Measurement Errors & Proportion
Absolute vs Relative Error
Sort each statement into the correct column: Absolute Error or Relative Error.
Measurement Tool & Precision
Draw a line from each measuring tool to the level of precision it typically provides.
Identify Absolute Error
Circle the correct absolute error for each measurement.
True value = 25.0 cm, Measured value = 25.3 cm
True value = 100 g, Measured value = 98.5 g
True value = 50.0 mL, Measured value = 50.8 mL
True value = 3.60 s, Measured value = 3.55 s
Calculate Percentage Error
Circle the correct percentage error. Use: percentage error = (absolute error ÷ true value) × 100%.
Absolute error = 0.5 cm, True value = 10.0 cm
Absolute error = 2 g, True value = 250 g
Absolute error = 0.3 s, True value = 15.0 s
Absolute error = 1.5 mL, True value = 50 mL
Scenario & Best Measuring Tool
Draw a line from each scenario to the most appropriate measuring tool.
Significant Figures
Circle the correct number of significant figures in each measurement.
0.00340 kg
12,500 m (trailing zeros not significant)
6.020 × 10³ mL
0.105 s
Reasonable vs Unreasonable Precision
Sort each measurement into the correct column: Reasonable Precision or Unreasonable Precision.
Upper and Lower Bounds
Circle the correct upper and lower bounds for each measurement.
A length is 24 cm, measured to the nearest cm. The lower and upper bounds are:
A mass is 3.5 kg, measured to the nearest 0.1 kg. The lower and upper bounds are:
A time is 12.0 s, measured to the nearest 0.1 s. The lower and upper bounds are:
A distance is 200 m, measured to the nearest 10 m. The lower and upper bounds are:
Error Propagation in Addition
When adding or subtracting measurements, absolute errors add. Circle the correct combined error.
Length A = 12.0 cm ± 0.2 cm, Length B = 8.0 cm ± 0.3 cm. Total length error is:
Mass of container = 150 g ± 2 g, Mass of contents = 340 g ± 3 g. Total mass error is:
Distance A = 4.5 km ± 0.1 km, Distance B = 3.2 km ± 0.1 km. Total distance error is:
Time 1 = 6.2 s ± 0.05 s, Time 2 = 3.8 s ± 0.05 s. Difference in time has error:
Error Propagation in Multiplication
When multiplying or dividing, percentage errors add. Circle the correct combined percentage error.
Length = 20 cm ± 2% and Width = 10 cm ± 3%. The percentage error in the area is:
Speed = distance ÷ time. Distance has 4% error, time has 1% error. Speed percentage error is:
Density = mass ÷ volume. Mass has 1% error, volume has 2% error. Density percentage error is:
Force = mass × acceleration. Mass has 2% error, acceleration has 3% error. Force percentage error is:
Error in Area Calculations
Circle the correct answer for each area-related error problem.
A rectangle is 8.0 cm ± 0.1 cm by 5.0 cm ± 0.1 cm. The percentage error in the area is approximately:
A square has side 12.0 cm ± 0.2 cm. The percentage error in its area is approximately:
A circle has radius 7.0 cm ± 0.1 cm. The percentage error in its area is approximately:
A triangle has base 10.0 cm ± 0.2 cm and height 6.0 cm ± 0.1 cm. The percentage error in its area is approximately:
Error in Volume Calculations
Circle the correct answer for each volume-related error problem.
A cube has side 10 cm ± 0.1 cm (1% error). The percentage error in its volume is approximately:
A cuboid is 5.0 cm ± 1% by 4.0 cm ± 2% by 3.0 cm ± 1%. The percentage error in its volume is approximately:
A cylinder has radius 3.0 cm ± 2% and height 10.0 cm ± 1%. The percentage error in its volume is approximately:
A sphere has radius 6.0 cm ± 0.5%. The percentage error in its volume is approximately:
Steps to Calculate Percentage Error
Put the steps for calculating percentage error in the correct order.
