Measurement Investigations 1

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

Achievement Objectives
GM4-1: Use appropriate scales, devices, and metric units for length, area, volume and capacity, weight (mass), temperature, angle, and time.
GM4-3: Use side or edge lengths to find the perimeters and areas of rectangles, parallelograms, and triangles and the volumes of cuboids.
Specific Learning Outcomes
  • Apply measurement knowledge to solving word problems.
  • Plan a mathematical investigation.
  • Take measurements and make calculations to complete an investigation.
  • Interpret the accuracy of the investigation.
Description of Mathematics

Measuring is about making a comparison between what is being measured and a suitable measurement unit. Central to the development of measuring skills is engaging in 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.

Opportunities for Adaptation and Differentiation

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
  • directly modelling measurement with scales, like rulers, before providing students with opportunities to copy the correct use of tools in pairs, and independently
  • clarifying the language around 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 45m2 as 45 square metres
  • modelling ways to collect, and organise, measurement data, such as using tables and spreadsheets
  • organising students into groups that include students with a varying levels of mathematical confidence and knowledge to encourage peer learning, scaffolding, extension, and productive learning conversations
  • providing opportunities for students to share ideas, question, clarify, and reflect in a range of whole-class, small-group, peer-to-peer, and student-to-teacher settings
  • allowing the use of digital tools
  • easing the calculation demands by providing appropriate opportunities for the use of calculators 

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 students to physically model solutions with manipulatives and digital representations, before requiring the abstract (mental) anticipation of measures
  • reducing the demands for a product, e.g. less calculations and words, and more diagrams and models.

Although each of the sessions are presented in a successive order, you could present them as measurement 'stations' for students to explore throughout a single (or several) lesson(s). If you choose to do this, ensure that you have explained each session, and modelled the processes involved, to enable student's successful and productive participation.

The contexts presented in this unit can be adapted to suit the interests and cultural backgrounds of your students, and to make links with other curriculum areas. 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.

Te reo Māori kupu such as ine (measure, measurement), mitarau (centimetre), mitamano (millimetre), and manomita (kilometre) could be introduced in this unit and used throughout other mathematical learning.

Required Resource Materials

For each session, present the problem and have the students work in pairs or small groups to decide what strategies would be best to complete the investigation. Scaffolding questions are provided within each session. If necessary, review and model appropriate strategies. After students have decided on their strategies, give them time to investigate the problem. Roam and support students as necessary. After a suitable period of time, gather the class together and have groups of students share their ideas. You might get groups to share with other groups, before choosing one group to share with the whole class. 

Consider providing a graphic organiser, such as a Think Mat, to encourage students to contribute and reflect on their ideas as a group. After groups have presented their groups, reflect and review as a class (e.g. What range of answers would be acceptable?), and confirm and discuss the accuracy of the answers presented. You might use any gaps demonstrated in knowledge as the focus of a subsequent teaching session.

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.)

  1. Scaffolding Questions:
    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?
  2. Ideally students will convert 2km = 2000m and divide that by the distance they calculated above.

Session 2

Investigation 2: Calculate the thickness of a page in your textbook.

  1. Scaffolding Questions:
    What measurements could you take to help complete this investigation?
    Can you measure the thickness of just one page accurately?  
  2. 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.
  3. 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. 

  1. Scaffolding Question:
    What measurements could you take to help complete this investigation?
  2. Discuss the accuracy of students’ answers.

Session 4

Investigation 4: Construct a box with a volume of 60cm3.  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?

  1. Scaffolding Questions:
    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.
  2. The students should decide on the best way of constructing their box.
    Can you draw a net for your box? (provide 1cm grid paper)
  3. Students should calculate the surface area of the box.
    How many faces does the box have?
    What is the area of each face?
  4. 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?

  1. Scaffolding Questions:
    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?
  2. Provide time for the students to carry out and write up their 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.
  3. What are the dimensions of containers with equal volumes?
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Level Four