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: y = −x² + 4. Maximum y =
Before any working — hunt first
Given
Numbers and units already in the sentence: 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: Maximum y =. Circle that word — that is the missing variable.
Find
We are required to find: Maximum y =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
Write y = 4 with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives Maximum y =.
In other words — as if you have never seen this before
Coefficient of x² is negative, so a maximum. It is the +4.
2. What do we already know?
A downward parabola. Vertex on the y-axis.
3. Which method — and why this one?
Coefficient of x² is negative, so a maximum. It is the +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: 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: Maximum y =. Circle that word — that is the missing variable.
Find: We are required to find: Maximum y =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: Write y = 4 with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives Maximum y =.
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
A downward parabola. Vertex on the y-axis.
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
Coefficient of x² is negative, so a maximum. It is the +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
Read q
Completed square already: max = 4 at x = 0.
y=4
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
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
Formula: y=4
Answer: 4
6. Check it like a marker
At x=0, y=4. At x=1, y=3, lower.
7. A trap first-timers fall into
−4 would be if the square were added with a plus in front of x².
1.2 · 10 Marks
5C2=
Notes: Combinations, order does not count. n=5, r=2.
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: 5C2 =
Before any working — hunt first
Given
Numbers and units already in the sentence: 5C · 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: 5C2 =. Circle that word — that is the missing variable.
Find
We are required to find: 5C2 =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
Write 10 with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives 5C2 =.
In other words — as if you have never seen this before
nCr=n!(r!(n−r)!). 5×42=10.
2. What do we already know?
Combinations, order does not count. n=5, r=2.
3. Which method — and why this one?
nCr=n!(r!(n−r)!). 5×42=10.
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: 5C · 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: 5C2 =. Circle that word — that is the missing variable.
Find: We are required to find: 5C2 =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: Write 10 with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives 5C2 =.
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
Combinations, order does not count. n=5, r=2.
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
nCr=n!(r!(n−r)!). 5×42=10.
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
Formula
5C2=(5×4)(2×1)=10.
10
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
10
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
10
Copy onto your answer sheet — in this order
Formula: 10
Answer: 10
6. Check it like a marker
5P2 = 20. Combinations are permutations divided by r!.
7. A trap first-timers fall into
20 is 5P2. The C means we do not care about order.
2.1 · 10 Marks · Inclination & circles
Equation of a circle centre (0;0), r=5
Formulas
tan θ=m
Inclination
(x − a)²+(y − b)²=r²
Circle centre (a;b)
Notes: Centre origin. r = 5 so r² = 25.
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: Equation of a circle centre (0;0), r=5
Before any working — hunt first
Given
Numbers and units already in the sentence: 0 · 0 · 5. 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: Equation of a circle centre (0;0), r=5. Circle that word — that is the missing variable.
Find
We are required to find: Equation of a circle centre (0;0), r=5. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
Write x²+y²=25 with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives Equation of a circle centre (0;0), r=5.
In other words — as if you have never seen this before
x²+y²=r².
2. What do we already know?
Centre origin. r = 5 so r² = 25.
3. Which method — and why this one?
x²+y²=r².
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 · 0 · 5. 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: Equation of a circle centre (0;0), r=5. Circle that word — that is the missing variable.
Find: We are required to find: Equation of a circle centre (0;0), r=5. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: Write x²+y²=25 with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives Equation of a circle centre (0;0), r=5.
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
Centre origin. r = 5 so r² = 25.
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
x²+y²=r².
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
Square the radius
r²=25. Write x²+y²=25.
x²+y²=25
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
x²+y²=25
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
x²+y²=25
Copy onto your answer sheet — in this order
Formula: x²+y²=25
Answer: x²+y²=25
6. Check it like a marker
Point (3;4): 9+16=25, on the circle. A 3-4-5 radius.
7. A trap first-timers fall into
= 5 forgets to square r. The formula uses r².
2.2 · 12 Marks · Dependent events & Venn
Independent: P(A)=0.4, P(B)=0.5. P(A and B)=
Formulas
P(A ∪ B)=P(A)+P(B)−P(A ∩ B)
Addition rule
Notes: Independent events. P(A)=0.4, P(B)=0.5.
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: Independent: P(A)=0.4, P(B)=0.5. P(A and B)=
Before any working — hunt first
Given
Numbers and units already in the sentence: 0.4 · 0.5. 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: P(A and B)=. Circle that word — that is the missing variable.
Find
We are required to find: P(A and B)=. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
Write 0.20 with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives P(A and B)=.
