Sec 2 Maths — the teaching behind every question

Every Secondary 2 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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18 topics. Each card ends with a button that takes your child straight into questions on that topic.

Algebraic fractionsAlgebraic manipulationDirect and inverse proportionExpansion and factorisationIndices and standard formLinear inequalitiesMap scalesMensurationProbabilityPythagorasQuadratic equationsQuadratic graphsSimilarity and congruenceSimultaneous equationsStatistics from frequency tablesStraight-line graphsTransformationsTrigonometric ratios

Algebraic fractions: the number under the line shares the whole top

Method: Divide every term of the top by the number under the line, then check the answer with a value of x

(10x + 20) ÷ 52x + 42x + 42x + 42x + 42x + 4
five equal shares of 2x + 4 make 10x + 20
  • A fraction line means share: everything on top is divided by what is underneath.
  • So the number under the line has to be shared into every term on top, not just the first one.
  • Work out the top first, then divide - that way you can see the number under the line has really been used.

One question, worked step by step

A canteen stall writes the cost of a set meal as the fraction (10x + 20)/5 dollars, where x is the number of extra side dishes. (a) What is the value of 10x + 20 when x = 6? (b) What is the value of the whole fraction then? (c) Tidy the fraction so the 5 is gone, and show it still gives the same value.
Step 1: Put x = 6 into the top of the fraction only, and leave the fraction line alone.10 × 6 + 20 = 80
Step 2: Now share that whole top by the 5 under the line.80 ÷ 5 = 16
Step 3: The same sharing tidies the algebra: each term of the top is divided by 5.10 ÷ 5 = 2 → 20 ÷ 5 = 4
Step 4: So the fraction is 2x + 4, and putting x = 6 into it gives the same 16 - that is the check.2 × 6 + 4 = 16
Where children go wrong: Sharing only the 10x by 5 and leaving the 20 alone - our own Sec 2 answers show this again and again, and the 5 under the line divides the whole top, so both terms must shrink.

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Algebraic manipulation: a number in front of a bracket reaches every term

Method: Multiply the outside number onto every term inside the bracket, keeping each sign, then check with a value

4 × (5x - 9)5x - 95x - 95x - 95x - 9
four identical crates, each holding 5x - 9 parts
  • A number written just before a bracket multiplies everything inside that bracket.
  • So 4(5x - 9) means four lots of 5x AND four lots of 9.
  • The minus sign inside stays with the 9 the whole way through the multiplying.

One question, worked step by step

A workshop packs 4 identical crates of parts. Each crate holds (5x - 9) parts, where x is the number of shelves in the store. (a) Write the total number of parts without a bracket. (b) What number of parts is taken away in your answer? (c) How many parts is that when x = 7?
Step 1: The 4 in front multiplies the first term inside the bracket.4 × 5 = 20
Step 2: The same 4 must multiply the 9 as well, and the minus stays where it is.4 × 9 = 36
Step 3: So the bracket is 20x - 36. Check it with x = 7 by working the bracket first.4 × (35 - 9) = 104
Step 4: The tidy form must give the same number, and it does.20 × 7 - 36 = 104
Where children go wrong: Multiplying only the first term - the 4 reaches the 5x and then the 9 is left untouched. Our Sec 2 bank records that exact slip, and the 4 belongs to the whole bracket, so both terms inside take it.

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Direct and inverse proportion: a fixed job, shared by fewer hands

Method: When both amounts grow together find the rate for one; when one grows as the other shrinks, multiply to find the total and share it after

8 painters6 painters
fewer painters share the same 72 painter-days, so each of them works more days
  • Two amounts are in proportion when their ratio stays the same - litres and kilometres, painters and days.
  • When both grow together, find what ONE costs or covers, then multiply.
  • When one grows while the other shrinks, the TOTAL stays fixed, so multiply first and share afterwards.

