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 (practice)Tue 13 October1 / 4
Averages and decimal multiplication: practice
Last week's three ideas, once more with your own hands. Estimate before and after calculating
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Last week, in three lines
1. Average = add everything, divide by how many (it levels out the bumps). Run it backwards: average × count = total. 2. Decimal × decimal Multiply as whole numbers, then give the answer as many decimal places as the two factors have together. (0.3 × 0.4: 3 × 4 = 12; ten times and ten times is 100 times, so divide by 100 → 0.12) 3. Estimating a product Multiply by less than 1 and it gets smaller; by more than 1 and it gets bigger.
First, where it goes wrong
"Multiplying always makes a number bigger." Only when you multiply by more than 1.
0.5 × 0.8: 5 × 8 = 40, and 1 + 1 = 2 decimal places → 0.40. 0.40 and 0.4 are the same number, so you may drop the last 0 and write 0.4. But 0.04 or 4.0 are different numbers. Estimate first: is the answer smaller than 0.8?
1 · Decimal × decimal: places and estimates
1Calculate 0.5 × 0.8. Also say whether you may drop a 0 at the end of the answer, and why.
Do 5 × 8 first. How many decimal places in all?
22.4 × 0.35
24 × 35 = 840. Ten times and a hundred times makes how many times?
time yourself
Maths A (practice)Tue 13 October2 / 4
Averages and decimal multiplication: practice
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3Find the area of a rectangle 0.6 m by 0.7 m with an area diagram. Divide a 1 m × 1 m square into a 10 × 10 grid, colour in 0.6 m by 0.7 m of it, and count how many 0.01 m² squares that is.
How many squares tall and how many wide is the coloured part?
43.6 × 2.5
50.06 × 0.15
Easy to miscount the zeros. After 6 × 15, count 2 + 2 places.
6Answer without calculating. Which of these come out smaller than 12? Choose all of them.
A 12 × 0.9 B 12 × 1.01 C 12 × 0.05 D 12 × 1.5
Look only at whether the multiplier is less or more than 1.
time yourself
Maths A (practice)Tue 13 October3 / 4
Averages and decimal multiplication: practice
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7Working 4.8 × 3.1 in columns gave the digits 1488. Where does the decimal point go? Make an estimate first, then answer.
Seeing 4.8 as about 5 and 3.1 as about 3, about how big is the answer?
2 · Averages (levelling out)
8A child read for 25, 40, 0, 35 and 30 minutes over five days. What is the average per day?
A day with 0 still counts as a day.
9The average of four kanji tests (out of 10) is 6.5. After a fifth test the average should be 7. What must the fifth score be?
Average × count = total. Find the 4-test total and the 5-test total that is needed.
time yourself
Maths A (practice)Tue 13 October4 / 4
Averages and decimal multiplication: practice
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10A child's travel time to school, Monday to Friday, was 12, 14, 13, 15 and 61 minutes (on Friday the train was late).
(1) What is the 5-day average?
(2) What is the average of the 4 days without the 61?
(3) To describe the "usual" journey, which of (1) and (2) fits better? Give a reason.
3 · Putting them together
11An iron bar weighs 0.45 kg per metre. How much does 2.8 m of it weigh? Draw an area diagram or a number line before you write the calculation.
12A car goes 12.5 km on 1 L of petrol. Over three days it used 2.4 L, 3.1 L and 2.3 L.
(1) How many litres a day did it use on average?
(2) How many km a day does that come to on average?
There are two ways for (2): use the average, or use the 3-day total and divide by 3.
time yourself
Costume 1 · ITWed 14 October1 / 4
Average internet speed
Mbps means "megabits arriving per second". Still "per one"
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Internet speed (made-up data)
Speed is how much data arrives per second. The unit Mbps (megabits per second) = how many Mb (megabits) arrive in 1 second. time (s) = file size (Mb) ÷ speed (Mbps)
Five measurements in the same place gave 82.4 / 91.1 / 68.7 / 88.2 / 79.6 Mbps. It differs each time with how busy the network is, so we "level it out" with an average. Today file sizes are in Mb too. MB (megabyte) is 1 MB = 8 Mb and appears only in question 7.
