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: Solve x² = 49
Before any working — hunt first
Given
Numbers and units already in the sentence: 49. 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: Solve x² = 49. Circle that word — that is the missing variable.
Find
We are required to find: Solve x² = 49. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
Write x = ±7 with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives Solve x² = 49.
In other words — as if you have never seen this before
Square root both sides. Remember the ± . 7² and (−7)² both equal 49.
2. What do we already know?
x squared equals 49.
3. Which method — and why this one?
Square root both sides. Remember the ± . 7² and (−7)² both equal 49.
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: 49. 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: Solve x² = 49. Circle that word — that is the missing variable.
Find: We are required to find: Solve x² = 49. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: Write x = ±7 with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives Solve x² = 49.
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
x squared equals 49.
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
Square root both sides. Remember the ± . 7² and (−7)² both equal 49.
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 root
x=±√49=±7.
x=±7
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
x=±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.
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=±7
Copy onto your answer sheet — in this order
Formula: x=±7
Answer: x=±7
6. Check it like a marker
7×7=49 and (−7)×(−7)=49.
7. A trap first-timers fall into
x = 7 only loses a mark. The paper wants both roots.
1.2 · 10 Marks · Inclination & circles
Distance between (0;0) and (3;4)
Formulas
tan θ=m
Inclination
(x − a)²+(y − b)²=r²
Circle centre (a;b)
Notes: A(0;0), B(3;4). Unknown: AB.
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: Distance between (0;0) and (3;4)
Before any working — hunt first
Given
Numbers and units already in the sentence: 0 · 0 · 3 · 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: Distance between (0;0) and (3;4). Circle that word — that is the missing variable.
Find
We are required to find: Distance between (0;0) and (3;4). Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
Write 5 with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives Distance between (0;0) and (3;4).
In other words — as if you have never seen this before
Distance formula is Pythagoras. 3-4-5 triangle.
2. What do we already know?
A(0;0), B(3;4). Unknown: AB.
3. Which method — and why this one?
Distance formula is Pythagoras. 3-4-5 triangle.
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 · 3 · 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: Distance between (0;0) and (3;4). Circle that word — that is the missing variable.
Find: We are required to find: Distance between (0;0) and (3;4). Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: Write 5 with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives Distance between (0;0) and (3;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
A(0;0), B(3;4). Unknown: AB.
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
Distance formula is Pythagoras. 3-4-5 triangle.
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
√(3²+4²)
√9+16=√25=5.
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
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
Formula: 5
Answer: 5
6. Check it like a marker
A famous triple. Sketch it.
7. A trap first-timers fall into
3+4=7 adds the legs instead of squaring.
2.1 · 10 Marks · Inclination & circles
Midpoint of (2; 4) and (6; 8)
Formulas
tan θ=m
Inclination
(x − a)²+(y − b)²=r²
Circle centre (a;b)
Notes: A(2;4), B(6;8). Unknown: midpoint M.
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: Midpoint of (2; 4) and (6; 8)
Before any working — hunt first
Given
Numbers and units already in the sentence: 2 · 4 · 6 · 8. 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: Midpoint of (2; 4) and (6; 8). Circle that word — that is the missing variable.
Find
We are required to find: Midpoint of (2; 4) and (6; 8). Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
Write (4; 6) with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives Midpoint of (2; 4) and (6; 8).
In other words — as if you have never seen this before
Average the x’s, average the y’s.
2. What do we already know?
A(2;4), B(6;8). Unknown: midpoint M.
3. Which method — and why this one?
Average the x’s, average the y’s.
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 · 6 · 8. 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: Midpoint of (2; 4) and (6; 8). Circle that word — that is the missing variable.
Find: We are required to find: Midpoint of (2; 4) and (6; 8). Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: Write (4; 6) with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives Midpoint of (2; 4) and (6; 8).
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(2;4), B(6;8). Unknown: midpoint M.
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
Average the x’s, average the y’s.