Map Scale & Real Distance
Draw a line from each map measurement to the correct real-world distance.
Scale Drawing Calculations
Circle the correct answer for each scale drawing problem.
A room is 6 m long. On a 1:50 scale drawing, the length should be:
On a 1:200 plan, a wall measures 3.5 cm. The real length of the wall is:
A model car is built at 1:24 scale. If the model is 18 cm long, the real car is:
An architect's plan uses 1:100 scale. A garden measuring 15 m by 8 m appears on the plan as:
Direct Proportion Problems
Circle the correct answer for each proportion problem.
If 5 kg of apples cost $12, then 8 kg of apples cost:
A car uses 7 L of fuel per 100 km. For a 350 km trip, the fuel needed is:
A recipe for 4 people uses 300 g of flour. For 10 people, the flour needed is:
A tap fills a 60 L tank in 20 minutes. At the same rate, it fills a 90 L tank in:
Inverse Proportion Problems
Circle the correct answer for each inverse proportion problem.
6 workers can complete a job in 10 days. How long would 4 workers take?
A car travelling at 80 km/h takes 3 hours for a journey. At 60 km/h, the journey takes:
12 pipes fill a pool in 6 hours. To fill it in 4 hours, the number of pipes needed is:
A gear with 40 teeth turning at 120 rpm drives a gear with 60 teeth. The driven gear turns at:
Direct vs Inverse Proportion
Sort each scenario into the correct column: Direct Proportion or Inverse Proportion.
Map Measurement with Error Propagation
Show all working and state your answer with its error range.
A map has scale 1:25,000. You measure a lake's length on the map as 8.4 cm ± 0.2 cm. Calculate the real length of the lake and the range of possible actual lengths due to the measurement error.
Cube Volume with Percentage Error
Show all working. Explain your reasoning clearly.
You measure a cube's side as 5.0 cm ± 0.1 cm. (a) Calculate the volume of the cube. (b) Calculate the percentage error in the side measurement. (c) Calculate the percentage error in the volume. (d) Explain why the percentage error in the volume (about 6%) is three times the percentage error in the side measurement (2%).
Reducing Measurement Error
Describe a practical method and explain the mathematics behind it.
Design a method to measure the thickness of a single sheet of paper as accurately as possible, given that a ruler only measures to the nearest millimetre. Explain how your method reduces the percentage error in the measurement.
Scaling Problem — Model to Real Life
Show all working for each part.
An architect builds a 1:150 scale model of a building. (a) The model is 24 cm tall. How tall is the real building in metres? (b) A window on the model is 0.8 cm wide. What is the real window width in metres? (c) The floor area on the model is 32 cm². What is the real floor area in m²? (Hint: area scales by the square of the scale factor.)
Comparing Measuring Methods
Evaluate the two methods and justify which is more accurate.
Two students measure the length of a corridor. Student A uses a 30 cm ruler (±0.5 mm) and measures in 10 segments. Student B uses a 5 m tape measure (±2 mm) and measures in 2 segments. (a) What is the maximum total error for each student? (b) Which student's method is more accurate and why? (c) Suggest how Student A could reduce their total error.
True or False — Measurement Properties
Circle True or False for each statement about measurement and error.
Doubling all measurements in a rectangle doubles the area.
Percentage error is always positive.
When multiplying two measurements, you add the absolute errors.
Using a more precise instrument always eliminates measurement error.
If a cube's side has 2% error, the volume has 6% error.
Real-World Proportion Problem
Read the problem carefully, show all working, and give your answer in context.
A school needs to mix cordial for sports day. The ratio of cordial concentrate to water is 1:4. (a) How much concentrate is needed for 15 litres of drink? (b) If they have 2 litres of concentrate, how many litres of drink can they make? (c) 120 students each need 250 mL. How much concentrate is needed in total?
Error in Indirect Measurement
Show all working and discuss the sources of error.
