Sec 1 Maths — the teaching behind every question

Every Secondary 1 topic: the method in one line, the picture to draw, and one question worked step by step before your child practises it.

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Letters in algebra Angles in a triangle Perimeter and area Regular polygons Points on a grid A share of a share An equation is a balance Below zero Number patterns Percentage Prime factors Ratio Rounding Speed Squares, cubes and roots The mean Formulae Volume and surface area

Letters in algebra: gather only the terms that match

Method: Keep each sign with its term, gather matching letters only, then substitute

ten a's and five b's: two separate piles, and they never become one pile

A letter stands for a number you do not know yet, so 9a means nine of the same thing.
Only terms with the same letter can be gathered, like 9a and 4a; a and b are different things and never merge.
The minus sign in front of a term belongs to that term, so it travels with it when you tidy up.

One question, worked step by step

A canteen helper jots 9a - 4a + 5a + 7b - 2b on the day's order sheet, where a stands for a plate of rice and b for a bowl of soup, and tidies it. (a) What number stands in front of a? (b) What number stands in front of b? (c) What is the value of the tidied order when a = 2 and b = 6?

Step 1: Gather the a-terms, keeping the minus sign with the 4a it belongs to.9 − 4 + 5 = 10

Step 2: Gather the b-terms the same way.7 − 2 = 5

Step 3: The tidied order is 10a + 5b. Put a = 2 and b = 6 into it.10 × 2 + 5 × 6 = 50

Step 4: Check the biggest coefficient is the a-terms, which is what the sheet says.10 > 5

Where children go wrong: Joining 10a and 5b into 15ab - a and b are not the same size, so terms with different letters can never be added together.

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Angles in a triangle: three angles, one total of 180 degrees

Method: Add the two angles you were given, then take the total away from 180

63 + 81 = 144, so the third angle is 180 - 144 = 36 degrees

Whatever shape a triangle is, its three inside angles always add up to 180 degrees.
So if you know two of them, the third is what is left of 180.
An angle outside the triangle that sits on a straight line with the inside angle makes 180 with it.

One question, worked step by step

A triangular metal bracket has corners A, B and C. The angle at A is 63 degrees and the angle at B is 81 degrees. (a) What do those two angles add up to? (b) What is the angle at C? (c) The side BC is extended past C. What is the outside angle formed there?

Step 1: Put the two angles you were given together.63 + 81 = 144

Step 2: The three angles hold 180 degrees, so take that total away.180 − 144 = 36

Step 3: The outside angle sits on a straight line with the 36, and a straight line holds 180.180 − 36 = 144

Step 4: Check the outside angle the short way: it equals the two far inside angles added.63 + 81 = 144

Where children go wrong: Handing in 144 degrees as the missing angle - 144 is the two angles you were given added together, so one more subtraction is still owed.

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Perimeter and area: the edge all round against the space inside

Method: Perimeter: add one length and one width, then double it. Area: multiply them

16 cm by 9 cm: 50 cm of strip goes round the edge, and 144 square cm lies inside

Perimeter is the walk right round the outside, so it is measured in centimetres.
Area is the amount of surface inside, so it is measured in square centimetres.
A rectangle has two lengths and two widths, and the two jobs give two different numbers.

One question, worked step by step

A piece of card for a display is 16 cm long and 9 cm wide. A gold strip is glued right round its edge and costs 60 cents for each centimetre. (a) How many centimetres of strip are needed? (b) What is the area of the card? (c) What does the strip cost?

Step 1: One length and one width together.16 + 9 = 25

Step 2: A rectangle has two of each, so double that to walk right round.25 × 2 = 50

Step 3: Area is length times width.16 × 9 = 144

Step 4: Strip costs 60 cents a centimetre, and 50 centimetres are needed.50 × 0.6 = 30

Step 5: Check the perimeter the long way, walking all four sides one at a time.16 + 9 + 16 + 9 = 50

Where children go wrong: Counting one length and one width and calling 25 cm the perimeter - a rectangle has two of each side, so that half-walk has to be doubled.

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Regular polygons: every turn round the centre is the same size

Method: Share 360 degrees among the sides to get the outside angle, then count the matching looks

one outside angle of a regular eight-sided panel: 360 split eight ways is 45 degrees

Walking right round a regular shape and turning at every corner brings you back where you started, a full 360 degrees.
Every corner turns the same amount, so the outside angle is 360 shared among the number of sides.
The shape looks exactly the same once at every corner, so the number of sides is also the number of matching looks in a full turn.

