Ashdale Secondary College · T2 Weeks 3–6 · 16 Lessons
L1 · T2 Week 3
What is Pythagoras' Theorem?
Identify the hypotenuse, write the theorem and verify right-angled triangles
⚡ Do Now 8 min · display on board as students enter
1
Write the formula for the area of a rectangle
Answer: A = l × w
2
Find the area: length = 8 cm, width = 3 cm
Answer: 24 cm²
3
What is 5²?
Answer: 25
4
What is 12²?
Answer: 144
5
What is √36?
Answer: 6
6
What is √100?
Answer: 10
📖 Key Vocabulary
Right angle
90° — shown by the small square □
Hypotenuse
Longest side — always OPPOSITE the right angle
Legs (a and b)
The two shorter sides — they FORM the right angle
📐 Formula / Rule
a² + b² = c² where c = hypotenuse
✏️ Worked Examples teacher-led · reveal steps one at a time
Verify that 3, 4, 5 is a right-angled triangle
→Longest side = 5 → c = 5, a = 3, b = 4
→a² + b² = 3² + 4² = 9 + 16 = 25
→c² = 5² = 25
→25 = 25 ✓ It IS right-angled
Verify that 6, 7, 8 is a right-angled triangle
→Longest side = 8 → c = 8, a = 6, b = 7
→a² + b² = 6² + 7² = 36 + 49 = 85
→c² = 8² = 64
→85 ≠ 64 ✗ NOT right-angled
📝 Practice Questions
1
What is 7²?
✓ 49
2
What is √81?
✓ 9
3
What is 11²?
✓ 121
4
Write Pythagoras' theorem from memory.
✓ a² + b² = c²
5
Does (8, 15, 17) form a right triangle? Show working.
✓ 64 + 225 = 289 = 17² ✓
6
Does (4, 5, 6) form a right triangle? Show working.
✓ 16 + 25 = 41 ≠ 36 ✗
7
Does (7, 24, 25) form a right triangle? Show working.
✓ 49 + 576 = 625 = 25² ✓
8
Does (9, 10, 14) form a right triangle? Show working.
✓ 81 + 100 = 181 ≠ 196 ✗
🎯 Exit Ticket
1
Write Pythagoras' theorem.
✓ a² + b² = c²
2
Does (6, 8, 10) form a right triangle? Show all working.
✓ 36 + 64 = 100 = 10² ✓ Yes
💡 Teacher note: Keep diagrams BIG. Re-draw for every example. Ahmed A — speak steps aloud as you write.
L2 · T2 Week 3
Finding the Hypotenuse
Use a² + b² = c² to find the hypotenuse, showing all working
⚡ Do Now 8 min · display on board as students enter
1
What is 6²?
Answer: 36
2
What is 9²?
Answer: 81
3
What is √49?
Answer: 7
4
What is √144?
Answer: 12
5
Write Pythagoras' theorem from memory.
Answer: a² + b² = c²
6
Which side is the hypotenuse? Sketch a right triangle and label it.
Answer:
📖 Key Vocabulary
Hypotenuse (c)
The side we are FINDING — longest, opposite 90°
a and b
The two known short sides — substitute these in
√ (square root)
The inverse of squaring — use the √ button on calculator
📐 Formula / Rule
c = √(a² + b²)
✏️ Worked Examples teacher-led · reveal steps one at a time
Find the hypotenuse when a = 3, b = 4
→c² = a² + b²
→c² = 3² + 4² = 9 + 16 = 25
→c = √25 = 5
Find the hypotenuse when a = 5, b = 7
→c² = 5² + 7²
→c² = 25 + 49 = 74
→c = √74 ≈ 8.60 (2 d.p.)
📝 Practice Questions
1
a = 6, b = 8. Find c.
✓ 10
2
a = 9, b = 12. Find c.
✓ 15
3
a = 5, b = 10. Find c (1 d.p.)