In other words — as if you have never seen this before
Independent ⇒ P(A and B) = P(A)P(B).
2. What do we already know?
Independent events. P(A)=0.4, P(B)=0.5.
3. Which method — and why this one?
Independent ⇒ P(A and B) = P(A)P(B).
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: 0.4 · 0.5. 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: P(A and B)=. Circle that word — that is the missing variable.
Find: We are required to find: P(A and B)=. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: Write 0.20 with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives P(A and B)=.
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
Independent events. P(A)=0.4, P(B)=0.5.
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
Independent ⇒ P(A and B) = P(A)P(B).
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
Product
0.4×0.5=0.20.
0.20
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
0.20
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.
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
0.20
Copy onto your answer sheet — in this order
Formula: 0.20
Answer: 0.20
6. Check it like a marker
0.90 would be A or B if they were exclusive, which they are not.
7. A trap first-timers fall into
Adding 0.4+0.5 is the exclusive-or shortcut, wrong for “and”.
5 · 2 Marks · Sine, cosine & area rules
cos θ = 12 in [0°; 360°]. Solutions:
Formulas
asin A=bsin B=csin C=2R
Sine rule
c²=a²+b²−2ab cos C
Cosine rule
Area=½ ab sin C
Area 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: cos θ = 12 in [0°; 360°]. Solutions:
Before any working — hunt first
Given
Numbers and units already in the sentence: 1/ · 2 in · 0° · 360°. 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: Solutions:. Circle that word — that is the missing variable.
Find
We are required to find: Solutions:. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: CAST: cosine positive in Q1 and Q4. Then substitute the given, one line at a time, to get Solutions:.
In other words — as if you have never seen this before
CAST: cosine positive in Q1 and Q4.
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 1/ · 2 in · 0° · 360°. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
CAST: cosine positive in Q1 and Q4.
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: 1/ · 2 in · 0° · 360°. 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: Solutions:. Circle that word — that is the missing variable.
Find: We are required to find: Solutions:. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: What we need to calculate: CAST: cosine positive in Q1 and Q4. Then substitute the given, one line at a time, to get Solutions:.
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: 1/ · 2 in · 0° · 360°. 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
CAST: cosine positive in Q1 and Q4.
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
CAST: cosine positive in Q1 and Q4.
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
60° and 300°
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
60° and 300°
Copy onto your answer sheet — in this order
Answer: 60° and 300°
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…”
6 · 3 Marks · Identities, reduction & equations
sin(180°−θ)=
Identities, reduction & equations
Formulas
sin²θ+cos²θ=1
Pythagorean identity
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(180°−θ) =
Before any working — hunt first
Given
Numbers and units already in the sentence: 180°. 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(180°−θ) =. Circle that word — that is the missing variable.
Find
We are required to find: sin(180°−θ) =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: Reduction: Q2 sine is positive. Then substitute the given, one line at a time, to get sin(180°−θ) =.
In other words — as if you have never seen this before
Reduction: Q2 sine is positive.
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 180°. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
Reduction: Q2 sine is positive.
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: 180°. 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(180°−θ) =. Circle that word — that is the missing variable.
Find: We are required to find: sin(180°−θ) =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: What we need to calculate: Reduction: Q2 sine is positive. Then substitute the given, one line at a time, to get sin(180°−θ) =.
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: 180°. 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
Reduction: Q2 sine is positive.
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
Reduction: Q2 sine is 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
sin θ
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 θ
Copy onto your answer sheet — in this order
Answer: sin θ
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 · Parabola, hyperbola & exponential
x² − 5x + 6 < 0. Solution is
Parabola, hyperbola & exponential
Formulas
y=a(x + p)²+q
Shifted parabola
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: x² − 5x + 6 < 0. Solution is
Before any working — hunt first
Given
Numbers and units already in the sentence: 5x · 6 · 0. 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: Solution is. Circle that word — that is the missing variable.
Find
We are required to find: Solution is. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: Smile-parabola negative between roots 2 and 3. Then substitute the given, one line at a time, to get Solution is.
In other words — as if you have never seen this before
Smile-parabola negative between roots 2 and 3.
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 5x · 6 · 0. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
Smile-parabola negative between roots 2 and 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: 5x · 6 · 0. 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: Solution is. Circle that word — that is the missing variable.
Find: We are required to find: Solution is. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: What we need to calculate: Smile-parabola negative between roots 2 and 3. Then substitute the given, one line at a time, to get Solution 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: 5x · 6 · 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.
