Comparison of Length: 1.0
Supporting Materials
Indicator of Progress
Students can compare and order the length of two or more objects, even when the comparisons cannot be made directly. They use the language of length comparisons to describe what they did and what they found out.
Initially, students directly compare lengths of two objects by matching one item against another (placing them side-by-side). This implies that they can distinguish length from other attributes such as colour or shape. Next, students directly compare lengths of more than two objects by matching the items against each other. Success is evident when students can arrange more than two items in order of length, even when the items cannot be directly compared.
Students also need to extend their use of words describing the attribute (length / long) to comparative (longer) and superlative words (longest). The practical activities here are intended to provide students with the experiences that enable them to develop the appropriate language. In this situation, the concept and language develop together, from real world activity. It is not possible to explain meanings such as length/long/longer verbally.
More about this. Key stages in development of the concept of length and associated language.
Illustration 1
At this stage, students can plan, implement and describe a strategy to order lines in the diagram by length. They can work out which is the longest of the following lines, the next longest, and so on. They can use words like longest, longer, shorter, shortest. This indicates they can compare the length of more than two items. Since lines cannot be put against one another, students must compare the lengths indirectly using some other object (such as string) as the basis of the comparison.
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Illustration 2
Examples of the types of tasks that would be illustrative of measurement and comparison concepts, aligned from the Mathematics Online Interview:
- Question 45 (a – c) - Comparing string and stick
- Question 46 (a) (b) - Measure straw with paper clips
- Question 49 (a – f) - Predict and check which is heavier
- Question 50 - Use teddies to measure the mass of the container
Teaching Strategies
The learning and teaching strategies for this indicator of progress are based around practical tasks from which students infer the nature of attributes (e.g. length) and simultaneously develop appropriate vocabulary.
Activity 1: Preliminary experiences which focus on the attribute provides preliminary experiences, focusing attention on the attribute of length.
Activity 2: Indirect comparisons requires indirect comparisons in which it is necessary to use a third object as the basis of the comparisons.
Activity 3: Using more than one attribute at a time requires students, at a slightly higher level, to contrast two attributes. Such activities should be undertaken as soon as students have a basic understanding of individual attributes, to further highlight their nature.
An aspect of comparisons at this level is to provide tasks that challenge the students' perceptions. An example would be lines where it looks like one is longer, but in fact it is the other one that is longer. This challenge to perception is one of the fundamental rationales for formal measurement.
Activity 1: Preliminary experiences which focus on the attribute
Students need to develop awareness of the attribute of length, to compare objects directly, and to use a variety of language (not just long and longer, but also short, shorter, longest, taller, wider, higher etc).
Examples of possible tasks are:
• What is the same and what is different? (Choose two objects that have multiple attributes: examples would be two bicycles, two drink bottles, two caps, two shoes). Students will suggest many responses. In the review of their responses, draw attention to those that compare lengths perhaps contrasting them with statements that compare colour, shape etc. In particular, model the language that is used. Challenge students who use general comparatives (this one is bigger) to be specific about what they are comparing.
• What are some words to compare two trees? This question is deliberately ambiguous. The intention is that students hear a variety of words to describe size or other characteristics of a tree, and to distinguish those related to length (wide, tall, high, height, etc.). Again, modelling the language associated with length is critical (e.g. long, high, wide, further around the trunk etc.).
Other preliminary experiences include those that invite direct comparisons such as:
• Find something in the room longer than this piece of string.
• Make or find something that is as tall as this tower.
Examples of preliminary experiences using other attributes are:
• Find something heavier than this chair. Find something that holds more water than this glass. Find something that is larger than my footprint.
The emphasis here is on comparisons chosen by the student and developing the sophistication of their language through modelling and practice in practical situations.
Activity 2: Indirect comparisons
The following tasks prompt the idea of using the length of a third object for comparisons:
• Cut a streamer so that it is longer than your hand but shorter than your foot. Draw students' attention to:
- the need to take care that the ends of objects being compared are aligned
- that comparisons are done on the whole length and
- that the streamer is held straight and taut.
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• Which of these two lines is longer? This goes one step further in that, short of cutting up the paper, the lines cannot be directly compared. The small size requires accuracy in the comparisons as well. The task challenges the students' perceptions since it is not obvious which is longer. |
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• Which is longer: my desk or the tables in the playground? (In fact anything that cannot be moved can be used for these comparisons). In this case, the students must use a third object to compare the two items. By Level 2, they will be able to measure both and compare the measurement, but not at this early stage. As with all other activities, the teacher will give a purpose for these questions by capitalising on daily classroom routines and events (e.g. we need the longest table we can get for a display).
• Which is longer: the length of a streamer that goes around a 1 litre milk carton or the height of the carton? This task can further challenge the students' perceptions, highlighting the need for accurate measurement and comparisons. Ask students to justify their choice.
Similar indirect comparison tasks can be posed for area, capacity and mass.
Activity 3: Using more than one attribute at a time
Students need to be able to work with more than one attribute at a time. For example:
• Find two objects that hold the same amount of water, but one is taller than the other. Find two objects that weigh the same, but one is taller than the other.
The invitation to compare different attributes of the same object provides an opportunity to contrast the attributes (in this case mass and capacity). It can also challenge the students' perceptions.
Note: It is not expected that students at this level know and use the words 'mass', 'capacity', 'weight' etc.