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time yourself

Maths A (new unit)Tue 6 October1 / 4

Averages (evening out), and decimal × decimal

An average is not the "middle" — it is "evening out". Multiplying is area, decimals too

Name Date

Idea 1 · an average is "evening out"

An average is "evening out": move from the high places to the low places until everything is the same number, and that number is the average.
average = add everything up ÷ how many　　(evening out never changes the total)
And back again: average × how many = total　　A day with 0 still counts as a day.

Move cubes from tall columns to short ones until they match: "evening out". (Made-up numbers.)

Worked example

A sixth person arrives with 12 acorns. What is the average of the six?
① The five had 30 in all (the figure above)
② Six people: 30 + 12 = 42
③ 42 ÷ 6 = 7 (6 rose to 7: 12 is more than 6, so it had to rise)　7 × 6 = 42 ✓

First, the most common mistake

"An average is the middle number."　No.
Five scores 1, 1, 1, 1, 16: total 20 ÷ 5 = 4. The middle one in order (the third) is 1.
Nobody scored the average, 4. The single 16 pulls the average up.

time yourself

Maths A (new unit)Tue 6 October2 / 4

Averages (evening out), and decimal × decimal

Name Date

1 · Averages: evening out

1Five people picked up 12, 7, 0, 15 and 6 acorns. What is the average?

The person with 0 still counts as a person.

2The average of tests 1 to 4 (out of 10) was 6.5. After test 5 the average should be 7. How many points does test 5 need?

Find the total of four tests, and the total that an average of 7 over five tests needs.

3In a class whose average score is 60, must there be someone who scored exactly 60? Choose "there must be" or "there need not be", and give the reason using the idea of "evening out".

Try a class of just two.

4Four boys averaged 50 and six girls averaged 60. What is the average of all ten? Also say in a few words why "(50 + 60) ÷ 2 = 55" is wrong.

Find each group's total first.

time yourself

Maths A (new unit)Tue 6 October3 / 4

Averages (evening out), and decimal × decimal

Name Date

Idea 2 · decimal × decimal

Multiplying is area. 2.3 × 1.4 is the area of a rectangle 1.4 high and 2.3 wide.
Method　2.3 times 10 → 23, 1.4 times 10 → 14. 23 × 14 = 322.
Times 10 times 10 is 100 times too big, so ÷ 100 → 3.22
Do not memorise where the point goes. Decide it from "how many times did I multiply, so what do I divide by".
Estimate　a bit more than 2 × 1 = 2. So 3.22 makes sense.

2.3 × 1.4 as an area diagram, split into 2 and 0.3 by 1 and 0.4. Split the numbers and only calculations you already know are left.

First, the most common mistake

"Multiplying always makes a number bigger."　Only when the multiplier is more than 1.
8 × 2 = 16 (bigger), 8 × 1 = 8 (the same), 8 × 0.5 = 4 (smaller).
Multiply by a number less than 1 and the answer is smaller than the number you started with.

5Draw an area diagram and find 0.3 × 0.4. (Use a square of side 1 cut into 10 × 10.)

time yourself

Maths A (new unit)Tue 6 October4 / 4

Averages (evening out), and decimal × decimal

Name Date

6Find 1.2 × 1.5 by "times 10, then divide by 100". Write three lines: "multiply by 10", "multiply", "bring back".

7Find 2.5 × 0.8.

Is it 20? 2? 0.2? Settle the size first with an estimate.

8Answer without calculating. For which □ is 6 × □ less than 6? Choose all of them.
　A 0.9　　B 1.1　　C 1　　D 0.05

9Find 0.06 × 0.7.

10A rectangular flower bed is 3.6 m by 2.5 m. What is its area in m²? Draw it before you write the calculation.

11Painting 1 m² of wall takes 0.45 L of paint. How much paint does 3.2 m² of wall need?

12Someone answered "0.4 × 0.5 = 2". Use an estimate or an area diagram to explain the mistake, and write the correct answer.