The same calculation can be quick or slow. Change the order (4 × 2.7 × 25 = 100 × 2.7), change the form (0.75 × 36 = 3/4 of 36), or split a number (12 × 2.5 = 24 + 6). Then check the answer is about right.
Two students work out 25% of 48. One writes a long multiplication; the other halves twice in their head. Both get 12. Which would you rather do in a shop?
Every calculation with fractions, decimals and percentages can be done the long way. A good strategy makes it quick, and it usually comes from one of three moves:
Change the order. Multiplying can be done in any order: 4×2.7×25=(4×25)×2.7=100×2.7.
Change the form.0.75=43=75%: use whichever suits the numbers.
Split a number.12×2.5=12×2+12×0.5=24+6.
Then check: is the answer about the right size?
Predict: Read the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
Try: Try it on the board and compare the result with your prediction.
Investigate: Work out why it happens, one step at a time.
Modify: Change the problem on purpose and see what happens.
Practise: A few on your own, to check you have it.
Make: Make your own problem using what you have learned. Expect a few tries; that is normal.
Make your prediction first. Help opens once you've locked it in.
PredictRead the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
25% of 48
What fraction is 25%?
TryTry it on the board and compare the result with your prediction.
A quarter of the bar
Look at where 50% and 25% sit on the bar.
Try it yourselfYour edits are saved in this browser.
The bar is 48. Set it to 25%. What fraction of the bar is shaded? Was your prediction right?
Use the board. This ticks when you get there.
InvestigateWork out why it happens, one step at a time.
Three moves
Here are the three moves, each with a check.
Worked exampleBefore each step, say what you think comes next.
4×2.7×25=(4×25)×2.7=270Change the order: 4 × 25 = 100. Check: 2.7 is about 3, and 100 × 3 = 300.
0.75×36=43×36=27Change the form: 36 ÷ 4 = 9, × 3 = 27. Check: a bit less than 36.
12×2.5=12×2+12×0.5=24+6=30Split a number: 2.5 = 2 + 0.5. Check: between 12 × 2 and 12 × 3.
Your working can be short notes. Then choose the reason for the key step.
Work out 4×2.7×25 without a calculator. Write your steps.
Your working
A model solution
4×25=100multiplying can be done in any order, so pair the 4 and the 25
100×2.7=270multiplying by 100 makes each digit 100 times bigger
Check your own working
Work out 0.75×36 without a calculator. Write your steps.
Your working
A model solution
0.75=43a decimal can be written as a fraction
41 of 36 is 9, so 43 of 36 is 27 36 divides by 4, so the fraction is quick
Check your own working
ModifyChange the problem on purpose and see what happens.
One change at a time
Each task changes one thing.
Need a hint?
75% is three quarters.
Try it yourselfYour edits are saved in this browser.
The bar is 36. Find 75% of it. Work it out in your head first.
Use the board. This ticks when you get there.
Work out 0.25×3.6×40 without a calculator.
Work out 53 of 45 without a calculator.
PractiseA few on your own, to check you have it.
Quick ways
Work out 8×1.25.
Find 20% of 35.
Work out 99×0.4. (Try 100×0.4 first.)
Review: a $40 bag is 15% off. What is the sale price, in dollars?
MakeMake your own problem using what you have learned. Expect a few tries; that is normal.
18 in your head
Then find two more, and say which was quickest and why.
Try it yourselfYour edits are saved in this browser.
Need a hint?
50% of what is 18?
Another hint
Or 10% of something, or 25% of something.
Choose an amount and a percentage you can do in your head that give exactly 18.
Use the board. This ticks when you get there.
For teachers
Students compare strategies for calculations with fractions, decimals and percentages: changing the order (any order for multiplying), changing the form (0.75 = 3/4), and splitting (the distributive law), with an estimate to check (AC9M7N06, node 4). About 40 minutes.
I can choose a quick strategy for a calculation and explain why it is quick.
I can use any order for multiplying, a friendly fraction, or splitting a number.
I can estimate to check that my answer is about right.
Before class
Do the Run board yourself first. Students need multiplying fractions and percentages (previous lessons).
The one idea: there is more than one right way, and the best one depends on the numbers. Students own their working: the reasoning checks never mark the free working, only the answer and the reason at the key step.
This node integrates everything in the unit; there is no single new misconception. Listen for students who use the long way every time, and ask what made the quick way possible.
Running the lesson
Predict: ask a student who chose halving to explain why it works.
Run: ask which other percentages are friendly fractions (10%, 20%, 50%, 75%).
Investigate: share two different routes for 4 × 2.7 × 25 and compare.
Modify: for 37% of 50, show that percentages can swap: 37% of 50 = 50% of 37.
Make: collect pairs that give 18 and sort them by how quick they are.
The hard part (on purpose)
Pausing to look for a quick route instead of starting the long way. Give ten seconds of no-writing time to spot a friendly number before anyone calculates.
When students get stuck
A student always multiplies the long way. Ask: Is any number here a friendly fraction, or a pair that makes 10 or 100? Fix: Look for 25 and 4, 0.5, 0.25, 0.75, 10% and 50% first.
A student splits a product into a sum but forgets to multiply every part (25 × 3 + 0.2). Ask: What is 3.2 made of? Does each part get multiplied by 25? Fix: Every part of the split is multiplied.
A student's answer is far off and they do not notice. Ask: Roughly how big should it be? Fix: Estimate first: round the numbers and compare.
Stretch and support
Stretch: Make a poster of your own efficiency rules: when to use a fraction, a decimal, or a percentage, with an example of each.
Support: Keep a table of friendly percentages and their fractions visible (10% = 1/10, 25% = 1/4, …).
Where it leads
Next: Calculating with negative fractions and decimals (AC9M8N04), where the same moves work with signs.
Where it leads: mental strategies in financial maths (AC9M7N09) and algebraic manipulation, where the distributive law becomes expanding brackets.
What each step is for
25% of 48: Committing to the quickest method for 25% of 48 shows whether they see 25% as a quarter or only as a decimal to multiply.
a quarter of the bar: Seeing 25% as a quarter of the bar connects the percentage to the friendly fraction that makes it quick.
three moves: Each move has a reason: any order for multiplying, a fraction that suits the numbers, and splitting with the distributive law. Students write their own way and choose the reason for the key step.
one change at a time: One change at a time: three quarters on the board, a longer product to reorder, a fraction of a whole number, then choosing a strategy for an awkward percentage.
quick ways: A reorder, a friendly percentage, a split near a round number and a review, mixed; students choose the strategy each time.
18 in your head: Building several percentage-of-amount pairs that give 18, each chosen to be easy in your head, makes strategy choice the task.
Now you can
Choose a quick strategy for a calculation and explain why it is quick.
Use any order for multiplying, a friendly fraction, or splitting a number.