Lessons With Cuisenaire Rods Notes On The Filmstrip ‘Numbers In Color’

Caleb Gattegno

Educational Solutions Worldwide Inc.

First American edition published in 1968. Reprinted in 2009. Copyright ÂŠ 1968-2009 Educational Solutions Worldwide Inc. Author: Caleb Gattegno All rights reserved ISBN 85348 111 X Educational Solutions Worldwide Inc. 2nd Floor 99 University Place, New York, N.Y. 10003-4555 www.EducationalSolutions.com

The Numbers In Color Filmstrip The Cuisenaire rods are too small to use for demonstration purposes with a large audience and it was therefore necessary to devise a means of increasing their size while retaining the essential properties of the material. A filmstrip was the obvious answer. The filmstrip provided an opportunity not only to show the material itself but to indicate methods of working with it that teachers and pupils could find useful. Each frame, as it is projected, can be explained and discussed. A considerable number of lessons can thus be derived from the limited space provided by a strip. Since the various situations presented by the frames are always available, they can be referred to whenever the occasion arises. The Numbers in Color strip can be used for: oral revision of knowledge gained after the actual material has been in use; or as an introduction to it, giving a few examples to which other examples can be freely added; or as the basis of lectures in which teachers show their colleagues what they have been able to achieve with the Cuisenaire material;

or to provide subject matter for discussion among teachers. A still more obvious use is in the training of teachers. Yet another is to reveal to interested adults the scope of a material which they may have up till then regarded as a mere toy or kindergarten aid. A filmstrip can thus serve in many ways as a means of giving access to the material itself. ***** In the notes that follow, the situations presented in the Numbers in Color filmstrip are reproduced and explanations given as to why these particular arrangements of rods were selected.

Table of Contents

Description Of The Material ........................................ 1 Study Of Whole Numbers ............................................9 Study Of Squares ....................................................... 31 Study Of Fractions ..................................................... 37 Theory Of Powers And Geometric Progressions ........ 55

Description Of The Material

1

Lessons With Cuisenaire Rods

Situation 1 shows the contents of a set of Cuisenaire rods as made in England. The set is seen to contain cuboids in white, black, red, pink, tan, light green, dark green, blue, yellow and orange. Pupils examining the material discover that only these ten colors appear. In showing this frame one should say no more than: Find the elements of this set. The viewers will notice the shape of each rod, which makes it possible to ask questions about the relative sizes of rods in the pile: which is the largest? which the smallest? A more careful examination will reveal other characteristics.

2

Description Of The Material

Situation 2 shows groups of rods of the same colors. Passing from the one to the other of these first two situations makes clear the principle of equivalence by color. The audience can be asked the colors of the rods, and to guess them in increasing or decreasing length. Situation 3 gives the order, increasing in one direction and decreasing in the other, of the selection of rods above. Thus if we take at random two rods a and b of the first set, it is clear that either a = b or a < b (a is smaller than b) or a > b (a is bigger than b). Situation 4 A single rod of each color is enough to represent the set in a particular way. To consider the set of equivalence classes (either by color or by length) we can take one rod of each kind to represent each class. This way of operating on

3

Lessons With Cuisenaire Rods

4

Description Of The Material

the set produces an interesting sub-set which will be used several times below.

Situation 5 deals with three samples of the sub-set above. a) In the first example the staircase on the right shows simply that the sub-set can be ordered. But since no two rods in the set taken at random can be equal to each other, one inevitably being either smaller or larger than the other, it is said that the order is strict, or that the sub-set is strictly ordered; i.e. strictly increasing as we move from the white to the orange and strictly decreasing as we move from the orange to the white. 5

