The Parents' Review
A Monthly Magazine of Home-Training and Culture
"Education is an atmosphere, a discipline, a life."
Algebra Through Geometry.
by P. G. O'Connell.
We have all met the individual, usually of limited intellect, who delights in asking, in ordinary society, such questions as, "If a lady's age be twenty years less than twice her age, what is her age?" That he is not only usually tolerated, but also often encouraged, is a tribute to the excellence of present day education. I have met him so often that I have been interested in studying the answers he receives. I have found, in my experience of him, that women usually answer him correctly, if they take the trouble to answer him at all, while men answer willingly enough, but incorrectly. I attribute this, not to any intellectual superiority on the part of women, but to the fact that the women he addresses have usually not been "taught" algebra, while the men have. It is a terrible handicap. It means that men have been taught as children to solve such "problems" by rule. The implication is that the lady's age is twenty, not because it necessarily must be twenty, but because the rule says that, in any such equation as x= 2x = 20, any term may be moved, with its sign changed, from one side of the equation to the other.
Is it to be wondered at that a child thus taught to believe that truth is truth, not by Divine right, but because it follows some rule enunciated by man, should gradually lose the power of correct thinking which it brought into the world with it as its birthright? The wonder is that the born mathematician survives the treatment, perceives the truth of the rule by rapidly thinking through a sufficient number of particular instances, and thereby unfortunately leads his teacher to believe that any boy could do it if he liked.
The aim of the teacher should be to make it interesting to the average child to do consciously what the born mathematician does by instinct, to develop the bump of mathematics, or mathematical faculty, which consists principally in a habit of testing the truth of any suggested generalization by thinking through a sufficient number of particular instances.
It is only in the lower branches that teaching by rule still lingers. Though writing for fairly advanced students, who may be expected to take an interest in mathematics for its own sake, the author of any treatise on the differential is careful to introduce a geometrical illustration as early as possible, thereby rendering his subject more interesting, because more obviously useful, and easier, because less abstract. But the writer of a work on algebra seems to have no such desire, though it ought to be doubly strong in his case, since he is writing for those whom it would be hardly safe to assume even a liking for arithmetic. After giving a number of those dry definitions with which knowledge is almost invariably hedged round, as if the object were to keep intruders out, he asks, symbolically, for some such interesting information as the value of four times X when X is five. To the unprejudiced beginner this must seem merely a cumbersome method of asking for the value of four times five. Thus the mighty symbol X is introduced rather as a producer of trouble and confusion than in its true character of a labour-saver.
If we take a hint from the writers on the differential, and commence with geometrical illustrations, we shall not only render algebra easier and more interesting, but also introduce the student to analytical geometry. In this diagram:--
5 * * * * *
4 * * * * *
3 * * * p *
2 * q * * *
1 * * * r *
* 1 2 3 4 5
it will be noticed that the fourth letter in the row marked 3 is a "p." The position of this "p" may be fully indicated by the two figures--(4,3)--provided we understand that the first figure denotes the number in the row and the second figure the number of the row. Similarly, the positions of the letters "q" and "r" may be fully indicated by the figures--(2, 2) and (4, 1). This method of indicating the position of a point by two numbers is used in geography, and therefore may be familiar, but the youthful mind should be allowed plenty of time to assimilate the idea. To assist the process, let us make a plan of the kitchen garden at the back of the house, marking the position of various objects. We first go out and measure the distance of each object from the wall on the left, and also its distance from the house. For the sake of brevity, let us call the "distance from the left hand wall" of any point the "x" of that point, and the "distance from the house" of any point on the "y" of that point. There is a wall at the end of the garden, running parallel to the house, at a distance of fifty-one yards from it. The y of every point at the foot of this wall is fifty-one. On our plan, the wall will naturally be represented by a straight line parallel to the line which represents the house, and at a distance of fifty-one somethings from it. If we represent every yard as one-tenth of an inch, the y of every point on the line will be 5'1 inches. Now we suppose that, instead of calling this line AB or CD, we call it the line "y = 5'1." This name is longer than AB or CD, but it has the advantage of conveying more information; it tells us the precise position of the line upon our plan.
The idea that a line can be represented by an equation is so strange and new to the youthful mind that some weeks should be allowed for it to sink in. During these weeks any number of gardens may be surveyed, any number of lines such as "x = 10" drawn, and the idea gained that any point (a, b) is the point of intersection of the lines "x = a" and "y = b."