To find the height of a tree, you stand 20 m from its base (measured ± 0.5 m) and use a clinometer to measure the angle of elevation as 35° ± 1°. Using height = distance × tan(angle): (a) Calculate the estimated height. (b) Calculate the height using the upper bounds (20.5 m and 36°) and lower bounds (19.5 m and 34°). (c) What is the range of possible heights? (Hint: tan 34° ≈ 0.6745, tan 35° ≈ 0.7002, tan 36° ≈ 0.7265.)
Measurement Precision Experiment
Investigate how measurement technique affects accuracy and precision.
- 1Choose an object at home (e.g. a book or table). Measure its length 10 times with a ruler, recording each result to the nearest millimetre. Calculate the mean, range, and percentage variation. What does this tell you about measurement uncertainty?
- 2Now measure the same object using a different method (e.g. a tape measure or string and ruler). Compare the two sets of results — which method was more precise? Which do you think was more accurate? Explain your reasoning.
- 3Measure the dimensions of a room to calculate its area. Then estimate the percentage error if each measurement is off by 1 cm. How does the room size affect the percentage error?
Map and Scaling Investigation
Explore proportional reasoning and scale in the real world.
- 1Find a map of your local area (printed or online). Use the scale to estimate the straight-line distance from your home to three different locations. Then check with a mapping app — how close were your estimates?
- 2Choose a small object (e.g. a toy car or figurine). Measure its dimensions and calculate the scale factor compared to the real object. If the real car is 4.5 m long and the model is 18 cm, what is the scale ratio?
- 3Draw a scale plan of your bedroom using a scale of 1:20. Include the positions of furniture. Calculate the real area of your room from your drawing and compare it to a direct measurement.
Percentage Error — Calculate and Interpret
Calculate and interpret percentage error in measurement contexts.
A student measures a desk as 1.42 m long. The true length is 1.45 m. Calculate: (a) Absolute error = |measured − true| (b) Relative error = absolute error ÷ true value (c) Percentage error = relative error × 100 (d) Is this level of error acceptable for furniture? For precision engineering?
A thermometer reads 36.2°C. The true temperature is 37.1°C. Calculate the percentage error. For medical purposes, is this acceptable?
Measurement Terms — Definitions
Draw a line from each measurement term to its correct definition.
Direct vs Inverse Proportion
Circle whether each relationship is direct or inverse proportion.
The faster you drive, the less time to travel a fixed distance
The more workers on a job, the less time to complete it
The further you drive, the more petrol you use
The longer a candle burns, the shorter it gets
Direct Proportion — Find the Constant
Identify the constant of proportionality in real-world direct proportion relationships.
A car uses 7.5 L of fuel per 100 km. (a) Is fuel consumption directly proportional to distance? (b) Write the equation F = kd and find k (in L/km). (c) How much fuel for a 420 km trip? (d) How far can you travel on a full 55 L tank?
A metal wire stretches directly proportional to the force applied (Hooke's Law). When 20 N is applied, the wire stretches 4 mm. (a) Find the constant k (in mm/N). (b) How much does it stretch under 35 N? (c) What force causes 10 mm stretch?
Classify the Proportion Type
Sort each equation: Direct Proportion, Inverse Proportion, or Neither.
Compound Proportion
Circle the correct answer for each compound proportion problem.
4 workers take 6 days. How long for 8 workers (same total work)?
A tap fills a tank in 3 hours. Two identical taps take:
If 5 people eat 30 meals in 6 days, how many meals do 8 people eat in the same time?
A job takes 12 workers 5 days. How many workers to finish in 3 days?
Error Propagation in Multi-Step Calculations
Investigate how errors compound through calculations.
A rectangle is measured as 8.4 cm × 5.3 cm (each to 1 dp). The actual dimensions are 8.43 cm × 5.27 cm. Calculate: (a) Area using measured values (b) Area using actual values (c) Absolute error in area (d) Percentage error in area. Compare this to the percentage error in the individual measurements.
