# BASIC OPERATIONS ON DIRECTED NUMBERS

**FIRST TERM **

**LEARNING NOTES**

**CLASS: JSS 2 (BASIC 8)**

**SCHEME OF WORK WITH LESSON NOTES **

**Subject: **

### MATHEMATICS

**Term:**

**FIRST TERM **

**Week:**

**WEEK 8**

**Class:**

**JSS 2 (BASIC 8)**

**Previous lesson: **

The pupils have previous knowledge of

that was taught as a topic during the last lesson.

**Topic :**

**BASIC OPERATIONS ON DIRECTED NUMBERS**

**Behavioural objectives:**

At the end of the lesson, the pupils should be able to

- solve mental sums on addition and subtraction of directed numbers (Revision)
- Work out solutions to simple sums on multiplication and division of directed numbers.
- Understand the simple concept of Inverse and identity

**Instructional Materials:**

- Wall charts
- Pictures
- Related Online Video
- Flash Cards

**Methods of Teaching:**

- Class Discussion
- Group Discussion
- Asking Questions
- Explanation
- Role Modelling
- Role Delegation

**Reference Materials:**

- Scheme of Work
- Online Information
- Textbooks
- Workbooks
- 9 Year Basic Education Curriculum
- Workbooks

**Content**

**BASIC OPERATIONS ON DIRECTED NUMBERS**

**TOPICS**

- Addition and subtraction of directed numbers (Revision)
- Multiplication and division of directed numbers.
- Inverse and identity.

** **

**ADDITION AND SUBTRACTION OF DIRECTED NUMBER LINE (REVISION)**

**Addition:** To add using the number line, we move to the right (forward) of the number line counting each step till the addition is complete. The number at the end is the result of the addition.

**Subtraction:** To subtract using the number line, we move to the left (backwards) of the number line counting each step till the subtraction is complete. The number at the end is the result of the subtraction.

- Try out the exercises on this worksheet using the first representation as an example.
- Use the number lines to solve each problem.

**MULTIPLICATION OF DIRECTED NUMBERS**

Multiplication is a short way of writing repeated additions. For example,

3 × 4 = 3 lots of 4

= 4 + 4 + 4

= 12

With directed numbers,

(+4) + (+4) + (+4) = 3 lots of (+4)

= 3 × (+4)

The multiplier is 3. It is positive. Thus,

(+3) × (+4) = (+4) + (+4) + (+4) = +12

The illustration above shows 1 x (+4) and (+3) × (+4) as movement on the number line. The movements are in the same direction from 0.

Similarly,

(-2) + (-2) + (-2) + (-2) + (-2)

= 5 lots of (-2)

= 5 × (-2)

The multiplier is 5. It is positive.

Thus, (+5) × (-2)

= (-2)+ (-2) + (-2) + (-2) + (-2)

= -10

This is illustrated below:

In general, (+a) × (+b) = + (a × b)

(+a) × (-b) = – (a × b)

(+a) ÷ (-b) = – (a ÷ b)

(-a) ÷ (-b) = + (a ÷ b)

**Example 1:**

- (+9) × (+4) = + (9 × 4) = +36
- (+17) × (-3) = – (17 × 3) = – 51
- (+ 12) × (+14) = + (12×14) = +18
- 3 × (-1.2) = -(3 × 1.2) = -3.6

NOTE: *Teachers should give more illustrations especially on (negative multiplier)*

(i) Negative multiplier

In general: (-a) × (-b) = -(a × b)

(-a) × (-b) = + (a × b)

**Example 2:**

Simplify the following:

- (-7) × (+4) = -(7 × 4) = -28
- (-5) × (-18)= + (5 × 18) = 90
- (-13) × (+25) = – (13 × 25) = – 215
- (-4) × (-2.2) = + (4 × 2.2) = + 8.8

**CLASS ACTIVITY**

Simplify the following:

(i) (-8) × (-12) × (-30)

(ii) (-9) × (-5) × (-3)

**DIVISION WITH DIRECTED NUMBER**

The rule involved in multiplying directed numbers is equally applicable for division. In division, note that:

(i) Two similar signs give a positive result.

(ii) Two unlike signs give a negative result.

