Notes of Lessons: Algebra, Class III

[We have thought that it might be of use to our readers (in their own families) to publish from month to month during the current year, Notes of Lessons prepared by students of the House of Education for the pupils of the Practising School. We should like to say, however, that such a Lesson is never given as a tour de force, but is always an illustration or an expansion of some part of the children’s regular studies (in the Parents’ Review School), some passage in one or other of their school books.—Ed.]

Group: Mathematics • Class III • Average age: 12 • Time: 30 minutes

By H. M. A. Bell
The Parents’ Review, 1905, pp. 385-387

Objects

I. To introduce a new branch of mathematics, touching on the two first simple rules.

II. To increase the power of attention and reasoning.

III. To encourage accuracy.

IV. To stimulate interest in a new subject.

Lesson

Step I.—Tell the children about the introduction of algebra: Arabs derived it from the Hindus, and it was from Arabs that Europeans first obtained their acquaintance with it. The first books on algebra were written in the fourth century. Algebra derived its name through the Italian and Spanish from the Arabic Al-jebr = the resetting of anything broken, hence combination, i.e., the combination of numbers and quantities. Algebra, the science or knowledge of numbers, of later growth than arithmetic, was at first merely a kind of universal arithmetic, symbols taking the place of numbers. It is now a distinct branch of mathematics.

Step II.—Ask the children for the different signs used in arithmetic and for their respective values, as:—

Equals, = stands for “is equal to” or “are equal to”; example, 3 + 2 = 5.

Plus, + put before a number means that what that symbol represents has to be added; as, 4 + 5 = 9. (Ask for examples of symbols with + between.)

Minus, – put before a number means that what that symbol represents has to be subtracted; as, 5 – 2 = 3. (Ask for examples of symbols with – between.)

When a symbol has neither + nor – written before it, + is always understood.

Step III.—Shew the difference between positive and negative signs, and how they are used, the positive before a positive number or one to be added, the negative before a negative number or one to be subtracted. All numbers are either positive or negative. (Ask for examples of each kind.) Shew from examples how, in considering negative numbers, we overstep the boundary of arithmetic and enter on algebra. Thus in arithmetic you cannot subtract 7 from 4 to give a sensible answer, but in algebra you can have negative answers.

Step IV.—Let the children work the following examples:—

1. A man, starting from a sign-post, walks on for 7 steps (positive) and then goes back 10 steps (negative) to pick up something. How far would he be from the post?

2. A boy gained 16 marks and lost 18. How many did he gain on the whole?

3. A owes B £6, and B owes A £8. How much does A owe B on the whole?

4. A cart was driven 15 miles along a road running south, the driver turned the horses round and drove 20 miles back. How far south was it then?

5. A boy had gone already 20 steps towards his school when he found that he had forgotten to buy a book at a shop which was 26 steps in the opposite direction. When he was at the shop how much nearer school was he than when he started?

Step V.—Ask the children if they know how algebra differs from arithmetic, i.e., that in algebra we use letters as well as numbers, and any letter may stand for any number. Thus a may = 1, 2, 3, 24, etc., and any other letter may have the same value. Give the following examples to be worked out:—

1. If a = 3, b = 6, and c = 2, find the value of:—

(1) a + 4

(2) b – 3

(3) c – 5

(4) a + b – c

2. If x= 6

(1) What is half x?

(2) What is ¹⁄₃ of x?

3. If x = 12

(1) What is twice x?

(2) What is six times x?

4. I have £x, you have £y, and someone else has £z. How many have we altogether?

5. How old are you now? How old will you be in x years?

6. If you are 15 years old now, how old were you v years ago?

7. Add together p, q, x, a, b.

8. Subtract a and b from x.