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
Averages
x=(2+6)2=4. y=(4+8)2=6.
(4; 6)
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; 6)
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; 6)
Copy onto your answer sheet — in this order
Formula: (4; 6)
Answer: (4; 6)
6. Check it like a marker
M is halfway: from 2 to 6 is +4, half is +2 → 4.
7. A trap first-timers fall into
(8;12) added without dividing by 2.
2.2 · 12 Marks · Dependent events & Venn
P(even) on a fair die
Formulas
P(A ∪ B)=P(A)+P(B)−P(A ∩ B)
Addition rule
Notes: Die faces 1–6. Even: 2,4,6. Three faces.
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: P(even) on a fair die
Before any working — hunt first
Given
Numbers and units already in the sentence: no numbers yet — only a name, a diagram, or a worded unknown. Copy each one into a column. A name without a number is not given.
Missing
After you copy the given, the letter that still has no number is missing. The paper is pointing at: P(even) on a fair die. Circle that word — that is the missing variable.
Find
We are required to find: P(even) on a fair die. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
Write 12 with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives P(even) on a fair die.
In other words — as if you have never seen this before
P=favourabletotal=36=12.
2. What do we already know?
Die faces 1–6. Even: 2,4,6. Three faces.
3. Which method — and why this one?
P=favourabletotal=36=12.
4. Step-by-step working
01 / 08
Step A — Read the question twice
Slow down. Circle the word that names the unknown. Underline every number and its unit. Do not grab a formula yet — first decide what the paper is actually asking.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
02 / 08
Step A2 — Hunt: Given, Missing, Find, Calculate
Given: Numbers and units already in the sentence: no numbers yet — only a name, a diagram, or a worded unknown. Copy each one into a column. A name without a number is not given.
Missing: After you copy the given, the letter that still has no number is missing. The paper is pointing at: P(even) on a fair die. Circle that word — that is the missing variable.
Find: We are required to find: P(even) on a fair die. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: Write 12 with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives P(even) on a fair die.
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
Die faces 1–6. Even: 2,4,6. Three faces.
Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
04 / 08
Step C — Name the method in words
P=favourabletotal=36=12.
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
Count
3 even out of 6 = 12.
12
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
12
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
12
Copy onto your answer sheet — in this order
Formula: 12
Answer: 12
6. Check it like a marker
Odd is also 12. They partition the sample space.
7. A trap first-timers fall into
16 is one single face, not the event “even”.
5 · 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…”
6 · 2 Marks · Parabola, hyperbola & exponential
Turning point of y = (x+1)² − 4 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: Turning point of y = (x+1)² − 4 is
Before any working — hunt first
Given
Numbers and units already in the sentence: 1 · 4 is. 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+1)² − 4 is. Circle that word — that is the missing variable.
Find
We are required to find: Turning point of y = (x+1)² − 4 is. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: Form a(x+p)²+q has TP (−p; q) = (−1; −4). Then substitute the given, one line at a time, to get Turning point of y = (x+1)² − 4 is.
In other words — as if you have never seen this before
Form a(x+p)²+q has TP (−p; q)=(−1; −4).
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 1 · 4 is. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
Form a(x+p)²+q has TP (−p; q)=(−1; −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 is. 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+1)² − 4 is. Circle that word — that is the missing variable.
Find: We are required to find: Turning point of y = (x+1)² − 4 is. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: What we need to calculate: Form a(x+p)²+q has TP (−p; q) = (−1; −4). Then substitute the given, one line at a time, to get Turning point of y = (x+1)² − 4 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: 1 · 4 is. Mixing cm with m, or minutes with seconds, loses a method mark.
Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
04 / 08
Step C — Name the method in words
Form a(x+p)²+q has TP (−p; q)=(−1; −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
Form a(x+p)²+q has TP (−p; q)=(−1; −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
(−1; −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
(−1; −4)
Copy onto your answer sheet — in this order
Answer: (−1; −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…”
7 · 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…”
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.
