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. 2.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 ?

Practice the Facts  Webs  Dice games  Card games  Flash cards

Facts with 9s: ? X 9 and 9 x ?  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

Practice the Facts  Calculator

Extend Nifty Nines To 10s, 100s, 1000s  4 x 90  9 x 60  5 x 900  9 x 700  6 x  9 x 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

Facts with 5s  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.

Practice Strategy Selection  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

Extend Clock Facts To 10s, 100s, 1000s  5 x 80  7 x 50  5 x 400  6 x 500  9 x  5 x 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

Facts with 0s  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 1s  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 3s  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.

Extend Threes Facts To 10s, 100s, 1000s  5 x 80  7 x 50  5 x 400  6 x 500  9 x  5 x 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

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

Extend Fours Facts To 10s, 100s, 1000s  4 x 40  7 x 40  8 x 400  4 x 600  8 x  4 x 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

The Last Nine 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 x 6 = 5 x x 6 6 x 8 = 5 x x 8 = 5 x 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?

Extend Last 9 Facts To 10s, 100s, 1000s  6 x 60  7 x 80  6 x 700  7 x 700  8 x  4 x 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

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: $ $ would have a “ball-park” estimate of $ $ or $

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: $ $ would be rounded to $ $ to get an estimate of $

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? Nova Scotia Prince Edward Island Saskatchewan Newfoundland New Brunswick

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

Visualization of Percent  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 25 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.

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

Visualization of 33 ⅓ % 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 Percent Find 33⅓% of:  $96  $45  $120  $339  $930  $6309

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

Visualization of 66 ⅔ 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 66 ⅔ Percent Find 66⅔% of:  $24  $60  $120  $360  $660

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

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!