This document provides information about integers, fractions, decimals, squares, and square roots for an 8th grade mathematics strand. It includes specific learning outcomes, definitions of key terms, examples of operations and conversions, and a quiz. The learner is expected to identify integers, represent and operate on them using a number line, and understand their real-world applications. Fractions, decimals, squares, and square roots are similarly defined and examples are provided of related operations and conversions. A quiz at the end assesses these concepts.

1. MATHEMATICS GRADE 8
STRAND 1.0: NUMBERS

2. SPECIFIC LEARNING OUTCOMES
•By the end of the sub- strand the learner should be able to;
•Identify integers in different situations
•Represent integers on a number line in different situations
•Carry out operations of addition and subtraction integers on
the number line in real life situations
•Use IT or print resources for learning more on integers and for
skills development
•Reflect on use of integers in real life situations.

3. SUB-Strand 1.1: INTEGER
• Integers are a set numbers that includes all the whole
numbers, both positive and negative along with zero. Integers
include positive numbers such as 1,2,3,4 and negative numbers
e.g -10,-20,-30 and also zero(0)

4. ADDITION AND SUBTRACTION OF INTEGERS

5. ADDITION
• Moving forward shows addition
• Moving backwards shows subtraction.
• For example
• GMT weather forecasters are tracking temperature changes in
Nairobi. The temperature on Day one was 50 and on day 2 the
temperature had increased by 30 . What was the temperature on day
2?
• Increase in temperature means we add the temperature
• 5+3=8
• On the 2nd day temperature was at 80

6. SUBTRACTION
•Suppose you owe your friend 7 dollars and
you give him 5 ksh , how much do you still
owe your friend?
•5-7=-2
•From the above equation we get to see that
you still owe your friend 2 ksh

7. USES OF INTEGERS IN REAL LIFE
•Temperatures-Commonly used to represent temperature
changes
•Finances-Positive integers represent money earned while
negative integers represent expenses or money spent
•Sports and games- In sports and games integers are used
to represent number of points scored
•Scores and grades-In education, integers are used to
represent scores and grades. e.g scoring 80% in a
Mathematics test

8. QUIZ
• 1. show the following subtractions using a number line and give the results
• a) 45-17 b)19-70 c) (13) – (-6)
• 2.On a certain day, a student measured temperature inside a deep freezer
and found that it was -30 C while the room temperature was 240 C. What was
the temperature difference between room temperature and the deep
freezer?
• 3. Show the following additions using a number line and give the results
• a) (2) + (3) b) (+8) + (+7) c) (-15) + (+12) d) (+6) + (+2) + (-21)
• 4.Is 6-4 and -6+4 give the same answer? give a reason
• 5.Mary walked four floors down from the 10th floor and then took a lift to 18th
floor. How many floors did she go through while in the lift?

9. SPECIFIC LEARNING OUTCOMES
•By the end of the sub- strand, the learner should be
able to;
•carry out combined operations on fractions in
different situations
•Work out operations on fractions in real life Situations
•use IT devices for learning more on fractions and for
enjoyment,
•promote use of fractions in real life situations

10. FRACTIONS
•Fractions is a part of a whole. Its written in the form
of a/b where both A and B are numbers and b should
not be equal to zero. Upper number which is A is
called numerator while B which is lower number is
called denominator

11. COMBINED OPERATIONS WITH FRACTIONS
•Combined operations involves the use of
BODMAS. i.e brackets of Division Subtraction,
division, multiplication, addition, subtraction.
•For example:
• 1
2+ 1
4 × 2
5=

12. • Step 1: Perform multiplication first.
• 1
4 × 2
5 =(1 × 2)
(4 × 5) = 2
20= 1
10
• Step 2: Add the fractions.
• 1
2 + 1
10
• To add these fractions, they need a common denominator. The common
denominator for 2 and 10 is 10.
• Step 3: Convert 1
2 to have a denominator of 10.
• 1
2= (1
2) × (5/5) = 5/10
• Now, the expression becomes: 5/10
• Step 4: Add the fractions with the same denominator.
• 5
10 + 1
10= 6/10
• Step 5: Simplify the fraction, if possible.
• 6
10can be simplified to 3
5
•
• Therefore, the final answer is3
5

