STRAND 1 NUMBERS.pptx GRADE 8 CBC FOR LEARNERS

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

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)

ADDITION AND SUBTRACTION OF
INTEGERS

ADDITION
Moving forward shows addition
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?
Solution
Increase in temperature means addition of the temperature
Therefore;
50 + 30 = 80
On the 2nd day, the temperature was at 80

IN PAIRS WORKOUT THE
FOLLOWING;
1. Ouma had 5 bananas, he was given 6 more
bananas by Njeri. How more bananas did
Ouma had?
2. A class has 32 learners 5 more learners
joined the class. How many learners in total
where there in class?
3. Juma had 12 exercise books, while Nekesa
had 9 exercise books. How many exercise
books did Juma and Nekesa had altogether?

SUBTRACTION
It is moving backwards or taking away.
For example
Suppose you owe your friend ksh 7 and
you give him ksh 5, how much do you still
owe your friend?
Solution
5 - 7= -2
From the above equation we get to see
that you still owe your friend ksh 2

IN PAIRS WORKOUT THE
FOLLOWING;
1.(a). 10 – 12 =
(b). 6 – 4 =
(c). 4 – 8 =
2. Munila had ksh. 20 in her pocket on
her way to the market she lost sh.
12. How much money did she
remained with?

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

ACTIVITY I
1. Use number line to solve the following problems.
(a) i. -10 – 3=
ii. 7 – 8 =
iii. 12 – 4 + 3=
iv. - 2 – 4 +3 =
(b) i. (+7) + (-9) =
ii. (-4) + (-3) =
iii. (-6) – (-6) =
iv. (+4) + (+5) =
2. Wanyonyi borrowed sh. 6 from a shop, he later received sh. 10 from his
mother which he used to pay the debt. Use number line to show the amount
of money that Wanyonyi remained with.

SELF CHECK ASSESSMENT I
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)

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

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
Example
2/3 in this case 2 is the numerator while 3 is
denominator.

OPERATIONS OF
FRACTIONS
Operations of fractions include;
Addition,
Subtraction,
Multiplication,
Division

ADDITION
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
Therefore;
1/3+1/4=7/12

IN GROUPS WORK OUT THE FOLLOWING
PROBLEMS AND PRESENT IN CLASS
1. 2/3 + 1/4 =
2. 3/5 + 2/5 =
3. 1/2 + 2/3 =
4. 3/4 + 1/6 =
5. 5/6 + 2/3 =
6. 4/7 +3/7 =
7. 2/5 + 1/10 =
8. 5/8 + 2/7 =
9. 3/9 + 1/3 =
10. 4/9 + 5/9 =

SUBTRACTION
Njoroge had 1
2 of the pizza, he gave out 1
3
of it. What fraction of pizza did he remaining
with?
SOLUTION
1
2 - 1
3=
Step 1: find the L.C.M of the denominators
2,3
Step 2: the L.C.M of 2,3 is 6
Find the L.C.M of
1 1 3x1 – 2x1 = 1
6

SUBTRACTION OF FRACTION:
WORK SHEET
1. 5/6 – 1/3=
2. 4/5 – 2/5=
3. ¾ - 1/8 =
4. 7/8 – ½ =
5. 2/3 – 1/4 =
6. 5/12 – 1/6 =
7. 2/9 – 2/3 =
8. 5/7 – 3/7=

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

MULTIPLICATION OF FRACTION:
WORK SHEET
1.2/3 X 2/5 =
2.3/5 X 4/8 =
3.½ X 2/3 =
4.¾ X 1/6 =
5.5/6 X 2/5 =
6. 4/7 X 3/7 =
7. 2/5 X 1/10 =
8. 5/8 X 1/8 =
9. 3/9 X 1/3 =
10. 4/9 X 4/10 =

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 will get 3 apples each.