The commonest mistake
"Average = halfway between the fastest and the slowest."
(91.1 + 68.7) ÷ 2 = 79.9, but the average of the five is 82.0. The two ends say nothing about where the three in between are. An average adds all of them and divides by how many.
1 · The average of five
1Find the average of the five measurements (Mbps).
2Find each of the five measurements' difference from the average (+ if faster, − if slower). What do the five differences add up to, and why?
time yourself
Costume 1 · ITWed 14 October2 / 4
Average internet speed
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3Throw away the slowest run (68.7) and find the average of the other four, rounded to one decimal place. Did the average go up or down, and why?
4A sixth run is made, and the six-run average should be exactly 83 Mbps. What speed must the sixth run be?
average × count = total.
2 · From speed to time taken
5A file is 205 Mb.
(1) At the average of 82.0 Mbps, how many seconds does it take?
(2) At the slowest run, 68.7 Mbps, how many seconds? Round to one decimal place.
What do you multiply by to make the divisor whole?
time yourself
Costume 1 · ITWed 14 October3 / 4
Average internet speed
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6A game update of 369 Mb arrived in exactly 4.5 seconds.
(1) How many Mb per second (how many Mbps) is that?
(2) At the same speed, how many Mb arrive in 1.8 seconds?
7A video file is 45 MB. Take 1 MB = 8 Mb.
(1) How many Mb is that?
(2) At 90 Mbps, how many seconds?
(3) A friend answered "45 ÷ 90 = 0.5 seconds". What is wrong?
Small b is a bit, capital B is a byte. A byte is 8 times bigger.
3 · Does the average match what it feels like?
8An advert says "up to 100 Mbps". The five-run average of 82.0 Mbps is how many times the advertised speed? What percentage is that?
how many times = compared amount ÷ base amount.
time yourself
Costume 1 · ITWed 14 October4 / 4
Average internet speed
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9One day, measuring once an hour, six runs gave 94.0, 92.5, 90.5, 4.5, 93.0 and 96.5 Mbps.
(1) What is the average of the six, in Mbps?
(2) What is the average of the five without the 4.5?
(3) Someone says "it always feels like 90 Mbps or more". Which of (1) and (2) matches that feeling? And what should be done about the 4.5 run?
10Home A's five-run average is 82.0 Mbps; home B's four-run average is 84.5 Mbps.
(1) Someone found the "average of the two averages", (82.0 + 84.5) ÷ 2 = 83.25. Why is that not right?
(2) Find the average of all nine runs, rounded to one decimal place.
Go back from each average to its total.
time yourself
Maths B (word problems · applications)Fri 16 October1 / 4
Dividing by decimals: word problems, applied
In every problem the diagram comes first, the calculation after. Find the "per one" first
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An answer with no diagram does not get a tickDraw a double number line or an area diagram before the calculation. It need not be neat.
Two steps, same picture: build the 1 first, then travel to the length you want. The numbers are made up.
Worked example: two steps
A ribbon costs 138 yen for 0.6 m. What do 0.9 m cost?
1. Draw it. 0.9 lies between 0.6 and 1 → the answer lies between 138 and "the price of 1 m"
2. Make it 1 m 138 ÷ 0.6 = 1380 ÷ 6 = 230 (yen)
3. For 0.9 m 230 × 0.9 = 207 → 138 < 207 < 230 ✓ Answer 207 yen First "per one", then "that many".
1 · Find the "per one"
1Cloth costs 270 yen for 1.5 m. What does 1 m cost? Draw the double number line before you write the calculation.
time yourself
Maths B (word problems · applications)Fri 16 October2 / 4
Dividing by decimals: word problems, applied
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2Meat costs 1,080 yen for 1.2 kg. What do 0.75 kg cost?
Two steps, as in the example.
3An iron chain of 5.2 m weighs 7.9 kg. How much does 1 m weigh? Round to two decimal places.
Work to three decimal places, then round.
4There are 15.8 L of water, to be poured into bottles that hold 2.5 L.
(1) How many bottles are filled with 2.5 L, and how many litres are left over?
(2) How many bottles are needed to hold all the water?
