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CHAPTER X.

ARITHMETIC (STANDARD 11.). In this standard Simple Multiplication and Short Division are introduced.

Copious exercises in Mental Arithmetic should precede and accompany the mechanical work in these rules.

In working such exercises as 96549 842, instead of multiplying by 4, as is usually done, the 2's line should be doubled. In the same way, instead of multiplying by 8, the 4's line should be doubled. Similarly when 3, 6, and 9 are used as multipliers, the 6's line should be double of the 3's line, and the 9's line treble the 3's line. All “short cuts” of this kind increase the inventiveness and intelligence of the class, and check the weariness due to mere mechanical work.

The use of the cipher in multiplying may be explained by reference to the 10 times multiplication table, in which it will be pointed out that 10 x1 = 10, 10 x 2 = 20, etc., so that the effect of multiplying 1, 2, etc., by 10 is to add 0 to the 1, 2, etc. The rule is thus deduced, that in multiplying by any number, say 456 by 10, we merely add 0 to the multiplicand.

In the same way, 1 x 100 100; 2 x 100 = 200, etc.; and to multiply any number by 100, we merely add 00 to the right of the multiplicand. Similar remarks will apply

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to 1000; and to the cipher occurring between integers in the multiplier.

In multiplying by 25, which is a fourth of 100, we add 00 to the multiplicand, and divide by 4; and in multiplying by 125, add 000 and divide by 8.

Besides the ordinary way of multiplying, say 8346 by 543, work the sum thus :

8346 X 500 = 4,173,000
8346 X

333,840
8346 X

3 = 25,038

40 =

8346 X 543 = 4,531,878 Every figure that can be saved in arithmetic should be so; thus, in 836 x 648, we have 836 x 8 = 6688; now 64 = 8 x 8 :: 836 x 64 = 8 times the preceding (to be put one place to the left); this gives

6,688 535,04

541,728 In Division, fractions should be thoroughly explained up to twelfths. Thus, 64 • 2 = of 64 = 32, etc. These should be illustrated by reference to the aliquot parts of ls., 28., 2s. 6d., 58., 108., and £l; as well as of 1 foot, 1 yard, and 1 lb.; and by reference to a line, rectangle, and square, divided into aliquot parts on the blackboard.

MODEL LESSON IN MULTIPLICATION BY ONE FIGURE.

First Stage (without carrying).

234

2

468

Point out that
Twice 4 units = 8 units

8
3 tens = 6 tens

60 2 hundreds = 4 hundreds 400

468 Second Stage (with carrying).

234

3

702 Point out that

3 times 4 units = 12 units = 1 ten, 2 units.
3 3 tens 9 tens and 1 ten 10 tens, or 100.
3 2 hundreds are 6 hundreds and 1 hundred =

7 hundreds.
Third Stage, with two figures (without carrying).

234 21

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234 468

4914 Point out that,

2 tens multiplied by 4 units = 8 tens.
2

3 tens = 6 hundreds.
2

2 hundreds 4 thousands. At first, the blank spaces on the right may be filled in by ciphers (O's).

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MODEL LESSON IN SIMPLE AND SHORT DIVISION.

First Stage (without carrying).

2) 648

324
Proof-

2
648 Like top line.

Here point out that 648 = 600 + 40 + 8.

600 = 2 = 300
40 : 2 20
8 • 2 = 4

648 ; 2 = 324 Second Stage (with carrying).

2)748

374
Here point out that 748 = 700 + 40 + 8.

700 • 2 = 300 and 100 left over.
140 = 2 = 70

4

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848 • 2 = 374 Third Stage (with final remainder).

2) 749

374 + 1 = 374 + 1 And so on with harder sums.

The following typical exercises in Mental Arithmetic for Standard II. will indicate to the teacher the way in which others

may

be invented :

EXERCISES IN MENTAL ARITHMETIC (STANDARD II.).

Notation and Numeration.
I. Express in words-
6,017
60,001

16,007
10,001
11,001

11,111
11,001
10,111

10,010 II. Name the figures which express

Eleven thousand one hundred and one.
A hundred and ten thousand.
A hundred thousand.
Ten thousand one hundred and ten.

40

Multiplication.
III. Give the squares of—
7 -
8
9

10
11
12
13

14
15
16
17

18
19
20
30

40
IV. Give the factors of-
81
108
110

120
121
132
144

169
V. Name at sight the products of-
6,170 x 10

90,017 X 20 61,701 x

67,019 X 50 6,179 X 60

11,110 x 90 7,014 x 80

61,794 X 6,178 x 100

1,679 X 1000 617 x 200

914 x 300 9,081 x 5000

1,794 x 6000 VI. If I pick up 6079 stones a day, how many can I pick up in a week ?

How many shillings in £456 168.?
How many pence in 1958. lld. ?
How many farthings in 4s. 5d.?
How many

inches in 19 feet 11 inches ? How many inches in 79 yards long? What is the rent of 199 acres at £12 per

acre ?
Find the price of 9 horses at £187 each.
A ship has 3791 tons; how much in 25 ships?
If 1 horse can draw 7967 lbs., what can 300 draw ?

70

Short Division,

VII. Divide 56,784 marbles among 12 boys.
In 473,684 shillings how many pounds ?
In 767,473 lbs. how many parcels of 7 lbs. each ?

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