Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: 2ˣ = 16. x =
Before any working — hunt first
Given
Numbers and units already in the sentence: 2 · 16. Copy each one into a column. A name without a number is not given.
Missing
After you copy the given, the letter that still has no number is missing. The paper is pointing at: x =. Circle that word — that is the missing variable.
Find
We are required to find: x =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
Write 16 = 2⁴ with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives x =.
In other words — as if you have never seen this before
Write 16 as a power of 2. 16 = 2⁴, so x = 4.
2. What do we already know?
Exponential: base 2, equals 16.
3. Which method — and why this one?
Write 16 as a power of 2. 16 = 2⁴, so x = 4.
4. Step-by-step working
01 / 05
Same base
16=2⁴. So 2ˣ=2⁴.
16=2⁴
Why this step: Write the formula before numbers. The memo awards a method mark for the law even if a digit slips later.
02 / 05
Equate exponents
x=4.
x=4
Why this step: One job per line so you can find a slip, and so a marker can follow the marks.
03 / 05
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
04 / 05
Step F — Box the answer with its unit
4
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
05 / 05
Step H — What the marker is looking for
On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
4
Copy onto your answer sheet — in this order
Formula: 16=2⁴
Answer: 4
6. Check it like a marker
2⁴=16. Yes. 2⁸=256, far too big.
7. A trap first-timers fall into
Do not write x = 162=8. You cannot divide when the unknown is an exponent.
1.2 · 12 Marks · Remainder & factor theorem
5P2=
Remainder & factor theorem
Formulas
f(a)=0⇔(x − a) | f(x)
Factor theorem
Notes: Permutations of 5 taken 2 at a time. Order matters.
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: 5P2 =
Before any working — hunt first
Given
Numbers and units already in the sentence: 5P · 2. Copy each one into a column. A name without a number is not given.
Missing
After you copy the given, the letter that still has no number is missing. The paper is pointing at: 5P2 =. Circle that word — that is the missing variable.
Find
We are required to find: 5P2 =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
Write nPr = n!/(n−r)! with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives 5P2 =.
In other words — as if you have never seen this before
nPr=n!(n−r)!=n(n−1)… down r factors. 5×4=20.
2. What do we already know?
Permutations of 5 taken 2 at a time. Order matters.
3. Which method — and why this one?
nPr=n!(n−r)!=n(n−1)… down r factors. 5×4=20.
4. Step-by-step working
01 / 05
Write the definition
5P2=5!3!=5×4×3×2×13×2×1=5×4.
nPr=n!(n−r)!
Why this step: Write the formula before numbers. The memo awards a method mark for the law even if a digit slips later.
02 / 05
Multiply
5×4=20.
20
Why this step: One job per line so you can find a slip, and so a marker can follow the marks.
03 / 05
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
04 / 05
Step F — Box the answer with its unit
20
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
05 / 05
Step H — What the marker is looking for
On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
20
Copy onto your answer sheet — in this order
Formula: nPr=n!(n−r)!
Answer: 20
6. Check it like a marker
5C2 would be 10 (order does not matter). 5! is 120. The paper asked P, not C or !.
7. A trap first-timers fall into
Do not pick 10 (that is 5C2) or 120 (that is 5!). P keeps order.
2.1 · 14 Marks · Compound angles
cos 2A=
Compound angles
Formulas
sin(A + B)=sin A cos B+cos A sin B
Sine compound
Notes: Double-angle identity for cosine. No numbers.
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: cos 2A =
Before any working — hunt first
Given
Numbers and units already in the sentence: 2A. Copy each one into a column. A name without a number is not given.
Missing
After you copy the given, the letter that still has no number is missing. The paper is pointing at: cos 2A =. Circle that word — that is the missing variable.
Find
We are required to find: cos 2A =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
Write cos 2A = cos²A − sin²A with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives cos 2A =.
In other words — as if you have never seen this before
Three equivalent forms: cos²A − sin²A, 2cos²A − 1, 1 − 2sin²A. Not 2 sin A cos A (that is sin 2A).
2. What do we already know?
Double-angle identity for cosine. No numbers.
3. Which method — and why this one?
Three equivalent forms: cos²A − sin²A, 2cos²A − 1, 1 − 2sin²A. Not 2 sin A cos A (that is sin 2A).
4. Step-by-step working
01 / 08
Step A — Read the question twice
Slow down. Circle the word that names the unknown. Underline every number and its unit. Do not grab a formula yet — first decide what the paper is actually asking.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
02 / 08
Step A2 — Hunt: Given, Missing, Find, Calculate
Given: Numbers and units already in the sentence: 2A. Copy each one into a column. A name without a number is not given.
Missing: After you copy the given, the letter that still has no number is missing. The paper is pointing at: cos 2A =. Circle that word — that is the missing variable.
Find: We are required to find: cos 2A =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: Write cos 2A = cos²A − sin²A with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives cos 2A =.
Why this step: Do not name the letters yet. First hunt: what already has a number (Given), what has no number (Missing), the word the paper wants (Find), and which formula still contains the missing letter (Calculate).
03 / 08
Step B — Write the given in a column
Double-angle identity for cosine. No numbers.
Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
04 / 08
Step C — Name the method in words
Three equivalent forms: cos²A − sin²A, 2cos²A − 1, 1 − 2sin²A. Not 2 sin A cos A (that is sin 2A).
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
05 / 08
Quote the identity
cos 2A=cos²A−sin²A.
cos 2A=cos²A−sin²A
Why this step: Write the formula before numbers. The memo awards a method mark for the law even if a digit slips later.
06 / 08
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
07 / 08
Step F — Box the answer with its unit
cos²A − sin²A
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
08 / 08
Step H — What the marker is looking for
On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
cos²A − sin²A
Copy onto your answer sheet — in this order
Formula: cos 2A=cos²A−sin²A
Answer: cos²A − sin²A
6. Check it like a marker
At A = 0: cos 0 = 1, and 1 − 0 = 1. The sin 2A form 2sinAcosA would give 0, which is wrong.
7. A trap first-timers fall into
2 sin A cos A is sin 2A, not cos 2A. sin²A − cos²A is −cos 2A.
2.2 · 14 Marks
r=−0.95 means
Notes: Pearson’s r = −0.95 for a bivariate data set.
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: r = −0.95 means
Before any working — hunt first
Given
Numbers and units already in the sentence: 0.95 means. Copy each one into a column. A name without a number is not given.
Missing
After you copy the given, the letter that still has no number is missing. The paper is pointing at: 95 means. Circle that word — that is the missing variable.
Find
We are required to find: 95 means. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
Write |r| = 0.95 with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives 95 means.
In other words — as if you have never seen this before
r near −1: strong negative linear. r does not prove causation.
2. What do we already know?
Pearson’s r = −0.95 for a bivariate data set.
3. Which method — and why this one?
r near −1: strong negative linear. r does not prove causation.
4. Step-by-step working
01 / 05
Read the sign
Negative: as x increases, y tends to decrease.
Why this step: One job per line so you can find a slip, and so a marker can follow the marks.
02 / 05
Read the size
|r|=0.95 is close to 1, so the linear pattern is strong.
|r|=0.95
Why this step: One job per line so you can find a slip, and so a marker can follow the marks.
03 / 05
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
04 / 05
Step F — Box the answer with its unit
strong negative linear correlation
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
05 / 05
Step H — What the marker is looking for
On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
strong negative linear correlation
Copy onto your answer sheet — in this order
Formula: |r|=0.95
Answer: strong negative linear correlation
6. Check it like a marker
Scatter would slope down, tightly around a line.
7. A trap first-timers fall into
Correlation is not causation. Do not say “x causes y to fall”.
5 · 4 Marks · Similarity & proportion
f'(x) = 0 for f(x) = x² − 6x + 1 at
Similarity & proportion
Formulas
AA similarity ⇒ corresponding sides proportional
AA similarity
Notes: f(x) = x² − 6x + 1. Unknown: the x where the gradient is zero (stationary / turning point).
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: f'(x) = 0 for f(x) = x² − 6x + 1 at
Before any working — hunt first
Given
Numbers and units already in the sentence: 0 for · 6x · 1 at. Copy each one into a column. A name without a number is not given.
Missing
After you copy the given, the letter that still has no number is missing. The paper is pointing at: f'(x) = 0 for f(x) = x² − 6x + 1 at. Circle that word — that is the missing variable.
