1. 52 x 232. 16% of $703. 1.6 x 354. 89 ÷ 235. 8 + 12 + 7.5 + 11

6. 5/8 of 967. 54 000 x 2008. 8100 ÷ 909. 1.1 x 0.910.75

1. 52 x 232. 16% of $703. 1.6 x 354. 89 ÷ 235. 8 + 12 + 7.5 +

11

6. 5/8 of 967. 54 000 x 2008. 8100 ÷ 909. 1.1 x 0.910.sqrt75

119611.55563 r 20 or 320/2328.5

6010 800 000900.99~8.5 (8.66 to 3sf)

Why do mental?• It’s an orderly start to the lesson.• Mental makes great revision.• It can be used to enable discovery learning.• It is fabulous for diagnostic assessments.• It makes good skills practice.• It is a real alternative to PPAs– which are history.• Mental strategies are derived from real deep

understandings of number.• It provides the opportunity to practice estimating.• Mental skills impress the oldies.• These skills may actually be useful in students’ private

and/or public lives.

A few words about PPAs“Classical PPA instruction is doomed to

relative failure in a world where arithmetic is almost universally done with calculators and where even the dimmest child will see that attaining skill in PPA has almost no value in non-academic pursuits.”

Ralston, A. 1999. Let’s abolish pencil-and-paper arithmetic, Journal of Computers in Mathematics and Science Teaching, 18, 2, 173-194

Orderly start to lesson1. 8 x 52. 12 x 63. 9 x 94. 11 x 125. 5 x 8

6. 6 x 97. 12 x 78. 7 x 79. 8 x 710. 10 x 3

Question: How important is it to know your times tables these days?

Strategy: Allow students to have their tables in front of them when they do mental. Then go fast!

Great revision1. (x + 2)(x – 4)2. 5(2m – 6)3. -5(2m – 6)4. 25 cents is how many

dollars?5. To rent a drinks machine

costs a flat $20 and 25 cents per drink sold. Write the equation for c – the total cost – in terms of n - the number of drinks sold.

6. How much would it all cost if 6 drinks were sold?

7. What’s c if n = 6?

8. What’s the c-intercept?

9. What’s the steady rate of increase?

10. What is m (the gradient)?Question: What mental Algebra skills

should your students have?

Discovery learning1. 16.5% of

$702. 10% of $703. 5% of $704. 1% of $705. 0.5% of $70

6. 52 x 237. 50 x 238. 100 x 239. 2 x 2310. 52.1 x 23

Question: Does doing things mentally mean you cannot write anything down?

Strategy: Ask students to share their strategies. The kids who catch on faster make great role models for the weaker kids.

Diagnostic assessmentWrite numerically:1. Five thousand and twenty2. Five thousand and two3. Five thousand two hundred4. Twelve thousand and thirty5. Twelve thousand three hundred6. Which is larger: 0.5 or 0.50?7. 106, 107, 108, 109, … write the next two

numbers8. What is of 60?9. What is 0.5 of 60?10. What is 50% of 60?

Question: What would you do if your students did poorly on these questions?

Skills practiceWrite in decimal

notation1. 3.05 x 102

2. 5.2 x 104

3. 9.002 x 102

4. 5.3 x 10-3

5. 4 x 10-6

Write in scientific notation

6. 890 0007. 50 0308. 89. 0.00510. 0.000 000 023

Question: What couldn’t you revise using mental?

Alternative to PPAs

1. 1.99 x 52. 8.99 x 53. 83 x 204. 375 – 2255. 62 – 44

6. 12 + 7.5 + 17 + 87. $3.95 + $6.958. 5.5m x 5m x

$20/m2

9. 1080 – 450 - 39010. $2.50 x 3 x 5

Question: Doing these types of ‘problems’ is hard – perhaps harder than doing them using a PPA - but does that mean we should not be doing them?

Strategies from deep understandings of number

1. 52.1 x 232. 50 + 23. 100 24. 100 x 235. 2300 2

6. 2 x 237. 0.1 x 238. of 239. 1150 + 46 +

2.310. 52.1 x 23Question (might be a bit uncomfortable): How

often do you use these strategies yourself? How deep is your understanding of number, partitioning, etc?

Estimating

1. sqrt752. 0.23 x 8803. 112 9.874. $562 879.32 95. 1/11 of 23

6. 0.4632 x $235.767. $42 671.08 less

28%8. 0.175g @

$8.99/kg9. 6.72m x 5.12m10. 52.1 x 23Question: Realistically, how often do you need

a precise answer (the sort that a calculator can provide)? How do people usually calculate in their private lives?

Impressing the oldies

1. 5 hrs 10 mins = ____ mins2. ₤8 6s 2d + ₤3 8s 11d 3. 5 yds @ ₤1 6s 2d per yd4. 564 miles @ 40 miles per hour5. 9 chains = ____ yds

Question: Do these represent the Good Old Days?

Usefulness1. Can I afford to rent this computer for 1 year?2. Is it worth travelling 12km to save 2c/L for

fuel?3. How much wood do I need for my pergola?4. How long will it take me to mark these exams

@ 23 minutes per paper?5. How any mini-frankfurters do I need for my

son’s party?Question: What Maths skills do people mostly

use in their private lives?Strategy: Get the kids to write questions for

you to use – or let them present the day’s mental.