Lessons With Cuisenaire Rods

In the left-hand staircase formed by the sub-set a new property is revealed: that the difference between consecutive rods is always the same and is equal to the length of a white rod. b) A third set has been arranged in sub-sets according to the principle of color affinity. This way of putting the rods together gives what Georges Cuisenaire calls the â€˜familiesâ€™: the red, the yellow, the blue, the black and the white. The choice of colors and families is an essential part of Cuisenaireâ€™s invention. By means of it a wide experience is immediately given to the pupils. While this frame is projected, the viewers may be asked to memorise the colors of the families in ascending and descending order. At least ten minutes may be needed for this. (For colorblind people this should be done by handling the rods, with recognition of the lengths to be associated with each color. Then the frame will confirm that the association is, for them, as easy and as secure as for the others.) The stairs with the white rods provide the opportunity for asking the following questions: How many white rods on top of each other do you think would be needed to produce the height of the red rod? of the light green rod? of the pink rod? of the yellow rod? etc. We can then return to the first staircase and give the names 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 to the rods, as measured by the white. The families are known by their color names and by their numerals: (1); (7); (2, 4, 8); (3, 6, 9); (5, 10).

6

Description Of The Material

With this introduction to the material we have drawn attention to the fundamental structures of sets of equivalence of classes of equivalence of order of ordered sets in two directions, increasing and strictly decreasing

strictly

and we have taught counting in both directions, from 1 to 10 and from 10 to 1.

7

Study Of Whole Numbers

Counting is a rhythmic activity. There is thus a danger that the mind will wander when numbers are being recited. Saying them backwards and skipping some regular ones prevents the process from being automatic. (Situation 5 may be reconsidered here.)

9

Lessons With Cuisenaire Rods

Situation 6 shows the first ten rods arranged in staircases so that the first step can be varied in height without altering the height of succeeding steps. In particular these staircases show odd as well as even numbers. All are arithmetical progressions and should be read in two directions. Some possible arithmetical progressions are contained in examples met already (for example, the series 2, 3, 4, 5 . . . is found within 1, 2, 3, 4 . . .). Others which have been omitted for lack of space can be made with the rods. See also in this connection Situation 28.

10

Study Of Whole Numbers

Situation 7 contains tables of decompositions of the first rods. It shows: 1

that the number of decompositions increases very rapidly with the length of the rod: 2 for the red 4 for the light green 8 for the pink 16 for the yellow, and so on.

2

that each set of decompositions is what we can now name 2, 3, 4 or 5 respectively (which eliminates the danger that 11

Lessons With Cuisenaire Rods

one color only will always be associated with one number since 5, for example, can be represented as much by light green and red, or simply white, as it is by the yellow). 3

that the reading of each line is an addition, and that it is possible, using the notation of multiplication when one rod is repeated, to read into these tables of decomposition that 5=2x2+1

or

5=3x1+2

12

or 5 = 2 + 3 x 1, etc.

Study Of Whole Numbers

Situation 8 studies the decompositions of 10, first by means of two rods (giving the complementary sub-sets in a set having 10 elements), then using a larger number of rods in each line. All lines of this table can also be read as divisions: e.g. 10 Ăˇ 3 and 10 Ăˇ 4. But the essential point in this frame is the associating of subtraction with the initial additions. When we find the rod that is needed to complete each line in order to reach a length equal to that of the rod in the first line, it is at the same time seen that the gesture can be made to the right or to the left (stressing that addition is commutative). On the other hand it is seen that if a number (or a length, or a rod) is given, every subtraction can be read as the inverse of an addition. In fact more can be found: selecting one rod as the first line, we can mentally see each of the other rods beneath it, to the right or to the left. We can then ask: what remains of the original rod when the part covered is equal to each of the other rods, taken in turn? Here what is cut off is actually visible. But because of recognition of complements, it is evident that the mind sees the difference on the first rod; in fact, the difference is as easily seen as the amount cut off. Using this link we can easily learn all subtractions up to 20 with the rods. Going beyond that, we should use an algorism; this is described in detail in the Teacherâ€™s Introduction (p. 42-43).