DIVISION OF FRACTIONS: WORK
SHEET
A. 2
/3 ÷ 5
/6 =
B. 3
/5 ÷ 2
/5 =
C. ½ ÷ 2
/3 =
D.¾ ÷ 1
/6 =
E. 5
/6 ÷ 2
/3 =
F. 4
/7 ÷ 3
/7 =
G. 2
/5 ÷ 1
/10 =
H.5
/8 ÷ 1
/8 =
I. 3
/9 ÷ 1
/3 =
J. 4
/9 ÷ 5
/9 =

COMBINED OPERATIONS WITH
FRACTIONS
Combined operations involves the
use of BODMAS. i.e. Brackets, Of,
Division , Multiplication, Addition,
Subtraction.

For example:
𝟏
𝟐+ 𝟏
𝟒 × 𝟐
𝟓=
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

IN GROUPS WORKOUT THE
FOLLOWING;
1. (a). 1
2 - 2
3 + 1
10 ÷ 1
3
(b) 3
4 - 1
5 – 1
2 of 1
10
2. Muema spent ¾ of his salary as follows
1
8 given to an orphanage home
¼ used for transport
The remainder was used to pay school fees.
Which fraction of the remainder was used to

APPLICATION OF FRACTIONS
IN REAL LIFE
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 Kenya, 1 shilling is
divided into 100 cents, so 1
4 shillings is equal to
25 cents.
Time: Fractions are often used to express time.
For example, 1
2 hour is equal to 30 minutes, and

ASSESSMENT
1. Work out
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. Work out
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?

CONTINUOUS ASSESSMENT
5. 11
2 ÷52
3 × 9
10
6. 3/5 of learners in Bora comprehensive school are girls the rest are boys.
One day 20 boys were absent. The number of girls that day was two times
that of boys. How many learners were present if no girl was absent?
7. Muremi spent 1/3 of his day reading story books, ¼ for leisure and 1/5 of
the remainder for playing football. He then spent the remaining time
sleeping. If he slept for 6 hours, how much more time did he spend on
reading story book than playing football?

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

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

USING LONG METHOD TO CONVERT A
FRACTION TO DECIMAL
Fraction: 1
4
solution
0.25
4 1 0
- 8
.20
.20
0

ASSESSMENT: CONVENT THE
FOLLOWING FRACTIONS INTO
DECIMALS
1. Convert 3/5 cups of flour into decimals for a cake recipe.
2. Convert 1/4 gallon of fuel into decimals to measure how far a
car can travel.
3. Convert 2/3 tablespoons of salt into decimals for a large batch
of soup.
4. Convert 5/8 meters of wire into decimals to cut the correct
length for electrical wiring.
5. Convert 7/10 of a shilling into decimals for calculating change
in a purchase.
6. Convert 1/2 milliliters of medicine into decimals for

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)

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

ASSESSMENT: USING RECURRING
DECIMALS IN REAL LIFE SITUATIONS
1. The recipe calls for 0.666.. cups of sugar. Convert this recurring
decimal to a fraction.
2. The architect needs to measure 0.454545... meters for a
specific dimension. Convert this recurring decimal to a fraction.
3. The interest rate for a loan is 0.333... percent. Convert this
recurring decimal to a fraction.
4. In a track race, the athlete's time is 0.272727... seconds.
Convert this recurring decimal to a fraction.
5. In a musical piece, the rhythm pattern repeats every 0.125125...
beats. Convert this recurring decimal to a fraction.

IN GROUP CONVERT THE FOLLWONG TO
FRACTIONS
1. 0.3666….
2. 0.0111…
3. 0.888…..
4. 0.909090….
5. 0.272727……
6. 0.777……..
7. 0.040404……

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

EXAMPLE
Round of 3.71589 correct to 1,2,3 decimal places
solution
Round to 1 decimal place: Look at the digit in the tenths place, which is 1.
Since the digit immediately to the right of 1 is less than 5, we leave 1
unchanged.
So, 3.71589 rounded to 1 decimal place is 3.7
Round to 2 decimal places: Look at the digit in the hundredths place,
which is 5. Since the digit immediately to the right of 5 is 8, which is
greater than or equal to 5, we round 5 up.
So, 3.71589 rounded to 2 decimal places is 3.72.
Round to 3 decimal places: Look at the digit in the thousandths place,
which is 8. Since the digit immediately to the right of 8 is 9, which is
greater than or equal to 5, we round 8 up.
So, 3.71589 rounded to 3 decimal places is 3.716.