9 · 3 Marks · Quadratic equations & nature of roots
First differences of a quadratic with a=2. Second difference =
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: First differences of a quadratic with a=2. Second difference =
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: Second difference =. Circle that word — that is the missing variable.
Find
We are required to find: Second difference =. 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 = 4. Then substitute the given, one line at a time, to get Second difference =.
In other words — as if you have never seen this before
2a = second difference = 4.
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?
2a = second difference = 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: 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: Second difference =. Circle that word — that is the missing variable.
Find: We are required to find: Second difference =. 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 = 4. Then substitute the given, one line at a time, to get Second difference =.
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
2a = second difference = 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
2a = second difference = 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
4
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
08 / 08
Step H — What the marker is looking for
On an NSC memo you usually earn: (1) a method mark for naming the right formula or idea in words, (2) an accuracy mark for substituting the given numbers with units, (3) an accuracy mark for the boxed answer with its unit. Write those three lines even if you are unsure of the arithmetic — blank pages score zero.
Why this step: First-timers chase the final number and leave the method blank. The memo still pays out for the idea and the substitution even when the last line is wrong.
5. The answer
4
Copy onto your answer sheet — in this order
Answer: 4
6. Check it like a marker
Does your answer appear as one of the options? Do the units match what was asked (m, N, J, °)? If you put the number back into the original equation, does it still hold? Estimate: is the size sensible?
7. A trap first-timers fall into
First-timers often grab the first formula they remember, mix units (cm with m, minutes with seconds), or drop a minus sign. Write the formula in words first: “this equals that, because…”
10 · 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).
11 · 3 Marks · Sine, cosine & area rules
cos θ = 12 in [0°; 360°]
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
Notes: cos θ = +12. Interval a full revolution.
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°]
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: cos θ = 12 in [0°; 360°]. Circle that word — that is the missing variable.
Find
We are required to find: cos θ = 12 in [0°; 360°]. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
Write α = 60° with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives cos θ = 12 in [0°; 360°].
In other words — as if you have never seen this before
CAST: cosine is positive in Q1 and Q4. Reference angle 60°.
2. What do we already know?
cos θ = +12. Interval a full revolution.
3. Which method — and why this one?
CAST: cosine is positive in Q1 and Q4. Reference angle 60°.
4. Step-by-step working
01 / 05
Reference
cos⁻¹(1/2)=60°.
α=60°
Why this step: One job per line so you can find a slip, and so a marker can follow the marks.
02 / 05
Quadrants
Q1: 60°. Q4: 360°−60°=300°.
60° and 300°
Why this step: Write the formula before numbers. The memo awards a method mark even if a digit slips later.
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
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.
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
60° and 300°
Copy onto your answer sheet — in this order
Formula: α=60°
Answer: 60° and 300°
6. Check it like a marker
cos 120° = −12, the other sign. Sketch the CAST diagram.
7. A trap first-timers fall into
60° and 120° is the sine pair. Cosine does not live in Q2 when it is positive.
12 · 4 Marks · Variance, ogives & five-number summary
Outlier fence: values below Q1 − 1.5 IQR
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: Outlier fence: values below Q1 − 1.5 IQR
Before any working — hunt first
Given
Numbers and units already in the sentence: 1 · 1.5 IQR. 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: 5 IQR. Circle that word — that is the missing variable.
Find
We are required to find: 5 IQR. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: 1. Then substitute the given, one line at a time, to get 5 IQR.
In other words — as if you have never seen this before
1.5×IQR rule.
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 1 · 1.5 IQR. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
1.5×IQR rule.
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 · 1.5 IQR. 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: 5 IQR. Circle that word — that is the missing variable.
Find: We are required to find: 5 IQR. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: What we need to calculate: 1. Then substitute the given, one line at a time, to get 5 IQR.
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 · 1.5 IQR. 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
1.5×IQR rule.
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
1.5×IQR rule.