13

Lessons With Cuisenaire Rods

Those who use the rods will see how to develop the content of this frame step by step, thus achieving mastery of subtraction taught simultaneously with addition. **** Situations 9-15 are concerned with multiplication, factors and rapid mental calculations. The examples chosen are simple but the work they suggest can easily be generalised, especially if used in conjunction with the Product Chart invented by Georges Cuisenaire (shown in Situation 11). Lengths formed from one or more rods cannot all be made up by using rods of one color only (excluding repetitions of the white rod) or by a chosen group of rods repeated a certain number of times. Those that cannot be so formed give the prime numbers. The others correspond to the numbers that have factors. Situation 9 shows with the first table of decompositions that 7 is prime. It also studies the factors of 4, 6, 8, 9, 10, 12, 24 and 25. It shows the lengths made up of rods of one color immediately below the length corresponding to each number: 4=2x2 6=2x3=3x2 8=2x4=4x2

14

Study Of Whole Numbers

9=3x3 10 = 2 x 5 = 5 x 2 12 = 2 x 6 = 6 x 2 =3x4=4x3 24 = 2 x 12 = 12 x 2 =3x8=8x3 =4x6=6x4 25 = 5 x 5 Situation 10 shows how to pass from the table of decompositions of any composite number (the example here being 12) to the 15

Lessons With Cuisenaire Rods

factors of that number, and conversely. The length 12, made up of 10 + 2, is also made up of 2 x 6, 3 x 4, 4 x 3 and 6 x 2. Putting six red rods side by side and two dark green rods side by side, we form two rectangles that are found to be congruent

on superimposing one on the other. Similarly, two rectangles made from four light green rods and from three pink rods are also congruent. The shortest dimension of the dark green rectangle is equal to the length of the red rod; the longest dimension of the red rectangle is equal to the length of the dark green rod. It can be seen from the frame what the other two arrangements of rods suggest.

16

Study Of Whole Numbers

So, if we represent with one cross of rods (in which the two dimensions are apparent) both the dark green and the red rectangles, or with another cross the light green and the pink rectangles, we can see how in each case we are also representing the original length orange and red end to end. For us to return from a cross to the original length, it is enough to build the rectangles whose dimensions are shown, and to put the rods thus obtained end to end. The cross is therefore a symbol for the composite number, since by following these instructions we can find that number. The factors of twelve, i.e. the rods used in the crosses, are put on one side as something further that can be deduced from the study of this situation. Situation 11 The Cuisenaire Product Chart. This shows the thirty-seven different products that can each be made by one or two crosses of rods. But here we have the symbols of the products only. (In order to fit the filmstrip, the lower part of the chart is reproduced in a second frame, on its side.) A discussion on how to use the chart is found in the text Numbers in Color and in the Teacherâ€™s Introduction. The Product Chart and sets of Product Cards are important items in this method, and although their mastery takes a little time, their use gives a considerable return for the time spent.

17

Lessons With Cuisenaire Rods

Situation 12 assumes that products are being learnt and gives an interesting exercise. Though this may be repeated many times it retains its freshness since it presents each time a new challenge to the pupil and frees thought in a way not commonly known.

18

Study Of Whole Numbers

A handful of rods is placed before the pupils and they are told to find what length would be made if all the rods were put end to end. No one may touch them; they must operate mentally.

Two ways of operating are suggested: â€˘

In the first case the rods of one color are noticed, counted and their product formed, this being remembered or added to what went before. In operating systematically from white to orange on all the colors that are present, the learners exhaust the pile. A count could be made of the number of operations involved and the time taken

19

following this method, and of the number of errors that occur in using it. â€˘

The procedure that seems the most satisfactory to adopt is to look not for products to employ, but for complementary numbers forming tens. This is illustrated. The eye sweeps over the field and picks out the number of orange rods (if there are any) in the set, counting them and memorising the result. The choice of pairs is then made at random, black with light green, tan with red, dark green with pink, white with blue. Sometimes the positions of the rods suggest new combinations: for example, 3 tan rods and 1 dark green (equal to 3 orange rods) might be taken together to give 30. The variety is considerable.

Lessons Study With Of Whole Cuisenaire Numbers Rods

Situation 13 displays three such handfuls of rods, and it will be of interest after the study of the last situation to attempt to apply both the methods just described, or variations of these, to the three heaps. The aim of the exercise is to show the range of mental activity that can be performed from the simple display of a handful of rods. It gives useful practice with number relations that are of value in everyday life. The game can be varied by asking how many dozens (or how many twenties) the pile contains, instead of tens as described above. In countries where the number twelve is important, this exercise gives a facility that is obviously useful. In Modern Mathematics with Numbers in Color, handfuls of rods are used for the study of sets and sub-sets. Situation 14 contains examples of important uses of the Cuisenaire rods for training in the understanding of products, of swift calculations and of the way in which one situation can be turned around in a large number of ways, always producing new information, another notation or new words.