ASSESSMENT
1. Round off each of the following correct to one
decimal place;
(a). 7.386 (b). 23.105 (c).
199.93
2. The masses of 3 objects are as follows in grams,
round off to two decimal places.
(a)1.97847g (b). 18.7631g (c).
7.2893g
3. Write 15.786349 to the nearest:
(a). Hundredths (b). Thousandths
(c). Ten Thousandths

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

LEARNING POINTS
Rules for Determining Significant Figures:
All non-zero digits are significant (e.g., 3.14 has three
significant figures).
Zeros between non-zero digits are significant (e.g., 405 has
three significant figures).
Leading zeros (zeros to the left of the first non-zero digit) are
not significant (e.g., 0.007 has one significant figure).
Zeros at the end of a number are significant only if one
number has a decimal point. (e.g. 700.0)
Trailing zeros in a number without a decimal point may or
may not be significant (e.g., 1200 may have two, three, or
four significant figures depending on the context).

EXAMPLE OF SIGNIFICANT
FIGURES
(i). 0.078
Solution
2 significant digits
2 and 8 are non-zero digits
(ii) 281010
solution
6 significant figures
The numbers 2,8 and 1 are non-zero digits.
The zero between 1 and 1 is significant being between two non-digits.
The last zero is after a decimal point thus it is significant.

ASSESSMENT: SIGNIFICANT
FIGURES
1. In a science lab, students measure the volume of a liquid
using a graduated cylinder. The volume is recorded as
23.423 mL. Determine the number of significant figures in
the measurement and express it to the correct number of
significant figures.
2. During a math's lesson a teacher experiment, the length of a
metal rod is measured to be 0.00572 meters. Determine the
number of significant figures in the measurement and
express it to the correct number of significant figures.
3. In engineering, a designer specifies the thickness of a sheet
of metal as 0.028 inches. Determine the number of
significant figures in the measurement and express it to the
correct number of significant figures.

ASSESSMENT: SIGNIFICANT
FIGURES
1. A weather forecaster predicts the temperature
for tomorrow as 22.9 degrees Celsius. Determine
the number of significant figures in the
prediction and express it to the correct number
of significant figures.
2. A company reports its annual revenue as
5,620,000 shillings. Determine the number of
significant figures in the revenue figure and
express it to the correct number of significant
figures.

EXPRESSING NUMBERS IN
STANDARD FORM
Standard form (also called
scientific notation) is a way to
express very large or very small
numbers:
A number can be expressed in
the form P x 10n where P is not
less than 1 but less than 10 and

EXAMPLE
The population of a certain county
is 6 300 000. Express the number
in standard form
6 300 000→6.3×106
So,
6 300 000 in standard form is
6.3×106.

ASSESSMENT
1. The distance between two cities is 3,200,000 meters. Express this
distance in standard form.
2. The concentration of a solution is 0.000025 moles per liter. Express this
concentration in standard form.
3. The capacity of a computer hard drive is 500,000,000,000 bytes.
Express this capacity in standard form.
4. The length of a DNA molecule is 0.00000000005 meters. Express this
length in standard form.
5. The distance between Earth and the Sun is approximately 93,000,000
miles. Express this distance in standard form.
6. A company's annual revenue is 5,600,000 shillings. Express this revenue
in standard form.