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
are outliers
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
are outliers
Copy onto your answer sheet — in this order
Answer: are outliers
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 · 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…”
14 · 2 Marks · Parabola, hyperbola & exponential
y=−x²+4. Maximum y=
Parabola, hyperbola & exponential
Formulas
y=a(x + p)²+q
Shifted parabola
Notes: A downward parabola. Vertex on the y-axis.
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 = −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.
15 · 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…”
16 · 2 Marks · Quadratic equations & nature of roots
Nature of roots of x² + 6x + 9 = 0
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: Nature of roots of x² + 6x + 9 = 0
Before any working — hunt first
Given
Numbers and units already in the sentence: 6x · 9 · 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 of x² + 6x + 9 = 0. Circle that word — that is the missing variable.
Find
We are required to find: Nature of roots of x² + 6x + 9 = 0. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: real, rational, equal. Then substitute the given, one line at a time, to get Nature of roots of x² + 6x + 9 = 0.
In other words — as if you have never seen this before
real, rational, equal
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 6x · 9 · 0. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
real, rational, equal
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: 6x · 9 · 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 of x² + 6x + 9 = 0. Circle that word — that is the missing variable.
Find: We are required to find: Nature of roots of x² + 6x + 9 = 0. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: What we need to calculate: real, rational, equal. Then substitute the given, one line at a time, to get Nature of roots of x² + 6x + 9 = 0.
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: 6x · 9 · 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
real, rational, equal
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 = 36 − 36 = 0. Equal real rational roots; it is a perfect square (x+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
real, rational, equal
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
real, rational, equal
Copy onto your answer sheet — in this order
Answer: real, rational, equal
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 · 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…”
18 · 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…”
19 · 2 Marks · Dependent events & Venn
P(A or B) for mutually exclusive events
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: P(A or B) for mutually exclusive events
Before any working — hunt first
Given
Numbers and units already in the sentence: no numbers yet — only a name, a diagram, or a worded unknown. Copy each one into a column. A name without a number is not given.
Missing
After you copy the given, the letter that still has no number is missing. The paper is pointing at: P(A or B) for mutually exclusive events. Circle that word — that is the missing variable.
Find
We are required to find: P(A or B) for mutually exclusive events. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: No overlap, so add. Then substitute the given, one line at a time, to get P(A or B) for mutually exclusive events.
In other words — as if you have never seen this before
No overlap, so add.
2. What do we already know?
Copy these onto the left of your page, one per line, with units: every number and its unit in the sentence. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
No overlap, so add.
4. Step-by-step working
01 / 08
Step A — Read the question twice
Slow down. Circle the word that names the unknown. Underline every number and its unit. Do not grab a formula yet — first decide what the paper is actually asking.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
02 / 08
Step A2 — Hunt: Given, Missing, Find, Calculate
Given: Numbers and units already in the sentence: no numbers yet — only a name, a diagram, or a worded unknown. Copy each one into a column. A name without a number is not given.
Missing: After you copy the given, the letter that still has no number is missing. The paper is pointing at: P(A or B) for mutually exclusive events. Circle that word — that is the missing variable.
Find: We are required to find: P(A or B) for mutually exclusive events. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: What we need to calculate: No overlap, so add. Then substitute the given, one line at a time, to get P(A or B) for mutually exclusive events.
Why this step: Do not name the letters yet. First hunt: what already has a number (Given), what has no number (Missing), the word the paper wants (Find), and which formula still contains the missing letter (Calculate).
03 / 08
Step B — Write the given in a column
Copy these onto the left of your page, one per line, with units: every number and its unit in the sentence. Mixing cm with m, or minutes with seconds, loses a method mark.
Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
04 / 08
Step C — Name the method in words
No overlap, so add.
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
No overlap, so add.
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
P(A)+P(B)
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
P(A)+P(B)
Copy onto your answer sheet — in this order
Answer: P(A)+P(B)
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 · 2 Marks · Quadratic equations & nature of roots
Nature of roots of x² + 4x + 4 = 0
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: Nature of roots of x² + 4x + 4 = 0
Before any working — hunt first
Given
Numbers and units already in the sentence: 4x · 4 · 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 of x² + 4x + 4 = 0. Circle that word — that is the missing variable.