✓ 11.2
4
a = 7, b = 7. Find c (2 d.p.)
✓ 9.90
5
a = 2.5, b = 6. Find c (2 d.p.)
✓ 6.50
6
a = 11, b = 4. Find c (2 d.p.)
✓ 11.70
7
a = 9, b = 40. Find c.
✓ 41
8
a = 8, b = 8. Find c (2 d.p.)
✓ 11.31
🎯 Exit Ticket
1
Find c when a = 8, b = 15.
✓ 17
2
Find c when a = 4, b = 9. Round to 2 d.p.
✓ 9.85
💡 Teacher note: Insist on every line of working: c² = ... on one line, c = ... on the next.
L3 · T2 Week 3
Finding a Short Side
Rearrange Pythagoras' theorem to find a missing leg
⚡ Do Now 8 min · display on board as students enter
1
Find c: a = 6, b = 8
Answer: 10
2
Find c: a = 5, b = 12
Answer: 13
3
What is √64?
Answer: 8
4
100 − 36 = ?
Answer: 64
5
169 − 25 = ?
Answer: 144
📖 Key Vocabulary
Rearranging
Moving parts of the formula to isolate what we want to find
a² = c² − b²
Subtract the known leg² FROM the hypotenuse²
Inverse operation
Square rooting undoes squaring
📐 Formula / Rule
a = √(c² − b²) or b = √(c² − a²)
✏️ Worked Examples teacher-led · reveal steps one at a time
Find the missing leg when c = 13, b = 5
→a² + b² = c²
→a² + 5² = 13²
→a² + 25 = 169
→a² = 169 − 25 = 144
→a = √144 = 12
Find the missing leg when c = 10, b = 6
→a² = c² − b²
→a² = 10² − 6² = 100 − 36 = 64
→a = √64 = 8
📝 Practice Questions
1
c = 10, b = 6. Find a.
✓ 8
2
c = 17, b = 8. Find a.
✓ 15
3
c = 15, b = 9. Find a.
✓ 12
4
c = 20, b = 12. Find a.
✓ 16
5
c = 11, b = 5. Find a (2 d.p.)
✓ 9.80
6
c = 8, b = 3. Find a (2 d.p.)
✓ 7.42
7
c = 25, b = 7. Find a.
✓ 24
8
c = 30, b = 18. Find a.
✓ 24
🎯 Exit Ticket
1
Find a when c = 26, b = 10.
✓ 24
2
Find b when c = 14, a = 9. Round to 2 d.p.
✓ 10.44
💡 Teacher note: Common error: students ADD instead of SUBTRACT. Say: 'We already have the big side — we go DOWN.' Leave both rearrangements on the board all lesson.
L4 · T2 Week 3
Pythagoras + Area & Perimeter
Use Pythagoras to find a missing dimension, then calculate area and perimeter
⚡ Do Now 8 min · display on board as students enter
1
Write the formula for area of a triangle.
Answer: A = ½ × b × h
2
Triangle: base 10 cm, height 6 cm. Area?
Answer: 30 cm²
3
Find c: a = 8, b = 15
Answer: 17
4
Find the missing leg: c = 13, b = 5
Answer: 12
📖 Key Vocabulary
Strategy
Find the unknown side with Pythagoras FIRST — then use area or perimeter
Exact form
Leave as a surd — e.g. √112 cm
Approximate form
Rounded decimal — e.g. 10.58 cm
📐 Formula / Rule
Step 1: Find unknown side (Pythagoras) → Step 2: Area = ½ × b × h → Step 3: P = sum of all sides
✏️ Worked Examples teacher-led · reveal steps one at a time
Right triangle: base 12 cm, hypotenuse 16 cm. Find height, area and perimeter.
