Mathematics Developmental Continuum P-10

Extending Work with Formal Units: 3.25

Supporting materials

 

Indicator of Progress

At this level students know the appropriate units from the metric system for measuring length, area, volume/capacity, and mass and have suitable benchmarks for estimating them. They understand how to use measurements for straightforward applications, such as determining the perimeter of rectangles, and for more extended measurement problems.

Before achieving this, students may have difficulty estimating measurements because they do not have mental or physical referents or benchmarks to use as points of reference. They may also struggle to keep track of the sequence of steps required to solve applied problems.

There are three phases for teaching measurement – in terms of the measurement phases, this indicator is about moving from ‘learning to measure’ to ‘learning to calculate’, with a focus on appropriate estimation and the application of measurement to real world problems.

Illustration 1: Estimation Skills

Students need some experience with measuring with units before they can be expected to estimate measurements in those units. Once they have done sufficient work with measuring to gain a sense of the magnitude of the relevant units, however, they should be expected to estimate first and then measure. Comparing the measurement to the estimate will help improve estimation skills.

Students with a set of personal benchmarks for reference should be able to estimate more effectively. They may reason that the desk is about 90cm wide because three A4 pieces of paper fit along it, and they know that an A4 piece of paper is about the length of a school ruler. A box might have an estimated capacity of 6 litres because a student can visualise about 6 milk cartons fitting into it.

 

Illustration 2: Applying measurement and estimation in real world situations

Students with a good understanding of attributes and appropriate measurement units should be able to apply that knowledge in real world situations. Students will need to work out relationships among different quantities, and multiplicative thinking is often involved. For example, to work out how many bottles of soft drink to buy for a class party, students will have to estimate how much one student will want to drink, then determine how many serves can be obtained from one bottle, and, consequently, calculate how many bottles are needed.

 

Illustration 3: Links to the Mathematics Online Interview

Examples of the types of tasks that would be illustrative of successful use of formal units and measuring instruments aligned from the Mathematics Online Interview:

  • Question 47 – Using the Ruler
  • Question 48 – Tearing the Streamer
  • Question 52 – Using formal units (Mass)
  • Question 53 – Using kitchen scales
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Teaching Strategies

Explicitly teaching students about some useful measurement benchmarks will give them physical reference measurements that they can use and/or visualise for estimation purposes, since not all students will recognise the usefulness of this on their own. Students should have as many opportunities as is practical to apply their measurement skills in real-world situations. 

Teachers can support students’ estimation skills by sharing their own estimation strategies when estimating and checking measurements. There are many opportunities to do this in daily classroom life.

The importance of estimation and measuring can be discussed at parent information events. Many parents with good practical estimation skills may not appreciate how much sharing their informal knowledge can help children with school work. They may also not appreciate that school maths is meant to include practical, useful skills for dealing with everyday things.

Activity 1: Benchmarks and Estimation gives some suggestions for suitable benchmarks that can be made explicit for students.

Activity 2: Language reviews pronunciation, derivation of terms and symbols for units.

Activity 3: Real-World Applications can be used to help students develop their measurement skills in practical situations around the school and in the community.

Activity 4: Just for Fun: Getting Your Money’s Worth is an example of a fun problem that requires estimation and that has a surprising answer.

Activity 5: Fermi Questions are extension problems that require students to combine measurement, estimation, and problem solving strategies.

Activity 1: Benchmarks and Estimation

Students should be encouraged to remember a set of benchmarks for various measures. The ability to call up mental pictures of these quantities is vital in estimation, unit conversion, and, more generally, in making sense of the world. Important measurements, with possible reference benchmarks, include

  • 1 cm (width of fingernail)
  • 1 m (long stride/pace or common height of belly-button)
  • 20 m (length of a cricket pitch)
  • 50 m (length of an Olympic sized swimming pool)
  • 100 m (running track or length of a hockey field)
  • 1 km (the distance a car travels in a minute in a 60km/h zone with no traffic lights OR  the distance you can walk in about 15 mins OR pick a local distance)
  • 1 cm2 i.e., 1 square centimetre (area of little fingernail, or area of the face of a centicube/MAB mini)
  • 1 m2 i.e., 1 square metre (a bit more than two double pages of The Age)
  • 1 ha (about the size of a football field or two hockey fields)
  • 1 km2 i.e., 1 square kilometre (use a local reference of a square region of land with boundaries that are 1km long)
  • 1 cm3 i.e., 1 cubic centimetre (volume of a centicube or MAB mini); 1 cubic centimetre of liquid has a capacity of 1 mL (a teaspoon contains about 5 mL)
  • 250 mL (a standard measuring cup; a mug is usually a little bigger than this and some disposable plastic cups are a bit smaller)
  • 1L (capacity of a regular 1L carton of milk or the size of an MAB 1000 block)
  • 1 m3 i.e., 1 cubic metre (volume of 4 big 240L wheelie bins; it is also a good idea to build a classroom model of a cubic metre using a kit, or dowels, or even rolled up newspaper and tape; an example is shown below/at right.)
  • 1 ML is 1 000 000 L or 1000 m3 (an Olympic swimming pool which is 50m long, 25m wide and 2m deep has a capacity of 2.5 ML)
  • 1 g (mass of a Smartie™ or M&M™ or the mass of one commercial plastic connecting centicubes; note that a piece of A4 photocopy paper weighs about 5g)
  • 1 kg (mass of 1 litre of water; this is an accurate relationship not an approximation)
  • 1 tonne (mass of a small car)

As students are making formal measurements they should be encouraged to estimate before measuring (once they have had some experience with the magnitude of the relevant units). Simple competitions for getting the 'best estimate' could be conducted, and the strategies used could be discussed. Suitable estimation competitions might include: the length of a displayed piece of string, the length of a school corridor, the mass of a chair, the capacity of container, the area of a hand-print.