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.
Learning point
When working out combined operation
involving decimals, the order of operations
start with bracket, division, multiplication,
addition and subtration (BODMAS)

EXAMPLE
Work out
7.5 – 0.7 x 0.5 + 3.6 ÷ 1.2
Use BODMAS
(i). Work out division 3.6 ÷ 1.2 = 3
(ii). Work out multiplication 0.7 x 0.5 =0.35
(iii). Work out addition 0.35 + 3 = 3.35
(iv). Work out subtraction 7.5 – 3.35 =
4.15
Answer = 4.15

ASSESSMENT
1. You have sh. 1 000 to spend at the grocery store. You buy
apples for sh. 200.75 per kilogram. If you buy 2.5 kilograms of
apples, 1.25 kilograms of bananas for sh 100.50 per kilogram,
and a half of bread for sh 32.5, how much money do you have
left?
2. A car travels 240 km on 90.5 liters of petrol. If the price of
petrol is sh 203.20 per litre, how much does it cost to travel 1
km?
3. A recipe calls for 1.5 cups of flour, 0.75 cups of sugar, and 0.5
cups of butter. If you want to double the recipe, how much of
each ingredient do you need?

ASSESSMENT
1. You are renovating your kitchen and need to purchase tiles for
the floor. Each tile costs sh 101.50, and you need 200 tiles to
cover the entire floor. If you have a 10% discount on your
purchase, how much do you pay in total?
2. A textbook has 116 pages and 2 cover page. Each page
weighs 0.85 g while each cover weighs 1.5 g. find the mass of
the text books.
3. You run 3.5 miles every morning for a week. On Monday, you
run at an average pace of 8.5 minutes per mile. On Tuesday,
you run at an average pace of 9 minutes per mile. If you want
to calculate your average pace for the week, how long does it

APPLYING DECIMALS TO REAL-LIFE
SITUATIONS:
Money: Prices at the store,
calculating change.
Measurements: Length, weight,
volume.
Science: Measuring temperature,
density, etc.

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!

ASSESSMENT
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

4. Workout
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

SUB STRAND 1.4:
SQUARES AND
SQUARE ROOTS

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

DISCUSS IN GROUPS HOW TO
FIND THE SQUARES OF NUMBERS
AND PRESENT YOUR FINDINGS IN
CLASS.
1. Find the square of 5.
4. Find the square of 15.5.
6. Find the square of 2.5.

SQUARES USING MATHEMATICAL
TABLES

PROCEDURE USING THE MATHEMATICAL
TABLE
Example 1: Use the mathematical table to find the square of 2.2
procedure
i. Look for 2.2 down the column headed x and move to the right
along the row up to where it intersects the column headed 0.
ii.Write the number in the position.
iii.The number 4.840
Example 2: square of 2.26 (using the table above)
Procedure
iii.The number is 5.108
KSH

ASSESSMENT
(Use tables of squares in this exercise)
1. Find the square of each of the
following:
(a). 3.94 (b). 5.42 (c).
53.25
2. Find the value of (3.143)2
3. A square farm measures 4.32m on
each side. calculate the are of farm in
metres.
4. A square design printed on a wall

SQUARE OF NUMBER GREATER
THAN 10
Consider (525)2
Solution
i. Write 525 in standard form, 5.25 x 102
ii. Find the square of 5.25 from the table of
squares,
iii.Find the square of 10 which is 10 x 10
iv.Multiply the value obtained in (ii) and
product in (iii).

SQUARE OF NUMBERS BETWEEN 0
TO 1
Using the mathematical table to solve square of 0.2394
Solution
i. Write 0.23942 in standard form (2.394 x 10-1)2
ii. Find the value of 2.3942 in the table of square
iii. Look for 2.3 down the column headed x, then move to the right along the rows
of 2.3 upto the point where it intersects to the column headed 9.
iv. Read and recorded the number in the position of intersection which is 5.712.
v. Move further the row to the ‘ADD’ column headed 4 and record the number as
the point of intersection as 19.
vi. Add 19 to the last digit of 5.712
vii. 5.712 + 0.019=5.731
viii. To obtain 5.731 which is the square of 2.394
ix. Therefore, 0.239422 = 5.731 x 0.1x0.1
= 0.005731

