Measuring toy cars
Three children measured some toy cars.
They used the ruler under each car to measure its length.
This resource can be used to provide evidence of students' understanding of measurement of length.
Y4 (Oct-Nov 2013)* Online |
Y5 (March 2010)** Print version |
||
a) | 5 cm | moderate |
difficult |
b) | 8 cm | easy | difficult |
c) | 6.5 cm (or 6 ½ cm) | moderate | difficult |
* Based an a sample of 48 Y4 students in Oct-Nov 2013
** Based on a representative sample of 159 Y5 students in March 2010
NOTE: Students found the online version notably easier than the print version.
This assessment resource involves being able to measure the length of pictures of everyday objects. Specifically it involves recognising that if you start anywhere other than zero you need to compensate (offset) for that when you read the end measure on the ruler (or scale). It also involves identifying and recording the half unit of measurement [question c)].
An important concept in measurement is that it is the space that the measure takes up that is the measure not the mark itself (the space between the marks). This idea is also fundamental to the definition of partitioning (which underlies the construction of sub-units).
A significant number of students read the scale correctly to get the "end measure", but did not start measuring length at "0" (offset error). For question a) this is particularly prevalent as the object aligns with "1 cm" which is a common starting point for counting numbers (Natural numbers).
For question b) this error is less likely as the object aligns with 2 cm. For this question, under half the students simply read the "end measure", and over half applied an offset of 1 cm (instead of 2 cm), suggesting they believe that measures should start at 1. The last question (c) involved an offset and a half measure. Again, a significant proportion of students did not start measure at 0, but did incorporate the 0.5 into their answer. Overall, about 60% of students incorporated the 0.5 into their measure.
Common response | Likely misconception | |
a) |
1.6 or 16 or 7 |
Developing a different incorrect measuring system |
a) |
11 |
Counting the marks on the ruler |
a) |
4 |
Counting the units between the start and end mark |
a) |
6 |
Not starting to measure at 0 (offset error) |
Developing a different incorrect measuring system
Students who could not interpret the scale on the ruler could start with non standard units (such as feet, hand spans, paces, thumbs, etc.) to build up an understanding about how length is measured (i.e., a combination (or count) of end to end, non-overlapping units). As they measure, students could make a mark to indicate the increment of the units - this would result in them building up a scale which could be compared to a ruler.
Counting the marks on the ruler or the units between the start and end marks
These students indicate some understanding of how length is measured, but need to explore the unit they are using. Getting students to build up their own (non-standard) scale for measuring length can help them understand how a ruler uses the combination (or count) of a given unit.
Not starting to measure at zero (offset error)
For students who do not begin measuring length at "0"; these students need to explore measuring physical objects and count up as they measure the units. For example use 1cm cubes in a line and build a scale by marking each added new cube. This way the count of the first cube results in the first measure which supports students to recognise where zero might be and that measuring starts from zero. Students need to find out that the units stack end to end on each other to accumulate (as with non-standard units) and that a measure is a combination of these "non-overlapping units". Students can then begin to explore what it means when the measure is halfway (or a portion) of their standard measure.
Not measuring to the nearest half unit
For students who did not measure accurately to the half unit, ask them how they could describe a measure if it is not a whole unit so they could let somebody else know how far along beyond the whole number the measure was. Have them explore other practical ways of measuring lengths that involve parts of a unit.
For students who correctly measured the objects to the nearest half unit, they could be asked how many millimetres their answers would be. They could also look at the relationship between kilometres, metres and centimetres and talk about the need to find smaller units to help measure the length/distance accurately, and how these smaller units relate to the larger units.
Students could also start by working out how many of the "sub units (millimetre)" there are in a centimetre, and explore partitioning of lengths (or number lines) into different numbers. They could also look at the relationship between kilometres, metres, centimetres, and millimetres, and identify the best (most appropriate) measure for a range of lengths/distances, See Millimetre, centimetre, metre or kilometre? or Units of measurement.