Converting Between Measurement Units: 4.0

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Teaching Strategies

Supporting Materials

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Students working

Students can convert within the metric measurement units for length, area, capacity, mass and time, and between units of mass, capacity and volume.

Initially, the metric system was designed so that 1 gram was defined as the mass of 1 cubic centimetre of pure water at 4 degrees Celsius and the litre was defined as the volume of 1 cubic decimetre (10cm × 10 cm × 10 cm). This allows conversion between these units.

Success depends on assembling a variety of pre-requisite knowledge:

knowledge of metric units and metric prefixes (milli, Mega etc.)
knowledge of decimal notation
ability to multiply and divide by powers of ten, including with decimals
elementary proportional reasoning
knowledge of approximate size and common uses of metric units.

Illustration 1: The range of difficulty of conversions

Conversions like 'How many cm in 1 m?' are straightforward questions of fact.

Mass and capacity units are the easiest to convert, in that the common units increase by factor of 1000 (. . . , mg, g, kg, tonne, . . .). For example, converting 2 tonnes to 2000 kilograms and then to 2000000 grams.

Length is similar, although cm introduces a level of complexity, as converting between successive units involves factors of 10, 100 and 1000 (mm, cm, m, km).

Time is different in that the units for time are not fully compatible with the base ten system. Tasks like 'How many hours in 2 weeks?' require multiplication or division by 24 and 7, not just by 10 and its powers.

Conversions involving whole numbers will be easier than conversions requiring decimal numbers. So for example, Write 5000 mg as kg is easier than Write 500 mg as kg or Write 5250 mg as kg. These are easier than Write 0.25 mg as kg which would require understanding of decimals beyond this level.

A later difficulty is converting between derived units such as from square centimetres to square metres.

Illustration 2: Difficulties with decimals

A major strength of the metric system is that it is based on the base ten system, as is our number system.

However, this means that students who do not fully understand the base ten and place value properties of the number system are greatly disadvantaged. One substantial difficulty is multiplying and dividing by powers of ten (10, 100, 1000, 10000 etc.). The other is knowing the meaning of decimal notation.

For more information: Comparing decimal numbers (Level 4.0) addresses these misconceptions about decimals.

Teaching Strategies

As with all measurement teaching, the pen-and-paper calculations outlined below need to be balanced with practical measurement tasks that strengthen understanding of the size of the units involved. Everyday school and home events provide a good supply of contexts for conversion of units.

Activity 1: Converting between units within the same attribute provides task that are are sharply focussed on the mathematical processes of conversion, but students also need an appreciation of the size of the units.
Activity 2: Converting between different attributes gives tasks which require the use of one measurement attribute to solve a problem involving a different attribute. Again pen-and-paper activities should be balanced with activities involving measurement of real items.
Activity 3: Design metric time is intended to highlight the benefits of the metric system, in an open-ended activity where students can stretch their imagination.
Activity 4: Multiple step conversions is a culminating activity, requiring multiple time conversions.

Activity 1: Converting between units within the same attribute

In the metric system it is usual to express the quantity of each unit between 1 and 1000 (cm is an exception), so fractions of one unit are expressed using a smaller unit.
If the number of a unit is too large, it is expressed using a larger unit.
It is generally preferred to use a decimal number of one unit, than to mix units.

The mixed examples below indicate the issues involved - teachers will make their own set of examples concentrating on certain units etc.

Write in the most appropriate unit:	Answer 1	Answer 2	Comment
0.5 kg	500 g		Examples should include a range of different number of explicit decimal places. Zeros are one of the most difficult aspects to deal with.
0.375 L	375 mL
0.25 L	250 mL
0.05 m	5 cm	50 mm
2500 mg	2kg 500 g	2.5 kg	Answer 2 is preferred in more advanced work, because it does not use units and subunits.
2 000 000 mm	2 km	2.000000 km	This involves multiple conversions: e.g. mm to m, then m to km. Later (progression point 5.25), students learn to retain the same number of significant figures to indicate accuracy of measurement as in answer 2.

Activity 2: Converting between different attributes

In a game show, you win a prize. You are offered 1 kg of $1 coins, or 1 square metre covered in 20c pieces. Which would you choose?

While not strictly converting between measurement units, this task requires conversions between mass, money and area. Such a task encourages estimation and creative strategies for solution, and develops proportional reasoning.

Describe the dimensions of a container that could hold 100L of honey. (Use fact that a cube with side 10 cm contains 1L).
Approximately how heavy is a 9L watering can when it is full of water? (Use fact that 1L of water weighs 1 kg).

These tasks use the properties of the metric system to convert between volume, mass and capacity. Although these conversions do not appear in the progression points until level 4.75, many younger students will already know them and can answer simple questions such as these.

Activity 3: Design metric time

Design a metric system for time and consider the benefits, difficulties and the changes that this might make to our lives.

This task allows consideration of the benefits of the metric system of units with its ease of conversions between units. Time is the only measurement attribute that is not a base ten system. This makes calculations between units different. (In fact, seconds are part of the metric system, but other units of time are not). More on metric units.

Students may be surprised to know that Australia did not always use the metric system. Many students are interested in the imperial units still used in the United States (e.g. for basketball players' heights and weights) and might be interested to try some conversions.

There are several ways in which students might approach this task. For example, a student might decide that the 'day' should be kept as a basic unit, and should be divided into 10 metric hours. How long would these metric hours be in today's units? Each metric hour might be divided into 10 decihours, and each decihour could be divided into 10 centihours. How long would these be in today's units. How long would a kilohour be in today's units?

And maybe the year should be divided into ten metric months and each of them divided into 10 metric weeks - how long would they be in today's units? What would we lose and what would we gain?

Activity 4: Multiple step conversions

This is a culminating activity, suitable to be done as a problem solving task. Since it may take a lot of time, students might do it at home as a project. Certainly calculators should be used, because the emphasis is not on arithmetic here, but on the overall plan for the solution. As always, follow up class discussion is essential to turn mathematical experiences into learning.

How many days have you been alive? Approximately how many seconds is that?

This task requires multiple conversions from years to days to hours to minutes to seconds. Students need to organise their work carefully, writing down the key steps. Allow them to use a calculator. Discussing the results helps give a meaning to large numbers.

References

Steinle, V., Stacey, K., Chambers, D. (2006) Teaching and Learning about Decimal Numbers V 3.1. http://extranet.edfac.unimelb.edu.au/DSME/decimals/index.htm

Mathematics Developmental Continuum P-10