04 / 08
Step C — Name the method in words
Smile-parabola negative between roots 2 and 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
Working
Smile-parabola negative between roots 2 and 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
(2; 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
(2; 3)
Copy onto your answer sheet — in this order
Answer: (2; 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…”
8 · 3 Marks · Nominal vs effective interest
12% p.a. compounded monthly. Monthly interest rate i =
Nominal vs effective interest
Formulas
1+ieff=(1 + inom/m)m
Effective rate
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: 12% p.a. compounded monthly. Monthly interest rate i =
Before any working — hunt first
Given
Numbers and units already in the sentence: 12%. 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: Monthly interest rate i =. Circle that word — that is the missing variable.
Find
We are required to find: Monthly interest rate i =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: i = 0. Then substitute the given, one line at a time, to get Monthly interest rate i =.
In other words — as if you have never seen this before
i=0.1212=0.01.
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 12%. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
i=0.1212=0.01.
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: 12%. 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: Monthly interest rate i =. Circle that word — that is the missing variable.
Find: We are required to find: Monthly interest rate i =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: What we need to calculate: i = 0. Then substitute the given, one line at a time, to get Monthly interest rate i =.
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: 12%. 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
i=0.1212=0.01.
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
i=0.1212=0.01.
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
0.01
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
0.01
Copy onto your answer sheet — in this order
Answer: 0.01
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 · 3 Marks · Proportion & midpoint theorem
Angle at the centre is 80°. Angle at the circumference on the same arc is
Proportion & midpoint theorem
Formulas
DE ∥ BC, AD=DB⇒AE=EC
Midpoint theorem
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 at the centre is 80°. Angle at the circumference on the same arc is
In other words — as if you have never seen this before
Angle at circumference is half the angle at the centre.
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 80°. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
Angle at circumference is half the angle at the centre.
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: 80°. 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
Angle at circumference is half the angle at the centre.
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
Angle at circumference is half the angle at the centre.
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
40°
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
40°
Copy onto your answer sheet — in this order
Answer: 40°
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 · 2 Marks · Quadratic equations & nature of roots
Roots of (x−4)(x+7)=0
Formulas
Δ=b²−4ac
Discriminant
Notes: Product of two brackets = 0. Unknown: 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: Roots of (x−4)(x+7)=0
Before any working — hunt first
Given
Numbers and units already in the sentence: 4 · 7 · 0. 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: Roots of (x−4)(x+7)=0. Circle that word — that is the missing variable.
Find
We are required to find: Roots of (x−4)(x+7)=0. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
Write x = 4 with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives Roots of (x−4)(x+7)=0.
In other words — as if you have never seen this before
Zero-product: a factor is 0. Set each bracket to zero.
2. What do we already know?
Product of two brackets = 0. Unknown: x.
3. Which method — and why this one?
Zero-product: a factor is 0. Set each bracket to zero.
4. Step-by-step working
01 / 05
First bracket
x−4=0→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.
02 / 05
Second bracket
x+7=0→x=−7.
x=−7
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 and −7
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 and −7
Copy onto your answer sheet — in this order
Formula: x=4
Answer: 4 and −7
6. Check it like a marker
FOIL back: x²+3x−28. At x=4: 16+12−28=0. At x=−7: 49−21−28=0.
7. A trap first-timers fall into
Swapping signs to −4 and 7 is the most common first-timer slip.
11 · 3 Marks · Dependent events & Venn
Independent P(A)=0.2, P(B)=0.5. P(A and B) =
Formulas
P(A ∪ B)=P(A)+P(B)−P(A ∩ B)
Addition 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: Independent P(A)=0.2, P(B)=0.5. P(A and B) =
Before any working — hunt first
Given
Numbers and units already in the sentence: 0.2 · 0.5. 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: P(A and B) =. Circle that word — that is the missing variable.
Find
We are required to find: P(A and B) =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: 0. Then substitute the given, one line at a time, to get P(A and B) =.
In other words — as if you have never seen this before
0.10
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 0.2 · 0.5. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
0.10
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 · 0.5. 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: P(A and B) =. Circle that word — that is the missing variable.
Find: We are required to find: P(A and B) =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: What we need to calculate: 0. Then substitute the given, one line at a time, to get P(A and B) =.
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 · 0.5. 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
0.10
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
Independent events: P(A ∩ B) = P(A)P(B) = 0.2 × 0.5 = 0.10. Do not add — adding is “or” for mutually exclusive.
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
0.10
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
0.10
Copy onto your answer sheet — in this order
Answer: 0.10
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…”
12 · 4 Marks · Quadratic equations & nature of roots
x² − 2x + 5 = 0. Nature of roots?