One question, worked step by step

A team of 8 painters takes 9 days to paint a block of flats. Everyone paints at the same steady rate and everyone works on every day of the job. (a) How many painter-days does the job need altogether? (b) If only 6 painters turn up, how many days will the job take? (c) How many painters would finish the same job in 4 days?
Step 1: The job is a fixed amount of work, and painters multiplied by days is that amount.8 × 9 = 72
Step 2: Share the 72 painter-days among the 6 who actually turned up.72 ÷ 6 = 12
Step 3: For a 4-day finish, share the same total by the 4 days instead.72 ÷ 4 = 18
Step 4: Check the direction: fewer painters, more days - 6 painters need longer than the 8 did.12 > 9
Where children go wrong: Sharing the 72 painter-days by the 8 who were there before instead of the 6 who turned up - our bank stores that slip as dividing by the old number of workers. The new team is the one that does the dividing.

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Expansion and factorisation: four little products, then collect the middle

Method: Multiply all four pairs, then collect the two terms that carry an x

x × xx squared↓x × 66x↓5 × x5x↓5 × 630↓x² + 11x + 30the two x terms joined
four products, and the two carrying an x are added in the middle
  • Two brackets side by side means every term in the first multiplies every term in the second.
  • Four little products come out: one is x squared, two carry an x and get added, one is a plain number.
  • The plain number at the end is always the product of the two numbers in the brackets.

One question, worked step by step

A rectangular notice board is x cm square. A 5 cm strip is added along one side and a 6 cm strip along the other, so the new area is (x + 5)(x + 6) square centimetres. (a) What number sits in front of x once the brackets are multiplied out? (b) What is the plain number at the end? (c) What is the area when x = 8?
Step 1: The two x terms come from the two cross products, and they are added.5 + 6 = 11
Step 2: The plain number is the two numbers multiplied.5 × 6 = 30
Step 3: So the expanded form is x² + 11x + 30. Check it at x = 8 by working the brackets first.(8 + 5) × (8 + 6) = 182
Step 4: The expanded form must give the same number at x = 8.8 × 8 + 11 × 8 + 30 = 182
Where children go wrong: Putting 5 × 6 into the middle - our Sec 2 answers show the two numbers multiplied where they should be added. The middle is the two x terms added, so it is 11, and the 30 is the separate plain number.

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Indices and standard form: multiplying adds, dividing takes away

Method: Use the index rule first, then check it by working out the values

255²1255³6255⁴
each step up the powers multiplies by 5, so each step down divides by 5
  • A power tells you how many of that number are multiplied together: 5 to the 4 means four 5s.
  • Multiplying adds the indices and dividing takes one index away, because the shared numbers cancel.
  • Standard form is one digit, a point, the rest of the digits, and a power of ten saying how far the point moved.

One question, worked step by step

A number puzzle card shows 5 to the power 4, and a second line divides it by 5 squared. (a) What is the value of 5 to the power 4? (b) What is the index when 5 to the 4 is divided by 5 squared? (c) A red blood cell measures 0.0043 mm across: what power of ten does that need in standard form?
Step 1: Write the power out and work it up step by step.5 × 5 × 5 × 5 = 625
Step 2: Dividing takes the indices away, so the answer is a power of 5 on its own.4 - 2 = 2
Step 3: Check that by doing the division with the values.625 ÷ 25 = 25
Step 4: For standard form the point moves right until one digit is in front, three places, so the power of ten is negative three.4.3 × 10^-3
Where children go wrong: Adding the indices when the powers are being divided - our bank records this as adding where the powers are divided. Dividing cancels shared numbers, so 5 to the 4 over 5 squared is 5 squared, not 5 to the 6.

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Linear inequalities: the largest whole number that still fits

Method: Solve it like an equation, then test the next whole number up before you answer

3+4+5+6+76 fit, 7 go over
6 crates fit; a 7th would bring the shelf to 36 kg against a 32 kg limit
  • An inequality says how big something may be, not exactly what it is.
  • You solve it just like an equation: move the numbers across, then share by the number in front of the letter.
  • When the answer must be a whole number of things, pick the largest whole number that still fits.