21

Lessons With Cuisenaire Rods

1 The first line shows how one product can be changed into another: 7x8=7x4x2=7x2x2x2 A cross thus becomes a tower. Reading to the left a tower becomes a cross. If the second factor of the cross is changed into a product of prime factors, we obtain the tower of each number that shows only its prime factors. Thus: 48 = 6 x 8 = 3 x 2 x 2 x 2 x 2

22

Study Of Whole Numbers

The decompositions of two numbers into their prime factors will yield two towers showing at once which are the common factors and which is the highest common factor (H.C.F.). Towers are also the source of powers since the height of the tower can be read as its index: 8 = 2 x 2 x 2 = 23, etc. A new notation appears: where two rods are one horizontal and one vertical, we read the pair as the number representing the horizontal rod taken to the power equal to the number representing the vertical rod. In Mathematics with Numbers in Color Book V this is considered in detail. A first step in the understanding of logarithms can be found here. (This subject is discussed further at the end of the notes to Situation 28.) 2 The second line shows what happens when we want to read a product using factors not shown in the cross. Thus 40 = 2 x 20 or 4 x 10 or 8 x 5. We can also say that if we double one factor we must divide the other by two. It works both ways. The number 72, for example, which presents more combinations, could now be tried. It was not possible to use this number in the filmstrip because of the space it would have taken up. 3 The third line shows examples of â€˜threesâ€™, or towers made of three rods. We can multiply the three numbers together and find the value (N) for which the tower stands. We find that the value does not depend on the order of the multiplication; thus in a tower of the three rods a, b and c:

23

N=axbxc

and also

N = (a x b) x c

=axcxb

= (a x c) x b

=cxbxa

= (c x b) x a

=bxaxc

= a x (c x b)

=bxcxa

= b x (a x c)

=cxaxb

= c x (a x b)

By removing one rod from the tower a, b, c we are effectively dividing the number represented by the tower by the number represented by that rod. Thus

= b x c indicates that when rod

a is removed, the number is divided by a, the result being what is left, b x c. Hence we find that by removing one rod, or a cross:

Lessons With Cuisenaire Rods

From this work many other relations can be established such as:

n x N (where n is an integer) = Try to solve

Constant practice with such exercises creates an immense store of arithmetical bonds which will be found useful on many occasions.

25

Lessons Study With Of Whole Cuisenaire Numbers Rods

Situation 15 gives an idea of how special towers are to be considered as powers, and leads to the recognition that the laws of powers are independent of the particular element considered. This illustration can also serve as groundwork for geometric progressions. See also the notes to Situation 28.

Situation 16 is to be considered with the next, and provides a basis for the introduction of different scales of numeration. In the illustration the example of 23 is considered, this number being formed in each row using rods of one color, with the remainder added. 26

Study Of Whole Numbers

Thus:

23 = 2 x 10 + 3 =2 x 9 + 5 =2 x 8 + 7 =3 x 7 + 2 =3 x 6 + 5 =4 x 5 + 3 = 1 x 42 + (1 x 4) + 3

The writing of the last expression shows that we do not allow ourselves to employ, using that notation, any figure that represents a number greater than the rod used repeatedly. A given length can therefore be written in various ways. If we consider the base to be the orange rod, we write the length chosen here as 23. If the base were the blue rod, we should write 25 â€” for this means 2 blue rods and a yellow. 27 is the writing for the length in the base 8; 32 (base 7); 35 (base 6); 43 (base 5); 113 (base 4). Find the expressions in base 3 and base 2. Once this situation is considered, we should work out examples using other initial lengths in order to gain experience with different bases. Operating on them will become an easy matter since the rules are the same in all bases, even if the writings look different. For example, 7 x 7 is equal to 100 (base 7). In Mathematics with Numbers in Color Book V these questions are treated for primary school learners.

