Printing: use your browser's print (Ctrl+P / ⌘P). The whole week is 20 A4 sheets (10 pieces of paper double-sided) — the same content as the single days.
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
NameDate
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
NameDate
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
NameDate
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
NameDate
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.
time yourself
Costume 1 · weatherWed 7 October1 / 4
What is a "normal" (30-year average)?
"It was hot" said with numbers, not feelings. "Per one" appears too
NameDate
The 30-year normal, and the difference from it
The 30-year normal (called "heinenchi" in Japan) = the average over 30 years of a place's temperature or rainfall. Every 10 years it is renewed with a new 30-year span.
How far is this year from normal → difference = this year's value − the normal (+ means above normal, − below) Say "it was hot" with a number, not a feeling.
The rule for this sheet (made up): a difference of +1.0 °C or more is "hot", −1.0 °C or less is "cool", anything between is "about normal".
Town A
2022
2023
2024
2025
2026
August mean temperature (°C)
27.9
28.3
27.2
26.8
28.9
Source: the August mean temperature of Town A (made up). The August normal of Town A is 27.4 °C.
1 · The normal and the difference
1Over 30 years the August mean temperatures of Town A add up to 822.0 °C. What is the August normal?
2Find the difference for 2026 (28.9 °C). By the rule, is it "hot", "cool" or "about normal"?
3Find the difference for 2025 (26.8 °C). (It comes out negative.)
time yourself
Costume 1 · weatherWed 7 October2 / 4
What is a "normal" (30-year average)?
NameDate
First, the most common mistake
"The normal is a typical year, so most years land close to it." Not so.
The normal is an evened-out number. None of the five years in the table is exactly 27.4 °C. Years that match the normal can be the rare ones.
4Find the difference for each of the five years in the table. How many were above the normal?
5Find the mean of the five years, rounded to one decimal place. How many °C above (or below) the normal is it?
Work to two decimal places, then round.
6A friend called 2023 (28.3 °C) "hot". By this sheet's rule, what is it? Answer with numbers.
7In a town whose normal is 27.4 °C, one year had a difference of +2.1 °C. What was that year's mean temperature?
time yourself
Costume 1 · weatherWed 7 October3 / 4
What is a "normal" (30-year average)?
NameDate
2 · Rainfall, and "per one"
8In August (31 days) this year Town A had 186 mm of rain. The August normal (made up) is 155 mm in total.
(1) How many mm per day this year? (2) How many mm per day is the normal?
(3) This year is how many times the normal?
Daily highest temperature (°C)
1
2
3
4
5
Town P
29.8
30.2
30.0
29.9
30.1
Town Q
24.5
35.5
30.0
33.0
27.0
Source: five days in one August (made up)
9Find the mean of the five highest temperatures for Town P and for Town Q. Are they the same? Then find "the hottest day − the coolest day" for each.
time yourself
Costume 1 · weatherWed 7 October4 / 4
What is a "normal" (30-year average)?
NameDate
10The normal is based on the total of 30 years. After 10 years the oldest 10 years are replaced by the newest 10.
The old 10 years total 273.0 °C and the new 10 years total 281.0 °C.
(1) Find the new 30-year normal, rounded to one decimal place. (The old 30-year total was 822.0 °C.)
(2) Find this year's difference for 28.9 °C against the new normal.
11Someone reasons: "Town A's August normal is 27.4 °C, so every August day is about 27 °C." Using question 9, write one thing an average alone cannot tell you.
12Looking only at the days it rained, what happens?
(1) In a normal August Town A has 9 rainy days and 155 mm in total. How many mm per rainy day? (Round to one decimal place.)
(2) This August it had 11 rainy days and 186 mm. How many mm per rainy day? (One decimal place again.)
(3) The total is bigger this year, yet what do (1) and (2) show when compared?
What "per one" means depends on what you count as one.
time yourself
Maths B (practice)Fri 9 October1 / 4
Dividing by a decimal, in words
Diagram first, then the calculation. An answer with no diagram is not marked right
NameDate
Today's four tools (last week continued)
① Make the divisor whole multiply both numbers by the same thing.
② Keep dividing if it does not come out even, put 0s after the remainder and continue.
③ Remainder its decimal point lines up with the point in the original number.
④ "□ times" problems first decide in a diagram which quantity is "1".
Before any calculating, this diagram. Top and bottom move in step. Whether the ? sits to the right or left of 90 tells you in advance if the answer is bigger or smaller.
Worked example
0.6 kg of beans cost 90 yen. What does 1 kg cost?