 

 

 

Activity 2: Language

Metric measurement words can be analysed, so that students understand them better. Divide the names into the basic unit name (e.g., metre, gram) and the prefix (milli, centi, kilo). Link the meaning of the prefixes with work on place value. There is a reference list of prefixes in Terminology of Units.

Note: Technically speaking, the kilogram is the base metric unit for mass rather than the gram, despite the fact that it has a prefix.

Students should use correct pronunciation (e.g., kilometre should be pronounced KILL-o-metre, so that “metre” sounds the same as in “centimetre” (i.e., NOT kil-OH-mitta). The metric unit names consist of a prefix and basic unit name, and this is reflected in the correct pronunciation.

For area and volume the correct terminology is “square centimetres” and “cubic metres”, NOT centimetres squared or metres cubed.

 

Activity 3: Real-World Applications

Students should be given opportunities to apply their measurement knowledge in a variety of practical situations. These should be real or realistic situations, and ideally should build on students’ interests or on contemporary topics. Some ideas are suggested below; most of these are intended as a springboard for a teacher’s own ideas, adapted for and relevant to the local situation.  There are many opportunities to do this in daily classroom life.

Around the School

Look for practical measurement situations that might arise around the school and classroom. 

  • What is the perimeter of the basketball court? This can lead to exploring quick methods for calculating the perimeter of rectangles.
  • If we want to decorate our classroom with streamers, how many packets of streamers will we need? (first how many metres, then how many packets)
  • How much soft drink will we need for a class party?
  • If we want to have a whole school assembly, what rooms would be big enough?
  • If some landscaping is about to be done, how much grass seed is going to be needed?
  • How many laps of our school oval would the Grand Prix cars have to do to make one lap of Albert Park?

In the Community

Make the most of situations that may arise in newspapers, contemporary events, as part of the local community’s environment and activities, or on class excursions.

  • What distance is covered on a local bushwalk? How many times around the school is that? How long will it take to walk?
  • Melbourne people are being asked to use no more than 155L of water per person per day. How does that relate to the size of a bathtub? How much water is in the local reservoir? How long will the reservoir last if everybody follows the 155L rule, but no more rain comes?
  • How much food do the elephants at the zoo eat? How much do we eat in comparison?
  • What length of electricity cable is going to be needed for the new sub-division?
  • Australian plague locusts eat about 0.2 gram of food per day. If our oval was covered in Australian plague locusts, with about 50 locusts on each square metre, how much grass would they eat? [NOTE: A swarm of one square kilometre can contain 50 million locusts consuming up to 10 tonnes of vegetation every 24 hours.]

Estimate as well as measure

The suggestions above are useful for teaching estimation of quantities, as well as measuring. Teachers should model their own estimation strategies, and involve students in authentic problems.  For example, a teacher who wants to know if all the Grade 4 students could line up down the hall in the pageant could share her strategy of estimating each student will need a metre, and then pacing out the hall.  Parents can also be encouraged to involve their children, when measurement and estimation is required. Many parents have excellent estimation abilities and strategies that they use in their daily lives.

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Activity 4: Just for Fun: Getting Your Money’s Worth

Congratulations!! You have won a prize: your choice of one of the following options. Which one would you choose? Estimate the value of each of the prizes, showing how you arrived at your estimate.

  • A cubic decimetre* container filled with 20 cent coins (packed as full as you can)
  • 1 kilogram of $1 coins
  • A one metre long line of $2 coins (lying flat and touching)
  • A square metre of 5 cent coins (the coins fill the square metre and are lying flat and touching)

* A decimetre is a tenth of a metre, so 10 cm. Note that  “deci” = “tenth”. A cubic decimetre is 10 × 10 × 10 cm3 which is one litre.  See Terminology of Units.

 

Activity 5: Fermi Questions

Fermi Questions are problems that initially seem impossible to answer but  by making a 'mathematical model' of the situation,  and using some sensible assumptions and estimates, it is possible to come up with a reasonable estimate. Fermi questions often involve big numbers, multiplicative thinking, problem-solving skills, and thought-provoking situations, and so are quite useful and interesting for students. Students may need to conduct some research to find necessary pieces of information. Fermi Questions can develop students’ mathematical communication skills as they explain and justify their answers, and are ideally suited for small group work.

Fermi Questions permit different methods of solution. These may therefore yield different answers, but teachers should be aware that while some of these different answers may be acceptable because the mathematical reasoning behind them is appropriate, others may involve faulty reasoning.

Some examples of Fermi Questions are given below. These examples have a measurement flavour and are estimated as being suitable for this level.

  • How many people could we fit in our classroom, standing on the floor?
  • If all the students in the school joined hands in a line, how long would the line stretch?
  • How long would it take our class to drink all the water in an Olympic Pool (if the water was pure!)?
  • If all the cars in Australia were parked along a road, how far would the line stretch?
  • How many students would it take to weigh the same as the mass of a blue whale?
  • How tall is the tallest building in our school?
  • Could all the people in the world fit into Victoria?
  • How many grains of rice are in a 10kg bag?
  • How many houses could we fit on our school oval?
  • How much rubbish do we throw out each year?
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