Arithmestics Makes Arithmetic Child’s Play
Caleb Gattegno
Educational Solutions Worldwide Inc.
First published in the United States of America in 1977. Reprinted in 2009. Copyright Š 1977-2009 Educational Solutions Worldwide Inc. Author: Caleb Gattegno All rights reserved ISBN 978-0-87825-019-6 Educational Solutions Worldwide Inc. 2nd Floor 99 University Place, New York, N.Y. 10003-4555 www.EducationalSolutions.com
Your set of colored rods was invented by a man called Georges Cuisenaire who lives in Thuin, Belgium. There are a great number of things you can build with these rods: houses, towers, ships, roads, bridges and many others. They can also be used to make patterns of various kinds, and some of these are shown in the pictures you will find in this book. Each picture tells you a particular story; as stories can never be told completely, you are left to explore and expand them as much as you can. You will be surprised to see how much you can discover using each picture as a starting point. As we do not want you to be restricted to the way we tell each story, we have suggested games and asked questions which will help you make your own discoveries in the world of numbers. The more you discover, the more you will understand and enjoy your school mathematics. The pictures in the following pages will introduce you to additions, decompositions, subtractions, short divisions, products and factors.
Table of Contents
The Story Of Twelve .....................................................1 The Story Of Eighteen ................................................. 9 Complementary Numbers .......................................... 17 The Story Of A Difference.......................................... 23 Re-Reading Additions As Divisions ........................... 33 Products And Factors.................................................41 Algebricks ....................................................................................... 47 Product Cards And Product Charts ............................................... 47 Publications.................................................................................... 47 Gattegno Mathematics Text-Books 1-7 ................................... 47 Workbooks 1-6 .........................................................................48 For Teachers And Parents..............................................................48 Now Johnny Can Do Arithmetic—C. Gattegno .......................48
The Story Of Twelve
1 Take your rods and make a table like the one on page 3. Replace one line with another; do it again. Can you add new lines forming patterns which do not appear in the table? Try to make some. You will see that almost any table could be extended in this way and that we should need a great number of rods to make all the combinations or patterns that are possible. 2 Look at the table you have made or at the one in the book. Shut your eyes and see whether you can remember one line, or two, or more. The names of the colors of the rods starting with the smallest rod are: white, red, light green, pink, yellow, dark green, black, tan, blue and orange. Use these color-names when you read the table. Look at the table for some time, then shut your eyes and call out as many lines as you can. Do this again several times. You will soon find a way of remembering them in any order. You will even be able to tell someone else new lines that could be added which would be part of the table in the picture.
1
Arithmestics Makes Arithmetic Child’s Play
3 Now make tables for the red, the light green, the pink and all the other rods up to the orange rod. With each table that you make repeat the game of finding new lines and naming them with your eyes shut. Each of these tables tells the story of one rod. 4 You can easily find out that the red rod is equivalent to a train made up of two white ones; and the light green to a train of three white ones; and the pink to a train of . . . and so on. The orange rod is equivalent to a train of two yellow rods or five red rods or ten white rods. Thus you find you can measure each rod using other rods as the unit. To read your tables, instead of using the color-names you can use the number-names which are found when all rods are measured by the white rod. Instead of calling the rods in the first line of the table ‘orange and red’, you can say ‘ten and two’ (and that is called twelve and written 12). So the second line can be read as five and seven or seven and five. Read in this way the rest of the story of twelve either from the table you have made, or from the one in the book. See if you can do it with your eyes shut using the number-names instead of the color-names 5 In your exercise book, write down your table using these signs:
1 for white
5 for yellow
2 for red
6 for dark green
10 for orange
3 for light green
7 for black
11 for orange and white
4 for pink
8 for tan
12 for orange and red
2
9 for blue
The Story Of Twelve
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3
Arithmestics Makes Arithmetic Child’s Play
Use the sign + to mean that the rods are put in a train or end to end (and read it plus). Use = to mean equivalent to. See whether these lines are in your table: 10 + 2
9+3
6+6
1 + 11
4+8
7+5
6 Write down, using the signs for the number-names, all the tables you have already made for each of the rods. In which table can you find the following: 2+7
3+4
1+8
5+2+1
7+2+1
Check your answers with the rods. Are 2 + 7 and 7 + 2 equivalent? 9 + 3 and 3 + 9 equivalent? 5 + 6 and 6 + 5 equivalent? Make a staircase with your rods beginning with the white one. Is it true that 10 = 9 + 1, 9 = 8 + 1, 8 = 7 + 1, 7 = 6 + 1, 6 = 5 + 1, 5 = 4 + 1, 4 = 3 + 1, 3 = 2 + 1, 2 = 1 + 1?
