Educational TechnologyAuthor(s): John ClarkeSource: Mathematics in School, Vol. 2, No. 4 (Jul., 1973), pp. 19-21Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30211025 .

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5. Cubic Numbers (i) Show that the sum of any three consecutive numbers is always divisible by 9 exactly. (ii) What can you discover about the sum of (a) four consecutive cubic numbers, (b) five consecutive cubic numbers?

6. What Comes Next? (by L. R. Goide) (i) Find the next number in the series 9, 32, 75, 144, 245, ...... (ii) Some numbers can be expressed in the form X+ y+Z'. Can you find X, Y, Z for the first three numbers of the series above, and so find a formula for the nth term of the series? (iii) Some numbers in the series are square numbers, e.g. 9 and 144. Find the next square number in the series. For what values of n is the nth term in the series a square number? Why are all square numbers in the series also multiples of 9?

7. Happy Birthday to You! Were you born on a lucky date? No horoscopes or crystal balls are needed to discover the answer to this question-just simple arithmetic!

Write down your birthday in figures: Add the three numbers together: Subtract 20 times the first number: Add 18 times the second number: Add your age on your birthday in 1976: Divide the total by 19:

Example 17 March 1934 = 17/3/1934

17+3+1934 1954

20x17 = 340 1614

18x3 = 54 1668

= 42 1710 1710 =

19x90 90

If there is no remainder, you were born on a lucky date! But if you find that there is a remainder, you might be lucky if you tried again. Better luck next time!

If you have difficulty in dividing by T U 19, write down the 19-times table that begins-+ 19x1 1 9 and you will see that the U-figure de- 19x 2 3 8 creases by one each time, and the T-figure 19x 3 5_71 increases by two each time. So in the U- column we have the sequence 9, 8, 7, ....

and in the T- column we have the sequence of odd numbers, 1, 3, 5, ... Complete the table as far as 19x9. You can check your table by treating 19xn as 20n-n = 10(2n-1) + (10-n). This explains why we get the sequences in the T and U columns. So, having checked with the use of this table that your birthday, and everybody else's birthday, is a lucky date, can you explain why this is so, and why the sum of the T-figure and the U-figure in the table forms the sequence of integers from 10 to 19?

Solutions on page 31

A Magazine for Pupils Student Mathematics is published once a year, in September. It is edited by Prof. W. W. Sawyer, and is suitable for all secondary pupils, but sixthformers.will probably enjoy it most. Single copies will be supplied upon receipt of 3 international postal coupons from The Secretary, Student Mathematics, Room 373, College of Education, 371 Bloor Street West, Toronto 181, Ontario, Canada. An addressed envelope (at least 9"x4") would be appreciated.

Educational Technology

by John Clarke, Dundee College of Education

In the May 1972 issue of Mathematics in School (Vol. 1, Number 4) my approach to the structuring of activities designed to enable children to measure was briefly explained. This article is intended to describe the method in greater detail and to illustrate the nature of the children's work at each stage.

First, let us look at the knowledge and skills required before a child can be said to have the skills which enable him to make a wide variety of measurements.

They are as follows:

1. an understanding of the property (attribute) to be measured (e.g. length, mass, area, speed, etc.)

2. an understanding of the nature of the measure to be used (e.g. centimetre, gram, square metre, metre/ second, etc.)

3. an ability to select a suitable measuring unit for the task in hand and an ability to estimate a given measurement

4. an understanding of the use of a main measure (relative to the task in hand) and its fractional parts (e.g. a centimetre can be considered either as a frac- tion of a metre or as a measure in its own right)

5. a knowledge of the basic units in everyday use 6. the relationship between the measures and the

number system. From these aspects we can list the main concepts, principles and skills as in the table below and this enables us to construct appropriate activities for child- ren to learn to measure.

Concepts Principles Skills

attributes establishing relationships

creating and interpreting records

measuring units iterative process whereby a measure is related to the attribute

the use of measuring instruments

main units relationship between main units and

multiples and

sub-multiples

estimating and

rounding off

the ability to relate the activity of

measuring and the use of the number system, to record or carry out

computations with measurements

cardinal number grouping consistently (i.e. working to a

given number 'base')

An understanding of qualitative aspects of measure- ment and of attributes of objects and events depends on experience and the development of related vocabu- lary. Take one example, the term "length". It might

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refer to the length of a book, a rule, a desk, a table, a room, a playground or the road from town A to town B. Qualitatively its meaning may vary each time accord- ing to the context in which it occurs as for instance with the length of the desk which is shorter than that of the table but longer than that of the rule. Quantita- tively however the length of the desk can be considered to be constant and hence measurable in a given number of units. Before such aspects of given attributes can become meaningful to children they need much experi- ence with the common items which surround them in their contrived environment in the classroom and from the world outside. Activities with sand and water, with string used to make comparisons of linear attributes, with balance scales, with activities which can be timed or involve the class shop, together with teacher inter- vention as required (Fig. 10), provide the experience necessary to develop the relevant vocabulary and associated concepts without which measuring activities are largely meaningless. The repetitive computational exercises involving measures, indulged in in the recent past, produced supposed abilities in measurement which bore little relationship to actual skill when the prob- lems became practical and the units and such concepts as total lengths or differences had to be applied.

