The Measurement of LENGTH

Transcription

1 The Measurement of LENGTH

2 What is Measuring? MEASUREMENT To measure, according to The Concise Oxford Dictionary is to ascertain extent or quantity of (thing) by comparison with fixed unit or with object of known size. We can extract 3 important ideas in that definition for consideration with regard to learning about measurement. 1. extent or quantity 2. comparison 3. fixed unit or object of known size. 1. The extent of quantity gives us the purpose of measuring - to find out how thick, how long, how heavy, etc. Measuring depends on pulling together ideas of shape and space as well as number, and making connections between them. 2. When we consider the idea of comparison we become aware of the approximation involved in measurement. Originally body measurements, such as feet, cubits, were used to measure length, but these were very inexact. However, we must be aware that the instruments we use for measurement still possess a degree of inexactitude - comparisons of a bundle of classroom rulers reinforce this. Classroom balances also give an approximate measurement and cannot be used for fine distinctions. Children need to be helped towards an understanding of how accurate the comparison needs to be in particular cases. In order to make them better able to achieve this we should provide opportunities and practice in estimation, approximation and making judgements about appropriateness and accuracy. 3. The notion of a fixed unit involves the idea of scale. So that besides knowing that Sarah needs bigger shoes than David, we know how much bigger in a scale relevant to the size of the shoes. The system of units we, as members of a society, use, has evolved because of convenience and history. Understanding the need for, and function of, standard units must come before learning about particular systems of units. Stage 1 Pre-measurement The Learning Process Must of a child s early experience and early language involves measuring in one dimension - how long, wide, tall, high, far, etc. This does not necessarily lead to a comprehensive way of measuring. One way of developing a sense of measuring is ordering a set of familiar objects by different attributes, such as weight, length, height, capacity, etc. This may well produce different orderings, which can be discussed. These kind of activities will add to the child s sense of measuring and reinforce the sameness as well as the differences. Although ordering by weight is different from ordering by length, the ordering itself is the same process of comparing each object with the others in the set. Much of the work at this stage is concerned with the language of measurement, of comparison longer/shorter, heavier/lighter, further/nearer, faster/slower, and so on. Such comparison does not necessitate the use of a scale. Extending into the comparison of more than two objects allows for the use of the superlative; longest, widest, etc. It should be noted that ordering of more than two objects relies upon the child understanding that is A is heavier than B, and B is heavier than C, then A is heavier than C. This

3 is necessary for any ordering activity. Direct Comparisons compare and order objects without measuring. (NC AT4 level 1) At this same stage children should experience matching activities which involve matching different sets by the same attribute, eg bears to chairs, beds, bowls and spoons. The concept of conservation may be forming at the same time. A child may seem to understand that an object is the same length whatever its orientation, but not yet appreciate that area stays the same despite cutting and rearranging, or weight remains constant in a ball of plasticine no matter what shape it takes. The conservation of length may not be as firm as appeared when a child is asked to compare the distance between self and teacher and the same distance when both have moved by the same amount. Here is another example of the need for a child to take many different routes towards the grasp of a concept. Stage 2 - Choosing an appropriate measuring device Once the sense of measuring is present, children are happy to choose an appropriate measuring device. At first this may be something for comparison, so the object being measured is seen, and described as, greater or smaller in size. Children may make a mark on a stick or similar to measure length. Such judgements could be made about measurements of area, with the use of questions such as How much paper do I need to cover this book? Is this piece of paper the right size, too small, too large? The idea of a reading or mathematics area in a classroom makes the notion of area concrete for a child. The children could investigate how many bottoms fill the reading area, and compare with how many pairs of knees. We may need to find another unit for measuring the mathematics area as the use could be different, with tables and equipment taking up some of the space. This highlights the need for the unit of measurement to be appropriate. Children need many varied experiences in order to understand the need for the unit to be appropriate. They also need experience of different kinds of activities to build up an understanding of what criteria to measure appropriateness against. Why are squares more suitable than circles for measuring the area of a rectangular table? Children need to discover the answer to such questions and not just be given squared paper for the activity. This has implications for resourcing the classroom and organising and accessing these resources. Stage 3 - Repeating the use of the unit The next step after choosing an appropriate unit is using it repeatedly. This does involve recognising that the whole is the sum of its parts and that units can combine into a system which can be matched against the object to be measured. If we look at what this means in a practical situation, we can consider the measurement of the capacity of a container. If a child chooses marbles as the appropriate unit and fills the container with these, he/she is accepting that the total number of marbles represents the total space in the container, however inexact. There is also an acceptance that marbles can be used to substitute for this total space we wish to measure and that we can count our marbles and talk about so many marbles-full. When comparing objects children will not necessarily use the same units if they are offered a choice of material. For example, a child, trying to establish which of two containers holds more, is likely to fill one with sand and the other with centicubes. Children must be given the opportunity to go through this stage, with discussion and appropriate questions, if they are to reach an understanding of the need for a standard unit.

4 Working towards using standard units When developing the concept of measurement as repeating a unit the following general strategy should be used. Children need Explanation Examples to experience a) Non-uniform These are units which are not exactly the Crayon, non-standard same, but vary according to certain factors. pencils, (arbitrary) Therefore a bed made to be 10 handspans body measurements. units. long will vary according to the person who measured and made it. b) Uniform non- These are units which do not vary in length Cuisenaire Rods, standard units and therefore do provide a repeated unit as Multilink, a basis for measuring and comparing. Playing cards c) Standard units Adapted SI units Metres, Kilograms When using standard units for any aspect of measurement children need plenty of experience of using units in different circumstances to develop their understanding of what is, and is not, a sensible unit to use in different contexts. Computations involving measurements in standard units will only be meaningful to children if they have had sufficient time to become quite used to the size and feel of these units.