ASSESSMENT
1. Find the square of each of the following;
(a)0.2584
(b)261.02
(c)315.7
(d)0.58
(e)0.0358
(f)0.000196
(g)1052.01

SQUARE ROOTS
Discuss in groups how to find the square
roots of numbers and present your
findings in class.
i. √2.1
ii. √2.13
iii.√2.134
iv.√48

SQUARE ROOTS USING TABLES
Learning point
To find the square root of a number
x, such that 1 ≤ x < 100, read
intersections of rows for the number
upto 1 decimal place and the column
for the 2nd decimal place from the
table of square root.

FROM THE TABLE OF SQUARE ROOT FIND THE
FOLLOWING;
Example 1: Use the mathematical table to find the square root of 2.1
procedure
iii.The number 1.4491
Example 2: square root of 2.26 (using the table above)
Procedure
iii.The number is 1.5033

ASSESSMENT
1. Finding square roots using tables,
a) √1.86
b) √42.57
c) √0.8236
2. The area of a square window is 2.25m2. Find
the length of each side.
3. The area of a square garden is 38.44m2. Find
the length of each side.
4. The area of a square tile is 9.26m2. What is the
length of each side?

SQUARE ROOTS OF NUMBER
GREATER THAN 100
Consider √836.3
Solution
i. Write √836.3 in standard form, √836.3 x 102
note √836.3 = √8.363 x √100
= √8.363 x 10
i. ii. Look for 8.3 down the column headed x and move to the right along
the row up to where it intersects the column headed 6.
ii.Write the number in the position. Move further along the row to the mean
difference column headed 3, note down the digit as intersection point.
Add the digit to the last digit of the reading at the intersection of 8.3 and
the column headed 6.
iii.Multiply the number obtain in (ii) by 10.
iv.Discuss and share your findings.

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

WORKING OUT SQUARES OF
NUMBERS USING A CALCULATOR
IN DIFFERENT SITUATIONS
Work out the square of 0.643 using calculator
i. Switch on the calculator
ii. Press 0.643
iii. Press x2
iv. press =
v. 0.413449 is displayed
vi. Therefore the square of 0.643 is 0.413449

ASSESSMENT
1. A square carpet has a length of 0.65m. Find its
area in m2
2. A farm in the shape of a square has a length of
0.962km. Find its area in cm2.
3. A square picture frame of length 37.4cm is
made up of 4 pieces. Each piece has a width of
0.4cm. Find the area enclosed by the frame in
cm2.
4. A square room has a length of 7m. Find its area
in cm2.
5. Use a calculator to find the square of the

WORKING OUT SQUARE ROOT OF
NUMBERS USING A CALCULATOR
Use a calculator to find the square root of
0.004219
i. Switch on your calculator
ii. Press √
iii.Press 0.004219
iv.0.064953829 is displayed on the screen
Therefore, the square root of 0.004219 is
0.064953829

ASSESSMENT
Use a calculator in this assessment
1. Find the square root of each of the
following;
(a). 0.2834 (b). 0.7292
(b). 0.005784 (d). 0.00009875
2. The area of a square farm is 1.44km2. Work
out its length.
3. A housing building project covers a square
piece of land of an area 174 724m2. calculate
the length of the side of the land.

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.

ASSESSMENT
1. 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.

SUB STRANDS 1.5: RATE,
RATIO PROPORTION AND
PERCENTAGE

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

RATES
Rates are a way of comparing two quantities that are
measured in different units.
Example
A labourer earns sh. 400 for working for 8 hrs a day.
Find the rate of earning per hour.
Solution
Rate =
=
= ksh 50 per hour
Amount of money paid
Number of hours
40
0
8

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).