One question, worked step by step

A storeroom shelf carries at most 32 kg. The shelf board itself weighs 8 kg and every crate of parts weighs 4 kg. (a) How much of the 32 kg is left for crates? (b) How many crates would use up exactly that much? (c) What is the greatest whole number of crates the shelf may carry?
Step 1: Take away the mass that is always on the shelf.32 - 8 = 24
Step 2: Each crate takes 4 kg, so share the 24 kg by 4.24 ÷ 4 = 6
Step 3: Now test one more crate, because the limit allows exactly 32 kg.4 × 7 + 8 = 36
Step 4: 36 kg is more than 32 kg, so 7 crates is refused and 6 is the greatest whole number.36 > 32
Where children go wrong: Rounding up to the next whole number - our bank stores that as rounding up although the inequality only allows values below the limit. The count has to be one the shelf really allows, and a 7th crate brings it to 36 kg against a 32 kg limit.

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Map scales: multiply by the scale, then change the units properly

Method: Multiply the map length by the second number of the scale, then divide down the units step by step

7 cm on the mapthe scale is 1 : 40 000↓× 40 000280 000 cm of ground↓÷ 1002 800 metres↓÷ 10002.8 kilometres
multiply once, then divide twice, and never guess a unit
  • A scale such as 1 : 40 000 means one centimetre on the map is 40 000 centimetres on the ground.
  • Both numbers of the scale are in centimetres, so the answer starts in centimetres whether you like it or not.
  • Centimetres to metres is divide by 100; metres to kilometres is divide by 1000.

One question, worked step by step

On a map of Singapore, 1 cm stands for 40 000 cm on the ground. A straight new road is 7 cm long on the map. (a) How many centimetres of ground does the road cover? (b) How many metres is that? (c) How many kilometres is that?
Step 1: Multiply the map length by the second number of the scale.7 × 40000 = 280000
Step 2: Change centimetres to metres: there are 100 cm in a metre, so divide.280000 ÷ 100 = 2800
Step 3: Change metres to kilometres: there are 1000 m in a kilometre.2800 ÷ 1000 = 2.8
Step 4: Sanity check: 7 cm on the map cannot be less than a kilometre on the ground, and 2.8 km is sensible for a road.2.8 > 1
Where children go wrong: Stopping at 280 000 cm - our bank records answers that stop at centimetres and never change the unit. A map answer is expected in metres or kilometres, and each change of unit is a division, not a guess.

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Mensuration: halve the diameter before the circle formulas

Method: Halve the diameter to get the radius, then put the radius into the formula the question asks for

90°a quarterof thewhole 360°
a quarter of the circle is 90 degrees of the 360, so a quarter of the area and a quarter of the round edge
  • The radius runs from the middle to the edge; the diameter runs right across, so it is twice the radius.
  • The distance round the edge is 2 × pi × radius, and the space inside is pi × radius × radius.
  • When a question says to use 22/7 for pi, use that and leave the answer in a neat form.

One question, worked step by step

A round cafe table top measures 4.2 m right across. Use pi = 22/7 throughout. (a) What is the radius? (b) How far is it round the edge? (c) What is the area of the top?
Step 1: 4.2 m is the whole way across, so halve it to get the radius both formulas want.4.2 ÷ 2 = 2.1
Step 2: Distance round the edge is 2 × pi × radius, using 22/7 for pi.2 × 22 ÷ 7 × 2.1 = 13.2
Step 3: Area is pi × radius × radius.22 ÷ 7 × 2.1 × 2.1 = 13.86
Step 4: Check the area against the square drawn around the table: the circle must be smaller than that square.4.2 × 4.2 = 17.64
Where children go wrong: Feeding the 4.2 m straight into the formulas - our Sec 2 bank stores this as using the diameter where the radius belongs. Both the edge and the area formulas want the radius, so the halving comes first.

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Probability: count the whole lot before the part you want

Method: Add every group for the bottom number, then count only the group the question asks for

7red5white8green
20 counters in the bag, and every one of them is equally likely to come out
  • Probability is how many of the things you want, out of everything there is.
  • The bottom number is always the whole lot, never just the part you were given.
  • The word not means every other group, so add them up and keep the same bottom number.