One question, worked step by step

A floor tile is a regular shape with eight equal sides. (a) What is the size of each outside angle? (b) How many times does the tile look exactly the same in one full turn? (c) A second tile is a regular triangle. What is its outside angle?

Step 1: Share the full turn among the eight sides.360 ÷ 8 = 45

Step 2: Check the eight turns come back to a full turn.45 × 8 = 360

Step 3: The tile looks the same once at every corner, so count the corners.8 sides = 8 matching looks

Step 4: For the triangle, share the full turn three ways instead.360 ÷ 3 = 120

Where children go wrong: Halving 360 degrees to get the turn of an eight-sided tile - 180 degrees is half a turn, not one turn split among eight sides.

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Points on a grid: a minus sign is still a distance

Method: Count both sides of zero, add them, then halve the gap for the middle

5 units from A up to zero, then 7 more to B: the gap crosses zero, so both parts are added

A coordinate is two numbers: the first says how far across, the second says how far up.
A minus sign means the point is on the other side of zero, so the gap between two points crosses zero and both parts count.
The middle point is half way along, so its coordinate is the left end plus half the gap.

One question, worked step by step

On a grid, post A stands at (-5, 4) and post B stands at (7, 4). A wire is to run straight from A to B. (a) How many units long is the wire? (b) What is the x-coordinate of the point exactly half way between them? (c) What is its y-coordinate?

Step 1: From -5 up to zero is 5 units, and from zero on to 7 is 7 more.5 + 7 = 12

Step 2: Half the gap is 6 units, and the middle is 6 units right of -5.12 ÷ 2 = 6

Step 3: Add that 6 to the left end to get the middle coordinate.−5 + 6 = 1

Step 4: Both posts stand at the same height, so the middle is at that same height.4 = 4, so the middle is at (1, 4)

Where children go wrong: Dropping the minus sign and working out 7 - 5 to get 2 units - the gap crosses zero, so the 5 on the left and the 7 on the right are added, not taken away.

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A share of a share: the second fraction is of the first part

Method: Find the first share, then take the second fraction of THAT answer

5 of the 9 parts are covered seats, and a fifth of those covered seats are reserved

A fraction of something means that something is split into equal parts and some are taken.
When two shares come one after the other, the second is worked out from the answer to the first, not from the whole.
A decimal like 0.2 is one fifth, so the same question can be asked two ways.

One question, worked step by step

A stadium holds 3600 seats. Five ninths of the seats are under cover. Of the covered seats, 0.2 are reserved for members. Each reserved seat is given a printed programme costing 1.50 dollars. (a) How many seats are under cover? (b) How many are reserved for members? (c) What do the programmes cost altogether?

Step 1: Split all the seats into nine equal parts.3600 ÷ 9 = 400

Step 2: Five of those parts are under cover.400 × 5 = 2000

Step 3: 0.2 is one fifth, so split the 2000 covered seats five ways.2000 ÷ 5 = 400

Step 4: Every reserved seat gets a programme at 1.50 dollars.400 × 1.5 = 600

Step 5: Check the second share is smaller than the first, as it must be.400 < 2000

Where children go wrong: Taking the fifth from all 3600 seats and getting 720 - the reserved seats are a share of the COVERED seats, not of the whole stadium.

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An equation is a balance: take off the fixed part first

Method: Take the fixed amount off the total, then share what is left among equal parts

8 dollars were in the tin before any cup was sold, so the other 132 dollars came from the 11 cups

An equation says two things are worth the same, so whatever you do to one side you do to the other.
A fixed amount that was there before anything happened is not part of the equal parts.
Take that fixed amount off first, and then the sharing out is simple.

One question, worked step by step

A drink stall sells cups at the same price all day. The money tin started with a float of 8 dollars. After 11 cups were sold the tin held 140 dollars. (a) How much came from the 11 cups? (b) What is the price of one cup? (c) The stall next door sells 15 cups at the same price, with a float of 8 dollars too. How much will its tin hold?

Step 1: Take the float off the total, because the float was there before any cup was sold.140 − 8 = 132

Step 2: Eleven cups are worth that 132 dollars, so share it eleven ways.132 ÷ 11 = 12

Step 3: Fifteen cups at 12 dollars each.15 × 12 = 180

Step 4: Add that stall's own float back on.180 + 8 = 188

Where children go wrong: Sharing the whole 140 dollars among the 11 cups - the 8 dollars was in the tin before anything was sold, so it has to come off first.