13. OPERATIONS OF FRACTIONS
•Operations in fractions include addition, subtraction,
multiplication, division
Addition
•1.When you mix two different colors of paint, you are
combining fractions to get a new color. For example, if you
mix 1
3cup of red paint with 1
4 cup of blue paint, you will
get 7
12cup of purple paint Find the l.c.m of ;
1/3+1/4=4/12+3/12 =7/12
•1/3+1/4=7/12

14. SUBTRACTION
•You have 1
2of a pizza and you eat 1
3 of the pizza. How
much pizza do you have in remaining? Find the l.c.m of
½-1/3 =3/6-2/6 =1/6
•
•1
2−
1
3=1
6

15. MULTIPLICATION
•You have1
8 of a cake and you cut it into 4 pieces.
How much cake is in each piece?
•1
8 cake ÷ 4 pieces=
•1
8 × 1
4=1/32

16. DIVISION
•You have 12 apples and you want to give them to 4 friends.
How many apples will each friend get?
•12
4=3
•They get 3 apples each
•Application of fractions
• Fractions are a fundamental part of mathematics, and they
are used in many different real life situations. Here are a few
examples of how fractions can be used in real life:

17. APPLICATION OF FRACTIONS
•Cooking: When you are cooking, you often need to use
fractions to measure ingredients. For example, a recipe
may call for 1
2 cup of flour or 1
4 cup of sugar.
•Money: Fractions are often used to express currency. For
example, in the United States, a dollar is divided into 100
cents, so 1
4dollar is equal to 25 cents.
•Time: Fractions are often used to express time. For
example, 1
2 hour is equal to 30 minutes, and 1
4hour is
equal to 15 minutes.

18. QUIZ
1. Evaluate
a). -1
4-(-1
2) b) 31
6-21
3+7
12
2.Express as Mixed numbers
a)8
3 b)38
9 c)523
9
3)Evaluate
a)2 1
5+10 2
7 b) 1
2 of 1
4 ÷ 1
8+3
4-1
8
4)It takes 1 ½ days to make a toy train. How many toys can be made in
14 days?
5)1 1
2 /52
3 × 9
10

19. SPECIFIC LEARNING OUTCOMES
• By the end of the sub- strand, the learner should be able to;
• Convert fractions to decimals in different situations
• Identify recurring decimals in different situations
• Convert recurring decimals into fractions in different situations
• Round off a decimal number to a required number of decimal places in
different situations
• Express numbers to a required significant figure in real life situations.
• Express numbers in standard form in different situations
• Carry out combined operations on decimals in different situations
• Apply decimals to real life situations

20. DECIMALS
a) Converting Fractions to Decimals:
Converting fractions to decimals involves dividing
the numerator (the top number) by the
denominator (the bottom number). For example:
Fraction: 1
4
Decimal: 1 ÷ 4 = 0.25

21. IDENTIFYING RECURRING DECIMALS
•Recurring decimals are decimals that have a
repeating pattern of digits. For instance:
•Decimal: 0.333... Recurring Pattern: 3
repeats infinitely (0.3 with 3 repeating)

22. CONVERTING RECURRING DECIMALS TO
FRACTIONS
• To convert recurring decimals to fractions, we set up an equation where "x" is
the recurring decimal and solve for "x":
• Decimal: 0.333...
• Equation: x = 0.333...
• Multiply both sides by 10 to shift the decimal: 10x = 3.333...
• Subtract the original equation from the shifted one: 10x - x = 3.333... -
0.333...
• Solve for x: 9x = 3
• Divide both sides by 9: x = 3/9 = 1/3

23. ROUNDING OFF DECIMALS
•When you need to round off a decimal to a
certain number of decimal places, look at the
digit immediately after the required place and
follow these rules:
•If it's 5 or more, increase the previous digit by 1.
•If it's less than 5, keep the previous digit
unchanged