→Step 1 — Height: h = √(16² − 12²) = √(256 − 144) = √112 ≈ 10.58 cm
→Step 2 — Area: A = ½ × 12 × 10.58 = 63.50 cm²
→Step 3 — Perimeter: P = 12 + 16 + 10.58 = 38.58 cm
Isosceles triangle: base 16 cm, equal sides 10 cm. Find the height then area.
→Step 1 — Split in half: base half = 8 cm, hyp = 10 cm
→Step 2 — Height: h² = 10² − 8² = 100 − 64 = 36 → h = 6 cm
→Step 3 — Area: A = ½ × 16 × 6 = 48 cm²
📝 Practice Questions
1
Right triangle: legs 9 cm and 12 cm. Find (a) hypotenuse (b) perimeter (c) area.
✓ (a) 15 cm (b) 36 cm (c) 54 cm²
2
Right triangle: hyp = 20 cm, one leg = 12 cm. Find the other leg then the area.
✓ leg = 16 cm, A = 96 cm²
3
Isosceles triangle: base = 10 cm, equal sides = 13 cm. Find height then area.
✓ h = 12 cm, A = 60 cm²
4
Right triangle: legs 7 cm and 7 cm. Find hyp (2 d.p.), perimeter (2 d.p.) and area.
✓ c ≈ 9.90 cm, P ≈ 23.90 cm, A = 24.5 cm²
🎯 Exit Ticket
1
Right triangle: legs 6 cm and 8 cm. Find the hypotenuse, then the area.
✓ c = 10 cm, A = 24 cm²
💡 Teacher note: Find unknown side FIRST is the key habit. Ahmed A — write each step on a new line.
L5 · T2 Week 3
Is it Right-Angled? (The Converse)
Use the converse of Pythagoras to test whether a triangle is right-angled
⚡ Do Now 8 min · display on board as students enter
1
Find c: a = 6, b = 8
Answer: 10
2
Find a: c = 25, b = 24
Answer: 7
3
What does the □ symbol mean on a diagram?
Answer: 90° right angle
4
What is 15²?
Answer: 225
📖 Key Vocabulary
Converse
The REVERSE of the theorem — use it to TEST a triangle
Pythagorean triad
Three whole numbers satisfying a² + b² = c²
📐 Formula / Rule
If a² + b² = c² → RIGHT-ANGLED ✓ If a² + b² ≠ c² → NOT right-angled ✗
✏️ Worked Examples teacher-led · reveal steps one at a time
Are sides 7, 24, 25 a right-angled triangle?
→Longest side = 25 → c = 25, a = 7, b = 24
→a² + b² = 49 + 576 = 625
→c² = 625
→625 = 625 ✓ RIGHT-ANGLED
Are sides 5, 7, 9 a right-angled triangle?
→Longest side = 9 → c = 9, a = 5, b = 7
→a² + b² = 25 + 49 = 74
→c² = 81
→74 ≠ 81 ✗ NOT right-angled
📝 Practice Questions
1
8, 15, 17 — right-angled?
✓ 64 + 225 = 289 = 17² ✓ Yes
2
6, 9, 11 — right-angled?
✓ 36 + 81 = 117 ≠ 121 ✗ No
3
20, 21, 29 — right-angled?
✓ 400 + 441 = 841 = 29² ✓ Yes
4
4, 7, 8 — right-angled?
✓ 16 + 49 = 65 ≠ 64 ✗ No
5
9, 40, 41 — right-angled?
✓ 81 + 1600 = 1681 = 41² ✓ Yes
6
10, 12, 15 — right-angled?
✓ 100 + 144 = 244 ≠ 225 ✗ No
🎯 Exit Ticket
1
Are sides 11, 60, 61 a right triangle? Show full working.
✓ 121 + 3600 = 3721 = 61² ✓ Yes
💡 Teacher note: Always use the LONGEST side as c — students who pick the wrong side get the wrong answer.