27

Lessons With Cuisenaire Rods

Situation 17 An example of working out the lengths of the rods using only red and white rods appears here. When we reach the pink, we use the red-red cross; and when we reach the tan colored rod we use the red-red-red â€˜threeâ€™. If we translate each scheme on the left into figures (using 1 to express presence and 0 to express absence) we can write; 1, 10, 11, 100, 101, 110, 111, 1000, 1001, 1010 as the lengths of the ten rods in the binary scale. Any length made of rods can be written using this notation and calculations can be performed in it, as in the ordinary denary notation.

28

Study Of Whole Numbers

If we use light green rods in place of the red, but in conjunction with one red or one white as appropriate, any length of rods can be made up and the corresponding written forms will contain the figures 0, 1, 2. It is clear that any length of rods can similarly be made up using rods of one color only with a rod from one of the shorter lengths.

29

Study Of Squares

Situation 18 The equivalence a2 - b2 = (a + b)(a - b) The difference between two squares can be seen by placing the smaller square on the larger, in one corner. When the rods of the larger not covered by the small square are rotated through 90째, a rectangle is formed whose dimensions are on the one hand the sum of, and on the other the difference between, the lengths of the sides of the two squares. In the first of the three examples shown, the small red square is placed on the larger dark green square. The dark green rods that are not covered by red rods are then swung round to form a rectangle with the uncovered portion of the remaining dark green rods. The longer dimension of this rectangle measures dark green plus red; the shorter dimension measures dark green minus red. In notation, the difference between those two particular squares is rendered thus: d2 - r2 = (d + r)(d - r)

31

Lessons With Cuisenaire Rods

What would the notation be for the other two examples illustrated? Situation 19 The equivalence (a + b)2 = a2 + b2 + 2ab This relationship obtains when we compare the square on a sum of two lengths with the sum of the two squares on each length. We note that the latter taken together are smaller than the first square by two equal rectangles which have for their dimensions one side from each of the two smaller squares. In the first example, squares built on the lengths of the pink and the dark green rods are placed on a square built on the 32

Study Of Squares

combined lengths of these two rods. In the illustration this third square is formed with orange rods since the pink and the dark green rods end to end are equivalent in length to an orange rod. It can be seen that the area of the square of orange rods is equivalent to that of the combined areas of the two smaller squares, plus the areas of the two uncovered rectangles

defined by the sides of the smaller squares. In this example the relationship can be expressed as: (d + p)2 = d2 + p2 + 2(d x p)

33

Lessons With Cuisenaire Rods

What would the notation be for the other two examples illustrated? Situation 20 The equivalence (a - b)2 = a2 + b2 - 2ab The square of a difference is harder to perceive but is, all the same, easily produced with the rods. The details of the actions needed to establish it appear in the illustration. It is necessary to note that a square can always be represented as the square of a difference, for example: 62 = (8 - 2)2 = (10 - 4)2 = (12 - 6)2 = (13 - 7)2

34

Study Of Squares

So a dark green square could be compared with any number of related pairs of squares. If we border the dark green square on two sides with tan rods and attempt to form a new square, we find that two tan rods are needed on one side and two on the other and that this actually produces an area equal to 82 + 22. Bordering the dark green square with orange rods in similar fashion, we should need four on one side and four on the other, producing an area equal to 102 + 42. But if we remove the borders from these new figures, we have to subtract in the first case 2 x (2 x 8), and in the second

35

Lessons With Cuisenaire Rods

2 x (4 x 10), re-obtaining both times the dark green square. So in this example, firstly the dark green square (or 62 ) can be considered as (8 - 2)2; secondly if we add two tan colored rectangles each 2 x 8 to the dark green square we increase the size of the figure to 82 + 22; thirdly, we can return to the original square by removing the two added rectangles. Consequently we can write that: 82 + 22 - 2 x (2 x 8) = (8 - 2)2 or in terms of the actual rods used in the example: (t - r)2 = t2 + r2 - 2(t x r) In the lower example we see that: 42 = (7 - 3)2 = 72 + 32 - 2 x (3 x 7)

