Grade 9 Mathematics Study Notes

Introduction to Numbers

In Grade 9, the Numbers strand covers: integers, cubes and cube roots, indices and logarithms, and proportions.

Sub-Strand 1.1 — Integers: what they are, and how to add, subtract, multiply, divide, and combine them.

Operations on integers — What integers are

An integer is any whole number — positive, negative, or zero. Fractions and decimals are not integers.

• Positive integers: +1, +2, +3, … (whole numbers above zero)
• Negative integers: −1, −2, −3, … (whole numbers below zero)
• Zero (0): neither positive nor negative

On a number line, positive integers sit to the right of zero; negative integers sit to the left.

Counters (circular tokens marked + or −) and number cards (small cards labelled + or −) are used to show integer operations.

Addition of positive integers to positive integers

Adding a positive number to a positive number gives a positive number.
That is: (+) + (+) = (+)

Worked examples

1. Work out: (+8) + (+5)
   Using counters with positive signs, place 8 counters then add 5 counters.
   (+8) + (+5) = +13
2. A training camp is located 2 400 m above the foot of a mountain. A trainee moved 1 732 m towards the top. How far is the view point from the foot?
   Moving up the mountain is positive.
   (+2 400) + (+1 732) = +4 132 m
   The view point is 4 132 m above the foot of the mountain.
3. A thermometer reads +25°C. The water is heated and the temperature changes by +10°.
   (+25) + (+10) = +35°C
4. Mercy had Ksh 2 000 in her bank account. She deposited Ksh 3 000 more.
   (+2 000) + (+3 000) = +Ksh 5 000

Addition of negative integers to negative integers

Adding a negative number to a negative number gives a negative number.
That is: (−) + (−) = (−)

Worked examples

1. Work out: (−7) + (−4)
   Using counters with negative signs, place 7 counters then add 4 counters.
   (−7) + (−4) = −11
2. The temperature of ice in a refrigerator was −8°C. The temperature was further reduced by 5°C. What was the final temperature?
   (−8) + (−5) = −13°C
3. A sea diver was 15 m below the water surface. He moved deeper by another 10 m. Taking distance below the surface as negative, how far did he reach?
   (−15) + (−10) = −25 m from the surface
4. Shiroya had borrowed Ksh 500. She borrowed another Ksh 300. How much had she borrowed in total?
   (−500) + (−300) = −800
5. A drilling record shows −27 m. The drill goes 64 m further. What is the new record?
   (−27) + (−64) = −91 m

Addition of negative to positive integers or positive to negative integers

Adding a positive integer to a negative integer — or a negative to a positive — is the same as subtracting the smaller number from the larger number, then giving the answer the sign of the larger number.

Worked examples

1. Work out: (+16) + (−19)
   The larger number is 19 and it has a (−) sign; the smaller number is 16.
   So (+16) + (−19) can be written as 19 − 16 = 3.
   We give the answer (−) because the larger number was (−).
   Answer: −3
2. Work out: (+21) + (−7)
   The larger number is 21 and it has a (+) sign.
   21 − 7 = 14.
   We give the answer (+) because the larger number was (+).
   Answer: +14
3. Work out: (+298) + (−672)
   The larger number is 672 and it has a (−) sign.
   672 − 298 = 374.
   Since the larger number was negative, the answer is −374.
4. Work out: (+6) + (−9)
   Select 6 positive counters and 9 negative counters. Pair them off.
   6 pairs cancel. 3 negative counters remain unpaired.
   Answer: −3
5. Work out: (+10) + (−4)
   Select 10 positive counters and 4 negative counters. Cross them pair by pair.
   4 pairs cancel. 6 positive counters remain.
   (+10) + (−4) = +6
6. One morning the temperature on the slopes of Mount Kenya was −6°C. At noon it had risen by 10°. What was the temperature at noon?
   The larger number is 10 and it has a (+) sign.
   10 − 6 = 4. Answer: +4°C
7. The electricity metre read 120 units. After one week the reading had dropped by 40 units.
   The larger number is 120 and it has a (+) sign.
   120 − 40 = 80. Answer: +80 units
8. Wasilwa had borrowed Ksh 3 545. When he earned his salary of Ksh 10 500 he settled the debt and kept the balance.
   The larger number is 10 500 and it has a (+) sign.
   10 500 − 3 545 = 6 955. Answer: +Ksh 6 955

Subtraction of Integers

To subtract integers, use these rules:

1. (+) − (+) = positive, if the first integer is greater than the second
2. (+) − (+) = negative, if the second integer is greater than the first
3. (−) − (−) = negative, if the first integer is greater than the second
4. (−) − (−) = positive, if the second integer is greater than the first

Worked examples

1. (+6) − (+4) = 6 − 4 = +2
2. (−4) − (+11) = −4 − 11 = −15
3. (+15) − (−8) = 15 + 8 = +23
4. (−25) − (−30) = −25 + 30 = +5
5. The temperature in a room was 26°C. It dropped by 4°C.
   (+26) − (+4) = +22°C
6. Miriam has Ksh 95. An exercise book costs Ksh 120. How much more does she need?
   95 − 120 = −25 (she needs Ksh 25 more)
7. 460 votes were cast; 28 were rejected. How many were valid?
   460 − 28 = 432
8. A vendor had 120 oranges. She sold 55 on day 1 and 60 on day 2. How many remained?
   120 − 55 − 60 = 5