L6 · T2 Week 3
Word Problems & Drawing Diagrams
Draw a labelled diagram from a word problem and solve using Pythagoras
⚡ Do Now 8 min · display on board as students enter
1
Find c: a = 5, b = 12
Answer: 13
2
Are 8, 15, 17 right-angled?
Answer: Yes: 64 + 225 = 289 = 17² ✓
3
A triangle has sides 3, 4 and 5. Which is the hypotenuse?
Answer: 5
📖 Key Vocabulary
4-Step Method
Read → Draw → Identify → Solve
Diagram
Always sketch and label — no diagram, no marks
Units
Always include cm / m / km in the answer
📐 Formula / Rule
READ the problem → DRAW and label → IDENTIFY what to find → SOLVE with Pythagoras
✏️ Worked Examples teacher-led · reveal steps one at a time
A ladder leans against a wall. Base is 1.5 m from wall, wall is 3.5 m high. How long is the ladder?
→Draw: right triangle — base = 1.5 m, vertical = 3.5 m, ladder = c
→c² = 1.5² + 3.5² = 2.25 + 12.25 = 14.5
→c = √14.5 ≈ 3.81 m
A field is 40 m long and 30 m wide. How long is the diagonal path?
→Draw: rectangle — diagonal is the hypotenuse
→c² = 40² + 30² = 1600 + 900 = 2500
→c = √2500 = 50 m
📝 Practice Questions
1
A ramp is 4 m long and reaches 1.5 m high. How far does it extend horizontally? (2 d.p.)
✓ √(16 − 2.25) = √13.75 ≈ 3.71 m
2
A screen is 55 cm wide and 31 cm tall. Find its diagonal. (2 d.p.)
✓ √(3025 + 961) = √3986 ≈ 63.13 cm
3
Two friends walk from the same corner — one 6 km east, one 8 km north. How far apart?
✓ √(36 + 64) = 10 km
4
A 12 m rope goes from the top of a 9 m pole to a peg. How far is the peg from the base?
✓ √(144 − 81) = √63 ≈ 7.94 m
🎯 Exit Ticket
1
A TV is 90 cm wide and 50 cm tall. Find the diagonal. Round to 1 d.p.
✓ √10600 ≈ 103.0 cm
💡 Teacher note: No diagram = no marks. Non-negotiable. Ahmed A — a rough sketch is fine.
L7 · T2 Week 3
Pythagoras — Mixed Practice
Apply all Pythagoras skills: hypotenuse, short side, converse and word problems
⚡ Do Now 8 min · display on board as students enter
1
Find c: a = 7, b = 24
Answer: 25
2
Find a: c = 17, b = 15
Answer: 8
3
Are 6, 8, 10 right-angled?
Answer: Yes: 36 + 64 = 100 ✓
4
Write the 4-step word problem method.
Answer: Read, Draw, Identify, Solve
📖 Key Vocabulary
Exact answer
Leave as a surd — e.g. √74
Approximate answer
Rounded decimal — e.g. 8.60 cm
📐 Formula / Rule
a² + b² = c² | a = √(c² − b²) | Converse: test if a² + b² = c²
✏️ Worked Examples teacher-led · reveal steps one at a time
Right triangle with legs 6 m and 6 m. Find hypotenuse, area and perimeter.
→c = √(6² + 6²) = √72 ≈ 8.49 m
→Area = ½ × 6 × 6 = 18 m²
→Perimeter = 6 + 6 + 8.49 = 20.49 m
📝 Practice Questions
1
Find c: a = 9, b = 12
✓ 15
2
Find a: c = 10, b = 8
✓ 6
3
Are 5, 12, 13 right-angled? Show working.
✓ 25 + 144 = 169 = 13² ✓ Yes
4
Ladder 5 m long, base 2 m from wall. How high does it reach?
✓ √(25 − 4) = √21 ≈ 4.58 m
5
Right triangle legs 9 cm, 12 cm: find hyp, area and perimeter.