Find
We are required to find: f'(x) = 0 for f(x) = x² − 6x + 1 at. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
Write f'(x) = 2x − 6 with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives f'(x) = 0 for f(x) = x² − 6x + 1 at.
In other words — as if you have never seen this before
Differentiate, then set the derivative equal to zero. That is where the tangent is flat.
2. What do we already know?
f(x) = x² − 6x + 1. Unknown: the x where the gradient is zero (stationary / turning point).
3. Which method — and why this one?
Differentiate, then set the derivative equal to zero. That is where the tangent is flat.
4. Step-by-step working
01 / 05
Differentiate
f'(x) = 2x − 6. (Power rule on x², and −6x becomes −6.)
f'(x)=2x−6
Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.
02 / 05
Set equal to zero
2x − 6 = 0 → 2x = 6 → x = 3. The turning point sits at x = 3 (a minimum, because 2>0).
x=3
Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.
03 / 05
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
04 / 05
Step F — Box the answer with its unit
x=3
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
05 / 05
Step H — What the marker is looking for
On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
x=3
Copy onto your answer sheet — in this order
Formula: f'(x)=2x−6
Answer: x=3
6. Check it like a marker
Vertex of a parabola x = −b2a=64? Wait: a=1, b=−6, −b2a=62=3. Same answer, different road.
7. A trap first-timers fall into
Picking x = 6, which is −b, or x = 1, a root-ish guess. Stationary is where f' is zero, not where f is zero.
6 · 2 Marks · Circles & tangents
Gradient of radius to (4; 0) from (0; 0) is
Formulas
(x − a)²+(y − b)²=r²
Circle
mradius×mtangent=−1
Tangent ⟂ radius
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: Gradient of radius to (4; 0) from (0; 0) is
In other words — as if you have never seen this before
0
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 4 · 0 · 0 · 0. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
0
4. Step-by-step working
01 / 07
Step A — Read the question twice
Slow down. Circle the word that names the unknown. Underline every number and its unit. Do not grab a formula yet — first decide what the paper is actually asking.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
02 / 07
Step B — Write the given in a column
Copy these onto the left of your page, one per line, with units: 4 · 0 · 0 · 0. Mixing cm with m, or minutes with seconds, loses a method mark.
Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
03 / 07
Step C — Name the method in words
0
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
04 / 07
Reason
Horizontal radius, m=0. Tangent then vertical.
Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.
05 / 07
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
06 / 07
Step F — Box the answer with its unit
0
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
07 / 07
Step H — What the marker is looking for
On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
0
Copy onto your answer sheet — in this order
Answer: 0
6. Check it like a marker
Does your answer appear as one of the options? Do the units match what was asked (m, N, J, °)? If you put the number back into the original equation, does it still hold? Estimate: is the size sensible?
7. A trap first-timers fall into
First-timers often grab the first formula they remember, mix units (cm with m, minutes with seconds), or drop a minus sign. Write the formula in words first: “this equals that, because…”
7 · 2 Marks · Area, sine and cosine rules
ddx (x³)=
Formulas
Area=½ ab sin C
Area rule
c²=a²+b²−2ab cos C
Cosine rule
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: ddx (x³) =
Before any working — hunt first
Given
Numbers and units already in the sentence: no numbers yet — only a name, a diagram, or a worded unknown. Copy each one into a column. A name without a number is not given.
Missing
After you copy the given, the letter that still has no number is missing. The paper is pointing at: ddx (x³) =. Circle that word — that is the missing variable.
Find
We are required to find: ddx (x³) =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: Power rule: n x^{n−1}. Then substitute the given, one line at a time, to get ddx (x³) =.
In other words — as if you have never seen this before
Power rule: n xⁿ⁻¹.
2. What do we already know?
Copy these onto the left of your page, one per line, with units: every number and its unit in the sentence. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
Power rule: n xⁿ⁻¹.
4. Step-by-step working
01 / 08
Step A — Read the question twice
Slow down. Circle the word that names the unknown. Underline every number and its unit. Do not grab a formula yet — first decide what the paper is actually asking.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
02 / 08
Step A2 — Hunt: Given, Missing, Find, Calculate
Given: Numbers and units already in the sentence: no numbers yet — only a name, a diagram, or a worded unknown. Copy each one into a column. A name without a number is not given.
Missing: After you copy the given, the letter that still has no number is missing. The paper is pointing at: ddx (x³) =. Circle that word — that is the missing variable.
Find: We are required to find: ddx (x³) =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: What we need to calculate: Power rule: n x^{n−1}. Then substitute the given, one line at a time, to get ddx (x³) =.
Why this step: Do not name the letters yet. First hunt: what already has a number (Given), what has no number (Missing), the word the paper wants (Find), and which formula still contains the missing letter (Calculate).
03 / 08
Step B — Write the given in a column
Copy these onto the left of your page, one per line, with units: every number and its unit in the sentence. Mixing cm with m, or minutes with seconds, loses a method mark.
Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
04 / 08
Step C — Name the method in words
Power rule: n xⁿ⁻¹.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
05 / 08
Working
Power rule: n xⁿ⁻¹.
Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.
06 / 08
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
07 / 08
Step F — Box the answer with its unit
3x²
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
08 / 08
Step H — What the marker is looking for
On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
3x²
Copy onto your answer sheet — in this order
Answer: 3x²
6. Check it like a marker
Does your answer appear as one of the options? Do the units match what was asked (m, N, J, °)? If you put the number back into the original equation, does it still hold? Estimate: is the size sensible?
7. A trap first-timers fall into
First-timers often grab the first formula they remember, mix units (cm with m, minutes with seconds), or drop a minus sign. Write the formula in words first: “this equals that, because…”
8 · 4 Marks
Stationary point of f(x)=x³−3x+1 is at x =
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: Stationary point of f(x)=x³−3x+1 is at x =
In other words — as if you have never seen this before
f'=3x²−3=0→x²=1→x=±1.
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 3x · 1 is. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
f'=3x²−3=0→x²=1→x=±1.
4. Step-by-step working
01 / 07
Step A — Read the question twice
Slow down. Circle the word that names the unknown. Underline every number and its unit. Do not grab a formula yet — first decide what the paper is actually asking.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
02 / 07
Step B — Write the given in a column
Copy these onto the left of your page, one per line, with units: 3x · 1 is. Mixing cm with m, or minutes with seconds, loses a method mark.
Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
03 / 07
Step C — Name the method in words
f'=3x²−3=0→x²=1→x=±1.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
04 / 07
Working
f'=3x²−3=0→x²=1→x=±1.
Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.
05 / 07
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
06 / 07
Step F — Box the answer with its unit
±1
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
07 / 07
Step H — What the marker is looking for
On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
±1
Copy onto your answer sheet — in this order
Answer: ±1
6. Check it like a marker
Does your answer appear as one of the options? Do the units match what was asked (m, N, J, °)? If you put the number back into the original equation, does it still hold? Estimate: is the size sensible?
7. A trap first-timers fall into
First-timers often grab the first formula they remember, mix units (cm with m, minutes with seconds), or drop a minus sign. Write the formula in words first: “this equals that, because…”
9 · 2 Marks
S_∞ of 16 + 8 + 4 + … =
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: S_∞ of 16 + 8 + 4 + … =
Before any working — hunt first
Given
Numbers and units already in the sentence: 16 · 8 · 4. Copy each one into a column. A name without a number is not given.
Missing
After you copy the given, the letter that still has no number is missing. The paper is pointing at: S_∞ of 16 + 8 + 4 + … =. Circle that word — that is the missing variable.
Find
We are required to find: S_∞ of 16 + 8 + 4 + … =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: 32. Then substitute the given, one line at a time, to get S_∞ of 16 + 8 + 4 + … =.
In other words — as if you have never seen this before
32
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 16 · 8 · 4. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
32
4. Step-by-step working
01 / 08
Step A — Read the question twice
Slow down. Circle the word that names the unknown. Underline every number and its unit. Do not grab a formula yet — first decide what the paper is actually asking.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
02 / 08
Step A2 — Hunt: Given, Missing, Find, Calculate
Given: Numbers and units already in the sentence: 16 · 8 · 4. Copy each one into a column. A name without a number is not given.
Missing: After you copy the given, the letter that still has no number is missing. The paper is pointing at: S_∞ of 16 + 8 + 4 + … =. Circle that word — that is the missing variable.