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4
The Story Of Twelve
7 There is another way of writing down the table for each rod, by using one letter as a sign for each color: w for white
p for pink
b for black
r for red
y for yellow
t for tan
g for light green
d for dark green
B for blue o for orange
So that now you can write the pattern this way: o+r b+y d+d etc. . . . Or this way: o+r=b+y o+r=d+d o+r=B+g etc. . . . This notation shows the relationship of the rods to each other regardless of which rod is used to measure the others by. The line d + d, for instance, will be equivalent to o + r whether you measure with the red rod or the white one. If the red rod is used
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Arithmestics Makes Arithmetic Child’s Play
for measuring, only the number-names would change from 6 + 6 = 12 to 3 + 3 = 6. Is it true that r+d=t
g+y=t
b+w+r=o
p + r + g =B w + t = B
g+d+g=o+r
Check your answers with the rods. 8 Now think of a white rod as a bar of chocolate. Then which rod will represent 3 bars, 8 bars? When you have 12 bars of chocolate, you also have one dozen. Can you find how many boys can have 3 bars each if they have 12 bars to share among them? If you have a dozen pencils, how many people can have 6 each (or half a dozen)? 9 Can you complete the following expressions:
•
w+ +w=p r+w+r= g=r +
y= +r p= +g +g=y
w+=g w+g+=y +r=p
•
3+ =4 2+1= +1+2=4
3= +2 1+2+ =5 5=3+ +1
3+2= 1+1+=3 2= +
•
b=r++r
d=g+w+
g+p+w=
6
The Story Of Twelve
•
y + +p = o r+r+r+r= 2r + 2g =
g+g+w+ =B =y+w =d+r+w 3w + r = = 3w + y o=y+w+
4+1+ =8 5+5= 9=3+2+ 7=2+2+
3+=6 =7+3 4+4=3+ 6+2+2=
Make up your own sums and find the answers.
7
4+2+=9 2+3+1+1= 2+1+4=+6 =4+4+1
Arithmestics Makes Arithmetic Child’s Play
8
The Story Of Eighteen
1 You can see that this story is not very different from the one before. But since 18 is a larger number, we shall have a much bigger table. There will also be much more to say and write, and many more questions to ask. Make this new table with your rods as you did before with the table of 12 and try to add new lines of your own. Write the table down in your exercise book using either figures or letters. 2 In this table, as in the last one, we can see that it is easy to make lengths greater than 10 by making a train of rods. You can see this clearly if you make a staircase (like the one on the right-hand side of the picture at the top of page 12, or like the figure on the right). You can give to each step of the staircase a number value if you measure with the white rod as the unit: 1, 2, 3 . . . 10. If you place an orange rod end to end with the white rod of the staircase you have a length called eleven (written 11); now place the orange rod end to end with the red to make twelve; then with the light green, the pink, the yellow, the dark green, the black, the tan and the blue to make thirteen, fourteen,
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Arithmestics Makes Arithmetic Child’s Play
fifteen, sixteen, seventeen, eighteen, nineteen which are written 13, 14, 15, 16, 17, 18, 19.