① Diagram (above). 1 kg is more than 0.6 kg → the answer must be more than 90
② Calculation 90 ÷ 0.6 ③ Both times 10 900 ÷ 6 = 150
④ Check 150 × 0.6 = 90 ✓ Answer 150 yen
1 · Estimate first, then calculate
1Answer without calculating. For which □ is 9 ÷ □ bigger than 9? Choose all of them.
A 0.8 B 1.5 C 0.09 D 1.01
time yourself
Maths B (practice)Fri 9 October2 / 4
Dividing by a decimal, in words
NameDate
First, the most common mistake
"8.9 ÷ 0.4 = 22 remainder 1" The decimal point in the remainder has been forgotten.
89 ÷ 4 = 22 remainder 1. But the remainder lines up with the point in the original 8.9, so it is 22 remainder 0.1 (0.4 × 22 + 0.1 = 8.9). With "remainder 1" the check would give 9.8, not 8.9.
2Find 7.8 ÷ 0.65.
Multiply both by what to make the divisor whole?
3Find 4.2 ÷ 0.8, continuing until it comes out exactly.
Put a 0 after the remainder of 42 ÷ 8 and carry on.
4Find 9.5 ÷ 0.6, rounding the quotient to one decimal place.
Work to two decimal places, then round.
5Find 8.9 ÷ 0.4 with a whole-number quotient, and write the remainder. Write the check too.
time yourself
Maths B (practice)Fri 9 October3 / 4
Dividing by a decimal, in words
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2 · "Times" and "per one" word problems
6B weighs 3.84 kg, which is 1.6 times A. How much does A weigh? Draw a diagram (a bar or a double number line) before the calculation.
With A as 1, B is 1.6.
72.4 kg of fertiliser covers a field of 3.6 m².
(1) How many m² does 1 kg cover?
(2) How many kg are used on 1 m²? Round to two decimal places.
Draw the double number line first.
In (1) and (2), the thing that becomes "1" is different.
8Tea, 3.2 L, is poured into bottles of 0.35 L each. How many bottles, and how many litres of tea are left over? Draw it.
time yourself
Maths B (practice)Fri 9 October4 / 4
Dividing by a decimal, in words
NameDate
9This year's yuzu weighed 5.5 kg, which is 1.25 times last year's. How many kg was it last year? Draw it.
10A rectangle has an area of 8.64 cm². It is 2.4 cm wide. How many cm tall is it? Draw it.
11A car goes 12.5 km on 1 L. How many litres does it need for 7.5 km? First say whether the answer is more or less than 1 L.
12Write only the calculation. Decide between × and ÷ with a double number line.
(1) String costs 120 yen a metre. What do 2.5 m cost?
(2) String costs 120 yen for 2.5 m. What does 1 m cost?
Same numbers, but what counts as "1" changes the calculation.
"How many yen more" or "how many times"? An average price is an evened-out price
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Prices, and "times"
Prices = how the price of the same thing changes over time.
"How many yen did it rise" (subtraction) is harder to compare across items than "how many times did it grow" (division). times = this year ÷ last year An average price evens out the different prices of different shops. Putting things per 1 kg lets you compare different amounts.
item
last year (yen)
this year (yen)
milk, 1 L
200
230
eggs, a box of 10
240
300
rice, 5 kg
2,000
2,400
bread, 1 loaf
140
154
Source: last year's and this year's receipts (made up). The basket holds these four.
1 · Receipts side by side
1A box of 10 eggs cost 240 yen last year and 300 yen this year. This year is how many times last year?
2In the table, which item (1) rose most in yen, and (2) rose most in "times"? Work out the "times" for all four.
3This year, five shops sold 1 L of milk at 228, 235, 230, 224 and 238 yen. What is the average price? If last year's average was 200 yen, this year is how many times last year (rounded to two decimal places)?
4Petrol cost 165.5 yen a litre last year. This year it is 1.2 times that. What does 1 L cost this year?
First estimate: more or less than 165.5 yen?
5This year 0.45 kg of ham was sold for 360 yen. (1) How much is 1 kg of ham? (2) Last year ham cost 700 yen a kilogram. This year is how many times last year (rounded to two decimal places)?
For (1), start with the double number line.
6Put the four items in one basket. Find last year's total and this year's, and how many times last year this year is, rounded to two decimal places.
7Over five years, an item's price each year became 1.02, 1.03, 1.01, 1.04 and 1.05 times the year before. Find the average of these five numbers.
8Shop A's item went from 200 to 230 yen, and Shop B's from 120 to 150 yen. Both rose by 30 yen. Is "they rose by the same" right? Compare in "times" and give the reason.
9Rice is sold in a 5 kg bag for 2,400 yen and a 10 kg bag for 4,500 yen. Find the price per kilogram of each and say which bag is cheaper.