Find: We are required to find: S_∞ of 16 + 8 + 4 + … =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: What we need to calculate: 32. Then substitute the given, one line at a time, to get S_∞ of 16 + 8 + 4 + … =.
Why this step: Do not name the letters yet. First hunt: what already has a number (Given), what has no number (Missing), the word the paper wants (Find), and which formula still contains the missing letter (Calculate).
03 / 08
Step B — Write the given in a column
Copy these onto the left of your page, one per line, with units: 16 · 8 · 4. Mixing cm with m, or minutes with seconds, loses a method mark.
Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
04 / 08
Step C — Name the method in words
32
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
05 / 08
Working
a=16, r=12. S=16(1−0.5)=32.
Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.
06 / 08
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
07 / 08
Step F — Box the answer with its unit
32
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
08 / 08
Step H — What the marker is looking for
On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
32
Copy onto your answer sheet — in this order
Answer: 32
6. Check it like a marker
Does your answer appear as one of the options? Do the units match what was asked (m, N, J, °)? If you put the number back into the original equation, does it still hold? Estimate: is the size sensible?
7. A trap first-timers fall into
First-timers often grab the first formula they remember, mix units (cm with m, minutes with seconds), or drop a minus sign. Write the formula in words first: “this equals that, because…”
10 · 4 Marks
f'(x)=0 for f(x)=x² − 6x + 5 at
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: f'(x)=0 for f(x)=x² − 6x + 5 at
Before any working — hunt first
Given
Numbers and units already in the sentence: 0 for · 6x · 5 at. Copy each one into a column. A name without a number is not given.
Missing
After you copy the given, the letter that still has no number is missing. The paper is pointing at: f'(x)=0 for f(x)=x² − 6x + 5 at. Circle that word — that is the missing variable.
Find
We are required to find: f'(x)=0 for f(x)=x² − 6x + 5 at. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: x = 3. Then substitute the given, one line at a time, to get f'(x)=0 for f(x)=x² − 6x + 5 at.
In other words — as if you have never seen this before
x=3
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 0 for · 6x · 5 at. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
x=3
4. Step-by-step working
01 / 08
Step A — Read the question twice
Slow down. Circle the word that names the unknown. Underline every number and its unit. Do not grab a formula yet — first decide what the paper is actually asking.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
02 / 08
Step A2 — Hunt: Given, Missing, Find, Calculate
Given: Numbers and units already in the sentence: 0 for · 6x · 5 at. Copy each one into a column. A name without a number is not given.
Missing: After you copy the given, the letter that still has no number is missing. The paper is pointing at: f'(x)=0 for f(x)=x² − 6x + 5 at. Circle that word — that is the missing variable.
Find: We are required to find: f'(x)=0 for f(x)=x² − 6x + 5 at. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: What we need to calculate: x = 3. Then substitute the given, one line at a time, to get f'(x)=0 for f(x)=x² − 6x + 5 at.
Why this step: Do not name the letters yet. First hunt: what already has a number (Given), what has no number (Missing), the word the paper wants (Find), and which formula still contains the missing letter (Calculate).
03 / 08
Step B — Write the given in a column
Copy these onto the left of your page, one per line, with units: 0 for · 6x · 5 at. Mixing cm with m, or minutes with seconds, loses a method mark.
Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
04 / 08
Step C — Name the method in words
x=3
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
05 / 08
Reason
f'=2x−6=0→x=3.
Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.
06 / 08
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
07 / 08
Step F — Box the answer with its unit
x=3
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
08 / 08
Step H — What the marker is looking for
On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
x=3
Copy onto your answer sheet — in this order
Answer: x=3
6. Check it like a marker
Does your answer appear as one of the options? Do the units match what was asked (m, N, J, °)? If you put the number back into the original equation, does it still hold? Estimate: is the size sensible?
7. A trap first-timers fall into
First-timers often grab the first formula they remember, mix units (cm with m, minutes with seconds), or drop a minus sign. Write the formula in words first: “this equals that, because…”
11 · 3 Marks
log₁₀ 100=
Notes: Base 10. Argument 100. Unknown: the exponent.
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: log₁₀ 100 =
Before any working — hunt first
Given
Numbers and units already in the sentence: 100. Copy each one into a column. A name without a number is not given.
Missing
After you copy the given, the letter that still has no number is missing. The paper is pointing at: log₁₀ 100 =. Circle that word — that is the missing variable.
Find
We are required to find: log₁₀ 100 =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
Write 10ˣ = 100 with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives log₁₀ 100 =.
In other words — as if you have never seen this before
log₁₀ 100 is the power you raise 10 to, to get 100. 10² = 100.
2. What do we already know?
Base 10. Argument 100. Unknown: the exponent.
3. Which method — and why this one?
log₁₀ 100 is the power you raise 10 to, to get 100. 10² = 100.
4. Step-by-step working
01 / 05
Rewrite as an exponential
log₁₀ 100=x⇔10ˣ=100.
10ˣ=100
Why this step: Write the formula before numbers. The memo awards a method mark for the law even if a digit slips later.
02 / 05
Recognise the power
100=10², so x=2.
x=2
Why this step: One job per line so you can find a slip, and so a marker can follow the marks.
03 / 05
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
04 / 05
Step F — Box the answer with its unit
2
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
05 / 05
Step H — What the marker is looking for
On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
2
Copy onto your answer sheet — in this order
Formula: 10ˣ=100
Answer: 2
6. Check it like a marker
10²=100. Yes.
7. A trap first-timers fall into
log₁₀ 100 is not 10, and not 1. Count the zeros of a power of ten: 100 has two zeros.
12 · 3 Marks · Regression & correlation
r=0.2 means
Formulas
ŷ=a+bx
Least-squares line
−1 ≤ r ≤ 1
Pearson’s r
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: r = 0.2 means
Before any working — hunt first
Given
Numbers and units already in the sentence: 0.2 means. Copy each one into a column. A name without a number is not given.
Missing
After you copy the given, the letter that still has no number is missing. The paper is pointing at: 2 means. Circle that word — that is the missing variable.
Find
We are required to find: 2 means. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: weak positive correlation. Then substitute the given, one line at a time, to get 2 means.
In other words — as if you have never seen this before
weak positive correlation
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 0.2 means. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
weak positive correlation
4. Step-by-step working
01 / 08
Step A — Read the question twice
Slow down. Circle the word that names the unknown. Underline every number and its unit. Do not grab a formula yet — first decide what the paper is actually asking.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
02 / 08
Step A2 — Hunt: Given, Missing, Find, Calculate
Given: Numbers and units already in the sentence: 0.2 means. Copy each one into a column. A name without a number is not given.
Missing: After you copy the given, the letter that still has no number is missing. The paper is pointing at: 2 means. Circle that word — that is the missing variable.
Find: We are required to find: 2 means. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: What we need to calculate: weak positive correlation. Then substitute the given, one line at a time, to get 2 means.
Why this step: Do not name the letters yet. First hunt: what already has a number (Given), what has no number (Missing), the word the paper wants (Find), and which formula still contains the missing letter (Calculate).
03 / 08
Step B — Write the given in a column
Copy these onto the left of your page, one per line, with units: 0.2 means. Mixing cm with m, or minutes with seconds, loses a method mark.
Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
04 / 08
Step C — Name the method in words
weak positive correlation
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
05 / 08
Reason
Close to 0, positive.
Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.
06 / 08
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
07 / 08
Step F — Box the answer with its unit
weak positive correlation
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
08 / 08
Step H — What the marker is looking for
On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
weak positive correlation
Copy onto your answer sheet — in this order
Answer: weak positive correlation
6. Check it like a marker
Does your answer appear as one of the options? Do the units match what was asked (m, N, J, °)? If you put the number back into the original equation, does it still hold? Estimate: is the size sensible?
7. A trap first-timers fall into
First-timers often grab the first formula they remember, mix units (cm with m, minutes with seconds), or drop a minus sign. Write the formula in words first: “this equals that, because…”
13 · 2 Marks
Present value of an annuity is used for
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: Present value of an annuity is used for
Before any working — hunt first
Given
Numbers and units already in the sentence: no numbers yet — only a name, a diagram, or a worded unknown. Copy each one into a column. A name without a number is not given.
Missing
After you copy the given, the letter that still has no number is missing. The paper is pointing at: Present value of an annuity is used for. Circle that word — that is the missing variable.
Find
We are required to find: Present value of an annuity is used for. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: P formula discounts future payments. Then substitute the given, one line at a time, to get Present value of an annuity is used for.