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Below zero: a fall takes away, a rise adds on

Method: Start at the reading, move in the direction of the change, and pass through zero carefully

7 below zero at six in the morning, 2 above by noon, then 3 below zero in the evening

Numbers below zero are still numbers, only on the other side of zero.
A fall moves you further down, and a rise moves you up, even when you are already below zero.
The size of a change is the distance travelled, so a rise from 7 below zero to 2 above is 9 degrees, not 5.

One question, worked step by step

A chest freezer reads 7 degrees below zero. It is switched off and the inside warms by 9 degrees. (a) What does it read then? (b) How many degrees did the reading travel to get there? (c) The lid is opened and it falls 5 degrees from that reading. What does it read now?

Step 1: A rise of 9 from 7 below zero takes the reading through zero.−7 + 9 = 2

Step 2: Count how far it travelled: 7 degrees up to zero, then 2 more above.7 + 2 = 9

Step 3: A fall of 5 from 2 above zero crosses back below zero.2 − 5 = −3

Step 4: Check the sign: the last reading is below zero, so it must carry a minus sign.−3 < 0

Where children go wrong: Working out 9 - 7 = 2 and calling that the travel - the reading crossed zero, so the 7 below and the 2 above are added to get 9 degrees.

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Number patterns: find the step, count the steps carefully

Method: Find the step between two terms, then count steps not terms

each day is 6 more stamps than the day before

A sequence follows one rule again and again, and that rule is usually the size of the step.
To go from the first day to the tenth day you take nine steps, because the first day is already there.
Adding a whole list of terms is quick: add the first and last, halve how many there are, then multiply.

One question, worked step by step

A stall stamps loyalty cards. It stamps 7 cards on day one, 13 on day two, 19 on day three and 25 on day four, and keeps the same daily increase. (a) By how many cards does the daily count grow? (b) How many cards does it stamp on day ten? (c) How many stamps has it given out in the first ten days altogether?

Step 1: Find the step by taking one day from the day before it.13 − 7 = 6

Step 2: Day ten is nine steps after day one, because day one is already counted.7 + 9 × 6 = 61

Step 3: For the total, add the first and last days together.7 + 61 = 68

Step 4: There are ten days, and pairs of days each make 68.10 ÷ 2 = 5

Step 5: Five pairs of 68 gives the whole total.5 × 68 = 340

Where children go wrong: Adding ten steps to day one and answering 67 - from day one to day ten there are only nine steps, because day one is already on the list.

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Percentage: the discount is not the price you pay

Method: Find 1 per cent first, then as many per cent as the question asks

42 dollars is taken off, which is 12 per cent, so 308 dollars is paid, which is 88 per cent

Per cent means out of a hundred, so 100 per cent is the whole price.
If you know what 1 per cent is worth, any other percentage is that one value multiplied.
A discount is the part taken off, so the price paid is what is left after it.

One question, worked step by step

A bicycle shop cuts 12 per cent off every bicycle in a closing-down sale. (a) How many dollars come off a bicycle priced at 350 dollars? (b) What does that bicycle then cost? (c) A child's bicycle priced at 150 dollars carries the same 12 per cent. What does it cost?

Step 1: Share the price into a hundred equal parts to find 1 per cent.350 ÷ 100 = 3.5

Step 2: Twelve of those parts come off.3.5 × 12 = 42

Step 3: Take the discount off to get the price paid.350 − 42 = 308

Step 4: For the child's bicycle, find its 1 per cent the same way.150 ÷ 100 = 1.5

Step 5: Twelve per cent of 150, then take it off.1.5 × 12 = 18

Step 6: The price paid on the child's bicycle.150 − 18 = 132

Where children go wrong: Stopping at the 42 dollars taken off - the question asks what the bicycle costs, so the discount still has to be taken away from the marked price.

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Prime factors: keep dividing until nothing but primes is left

Method: Divide again and again by the smallest prime that goes in, then read the powers off

45 goes down the ladder as two 3s and one 5: 3 times 3 times 5 is 45

A prime can only be divided by 1 and itself, so it is the end of a factor ladder.
Breaking a number fully into primes means keeping on dividing until every number left is prime.
The count of factors comes from the powers: add one to each power, then multiply.

One question, worked step by step

A pupil breaks 45 up completely into primes. (a) How many 3s turn up? (b) How many 5s? (c) How many factors does 45 have altogether?