✓ c=15, A=54 cm², P=36 cm
6
a = 7, b = 9. Find c (2 d.p.)
✓ 11.40
7
c = 20, b = 13. Find a (2 d.p.)
✓ 15.20
8
Are 8, 9, 12 right-angled?
✓ 64 + 81 = 145 ≠ 144 ✗ No
🎯 Exit Ticket
1
Find c when a = 20, b = 21.
✓ 29
2
A rope of 10 m is pegged 6 m from a pole. How high up the pole does it reach?
✓ 8 m
💡 Teacher note: Circulate actively. Use levelled prompts: 'What do you know? What are you finding?'
L8 · T2 Week 3
Pythagoras — Consolidation & Test Prep
Review all Pythagoras skills and prepare for the Measurement Test
⚡ Do Now 8 min · display on board as students enter
1
Find c: a = 8, b = 6
Answer: 10
2
Find a: c = 13, b = 5
Answer: 12
3
Are 9, 40, 41 right-angled?
Answer: Yes ✓
4
Ladder 5 m, base 2 m from wall. Height?
Answer: √21 ≈ 4.58 m
5
Right triangle legs 3 m, 4 m: perimeter?
Answer: 12 m
📖 Key Vocabulary
Measurement Test
NEXT WEEK — covers area, perimeter, composite shapes and Pythagoras
📐 Formula / Rule
a² + b² = c² | a = √(c²−b²) | Converse | Word problems with diagram
📝 Practice Questions
1
a = 3, b = 4 → c = ?
✓ 5
2
a = 5, b = 12 → c = ?
✓ 13
3
c = 10, b = 6 → a = ?
✓ 8
4
c = 26, b = 10 → a = ?
✓ 24
5
a = 7, b = 9 → c (2 d.p.)
✓ 11.40
6
c = 14, b = 11 → a (2 d.p.)
✓ 8.66
7
Right tri legs 9, 12 cm: find hyp, area, perimeter.
✓ c=15, A=54 cm², P=36 cm
8
Ladder 5 m, base 2 m from wall. How high does it reach?
✓ √21 ≈ 4.58 m
🎯 Exit Ticket
1
What topics are on the Measurement Test? List all of them.
✓ Area, perimeter, composite shapes, Pythagoras, converse, word problems
💡 Teacher note: REMINDER: Measurement Test is Lesson 10 (T2 Week 4, second lesson). Write test topics on board at end of this lesson.
L9 · T2 Week 4
Ways to Earn Money
Name and calculate different types of income: wages, salary, commission, piecework
⚡ Do Now 8 min · display on board as students enter
✏️ Worked Examples teacher-led · reveal steps one at a time
After test — Sam earns $22/hr and works 38 hours. Weekly pay?
→Pay = $22 × 38 = $836
📝 Practice Questions
1
$18/hr × 40 hrs = ?
✓ $720
2
Salary $52,000/year. Weekly pay?
✓ $1,000
3
$23.50/hr × 6 hrs = ?
✓ $141
4
Salary $78,000/year. Fortnightly pay?
✓ $3,000
🎯 Exit Ticket
1
You earn $20/hr. You work 9 hours. How much do you earn?
✓ $180
💡 Teacher note: TEST: seating plan strictly enforced, clear desks, formula sheet provided. Ahmed A — extra time if needed. After test: keep intro light and engaging.
L11 · T2 Week 4
Wages & Penalty Rates
Calculate wages including time and a half, double time and mixed weeks
⚡ Do Now 8 min · display on board as students enter
1
$18 × 6 = ?
Answer: $108
2
$18 × 1.5 = ?
Answer: $27
3
$18 × 2 = ?
Answer: $36
4
$22.50 × 8 = ?
Answer: $180
5
What does 'time and a half' mean? Write in your own words.