Formulas
Δ=b²−4ac
Discriminant
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: x² − 2x + 5 = 0. Nature of roots?
Before any working — hunt first
Given
Numbers and units already in the sentence: 2x · 5 · 0. 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: Nature of roots. Circle that word — that is the missing variable.
Find
We are required to find: Nature of roots. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: Δ=4−20=−16<0. Then substitute the given, one line at a time, to get Nature of roots.
In other words — as if you have never seen this before
Δ=4−20=−16<0.
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 2x · 5 · 0. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
Δ=4−20=−16<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: 2x · 5 · 0. 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: Nature of roots. Circle that word — that is the missing variable.
Find: We are required to find: Nature of roots. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: What we need to calculate: Δ=4−20=−16<0. Then substitute the given, one line at a time, to get Nature of roots.
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: 2x · 5 · 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.
04 / 08
Step C — Name the method in words
Δ=4−20=−16<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
Δ=4−20=−16<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
non-real
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
non-real
Copy onto your answer sheet — in this order
Answer: non-real
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 · 4 Marks · Proportion & midpoint theorem
Circle centre (2; −3), r = 5. Equation:
Proportion & midpoint theorem
Formulas
DE ∥ BC, AD=DB⇒AE=EC
Midpoint theorem
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: Circle centre (2; −3), r = 5. Equation:
Before any working — hunt first
Given
Numbers and units already in the sentence: 2 · 3 · 5. 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: Equation:. Circle that word — that is the missing variable.
Find
We are required to find: Equation:. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: r² = 25. Then substitute the given, one line at a time, to get Equation:.
In other words — as if you have never seen this before
r²=25. Centre (2; −3)→(y+3)².
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 2 · 3 · 5. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
r²=25. Centre (2; −3)→(y+3)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 · 3 · 5. 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: Equation:. Circle that word — that is the missing variable.
Find: We are required to find: Equation:. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: What we need to calculate: r² = 25. Then substitute the given, one line at a time, to get Equation:.
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 · 3 · 5. 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
r²=25. Centre (2; −3)→(y+3)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
r²=25. Centre (2; −3)→(y+3)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
(x−2)²+(y+3)²=25
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−2)²+(y+3)²=25
Copy onto your answer sheet — in this order
Answer: (x−2)²+(y+3)²=25
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 · 2 Marks · Exponents & surds
√48 simplified
Exponents & surds
Formulas
am/ⁿ=ⁿ√aᵐ
Rational exponent
√a·√b=√ab
Product of surds
Notes: √48. Unknown: simplest surd form (no square factor left under the radical).
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: √48 simplified
Before any working — hunt first
Given
Numbers and units already in the sentence: 48 simplified. 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: √48 simplified. Circle that word — that is the missing variable.
Find
We are required to find: √48 simplified. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
Write 48 = 16 × 3 with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives √48 simplified.
In other words — as if you have never seen this before
Pull out the biggest square factor. 48 = 16 × 3, and 16 is 4².
2. What do we already know?
√48. Unknown: simplest surd form (no square factor left under the radical).
3. Which method — and why this one?
Pull out the biggest square factor. 48 = 16 × 3, and 16 is 4².
4. Step-by-step working
01 / 05
Factor 48
48 = 16 × 3, and 16 is a perfect square. (You could use 4 × 12, but then you would simplify again.)
48=16×3
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.
02 / 05
Take the square out
√48=√16×3=√16×√3=4√3.
√48=4√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
4√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
4√3
Copy onto your answer sheet — in this order
Formula: 48=16×3
Answer: 4√3
6. Check it like a marker
4√3 squared: 16 × 3 = 48. Lands back. 16√3 would square to a much bigger number.
7. A trap first-timers fall into
Stopping at √16 × √3 without taking √16. Or writing 3√4, which is not simplified (√4 is still 2).
15 · 2 Marks · Exponents & surds
√75 simplified
Exponents & surds
Formulas
am/ⁿ=ⁿ√aᵐ
Rational exponent
√a·√b=√ab
Product of surds
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: √75 simplified
Before any working — hunt first
Given
Numbers and units already in the sentence: 75 simplified. 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: √75 simplified. Circle that word — that is the missing variable.
Find
We are required to find: √75 simplified. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: 5√3. Then substitute the given, one line at a time, to get √75 simplified.
In other words — as if you have never seen this before
5√3
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 75 simplified. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
5√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: 75 simplified. 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: √75 simplified. Circle that word — that is the missing variable.