5 Length: using body measurements The old measurement of cubit, yard and fathom are interesting. For the true values of these measures an adult should be used; nevertheless it is fun, and an important stage in measurement understanding, to use such non-uniform, non-standard units. Span Cubit Yard Distance from the tip of the little finger to the tip of the thumb when the hand is outstretched. Two spans make a cubit. Distance from the tip of the longest finger to the elbow. Two cubits make one yard. This relates back to the practice of measuring cloth from the nose to the fingertip. Fathom Two yards make one fathom. This is very approximately equal to the height of a grown person. Cut out handspans, footprints and cubits. Use them to measure the length of objects and to search for relationships. Encourage the children to ESTIMATE before measuring with any unit.

6 The Metre Introducing Standard Measures for Length Give opportunities for using a plain metre stick for children to develop a feel for its length. Find objects that are longer than, shorter than, and about the same length as a metre. This could be linked with work on body measurements. Gradually introduce calibrated tapes and rules, and discuss the subdivisions of the metre. A decimetre (10 cm) strip is useful for the children to make and use as the children can handle it easily. A centimetre is often too small to be an appropriate unit for measuring desks, cupboards etc, and a metre can be too big to be useful in some contexts. Once the children have had sufficient experience and need to refine their measurement skills the centimetre can be used, and eventually the millimetre where appropriate.

7 The use of rulers often causes problems for children. Always check that children have appropriate rulers for the task is marked just in cm or marked with cm and mm. Watch for rulers which are marked in inches cm, these prove particularly problematic for lower juniors. It is always useful to check that children know where to start measuring from (not all rulers are marked from zero), and how to read the length being measured.

8 Perimeter Perimeter is the length of a shape s boundary. (Distance around the outside edge). For some lower juniors the awareness of boundaries may need further development through: a) Walking around edges and boundaries of large objects eg carpet PE mats, playground markings b) Surrounding everyday objects with counters eg Remove the object so that its boundary is clearly defined. c) Drawing around familiar classroom objects using crayons or pens on paper. Can children match the objects to their outlines? 1 An awareness of edges and boundaries does not is aware of perimeter. It is only when some form of introduced that the idea of perimeter having a length necessarily mean a child measurement is arises. Direct comparison between more than two perimeters 1. Take three books ordered in size.

9 Use some suitable materials to measure around the cover of each book. Compare the three lengths ie your ribbons. What do you notice about the three lengths? Which length was used to measure the perimeter of which book? Is there a relationship between the lengths of ribbon and the length of the perimeter of the books? 2. Choose three large plane shapes and estimate the order of their perimeters. Check your results.

10 What sort of materials can be used to measure perimeter and what practical problems might be present? String: String is amongst the most common media for perimeter measurement but many children find it difficult to manipulate and to cut accurately and will therefore need help from another child. Paper streamers: Paper streamers are useful as they can be folded around corners to achieve an accurate measurement particularly in an awkward measuring situation. Wool: Wool is also popular for measuring perimeters but it tends to stretch which makes it difficult to use accurately. It will often snap as the children are using it. Match boxes. blocks and bricks: These are fairly easy to use when measuring regularly shaped objects, but how do you explain the bits left over?

11 The corners also present another problem: The above picture shows the correct measuring technique. This shows incorrect measuring as there is extra measurement at the corners.

12 Using standard units of measurement for perimeter Using Cuisenaire Rods (see attached sheet) 2. Using squared cm paper. - How many different rectangles can you draw which each have a perimeter of 24 cm? - How many different shapes can you draw which each have a perimeter of 24 cm? 3. Using geoboards. - Using a geoboard to make as many shapes as possible which each use 7 pins in their boundary. How does the perimeter vary? Which shape gives the longest/shortest perimeter? 4. Using scale. - Use a large scale map to find the perimeters of your school playground/field etc. If possible check by measuring the item in metres. - Use a large scale map to find the perimeter of islands. - Can the children draw an imaginary island using the scale 1cm : 1m. What is the perimeter of their island?

13 CUISENAIRE RODS 1. Join two rods of the same colour. Find the perimeter. Can you find different perimeters? Why are they different? 2. What designs can you make using 1 red, 1 yellow, and 1 dark green? Find the perimeters. 3. Using 1 red, 2 green and 1 purple can you make designs with perimeters of 14, 16, 19, 20, 22 and 26 units? 4. Make up a perimeter puzzle for a friend to solve. 5. How many rectangles can you make with a perimeter of 24 units? Now try 12, 20, 28, and 32 units. Record the areas of your rectangle.

14 Circumference Length of the distance around a circle (perimeter) is called circumference. This can be measured in various ways using circular objects such as tins, wheels etc. a) Wrapping a tape measure carefully around and reading the measurement where the overlap crosses the mark at the beginning of the tape. b) Using string, or a strip of cm paper, to wrap around a tin until it overlaps. c) Put a mark on the rim of the tin/circular object and place this beside a mark on a strip of paper. Roll the tin until the mark touches the paper again and make a second mark. The distance between the two marks is the circumference of the tin. Relationships can be explored between the diameter and circumference of the circular object being measured. For most purposes in primary it is approximate enough for the children to work on the basis of the circumference being three times the length of the diameter. Some children may want to explore the value of it (3.14 to two decimal places) and check the formula that circumference (c) it x diameter or using c = 2 л r where r is the length of the radius.