Let us now consider the three points (2, 1), (4, 2), (6, 3). We notice that the x of each point is double of its y. We also notice that the three points lie on the same straight line. If we draw this straight line we shall find that the x of any point on it is double of the y of that point, and that any point we may mention, whose x is double of its y, lies on this straight line, Thus the equation x = 2y represents a straight line.
The geometrical truths here exemplified are necessary facts, and most pupils will recognize them to be true after careful contemplation of a number of particular instances: but, if an explanation should be required, I suggest the following:--Draw, or imagine to be drawn, all the lines x = 2, x = 4, x = 6, etc., up to x = 50; also all the lines y = 1, y = 2, y = 3, etc., up to y = 25; join the points (2, 1), (4, 2), (6, 3), etc., up to (50, 25). In doing this, we are not assuming that these points lie on the same straight line; we are merely drawing the diagonals of twenty-four oblongs, and thus forming a number of triangles, which are obviously equal in all respects, so that every diagonal makes the same angle with some line parallel to the house. When this has been realized, it is easy for the mind to realize that all these diagonals must necessarily form parts of the same straight line. I have found this method of "explanation" very efficacious in the case of so-called dull boys; to show that it applies to every point on the line, it is only necessary to point out that the quantity we call "1" may be as small as we please.
Let us now consider the line which represents the house. No point on it is at any distance from it, so we may call it the line "y=0." In the same way, the line which represents the wall on the left may be called "x=0."
Let us now find a few points of which it is true that y=3x - 2. This is merely "substitution" in a new dress. Some children will find points unaided; others will need some such hint as--"If x be 1, what will y be?" If a child cannot answer this question, it is not because he cannot make the calculation. It may be that he does not understand that 3x means three times x, or that the symbol for minus means that the 2, which follows it, is to be subtracted. Possibly he is tongue-tied because he suspects a trap. It is absurd to suppose that he does not know that three times one are three, and that two from three leaves one, though he may have some difficulty in connecting this knowledge with the symbolical expression 3x - 2.
But pupils will seldom have any difficulty in finding several points of which it is true that y=3x - 2. They will then notice that all these points lie on a straight line; and, when they have considered a sufficient number of particular instances of equations of the form y=mx-0, they will be in a position to make the generalization that any such equation must represent a straight line.
If we draw the line y=3x-2 we find that it cuts the line x=0 at a point below the line y=0. From the equation y=3x-2 we find that, when x=0, y=-2. This shows us how to interpret, geometrically, negative values of x and y. Pupils may now be asked to draw such lines as y=-2 or x=-3, provided they realize that the meaning we attach to negative values is not an arbitrary one, but one that suggests itself naturally. If they do not realize this easily, let them find the points of intersection of several pairs of lines, such x=1, and y=x-2, or y=1 and x=y-3, the point of intersection having x or y negative in each case. To have gained a conception of negative distances will be a great advantage to the pupil later on.
In finding the equation to a straight line drawn at random on the plane, pupils will almost certainly require a little help; I will therefore give one method. Draw any straight line A B at random. Measure the x of the point in which A B cuts the line y=0. Let us call this point C, and suppose that its x is 1?1. From that part of the line y = 0,which lies to the right of C, cut off C D, making C D = 1. Through D draw the line x = 2?1, cutting A B at P. Measure the y of P, remembering that it may be negative. Suppose it is 2?2 then D P is obviously 2?2 times C D. But D P is the y of P and C D is its x --- 1?1. Therefore it is true of the point P that y = 2?2 times (x --- 1?1), which is usually written y=2?2 (x --- 1?1). A little consideration will show that this is true of any point on the line, and therefore is the equation to the line, that is the equation name of the line. This will require consideration, but it is not a new geometrical truth.
I have now introduced the bracket ( ) for the first time, and I recommend strongly that no new symbol should ever be introduced before it is necessary to use it in practical work. A new labour-saving symbol is like a new machine, extremely interesting when you realize its usefulness, merely strange when you see it lying idle.
To show that 2?2 times (x --- 1?1) = 2?2x---2?42 it will be best to take simpler cases first. This diagram:--
* * * * * * * * *
* * * * * * * * *
shows that two nines are equivalent to two tens all but two, that is, that 2 (10 --- 1) = 20 --- 2. Many similar diagrams should be drawn before a child is asked to realize that a(x---b)=ax---ab; for the average child cannot generalize without first considering a large number of particular instances.