The remainder's decimal point lines up with the one in 15.8.
time yourself
Maths B (word problems · applications)Fri 16 October3 / 4
Dividing by decimals: word problems, applied
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2 · How many times
5String A is 2.4 m, string B is 1.5 m.
(1) A is how many times B?
(2) B is how many times A?
Draw a diagram, and write what is taken as 1.
The answer to (2) is less than 1.
6This year my brother is 1.32 m tall, which is 1.1 times last year's height. How tall was he last year? Draw a diagram and write an equation with □ before solving it.
□ × 1.1 = 1.32.
3 · Together with averages
7A home's water use over five days was 118.4, 124.6, 109.2, 131.8 and 116.0 L.
(1) What is the average use per day, in L?
(2) The average daily amount fills how many 1.5 L bottles?
time yourself
Maths B (word problems · applications)Fri 16 October4 / 4
Dividing by decimals: word problems, applied
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8A car goes 45 km on 3.6 L. If it uses 2.4 L of petrol a day on average, how many km does it go in 5 days? Draw it and go step by step: per litre → per day → 5 days.
4 · Comparing, and working backwards
9Shop A sells mandarins at 94 yen for 0.4 kg, shop B at 84 yen for 0.35 kg. Which is cheaper per kg? By how many yen per kg?
10A piece of timber 0.75 m long weighs 1.8 kg. For the same timber weighing 6 kg, how long is it?
Treat "the weight of 1 m" like a unit price. length = weight ÷ (weight of 1 m).
The average speed of a round trip is not the average of the speeds. Let the calculation show it
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Speed, in brief
speed = distance ÷ time (the distance covered per hour, per second). 40 km/h is the speed that covers 40 km in 1 hour.
distance = speed × time time = distance ÷ speed average speed = total distance ÷ total time. It has the same shape as Wednesday's "average = total ÷ count".
1 · Guess, then check
1A driver goes out at 40 km/h and comes back at 60 km/h. What do you think the average speed of the whole round trip is? Write your guess first (it is fine if it is wrong).
The round trip as one strip, each leg as wide as the time it took. Fill in the boxes. The numbers are made up.
2Take each leg as 120 km and check. To fill the boxes in the figure, find:
(1) the time the way out took, in hours;
(2) the time the way back took, in hours.
3How many km is the round trip, and how many hours did it take in all? What is the average speed of the round trip, in km/h?
4Compare your guess in question 1 with the answer to question 3. If they differ, explain in words why, looking at the time spent on the way out and on the way back.
5Now suppose the way out and the way back each took 2 hours, at 40 km/h out and 60 km/h back. What is the average speed of the round trip? What is different from question 3?
2 · In other settings
6On a bicycle, a 4.5 km climb is ridden at 9 km/h and a 4.5 km descent at 18 km/h.
(1) How many hours did the climb and the descent each take?
(2) What is the average speed overall, in km/h?
(3) In a few words, why must you not just do (9 + 18) ÷ 2 = 13.5?
7A 60 m run was split into two halves of 30 m. The first 30 m took 6 seconds and the second 30 m took 4 seconds.
(1) What was the speed in each half, in m/s?
(2) What was the average speed over the whole 60 m, in m/s?
(3) Compare the halfway value of the two speeds in (1), 6.25, with (2), and write what you notice.
3 · Working backwards from the average
8The way out was 90 km at 30 km/h. To make the average speed of the round trip 40 km/h, at what speed must the way back be driven? Draw a diagram and work from the total distance and the total time allowed.
9Corrections 1. From this week's notebook (Tue, Wed, Fri) pick one question you got wrong (or one with no diagram that was not ticked).
(a) which day and number; (b) circle the reason: calculation / diagram / setting up the equation / units / misreading;
(c) redo it with a diagram; (d) a calculation that checks the answer.
5 · One puzzle problem
10[Meeting puzzle for travellers] Two people set off at the same time from P and Q, 2.1 km apart, and walk straight towards each other. The one from P walks 80 m a minute, the one from Q 60 m a minute. Draw a line diagram first, then answer.
(1) How many minutes after setting off do they meet?
(2) How many metres from P do they meet?