Answer: 1.5× normal rate
📖 Key Vocabulary
Normal time
×1 — standard hourly rate
Time and a half
×1.5 — common for Saturday shifts
Double time
×2 — common for Sunday and public holidays
Double time and a half
×2.5 — public holiday premium
📐 Formula / Rule
Penalty pay = hourly rate × multiplier × hours worked
✏️ Worked Examples teacher-led · reveal steps one at a time
Jake earns $17/hr. Mon–Fri 8 hrs/day, Sat 4 hrs (T&H), Sun 3 hrs (DT). Total?
Mia $19.50/hr. Tue–Sat 8 hrs. Sat = time & half. Total?
✓ $624 + $234 = $858
6
$24/hr. 3 hrs double time and a half. Pay?
✓ $24 × 2.5 × 3 = $180
🎯 Exit Ticket
1
You earn $21/hr. 6 hours on Sunday at double time. How much?
✓ $21 × 2 × 6 = $252
2
What multiplier is used for time and a half?
✓ 1.5
💡 Teacher note: Real-world connection: which shifts pay the most and why? The 'Your Roster' activity works well here.
L12 · T2 Week 4
Payslips — Reading & Calculating
Read a payslip and calculate gross pay, tax, super and net pay
⚡ Do Now 8 min · display on board as students enter
1
$850 − $125 = ?
Answer: $725
2
10% of $960 = ?
Answer: $96
3
11% of $840 = ?
Answer: $92.40
4
$1,200 − $312 − $96 = ?
Answer: $792
📖 Key Vocabulary
Gross Pay
Total earnings BEFORE any deductions
Tax
Money withheld for the government (ATO)
Superannuation (Super)
11% employer contribution — goes into your retirement fund
Net Pay
What you actually receive — gross minus deductions
YTD
Year to Date — total earned since 1 July
📐 Formula / Rule
Net Pay = Gross − Tax − Other Deductions | Super = 11% × Gross (employer pays this separately)
✏️ Worked Examples teacher-led · reveal steps one at a time
Gross = $1,400. Tax = $285. Super = 11% of gross. Find net pay and super.
→Super = 11% × $1,400 = $154 (employer pays this on top)
→Net Pay = $1,400 − $285 = $1,115
📝 Practice Questions
1
Gross=$960, Tax=$142, Other=$20. Net pay?
✓ $798
2
Gross=$2,100, Tax=$385. Net pay?
✓ $1,715
3
Gross=$780. Super=11%. How much super does employer pay?
✓ $85.80
4
Net=$1,260. Tax=$194. What was the gross?
✓ $1,454
5
Jake: 38 hrs at $23/hr. Tax=18% of gross. Find gross, tax, net.
✓ Gross=$874, Tax=$157.32, Net=$716.68
🎯 Exit Ticket
1
What is the difference between gross and net pay?
✓ Gross = before deductions, Net = take-home after tax etc.
2
Gross=$1,050, Tax=$210, Super=11%. Find net pay and employer super.
✓ Net=$840, Super=$115.50
💡 Teacher note: Key misconception: super is paid BY the employer ON TOP of gross — not deducted from wages. Say: 'It's a bonus put into a retirement account you can't touch yet.'
L13 · T2 Week 5
Commission & Piecework
Calculate commission (with and without base pay) and piecework earnings
⚡ Do Now 8 min · display on board as students enter
1
4% of $30,000 = ?
Answer: $1,200
2
2.5% of $12,000 = ?
Answer: $300
3
$3.20 × 45 = ?
Answer: $144
4
Gross=$920, Tax=$158. Net pay?
Answer: $762
📖 Key Vocabulary
Base + commission
Fixed amount plus a % of sales on top
Piecework rate
Fixed $ per item or task completed
Compare income
Calculate totals for each option — then decide
📐 Formula / Rule
Commission = rate% × total sales | Piecework = rate per item × number of items
✏️ Worked Examples teacher-led · reveal steps one at a time
Real estate agent: base $800/week + 1.5% commission. Sold a house for $620,000.