One question, worked step by step

A cloth bag holds counters for a class game: 7 red, 5 white and 8 green, all the same size. One counter is taken out without looking. (a) How many counters are in the bag? (b) What is the probability that it is red? Give a decimal. (c) What is the probability that it is NOT green? Give a decimal.
Step 1: Add the three groups to get the bottom number of the probability.7 + 5 + 8 = 20
Step 2: Red is one group out of the whole bag, as a decimal.7 ÷ 20 = 0.35
Step 3: Not green means the red and the white together, because those are the other groups.7 + 5 = 12
Step 4: Share that by the same whole bag.12 ÷ 20 = 0.6
Where children go wrong: Counting the colour the question rules out - our bank records answers that count the green ones when the question says not green. The favourable group is red and white together, 12 of the 20.

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Pythagoras: square, add, then undo the square

Method: Square the two shorter sides, add them, then take the square root for the longest side

35²1225↓12²144↓add them1369↓√136937 m
two squares added give 1369, and the square root turns that back into 37 m
  • In a right-angled triangle the two shorter sides squared add up to the longest side squared.
  • So squaring and adding gives you the SQUARE of the longest side, not the side itself.
  • The square root is the extra step that turns that square back into a length.

One question, worked step by step

A rectangular field is 35 m long and 12 m wide, and a straight track is laid from one corner right across to the opposite corner. (a) What is the square of the 35 m side? (b) What is the square of the 12 m side? (c) What is the square of the track? (d) How long is the track?
Step 1: Square the long side of the field.35 × 35 = 1225
Step 2: Square the short side.12 × 12 = 144
Step 3: The track is the longest side, so its square is the two squares added.1225 + 144 = 1369
Step 4: 1369 is the square of the track, so take the square root - that is the length itself.37 × 37 = 1369
Where children go wrong: Stopping at 1369 - our bank stores answers that stop at the square and never undo the squaring. 1369 is the square of the track, so the last step is the square root, and 37 × 37 = 1369.

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Quadratic equations: each bracket gives one root

Method: Set each bracket to zero on its own, and move the number across with the opposite sign

the product is zerox + 9x - 3
one of these two parts must be zero, so there are two answers
  • For a product to be zero, at least one of the things being multiplied must itself be zero.
  • So each bracket is set to zero on its own, and each one gives a root.
  • The number leaves the bracket with the opposite sign: x + 9 = 0 means x = -9.

One question, worked step by step

A number puzzle card reads (x + 9)(x - 3) = 0. (a) What value of x makes the first bracket zero? (b) What value makes the second bracket zero? (c) What is the product of the two roots?
Step 1: The first bracket is zero when x takes the 9 across with the other sign.-9 + 9 = 0
Step 2: The second bracket is zero when x is 3.3 - 3 = 0
Step 3: Check each root in the other bracket: it must NOT be zero there.3 + 9 = 12
Step 4: The product of the two roots.-9 × 3 = -27
Where children go wrong: Copying the sign straight out of the bracket - our bank records roots that keep the bracket's sign. The bracket (x + 9) is zero at x = -9, because the number comes out with the opposite sign.

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Quadratic graphs: find the middle x, then the lowest y

Method: Put x = 0 for the crossing point, then find the middle x and put it in for the lowest value

5x=02x=11x=22x=35x=4
the curve dips to 1 at x = 2, then climbs back to 5 by x = 4
  • Every point on the curve is a pair of numbers that fits the rule: x goes in, y comes out.
  • A U-shaped curve has one lowest point, and it sits at one particular x.
  • That x is half of the number in front of x, with its sign flipped: for -4x it is 2.

One question, worked step by step

A curve is drawn from the rule y = x² - 4x + 5. (a) What is y when x = 0? (b) At what x does the curve sit lowest? (c) What is the least value of y?
Step 1: Put x = 0 in to find where the curve crosses the up-down axis.0 - 0 + 5 = 5
Step 2: The lowest point sits at half of the 4, with the sign flipped.4 ÷ 2 = 2
Step 3: Put x = 2 into the rule, finishing with the constant.2 × 2 - 4 × 2 + 5 = 1
Step 4: Check one step to the side is higher, which proves it is a lowest point.3 × 3 - 4 × 3 + 5 = 2
Where children go wrong: Leaving the constant out of the working - our bank stores answers that value the squared term and the x term and never finish with the constant. 2 × 2 - 4 × 2 is only -4, and the least value comes out as 1 after the 5 goes on.