24. Expressing Numbers to Required Significant
Figures
•Significant figures are the meaningful digits in a
number:
•If you measure something with a ruler marked in
millimeters, your measurement might be 12.34
mm, which has 4 significant figures.
•If you're given a value like 15.00, it has 2
significant figures

25. Expressing Numbers in Standard Form
•Standard form (also called scientific notation)
is a way to express very large or very small
numbers:
•Number: 6,300,000 Standard Form: 6.3 × 10^6

26. COMBINED OPERATIONS ON DECIMALS
•You can perform operations like addition,
subtraction, multiplication, and division on
decimals just like with whole numbers.
Remember to line up the decimal points when
adding or subtracting

27. APPLYING DECIMALS TO REAL-LIFE SITUATIONS:
•Decimals are used in everyday life in various
ways:
•Money: Prices at the store, calculating change.
•Measurements: Length, weight, volume.
•Science: Measuring temperature, density, etc

28. PROMOTING USE OF DECIMALS IN REAL LIFE
• Decimals are essential for accurate measurements and
calculations. Encourage their use in situations like cooking,
shopping, DIY projects, and scientific experiments.
Understanding decimals helps you make precise decisions
and communicate information effectively.
•Remember, decimals are a key part of understanding the
world around us and making accurate calculations in various
situations!

29. QUIZ
Express each of the following as single decimal fractions
a)7
11 b) 5
6 c) 5
100
2.Round off the following numbers to 2 and 3 decimal
places
a)0.139789 b)0.0431285 c)5.108946
3.Express each of the following to standard form
a)369.4 b)0.0289 c)509.78

30. QUIZ
4.Evaluate
a)8.783+11.031-22.003 b)16.804-17.569+0.708
c)495.001-548.8
d)266×26.04 e)29.6×6.21
f)109.8006÷12
5. a) A family consumes 4.5 liters of milk every day, how much milk do
they consume during the month of April?
b) Joshua bought 8kg of sugar and 7 kg of rice at ksh44.50 and
ksh35.35 per kilogram respectively. How much money did she spend?

31. SPECIFIC LEARNING OUTCOMES
• By the end of the sub- strand the learner should be able to;
• work out the squares of numbers from tables in different situations
• work out the square roots of numbers from tables in different
situations
• work out squares and square roots of numbers using a calculator in
different situations
• use IT or other materials to learn more on squares and square
roots of numbers and for fun.
• enjoy using squares and square roots in real life situations

32. SQUARE AND SQUARE ROOTS
Squares Using tables
find squares using tables
a)4.25=18.06
b)3.17=10.05
c)5.94=35.28

33. SQUARE ROOTS USING TABLES
Finding square roots using tables, one reads
direct from the table
√1.86=1.363
√42.57=6.53
√0.8236= 0.908

34. SQUARE AND SQUARE ROOTS USING CALCULATOR
1.Measuring area
You are building a square garden, and you want to
calculate the area of the garden based on the length of
one side. Find the square of the length of each side to
determine the area.
Example: if the length of one side is 27 meters, calculate
the area.
•Square of 27 meters = 272 = 729 square meters

35. SPORTS AND GAMES
You are analyzing sports performance, and you want to
compare the scores of players or teams. Find the squares of
their scores to assess variations and trends.
Example: If the scores of two players are 12 and 15, calculate
the squares of their scores.
Solution:
Square of 12 = 12^2 = 144
Square of 15 = 15^2 = 225

36. SQUAREROOTS
• .Situation: Distance and Time
• You are measuring the distance covered in a specific time interval during a
race. Find the square root of the distance covered to determine the average
speed.
• Example: If the distance covered is 144 meters in a race, find the average
speed.
• Square root of 144= √144 = 12 meters