In other words — as if you have never seen this before
P formula discounts future payments.
2. What do we already know?
Copy these onto the left of your page, one per line, with units: every number and its unit in the sentence. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
P formula discounts future payments.
4. Step-by-step working
01 / 08
Step A — Read the question twice
Slow down. Circle the word that names the unknown. Underline every number and its unit. Do not grab a formula yet — first decide what the paper is actually asking.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
02 / 08
Step A2 — Hunt: Given, Missing, Find, Calculate
Given: Numbers and units already in the sentence: no numbers yet — only a name, a diagram, or a worded unknown. Copy each one into a column. A name without a number is not given.
Missing: After you copy the given, the letter that still has no number is missing. The paper is pointing at: Present value of an annuity is used for. Circle that word — that is the missing variable.
Find: We are required to find: Present value of an annuity is used for. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: What we need to calculate: P formula discounts future payments. Then substitute the given, one line at a time, to get Present value of an annuity is used for.
Why this step: Do not name the letters yet. First hunt: what already has a number (Given), what has no number (Missing), the word the paper wants (Find), and which formula still contains the missing letter (Calculate).
03 / 08
Step B — Write the given in a column
Copy these onto the left of your page, one per line, with units: every number and its unit in the sentence. Mixing cm with m, or minutes with seconds, loses a method mark.
Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
04 / 08
Step C — Name the method in words
P formula discounts future payments.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
05 / 08
Working
P formula discounts future payments.
Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.
06 / 08
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
07 / 08
Step F — Box the answer with its unit
loanshow much you still owe
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
08 / 08
Step H — What the marker is looking for
On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
loanshow much you still owe
Copy onto your answer sheet — in this order
Answer: loanshow much you still owe
6. Check it like a marker
Does your answer appear as one of the options? Do the units match what was asked (m, N, J, °)? If you put the number back into the original equation, does it still hold? Estimate: is the size sensible?
7. A trap first-timers fall into
First-timers often grab the first formula they remember, mix units (cm with m, minutes with seconds), or drop a minus sign. Write the formula in words first: “this equals that, because…”
14 · 3 Marks · Compound angles
sin 75°=
Compound angles
Formulas
sin(A + B)=sin A cos B+cos A sin B
Sine compound
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: sin 75° =
Before any working — hunt first
Given
Numbers and units already in the sentence: 75°. Copy each one into a column. A name without a number is not given.
Missing
After you copy the given, the letter that still has no number is missing. The paper is pointing at: sin 75° =. Circle that word — that is the missing variable.
Find
We are required to find: sin 75° =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: sin(45+30)=sin45cos30+cos45sin30. Then substitute the given, one line at a time, to get sin 75° =.
In other words — as if you have never seen this before
sin(45+30)=sin45cos30+cos45sin30.
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 75°. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
sin(45+30)=sin45cos30+cos45sin30.
4. Step-by-step working
01 / 08
Step A — Read the question twice
Slow down. Circle the word that names the unknown. Underline every number and its unit. Do not grab a formula yet — first decide what the paper is actually asking.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
02 / 08
Step A2 — Hunt: Given, Missing, Find, Calculate
Given: Numbers and units already in the sentence: 75°. Copy each one into a column. A name without a number is not given.
Missing: After you copy the given, the letter that still has no number is missing. The paper is pointing at: sin 75° =. Circle that word — that is the missing variable.
Find: We are required to find: sin 75° =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: What we need to calculate: sin(45+30)=sin45cos30+cos45sin30. Then substitute the given, one line at a time, to get sin 75° =.
Why this step: Do not name the letters yet. First hunt: what already has a number (Given), what has no number (Missing), the word the paper wants (Find), and which formula still contains the missing letter (Calculate).
03 / 08
Step B — Write the given in a column
Copy these onto the left of your page, one per line, with units: 75°. Mixing cm with m, or minutes with seconds, loses a method mark.
Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
04 / 08
Step C — Name the method in words
sin(45+30)=sin45cos30+cos45sin30.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
05 / 08
Working
sin(45+30)=sin45cos30+cos45sin30.
Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.
06 / 08
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
07 / 08
Step F — Box the answer with its unit
(√6+√2)4
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
08 / 08
Step H — What the marker is looking for
On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
(√6+√2)4
Copy onto your answer sheet — in this order
Answer: (√6+√2)4
6. Check it like a marker
Does your answer appear as one of the options? Do the units match what was asked (m, N, J, °)? If you put the number back into the original equation, does it still hold? Estimate: is the size sensible?
7. A trap first-timers fall into
First-timers often grab the first formula they remember, mix units (cm with m, minutes with seconds), or drop a minus sign. Write the formula in words first: “this equals that, because…”
15 · 3 Marks · Compound angles
sin(A − B) expands to
Compound angles
Formulas
sin(A + B)=sin A cos B+cos A sin B
Sine compound
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: sin(A − B) expands to
Before any working — hunt first
Given
Numbers and units already in the sentence: no numbers yet — only a name, a diagram, or a worded unknown. Copy each one into a column. A name without a number is not given.
Missing
After you copy the given, the letter that still has no number is missing. The paper is pointing at: sin(A − B) expands to. Circle that word — that is the missing variable.
Find
We are required to find: sin(A − B) expands to. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: sinA cosB − cosA sinB. Then substitute the given, one line at a time, to get sin(A − B) expands to.
In other words — as if you have never seen this before
sinA cosB − cosA sinB
2. What do we already know?
Copy these onto the left of your page, one per line, with units: every number and its unit in the sentence. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
sinA cosB − cosA sinB
4. Step-by-step working
01 / 08
Step A — Read the question twice
Slow down. Circle the word that names the unknown. Underline every number and its unit. Do not grab a formula yet — first decide what the paper is actually asking.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
02 / 08
Step A2 — Hunt: Given, Missing, Find, Calculate
Given: Numbers and units already in the sentence: no numbers yet — only a name, a diagram, or a worded unknown. Copy each one into a column. A name without a number is not given.
Missing: After you copy the given, the letter that still has no number is missing. The paper is pointing at: sin(A − B) expands to. Circle that word — that is the missing variable.
Find: We are required to find: sin(A − B) expands to. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: What we need to calculate: sinA cosB − cosA sinB. Then substitute the given, one line at a time, to get sin(A − B) expands to.
Why this step: Do not name the letters yet. First hunt: what already has a number (Given), what has no number (Missing), the word the paper wants (Find), and which formula still contains the missing letter (Calculate).
03 / 08
Step B — Write the given in a column
Copy these onto the left of your page, one per line, with units: every number and its unit in the sentence. Mixing cm with m, or minutes with seconds, loses a method mark.
Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
04 / 08
Step C — Name the method in words
sinA cosB − cosA sinB
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
05 / 08
Working
Compound-angle sine with a minus: the minus stays in the middle. Plus would be sin(A+B).
Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.
06 / 08
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
07 / 08
Step F — Box the answer with its unit
sinA cosB − cosA sinB
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
08 / 08
Step H — What the marker is looking for
On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
sinA cosB − cosA sinB
Copy onto your answer sheet — in this order
Answer: sinA cosB − cosA sinB
6. Check it like a marker
Does your answer appear as one of the options? Do the units match what was asked (m, N, J, °)? If you put the number back into the original equation, does it still hold? Estimate: is the size sensible?
7. A trap first-timers fall into
First-timers often grab the first formula they remember, mix units (cm with m, minutes with seconds), or drop a minus sign. Write the formula in words first: “this equals that, because…”
16 · 2 Marks
log₁₀ 0.01=
Notes: log base 10 of 0.01.
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: log₁₀ 0.01 =
Before any working — hunt first
Given
Numbers and units already in the sentence: 0.01. Copy each one into a column. A name without a number is not given.
Missing
After you copy the given, the letter that still has no number is missing. The paper is pointing at: 01 =. Circle that word — that is the missing variable.
Find
We are required to find: 01 =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
Write −2 with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives 01 =.
In other words — as if you have never seen this before
10x=0.01=10⁻², so x=−2.
2. What do we already know?
log base 10 of 0.01.
3. Which method — and why this one?
10x=0.01=10⁻², so x=−2.
4. Step-by-step working
01 / 08
Step A — Read the question twice
Slow down. Circle the word that names the unknown. Underline every number and its unit. Do not grab a formula yet — first decide what the paper is actually asking.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
02 / 08
Step A2 — Hunt: Given, Missing, Find, Calculate
Given: Numbers and units already in the sentence: 0.01. Copy each one into a column. A name without a number is not given.