Proportion — Scale Drawings
Apply proportion to interpret and create scale drawings.
A floor plan uses a scale of 1:100. On the plan, the lounge room is 5.2 cm × 3.8 cm. Find the actual dimensions of the room in metres. What is the actual floor area?
An architect wants to draw a room that is actually 7.5 m × 4.2 m on a plan with scale 1:50. What dimensions should be drawn on the plan in cm?
A map has scale 1:250,000. The measured distance between two towns on the map is 6.8 cm. What is the real distance in km?
Scale Notation
Draw a line from each scale to its correct meaning.
Bounds of Accuracy
Circle the correct upper and lower bounds for each measurement.
Distance measured as 240 m (to nearest 10 m). Bounds:
Mass 3.6 kg (to 1 dp). Bounds:
Time 15 s (to nearest second). Bounds:
Inverse Proportion — Applications
Model and solve inverse proportion problems.
The brightness of a light source follows the inverse square law: E = k/d². If E = 80 lux at d = 2 m, find: (a) The constant k. (b) The illuminance at d = 4 m. (c) At what distance is E = 5 lux?
The pressure of a fixed amount of gas is inversely proportional to its volume (Boyle's Law: PV = k). If pressure is 120 kPa at volume 5 L, find: (a) k, (b) pressure when volume is 3 L, (c) volume when pressure is 200 kPa.
Proportion in Cooking and Recipes
Explore proportion in everyday cooking.
- 1Find a recipe that serves 4 people. Scale it up to serve 7 people. Identify which ingredients scale directly and which might not (e.g. spices, baking powder).
- 2Make a simple recipe (e.g. pancakes or biscuits) and deliberately measure one ingredient incorrectly by 20%. Record what happens. Calculate the percentage error in the final product.
- 3Research 'baker's percentages' — how bakers express all ingredients as a percentage of the flour weight. Try writing a bread recipe using baker's percentages.
Identify the Type of Variation
Sort each situation into the correct type of variation.
Significant Figures in Measurement
Apply significant figure rules to multi-step calculations.
A cylinder has measured radius 4.5 cm and height 12.3 cm. Calculate the volume using the full calculator value, then round appropriately. How many significant figures should the final answer have? Why?
A student calculates speed = distance ÷ time = 142.7 m ÷ 8.5 s. The measurements have 4 and 2 significant figures respectively. What should the final answer be? Explain the rule.
Sources of Measurement Error
After conducting a measurement lab, tally each type of error observed.
| Item | Tally | Total |
|---|---|---|
Human parallax error | ||
Instrument calibration error | ||
Environmental interference | ||
Random reading variation | ||
Zero error on instrument |
Proportional Reasoning — Design Problem
Apply proportional reasoning to a design or engineering problem.
A gear system has a driving gear with 24 teeth meshing with a driven gear of 36 teeth. (a) What is the gear ratio? (b) If the driving gear rotates at 200 rpm, how fast does the driven gear rotate? (c) If the load requires 50 Nm of torque, what torque must the motor provide? (Torques are inversely proportional to gear ratio.)
Error Terminology — Match the Definition
Draw a line from each error term to its correct definition.
Significant Figures in Science
Apply significant figure rules to scientific measurements.
Explain what 'significant figures' means and state the rules for counting them.
Round each value to 3 significant figures: (a) 3.14159 (b) 0.002847 (c) 12,345 (d) 1.005
A measurement is recorded as 3.70 m. Why is the trailing zero important?
When adding 1.2 m + 3.45 m + 0.8 m, how many decimal places should your answer have? Explain.
Direct or Inverse Proportion?
Circle whether the relationship is direct or inverse proportion.
Speed and time for a fixed distance
Number of workers and time to complete a job
Distance travelled and fuel used
Price per item and number of items bought for a fixed budget
Side length and area of a square
Classify Measurement Errors
Sort each scenario into the type of error it represents.