Find: We are required to find: √75 simplified. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: What we need to calculate: 5√3. Then substitute the given, one line at a time, to get √75 simplified.
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 simplified. 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
5√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
Working
√75 = √(25 × 3) = 5√3. Pull out the biggest square factor (25), leave 3 under the radical.
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
5√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
5√3
Copy onto your answer sheet — in this order
Answer: 5√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…”
16 · 2 Marks · Quadratic equations & nature of roots
Quadratic pattern. Second difference constantly 6. a =
Formulas
Δ=b²−4ac
Discriminant
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: Quadratic pattern. Second difference constantly 6. a =
Before any working — hunt first
Given
Numbers and units already in the sentence: 6. 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: a =. Circle that word — that is the missing variable.
Find
We are required to find: a =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: 2a = second difference, a = 3. Then substitute the given, one line at a time, to get a =.
In other words — as if you have never seen this before
2a = second difference, a = 3.
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 6. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
2a = second difference, a = 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: 6. 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: a =. Circle that word — that is the missing variable.
Find: We are required to find: a =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: What we need to calculate: 2a = second difference, a = 3. Then substitute the given, one line at a time, to get a =.
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: 6. 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
2a = second difference, a = 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
Working
2a = second difference, a = 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
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…”
17 · 2 Marks · Exponents & surds
√32 simplified
Exponents & surds
Formulas
am/ⁿ=ⁿ√aᵐ
Rational exponent
√a·√b=√ab
Product of surds
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: √32 simplified
Before any working — hunt first
Given
Numbers and units already in the sentence: 32 simplified. 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: √32 simplified. Circle that word — that is the missing variable.
Find
We are required to find: √32 simplified. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: 4√2. Then substitute the given, one line at a time, to get √32 simplified.
In other words — as if you have never seen this before
4√2
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 32 simplified. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
4√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: 32 simplified. 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: √32 simplified. Circle that word — that is the missing variable.
Find: We are required to find: √32 simplified. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: What we need to calculate: 4√2. Then substitute the given, one line at a time, to get √32 simplified.
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: 32 simplified. 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√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
Reason
√16×2=4√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
4√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
4√2
Copy onto your answer sheet — in this order
Answer: 4√2
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 · 3 Marks · Identities, reduction & equations
cos θ = 0 in [0°; 360°]
Identities, reduction & equations
Formulas
sin²θ+cos²θ=1
Pythagorean identity
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 θ = 0 in [0°; 360°]
Before any working — hunt first
Given
Numbers and units already in the sentence: 0 in · 0° · 360°. 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 θ = 0 in [0°; 360°]. Circle that word — that is the missing variable.
Find
We are required to find: cos θ = 0 in [0°; 360°]. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: 90° and 270°. Then substitute the given, one line at a time, to get cos θ = 0 in [0°; 360°].
In other words — as if you have never seen this before
90° and 270°
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 0 in · 0° · 360°. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
90° and 270°
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 in · 0° · 360°. 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 θ = 0 in [0°; 360°]. Circle that word — that is the missing variable.
Find: We are required to find: cos θ = 0 in [0°; 360°]. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: What we need to calculate: 90° and 270°. Then substitute the given, one line at a time, to get cos θ = 0 in [0°; 360°].
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 in · 0° · 360°. 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
90° and 270°
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
Cosine is the x-coordinate on the unit circle. x = 0 at the top (90°) and the bottom (270°).
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
90° and 270°
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
90° and 270°
Copy onto your answer sheet — in this order
Answer: 90° and 270°
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…”
19 · 3 Marks · Proportion & midpoint theorem
Angle at circumference is 36°. Angle at centre on same arc
Proportion & midpoint theorem
Formulas
DE ∥ BC, AD=DB⇒AE=EC
Midpoint theorem
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 at circumference is 36°. Angle at centre on same arc
Before any working — hunt first
Given
Numbers and units already in the sentence: 36°. 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 at centre on same arc. Circle that word — that is the missing variable.
Find
We are required to find: Angle at centre on same arc. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: 72°. Then substitute the given, one line at a time, to get Angle at centre on same arc.
In other words — as if you have never seen this before
72°
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 36°. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
72°
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: 36°. 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 at centre on same arc. Circle that word — that is the missing variable.
Find: We are required to find: Angle at centre on same arc. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: What we need to calculate: 72°. Then substitute the given, one line at a time, to get Angle at centre on same arc.
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: 36°. 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
72°
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
Double.