Missing: After you copy the given, the letter that still has no number is missing. The paper is pointing at: 01 =. Circle that word — that is the missing variable.
Find: We are required to find: 01 =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: Write −2 with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives 01 =.
Why this step: Do not name the letters yet. First hunt: what already has a number (Given), what has no number (Missing), the word the paper wants (Find), and which formula still contains the missing letter (Calculate).
03 / 08
Step B — Write the given in a column
log base 10 of 0.01.
Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
04 / 08
Step C — Name the method in words
10x=0.01=10⁻², so x=−2.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
05 / 08
Rewrite
0.01=1100=10⁻².
−2
Why this step: Write the formula before numbers. The memo awards a method mark even if a digit slips later.
06 / 08
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
07 / 08
Step F — Box the answer with its unit
−2
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
08 / 08
Step H — What the marker is looking for
On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
−2
Copy onto your answer sheet — in this order
Formula: −2
Answer: −2
6. Check it like a marker
10⁻² is 0.01. Count two decimal places to the right of 1.
7. A trap first-timers fall into
+2 is log 100, not log 0.01.
17 · 2 Marks
log₁₀ 1000=
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: log₁₀ 1000 =
Before any working — hunt first
Given
Numbers and units already in the sentence: 1000. Copy each one into a column. A name without a number is not given.
Missing
After you copy the given, the letter that still has no number is missing. The paper is pointing at: log₁₀ 1000 =. Circle that word — that is the missing variable.
Find
We are required to find: log₁₀ 1000 =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: 10³ = 1000. Then substitute the given, one line at a time, to get log₁₀ 1000 =.
In other words — as if you have never seen this before
10³=1000.
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 1000. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
10³=1000.
4. Step-by-step working
01 / 08
Step A — Read the question twice
Slow down. Circle the word that names the unknown. Underline every number and its unit. Do not grab a formula yet — first decide what the paper is actually asking.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
02 / 08
Step A2 — Hunt: Given, Missing, Find, Calculate
Given: Numbers and units already in the sentence: 1000. Copy each one into a column. A name without a number is not given.
Missing: After you copy the given, the letter that still has no number is missing. The paper is pointing at: log₁₀ 1000 =. Circle that word — that is the missing variable.
Find: We are required to find: log₁₀ 1000 =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: What we need to calculate: 10³ = 1000. Then substitute the given, one line at a time, to get log₁₀ 1000 =.
Why this step: Do not name the letters yet. First hunt: what already has a number (Given), what has no number (Missing), the word the paper wants (Find), and which formula still contains the missing letter (Calculate).
03 / 08
Step B — Write the given in a column
Copy these onto the left of your page, one per line, with units: 1000. Mixing cm with m, or minutes with seconds, loses a method mark.
Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
04 / 08
Step C — Name the method in words
10³=1000.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
05 / 08
Working
10³=1000.
Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.
06 / 08
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
07 / 08
Step F — Box the answer with its unit
3
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
08 / 08
Step H — What the marker is looking for
On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
3
Copy onto your answer sheet — in this order
Answer: 3
6. Check it like a marker
Does your answer appear as one of the options? Do the units match what was asked (m, N, J, °)? If you put the number back into the original equation, does it still hold? Estimate: is the size sensible?
7. A trap first-timers fall into
First-timers often grab the first formula they remember, mix units (cm with m, minutes with seconds), or drop a minus sign. Write the formula in words first: “this equals that, because…”
18 · 2 Marks · First principles of differentiation
f(x)=x²−6x. Stationary x=
Formulas
f'(x)=limh→0[f(x+h) − f(x)]h
Definition of the derivative
Notes: f(x)=x²−6x. Unknown: x where f'=0.
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: f(x)=x²−6x. Stationary x =
Before any working — hunt first
Given
Numbers and units already in the sentence: 6x. Copy each one into a column. A name without a number is not given.
Missing
After you copy the given, the letter that still has no number is missing. The paper is pointing at: Stationary x =. Circle that word — that is the missing variable.
Find
We are required to find: Stationary x =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
Write f' = 2x−6 with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives Stationary x =.
In other words — as if you have never seen this before
Stationary ⇒ f'(x) = 0. Differentiate first.
2. What do we already know?
f(x)=x²−6x. Unknown: x where f'=0.
3. Which method — and why this one?
Stationary ⇒ f'(x) = 0. Differentiate first.
4. Step-by-step working
01 / 05
Differentiate
f'=2x−6.
f'=2x−6
Why this step: Write the formula before numbers. The memo awards a method mark even if a digit slips later.
02 / 05
Set to zero
2x−6=0→x=3.
x=3
Why this step: One job per line so you can find a slip, and so a marker can follow the marks.
03 / 05
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
04 / 05
Step F — Box the answer with its unit
3
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
05 / 05
Step H — What the marker is looking for
On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
3
Copy onto your answer sheet — in this order
Formula: f'=2x−6
Answer: 3
6. Check it like a marker
f''=2 > 0, a minimum. f(3)=9−18=−9.
7. A trap first-timers fall into
x = 6 is where f(x)=0 (a root), not where the gradient is zero.
19 · 3 Marks
log₃ 81=
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: log₃ 81 =
Before any working — hunt first
Given
Numbers and units already in the sentence: 81. Copy each one into a column. A name without a number is not given.
Missing
After you copy the given, the letter that still has no number is missing. The paper is pointing at: log₃ 81 =. Circle that word — that is the missing variable.
Find
We are required to find: log₃ 81 =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: 4. Then substitute the given, one line at a time, to get log₃ 81 =.
In other words — as if you have never seen this before
4
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 81. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
4
4. Step-by-step working
01 / 08
Step A — Read the question twice
Slow down. Circle the word that names the unknown. Underline every number and its unit. Do not grab a formula yet — first decide what the paper is actually asking.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
02 / 08
Step A2 — Hunt: Given, Missing, Find, Calculate
Given: Numbers and units already in the sentence: 81. Copy each one into a column. A name without a number is not given.
Missing: After you copy the given, the letter that still has no number is missing. The paper is pointing at: log₃ 81 =. Circle that word — that is the missing variable.
Find: We are required to find: log₃ 81 =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: What we need to calculate: 4. Then substitute the given, one line at a time, to get log₃ 81 =.
Why this step: Do not name the letters yet. First hunt: what already has a number (Given), what has no number (Missing), the word the paper wants (Find), and which formula still contains the missing letter (Calculate).
03 / 08
Step B — Write the given in a column
Copy these onto the left of your page, one per line, with units: 81. Mixing cm with m, or minutes with seconds, loses a method mark.
Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
04 / 08
Step C — Name the method in words
4
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
05 / 08
Working
3⁴ = 81, so the log is 4. Ask: “3 to the what equals 81?”
Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.
06 / 08
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
07 / 08
Step F — Box the answer with its unit
4
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
08 / 08
Step H — What the marker is looking for
On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
4
Copy onto your answer sheet — in this order
Answer: 4
6. Check it like a marker
Does your answer appear as one of the options? Do the units match what was asked (m, N, J, °)? If you put the number back into the original equation, does it still hold? Estimate: is the size sensible?
7. A trap first-timers fall into
First-timers often grab the first formula they remember, mix units (cm with m, minutes with seconds), or drop a minus sign. Write the formula in words first: “this equals that, because…”
20 · 3 Marks
Angle of elevation is measured from the
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: Angle of elevation is measured from the
Before any working — hunt first
Given
Numbers and units already in the sentence: no numbers yet — only a name, a diagram, or a worded unknown. Copy each one into a column. A name without a number is not given.
Missing
After you copy the given, the letter that still has no number is missing. The paper is pointing at: Angle of elevation is measured from the. Circle that word — that is the missing variable.
Find
We are required to find: Angle of elevation is measured from the. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: Elevation from horizontal. Then substitute the given, one line at a time, to get Angle of elevation is measured from the.
In other words — as if you have never seen this before
Elevation from horizontal.
2. What do we already know?
Copy these onto the left of your page, one per line, with units: every number and its unit in the sentence. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
Elevation from horizontal.
