Ngā Ine · Measurement
Area, Pythagoras, speed–distance–time, metric units, and significant figures · Years 9–10
Ngā Whāinga Akoranga · Learning Intentions
- Calculate perimeter, circumference, and area of 2D shapes including circles, parallelograms, and trapezoids.
- Apply Pythagoras' theorem to find unknown side lengths in right-angled triangles.
- Use the speed–distance–time formula to solve practical problems.
- Convert between metric units and represent numbers in scientific notation.
- Identify the number of significant figures in a measurement and apply appropriate precision to solutions.
Paearu Angitu · Success Criteria
- I can find the area and circumference of a circle using A = πr² and C = 2πr.
- I can use Pythagoras' theorem (a² + b² = c²) to find an unknown side or verify a right angle.
- I can rearrange speed = distance ÷ time to find any of the three quantities.
- I can write numbers in scientific notation and convert between metric prefixes.
- I can count significant figures and express answers with appropriate precision.
Hononga Marautanga · Curriculum Connections
This activity addresses the following Mathematics curriculum statements (Draft 2025, Phase 4 — Years 9, 10):
- Area and perimeter: Finding perimeter of 2D shapes, circumference of circles, area of parallelograms, trapezoids, and composite shapes.
- Circles: The area of a circle is given by A = πr². Surface area of solid objects.
- Pythagoras — verify: Using Pythagoras' theorem to verify that given side lengths satisfy the theorem (a² + b² = c²).
- Pythagoras — find sides: Using Pythagoras' theorem to find the length of an unknown side and check if a triangle is right-angled.
- Scaling shapes: Resizing a shape changes its perimeter, area, or volume proportionally according to scale factor.
- Speed–distance–time: Finding speed, distance, or time, given any two of the measurements.
- Speed–distance–time: Finding distance given speed and time; finding time given distance and speed.
- Duration: Reasoning about duration using different units of time, including decimal fractions of milliseconds.
- Duration: Decimal measures are used for very small durations; the rest of time measurement is non-decimal.
- Metric units: Converting between metric units and using appropriate prefixes (kilo-, centi-, milli-, etc.).
- Units: Selecting and using appropriate measurement units for a given context, converting between metric units.
- Measurement: Estimating, calculating, converting, and accurately representing measurements.
- Significant figures: Rules for identifying significant figures — non-zero digits, zeros between digits, trailing zeros with decimal point.
- Significant figures: The number of significant figures is the number of digits that contribute to the precision of the measurement.
- Precision: A solution cannot be more precise than the least precise number used in the calculation.
- Estimation: Using rounding and estimation to predict results and check reasonableness.
- Scientific notation: A number in scientific notation has the form a × 10ᵏ, where 1 ≤ a < 10 and k is an integer.
- Scientific notation: Recording, comparing, ordering, and calculating with numbers in scientific notation.
- Scientific notation: Recording, comparing, and ordering whole and decimal numbers using scientific notation.
- Irrational numbers: Non-repeating, infinite decimals are irrational (e.g. π, √2); they are represented by special symbols.
- Exponents: Generalising about exponents of 0 and 1; operations with integer exponents.
- Number operations: Adding, subtracting, multiplying, and dividing positive and negative numbers including fractions and decimals.
- Percentages: Percentages are a way of expressing a fraction of 100; used for proportional comparisons.
- Fractions and percentages: Finding a fraction or percentage of a number; finding the whole given a part.
- Proportional change: Applying a proportional increase or decrease; calculating percentage increase or decrease.
- Proportional relationships: Increasing or decreasing by a given proportion; representing proportional relationships.
- Financial mathematics: Applying percentage mark-ups and discounts; calculating simple interest and GST on dollar amounts.