37. SQUAREROOTS
•You are conducting an experiment that involves
measuring the time taken for an object to fall from a
certain height. Find the square root of the time to
determine the object's speed.
•Example: If the time taken for the object to fall is 1225
seconds, find its speed.
•Square root of 1225 seconds = √1225 = 35 seconds
(This gives the time taken for the object to fall, not the
speed)

38. APPLICATION OF SQUARES AND SQUARE
ROOTS
• Here are some specific examples of how students can enjoy using squares and
square roots in real-life situations:
• Playing games: For example, students could play a game where they have to roll
a die and then find the square of the number that they rolled.
• Using manipulatives: Manipulatives can be used to help students visualize
squares and square roots include , students could use square tiles to build a
square and then find the area of the square.
• Creating art projects: Students can create art projects that incorporate squares
and square roots.

39. QUIZ
• Find the square root of each of the following numbers using tables
• a) 5.38 b)6.142 c)7.358
• 2.Use tables to find squares of the following numbers
• a)2.78 b)9.32 c)3.97 d)8.02
• 3.Find square root of each of the following numbers using calculator
• a) 76,176 b)4356 c)15.625
• 4.if a=3, b=4.7 and c=6.4 find the value of √(a2b2/c2)
• 5.The area of a triangle whose height is equal to the length of its base is
40.5cm2. Calculate the length of the base.

40. SPECIFIC LEARNING OUTCOMES
• By the end of the sub- strand the learner should be able to;
• Identify rates in different situations
• Work out rates in real life situations
• Express fractions as ratios in real life situations
• Compare two or more ratios in different situations
• Divide quantities in given ratios in real life situations
• Work out ratios in different situations
• Work out increase and decrease of quantities using ratios in real life situations
• Work out percentage change of given quantities in real life situations.
• Identify direct and indirect proportions in real life situations
• Work out direct and indirect proportions in real life situations

41. Rates, Ratio, Percentages and
Proportion

42. RATES
•Rates are a way of comparing two quantities that
are measured in different units. For example if a
car takes 2 hours to travel distance of 160 km,
then we say that it’s travelling at an average
speed of 80km/hr.

43. APPLICATION OF RATES
• Here are some examples of rates in different situations:
• Speed: Speed is the rate at which something moves. It is measured in distance
per time, such as miles per hour (mph) or kilometers per hour (km/h).
• Acceleration: Acceleration is the rate at which speed changes. It is measured
in distance per time squared, such Fuel efficiency: Fuel efficiency is the rate at
which a car uses fuel. It is measured in miles per gallon as meters per second
squared (m/s²).
• (mpg) or kilometers per liter (km/l).
• Work rate: Work rate is the rate at which work is done. It is measured in units
of work per time, such as watts (W).

44. EXAMPLES OF RATES IN REAL LIFE
• 1.If you cover 200 kilometers in 4 hours, the rate of speed is 200 km / 4 hours
=50
• 2. Determine the rate of interest on a loan or investment. For example, if you
earn $500 in interest on a $10,000 investment in one year what is the
interest rate?
• $500/$10000=0.05
• Interest rate is 0.05
• 3.A tenant paid ksh. 36,000 to his landlord in one year. What was his rate of
payment per month?
• 36000
12= 3,000 The tenant pays 3000 per month

45. Express ratios as fractions
• Ratios are ways of comparing two similar quantities
• i)If Joel is 12 years old and his brother William is 21 years old, Joel’s age is
12
21 of William’s age and their ages are said to be in the ratio of 12 to 21
written as 12:21
• ii)If you spend 2 hours out of 3 hours studying, the fraction is 2
3. As a ratio, it
can be expressed as "2 out of 3 hours," or "2:3."
• iii) If a recipe calls for 2 cups of flour out of a total of 5 cups of flour, the
fraction is 2
5. As a ratio, it can be expressed as "2 out of 5 cups of flour," or
"2:5."