Make tables with your rods for 13, for 14 . . . up to 19. In the table of 18 you have made (or in the picture) can you find the following? 18 = 12 + 6
18 = 13 + 5
Write down, in figures, some other lines of the table of 18. 3
From the table you can see that 5 + 7 + 6 = 18
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10
The Story Of Eighteen
If you change the order of the figures you can write 5 + 6 + 7 = 18; if you change it again you can write 6 + 7 + 5 = 18, and so on. Can you do the same with other lines in the table? 4 In one line you have (1 + 2 + 3) + (1 + 2 + 3) + (1 + 2 + 3). Which is it? Each group of numbers in the brackets can be represented by one rod. Which? Is it possible to make a line with these same rods putting them like this: (1 + 1 + 1) + (2 + 2 + 2)+(3 + 3 + 3) Represent each group of numbers in the brackets with a single rod and see whether the result is also equivalent to 18 5 In another line you have (3 + 1 + 5) + (3 + 1 + 5). Which is it? Represent each brackets by one rod equivalent to that length. What line do you get? Is it already in the table? If you again change the order of the numbers and put (1 + 1) + (3 + 3) + (5 + 5) and then use a single rod for each brackets, which line do you get? 6 In the eight lines of the table which follow the first line, you see that there are two equal rods at the ends and one rod in the middle. If we read each line as
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Arithmestics Makes Arithmetic Child’s Play
two blue two tan and one red two black and one pink two dark green and one dark green (or three dark green) two yellow and one tan two pink and one orange two light green and two dark green two tan and two white we find interesting patterns which can be easily remembered. Shut your eyes and try to repeat the names of the colors used in these and other lines you can remember in the table. Can you write these down using letters as signs for the colors? 7 We can, of course, make many lengths longer than 19 by putting rods end to end. Taking your staircase again, you will find that by putting an orange rod end to end with the orange rod in the staircase you make a length called ‘twenty’ (written 20). Now place two orange rods (or 20) end to end with the white one in the staircase to make twenty-one (or 21); then move these orange rods down the staircase so that they are end to end with the red, then the light green and so on down to the blue rod to make 22, 23, . . . 29. When they are placed end to end with the orange—so that there are three orange rods in the train—you make the length called ‘thirty’ (30). Now place the three orange rods end to end with the white one to make 31, and move them down the staircase to make 32, 33, . . . 39. End to end with the orange—so that there are four orange rods in the train—you make the length forty (40). Five, six, seven, eight,
12
The Story Of Eighteen
nine orange rods end to end make fifty, sixty, seventy, eighty, ninety and are written 50, 60, 70, 80, 90 respectively. Can you form lengths with your rods which are equivalent to 27, 34, 45, 53, 69, 71, 82, 98 Make up your own lengths and write down the numbers they represent. If nine orange rods make 90, how many orange rods would you need in a train to make one hundred (or 100)? 8 Make tables for 21, 25, 27, 36 and find out as you did with the tables for 12 and 18 what you think is interesting about them. Write down what you find. Do it using figures and then letters. 9 What are the answers to 10 + 3 + 7 + 2 =
3+5+4+9=
5+6+7+8+9=
1+2+3+4+5+6+7+8+9=
Think up your own sums and then find the answers. In books you will often find sums written this way—known as the vertical notation. You will find you can do them just as easily.
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Arithmestics Makes Arithmetic Child’s Play
Again make up your own sums and find the answers, this time using the vertical notation. 10 In the tables of 12 and 18, and in some of the other tables you have made, you will have noticed that some of the lines contain rods of one color only; for instance, in the table of 18, you can find the lines blue + blue (or 2 blue rods) and dark green + dark green + dark green (or 3 dark green rods). Can you find other such lines in the pictures? Which? Are there any other lines you can make using rods of one color? You can write down what you see in this way: for 2 blue rods, put 2 × 9 = 18; the sign × is read times. Write down the following: 6 times 3 = 18
9 times 2 = 18
3 times 6 = 18.
Find the lines these expressions correspond to in the picture. 11
Look at the table of 12.
Can you see the lines representing 2 × 6? 4 × 3? Can you find other lines that can be expressed in the same way? Write them down.
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Arithmestics Makes Arithmetic Child’s Play
12 Can you give the answers to these questions, using your rods to work them out if you find it necessary? 2×3= 5×2= 7×2=
4×2= =3×3 4=2× 12 = 6 × 3 × = 12 3×5= 18 = 3 × = 9 × 16 = 2 × = × 4.
Make up your own expressions, then give the answers.