4. Step-by-step working
01 / 08
Step A — Read the question twice
Slow down. Circle the word that names the unknown. Underline every number and its unit. Do not grab a formula yet — first decide what the paper is actually asking.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
02 / 08
Step A2 — Hunt: Given, Missing, Find, Calculate
Given: Numbers and units already in the sentence: no numbers yet — only a name, a diagram, or a worded unknown. Copy each one into a column. A name without a number is not given.
Missing: After you copy the given, the letter that still has no number is missing. The paper is pointing at: Angle of elevation is measured from the. Circle that word — that is the missing variable.
Find: We are required to find: Angle of elevation is measured from the. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: What we need to calculate: Elevation from horizontal. Then substitute the given, one line at a time, to get Angle of elevation is measured from the.
Why this step: Do not name the letters yet. First hunt: what already has a number (Given), what has no number (Missing), the word the paper wants (Find), and which formula still contains the missing letter (Calculate).
03 / 08
Step B — Write the given in a column
Copy these onto the left of your page, one per line, with units: every number and its unit in the sentence. Mixing cm with m, or minutes with seconds, loses a method mark.
Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
04 / 08
Step C — Name the method in words
Elevation from horizontal.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
05 / 08
Working
Elevation from horizontal.
Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.
06 / 08
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
07 / 08
Step F — Box the answer with its unit
horizontal up to the line of sight
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
08 / 08
Step H — What the marker is looking for
On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
horizontal up to the line of sight
Copy onto your answer sheet — in this order
Answer: horizontal up to the line of sight
6. Check it like a marker
Does your answer appear as one of the options? Do the units match what was asked (m, N, J, °)? If you put the number back into the original equation, does it still hold? Estimate: is the size sensible?
7. A trap first-timers fall into
First-timers often grab the first formula they remember, mix units (cm with m, minutes with seconds), or drop a minus sign. Write the formula in words first: “this equals that, because…”
21 · 3 Marks
S_∞ of 2 + 4 + 8 + 16 + … is
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: S_∞ of 2 + 4 + 8 + 16 + … is
Before any working — hunt first
Given
Numbers and units already in the sentence: 2 · 4 · 8 · 16. Copy each one into a column. A name without a number is not given.
Missing
After you copy the given, the letter that still has no number is missing. The paper is pointing at: S_∞ of 2 + 4 + 8 + 16 + … is. Circle that word — that is the missing variable.
Find
We are required to find: S_∞ of 2 + 4 + 8 + 16 + … is. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: r = 2, |r| ≥ 1. Then substitute the given, one line at a time, to get S_∞ of 2 + 4 + 8 + 16 + … is.
In other words — as if you have never seen this before
r=2, |r| ≥ 1. Do not write 2(1−2)=−2.
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 2 · 4 · 8 · 16. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
r=2, |r| ≥ 1. Do not write 2(1−2)=−2.
4. Step-by-step working
01 / 08
Step A — Read the question twice
Slow down. Circle the word that names the unknown. Underline every number and its unit. Do not grab a formula yet — first decide what the paper is actually asking.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
02 / 08
Step A2 — Hunt: Given, Missing, Find, Calculate
Given: Numbers and units already in the sentence: 2 · 4 · 8 · 16. Copy each one into a column. A name without a number is not given.
Missing: After you copy the given, the letter that still has no number is missing. The paper is pointing at: S_∞ of 2 + 4 + 8 + 16 + … is. Circle that word — that is the missing variable.
Find: We are required to find: S_∞ of 2 + 4 + 8 + 16 + … is. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: What we need to calculate: r = 2, |r| ≥ 1. Then substitute the given, one line at a time, to get S_∞ of 2 + 4 + 8 + 16 + … is.
Why this step: Do not name the letters yet. First hunt: what already has a number (Given), what has no number (Missing), the word the paper wants (Find), and which formula still contains the missing letter (Calculate).
03 / 08
Step B — Write the given in a column
Copy these onto the left of your page, one per line, with units: 2 · 4 · 8 · 16. Mixing cm with m, or minutes with seconds, loses a method mark.
Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
04 / 08
Step C — Name the method in words
r=2, |r| ≥ 1. Do not write 2(1−2)=−2.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
05 / 08
Working
r=2, |r| ≥ 1. Do not write 2(1−2)=−2.
Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.
06 / 08
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
07 / 08
Step F — Box the answer with its unit
does not exist (diverges)
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
08 / 08
Step H — What the marker is looking for
On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
does not exist (diverges)
Copy onto your answer sheet — in this order
Answer: does not exist (diverges)
6. Check it like a marker
Does your answer appear as one of the options? Do the units match what was asked (m, N, J, °)? If you put the number back into the original equation, does it still hold? Estimate: is the size sensible?
7. A trap first-timers fall into
First-timers often grab the first formula they remember, mix units (cm with m, minutes with seconds), or drop a minus sign. Write the formula in words first: “this equals that, because…”
22 · 3 Marks · Compound angles
ddx not on Paper 2. Which topic is Paper 2?
Compound angles
Formulas
sin(A + B)=sin A cos B+cos A sin B
Sine compound
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: ddx not on Paper 2. Which topic is Paper 2?
Before any working — hunt first
Given
Numbers and units already in the sentence: 2 · 2. Copy each one into a column. A name without a number is not given.
Missing
After you copy the given, the letter that still has no number is missing. The paper is pointing at: Which topic is Paper 2. Circle that word — that is the missing variable.
Find
We are required to find: Which topic is Paper 2. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: Trigonometry. Then substitute the given, one line at a time, to get Which topic is Paper 2.
In other words — as if you have never seen this before
Trigonometry
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 2 · 2. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
Trigonometry
4. Step-by-step working
01 / 08
Step A — Read the question twice
Slow down. Circle the word that names the unknown. Underline every number and its unit. Do not grab a formula yet — first decide what the paper is actually asking.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
02 / 08
Step A2 — Hunt: Given, Missing, Find, Calculate
Given: Numbers and units already in the sentence: 2 · 2. Copy each one into a column. A name without a number is not given.
Missing: After you copy the given, the letter that still has no number is missing. The paper is pointing at: Which topic is Paper 2. Circle that word — that is the missing variable.
Find: We are required to find: Which topic is Paper 2. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: What we need to calculate: Trigonometry. Then substitute the given, one line at a time, to get Which topic is Paper 2.
Why this step: Do not name the letters yet. First hunt: what already has a number (Given), what has no number (Missing), the word the paper wants (Find), and which formula still contains the missing letter (Calculate).
03 / 08
Step B — Write the given in a column
Copy these onto the left of your page, one per line, with units: 2 · 2. Mixing cm with m, or minutes with seconds, loses a method mark.
Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
04 / 08
Step C — Name the method in words
Trigonometry
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
05 / 08
Reason
Calculus and finance sit on Paper 1.
Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.
06 / 08
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
07 / 08
Step F — Box the answer with its unit
Trigonometry
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
08 / 08
Step H — What the marker is looking for
On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
Trigonometry
Copy onto your answer sheet — in this order
Answer: Trigonometry
6. Check it like a marker
Does your answer appear as one of the options? Do the units match what was asked (m, N, J, °)? If you put the number back into the original equation, does it still hold? Estimate: is the size sensible?
7. A trap first-timers fall into
First-timers often grab the first formula they remember, mix units (cm with m, minutes with seconds), or drop a minus sign. Write the formula in words first: “this equals that, because…”
23 · 4 Marks · Annuities & outstanding balance
Outstanding balance after k payments uses P on the remaining n−k periods
Annuities & outstanding balance
Formulas
F=x[((1+i)ⁿ − 1)/i]
Future value of an annuity
P=x[1 − (1+i)⁻ⁿ]i
Present value of an annuity
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: Outstanding balance after k payments uses P on the remaining n−k periods
In other words — as if you have never seen this before
Present-value of remaining instalments.
2. What do we already know?
Copy these onto the left of your page, one per line, with units: every number and its unit in the sentence. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
Present-value of remaining instalments.
4. Step-by-step working
01 / 07
Step A — Read the question twice
Slow down. Circle the word that names the unknown. Underline every number and its unit. Do not grab a formula yet — first decide what the paper is actually asking.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
02 / 07
Step B — Write the given in a column
Copy these onto the left of your page, one per line, with units: every number and its unit in the sentence. Mixing cm with m, or minutes with seconds, loses a method mark.
Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
03 / 07
Step C — Name the method in words
Present-value of remaining instalments.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
04 / 07
Working
Present-value of remaining instalments.
Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.