Multiplication of integers

Use the P-N triangle to remember the sign rules:

• a positive number × a negative number = a negative number: (+) × (−) = (−)
• a negative number × a positive number = a negative number: (−) × (+) = (−)
• a negative number × a negative number = a positive number: (−) × (−) = (+)
• a positive number × a positive number = a positive number: (+) × (+) = (+)

Worked examples

1. Work out: (+16) × (+5)
   The working is (+) number × (+) number = (+) number.
   (+16) × (+5) = +80
2. Work out: (−20) × (−3)
   The working is (−) number × (−) number = (+) number.
   (−20) × (−3) = +60
3. Work out: (−7) × (+15)
   The working is (−) number × (+) number = (−) number.
   (−7) × (+15) = −105
4. Work out: (+52) × (−4)
   The working is (+) number × (−) number = (−) number.
   (+52) × (−4) = −208
5. Rice is sold at a profit of Ksh 20 per kg. Find the overall profit if 150 kg is sold.
   (+20) × (+150) = +Ksh 3 000
6. A painting company bought 300 tins of paint at Ksh 200 each.
   (+200) × (+300) = +Ksh 60 000

Division of Integers

Division uses the same sign rules as multiplication:

• a positive number ÷ a negative number = a negative number: (+) ÷ (−) = (−)
• a negative number ÷ a positive number = a negative number: (−) ÷ (+) = (−)
• a negative number ÷ a negative number = a positive number: (−) ÷ (−) = (+)
• a positive number ÷ a positive number = a positive number: (+) ÷ (+) = (+)

Worked examples

1. Work out: (+120) ÷ (+4)
   The working is (+) number ÷ (+) number = (+) number.
   (+120) ÷ (+4) = +30
2. Work out: (−10) ÷ (−2)
   The working is (−) number ÷ (−) number = (+) number.
   (−10) ÷ (−2) = +5
3. Work out: (−320) ÷ (+16)
   The working is (−) number ÷ (+) number = (−) number.
   (−320) ÷ (+16) = −20
4. Work out: (+143) ÷ (−11)
   The working is (+) number ÷ (−) number = (−) number.
   (+143) ÷ (−11) = −13
5. A frozen solid was at −200°C. After heating, the final temperature was a quarter of the initial temperature.
   (−200) ÷ (+4) = −50°C
6. Sally bought 218 m of cloth and divided it into 2 m pieces.
   (+218) ÷ (+2) = +109 pieces
7. Galgallo has 200 hectares and divides it into 5-hectare pieces.
   (+200) ÷ (+5) = +40 pieces

Combined operations on integers

When an expression has more than one operation, follow this order: brackets, division, multiplication, addition and subtraction.

Worked example

Evaluate: 288 ÷ 24 − (3 + 8) + 3 × (−9)

Applying combined operations to real-life problems

1. Temperature — patient with fever

A patient's temperature was 39°C. After three hours it fell by 3°C then increased by 1°C. Find the final temperature.

1. After the fall: (+39) + (−3) = +36°C
2. After the rise: (+36) + (+1) = +37°C

2. Maize store — Peter

350 bags (90 kg each). Sold 120 bags day 1, 50 bags day 2. Bought 80 bags day 3. Shared all remaining maize equally among 5 customers.

1. Bags remaining: 350 − 120 − 50 + 80 = 260 bags
2. Total mass: 260 × 90 = 23 400 kg
3. Each customer received: 23 400 ÷ 5 = 4 680 kg

3. Poultry farm — James

460 eggs collected; 40 cracked in transport. Good eggs sold at Ksh 450 per tray (30 eggs = 1 tray). Each cracked egg sold at Ksh 10. James spent Ksh 2 500 on feed and donated Ksh 1 000.

1. Good eggs: 460 − 40 = 420
2. Number of trays: 420 ÷ 30 = 14 trays
3. Revenue from good eggs: 14 × 450 = Ksh 6 300
4. Revenue from cracked eggs: 40 × 10 = Ksh 400
5. Total revenue: 6 300 + 400 = Ksh 6 700
6. Amount deposited: 6 700 − 2 500 − 1 000 = Ksh 3 200

4. Examination scores

Correct answer: +2 marks. Wrong answer: −1 mark. Not attempted: 0 marks.

1. A student did 40 questions and got 35 correct (5 wrong):
   (35 × 2) + (5 × (−1)) = 70 − 5 = 65 marks
2. A student did all 50 but got half wrong (25 correct, 25 wrong):
   (25 × 2) + (25 × (−1)) = 50 − 25 = 25 marks

5. Minibus journey

A minibus had 35 passengers. At stop 1: 15 alighted, 10 boarded. At stop 2: 5 alighted, 8 boarded. No further stops.

1. After stop 1: 35 − 15 + 10 = 30 passengers
2. After stop 2: 30 − 5 + 8 = 33 passengers arrived at the destination

Key Points — Sub-Strand 1.1: Integers

Integers in everyday life

• Temperature: below zero = negative (e.g. −8°C in a freezer)
• Bank balance: deposits are positive (+); debts are negative (−)
• Altitude and depth: above ground or sea level (+); below (−)
• Exam scores: correct marks are positive; penalty marks are negative
• Business: profit is positive (+); loss is negative (−)