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Arithmestics Makes Arithmetic Child’s Play
Complementary16Numbers Â
Complementary Numbers
1 If you look at the top picture opposite, you will notice that each line of the table of 10 is made up of two rods only. In this way you can find pairs of numbers which together add up to 10. These two numbers are called complements in 10 of each other. Read the table in the picture (or make one for yourself with your rods) using first the color-names and then the number-names of the rods when the white one is the unit. Can you say which number is the complement of 9, of 8, of 7, of 6, of 5, of 4, of 3, of 2, of 1 in 10? In the bottom picture the order of the rods has been changed. Complete each side of the pattern by adding a single rod to each line. Now remove the rods you added and see if you can tell quickly which rod is needed to complete every line. Compare your answers with what you can see in the top picture. Now try to answer the following questions in writing:
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Arithmestics Makes Arithmetic Child’s Play
10 = + 7 5+
10 = 2 +
= 10
+ 9 = 10
2 Reproduce the table at the top of the page using your rods. Add an orange rod at the top to complete the pattern of interlocking staircases. Now move the staircase on the right down so that its white rod is end to end with the tan rod of the staircase on the left. When you have done this you will have found the table of the complements in 9. Can you answer the following p+=B
d+g=
B=+w
9=4+
+8=9
2+7=
Now move the staircase on the right so that its white rod is, in turn, end to end with the black, the dark green, the yellow, the pink, the light green and finally the red rod in the staircase on the left. Find out each time which number you have formed the complements in and write down, using letters or figures, as many relationships as you can. 3 Instead of moving the staircase on the right down, you can move it up so that you obtain the complements in numbers larger than 10. For instance, if you put the orange rod of one staircase end to end with the white rod of the other, you get the complements in 11. When you come to make the table of complements in 12, you will see that the lines representing 11 + 1 and 1 + 11 are missing, since there is no single rod for 11.
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18
However, you can either use an orange and white train for this length or imagine the single rod that would be needed to fill up the gap. By moving up the staircase on the right, you can obtain the complements in numbers up to 19. You would now be able to complete the following: 5 + = 12
7 + = 11
9+4=
14 = + 8
3 + = 17
19 = 12 +
15 + = 18
1 + = 16
11 + = 17
4 There is another way of looking at these tables of complements. You can say: ‘What is the difference between 19 and 12? or between 17 and 6?’ This is the way these differences are expressed in writing: 19 – 12 =
17 – 6 =
Where the sign – is read minus. Can you find the answers to 15 – 13 =
18 – 5 =
19 – 11 =
10 – 4 =
19 – 3 =
16 – 2 =
These expressions are also called subtractions.
19
Complementary Numbers Arithmestics Makes Arithmetic Child’s Play
5 Make a table in your exercise book in which you write down all the subtractions you can think of; now find the answers to them. You can, at first, use your rods to help you, but after a while you should try to find the answer either by placing the rods which represent the terms of the subtraction side by side and guessing the rod that would fill up the gap, or without using any rods at all. 6 In books, you will sometimes find that subtractions are written in this way:
instead of the 19 – 12 = 7 notation you have been using. This is again called vertical notation. Make up some subtractions of your own using the vertical notation, then give the answers. 7 If you have to find the answers to subtractions involving larger numbers, you can make the working out easier by reducing them to smaller and easier ones. For instance, you could be asked to give the answer to 46 – 33 = Make, using your rods, the trains which represent these numbers and arrange them side by side. The answer you want is represented by the difference in length between the two trains.
20
If you remove an orange rod from each, does the difference change? If you remove a second and a third orange rod from each, does the difference change? You can see that when you have removed all these orange rods you are left with two trains which represent the numbers 16 and 3. Having worked through sections 4 and 5 above you will know that 16 – 3 = 13. This subtraction, which you have carried out with your rods, can be expressed in writing in the following way:
The sign ~ (read as ‘equivalent to’) shows that the difference between each pair of numbers is kept at the same value, when you subtract the same number (here 10 each time) from both the terms. Would it be right to use the sign ~ if you added the same number to each of the terms of a subtraction? If you are not sure of the answer to this, read the above expressions from right to left. You can of course do all this in one step by subtracting 30 (that is, taking 3 orange rods from your trains) from the terms of your original subtraction. In writing
21
Complementary Numbers Arithmestics Makes Arithmetic Child’s Play
8 Find the answers to the following subtractions:
Always begin by trying to find the largest number of tens you can subtract so as to give you an easier pair of terms to work with. Try to use your rods only when you cannot do it directly in your exercise book.