Find
We are required to find: Nature of roots of x² + 4x + 4 = 0. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: real, rational, equal. Then substitute the given, one line at a time, to get Nature of roots of x² + 4x + 4 = 0.
In other words — as if you have never seen this before
real, rational, equal
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 4x · 4 · 0. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
real, rational, equal
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: 4x · 4 · 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 of x² + 4x + 4 = 0. Circle that word — that is the missing variable.
Find: We are required to find: Nature of roots of x² + 4x + 4 = 0. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: What we need to calculate: real, rational, equal. Then substitute the given, one line at a time, to get Nature of roots of x² + 4x + 4 = 0.
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: 4x · 4 · 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
real, rational, equal
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
Δ=0, perfect square.
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
real, rational, equal
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
real, rational, equal
Copy onto your answer sheet — in this order
Answer: real, rational, equal
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 · Quadratic equations & nature of roots
For ax²+bx+c=0, if Δ>0 and not a square, roots are
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: For ax²+bx+c=0, if Δ>0 and not a square, roots are
In other words — as if you have never seen this before
Positive non-square discriminant.
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 0 · 0 and. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
Positive non-square discriminant.
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: 0 · 0 and. 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
Positive non-square discriminant.
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
Positive non-square discriminant.
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
real irrational unequal
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
real irrational unequal
Copy onto your answer sheet — in this order
Answer: real irrational unequal
6. Check it like a marker
Does your answer appear as one of the options? Do the units match what was asked (m, N, J, °)? If you put the number back into the original equation, does it still hold? Estimate: is the size sensible?
7. A trap first-timers fall into
First-timers often grab the first formula they remember, mix units (cm with m, minutes with seconds), or drop a minus sign. Write the formula in words first: “this equals that, because…”
22 · 3 Marks · Parabola, hyperbola & exponential
Turning point of y = (x − 3)² + 2
Parabola, hyperbola & exponential
Formulas
y=a(x + p)²+q
Shifted parabola
Notes: y = (x − 3)² + 2. Completed-squareturning-point form a(x − p)² + q.
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 + 2
Before any working — hunt first
Given
Numbers and units already in the sentence: 3 · 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 of y = (x − 3)² + 2. Circle that word — that is the missing variable.
Find
We are required to find: Turning point of y = (x − 3)² + 2. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
Write TP = (p; q) = (3; 2) 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 + 2.
In other words — as if you have never seen this before
The turning point is (p; q). Here p = 3 (opposite of the sign inside), q = 2.
2. What do we already know?
y = (x − 3)² + 2. Completed-squareturning-point form a(x − p)² + q.
3. Which method — and why this one?
The turning point is (p; q). Here p = 3 (opposite of the sign inside), q = 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: 3 · 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 of y = (x − 3)² + 2. Circle that word — that is the missing variable.
Find: We are required to find: Turning point of y = (x − 3)² + 2. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: Write TP = (p; q) = (3; 2) 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 + 2.
Why this step: Do not name the letters yet. First hunt: what already has a number (Given), what has no number (Missing), the word the paper wants (Find), and which formula still contains the missing letter (Calculate).
03 / 08
Step B — Write the given in a column
y = (x − 3)² + 2. Completed-squareturning-point form a(x − p)² + q.
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
The turning point is (p; q). Here p = 3 (opposite of the sign inside), q = 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
Read (p; q)
a(x − p)²+q has TP (p; q). We have (x − 3)²+2, so p=3, q=2. TP=(3; 2).
TP=(p; q)=(3; 2)
Why this step: The minus inside the bracket is already “x minus p”. Do not flip 3 into −3 — that is the (x + 3)² story.