Putting both per 1 kg lets you compare bags of different sizes.
10[Corrections] Pick one question from this week's notebook that you got wrong. Write all of the following.
1) the question (day and number) 2) circle why it went wrong: calculation / no diagram / calculation set-up / no estimate
3) do it again correctly 4) what you will do first next time the same question comes up
11[Difference puzzle] Sweets are handed out to children. With 6 each, 15 are left over; with 9 each, 3 are short. How many children, and how many sweets? Draw a bar diagram showing the "left over" and "short" first.
When each child gets 9 instead of 6, how many more sweets are needed in all?
time yourself (optional)
Stretch (not recorded)From Sun 11 October, any time1 / 4
Stretch — "per one" in a paddy, water, a magnet and a factory
Not recorded. Getting it wrong is fine. Some numbers in the sources are not needed. All numbers are made up
NameDate
1 · Two paddy fields [social studies: rice farming]
Source 1
A (made-up) farm has two paddy fields planted with the same kind of rice. Four family members planted them. In autumn the weight of rice harvested and the weight of fertiliser spread on each field were recorded. Area is measured in a (ares).
field
area
fertiliser
rice harvested
A
8.5 a
25.5 kg
459 kg
B
6.4 a
25.6 kg
371.2 kg
Source 2 — this year's records (made up)
1Which field gave more rice this year (how many kg)? And how many kg of rice per 1 a did A and B each give?
Always draw. Put a number line for the area above a number line for the rice.
2Next year field B will be made 1.5 times as large. If the rice per 1 a stays the same as this year, how many kg of rice will it give?
3"B used more fertiliser, so B gave more rice. So if you double the fertiliser, you double the rice." Is that right? Use the "per 1 a" numbers to give your view.
time yourself (optional)
Stretch (not recorded)From Sun 11 October, any time2 / 4
Stretch — "per one" in a paddy, water, a magnet and a factory
NameDate
2 · How much dissolves [science: dissolving]
Source 1
How many grams of salt and of alum dissolve in 100 mL of water was tested at different water temperatures (made up). A 300 mL beaker was used, and each mixture was stirred for 2 minutes.
Source 2 — how much dissolves in 100 mL of water (made-up numbers).
4How many grams of alum dissolve in 100 mL of water at 60 °C? How many times as much is that as at 20 °C?
5How many grams of alum dissolve in 350 mL of water at 60 °C?
6"Salt water is like alum water: cool it from 60 °C to 20 °C and what can no longer stay dissolved comes out. So cooling salt water gets you lots of salt." Is that right? For 100 mL of water, compare the amount that comes out for salt and for alum, and give your view.
time yourself (optional)
Stretch (not recorded)From Sun 11 October, any time3 / 4
Stretch — "per one" in a paddy, water, a magnet and a factory
NameDate
3 · How strong a magnet [science: electromagnets]
Source 1
Electromagnets were made by winding enamelled wire 50, 100 and 200 times around a 6 cm iron nail and connecting each to one battery. The number of paper clips that stuck was counted in 4 trials each (made up).
turns
trial 1
trial 2
trial 3
trial 4
50
6
7
7
6
100
13
12
14
13
200
24
26
25
25
Source 2 — paper clips that stuck
7In which coil, in which trial, did the most clips stick? How many?
8Work out the mean of the 4 trials for the 50-, 100- and 200-turn coils.
9On average, how many clips stick per 1 turn? Answer for each of the three coils.
10"0.125 clips per turn, so a 400-turn coil will hold exactly 50. You do not need an experiment — the calculation tells you." Is that right? Use your numbers from questions 8 and 9 to give your view.
time yourself (optional)
Stretch (not recorded)From Sun 11 October, any time4 / 4
Stretch — "per one" in a paddy, water, a magnet and a factory
NameDate
4 · A bicycle factory's week [social studies: manufacturing]
Source 1
A (made-up) factory, with 36 workers, makes bicycles. In one week the machines ran for 8 hours each on Monday, Tuesday and Wednesday, 4.4 hours on Thursday and 7.5 hours on Friday.
Source 2 — bicycles made each day (made-up numbers).
11Which day had the fewest bicycles, and how many?
12What was the mean number of bicycles per day that week?
13Work out the bicycles per hour of machine time for Monday and for Thursday.
Thursday is 66 ÷ 4.4. Multiply both numbers by 10: 660 ÷ 44.
14"Only 66 were made on Thursday. Work went more slowly on Thursday than on the other days." Is that right? Use your numbers from question 13 to give your view.
kids/learn — weekly worksheet. The answers have a page of their own.