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22
The Story Of A Difference
1 Look at the picture on page 26. Find out which rod is needed to make up the difference between each pair of rods. What can you say about these differences? Is the following statement true and does it agree with what you have found? 3 = 4 – 1 = 5 – 2 = 6 – 3 = 7 – 4 = 8 – 5 = 9 – 6 = 10 – 7 = 11 – 8 Can you find other pairs of numbers whose difference is 3? Obviously you do not have to stop at 11 – 8; in fact, you can keep on finding pairs of numbers Examples: 43 – 40
97 – 94
256 – 253
1013 – 1010
Write down some further examples of your own. 2 Make tables of some of the pairs of numbers whose difference is 4, 7, 10 and 16. Is there anything special about the pairs with a difference of 10?
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Arithmestics Makes Arithmetic Child’s Play
3 You will by now be able to see that subtractions form families: we see that there is in each family an infinite number of subtractions which show the same difference. These families are called families of equivalent differences. You had some experience with families of equivalent differences in the previous chapter when you transformed 46 – 33 into 16 – 3. We shall now extend this idea a little further. 4 If you were asked to find the answer to 72 – 47, you could start by subtracting 40 from each term; this you know would not change the value of the difference. But would the new subtraction thus obtained, 32 – 7, be really any easier to solve than the original one? Remember that you are asked to find the answer mentally without using your rods. But, if you added 3 to each of the terms wouldn’t you obtain a new subtraction equivalent to the one you had before? If you are not sure of this, check with your rods: place the trains which correspond to 32 and 7 side by side, measure their difference, then add a light green rod to each of these trains and see whether the value of the difference changes. In writing, this would be expressed as
~
~
From the last expression, the answer is easily seen to be 25.
24
5
Can you give the answers to the following?
When you either add a number to, or subtract it from, both terms of a subtraction, the difference between the terms is not altered; make use of this fact, which you have discovered for yourself above, to form subtractions equivalent to the ones given, stopping when you reach one which to you seems easier than the others.
25
Arithmestics Makes Arithmetic Child’s Play
26
The Story Of A Difference
6 What you have done in section 5 above need not be limited to small numbers. Use the same technique of adding equal amounts to, or subtracting them from, both the terms to obtain easier equivalent expressions for the following subtractions:
Make up some more subtractions of your own and find the answers to them. 7 Which in each of the following pairs of equivalent subtractions do you find the easier to work out?
~
~
The subtractions on the right hand side of each of the above pairs are of a type which can be worked out in a single step; this step can, in fact be taken at once if their terms are read out aloud. If we were able to transform any subtraction we were given into an equivalent expression of this type, then we should have found subtractions that were really the simplest possible. You can learn how to transform subtractions in this way using your knowledge of the complements in 10 and in 9.
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Arithmestics Makes Arithmetic Child’s Play
8 Let us take 10,214 – 7,648 as our example. If we add 2 to each of the numbers, they become 10,216 – 7,650. If we now add 50 to each, they become 10,266 – 7,700. If we then add 300 to each, they become 10,566 – 8,000. Finally, we can add 2,000 to each to obtain 12,566 – 10,000. Our aim has been achieved. Rewriting in vertical notation what we have done, we obtain
~
~
~
~
The numbers added at each step are shown on top of the subtractions. Write down a few subtractions of your own and practice this pattern of transforming the second term into a number of the form 10000 . . . . Repeat this until you are quite sure that you will transform any subtraction into its easiest form. Once you are able to transform subtractions by writing down each step in this way see if you can do it mentally, showing only the first and last of the equivalent expressions, as shown below:
~
28
The Story Of A Difference
9 If you look closely at the second term of the given subtraction and then at the number shown at the top of the last equivalent expression (in our examples above these would be 7,648 and 2,352) you will notice that the last figures on the right-hand side of each number are complements in 10 of one another (in this case 8 and 2), while the remaining pairs are complements in 9 (4 and 5, 6 and 3, 7 and 2). Can you see why this is so? This provides us with a means of speeding up the operation, saving us the trouble of writing down the necessary equivalent expressions and allowing us to do all the work mentally. In this way, all that we need to write is
To do this we would need to think as follows: What do I need to bring 8 up to 10? The answer is 2. Add this 2 to the 4 in the top term; this gives the answer 6, which can be written down. In the remaining columns, complements in 9 are needed. What is the complement of 4 in 9? The answer is 5. Add this 5 to the 1 of the top term; this gives the answer 6, which can be written down. What do I need to bring 6 up to 9? 3; 2 + 3 = 5; put down 5. Finally, what do I need to bring 7 up to 9? 2; 0 + 2 – 2; put down 2.