How many metres closer do they get in 1 minute? How many metres is 2.1 km?
time yourself (optional)
Stretch (not recorded)From Sun 18 October, any time1 / 4
Stretch — averages and "per one"
Not recorded. Getting it wrong is fine. Some numbers in the sources are not needed
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1 · Timing ten swings [science: pendulums]
Source 1
One swing there and back is so short that timing it once gives a large error. So the time for ten swings was measured three times (a made-up experiment), and one swing is worked out from the records. Every weight was 20 g.
pendulum
length
1st try (s)
2nd try (s)
3rd try (s)
A
25 cm
10.2
9.9
9.9
B
64 cm
16.2
15.9
15.9
C
100 cm
20.4
19.8
19.8
Source 2 — time taken for ten swings (made up)
1Which single measurement of ten swings took the longest: which pendulum, which try, and how many seconds?
2Even out the three times for B (take the mean) to get the time for ten swings. From that, how many seconds is one swing of B?
3Counting from the moment it is released, how many swings does pendulum B make in one minute (60 seconds)? What does the part after the decimal point mean for the pendulum?
4Work out the time for one swing of A and of C. Then: a friend says "if you make the pendulum twice as long, one swing takes twice as long". Is that right? Explain with the numbers for A and C.
time yourself (optional)
Stretch (not recorded)From Sun 18 October, any time2 / 4
In a (made-up) country, the share of the food eaten there that was also made there is called food self-sufficiency. Here we use weight. The country has 4.3 million ha of farmland. Self-sufficiency (%) = amount made in the country ÷ amount eaten in the country × 100
food
made here (×10,000 t)
eaten here (×10,000 t)
rice
7.2
7.5
wheat
0.9
6.0
soybeans
0.24
3.0
vegetables
13.5
15.0
wheat (ten years ago)
0.6
5.0
Source 2 — this year, and wheat ten years ago (made up)
5This year, which food was eaten in the largest amount? How many ×10,000 t?
6What is the self-sufficiency of rice and of soybeans, in %?
7Wheat self-sufficiency: from what % ten years ago to what % now?
8A friend says: "In ten years the wheat made went up 1.5 times. So the self-sufficiency must have gone up 1.5 times too." Is that right? Explain with numbers from the table.
time yourself (optional)
Stretch (not recorded)From Sun 18 October, any time3 / 4
Stretch — averages and "per one"
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3 · Solar panels on a roof [science and social studies: electricity and energy]
Source 1
A (made-up) school put 12.5 m² of solar panels on its roof. The electricity made by 1 m² of panel in a day (in kWh) was measured from Monday to Friday. The school uses 7.5 kWh of electricity a day. The roof slopes at 30°.
Source 2 — the electricity made by 1 m² of panel in a day. The numbers are made up.
9On which day was the least electricity made, and how many kWh was it?
10What is the mean electricity made by 1 m² of panel in a day over the five days, in kWh?
11For the whole 12.5 m² of panels, what is the mean amount made in a day, in kWh?
12A friend says: "The mean amount made (question 11) is more than the 7.5 kWh the school uses a day, so on all five days the school did not need to buy electricity." Is that right? Explain with numbers for particular days.
time yourself (optional)
Stretch (not recorded)From Sun 18 October, any time4 / 4
Stretch — averages and "per one"
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4 · A train timetable [social studies: transport]
Source 1
The timetable of a train on a (made-up) line. A to B is 180 km, B to C is 108 km. The train waits a while at B to let a faster train go by. Its top speed is 260 km/h.
station
arrives
leaves
A
—
8:00
B
8:45
8:54
C
9:30
—
Source 2 — the timetable (made up)
13How many minutes does the train stand at station B?
14For A to B and for B to C, what is the speed while the train is running, in km/h?
Turn minutes into hours by dividing by 60 (30 min = 0.5 h).
15From A to C, including the time stopped at B, what is the mean speed in km/h?
16On another day the train ran the 180 km from A to B at 240 km/h on the way there and, because of snow, at 120 km/h on the way back. A friend says: "(240 + 120) ÷ 2 = 180, so the mean speed for the round trip is 180 km/h." Is that right? Work out the round-trip mean speed and give your reason.
The whole round-trip distance ÷ the whole time. Find the time there and the time back separately. Draw it.
kids/learn — weekly worksheet. The answers have a page of their own.