Propagation of Errors
Investigate how errors compound through calculations.
A rectangle is measured as 5.2 ± 0.1 cm by 8.4 ± 0.1 cm. Find the range of possible areas. What is the maximum absolute error in the area?
If each of 10 measurements has a maximum error of 2 mm, what is the maximum total error when you add all 10 measurements?
A scale is accurate to ±0.5 kg. How does repeated use affect trust in the combined weight of 5 items? Explain.
Types of Measurement Errors Found
Tally each type of measurement error identified in your class exercises.
| Item | Tally | Total |
|---|---|---|
Rounding error | ||
Systematic error | ||
Random error | ||
Unit conversion error | ||
Sig fig error |
Scale Drawings and Maps
Apply ratio and proportion to scale drawings.
A map has a scale of 1:50,000. Two towns are 8.5 cm apart on the map. What is the real distance in km?
A building is 25 m tall. You want to draw it at a scale of 1:200. What height should it be on your drawing?
You draw a floor plan at 1:100. The living room is drawn as 4.2 cm × 3.6 cm. What are the real dimensions?
Create a simple scale drawing of your bedroom using an appropriate scale. Label the scale and all dimensions.
Measurement and Proportion Around the Home
Apply measurement and proportion to everyday home tasks.
- 1Measure the dimensions of a room in your house. Calculate the floor area and the amount of flooring or paint needed (with 10% extra for waste).
- 2Find a recipe that serves 4 people. Scale it up to serve 11 people, showing all proportion calculations.
- 3Research the 'tolerance' used in a manufacturing process (e.g. car parts, electronics). Write about why tolerance matters.
- 4Measure your arm span and height. Calculate the ratio. Is it close to 1:1? Research da Vinci's 'Vitruvian Man' proportions.
- 5Find a map of your local area. Identify the scale and calculate the real-world distance between two landmarks.
Direct Proportion — Equations and Graphs
Write and interpret direct proportion relationships.
A car uses 8 L of fuel per 100 km. Write an equation for fuel used (F) in terms of distance (d). How much fuel for 340 km?
The graph of a direct proportion is a straight line through the origin. Explain why this is so.
If y is directly proportional to x and y = 15 when x = 3, find y when x = 8.
If y is inversely proportional to x and y = 12 when x = 4, find y when x = 6.
Percentage Error in Real Measurements
Calculate and compare percentage errors in practical contexts.
A student measures the length of a table as 1.48 m. The actual length is 1.50 m. Calculate the absolute error and percentage error.
A digital scale has an accuracy of ±0.5 g. When weighing a 50 g object, what is the maximum percentage error?
Discuss: is a 5% error acceptable when measuring: (a) ingredients in cooking, (b) a drug dose, (c) a room for new furniture? Explain your reasoning.
Proportional Contexts — Direct or Inverse?
Sort each context into the correct column.
Rates and Unit Rates
Calculate and compare rates in real-world contexts.
Car A uses 8 L per 100 km. Car B uses 6.5 L per 100 km. Over 15,000 km, how much more fuel does Car A use? At $2.10/L, what is the extra annual cost?
Supermarket A sells 2 kg of rice for $4.80. Supermarket B sells 5 kg for $11.50. Find the unit rate for each. Which is better value?
A tap drips at a rate of 12 mL per minute. How many litres per day? Per year? How much does this cost if water is charged at $3.50/kL?
Measurement Precision — Which Instrument?
Sort each measurement task to the most appropriate measuring instrument.
Currency Exchange and Commission
Apply ratio and proportion to currency conversion.
The AUD/USD exchange rate is 0.64. Convert $1,500 AUD to USD.
A currency exchange charges 2.5% commission. How much USD do you actually receive for $1,500 AUD after commission?
On a return trip, you convert 800 USD back to AUD at a rate of 0.65 (with 2.5% commission). How much AUD do you receive?
Why do banks offer different buy and sell rates for foreign currency? What is the 'spread'?