EXAMPLES OF RATES IN REAL LIFE
1. If you cover 200 kilometers in 4 hours, the
rate of speed is 200 km / 4 hours
=50km/h
2. A tenant paid ksh. 36,000 to his landlord
in one year. What was his rate of payment
per month?
36000
12= 3,000

ASSESSMENT
1. A farmer needs 120 bags of fertilizers to plant 30 hectares of land. Find
the number of bags he will need for I hectare.
2. A manufacturing company produces 8 400 sheets on iron in 10 days.
Calculate the rate of production per day.
3. The engine of a bus consumes 40 litres to cover a distance of 200km.
Find its rate of consumption per litre.
4. A grade 8 typed 6 000 words in one hour. Find the rate of typing per
minute.
5. Work out the rate of each of the following
(a). A runner took a 200 m dash in 25 seconds
(b). A school charges a fee of sh. 25 000 for three school terms.

EXPRESS THE 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

EXPRESS RATIOS AS
FRACTIONS
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."

EXAMPLE
A solution consists 3/5 of liquid A and the rest liquid B.
(a). Express the liquid A as a fraction of liquid B in the mixture.
Solution
The solution consists (1 – 3/5) of liquid B
= 2/5 of liquid B
(a) Liquid A/liquid B = 3/5 ÷ 2/5
= 3/5 x 5/2
= 3/2
(b) The ration A:B = 3:2

ASSESSMENT
1.If the ratio of boys to girls in a class is
3:5, and there are 24 boys, how many
girls are there?
2.If the ratio of red marbles to blue
marbles in a jar is 2:5, and there are 70
blue marbles, how many red marbles are
there?
3.Sarah and Emily share some money in
the ratio 3:2. If Sarah receives sh 30 000

ASSESSMENT
1. A recipe for pancakes requires flour and milk in the ratio 4:3. If
you need 2 cups of milk, how many cups of flour do you need?
2. In a bag of candy, the ratio of red to green to blue candies is
3:4:5. If there are 84 green candies, how many candies are
there in total?
3. The ratio of the lengths of two ropes is 5:3. If the longer rope
is 30 meters, what is the length of the shorter rope?
4. If it takes 5 hours to paint a fence that is 20 meters long, how
long will it take to paint a similar fence that is 30 meters long?

COMPARING RATIOS
Comparing ratios shows which one is greater
than, less than, or equal to
Example
In order to compare ratios, they have to be
expressed 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

ASSESSMENT
1.Compare the ratios 3:4 and 5:6. Which
one is greater?
2.In a class, the ratio of boys to girls is 2:3,
and the ratio of girls to total students is
5:12. What is the ratio of boys to total
students?
3.A recipe for lemonade calls for mixing
water and lemon juice in the ratio 3:2. If
you use 6 cups of water, how many cups
of lemon juice should you use to maintain

ASSESSMENT
1. In a bag of marbles, the ratio of red marbles
to green marbles is 4:7. If there are 28 green
marbles, how many red marbles are there?
2. If 15 workers can complete a task in 10 days,
how many days will it take for 25 workers to
complete the same task?
3. If 6 meters of fabric can make 4 shirts, how
many shirts can be made from 18 meters of

DIVIDING QUANTITIES GIVEN
RATIO
In a company, the total profit is sh200,000, 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 * 200,000
= 50,000.
4 parts ( 4
12of the total) for Partner B: 4/12 * 200,000
= 60,666.67.
5

LEARNING POINT
If a quantity is divided in the ratio a:b,the resulting
fractions of the quantity are: a for a and b
for b
a + b a + b
For the ratio a:b:c, the resulting fraction of
quantities are:
For a a ,for b, b and c
for c.
a + b + c a + b + c a + b
+ c

EXAMPLE
Twenty apples were shared between peace and
Rachael in the ratio 2:3.find out how many apples
each got?
Solution
Expressing each part of the ratio as a fraction of
the total, peace got 2 and Rachael got 3 of the
apples.
5 5
Therefore, Peace got 2 x 20 = 8 apples
5
Then Rachael got 3 x 20 = 12 apples