Wāhanga 1 · Area, Perimeter, and Circles
C = 2πr | C = πd
A = πr²
r = radius, d = diameter, π ≈ 3.14159
A = base × height
A = bh
height must be perpendicular to base
A = ½(a + b) × h
a and b are the two parallel sides
| Shape | Given | Find | Working | Answer |
|---|---|---|---|---|
| Circle | r = 5 cm | Area and Circumference | ||
| Circle | d = 14 m | Area and Circumference | ||
| Parallelogram | b = 8 m, h = 5 m | Area | ||
| Trapezoid | a = 6, b = 10, h = 4 m | Area |
- Perimeter / lengths scale by k
- Area scales by k²
- Volume scales by k³
Example: If a circle's radius doubles (k = 2), its area becomes 4 times larger (k² = 4).
Wāhanga 2 · Pythagoras' Theorem
In any right-angled triangle, the square of the hypotenuse (longest side, opposite the right angle) equals the sum of the squares of the other two sides.
a and b are the two shorter sides (legs); c is the hypotenuse
If a = 3, b = 4: c² = 3² + 4² = 9 + 16 = 25
c = √25 = 5
If c = 13, b = 5: a² = 13² − 5² = 169 − 25 = 144
a = √144 = 12
Verifying a right angle: If a² + b² = c², the triangle IS right-angled. If not, it isn't.
| Problem | Working | Answer |
|---|---|---|
| Find the hypotenuse: a = 6 m, b = 8 m | ||
| Find side a: b = 12 cm, c = 15 cm | ||
| Is this a right triangle? Sides 5, 12, 13 | ||
| A waka ramp is 5 m long and rises 3 m vertically. How far does it extend horizontally? |
Wāhanga 3 · Speed, Distance, and Time
Three quantities are always linked by the relationship below. Knowing any two allows you to find the third.
s = d ÷ t
d = s × t
t = d ÷ s
Units must be consistent: if speed is km/h, distance must be km and time in hours.
| Problem | Formula | Working | Answer + units |
|---|---|---|---|
| A waka travels 60 km in 2.5 hours. Find its speed. | |||
| A car travels at 80 km/h for 45 minutes. Find the distance. | |||
| A runner completes 10 km at 12 km/h. How long does it take (in minutes)? | |||
| A computer signal travels at 3 × 10⁸ m/s. How far in 2 milliseconds? |
Wāhanga 4 · Metric Units, Significant Figures, and Scientific Notation
Metric prefixes — memorise these:
| Prefix | Symbol | Meaning | Example | In metres |
|---|---|---|---|---|
| kilo- | k | × 1 000 | kilometre (km) | 1 km = 1 000 m |
| — (base) | — | × 1 | metre (m) | 1 m |
| centi- | c | ÷ 100 | centimetre (cm) | 1 cm = 0.01 m |
| milli- | m | ÷ 1 000 | millimetre (mm) | 1 mm = 0.001 m |
| mega- | M | × 1 000 000 | megawatt (MW) | — |
Significant figures (sig figs):
The number of significant figures in a measurement shows how precise it is. Rules:
- All non-zero digits are significant (e.g. 345 has 3 sig figs)
- Zeros between non-zero digits are significant (e.g. 4005 has 4 sig figs)
- Trailing zeros after a decimal point are significant (e.g. 2.50 has 3 sig figs)
- Leading zeros are NOT significant (e.g. 0.0034 has 2 sig figs)
- A solution cannot be more precise than the least precise input used
Scientific notation — a × 10ᵏ:
3 200 000 = 3.2 × 10⁶
Earth–Sun distance: 1.496 × 10¹¹ m
0.000045 = 4.5 × 10⁻⁵
Diameter of a cell: 1 × 10⁻⁵ m
| Question | Answer |
|---|---|
| Write 56 700 000 in scientific notation | |
| Write 0.000 089 in scientific notation | |
| How many significant figures in 0.004 050? | |
| Convert: 4.5 km to metres | |
| Convert: 850 mm to centimetres | |
| Calculate (2.4 × 10³) × (3 × 10²). Give answer in scientific notation. |
Wāhanga 5 · Percentages and Proportional Reasoning
Convert % to decimal then multiply:
25% of 80 = 0.25 × 80 = 20
Or: 80 × 25 ÷ 100
If 30% of a number is 12, find the whole:
whole = 12 ÷ 0.30 = 40