Step 1: 45 is odd, so the twos are no use. Try the next prime.45 ÷ 3 = 15

Step 2: 15 still divides by 3.15 ÷ 3 = 5

Step 3: 5 is prime, so the ladder ends. So 45 is two 3s and one 5. Add one to the power of 3.2 + 1 = 3

Step 4: Add one to the power of 5.1 + 1 = 2

Step 5: Multiply the two results to count the factors.3 × 2 = 6

Step 6: Check by listing them: 1 and 45, 3 and 15, 5 and 9 - that is three pairs.3 × 2 = 6

Where children go wrong: Adding the two powers, 2 + 1, and answering 3 factors - add one to each power first, then multiply, which gives 6.

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Ratio: count all the parts before you share anything

Method: Add the ratio numbers to get the parts, find one part, then multiply

3 parts, 4 parts and 5 parts make 12 parts, and each part is 20 stickers

A ratio tells you how many parts each person gets, not how many things.
The whole is shared among all the parts added together, so add the ratio numbers first.
Once you know what one part is worth, every share is that value multiplied by its own ratio number.

One question, worked step by step

A class raffle prize is a sheet of 240 stickers, shared between three winners in the ratio 3 : 4 : 5. (a) How many parts make up the whole sharing? (b) How many stickers is one part worth? (c) How many stickers does the winner with the largest share get?

Step 1: Add the ratio numbers to find how many parts the sheet is split into.3 + 4 + 5 = 12

Step 2: Share the 240 stickers among those 12 parts.240 ÷ 12 = 20

Step 3: The largest share is the 5 parts.5 × 20 = 100

Step 4: Check by rebuilding the whole sheet from the three shares.60 + 80 + 100 = 240

Where children go wrong: Sharing 240 by 5 and answering 48 for the largest - the 5 is one share out of twelve parts, so the dividing number is the parts total.

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Rounding: one digit decides, and it is the one on the right

Method: Name the place you are rounding to, look at the digit just right of it, then keep or raise

the crowd is 482 from 37000 but 518 from 36000, so 37000 is nearer

Rounding replaces a number with a nearby friendly one, so the answer is always a little off.
Only one digit matters: the digit one place to the right of the place you are keeping.
If that digit is 5 or more the kept digit goes up one; if it is under 5 the kept digit stays.

One question, worked step by step

A funfair gate counts 36518 visitors in a day. Each ticket costs 18 dollars. (a) Write the count to the nearest thousand. (b) Write it to the nearest hundred. (c) Using the count rounded to the nearest thousand, estimate the money taken at the gate.

Step 1: Nearest thousand: the digit to the right is the 5 in the hundreds place, which is 5 or more, so the thousands go up.37000 − 36518 = 482

Step 2: Nearest hundred: the digit to the right is the 1 in the tens place, which is under 5, so the hundreds stay.36518 − 36500 = 18

Step 3: Use the rounded count, 37 thousand, and multiply by the ticket price.37 × 18 = 666

Step 4: That answer is in thousands, so put the three zeros back.666 × 1000 = 666000

Where children go wrong: Looking at the thousands digit when rounding to the nearest hundred - only the digit just to the right of the place you keep decides, and here that is the tens digit.

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Speed: distance shared out by the hours it took

Method: Speed is distance divided by time, then choose multiply or divide for what is asked

195 km in the first three hours is 65 km every hour, so one more hour adds 65 km

Speed says how far something goes in one hour, so it is distance divided by time.
Once you know the distance in one hour, more hours means more distance.
If the distance is given and the time is asked for, divide the distance by the speed.

One question, worked step by step

A delivery van covers 195 km of road in 3 hours without stopping. (a) What is its average speed in km/h? (b) How far does it travel in 4 hours at that speed? (c) How long does it need to cover 455 km at the same speed?

Step 1: Share the distance among the hours it took.195 ÷ 3 = 65

Step 2: Four hours at 65 km each.65 × 4 = 260

Step 3: For a distance, ask how many 65s fit into it.455 ÷ 65 = 7

Step 4: Check the answers grow with more distance and more time.260 < 455

Where children go wrong: Adding the distance and the time, or multiplying them, instead of dividing - speed is distance per hour, so the hours share out the distance.

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Squares, cubes and roots: a cube root undoes a cube

Method: Multiply the edge by itself for a square, three times for a cube; a root asks what the edge was

a cube with a 130 cm edge: 130 times 130 times 130 gives 2197000 cubic cm

Squaring means multiplying a number by itself, so it is never the same as doubling it.
Cubing means multiplying it by itself three times, which is what fills a cube-shaped space.
A cube root works backwards: it finds the edge that would give this volume.