**EXAMPLE 1**

Divide: (a) -63 by 7

(b) 125 by 5

(c) -6 by -18

Solution:

(a) -63 ÷ 7 = −63+7 = -9

(b) 125 ÷ 5 = +125+5 = +25

(c) -6 ÷ -18 = −6−18 = +13

**Example 2: **

Simplify (−6)×(−5)−10

Solution:

(−6)×(−5)−10=+30+10=−(3010)=−3

**CLASS ACTIVITY**

- Divide these expressions:

(a) -3 by 24

(b) 4 by -32

(c) 28 by 7

(d) -20 by -4

- Simplify 4×(−3)−2×−6

**INVERSE AND IDENTITY**

**ADDITIVE INVERSE**

Definition of Additive Inverse

The additive inverse of a number is its mirror image on the number line. The additive inverse of -5 is 5 and the additive inverse of 5 is -5.

If a number is positive, its additive inverse is negative, and if a number is negative, its additive inverse is negative.

Additive inverse can also be defined as what you add to a number to get 0, i.e. 5 + (-5) = 0.

Additive identity is the value obtained when a number is added to its inverse. Thus additive identity is 0.

**MULTIPLICATIVE INVERSE**

The multiplicative inverse is what you multiply a number by to get 1. So, a number’s multiplicative inverse is 1 divided by the number.

The multiplicative inverse of 2 is 1 ÷ 2 = 12

The multiplicative inverse of 7 is 1 ÷ 7 = 17

To check, 2⋅12=2⋅12=22=1 and

7⋅17=7⋅17=77=1.

Multiplicative identity is 1.

**CLASS ACTIVITY**

- State the additive identity of:

(a) +14; (b) –23 (c) – ½

- State the multiplicative inverse of:

(a) +5; (b) – ¼ (c) +2/7

**ASSIGNMENT**

- Copy and complete the multiplication table in the figure below.

1^{st}no. |
||||||||

× | -6 | -4 | -2 | +2 | +4 | +6 | ||

+6 | ||||||||

+4 | ||||||||

2^{nd} no. |
+2 | |||||||

0 | ||||||||

-2 | ||||||||

-4 | ||||||||

-6 | ||||||||

- Copy and complete the division table in the figure below.

1^{st}no. |
||||||||

÷ | -6 | -4 | -2 | +2 | +4 | +6 | ||

+6 | ||||||||

+4 | ||||||||

2^{nd} no. |
+2 | |||||||

0 | ||||||||

-2 | ||||||||

-4 | ||||||||

-6 |

**Presentation**

The topic is presented step by step

Step 1:

The class teacher revises the previous topics

Step 2.

He introduces the new topic

Step 3:

The class teacher allows the pupils to give their own examples and he corrects them when the needs arise

### Evaluation

- A market trader asks #5000 for some cloth. A woman offers #1 200. After bargaining, they agree a price half-way between the two starting prices.(a) How much does the woman pay?
(b) What discount did she get by bargaining?

- The cash price of a used car is #500,000. To pay by hire requires a 10% deposit and 36 monthly payments of #24 420. Calculate the total amount of money that will be paid back by the hirer
- A bicycle can be bought either in cash for #247000 or by paying 52 weekly payments of #5400.Which amount is the higher price
- A history book costs #850. The writers of the book get 10% of the price of each book sold. How much will they get if it sells 15 628 copies in one year?
- A villager bought 11 goats for #76 000. A year later he sold them at a profit of 32%. What was the average selling price per goat?
- What is the simple interest on #120, 000 at the rate of 5% per annum for 12 years?
- The price of a car is #2 500 000. If 15% VAT is payable on the purchase. How much does a prospective buyer pay?
- An estate agent got 15% as commission on rental of #450 000. How much did he get?
- A customer deposits a cheque for #500 000. Her bank charges 12% commission for clearing the cheque. Calculate how much money is credited to her account.
- By selling goods for #5 350 a trader makes a profit of 7%. She reduces her prices to #5 150. What is her percentage profit now?

**Conclusion**

The class teacher wraps up or concludes the lesson by giving out a short note to summarize the topic that he or she has just taught.

The class teacher also goes round to make sure that the notes are well copied or well written by the pupils.

He or she makes the necessary corrections when and where the needs arise.

** **