05 / 07
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
06 / 07
Step F — Box the answer with its unit
true
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
07 / 07
Step H — What the marker is looking for
On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
true
Copy onto your answer sheet — in this order
Answer: true
6. Check it like a marker
Does your answer appear as one of the options? Do the units match what was asked (m, N, J, °)? If you put the number back into the original equation, does it still hold? Estimate: is the size sensible?
7. A trap first-timers fall into
First-timers often grab the first formula they remember, mix units (cm with m, minutes with seconds), or drop a minus sign. Write the formula in words first: “this equals that, because…”
24 · 3 Marks
S_∞ of 9 + 3 + 1 + … =
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: S_∞ of 9 + 3 + 1 + … =
Before any working — hunt first
Given
Numbers and units already in the sentence: 9 · 3 · 1. Copy each one into a column. A name without a number is not given.
Missing
After you copy the given, the letter that still has no number is missing. The paper is pointing at: S_∞ of 9 + 3 + 1 + … =. Circle that word — that is the missing variable.
Find
We are required to find: S_∞ of 9 + 3 + 1 + … =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: a=9, r=13. Then substitute the given, one line at a time, to get S_∞ of 9 + 3 + 1 + … =.
In other words — as if you have never seen this before
a=9, r=13. S=a(1−r)=9(2/3)=13.5.
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 9 · 3 · 1. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
a=9, r=13. S=a(1−r)=9(2/3)=13.5.
4. Step-by-step working
01 / 08
Step A — Read the question twice
Slow down. Circle the word that names the unknown. Underline every number and its unit. Do not grab a formula yet — first decide what the paper is actually asking.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
02 / 08
Step A2 — Hunt: Given, Missing, Find, Calculate
Given: Numbers and units already in the sentence: 9 · 3 · 1. Copy each one into a column. A name without a number is not given.
Missing: After you copy the given, the letter that still has no number is missing. The paper is pointing at: S_∞ of 9 + 3 + 1 + … =. Circle that word — that is the missing variable.
Find: We are required to find: S_∞ of 9 + 3 + 1 + … =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: What we need to calculate: a=9, r=13. Then substitute the given, one line at a time, to get S_∞ of 9 + 3 + 1 + … =.
Why this step: Do not name the letters yet. First hunt: what already has a number (Given), what has no number (Missing), the word the paper wants (Find), and which formula still contains the missing letter (Calculate).
03 / 08
Step B — Write the given in a column
Copy these onto the left of your page, one per line, with units: 9 · 3 · 1. Mixing cm with m, or minutes with seconds, loses a method mark.
Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
04 / 08
Step C — Name the method in words
a=9, r=13. S=a(1−r)=9(2/3)=13.5.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
05 / 08
Working
a=9, r=13. S=a(1−r)=9(2/3)=13.5.
Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.
06 / 08
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
07 / 08
Step F — Box the answer with its unit
13.5
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
08 / 08
Step H — What the marker is looking for
On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
13.5
Copy onto your answer sheet — in this order
Answer: 13.5
6. Check it like a marker
Does your answer appear as one of the options? Do the units match what was asked (m, N, J, °)? If you put the number back into the original equation, does it still hold? Estimate: is the size sensible?
7. A trap first-timers fall into
First-timers often grab the first formula they remember, mix units (cm with m, minutes with seconds), or drop a minus sign. Write the formula in words first: “this equals that, because…”
25 · 4 Marks · Area, sine and cosine rules
Area of Δ with a=6, b=10, included angle 90° is
Formulas
Area=½ ab sin C
Area rule
c²=a²+b²−2ab cos C
Cosine rule
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: Area of Δ with a=6, b=10, included angle 90° is
In other words — as if you have never seen this before
½ab sin90=½×6×10×1=30.
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 6 · 10 · 90°. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
½ab sin90=½×6×10×1=30.
4. Step-by-step working
01 / 07
Step A — Read the question twice
Slow down. Circle the word that names the unknown. Underline every number and its unit. Do not grab a formula yet — first decide what the paper is actually asking.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
02 / 07
Step B — Write the given in a column
Copy these onto the left of your page, one per line, with units: 6 · 10 · 90°. Mixing cm with m, or minutes with seconds, loses a method mark.
Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
03 / 07
Step C — Name the method in words
½ab sin90=½×6×10×1=30.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
04 / 07
Working
½ab sin90=½×6×10×1=30.
Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.
05 / 07
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
06 / 07
Step F — Box the answer with its unit
30
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
07 / 07
Step H — What the marker is looking for
On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
30
Copy onto your answer sheet — in this order
Answer: 30
6. Check it like a marker
Does your answer appear as one of the options? Do the units match what was asked (m, N, J, °)? If you put the number back into the original equation, does it still hold? Estimate: is the size sensible?
7. A trap first-timers fall into
First-timers often grab the first formula they remember, mix units (cm with m, minutes with seconds), or drop a minus sign. Write the formula in words first: “this equals that, because…”
26 · 2 Marks · First principles of differentiation
f(x)=x². From first principles f'(x)=
Formulas
f'(x)=limh→0[f(x+h) − f(x)]h
Definition of the derivative
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: f(x)=x². From first principles f'(x) =
Before any working — hunt first
Given
Numbers and units already in the sentence: no numbers yet — only a name, a diagram, or a worded unknown. Copy each one into a column. A name without a number is not given.
Missing
After you copy the given, the letter that still has no number is missing. The paper is pointing at: From first principles f'(x) =. Circle that word — that is the missing variable.
Find
We are required to find: From first principles f'(x) =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: [2xh + h²]h. Then substitute the given, one line at a time, to get From first principles f'(x) =.
In other words — as if you have never seen this before
[2xh + h²]h→2x as h→0.
2. What do we already know?
Copy these onto the left of your page, one per line, with units: every number and its unit in the sentence. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
[2xh + h²]h→2x as h→0.
4. Step-by-step working
01 / 08
Step A — Read the question twice
Slow down. Circle the word that names the unknown. Underline every number and its unit. Do not grab a formula yet — first decide what the paper is actually asking.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
02 / 08
Step A2 — Hunt: Given, Missing, Find, Calculate
Given: Numbers and units already in the sentence: no numbers yet — only a name, a diagram, or a worded unknown. Copy each one into a column. A name without a number is not given.
Missing: After you copy the given, the letter that still has no number is missing. The paper is pointing at: From first principles f'(x) =. Circle that word — that is the missing variable.
Find: We are required to find: From first principles f'(x) =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: What we need to calculate: [2xh + h²]h. Then substitute the given, one line at a time, to get From first principles f'(x) =.
Why this step: Do not name the letters yet. First hunt: what already has a number (Given), what has no number (Missing), the word the paper wants (Find), and which formula still contains the missing letter (Calculate).
03 / 08
Step B — Write the given in a column
Copy these onto the left of your page, one per line, with units: every number and its unit in the sentence. Mixing cm with m, or minutes with seconds, loses a method mark.
Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
04 / 08
Step C — Name the method in words
[2xh + h²]h→2x as h→0.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
05 / 08
Working
[2xh + h²]h→2x as h→0.
Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.
06 / 08
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
07 / 08
Step F — Box the answer with its unit
2x
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
08 / 08
Step H — What the marker is looking for
On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
2x
Copy onto your answer sheet — in this order
Answer: 2x
6. Check it like a marker
Does your answer appear as one of the options? Do the units match what was asked (m, N, J, °)? If you put the number back into the original equation, does it still hold? Estimate: is the size sensible?
7. A trap first-timers fall into
First-timers often grab the first formula they remember, mix units (cm with m, minutes with seconds), or drop a minus sign. Write the formula in words first: “this equals that, because…”
27 · 3 Marks · Compound angles
tan θ = m. If m is undefined the line is
Compound angles
Formulas
sin(A + B)=sin A cos B+cos A sin B
Sine compound
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: tan θ = m. If m is undefined the line is
Before any working — hunt first
Given
Numbers and units already in the sentence: no numbers yet — only a name, a diagram, or a worded unknown. Copy each one into a column. A name without a number is not given.
Missing
After you copy the given, the letter that still has no number is missing. The paper is pointing at: If m is undefined the line is. Circle that word — that is the missing variable.
Find
We are required to find: If m is undefined the line is. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: vertical. Then substitute the given, one line at a time, to get If m is undefined the line is.