One question, worked step by step

A cube-shaped tank holds 2197 litres of water. (a) How many cubic centimetres is that? (b) What is the length of one edge of the tank in centimetres? (c) What is that edge in metres?

Step 1: One litre is a thousand cubic centimetres.2197 × 1000 = 2197000

Step 2: Square a likely edge to start closing in on the volume.130 × 130 = 16900

Step 3: Multiply that face by the edge once more to fill the cube.16900 × 130 = 2197000

Step 4: So the edge is 130 cm. Change it to metres, since 100 cm makes a metre.130 ÷ 100 = 1.3

Where children go wrong: Halving the volume to find the edge - a cube root is the number that gives the volume when multiplied by itself three times, so it is far smaller than half.

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The mean: share the total out, and count every reading

Method: Add every reading, then divide by how many there are - recount them after any change

five days add to 85 visitors, so every day levels out at 17

The mean is the number every reading would get if they were all made equal.
So the total has to be shared out among the readings, never handed in as the answer.
When one more reading is added, both the total and the count change, so the mean is worked out again.

One question, worked step by step

A bookshop counts the visitors through its door over five days: 16, 24, 11, 19 and 15. (a) How many visitors came in those five days? (b) What is the mean number of visitors a day? (c) A sixth day brings 23 visitors. What is the mean over all six days now?

Step 1: Add the first two days.16 + 24 = 40

Step 2: Add the third.40 + 11 = 51

Step 3: Add the fourth.51 + 19 = 70

Step 4: Add the fifth to finish the total.70 + 15 = 85

Step 5: Share the total among the five days.85 ÷ 5 = 17

Step 6: The sixth day adds 23 visitors to the total.85 + 23 = 108

Step 7: Now there are six days, so count them before dividing.108 ÷ 6 = 18

Where children go wrong: Handing in the total of 85 as the mean - the total still has to be shared out among the five days.

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Formulae: put the numbers in, then work out the fixed part once

Method: Replace each letter with its number, do the multiplying, then add the fixed charge

30 dollars for taking the bicycle in, plus 22 dollars for each of the 7 hours

A formula is a rule written with letters, so it works for any numbers you put in.
Put the number in place of the letter, then follow the usual order: multiply before adding.
A fixed charge is paid once; an hourly rate is paid for every hour, so they are not treated the same.

One question, worked step by step

A bicycle workshop charges by the formula C = 30 + 22h dollars, where h is the number of hours of work. (a) What does it charge for a 4-hour repair? (b) What does it charge for a repair lasting 2.5 hours? (c) One repair cost 184 dollars. How many hours did it take?

Step 1: Put h = 4 into the formula, so the hourly part is 22 times 4.22 × 4 = 88

Step 2: Add the 30 charged for taking the bicycle in.88 + 30 = 118

Step 3: For 2.5 hours, the hourly part is 22 times 2.5.22 × 2.5 = 55

Step 4: Add the same 30 once.55 + 30 = 85

Step 5: Working backwards, take the 30 off the 184 first.184 − 30 = 154

Step 6: Then ask how many 22s fit into 154.154 ÷ 22 = 7

Where children go wrong: Multiplying the 30 by the hours as well - 30 is charged once for taking the bicycle in, and only the 22 is charged for each hour.

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Volume and surface area: the space inside against the paper outside

Method: Volume: three sides multiplied. Surface area: the three faces added, then doubled

24 cm by 10 cm by 6 cm: 1440 cubic cm inside, and 888 square cm of paper outside

Volume is the space inside a box, so it is measured in cubic centimetres.
Surface area is the paper needed to cover the box, so it is measured in square centimetres.
A box has three different faces, and each one appears twice on the outside.

One question, worked step by step

A shoebox is 24 cm long, 10 cm wide and 6 cm high. (a) What is its volume? (b) What is its total surface area? (c) Wrapping paper costs 5 cents for each square centimetre. What does it cost to cover the box?

Step 1: Volume: length times width, then times height.24 × 10 = 240

Step 2: Multiply by the height to fill the box.240 × 6 = 1440

Step 3: For the paper, find the second kind of face.24 × 6 = 144

Step 4: And the third kind of face.10 × 6 = 60

Step 5: Add the three different faces together.240 + 144 = 384

Step 6: Add the last face.384 + 60 = 444

Step 7: Every face appears twice on a closed box, so double it.444 × 2 = 888

Step 8: Paper at 5 cents a square centimetre.888 × 0.05 = 44.4

Where children go wrong: Giving the volume of 1440 when the paper is asked for - the wrapping covers the six faces on the outside, never the space inside the box.

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