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
72°
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
72°
Copy onto your answer sheet — in this order
Answer: 72°
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 · Nominal vs effective interest
R8000 depreciates at 10% p.a. reducing balance for 2 years. Value =
Nominal vs effective interest
Formulas
1+ieff=(1 + inom/m)m
Effective rate
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: R8000 depreciates at 10% p.a. reducing balance for 2 years. Value =
Before any working — hunt first
Given
Numbers and units already in the sentence: R8000 depreciates · 10% · 2 years. 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: Value =. Circle that word — that is the missing variable.
Find
We are required to find: Value =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: 8000(0. Then substitute the given, one line at a time, to get Value =.
In other words — as if you have never seen this before
8000(0.9)²=6480.
2. What do we already know?
Copy these onto the left of your page, one per line, with units: R8000 depreciates · 10% · 2 years. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
8000(0.9)²=6480.
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: R8000 depreciates · 10% · 2 years. 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: Value =. Circle that word — that is the missing variable.
Find: We are required to find: Value =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: What we need to calculate: 8000(0. Then substitute the given, one line at a time, to get Value =.
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: R8000 depreciates · 10% · 2 years. 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
8000(0.9)²=6480.
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
8000(0.9)²=6480.
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
R6480
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
R6480
Copy onto your answer sheet — in this order
Answer: R6480
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 · 2 Marks · Exponents & surds
√18 simplified
Exponents & surds
Formulas
am/ⁿ=ⁿ√aᵐ
Rational exponent
√a·√b=√ab
Product of surds
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: √18 simplified
Before any working — hunt first
Given
Numbers and units already in the sentence: 18 simplified. 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: √18 simplified. Circle that word — that is the missing variable.
Find
We are required to find: √18 simplified. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: √(9×2)=3√2. Then substitute the given, one line at a time, to get √18 simplified.
In other words — as if you have never seen this before
√9×2=3√2.
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 18 simplified. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
√9×2=3√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: 18 simplified. 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: √18 simplified. Circle that word — that is the missing variable.
Find: We are required to find: √18 simplified. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: What we need to calculate: √(9×2)=3√2. Then substitute the given, one line at a time, to get √18 simplified.
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: 18 simplified. 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
√9×2=3√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
√9×2=3√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
3√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
3√2
Copy onto your answer sheet — in this order
Answer: 3√2
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 · 4 Marks · Variance, ogives & five-number summary
Five scores: 2, 4, 4, 6, 9. Mean is
Variance, ogives & five-number summary
Formulas
x̄=Σxn
Mean
σ²=Σ(x − x̄)²n
Population variance (CAPS uses n)
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: Five scores: 2, 4, 4, 6, 9. Mean is
Before any working — hunt first
Given
Numbers and units already in the sentence: 2 · 4 · 4 · 6 · 9. 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: Mean is. Circle that word — that is the missing variable.
Find
We are required to find: Mean is. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: Σx = 25, n = 5, mean = 5. Then substitute the given, one line at a time, to get Mean is.
In other words — as if you have never seen this before
Σx=25, n=5, mean=5.
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 2 · 4 · 4 · 6 · 9. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
Σx=25, n=5, mean=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: 2 · 4 · 4 · 6 · 9. 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: Mean is. Circle that word — that is the missing variable.
Find: We are required to find: Mean is. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: What we need to calculate: Σx = 25, n = 5, mean = 5. Then substitute the given, one line at a time, to get Mean 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 · 4 · 6 · 9. 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
Σx=25, n=5, mean=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
Σx=25, n=5, mean=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
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
5
Copy onto your answer sheet — in this order
Answer: 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…”
23 · 4 Marks · Parabola, hyperbola & exponential
Range of y = (x−1)² + 4
Parabola, hyperbola & exponential
Formulas
y=a(x + p)²+q
Shifted parabola
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: Range of y = (x−1)² + 4
Before any working — hunt first
Given
Numbers and units already in the sentence: 1 · 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: Range of y = (x−1)² + 4. Circle that word — that is the missing variable.
Find
We are required to find: Range of y = (x−1)² + 4. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: Parabola opens up, minimum 4. Then substitute the given, one line at a time, to get Range of y = (x−1)² + 4.
In other words — as if you have never seen this before
Parabola opens up, minimum 4.
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 1 · 4. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
Parabola opens up, minimum 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: 1 · 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: Range of y = (x−1)² + 4. Circle that word — that is the missing variable.
Find: We are required to find: Range of y = (x−1)² + 4. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: What we need to calculate: Parabola opens up, minimum 4. Then substitute the given, one line at a time, to get Range of y = (x−1)² + 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: 1 · 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
Parabola opens up, minimum 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
Parabola opens up, minimum 4.