% change = (change ÷ original) × 100
Price rises from $80 to $96: change = 16, % = 16 ÷ 80 × 100 = 20% increase
New Zealand GST = 15%
- Price + GST: × 1.15
- Price before GST: ÷ 1.15
- Simple interest: I = P × r × t
| Problem | Method | Answer |
|---|---|---|
| Find 35% of $240 | ||
| A jacket costs $115 including GST. Find the pre-GST price. | ||
| 40% of a batch of mussels weigh 2.4 kg. Find the total weight. | ||
| A map has scale 1 : 50 000. If two pā are 3.2 cm apart on the map, find the real distance in km. | ||
| Simple interest: $2 000 at 4% p.a. for 3 years. Find the total interest. |
Tūhono Aotearoa · New Zealand Connections
Traditional Māori waka builders used geometric knowledge to design hulls. A waka ama hull cross-section often forms a right-angled triangle — Pythagoras' theorem would have been applied through practical measurement to ensure correct proportions for stability and speed.
A rescue helicopter travelling from Christchurch to Aoraki at 250 km/h covers approximately 330 km. Using speed = distance ÷ time: flight time ≈ 1 hour 19 minutes. Metric conversion and sig figs are essential in aviation for safety.
The Maramataka (Māori lunar calendar) uses circular reasoning for planting and harvesting. The moon's orbit traces a near-circle. Calculating the circumference of the moon's orbit (radius ≈ 384,400 km) uses C = 2πr — a real-world circle measurement affecting agricultural tradition.
New Zealand manages approximately 1.0 × 10⁶ km² of ocean (the Exclusive Economic Zone). Fish biomass estimates use scientific notation: e.g. the orange roughy quota may be ~1.6 × 10⁷ kg. Scientific notation allows scientists and kaitiakitanga managers to work with these enormous quantities efficiently.
Whakaaro Hōhonu · Deeper Thinking
- A circular garden (radius 3.5 m) is to be surrounded by a fence, then the grass inside re-seeded. Calculate the cost of fencing at $12/m and re-seeding at $8/m². Give your answer to the nearest dollar.
- A phone screen is 14.2 cm × 6.8 cm. If the manufacturer increases each dimension by 20%, by what percentage does the total screen area increase? (Use your scaling rule — don't just add 20%.)
- A signficant figures question: your ruler can measure to the nearest mm (3 sig figs). You calculate the area of a rectangle as 23.4 × 11.7 = 273.78 cm². Should you report 273.78 cm²? Explain using precision rules.
- Extension: The speed of light is 3.00 × 10⁸ m/s. The distance from the Sun to Neptune is approximately 4.50 × 10¹² m. How long does light take to reach Neptune from the Sun? Give your answer in hours, in scientific notation, to 3 sig figs.
Kuputaka · Key Vocabulary
Kaiako Planning Snapshot
Resources already provided. What to print: this handout (2 pages). All activities are self-contained — no additional photocopying required.
Classroom use: Works as a starter, formative check, or paired investigation. Linked next step: Te Wānanga for a differentiated lesson plan, or save to My Kete.
NZ pedagogy basis: Aligned to Te Mātaiaho (Draft 2025) and Tātaiako cultural competencies. Suitable for all NZ kura and schools.
Mō ngā kaiako — For teachers and ākonga:
Inclusion: ESOL / ELL ākonga — bilingual headings and visual supports built in. UDL / neurodiverse learners — chunked sections, multiple representation formats. ADHD-friendly: short, scaffolded tasks with clear structure.
Differentiation: Entry-level tasks use supported sentence frames; on-level tasks are open-ended; extension tasks require abstract reasoning. Scaffold down with word banks; stretch by removing structured supports.