36

Study Of Fractions

The next five situations illustrated convey little to the reader or viewer without some explanation. It is necessary to accept as a fact that each pair of rods can give rise to relationships. One, in particular, that a pair of rods can express is the fraction. We may describe this in another way by saying that in measuring one rod against another the result (made objective in the pair of rods with which we mentally associate this relation) will be, by definition, a fraction. As either of the two rods in a pair can be chosen as the measuring rod, the fraction can be named only when the order of presentation is decided â€” a fraction is an ordered pair. At the same time, if the order is reversed, the fraction produced is the reciprocal of the first fraction. Situation 21 gives an opportunity to read (and write) fractions. The staircase of Situation 5 is brought back here. We can now ask: If a particular rod is 1, what are the values of the other rods? This gives fractions 37

Lessons With Cuisenaire Rods

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

,

, ,

,

,

,

,

,

...... ,

,

,

38

,

,

,

Study Of Fractions

The illustration shows only

and

,

, , , , ,

,

,

Two sets of rod pairs at the top of the illustration show two fractions and their reciprocals: ,

and

,

This gives an opportunity to practise combining reciprocals. That is to say, we can recognise that since , then the reciprocal of the reciprocal of

is the reciprocal of is

; and we can

recognise what follows from this: that fractions and their reciprocals form two distinct classes by the parity of the iteration of reciprocals from the first onwards. Naturally when this is understood we can introduce the notation, starting with and

39

Lessons With Cuisenaire Rods

,

etc.

which opens a field of experience where the new notation gains its meaning. Thus the reciprocal of is

and

40

Study Of Fractions

Situation 22 Equivalence â€” A Another way of presenting the class of equivalence of a fraction is shown in this illustration. Beginning with a red on light green (which, if both are measured by the white, gives ), we again find if two red rods are placed end to end against two light greens. In this new case, if the red is the measuring rod, we get two over three. If we now add one more red and one more light green respectively and measure with the light green, we again find two over three. And so on. Thus it is seen that the rational number is formed of the class:

This equivalence we shall call Equivalence â€” A Similarly in the second column which shows the fractions:

the same relation is found when each pair of terms is measured by different rods. We can conclude that it is possible to multiply and divide the two terms of a fraction by the same number without changing its value. Furthermore if two of these fractions are chosen at random and we add the two numerators and the two denominators, once

41

Lessons With Cuisenaire Rods

again an equivalent fraction results. This is equally true if we subtract instead of add. Thus in beginning with

we can find a whole series of

equivalences by this method, and we can write them as:

(where the operations carried out above and below are the same. See also Mathematics with Numbers in Color Book IV.) Situation 23 gives examples of addition and subtraction of two fractions having the same denominators

and how to work out the addition when the starting pair is formed of fractions having different denominators. If the denominator of one fraction is a multiple of that of the second, then, as in the three middle lines of the illustration, the first is maintained unchanged while the second is changed into its equivalent.

42

Study Of Fractions

The example worked out is: or

or

The last example worked out is:

which is replaced by

and the answer is

43

.

Lessons With Cuisenaire Rods

The illustration only shows the steps of what is commonly known as the addition of fractions. The foundation of this work with the colored rods is to be found in the Teacher’s Introduction and Book IV.

Situation 24 Equivalence — M This illustration tells a complex story. The left-hand side shows, from top to bottom, the red and the yellow rods — first close to each other, later separated - denoting that the red is yellow or the yellow

of the red.

44

— of the

Study Of Fractions

The relationship remains independent of the distance between the rods. The next column shows that if we insert a light green rod between the red and the yellow, we can still say the red is the yellow, but it also indicated that the red is which is itself

of

of the green

of the yellow.

If we bring them closer, as is done in the next line, we can read that the red is

of

of the yellow as well as being still

of the

yellow. For the third line read the question: What is upwards we find:

of

is equivalent to

of

? Moving

since these are two

names for what the red is of the yellow. In the last column, inserting the black as well, we have the three steps leading to

of

of

; reading upwards from the bottom

line we find that this is again equal to

.

Thus we create a new class of equivalence for word â€˜ofâ€™. We shall call this Equivalence - M.

45

through the

Lessons With Cuisenaire Rods

Situation 25 Multiplication of fractions This shows how to work out the general problem: find a fraction equivalent to a fraction of a fraction. The procedure is worked out on of

and on

of

.