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
y ≥ 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
y ≥ 4
Copy onto your answer sheet — in this order
Answer: y ≥ 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…”
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: Independent P(A)=0.4, P(B)=0.5. P(A and B) =
Before any working — hunt first
Given
Numbers and units already in the sentence: 0.4 · 0.5. 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: P(A and B) =. Circle that word — that is the missing variable.
Find
We are required to find: P(A and B) =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
Write P(A ∩ B) = P(A)P(B) with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives P(A and B) =.
In other words — as if you have never seen this before
Independent means “one does not change the other.” Then P(A and B) = P(A) × P(B). Adding is “or” for mutually exclusive events.
Independent means “one does not change the other.” Then P(A and B) = P(A) × P(B). Adding is “or” for mutually exclusive events.
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: 0.4 · 0.5. 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: P(A and B) =. Circle that word — that is the missing variable.
Find: We are required to find: P(A and B) =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: Write P(A ∩ B) = P(A)P(B) with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives P(A and B) =.
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).
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
Independent means “one does not change the other.” Then P(A and B) = P(A) × P(B). Adding is “or” for mutually exclusive events.
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
Multiply
P(A ∩ B)=0.4×0.5=0.20.
P(A ∩ B)=P(A)P(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
0.20
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.
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
0.20
Copy onto your answer sheet — in this order
Formula: P(A ∩ B)=P(A)P(B)
Answer: 0.20
6. Check it like a marker
0.20 is smaller than both 0.4 and 0.5, as an “and” must be. 0.90 would be the “or” if they were mutually exclusive — they are not, and that was not asked.
7. A trap first-timers fall into
Adding 0.4+0.5=0.90. That is P(A or B) for mutually exclusive events, a different sentence.
25 · 3 Marks · Proportion & midpoint theorem
Midpoints of two sides of a triangle are joined. The third side is 10 cm. The join is
Proportion & midpoint theorem
Formulas
DE ∥ BC, AD=DB⇒AE=EC
Midpoint theorem
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: Midpoints of two sides of a triangle are joined. The third side is 10 cm. The join is
Before any working — hunt first
Given
Numbers and units already in the sentence: 10 cm. 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: The join is. Circle that word — that is the missing variable.
Find
We are required to find: The join is. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: Midpoint theorem: half the third side. Then substitute the given, one line at a time, to get The join is.
In other words — as if you have never seen this before
Midpoint theorem: half the third side.
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 10 cm. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
Midpoint theorem: half the third side.
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: 10 cm. 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: The join is. Circle that word — that is the missing variable.
Find: We are required to find: The join is. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: What we need to calculate: Midpoint theorem: half the third side. Then substitute the given, one line at a time, to get The join 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: 10 cm. 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
Midpoint theorem: half the third side.
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
Midpoint theorem: half the third side.
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
5 cm
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
5 cm
Copy onto your answer sheet — in this order
Answer: 5 cm
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 · Identities, reduction & equations
sin θ = 12 in [0°; 180°]
Identities, reduction & equations
Formulas
sin²θ+cos²θ=1
Pythagorean identity
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 θ = 12 in [0°; 180°]
Before any working — hunt first
Given
Numbers and units already in the sentence: 1/ · 2 in · 0° · 180°. 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 θ = 12 in [0°; 180°]. Circle that word — that is the missing variable.
Find
We are required to find: sin θ = 12 in [0°; 180°]. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: 30° and 150°. Then substitute the given, one line at a time, to get sin θ = 12 in [0°; 180°].
In other words — as if you have never seen this before
30° and 150°
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 1/ · 2 in · 0° · 180°. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
30° and 150°
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: 1/ · 2 in · 0° · 180°. 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 θ = 12 in [0°; 180°]. Circle that word — that is the missing variable.
Find: We are required to find: sin θ = 12 in [0°; 180°]. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: What we need to calculate: 30° and 150°. Then substitute the given, one line at a time, to get sin θ = 12 in [0°; 180°].
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: 1/ · 2 in · 0° · 180°. 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
30° and 150°
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
Sine is positive in Q1 and Q2. Reference 30°. Q2: 180° − 30° = 150°.
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
30° and 150°
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
30° and 150°
Copy onto your answer sheet — in this order
Answer: 30° and 150°
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 · Sine, cosine & area rules
In △ABC, A = 30°, a = 5, B = 45°. b =
Formulas
asin A=bsin B=csin C=2R
Sine rule
c²=a²+b²−2ab cos C
Cosine rule
Area=½ ab sin C
Area 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: In △ABC, A = 30°, a = 5, B = 45°. b =
Before any working — hunt first
Given
Numbers and units already in the sentence: 30° · 5 · 45°. 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: b =. Circle that word — that is the missing variable.
