Mathematics Developmental Continuum
The Idea of a Unit: 2.0
Supporting material
Indicator of Progress
Students can plan, implement and describe a strategy to use a unit repeatedly to compare one object with another, or to 'measure' the object. The students can also use a unit to compare two objects indirectly.
There are several preliminary stages. First, students develop the idea of a unit; that something that can be used over and over to 'measure' an object. Next, they use selected units repeatedly to compare or quantify the length (or width or height etc.) area, volume, mass and capacity of objects. At this level, informal units are used to develop fundamental principles of measurement. Beyond Level 2, the need for formal units of agreed size is established.
This development is explained in detail for length. Similar explanations and tasks can be used for the attributes of area, volume, mass and capacity. Click here for more information on this indicator of progress.
Illustration 1: Examples of informal units
Handspan is an informal unit for measuring length: e.g. How wide is your desk measured in handspans?
Pace is an informal unit for measuring length: e.g. Use your pace as a unit to find which is the longer court.
Both of these units are also used in practical measuring tasks by adults. For example, I know that my handspan is about 20 cm and my pace is about 1m. At higher levels, students will learn to use these benchmarks for estimation.
At this level, it is not intended that students know the approximate length of these units in cm. The handspan itself is being used as a unit, without reference to cm. Students will also use other informal units (e.g. the length of a ribbon, pencil case, etc.).
Illustration 2: A task using an informal unit
Ask the students to compare the length of the tent pole and the bicycle pump, which are both photographed on a deck. Review the strategies used. In particular, ask students who use the number of the floorboards (plus spaces) as the way of comparing the two objects to share their strategy. These students have used the floorboards as an informal unit.
Illustration 3
Examples of the types of tasks that would be illustrative of the idea of a unit, aligned from the Mathematics Online Interview:
- Question 46 - Measure straw with paper clips
- Question 50 - Use teddies to measure the mass of the container
Teaching Strategies
The general teaching strategy is to build up these fundamental principles of measurement with informal units that are familiar to the students. When the general principles of using units are strongly established, then formal units (such as centimetres and metres) are introduced. Because they are new to students, using formal units adds to the cognitive demand of the situation, so they are not used in these initial stages.
The teaching strategies for this development are based around two types of experiences:
Activity 1: Preliminary experiences using a unit provides preliminary experiences with an iterable unit (i.e. using a unit over and over again);
Activity 2: Using a unit to measure provides experiences that use an informal unit to highlight fundamental principles of measurement.
Key challenges for students are:
- to determine an appropriate starting point,
- to choose an appropriate unit,
- to take care with counting,
- to ensure that the unit is used end-to-end (without overlap or gaps), and
- that the length is measured in a straight line (the shortest distance).
These are fundamental principles of measurement.
This development is explained in detail for the attribute of length, with suggestions included for area, volume, mass and capacity.
Activity 1: Preliminary experiences using a unit
Estimate the length of your desk using paperclips. Now check.
Draw a line that you think is 5 paperclips long. Now check.
Find something that is about 2 shoes long.
Find something that is longer than 2 shoes but shorter than 3.
Either the same object can be used repeatedly (e.g. the same shoe moved along) or a row of objects of the same size (e.g. similar paperclips) can be used.
Other preliminary experiences
- Find something that is about as heavy as 2 books of the same size (mass).
- Find something that holds as much as 3 paper cups (capacity).
- Find something that is about the size of 4 A4 sheets of paper (area).
- Find something that holds more than 2 milk cartons but less than 3 milk cartons (capacity).
Activity 2: Using a unit to measure
Ming and Monica measured the basketball court. Ming said it was 20 rulers, Monica said that it was 19 ½ rulers. How could this happen?
This is a potentially very powerful task. The types of suggestions that students can make include:
- Ming overlapped the rulers.
- Ming did not measure in a straight line.
- Monica left gaps between the rulers.
- Ming rounded his answer.
- Ming measured from the outside of the lines and Monica measured from the inside.
- They measured different courts.
- They used rulers of slightly different lengths.
In the discussion of the task, emphasise that some of these are errors that can be avoided.
It is possible to pose similar tasks for the other measurement attributes.
References
Early Numeracy Research Project - (http://www.education.vic.gov.au/studentlearning/teachingresources/maths/enrp/default.htm)