By Equivalence - A =

and

=

46

Study Of Fractions

so

of

and that is

=

of

by Equivalence - M

The working for the second example since we need only alter

of

is slightly different

to obtain a situation where the 9 can

be dropped: of

=

It is customary to replace the word ‘of’ by the sign ‘x’ and say ‘multiplied by’ instead of ‘of’. (C. f. Book IV and Modern Mathematics with Numbers in Color.) Situation 26 Division of fractions This tells an even more complicated story. Starting with

and

we get

for the fraction equivalent to

of

Now, if we take

of

, or

of

47

, we get

or

.

Lessons With Cuisenaire Rods

So if we ‘multiply’

first by

and then by

definition, then, the product ( ‘quotient’ of

by

.

For we can see that

and we can write that

48

x

we regain

. By

) is equivalent to the

Study Of Fractions

or

by analogy with: from

6=2x3

we can write

6รท2=3

or

The steps are always the same whatever the fractions, and we have found the well-known general rule for the division of fractions.

49

Lessons With Cuisenaire Rods

Situation 27 Permutations This gives tables of permutations of 1, 2, 3 and 4 distinct elements. They contain respectively 1, 1 x 2, 1 x 2 x 3, and 1 x 2 x 3 x 4 different lines. How many lines would there be in a table of permutation of five distinct elements? By passing from one to the other of these tables we can see how to find the rule for the calculation of the permutations with n different elements. This frame may also be used for the decomposition of groups of permutations into sub-groups, making use of interchanges of two terms as operations allowing, or not, the passage from one known permutation to another (see Mathematics with Numbers in Color Book VII). Situation 28 Arithmetical progressions This presents a lesson which can easily be given orally to children of various ages. The construction of the staircase in this illustration shows the structure of an arithmetical progression:

50

Study Of Fractions

we may commence with any term whatever (here a light green rod) and what follows is obtained by always adding a constant term (here the red rod). The height attained at each step, from the lowest level upwards, gives the terms of the progression: ho h1 = h o + d h2 = h1 + d = ho + 2d h3 = h2 + d = ho + 3d hn = hn - 1 + d = ho + nd where h is the height of a step and d the difference between two steps.

51

Lessons With Cuisenaire Rods

Four types of problems can be dealt with from this illustration involving the finding of: 1 the level or height of a given step, knowing the row (or step number), the height of each step and the height of the first step: hn = ho + nd 2 how many steps must be climbed to reach a certain height, if the number of rows, the height of each step, and the height of the first step are known: n = (hn - ho ÷ d 3 the height of the first step, if the level, number of rows, and heights of the equal steps are known: ho = hn - nd 4 the heights of the equal steps, given the level, the number of rows and the height of the first step: d = (hn – ho) ÷ n The remainder of the illustration shows a way of finding the sum of the terms of an arithmetical progression (a). It is obvious that the sum of two terms at equal distances from the two ends is constant for a given progression (b). Moreover twice the sum of all the terms of the progression (each multiplied by 1) is equal to the number which measures the area of the rectangle formed by joining up the two staircases (c). Expressing this in writing we obtain: a) ∑ = ho + h1 + h2 + . . . . . . + hn ; (n + 1 terms) b) ∑ = hn + hn - 1 + hn - 2 + . . . . . . + ho (n + 1 terms) c) ho + hn = h1 + hn - 1 = h2 + hn - 2 = . . . . . .

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Study Of Fractions

d) 2âˆ‘ = (n + l)(ho + hn) = (n + 1) (2ho + nd) Here again there are four types of problems corresponding to the four ways in which this formula can be written (see also Book VII).

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Theory Of Powers And Geometric Progressions

Note If we go back to Situation 15 we see that similar steps are followed with rods of different colors. In one column we see rods; in the next column, crosses representing the square of each of these rods; then the cubes, and finally the fourth powers. But here, instead of adding a constant amount, we are multiplying by a constant amount. The work can be extended by imagining the successive powers. The points to be noticed are that: 1 the method of forming the successive powers is independent of the value of the rod used a, a2, a3, a4, a5, . . . b, b2, b3, b4, b5, . . . c, c2, c3, c4, c5, . . .