Find
We are required to find: b =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: bsin45=5sin30. Then substitute the given, one line at a time, to get b =.
In other words — as if you have never seen this before
bsin45=5sin30→b=5×(√2/2)(1/2)=5√2.
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 30° · 5 · 45°. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
bsin45=5sin30→b=5×(√2/2)(1/2)=5√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: 30° · 5 · 45°. 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: b =. Circle that word — that is the missing variable.
Find: We are required to find: b =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: What we need to calculate: bsin45=5sin30. Then substitute the given, one line at a time, to get b =.
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: 30° · 5 · 45°. 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
bsin45=5sin30→b=5×(√2/2)(1/2)=5√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
bsin45=5sin30→b=5×(√2/2)(1/2)=5√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
5√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
5√2
Copy onto your answer sheet — in this order
Answer: 5√2
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 · 2 Marks · Inclination & circles
Turning point of y = (x−3)² + 5
Formulas
tan θ=m
Inclination
(x − a)²+(y − b)²=r²
Circle centre (a;b)
Notes: Completed square y = (x − p)² + q with p = 3, q = 5.
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: Turning point of y = (x−3)² + 5
Before any working — hunt first
Given
Numbers and units already in the sentence: 3 · 5. 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: Turning point of y = (x−3)² + 5. Circle that word — that is the missing variable.
Find
We are required to find: Turning point of y = (x−3)² + 5. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
Write (3; 5) with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives Turning point of y = (x−3)² + 5.
In other words — as if you have never seen this before
TP is (p; q) = (3; 5). The square is zero at x = 3.
2. What do we already know?
Completed square y = (x − p)² + q with p = 3, q = 5.
3. Which method — and why this one?
TP is (p; q) = (3; 5). The square is zero at 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: 3 · 5. 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: Turning point of y = (x−3)² + 5. Circle that word — that is the missing variable.
Find: We are required to find: Turning point of y = (x−3)² + 5. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: Write (3; 5) with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives Turning point of y = (x−3)² + 5.
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
Completed square y = (x − p)² + q with p = 3, q = 5.
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
TP is (p; q) = (3; 5). The square is zero at 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
Read p and q
x−3=0→x=3. Minimum y=5.
(3; 5)
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
(3; 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
(3; 5)
Copy onto your answer sheet — in this order
Formula: (3; 5)
Answer: (3; 5)
6. Check it like a marker
At x=3, y=0+5=5. At x=4, y=1+5=6, higher, so a minimum.
7. A trap first-timers fall into
(−3; 5) copies the minus sign from the bracket. The root of x−3 is +3.
29 · 2 Marks
2⁴=
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: 2⁴ =
Before any working — hunt first
Given
Numbers and units already in the sentence: 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: 2⁴ =. Circle that word — that is the missing variable.
Find
We are required to find: 2⁴ =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: 2×2×2×2 = 16. Then substitute the given, one line at a time, to get 2⁴ =.
In other words — as if you have never seen this before
2×2×2×2=16.
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 2. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
2×2×2×2=16.
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. 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: 2⁴ =. Circle that word — that is the missing variable.
Find: We are required to find: 2⁴ =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: What we need to calculate: 2×2×2×2 = 16. Then substitute the given, one line at a time, to get 2⁴ =.
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. 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
2×2×2×2=16.
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
2×2×2×2=16.
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
16
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
16
Copy onto your answer sheet — in this order
Answer: 16
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 · 3 Marks · Inclination & circles
y=(x−1)²+2. Turning point?
Formulas
tan θ=m
Inclination
(x − a)²+(y − b)²=r²
Circle centre (a;b)
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: y = (x−1)² + 2. Turning point?
Before any working — hunt first
Given
Numbers and units already in the sentence: 1 · 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: Turning point. Circle that word — that is the missing variable.
Find
We are required to find: Turning point. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: (1; 2). Then substitute the given, one line at a time, to get Turning point.
In other words — as if you have never seen this before
(1; 2)
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 1 · 2. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
(1; 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: 1 · 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: Turning point. Circle that word — that is the missing variable.
Find: We are required to find: Turning point. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: What we need to calculate: (1; 2). Then substitute the given, one line at a time, to get Turning point.
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: 1 · 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
(1; 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
Reason
TP (p; q) with opposite sign on p.
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
(1; 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
(1; 2)
Copy onto your answer sheet — in this order
Answer: (1; 2)
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…”