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Similarity and congruence: one factor for every side, twice for an area

Method: Find the side factor by comparing matching sides, then square it for areas

flagscreen
the same shape with every side four times as long, so the picture covers sixteen times the space
  • Similar shapes have the same angles and every side multiplied by the same number.
  • Find that number first, by comparing two sides that match each other.
  • Areas need the number twice, because a length covers space in two directions at once.

One question, worked step by step

A small flag is 4 cm by 11 cm. A projector throws it on a screen as a similar picture, with the 4 cm side showing as 16 cm. (a) What is the scale factor? (b) How long is the 11 cm side on the screen? (c) The small flag covers 44 square centimetres: how much does the screen picture cover?
Step 1: Compare the two sides that match to find the factor.16 ÷ 4 = 4
Step 2: Every length takes the same factor.11 × 4 = 44
Step 3: For an area the factor is used twice.4 × 4 = 16
Step 4: So the screen picture covers sixteen times the small flag's area.44 × 16 = 704
Where children go wrong: Multiplying the area by 4 instead of 16 - our bank stores areas treated as if they grew in step with the lengths. The sides quadruple, so a square on them covers 4 × 4 = 16 times the space.

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Simultaneous equations: make one letter disappear first

Method: Check the signs on the letter you want gone, add or subtract the equations to remove it, then put the letter found into an easier line

4x + y = 31y has a plus↓2x - y = 11y has a minus↓add the two lines6x = 42↓x = 7then y = 3
a plus y and a minus y add to nothing, so x is left on its own
  • Two equations with two letters can be solved together if one letter is made to disappear.
  • Look at the signs: opposite signs mean ADD the equations, and the letter cancels itself out.
  • Once one letter is found, put it into either equation, then check both equations with the pair.

One question, worked step by step

Two rules describe the same two unknown weights in kilograms: 4x + y = 31 and 2x - y = 11. The y terms have opposite signs. (a) After adding the two lines, what does 6x equal? (b) What is x? (c) What is y?
Step 1: The plus y and the minus y add to nothing, so add the two right-hand sides too.31 + 11 = 42
Step 2: 6x is 42, so share it by 6.42 ÷ 6 = 7
Step 3: Put x = 7 into 4x + y = 31 and take the 4x part away.31 - 4 × 7 = 3
Step 4: Check the pair in the other equation.2 × 7 - 3 = 11
Where children go wrong: Subtracting the lines when the y terms are already opposite - our bank records this as subtracting when adding removes the unwanted letter. A plus y and a minus y vanish when you add, and subtracting would bring the y back.

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Statistics from frequency tables: the rows are answers, not people

Method: Multiply each answer by its count, add those amounts, then share by the counts added up

907142
20 pupils borrowed 15 books between them, so the mean is 0.75 books each
  • A frequency table folds a long list into: how many times each answer happened.
  • The number of rows is not the number of people, so add the count column for that.
  • Each row is worth its answer times its count, and the mean shares the total by the people.

One question, worked step by step

The school library records how many books each pupil in one class borrowed last term. 0 books: 9 pupils. 1 book: 7 pupils. 2 books: 4 pupils. (a) How many pupils are in the class? (b) How many books were borrowed altogether? (c) What is the mean number of books per pupil?
Step 1: Add the count column, not the rows, to get the number of pupils.9 + 7 + 4 = 20
Step 2: Each row is its answer times its count, starting with the 1-book row.1 × 7 = 7
Step 3: The 2-book row alone, since there is no 3-book row here.2 × 4 = 8
Step 4: Add the three amounts to get the books borrowed.0 + 7 + 8 = 15
Step 5: The mean shares the books by the pupils, never by the rows.15 ÷ 20 = 0.75
Where children go wrong: Sharing the total by the number of rows - our bank stores exactly this, the rows read as the number of pupils. Here that would give 15 shared by 3 rows, which is five books each when nobody borrowed more than 2. The mean is shared by the 20 pupils.

