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In a previous blog I talked about gaining a sense of measure and the importance of accuracy in the way we talk about measuring. This can be developed through the activity of pacing off.
Pacing off is something we all do quite regularly, often without realising. You measure something without actually reaching for a standard measuring tool, but instead use another object to make an iterative comparison. For example: measuring the width of a shelf in pencil lengths, the height of a table in hands, or the volume of a jug in cups. The process of iteratively measuring allows us to recognise the unit of measurement – pencil lengths, hands, cups – and relate lengths, volumes etc. to them without relying on direct comparison. It also gives us a mental tool for estimating; consider phrases you hear such as ‘it’s about three hands wide’, ‘about an arm’s length’, etc.
This is something young children do in school too. Some argue that we’d be much better just handing pupils a ruler, scales or measuring jug and teaching them how to read a scale, but this process is actually so much more than just about measuring something in a scientifically recognised unit.
One of the joys of working at Cambridge Maths is having the time and resources to be able to learn how areas of mathematics relate to and support each other, and we have found that pacing off is crucial to a number of developments. Through pacing off, children gain a greater sense of what they are really measuring. If you’re measuring the length of an object you need to be able to lay a series of objects end to end to form a linear measure. To measure area you need to be able to fill the ‘inside’ of a 2D shape with buttons, tiles or squares. Volume can be measured through pouring cups of rice or stacking cubes. This physical action reiterates the qualities of the actual measure.
Pacing off allows children to understand the need for different scales of unit: you may measure the height of your book in paper clips but measuring the length of the school corridor with the same unit seems quite preposterous (although now I’ve said it…hmmm!). It helps give a sense of scale.
Crucially, pacing off helps one-to-one correspondence when counting and develops a sense of the number line. Imagine we if we only had one straw to use to measure multiple children’s heights. How could we make the process efficient? By constructing a scale on the wall life is made quite easy. But how do we do this? Where does zero go? Where do we place the numbers? Even then, what if someone’s height lies between two of the numbers? Fractions and/or decimals as positions on the number line are given a practical purpose.
The need for standard units can be made apparent when pupils measure the same (largish) object using different and non-standard measures. Looking at the varying results can lead to an interesting discussion about why they are different.
Pacing off in standard units e.g. in metre sticks or litre jugs also gives us a much greater reference for when we need to estimate measurements. If you’ve spent time measuring various volumes in litres and I put some objects in front of you you’ll have a much better estimate of their volumes; similarly when you answer ‘life like’ volume questions you’ll have a sense of whether your answer is sensible.
As we write the framework, activities such as pacing off form critical junctions between various areas in mathematics. We’re able to see beyond the immediate content and recognise how certain activities develop multiple themes. Making these explicit to teachers and pupils helps cultivate their understanding of the connections that make up mathematics.
SOMETHING TO TRY:
KS1/KS2: Give every small group of pupils a box of measuring tools; paperclips, straws, bamboo canes etc. Ask them to use the equipment to measure the height of their desk, width of the room, length of the corridor.
Discuss as a class/in groups the findings. Consider the following questions;
- What happens if you just give a number answer? Why?
- What happens if you use the smaller units? Why?
- What is the most sensible unit to use in each case? Why?
- Can you mix the units? How?
KS3: Estimate a square metre. Consider what it really means. How many people could stand in a square metre? How many square cm in one square metre? How do you know this? How long would it take to fill a square metre with square centimetres if you put one in every second? What about if you were filling a cubic metre cube with cube centimetres?
KS4: What do you measure in morgens? What’s a modern day equivalent?
KS5: Investigate decimal time. How do you write 4am in decimal time? How many decimal seconds in a standard hour?