In other words — as if you have never seen this before
vertical
2. What do we already know?
Copy these onto the left of your page, one per line, with units: every number and its unit in the sentence. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
vertical
4. Step-by-step working
01 / 08
Step A — Read the question twice
Slow down. Circle the word that names the unknown. Underline every number and its unit. Do not grab a formula yet — first decide what the paper is actually asking.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
02 / 08
Step A2 — Hunt: Given, Missing, Find, Calculate
Given: Numbers and units already in the sentence: no numbers yet — only a name, a diagram, or a worded unknown. Copy each one into a column. A name without a number is not given.
Missing: After you copy the given, the letter that still has no number is missing. The paper is pointing at: If m is undefined the line is. Circle that word — that is the missing variable.
Find: We are required to find: If m is undefined the line is. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: What we need to calculate: vertical. Then substitute the given, one line at a time, to get If m is undefined the line is.
Why this step: Do not name the letters yet. First hunt: what already has a number (Given), what has no number (Missing), the word the paper wants (Find), and which formula still contains the missing letter (Calculate).
03 / 08
Step B — Write the given in a column
Copy these onto the left of your page, one per line, with units: every number and its unit in the sentence. Mixing cm with m, or minutes with seconds, loses a method mark.
Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
04 / 08
Step C — Name the method in words
vertical
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
05 / 08
Reason
No defined gradient.
Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.
06 / 08
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
07 / 08
Step F — Box the answer with its unit
vertical
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
08 / 08
Step H — What the marker is looking for
On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
vertical
Copy onto your answer sheet — in this order
Answer: vertical
6. Check it like a marker
Does your answer appear as one of the options? Do the units match what was asked (m, N, J, °)? If you put the number back into the original equation, does it still hold? Estimate: is the size sensible?
7. A trap first-timers fall into
First-timers often grab the first formula they remember, mix units (cm with m, minutes with seconds), or drop a minus sign. Write the formula in words first: “this equals that, because…”
28 · 3 Marks · Area, sine and cosine rules
If ΔABC ∼ ΔDEF with ratio 2:3, area ratio is
Formulas
Area=½ ab sin C
Area rule
c²=a²+b²−2ab cos C
Cosine rule
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: If ΔABC ∼ ΔDEF with ratio 2:3, area ratio is
In other words — as if you have never seen this before
Areas scale with the square of corresponding sides.
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 2 · 3. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
Areas scale with the square of corresponding sides.
4. Step-by-step working
01 / 07
Step A — Read the question twice
Slow down. Circle the word that names the unknown. Underline every number and its unit. Do not grab a formula yet — first decide what the paper is actually asking.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
02 / 07
Step B — Write the given in a column
Copy these onto the left of your page, one per line, with units: 2 · 3. Mixing cm with m, or minutes with seconds, loses a method mark.
Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
03 / 07
Step C — Name the method in words
Areas scale with the square of corresponding sides.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
04 / 07
Working
Areas scale with the square of corresponding sides.
Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.
05 / 07
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
06 / 07
Step F — Box the answer with its unit
4:9
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
07 / 07
Step H — What the marker is looking for
On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
4:9
Copy onto your answer sheet — in this order
Answer: 4:9
6. Check it like a marker
Does your answer appear as one of the options? Do the units match what was asked (m, N, J, °)? If you put the number back into the original equation, does it still hold? Estimate: is the size sensible?
7. A trap first-timers fall into
First-timers often grab the first formula they remember, mix units (cm with m, minutes with seconds), or drop a minus sign. Write the formula in words first: “this equals that, because…”
29 · 3 Marks · Similarity & proportion
AA similarity requires
Similarity & proportion
Formulas
AA similarity ⇒ corresponding sides proportional
AA similarity
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: AA similarity requires
Before any working — hunt first
Given
Numbers and units already in the sentence: no numbers yet — only a name, a diagram, or a worded unknown. Copy each one into a column. A name without a number is not given.
Missing
After you copy the given, the letter that still has no number is missing. The paper is pointing at: AA similarity requires. Circle that word — that is the missing variable.
Find
We are required to find: AA similarity requires. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: two equal corresponding angles. Then substitute the given, one line at a time, to get AA similarity requires.
In other words — as if you have never seen this before
two equal corresponding angles
2. What do we already know?
Copy these onto the left of your page, one per line, with units: every number and its unit in the sentence. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
two equal corresponding angles
4. Step-by-step working
01 / 08
Step A — Read the question twice
Slow down. Circle the word that names the unknown. Underline every number and its unit. Do not grab a formula yet — first decide what the paper is actually asking.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
02 / 08
Step A2 — Hunt: Given, Missing, Find, Calculate
Given: Numbers and units already in the sentence: no numbers yet — only a name, a diagram, or a worded unknown. Copy each one into a column. A name without a number is not given.
Missing: After you copy the given, the letter that still has no number is missing. The paper is pointing at: AA similarity requires. Circle that word — that is the missing variable.
Find: We are required to find: AA similarity requires. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: What we need to calculate: two equal corresponding angles. Then substitute the given, one line at a time, to get AA similarity requires.
Why this step: Do not name the letters yet. First hunt: what already has a number (Given), what has no number (Missing), the word the paper wants (Find), and which formula still contains the missing letter (Calculate).
03 / 08
Step B — Write the given in a column
Copy these onto the left of your page, one per line, with units: every number and its unit in the sentence. Mixing cm with m, or minutes with seconds, loses a method mark.
Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
04 / 08
Step C — Name the method in words
two equal corresponding angles
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
05 / 08
Reason
AA is sufficient.
Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.
06 / 08
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
07 / 08
Step F — Box the answer with its unit
two equal corresponding angles
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
08 / 08
Step H — What the marker is looking for
On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
two equal corresponding angles
Copy onto your answer sheet — in this order
Answer: two equal corresponding angles
6. Check it like a marker
Does your answer appear as one of the options? Do the units match what was asked (m, N, J, °)? If you put the number back into the original equation, does it still hold? Estimate: is the size sensible?
7. A trap first-timers fall into
First-timers often grab the first formula they remember, mix units (cm with m, minutes with seconds), or drop a minus sign. Write the formula in words first: “this equals that, because…”
30 · 2 Marks · Compound angles
sin 15° in surd form uses
Compound angles
Formulas
sin(A + B)=sin A cos B+cos A sin B
Sine compound
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: sin 15° in surd form uses
Before any working — hunt first
Given
Numbers and units already in the sentence: 15°. Copy each one into a column. A name without a number is not given.
Missing
After you copy the given, the letter that still has no number is missing. The paper is pointing at: sin 15° in surd form uses. Circle that word — that is the missing variable.
Find
We are required to find: sin 15° in surd form uses. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: sin(45−30). Then substitute the given, one line at a time, to get sin 15° in surd form uses.
In other words — as if you have never seen this before
sin(45−30)
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 15°. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
sin(45−30)
4. Step-by-step working
01 / 08
Step A — Read the question twice
Slow down. Circle the word that names the unknown. Underline every number and its unit. Do not grab a formula yet — first decide what the paper is actually asking.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
02 / 08
Step A2 — Hunt: Given, Missing, Find, Calculate
Given: Numbers and units already in the sentence: 15°. Copy each one into a column. A name without a number is not given.
Missing: After you copy the given, the letter that still has no number is missing. The paper is pointing at: sin 15° in surd form uses. Circle that word — that is the missing variable.
Find: We are required to find: sin 15° in surd form uses. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: What we need to calculate: sin(45−30). Then substitute the given, one line at a time, to get sin 15° in surd form uses.
Why this step: Do not name the letters yet. First hunt: what already has a number (Given), what has no number (Missing), the word the paper wants (Find), and which formula still contains the missing letter (Calculate).
03 / 08
Step B — Write the given in a column
Copy these onto the left of your page, one per line, with units: 15°. Mixing cm with m, or minutes with seconds, loses a method mark.
Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
04 / 08
Step C — Name the method in words
sin(45−30)
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
05 / 08
Reason
Compound angle.
Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.
06 / 08
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
07 / 08
Step F — Box the answer with its unit
sin(45−30)
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
08 / 08
Step H — What the marker is looking for
On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
sin(45−30)
Copy onto your answer sheet — in this order
Answer: sin(45−30)
6. Check it like a marker
Does your answer appear as one of the options? Do the units match what was asked (m, N, J, °)? If you put the number back into the original equation, does it still hold? Estimate: is the size sensible?
7. A trap first-timers fall into
First-timers often grab the first formula they remember, mix units (cm with m, minutes with seconds), or drop a minus sign. Write the formula in words first: “this equals that, because…”