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Straight-line graphs: rise over run, then the fixed part

Method: Share the change in y by the change in x for the gradient, then take the gradient's share out of one point for the fixed part

(3 km, $9)one point from the list↓(6 km, $15)the other point↓rise ÷ run6 ÷ 3 = 2↓y = 2x + 3so 10 km costs $23
two points, one gradient, and then the fixed charge that every delivery carries
  • A straight line says: every step across adds the same amount up.
  • That amount is the gradient, found by sharing the change in y by the change in x.
  • The fixed part c is what is left when the gradient's share is taken out of one point.

One question, worked step by step

A courier charges on a straight-line scale: a 3 km delivery costs $9 and a 6 km delivery costs $15. (a) How much does the charge go up for each extra kilometre? (b) What is the fixed charge every delivery carries? (c) What would a 10 km delivery cost?
Step 1: Find how much the charge changed between the two deliveries.15 - 9 = 6
Step 2: Find how much the distance changed.6 - 3 = 3
Step 3: The gradient is the change in charge shared by the change in distance.6 ÷ 3 = 2
Step 4: Take the gradient's share out of the $9 to find the fixed part.9 - 2 × 3 = 3
Step 5: Now use the rule for 10 km, and check it gives $15 back at 6 km.2 × 10 + 3 = 23
Where children go wrong: Handing in the rise as the gradient - our bank records the rise reported without being shared by the run. 6 dollars is how much the charge changed, and it still has to be shared by the 3 km the distance changed.

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Transformations: each move changes one thing at a time

Method: Name which number each move is allowed to change, then do one move at a time

BeforeQ(-8, 5)then things changeAfterQ'(8, 5) after the mirrorQ''(10, -4) after the slide
the mirror flips the sideways number, then the slide moves the point again
  • A transformation moves a shape without changing its size, except an enlargement.
  • A mirror standing on the y-axis flips only the left-right number; a mirror lying on the x-axis flips only the up-down number.
  • A translation is a slide written as a vector: the first number is across, the second is up or down.

One question, worked step by step

One corner of a logo sits at Q(-8, 5) on a grid. It is first reflected in the y-axis, and the picture that comes out is then translated by the vector (2, -9). (a) What is the sideways number after the reflection? (b) What are the coordinates after the reflection? (c) What are the coordinates of the final point?
Step 1: A mirror standing on the y-axis flips only the sideways number, so -8 becomes positive.-8 → 8
Step 2: The up-down number is not touched by that mirror, so it stays at 5.5 stays 5
Step 3: The vector's first number slides the point right.8 + 2 = 10
Step 4: The vector's second number slides it down, because it is negative.5 - 9 = -4
Where children go wrong: Flipping both numbers at the mirror - our bank records both coordinates changed although the mirror line is vertical. A mirror standing across the y-axis only flips the left-right number, and the up-down number stays exactly where it was.

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Trigonometric ratios: name the sides from the angle first

Method: Name opposite, adjacent and longest side from the angle you are given, then write the ratio before multiplying

base 40 cmheight 9 cm
the 40 cm side touches the angle and the 9 cm side faces it, so tan is 9 over 40
  • In a right-angled triangle the three ratios each compare two sides.
  • Name the sides from the angle, not from how they look: the opposite side faces the angle and the adjacent side touches it.
  • Tangent compares the two shorter sides - the one facing the angle over the one beside it - so the longest side is not used at all.

One question, worked step by step

A triangular pennant is cut with a 40 cm edge along the top and a 9 cm edge hanging straight down from its right-hand end, meeting at a right angle. The third edge joins the two free ends, and the angle at the left-hand corner is called theta. (a) What is the square of the 40 cm edge? (b) What is the square of the 9 cm edge? (c) How long is the third edge? (d) What is tan theta?
Step 1: Square the edge that touches the angle.40 × 40 = 1600
Step 2: Square the edge that faces the angle.9 × 9 = 81
Step 3: Add the two squares: that gives the square of the third edge.1600 + 81 = 1681
Step 4: Undo the squaring to get the third edge itself.41 × 41 = 1681
Step 5: Tangent is the side facing the angle over the side beside it, so it is 9 over 40 - written as a decimal.9 ÷ 40 = 0.225
Where children go wrong: Writing the tangent upside down - our bank records the two shorter sides swapped in the tangent ratio. Tangent is the side FACING the angle over the side beside it, so it is 9 over 40, not 40 over 9.

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