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
Formula: TP=(p; q)=(3; 2)
Answer: (3; 2)
6. Check it like a marker
At x = 3, y = 0 + 2 = 2. Neighbours: x = 2 or 4 give y = 1 + 2 = 3, which is higher, so (3; 2) is a minimum. Good.
7. A trap first-timers fall into
Writing (−3; 2) because “the sign flips”. It already flipped in the form (x − p).
23 · 2 Marks · Parabola, hyperbola & exponential
y = 2^x. y when x = 3
Parabola, hyperbola & exponential
Formulas
y=a(x + p)²+q
Shifted parabola
Notes: Exponential base 2. x=3.
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 = 2^x. y when x = 3
Before any working — hunt first
Given
Numbers and units already in the sentence: 2 · 3. 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: y when x = 3. Circle that word — that is the missing variable.
Find
We are required to find: y when x = 3. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
Write 8 with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives y when x = 3.
In other words — as if you have never seen this before
2x means multiply 2, x times. 2³=8, not 2×3.
2. What do we already know?
Exponential base 2. x=3.
3. Which method — and why this one?
2x means multiply 2, x times. 2³=8, not 2×3.
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: 2 · 3. 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: y when x = 3. Circle that word — that is the missing variable.
Find: We are required to find: y when x = 3. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: Write 8 with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives y when x = 3.
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
Exponential base 2. x=3.
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
2x means multiply 2, x times. 2³=8, not 2×3.
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
Expand
2×2×2=8.
8
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
8
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
8
Copy onto your answer sheet — in this order
Formula: 8
Answer: 8
6. Check it like a marker
Table: x=0 →1, x=1 →2, x=2 →4, x=3 →8.
7. A trap first-timers fall into
6 is 2×3. Exponents are not multiplication.
24 · 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…”
25 · 3 Marks · Identities, reduction & equations
Inclination of a line with m = 1 is
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: Inclination of a line with m = 1 is
Before any working — hunt first
Given
Numbers and units already in the sentence: 1 is. 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: Inclination of a line with m = 1 is. Circle that word — that is the missing variable.
Find
We are required to find: Inclination of a line with m = 1 is. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: tan θ = m = 1. Then substitute the given, one line at a time, to get Inclination of a line with m = 1 is.
In other words — as if you have never seen this before
tan θ=m=1→θ=45°.
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 1 is. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
tan θ=m=1→θ=45°.
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 is. 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: Inclination of a line with m = 1 is. Circle that word — that is the missing variable.
Find: We are required to find: Inclination of a line with m = 1 is. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: What we need to calculate: tan θ = m = 1. Then substitute the given, one line at a time, to get Inclination of a line with m = 1 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: 1 is. Mixing cm with m, or minutes with seconds, loses a method mark.
Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
04 / 08
Step C — Name the method in words
tan θ=m=1→θ=45°.
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
tan θ=m=1→θ=45°.
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
45°
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
45°
Copy onto your answer sheet — in this order
Answer: 45°
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 · 3 Marks · Parabola, hyperbola & exponential
TP of y = −(x+3)² + 1
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: TP of y = −(x+3)² + 1
Before any working — hunt first
Given
Numbers and units already in the sentence: 3 · 1. Copy each one into a column. A name without a number is not given.
Missing
After you copy the given, the letter that still has no number is missing. The paper is pointing at: TP of y = −(x+3)² + 1. Circle that word — that is the missing variable.
Find
We are required to find: TP of y = −(x+3)² + 1. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: (−3; 1). Then substitute the given, one line at a time, to get TP of y = −(x+3)² + 1.
In other words — as if you have never seen this before
(−3; 1)
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 3 · 1. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
(−3; 1)
4. Step-by-step working
01 / 08
Step A — Read the question twice
Slow down. Circle the word that names the unknown. Underline every number and its unit. Do not grab a formula yet — first decide what the paper is actually asking.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
02 / 08
Step A2 — Hunt: Given, Missing, Find, Calculate
Given: Numbers and units already in the sentence: 3 · 1. Copy each one into a column. A name without a number is not given.