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Lessons With Cuisenaire Rods

2 if the towers of rods from a given line are placed on top of each other, new powers are formed whose value is the power obtained by adding the number of rods in each tower a2 x a3 = a5 a x a4 = a 5 a3 x a5 = a8 3 if we remove a rod or rods from a cross or a tower, the value of the number represented is divided by the value of the tower removed, and a tower representing the result will contain a number of rods equal to the difference between that in the original tower and that of the tower removed a7 รท a3 = a4 a9 รท a6 = a3 a8 รท a5 = a3 4

any one power can be formed in many different ways a9 = a8 x a = a7 x a2 = a6 x a3 = a5 x a4

If instead of selecting two equal rods we make a cross of unlike rods and form powers of it, it will be seen that (a x b)3 can be proved to be equivalent to a3 x b3, etc. Similarly it will be seen that (a2)3 = a6 = (a3)2.

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Theory Of Powers And Geometric Progressions

Starting with a, and multiplying successively by b, we obtain a, a x b, a x b2, a x b3 which form the terms of a geometric progression. We can now return to the notation mentioned in the notes on Situation 14. If we place one rod horizontally and another vertically to represent the height of a tower and hence the power to which the first rod is raised, we can consider that we have a notation for powers. If we act on the vertical rod by adding to or subtracting from it, we indicate that we multiply or divide the value of the initial tower by the value of a second tower on the same base whose height is equal to the amount added to or subtracted from the vertical rod. The height of the vertical rod can be doubled or trebled. Conversely we could divide it by three or two. In so doing we should be passing from one pair to other pairs representing the cube root or the square root respectively of the first. In other words, we are now in a position to consider fractional indices.

Example:

= 512

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Lessons With Cuisenaire Rods

= 23 = 8

= 21 = 2

= 22 = 4

= 26 = 64

By becoming familiar with these and similar expressions, we have little left to be found out regarding how the algebra of integral indices extends to that of fractional indices. Situation 29 Study of Rectangular Prisms Here the rods are used to show that a variety of prisms with different dimensions can have the same volume. If we read the dimensions of a prism as length L, breadth B and height H, then a single pink rod, which is a prism, can be arranged as follows: L 1 4 1

B 1 1 4

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H 4 1 1

Theory Of Powers And Geometric Progressions

Using more than one rod we can find a number of positions

forming prisms and write their dimensions in as many ways. Thus two pink rods give: L 1 1 8

B 1 8 1

H 8 1 1

L 1 1 2 2 4 4

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B 2 4 1 4 1 2

H 4 2 4 1 2 1

Lessons With Cuisenaire Rods

or (1 x 3) + (1 x 6) = 9 different arrangements. Four light green rods give: L 1 1 12

B 12 1 1

H 1 12 1

L 2 2 3

B 2 3 2

H 3 2 2

L 4 4 3 3 1 1

B 1 3 1 4 4 3

H 3 1 4 1 3 4

L 1 1 2 2 6 6

B 2 6 1 6 1 2

H 6 2 6 1 2 1

or (2 x 3) + (2 x 6) = 18 different arrangements. Some of the arrangements of the two pink and the four light green rods are not shown on this frame.

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Theory Of Powers And Geometric Progressions

Situation 30 Surface areas of figures formed from cuboids This begins with the red rod, and it is easy to find that its total surface area is 10 sq. cm. The yellow is shown next and its total area is 22 sq. cm. The next six figures combine in different arrangements two out of three rods presented, but show only a few of the many possibilities. These may be set as problems, the requirement in every case being to find the total surface area of the combined body.

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Lessons With Cuisenaire Rods

With all three rods used together we can make many more spatial figures and ask for the surface area. Adding an orange rod necessitates that larger numbers be introduced in the calculations but does not otherwise increase the difficulty of the procedure. When actual rods are in use we can ask a great variety of questions on areas and volumes.

In addition to the arrangements shown in the filmstrip it would have been possible to photograph many other situations with the rods. Pythagorasâ€™ Theorem, or the sum of the squares of the terms of an arithmetical progression, are but two.

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