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29
The Story Of A Difference Arithmestics Makes Arithmetic Child’s Play
If I remember that by doing this I am transforming 7,648 into 10,000 (that is, changing a four-figure number into a five-figure one, of which the first figure on the left is 1), I see that the operation is completed since the only subtraction 1 – 1 gives 0. 10
Give the answers to the following:
Sometimes when a member of the top term is increased by a number needed to bring the corresponding member of the lower term up to 9 (or 10 if it is in the right-hand column), this results in a number greater than 9. For instance, in the last example above 742,604 – 256,277 you say: what do I need to bring 2 (the 2 of 277) up to 9? The answer is 7; 6 + 7 = 13. What you do is to put down the 3 and add the 1 on to the next figure on the left of the top term. In this case, 2 + 1 = 3. Thus, when you say, what do I need to bring 6 (of 6,277) up to 9? 3, you must remember to add 3 to 3, that is, 3 + (2 + 1) in order to obtain the correct figure, 6, to be put down.
30
Table 15
Table 17
32
Re-Reading Additions As Divisions
1 In the tables of 17 and 15 on the opposite page, you can see that we have tried to form each line with rods of one color only, using as many rods as would fit into the length. Was it possible to make up the length exactly in every case or did we sometimes have to add a rod of a different color to complete the pattern? Was it possible at all for 17? Was it possible in some cases for 15? Which cases? 2 Look at the table of 17. You can see that the second line is made up of 8 red rods + 1 white rod, that the third line is made up of 5 green rods + 1 red rod, and so on. Can you write down what you see in the table using both letters and figures, in the following manner: o + b = 8r + w
or
17 = (8 Ă— 2) + 1
o + b =5g + r
or
17 = (5 Ă— 3) + 2
33 Â
Note that the last line of the table can be read in two ways; either as o + b = o + p + g or as o + b =(o + p) + g
17 = 10 + 4 + 3 17 = 14 + 3
Write down in the same way what you see in the table of 15. 3 There is another way of looking at the tables. You could say: ‘How many red rods are needed to make up the length orange + black?’ 8 are needed, leaving a gap which can be filled with a white rod. Similarly you could say: ‘How many light green rods fit into the length of 17?’ The answer would be 5, leaving a difference of a red rod. This ‘difference’ or gap at the end of each line is called the remainder. Re-read each line of the table in this way. When you reach the last line you should say ‘How many times does the length of 14 fit into the length of 17?’ Now read the table of 15 in the same way. Are there any lines for which you do not have a remainder? Which? 4 This way of looking at the tables is usually written down as follows: 7 ÷ 2 = 8 r.1 which is read ‘17 divided by 2 is equivalent to 8, remainder 1’ instead of ‘how many twos fit into 17?’
34
Arithmestics Makes Arithmetic Child’s Play
You can now read each line of the tables as divisions and write them down giving the answer (called the quotient) and the remainder. Example: 15 ÷ 2 = 7 r.l 5 Give the answers to the following short divisions, using the rods if necessary: 13 ÷ 3 =
11 ÷ 2 =
9 ÷ 4 = 15 ÷ 7 =
16 ÷ 3 = 48 ÷ 12 =
21 ÷ 7 =
17 ÷ 8 =
19 ÷ 6 = 22 ÷ 11 = 15 ÷ 6 = 63 ÷ 6 =
23 ÷ 10 = 21 ÷ 12 = 23 ÷ 8 = 20 ÷ 9 = 20 ÷ 10 = 74 ÷ 8 = 6 Try to find the answers to the following short divisions without using the rods: 9÷2=
7÷3=
8 ÷ 5 = 10 ÷ 3 =
13 ÷ 2 = 54 ÷ 15 =
8÷3=
11 ÷ 2 =
13 ÷ 4 = 14 ÷ 9 =
24 ÷ 6 = 47 ÷ 12 =
13 ÷ 10 = 11 ÷ 9 =
18 ÷ 7 = 12 ÷ 5 =
36 ÷ 5 = 82 ÷ 20 =
7 In books you will sometimes find the vertical notation used for divisions. For example 9 ÷ 2 = 4 r.l would be written
Re-write some of the divisions given above, or make up your own, using the vertical notation.