Missing: After you copy the given, the letter that still has no number is missing. The paper is pointing at: TP of y = −(x+3)² + 1. Circle that word — that is the missing variable.
Find: We are required to find: TP of y = −(x+3)² + 1. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: What we need to calculate: (−3; 1). Then substitute the given, one line at a time, to get TP of y = −(x+3)² + 1.
Why this step: Do not name the letters yet. First hunt: what already has a number (Given), what has no number (Missing), the word the paper wants (Find), and which formula still contains the missing letter (Calculate).
03 / 08
Step B — Write the given in a column
Copy these onto the left of your page, one per line, with units: 3 · 1. Mixing cm with m, or minutes with seconds, loses a method mark.
Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
04 / 08
Step C — Name the method in words
(−3; 1)
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
05 / 08
Reason
a(x−(−3))²+1.
Why this step: We only do one job per line so a marker can follow the marks — and so you can find a slip without starting over.
06 / 08
Step G — Sense-check the size
Ask: is this the kind of number a Grade 10–12 paper would want? A speed of 3 000 ms for a learner walking, or a probability bigger than 1, means a unit or algebra slip. Estimate first: round the numbers and do the sum in your head.
Why this step: Markers write ‘unrealistic’ in the margin when the method is right but the size is wild. A 10-second head check often finds a missing 10² or a unit that never changed.
07 / 08
Step F — Box the answer with its unit
(−3; 1)
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
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; 1)
Copy onto your answer sheet — in this order
Answer: (−3; 1)
6. Check it like a marker
Does your answer appear as one of the options? Do the units match what was asked (m, N, J, °)? If you put the number back into the original equation, does it still hold? Estimate: is the size sensible?
7. A trap first-timers fall into
First-timers often grab the first formula they remember, mix units (cm with m, minutes with seconds), or drop a minus sign. Write the formula in words first: “this equals that, because…”
27 · 2 Marks · Identities, reduction & equations
Area of triangle a=6, b=5, included angle 30°
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: Area of triangle a=6, b=5, included angle 30°
In other words — as if you have never seen this before
½ab sin C=½×6×5×0.5=7.5.
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 6 · 5 · 30°. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
½ab sin C=½×6×5×0.5=7.5.
4. Step-by-step working
01 / 07
Step A — Read the question twice
Slow down. Circle the word that names the unknown. Underline every number and its unit. Do not grab a formula yet — first decide what the paper is actually asking.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
02 / 07
Step B — Write the given in a column
Copy these onto the left of your page, one per line, with units: 6 · 5 · 30°. Mixing cm with m, or minutes with seconds, loses a method mark.
Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
03 / 07
Step C — Name the method in words
½ab sin C=½×6×5×0.5=7.5.
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.
04 / 07
Working
½ab sin C=½×6×5×0.5=7.5.
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.
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
7.5
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.
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
7.5
Copy onto your answer sheet — in this order
Answer: 7.5
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…”
28 · 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…”
29 · 3 Marks · Identities, reduction & equations
tan 45°=
Identities, reduction & equations
Formulas
sin²θ+cos²θ=1
Pythagorean identity
Notes: Special angle 45°.
How we got there
Read this like a teacher at the board. We go slowly, name every number, and say why we take each step.
Explain in another language · 11
1. What is the question asking?
Read the question twice. Circle the unknown. Underline every number and unit. The paper is asking: tan 45° =
Before any working — hunt first
Given
Numbers and units already in the sentence: 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: tan 45° =. Circle that word — that is the missing variable.
Find
We are required to find: tan 45° =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
Write 1 with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives tan 45° =.
In other words — as if you have never seen this before
At 45°, opposite = adjacent, so tan = 1.
2. What do we already know?
Special angle 45°.
3. Which method — and why this one?
At 45°, opposite = adjacent, so tan = 1.