46. COMPARING RATIOS
•Comparing ratios shows which one is greater than, less
than,or equal to
•In order to compare ratios, they have to be expessed as
fractions first. i.e 2:3=2
3..The resulting fractions can be
compared. For example, which is greater 2:3 or 4:5?
•2:3=2
3,4:5=4
5
•2
3=10
15,4
5=12
15therefore 4
5 > 2
3

47. DIVIDING QUANTITIES GIVEN RATIO
• In a company, the total profit is $20,000, and it needs to be
divided between three partners in the ratio of 3:4:5. To divide the
profit according to the ratio, you would allocate 3 parts to one
partner, 4 parts to another, and 5 parts to the third partner. In this
case, you would divide the profit as follows:
• 3 parts (3
12 of the total) for Partner A: 3/12 * $20,000 = $5,000.
• 4 parts ( 4
12of the total) for Partner B: 4/12 * $20,000 =
$6,666.67.
• 5 parts (5
12of the total) for Partner C: 5/12 * $20,000 =
$8,333.33.

48. APPLICATION OF RATIO IN REAL LIFE
• i)For example, if one person can lift 100 pounds in 1 minute and the other
person can lift 125 pounds in 1 minute, then the ratio of their work rates is
100/125 = 4/5. This means that the first person can lift 4 pounds per minute
for every 5 pounds per minute that the second person can lift.
• Ratio=100
125=4
5 which is the same as 4:5
• ii)Ratio of boys to girls in a school is 2:3 if there are 160 boys, how many girls
are there?
• number of girls = (ratio of girls)/ (ratio of boys + ratio of girls) * number of
boys
• number of girls = (3/5)/(2/5 + 3/5) * 160 = 96

49. INCREASE AND DECREASE OF RATIO
•To increase or decrease quantity of given ratio, we
express the ratio as a fraction and multiply it with
the quantity

50. INCREASE
• Price of a pen is adjusted in ratio 6:5. If the original price was ksh. 50, What’s
the new price?
6:5=6
5
• 6
5 × 50 = 60
• The new price of the pen is ksh. 60

51. Decrease
• Decrease 45 in the ratio of 7:9
• 7:9 = 7
9
• 7
9 × 45 = 35
• Suppose the price of a product increased from $50 to $70. To find the
percentage increase:
• Percentage Increase = ((70−50)
50) * 100 = (20
50) * 100 = 40%
•
• The price of the product increased by 40%.

52. Direct and indirect Proportions
•1.Direct proportion: A direct proportion is a relationship
between two quantities where the product of the two
quantities is always constant.
•The amount of fuel you use is directly proportional to the
distance you travel.
•The amount of light you produce is directly proportional to
the amount of electricity you use.
•The force you exert is directly proportional to the mass of
the object you are trying to move.

53. INDIRECT PROPORTION
•An indirect proportion is a relationship between two
quantities where the product of the two quantities is always
inverse
•The time it takes to complete a task is inversely proportional
to the number of people working on the task
•The pressure exerted by a fluid is inversely proportional to
the area of the surface it is acting on.
•The amount of money you save is inversely proportional to
the amount of money you spend.

54. ASSESSMENT
Evaluate the following using number line.
i) -6+ -3 +11
ii) 12 - -7-13
iii) 8 + 3 -7
Ability to carry out combined operations on fractions
Ability to convert fractions to decimals
Ability to identify and convert recurring decimals into fractions
Ability to round off a decimal number to a required number of decimal places
Ability to express numbers in standard form

55. ASSESSMENT
Ability to round off a decimal number to a required number of decimal places
Ability to express numbers in standard form
Ability to carry out combined operations on decimalsAbility to carry out
combined operations on decimals
Ability to work out squares and square roots of numbers using Mathematical
tables and a calculator
Ability to carry out combined operations on decimalsAbility to carry out
combined operations on decimals
Ability to work out squares and square roots of numbers using Mathematical
tables and a calculator

56. ASSESSMENT
•Ability to identify and work out rates
• Ability to express fractions as ratios
• Ability to compare two or more ratios
•Ability to divide quantities in given ratios
• Ability to work out percentage increase and decrease of
quantities
• Ability to identify and work out direct and indirect
proportions