35
Re-Reading Additions As Divisions
8 Numbers which always show a remainder when divided by any other number (except 1) are called prime; the others are called composite. Thus 17 is an example of a prime number and 15 of a composite number. Can you say which of the following numbers are prime and which are composite? 12, 7, 16, 3, 9, 19, 25, 11. Make a table of all the numbers you know up to 100 and find those that are prime. 9 Since we have found that repeated addition (that is seeing how many times a certain number fits into another one) can be re-read as division, we can think of division as being repeated subtractions. For instance, if we wanted to divide 15 by 5, we can see that it could be done by subtracting 5 three times from 15: 15 – 5 = 10 10 – 5 = 5
5–5=0
No remainder
9–4=5
5 – 4 = 1 Remainder 1.
Similarly, for 17 ÷ 4: 17 – 4 = 13 13 – 4 = 9
Re-read several of the lines of the tables of 17 and 15 as repeated subtractions, then make up some of your own; write down the number of subtractions you have carried out in each case and the remainder, if any. Note that this remainder must always be smaller than the number you are subtracting—otherwise you could have carried out a further subtraction still.
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Arithmestics Makes Arithmetic Child’s Play
10 When numbers of three figures or more are present in divisions, we talk of long divisions. The following are examples of long division:
These operations can be carried out using the process of repeated subtractions studied above. For the first example you could, therefore, subtract 147 from 2,949 over and over again to see how many times it would go into 2,949.
. . . and so on. This, however, would be a very long and boring operation. To make it easier and quicker, you can look for a bigger number to subtract from 2,949. You could, for instance, multiply 147 by 10. 147 × 10 = 1,470. Subtracting this number from 2,949 you would get
37
Re-Reading Additions As Divisions
You can see that by subtracting 1,470 again you obtain 9 as the remainder and your working out of the problem would be complete. The operation, set out in full, would look like this:
Or: 2,949 ÷ 147 = 20 r. 9 You could, of course, have carried the operation out in a single step if you had first doubled 147, then multiplied the result by 10. 147 × 2 = 294. 294 × 10 = 2,940.
Can you find the answers to the following operations using the process of repeated subtraction?
6,327 ÷ 31
18,436 ÷ 253
3,942 ÷ 15
5,748,629 ÷ 417 38
Arithmestics Makes Arithmetic Child’s Play
Always try to obtain as big a number as you can before carrying out any subtractions as this will reduce the number of steps.
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39
40
Products And Factors
1 Looking at the pictures on the opposite page, you will see that except for the top line of the patterns, each is made up of rods of one particular color. For instance, in the pattern for the dark green rod there is a line made up of two light green rods and a line of three red rods. The numbers that these lengths represent are said to be composite, and a study of patterns such as these will lead us to some new discoveries. You already know (p. 11) how to express in writing what these patterns tell you. For instance you can put down 6 = 2 × 3 and 6 = 3 × 2 which are read as ‘six equals two times three’ or as ‘six equals two multiplied by three’, and ‘six equals three times two’ or ‘six equals three multiplied by two’. Expressions in which the sign × appears are called products and the operation multiplication. 2 The numbers which are multiplied together to form another are called factors of that number. Thus 2 and 3 are factors of 6. So are also 1 and 6.
41
Arithmestics Makes Arithmetic Child’s Play
What are the factors of 12, of 4, of 9, of 16, of 10? Make a list of some of the composite numbers you know and find their factors. 3 We shall now find out how products can be represented using the rods. Make the pattern of 12 as you see it in the picture. Take the pink rods and rearrange them side by side instead of end to end. Rearrange the light green rods in the same way. See the drawing below.
What can you say about the rectangles you have just formed? Place one rectangle on top of the other. Does it exactly fit on top of the other one? Could you say that they were equal? You can see that the length of one of the sides of the rectangles is equivalent to the length of a light green rod, and the other to the length of a pink rod. We can therefore represent the product as a cross, as shown in the drawing on the right.