Since 2?2 times 1?1 is 2?42 the equation to A B may be written: y=2?2x---2?42. It is not absolutely necessary to perform the multiplication, for we can obtain the result by measuring the y of the point in which A B cuts the line x=0, since this is the value of y when x=0; but, unless our drawing has been very accurate indeed, we shall find that the measured distance is at least ?02 too long or too short. It is therefore worth while to perform the multiplication in order to test the accuracy of our drawing. This desire to test the accuracy of one's drawing is particularly strong in youth, and I think it foolish not to utilize it in teaching arithmetic. If you tell a boy to multiply 5?73 by 4?29 and divide the result by 2?84, you merely set him an uncongenial task; but if it be necessary to perform the operation in order to test the accuracy of a drawing, the boy will beg to be allowed to do it.
The folly of attempting to teach the particular through the general is often strikingly illustrated in the case of the rule for the position of the decimal dot in products. Pupils who have been taught this rule, when multiplying together two such numbers as 91?11,025 and 1?984, will often get an answer less than 90, or greater than 200, though a glance at the figures shows that the product must lie between these limits. It is a habit of glancing at the figures that we ought to encourage.
In finding, first graphically and then analytically, the point of intersection of y=-3x and x=-2, pupils will notice that the product of -2 and -3 is positive. Other examples will lead to the generalization that the product of two negative quantities is always positive.
It seems so obvious to us that, if y-x=3x, y must be equal to 4x, that we are apt to forget that the statement is a generalization, which the average child can only realize by thinking through a sufficient number of particular instances. If we ask him to find points of which it is true that y-x=3x, he will easily see that, when x=1, y=4, when x=2, y=8, etc., and he will soon notice that y is necessarily equal to 4x. In the same way, he must be led to notice that, if y-5x + 2=3x, y must be equal to 8x-2. A large number of such examples will lead him to generalize still further.
In finding the points of intersection of such lines with the line y=0, and with each other, pupils will get practice in solving simple equations, mentally if possible, but always with the definite object of testing the accuracy of their drawing.
We may introduce multiplication by asking pupils to find points on such a line as y=(x-3) (x-4)-x2, or y=(2x-3) (3x-2)-6x2. Facility in mentally finding the product of any two binomial expressions may be gained by practice in mentally finding the product of two such numbers is 23 and 34, considered as twenty thirties, three thirties, four twenties and four threes. This facility having been gained, longer multiplication sums will present no difficulty. Pupils should test their results by seeing whether the original equation is true of a few points found from the simplified equation.
It is easy to see that division may be similarly introduced. Mental division should be encouraged, the use of pencil and paper being treated as merely as an aid to the memory.
Pupils may now test the truth of Euclid I. 47, by finding, graphically, the x and y of points on the circle whose centre is (0,0) and radius 5. They will not be able to realize that, at every point of the circle, x2 + y2 is necessarily equal to 25, until they have studied some proof. As an easy proof to understand, I recommend the following, which assumes only a knowledge of the meaning of the words--square, triangle, and right angle. Draw any right angle ACB, and, on AB, on the same sideas C, draw the square ABEL. Draw ED at right angles to BC, produced if necessary. Cut out the square ABEL. Cut out the triangle ACB, and turn it round A, till B falls on L. Cut out the triangle BDE, and turn it round E, till B falls on L. The new figure, thus formed, is obviously made up of two squares, the squares on AC and ED, that is, on AC and BC.
If pupils be now asked to find, graphically, the points in which the circle x2 + y2=25 is cut by the line y=3, they will notice that there are two points. They will therefore expect to get two values of x, when they substitute 3 for y in the equation, and, when they get as far as x2=16, they will not have the usual difficulty in seeing that x may be equal to -4. In finding where the same circle is cut by the lines y=x, y=2x, y=3x, they will have to solve slightly harder equations. They may now be introduced to such an equation as (x-1)2 + (y-2)2=16, if necessary, led to notice that it is only true of points at a distance 4 from (1, 2), and therefore is the equation to a circle. In finding the points in which such a circle is cut by lines as y=3, they will learn to solve slightly harder equations. They may now be asked whether (y-2)2 + x2-2x=15 represents a circle, or not. If they have sufficiently practised mental multiplication, they will easily see that it does, being readily able to notice, mentally, and not by rule, that x3-2x may be written (x-1)2-1.
By asking where the circle x2-8x + y1=9 is cut by the lines y=x, y=2x, y=x-4, etc., we may introduce equations of gradually increasing difficulty, all of which may be first solved graphically.
Geometry and Algebra are sister sciences, and it is impossible to understand much of either without the assistance of the other; is it not therefore naturally wise to study them together from the first?
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