Relations are used in this initial stage. Such relations as "is shorter than" (Fig. 1), "is heavier than", "is taller than" (Fig. 2), "costs more than", "covers less than", help to establish relationships existing between objects or events. Attributes such as length, width, height, weight, area, volume, etc., begin to be under- stood and the arbitrary nature of such relationships becomes evident as a child's experience grows. This is particularly true if these kinds of ordering experiences are enjoyed more or less simultaneously with others involving ordinal number, spatial and other properties 20

of objects or events such as roughness, age, alphabetical order of letters, sequence of months, etc.

When a child can make a range of comparisons with reasonable ease he is ready to quantify measurements. It is at this stage that the development of an under- standing of the nature of a measure is essential. A grasp of the notion of the invariability of a measure in use for a given exercise is vital and, in my belief, should be achieved through the use of arbitrary units which are manifestly the same. Long lists of such items for the different forms of measurement important in the primary school curriculum can be given: a few will suffice. For linear measurement-Cuisenaire or Colour Factor rods (Fig. 3), exercise or text books, pencils, beads on string; for weight-marbles, cotton reels (Fig. 4), beads, wooden cubes (Fig. 6-cubes were used to calibrate the scale), nails; for area-newspaper or maga- zine sheets (Fig. 5), exercise or text books; for volume- marbles, matchboxes, cubes-the list is almost endless.

On consideration, three aspects of the above items are immediately evident. First, with few exceptions, the above will be found in almost every primary school: secondly, for any one exercise a number of identical items will readily be available: thirdly, they are all very easily counted. So, there should be no difficulty in asking children to create exercises of their own design using such materials during which the teacher can intervene when necessary to encourage appreciation of such aspects as the invariability of the measure, the choice of a suitable measure (which enables one to use numbers within one's comprehension) and the nature of the iterative process which is the basis of all measure- ment. When the latter idea is established the basis for the ruler, the centimetre tape, the compression scale or the spring balance (Fig. 6), the graduated measure, i.e. the more common measuring instruments, becomes

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QI

n11 f~flt

Q

evident and the time approaches when these can be used with true understanding.

The activities discussed so far may well have been used with children in the infants school, that is at a time when the numbers which must be used as a result of the introduction of the metric system to this country are almost meaningless. Twenty, the maximum number which many children of infant school age can manipu- late, is of little use when associated with centimetres. Twenty centimetres is less than eight inches. Equally, one metre is of little use as it is probably longer than any dimension of the average piece of furniture or item of equipment found in most classrooms. So, the infant teacher may well have a problem. Arbitrary units can help to solve this difficulty. All computations involving numbers of a manageable size for young children can be carried out in a fully meaningful way until the day comes when the metric units and the numbers required can be handled (Fig. 7). Sets of articles to be measured can also be easily prepared and used to control the activities. Sets of sticks for linear measurements (Figs. 1, 3, 7, 8), small parcels for weight (Figs. 4, 6), sheets of cardboard for area, containers for volume, and a variety of activities such as bead threading or the placing of pegs in pegboard to be timed, will all help to solve the problems of over-sized classes and those raised by allowing children to develop and learn at their own pace.

Arbitrary units also help in developing the concept of a main unit. Cuisenaire or Colour Factor rods were designed to assist children to understand the relation- ship between a whole and its parts and any rod can clearly represent the whole. Fifteen centimetres can thus be measured as 14 orange rods, 3 yellow rods, 1' dark blue rods, 2" violet rods, etc., and as the rods can also be used as measures of volume, weight and area, it

will readily be appreciated what a powerful learning tool they can become in the hands of imaginative teachers and children (Fig. 8).

Games can be played with dice with the objective of developing the notion of a "base", i.e. that a given number of one value can be exchanged for one of the next greater value (Fig. 9). The "values" can be repre- sented by rods, geometric shapes (Fig. 10), or by objects with purely arbitrary relationships, e.g. cubes or beads of equal size but of different colours, the colours representing the values. Such activities are developing and consolidating ideas and principles common to measurement and number, a factor which is now of particular importance to us since the intro- duction of the use of metric measures. Any item can of course be considered to be the main unit and the remainder become either multiples or sub-multiples. In recording, the latter can be shown to be fractional parts by the use of any marker-an oblique stroke, a vertical line, a geometric figure or, eventually, a "point". Whether this "point" will be a "decimal point" will of course depend on whether or not the base is ten. If the base for different exercises is varied, so much the better for then a true appreciation will develop of the use of the decimal point as the indicator of the units place and, hence, those places representing fractional parts. If the bases are varied, the values of the fractions be- come clear and "tenths" and "hundredths" are no longer mere names.

This article is only intended to give an outline of an approach to measurement which is designed to build on a real understanding of the developing process at each stage. Fuller details are contained in Relations and Measurement, an occasional publication (cost 40p-45p including postage) available from the College of Educa- tion, Dundee.

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