**Find 66⅔% of:**$24 $60 $120 $360 $660 Visualization of 66⅔ Percent**Estimate:**67% of $27 65% of $90 68% of $116 65% of $326 67% of $894 Estimating Percent**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!

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