Measuring and estimating are two related and necessary skills. maths consultant and writer Geoffrey Faux offers ideas on how infants can develop both these areas and relate one to the other.
MAKING THE RULES
In a faraway country, the King, who was very large and fed up with people saying how enormous he was, decreed that all the rulers and tape measures and other things for measuring length must be destroyed. But the townspeople needed to have some rulers secretly made so that they could make clothes, measure rooms, make doors that would fit and... what else? Can you make some rulers for the townspeople so that they can do all their measuring jobs?
THE CROOKED TOWN
Life without the ability to measure - or at least to make good estimates - would be frustrating. Ask your pupils to imagine they have moved into a town that has been built and furnished by someone with poor or non-existent measuring skills. Turn over to pages 34 and 35 to show them what it might look like.
National Numeracy Strategy Objectives, Section 5,Supplement of examples Yrs 1,2 amp; 3. Bold items are covered in this project.
Page 72 Understand and use the vocabulary related to length, mass and capacity; begin to know the relationships between standard metric units.
Measure and compare * by direct, side by side, comparisons
* using non standard units
* using standard units
Page 73 Suggest suitable units to estimate or measure length, mass or capacity.
Page 76 Suggest and use simple measuring equipment, reading and interpreting number scales with some accuracy.
THINGS TO DO
Ask pupils to:
* Imagine they are the King's subjects. Order some material to make new clothes for the King, or themselves. They need a way of telling how much they will need in a way that a shopkeeper can understand.
* Find the tallest child in the class, then send the child's height by note or e-mail to another class, using a unit that the other class can also use. How can they be sure the measurement will be the same?
* Work out the length of the corridor (or hall). Use a unit that another class can use to check their measurement.
* Has the King really escaped being measured? Or can they still measure him?
Show children the picture on the left. Tell pupils the story of the King who wanted to get rid of all the measuring devices in his kingdom. Ask your class to imagine they are in that kingdom. They have no rulers, no trundle wheels, no tape measures. But they need to be able to measure because they have some jobs to do.
MADE FOR MEASURE
THERE ARE THOUSANDS OF BOOKS IN OUR LIBRARY. THERE CAN'T BE - BUT THERE MUST BE MORE THAN A HUNDRED.
What about the books in your own school library? More than how many? Less than how many? What do you think?
HOW MUCH WILL ALL THE PEOPLE IN THE QUEUE WEIGH?
Caitlyn and Nico wonder how much all the people in the queue weigh put together. What do you think? What's the most it will be? What's the least it can be?
HOW HIGH IS THE ROOF FROM THE GROUND?
This is the Coco Circus School. The children are using their skills to retrieve a ball from the roof. How high is the roof from the ground? What's the highest it could be? And the lowest? What about your own school?
National Numeracy Strategy Objectives, Section 5, Supplement of examples Yrs 1,2 amp; 3.
Bold items are covered in this project.
Page 16 Understand and use the vocabulary of estimation and approximation, and give sensible estimates for a number of objects Page 72 Measure and compare * by direct, side by side, comparisons
* using non standard units
* using standard units
Page 74 Suggest suitable units to estimate or measure length, mass or capacity
* Use the pictures on pages 36 and 37 to discuss upper and lower limits with pupils.
* Can they, in each case, narrow the gap to two numbers that all the class can agree with?
* Extend the discussion to other examples in your own classroom and your own school - how many centimetre cubes can pupils get into a box? If there's no clock, can they estimate the limits of what time it might be - it's got to be before 11 o'clock break, but it must be after 9.30am. How closely can we narrow it down?
Key stage 1 measuring
Infants first meet numbers through counting. They then go on to make the connection between counting - where there are no gaps between the whole numbers - and measuring, where there are always gaps between the marks on a scale. Counting starts at one, measuring starts at zero.
Measuring in KS2 makes the important bridge between whole numbers and decimals.
In school, the aim is to give lots of practical experience of both counting and measuring. Estimation is an important aid here - and the more practical experience of measuring and estimating we can give, the better children will become at developing a "personal frame of reference".
The Personal Frame of Reference What does this mean? It's best explained by thinking about your own frame of reference, built up from experience and always useful - for instance:
* Do you know, just by lifting it up, that a kettle has enough water to make two cups of coffee?
* As you drive or take public transport through the usual delays and blockages, do you know to within five minutes when you will arrive at school, without constantly checking your watch?
* How good are you at estimating costs when shopping? When was the last time you were surprised by a receipt?
* Do you know how far it is from home to the newsagent? Do you know this as time taken or distance travelled?
You exercise your set of estimation skills, almost subconsciously, all the time. This project helps children to develop their own set.
Notes on pages 32-33
The search for a unit as "standard" is a challenge for children. Allow discussion. Don't reject ideas. Children will probably come up with such "standard measures" as hand spans and length of pace. Discuss how "standard" these are.
* For small measures, children may suggest using a coin - this really is a standard measure.
* Extend to making rulers - strips of paper marked with home-grown units - for example, "rulers" with coins drawn on them. (Hide the class rulers.) Notes for pages 34-35
Two important mathematical skills are: being able to say what is seen and being able to see what is said.
Ask the children to say what mis-measuring they see in the drawing and tell other children why they know the measuring has been faulty.
Notes for pages 36- 37
In measurement there will always be an upper and a lower bound. This is an important idea. It has enabled mathematicians to solve some very difficult problems. The ancient Greeks used it. They imagined a circle inside a hexagon and a hexagon inside the circle and saw that the area of a circle was between two hexagons; then between, two octagons, two decagons... They found the area of a circle, using this idea, very accurately. See illustration, above right.
These pages can be used to practise the important skill of narrowing the gap between an upper and lower bound for an estimation - so when we guess the weight of the cake we know it is certainly more than five grams but less then ten kilos. The better at estimation we are, the narrower these limits. As the limits move together, they "trap" the actual quantity or measurement between them. We may reach a point where the upper and lower bounds are very close. In defining and narrowing this space the children are acting as mathematicians.
Acknowledgments Thanks to Ken Saunders for his work on measurement. "The Enormous Ruler" on pages 32 and 33 was inspired by the story about the giant Ingens, which was published in a journal of the Association of Teachers of Mathematics, 7 Shaftesbury Street, Derby DE23 8YB. Thanks also to Janet Ainley for her ideas on personal estimation frames in "Is there any mathematics in measurement?" a chapter in "Teaching and Learning School Mathematics", edited by Pimm amp; Love, published by Hodder amp; Stoughton.