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42
Products And Factors
4 If the light green—pink cross represents the product 3 × 4, what would a pink—light green cross represent? Are the two products equivalent? If you now placed the dark green rods in the pattern of 12 side by side to form one rectangle, and then the red rods to form another, do you think these two rectangles would be equal? Which rods would represent their sides? Could you form crosses with these rods? The crosses you should now have formed are expressed in the following statement: 12 = 3 × 4 = 4 × 3 = 2 × 6 = 6 × 2 Can you now form crosses to represent the products for 4, for 6, for 9, for 10 and for 16? 5 You could, on the other hand, make a cross with any two rods of your choice, and then try to find the number corresponding to this product. Let us suppose you make a
43
Arithmestics Makes Arithmetic Child’s Play
yellow—pink cross. You can obtain a rectangle if you place other yellow rods next to your first yellow rod until you have as many as you need to fit the length of the pink rod. In this case your rectangle would contain 4 yellow rods. These yellow rods, when placed end to end, form the train which represents 20, as you can see by checking with orange rods. Therefore, your pink—yellow cross stands for the product 4 × 5 which is equal to 20. Make some other crosses with your rods and find the products they stand for. 6 The pictures at the foot of the page opposite page 25 show another way in which products can be represented. The circles, colored to match the colors of the rods, represent the rods which form crosses. The first picture on the left shows 2 red circles, and stands for the red—red cross and the product 2 × 2. Which products do the second and the third pictures represent? The last two pictures show two pairs of colored circles, because the numbers they represent have 4 factors. The circles which are opposite each other are paired together to form the crosses. What numbers do these pictures represent and what are their factors? The Product Cards are a pack of 37 cards each showing one such group of colored circles. These can be used to play a number of card games, which will increase your knowledge of factors and products. 7 Find the answers to the following multiplications, using your rods if you find it necessary:
44
Products And Factors
5×5=
7×3=
4×8=
3×9=
4×7=
6×6=
8×4=
7×6=
8×5=
7×8=
5×7=
9×6=
8×3=
7×7=
9×9=
4 × 9 = 10 × 10 =
7×9=
5×6=
9×8=
Give the factors of 15, 16, 18, 20, 21, 22, 24, 25, 28, 36, 42, 49, 50, 56, 63, 64, 72, 77, 81, 84. Use your rods whenever you are not sure of the answer. 8 Multiplying by 2 is called doubling. Start with 1 and double it; double it again, and again, and again. Go on doubling as far as you can. Try to do it without using the rods. Dividing by 2 is called halving. Start with the biggest number you have obtained by doubling and halve it again and again. Do you eventually get back to 1? 9 In the same way, start with 3, with 5 and with 7 and go on doubling as you did with 2 above. When you have gone as far as you can, reverse the operation and keep on halving until you get back to the number you started with. Find the answers to the following:
32 × 2 =
14 × 2 =
48 × 2 =
8×2=
40 × 2 =
160 × 2 =
6×2=
64 × 2 =
56 × 2 =
45
Arithmestics Makes Arithmetic Child’s Play
This is how we write down what we do when we halve: of 8 = 4 which is read one half of 8 equals 4. Find the answers to
10
of 32 =
of 20 =
of 96 =
of 4 =
of 56 =
of 80 =
of 24 =
of 48 =
of 28 =
Can you tell which is bigger? 2 × 20 or
of 96
of 112 or 2 × 24 2 × 16 or of 96 or
of 56 of 80
46
Products And Factors
Listed below are the materials mentioned in the previous pages, followed by a list of some of the books associated with this approach to the teaching of mathematics.
Algebricks Obtainable as a set of 292 colored rods with separate compartments for each color in a strong plastic box. Suitable for 1-4 students.
Product Cards And Product Charts The cards and the charts (one set can equip 3-9 students) can be used for a number of games, as indicated on page 26.
Publications Gattegno Mathematics Text-Books 1-7 1
Qualitative Arithmetic The Study of Numbers from 1 to 20
2
3
Applied Arithmetic
4 Fractions, Decimals, Percentages
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47
Study of Numbers up to 1000 The Four Operations
Products And Factors Arithmestics Makes Arithmetic Child’s Play
5
Study of Numbers
7
Algebra and Geometry
6 Applied Mathematics
Workbooks 1-6 Containing over 1400 problems for first year students 1
Numbers up to 5
2
Numbers up to 15
3
Numbers up to 20
4 Numbers up to 100
5
Exercises
6 More Exercises
For Teachers And Parents Now Johnny Can Do Arithmetic—C. Gattegno A valuable source book providing an outline for the teaching of Mathematics and notes on the experiences and insights that formed the basis for Dr. Gattegno’s approach to the subject by way of colored rods.
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