4. Step-by-step working
01 / 08
Step A — Read the question twice
Slow down. Circle the word that names the unknown. Underline every number and its unit. Do not grab a formula yet — first decide what the paper is actually asking.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
02 / 08
Step A2 — Hunt: Given, Missing, Find, Calculate
Given: Numbers and units already in the sentence: 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: tan 45° =. Circle that word — that is the missing variable.
Find: We are required to find: tan 45° =. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: Write 1 with empty letters. Cross out every letter that already has a number. Calculate the letter that is still standing — that gives tan 45° =.
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
Special angle 45°.
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
At 45°, opposite = adjacent, so tan = 1.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
05 / 08
Isosceles right triangle
oppadj=11=1.
1
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
1
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
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
Copy onto your answer sheet — in this order
Formula: 1
Answer: 1
6. Check it like a marker
sin 45 = cos 45, so their ratio is 1.
7. A trap first-timers fall into
undefined is tan 90°. 45° is safe.
30 · 3 Marks · Inclination & circles
TP of y = 2(x − 4)² + 1
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: TP of y = 2(x − 4)² + 1
Before any working — hunt first
Given
Numbers and units already in the sentence: 2 · 4 · 1. Copy each one into a column. A name without a number is not given.
Missing
After you copy the given, the letter that still has no number is missing. The paper is pointing at: TP of y = 2(x − 4)² + 1. Circle that word — that is the missing variable.
Find
We are required to find: TP of y = 2(x − 4)² + 1. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate
What we need to calculate: (4; 1). Then substitute the given, one line at a time, to get TP of y = 2(x − 4)² + 1.
In other words — as if you have never seen this before
(4; 1)
2. What do we already know?
Copy these onto the left of your page, one per line, with units: 2 · 4 · 1. Mixing cm with m, or minutes with seconds, loses a method mark.
3. Which method — and why this one?
(4; 1)
4. Step-by-step working
01 / 08
Step A — Read the question twice
Slow down. Circle the word that names the unknown. Underline every number and its unit. Do not grab a formula yet — first decide what the paper is actually asking.
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
02 / 08
Step A2 — Hunt: Given, Missing, Find, Calculate
Given: Numbers and units already in the sentence: 2 · 4 · 1. Copy each one into a column. A name without a number is not given.
Missing: After you copy the given, the letter that still has no number is missing. The paper is pointing at: TP of y = 2(x − 4)² + 1. Circle that word — that is the missing variable.
Find: We are required to find: TP of y = 2(x − 4)² + 1. Write that as a symbol on the left, with a question mark, before you touch a formula.
Calculate: What we need to calculate: (4; 1). Then substitute the given, one line at a time, to get TP of y = 2(x − 4)² + 1.
Why this step: Do not name the letters yet. First hunt: what already has a number (Given), what has no number (Missing), the word the paper wants (Find), and which formula still contains the missing letter (Calculate).
03 / 08
Step B — Write the given in a column
Copy these onto the left of your page, one per line, with units: 2 · 4 · 1. Mixing cm with m, or minutes with seconds, loses a method mark.
Why this step: A column with units stops you mixing cm with m, or minutes with seconds. Markers can also see you started correctly.
04 / 08
Step C — Name the method in words
(4; 1)
Why this step: If you can say the method in a sentence, you will pick the right formula. First-timers skip this and grab the first equation they remember.
05 / 08
Working
Form a(x − p)² + q has turning point (p; q) = (4; 1). The 2 stretches, it does not move the point.
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; 1)
Why this step: A number without a unit is half a mark on a Physics paper, and a marker can miss a Maths answer that is buried in working.
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; 1)
Copy onto your answer sheet — in this order
Answer: (4; 1)
6. Check it like a marker
Does your answer appear as one of the options? Do the units match what was asked (m, N, J, °)? If you put the number back into the original equation, does it still hold? Estimate: is the size sensible?
7. A trap first-timers fall into
First-timers often grab the first formula they remember, mix units (cm with m, minutes with seconds), or drop a minus sign. Write the formula in words first: “this equals that, because…”