In this unit students complete a number of practical measuring investigations, with an emphasis on accuracy of measuring and communication of their findings.

- Plan a mathematical investigation in a group.
- Take measurements and make calculations to complete an investigation.
- Interpret the accuracy of the investigation.

Measuring is about making a comparison between what is being measured and a suitable measurement unit. Central to the development of measuring skills is lots of practical measuring experience. Also important is the reality that measurement is never exact. As measurement involves continuous quantities even the most careful measurements are only approximations.

An analysis of the process of measuring suggests that there are five successive stages. Students learn to measure by first becoming aware of the physical attributes of objects and therefore perceiving what is to be measured. When students have perceived a property to be measured they then compare object by matching, without the use of other tools of measurement. This comparison leads to the need for a measurement unit. Initially the unit may be chosen by the student from everyday objects. The use of informal or non-standard measuring units leads to the need for standard units for better precision and unambiguous communication.

This sequence is quite general and can apply to the measurement of any attribute. In fact, we believe that one of the broad aims of teaching about measurement is to help students develop an overall picture for coping with any measurement situation.

The investigations in this unit of work require the students to both use and apply standard metric measures. The students are also required to justify the level of accuracy appropriate for each investigation.

The learning opportunities in this unit can be differentiated by providing or removing support to students and by varying the task requirements. Ways to support students include:

- Making measurement tools available for practical use.
- Direct modelling of measurement with scales, like rulers, with opportunities for students to copy correct use of tools.
- Clarifying the language of measurement units, such as “square centimetre” as an area that is 1cm x 1cm, and “cubic centimetre” as a space (volume) that is 1cm x 1cm x 1cm.
- Clarifying the meaning of symbols, e.g., 45cm as 45 centimetres, and 45m
^{2}as 45 square metres. - Modelling ways to collect, and organise, measurement data, such as tables.
- Easing the calculation demands by providing calculators where appropriate.

Tasks can be varied in many ways including:

- Reducing the complexity of the numbers involved, e.g. whole number versus fraction dimensions. The choice of measurement units influences the difficulty of calculation as well as the level of precision.
- Allowing physical solutions with manipulatives before requiring abstract (in the head) anticipation of measures.
- Reducing the demands for a product, e.g. less calculations and words, and more diagrams and models.

The context for this unit can be adapted to suit the interests and cultural backgrounds of your students. The investigations can be framed using story shells, such as constructing boxes to hold fudge for a fundraiser, measuring the tennis court to get fit for the cross country, or working out how many flyers will fit in a delivery bag. Students might pose their own measurement challenges that are significant to them.

- Measuring tapes
- Rulers
- Micrometer (if available)
- Cardboard
- 1cm grid paper

**Session 1**

*Investigation 1*

*How many times would you have to walk around the tennis courts to cover a distance of 2 kilometres?* (If a court is not available choose an appropriate area close to the mathematics classroom.)

- The students might work in pairs or small groups to decide what strategies would be best to complete the investigation.

*What measurements could you take to help complete this investigation?*

How far would you walk in one time round the perimeter of the tennis court?

*What measurements do you need to take to work out the perimeter of the tennis court?*

*What units are most appropriate to use?*

*How accurate do your measurements need to be?*

*Once you find the perimeter, how will you work out how many courts are needed to walk 2 kilometres?*

Ideally students will convert 2km = 2000m and divide that by the distance they calculated above. - Allow students to carry out their chosen strategies.
- Discuss the accuracy of their answers.

*What range of answers would be acceptable?*

**Session 2**

*Investigation 2*

Calculate the thickness of a page in your textbook.

- The students should work in pairs or small groups to decide what strategies would be best to complete the investigation

*What measurements could you take to help complete this investigation?*

*Can you measure the thickness of just one page accurately?*

Students should realise that measuring a stack of pages is more realistic. For example, their chosen book might have 180 single pages and those pages might measure 18mm across.

*How will you work out the thickness of a single page?*

18 ÷ 180 = 0.1mm (one tenth of a millimetre) That is at the upper range for a sheet of paper. - Allow students to carry out their strategies for measuring the paper

*What units are appropriate for the answer?* - Share and discuss the accuracy of their answers.

*What range of answers would be appropriate?* - If possible, use a micrometer to check the answer.

**Session 3**

*Investigation 3*

The class set of mathematics textbooks are to be covered with plastic film. The film comes in rolls that are 600mm wide. Determine how many 10m rolls of film will need to be bought.

- The students should work in pairs or small groups to decide what strategies would be best to complete the investigation

*What measurements could you take to help complete this investigation?* - Discuss the accuracy of students’ answers.

**Session 4**

*Investigation 4*

*Construct a box with a volume of 60cm*^{3}^{.} The dimensions of the box should be whole numbers of centimetres. Calculate the surface area of the box. Which dimension would give the minimum surface area?

- The students should work in small groups to decide on the dimensions of their box.

*Can you find a set of three whole numbers measurements that will multiply to give 60 cubic centimetres?*

Note that the prime factorisation of 60 is 2x2x3x5. Any two of the numbers can be multiplied to give one dimension of the box and they other two numbers provide the other two dimensions. Therefore, the possibilities are 3cm x 4cm x 5cm; 2cm x 5cm x 6cm; or 2cm x 2cm x 15cm. - The students should decide on the best way of constructing their box.

*Can you draw a net for your box?*(provide 1cm grid paper) - Students should calculate the surface area of the box.

*How many faces does the box have?*

What is the area of each face? - Discuss the value of the surface area for boxes of different dimensions.

*Which dimensions would give the minimum surface area?*

**Session 5 (extension)**

*Investigation 5*

*Two vans are selling hot chips at the local A&P Show. Both vans use the same size scoop to serve a measure of chips. Mr Grease twists his paper to make a cone for his chips, and Mrs Crisp uses cylindrical containers. Why do customers think that Mr Grease is more generous?*

- Allow the students to plan the investigation.

*What information would you need to complete this investigation?*

What volume of chips would a scoop hold?

What assumptions should you make about the two containers? - Students should try the investigation.

*What are the formulae for finding the volumes of both containers?*

Volume of cylinder = area of circular base times height.

Volume of cone = 1/3 times circular area of base times height. - Students should write up their investigation showing all calculations.

*What are the dimensions of containers with equal volumes?* - Discuss the results of the investigation.