Planning tool
Year levels
Strands
Expected level of development
Australian Curriculum Mathematics V9: AC9M9M04
Numeracy Progression: Understanding units of measurement: P10, Proportional thinking: P6
At this level, students recognise that all measurements are estimates. Prior learning about place value and rounding comes to the fore as students consider the level of accuracy required in different contexts. Facilitate a class brainstorm to place a range of scenarios on a continuum that stretches from high to low requirements for accuracy. For example, compare the needs of NASA engineers designing spaceships to travel to Mars, to the drawing up a ‘foursquare’ court in the schoolyard.
Provide opportunities for students to explore and describe the diminishing returns of adding additional places of accuracy and determine a suitable level of accuracy for everyday measuring activities. They should state the absolute error of a given measuring device and use this to calculate the relative error. Ensure students understand that the absolute error is the amount of error that could exist in a measurement. Put simply, it is the discrepancy between the actual value and the measured value.
This connection of understanding naturally leads to discussions on the accuracy of the measuring devices, so have students explore different contexts with different devices. Students should use the relative error to compare the accuracy of different measurements. Ask questions: would it be more accurate to measure your weight using a device with ± 0.1 kg absolute error or ± 1 Kg or 0.001 Kg, and in which contexts would different accuracies be acceptable?
Students should also be aware of the different possible sources of error, other than the inherent limitations of measuring devices, Introduce the concept of ‘calibration’ and that miscalibration of devices is common and can affect its measurements. Furthermore, there is always a level of random error that naturally occurs with any observations. Finally, an error may be the result of using a previously rounded measurement earlier in a calculation. It is important to explicitly demonstrate this for students as this may lead to significant errors in complex calculations across other subject areas, such as Science and Business and Economics.
Teaching and learning summary:
 Recognise that all measurements are estimates.
 Consider the level of accuracy required in a variety of contexts.
 Calculate the absolute and relative error.
 Compare the accuracy of different measurements using the percentage error.
 Discuss sources of errors.
 Consider impact of using a rounded measurement in repeated calculations.
Students:
 can explain why all measurements are estimates
 determine the appropriate number of decimal places to obtain a suitable level of accuracy for a given context
 calculate the absolute and relative error of a measurement and use these to comment on the level of accuracy
 identify a potential source of error in a given measurement
 understand the negative effect of using a rounded measurement in repeated calculations.
Some students may:
 not have a solid understanding of place value and/or rounding. These are foundational for this topic and may need to be explicitly revisited for some students.
 find it difficult to accept that all measurements are estimates. This concept is somewhat philosophical and may clash with students’ physical interactions with the world around them.
To develop cognitive understanding here have students mathematically investigate error in class for simple calculations, such as measuring height or weight. For example, if students weigh themselves on a set of scales that is accurate to the nearest 100 grams, then student whose actual weight is 51.46 kilograms will be measured at 51.5 kilograms by the scales. In this example the absolute error is 51.5 – 51.46 = 0.04 kgs. This value can be used to calculate the relative error, which is equal to the absolute error / measurement being taken (0.04 / 51.5 = 0.08%). Have students consider the difference in measurement if this scenario were changed; that is, if the scales rounded to the nearest 1 Kg or 0.001 Kg.
The Learning from home activities are designed to be used flexibly by teachers, parents and carers, as well as the students themselves. They can be used in a number of ways including to consolidate and extend learning done at school or for home schooling.
Learning intention
 I can estimate measurements and comparing these to the actual result.
 I can calculate the error of a measurement.
 I can comment on the accuracy of my estimates.
Why are we learning about this?
Life is busy and we don’t always have time to measure things perfectly. Before going to the hardware store to buy paint for your bedroom walls or soil for a garden bed you first should estimate the quantity you need. This information becomes powerful (particularly if you run a painting or gardening business) as it helps you to make the right decision at the store.
But it is not the only information we should keep in mind. Whenever we make estimates there is always some error. If we keep track of how big or small this error might be, we can finetune or make reasonable estimates to be more precise. This enhances our ability to make good decisions.
What to do
Part A
You’ve decided to redecorate your rectangular room and the first step is to paint the walls with a fresh colour. Your parents have told you that they might be able to cover the cost of the paint if you provide them with a relatively accurate estimate of the cost.
After searching in the usual places, you realise that the tape measure has been misplaced. You go to the garden and find a straight stick that looks like it’s about 1 metre. Voila, you now have a measuring device! You use the stick to approximate the dimensions of the room and determine that the length of the room is between 4–5 stick lengths, the width is between 3–4 sticks and the height of each wall is between 2–3 lengths.
 Based on these measurements provide a reasonable estimate for the area of the four walls to be painted.
 What is the largest possible error from your estimate to the true area?
 Describe all the factors that affect the accuracy of your estimate.
 Through an online search you discover that a typical paint covers 11 m^{2} per litre.
 Estimate the number of litres of paint you need to cover your bedroom walls.
 Challenge – have you considered how many coats of paint will be required to get a good finish? This can vary greatly depending on the type and colour of paint you choose. Considering the possible errors in your measurements, how many litres under/over might your estimate be from the true value?
Part B
Your parents agree that painting your room is a great idea (they like your colour choice too!), but they are a little sceptical of your gardenstick estimates. They are prepared to fund this project if you come back with more accurate measurements. You explain that you couldn’t find the tape measure. Mum rushes out of the room and comes back with the tape measure. You give her a cheeky look and head back to your room.
The tape measure gives more accurate dimensions of the room. You measure the length as 4600 mm, the width as 3550 mm and the height as 2400 mm.
 Calculate the actual area of the four walls you will paint.
 What is the difference between your actual area calculations and your original estimates?
 Determine the percentage error of your estimate?
 How many litres of paint do you need to complete this project? (Remember to consider the number of coats!)
 What is the discrepancy between the number of litres you estimated and the number of litres you actually need?
 Research an approximate cost per litre for the paint you’d like to use. If your parents had provided you with money based on your original estimate, would they have given you too much or too little? State the value of this.
 Prepare a cost proposal to present to your parents. This should include your calculations to prove to them that this is the true cost. Remember, to set this out systematically so it is easy for you to communicate this and for your parents to understand.
Success criteria
 I can estimate measurements and comparing these to the actual result.
 I can successfully calculate the error of a measurement.
 I can communicate the accuracy of my estimates.
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Teaching strategies
A collection of evidencebased teaching strategies applicable to this topic. Note we have not included an exhaustive list and acknowledge that some strategies such as differentiation apply to all topics. The selected teaching strategies are suggested as particularly relevant, however you may decide to include other strategies as well.

Worked examples
A worked example is not just a preworked question that is given to the students. There are several types of worked examples and ways of using them.
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Explicit teaching
Explicit teaching is about making the learning intentions and success criteria clear, with the teacher using examples and working though problems, setting relevant learning tasks and checking student understanding and providing feedback.
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Teaching resources
A range of resources to support you to build your student's understanding of these concepts, their skills and procedures. The resources incorporate a variety of teaching strategies.

Percent error
Students are exposed to situations involving percent error, use the language of percent error and practise multistep problems involving percent error.
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Practice: percent error
A series of questions on percent error in everyday contexts.
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Area problem: percent error
A student interactive determining the area under a curve on a graph using estimations and any number of rectangles. Students can change variables to see the difference between the exact or absolute answer and the approximate or relative answer.
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Solve problems involving percentages
A teacher resource to provide insight into students’ familiarity and misconceptions on problems using percentages, including percentage error. This resource gives further teaching ideas and teaching activities.
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