Expected level of development
Australian Curriculum Mathematics V9: AC9M10N01
Numeracy Progression: Number patterns and algebraic thinking: P9
At this level, students make approximations using real numbers in repeated calculations. Students recognise the effect of the result of a problem comprising multiple steps of calculations when approximations or rounding has been used at each of those steps.
Repeated approximations can be compared to the actual result in multiple ways, such as estimating the approximate perimeter of your school property and comparing it to the perimeter calculated using digital geo-spatial technologies (Google Maps).
Other contexts can also be used – estimating the surface area of a cube or cone, estimating the year accumulation of casual hours worked at a supermarket, estimating where to hang a picture on a wall visually or using a measuring tape. It is important to review the concept of orders of magnitude so that students consider what constitutes a reasonable estimation. When estimating large numbers, being within a single order of magnitude is a reasonable result.
The other area where this topic could be explored is within the context of real-world examples where approximations would be not suitable. This helps students to know why they are learning about rounding or correcting to a specific number of decimal places. Chemistry is a perfect example, where exact measurements need to be calculated for a reaction to happen. Exact measurements are also important when baking a sponge cake or landing an aeroplane.
Teachers can develop a mini-project around Fermi estimation, where each student develops their own question and then estimates its result. Students present to the class and have their peers note down a prediction before each presentation. This can be made competitive by finding the closest on average after all presentations.
Teaching and learning summary:
- Teach students the effect of using approximations in repeated calculations versus using more accurate representations (for example, using three decimal places rather than a whole number).
- Review orders of magnitude and rounding.
- Discuss the situations in which precise calculations are required and those when estimations will suffice.
- Explore the impact of multi-step estimates on the overall accuracy.
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Teaching strategies
A collection of evidence-based 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.
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Collaborative learning
For group work to be effective students need to be taught explicitly how to work together in different settings, such as pairs or larger groups, and they need to practise these skills.
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Worked examples
A worked example is not just a pre-worked question that is given to the students. There are several types of worked examples and ways of using them.
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Mathematics investigation
By giving students meaningful problems to solve they are engaged and can apply their learning, thereby deepening their understanding.
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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.
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More and less?
Students investigate physical quantities of different kinds and make judgements on whether the estimations are reasonable or correct. Students consider why estimates were made and how close the estimates would be to real values.
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Approximating pi
This exercise uses geometry, approximation and refinement using a series of values to find the value of pi (π). Students see how approximation can be used to find reasonable solutions.
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Archimedes and numerical roots
This mathematical investigation for students includes notes and solutions for the teacher and requires students to find the lengths of polygons using square roots.
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Approximating square roots
Use this interactive number checker to help with mathematical investigations requiring solutions to square roots, or set up an activity for students to approximate square roots using a number line.
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Truncating and rounding numbers
Students can use this applet to practise truncating and rounding.
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Real numbers
This resource is a teacher-facing guide that provides a whole range of information on how to teach real numbers.
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Assessment
By the end of Year 10, students recognise the effect of approximations of real numbers in repeated calculations.
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Standard index form matching
A game where students must match the corresponding numbers and their representation in index form.
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Does this sound about right?
This assessment is a mathematical investigation into reasonableness, estimation and actual results.
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Approximating pi
This exercise can be used as an assessment task. It uses geometry, approximation and refinement using a series of values to find the value of pi. Students see how approximation can be used to find reasonable solutions.
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Rounding
This lesson plan and assessment cover reasonable rounding in many different contexts. The assessment questions allow students to see the situations where rounding makes sense and when it doesn’t.
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