ASSESSMENT
1. The angles of quadrilateral are in the ratio 6:3:2:4.calculate
the size of the angles.
2. A mixture of cement, sand and ballast is the ratio 1:4:5.find
the mass of each component in kilograms in a mixture of 1
tonne
3. A father shared 20 hectares of his land to his two children in
the ratio 2:3. find the number of hectares each child got.
4. A grade 8 learner was given shs. 240, divide the amount in
the following ratio
(a). 2: 3 (b). 3:5 (c). 4.1
5. An alloy is made of copper and aluminum in the ratio 3: 8. if
the mass of the alloy is 2.2kg. Find the mass of each metal

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

INCREASE AND DECREASE OF
RATIO IN REAL LIFE SITUATIONS
Learning Point
To increase or decrease quantity
of given ratio, we express the ratio
as a fraction and multiply it with
the quantity

INCREASE IN RATIO
EXAMPLE
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

ASSESSMENT
1. The number of learners in a class increase in the ratio 5:3. if
the original number was 30. find the new number of learners.
2. Mutwiri has a mass of 60 kg, his mass increased in the ratio
4:3. What was his new mass?
3. A farmer had 300 goats on his farm, he increase the number
of goats in the ratio 5:4. Calculate the new number of goats
in the farm.
4. Achieng’ drove a car at a speed 80km/h. on her return
journey she increase the speed in the ratio 3: 2. Calculate her
speed on return journey.
5. Fatuma harvested 120bags of maize in the year 2021, the
following year the number of harvest increase in the ratio

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

ASSESSMENT
1. The population of a certain town decreased in the ratio 7:8
because of corona pandemic. If the population was 600 000
before. Find the population now.
2. A family spending kshs. 18,000 per month. If the monthly
spending decreased in the ratio 4:5. Calculate the new
monthly spending.
3. The milk production in a dairy farm was 360litres a day. After
one month the production decreased in the ratio 5:9. Find
the new dairy production per day.
4. Decrease each of the following quantities using the ratio
given in the brackets;
(a). 100 m (3:2) (b). 240 litre (4:3), (c). Ksh. 10 000 (5:8)

PERCENTAGES IN RATIO
Learning point
Increase or decrease in quantities can be
expressed as a percentage of the original
quantities.
Percentage increase = Increase x 100
Original
Percentage decrease = Decrease x 100
Original

EXAMPLE
A box contains 25 pens. If five more pens are added into the
box, find the percentage increase in the number of pens in the
box.
Solution
Percentage increase = number of pens added x 100%
initial number of pens
= 5 x 100%
25
= 20%

ASSESSMENT
1. The ratio of boys to girls in a class is 3:5, what percentage of
the class are boys?
2. The ratio of apples to oranges in a basket is 4:7, what
percentage of the fruits are oranges?
3. In a bag of candies, the ratio of red to green candies is 2:5.
What percentage of the candies are green?
4. The ratio of students who passed an exam to those who
failed is 7:3. What percentage of the students passed the
exam?
5. The price of a product increases by 20% and the original
price was kshs 50, what is the new price?
6. The length of a rectangle is increased by 25%, and the

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.

EXAMPLE
The table below shows the cost of various number
of cups.
a) Find the ratio of the number of cups to the cost for each pair.
b) Share whether the number of cups is directly proportional to the cost.
Solution
(a). 1:60 4: 240 = 1: 60
2:120 = 1: 60 5:300 = 1: 60
3:180 = 1: 60
Thus the number of cups is proportional to the cost
Number of
cups
1 2 3 4 5
Cost (ksh.) 60 120 180 240 30
0

ASSESSMENT
1. The following tables shows amount of chicken mash (kg)consumed in
a day by a number of chicken:
Find the ratio of any two qualities in the first row.
2. Consider the following table of quantities x and y
(a)Find the ratio of y to x in each pair of given quantities
(b)What do you notice
(c)Fill in the blanks
Number of chicks 12 24 36
Amount of chicken mash(kg) 1 2 3
x 4 9 12 24 30 -
y 12 27 36 _ _ 120

3. Six mangoes cost Ksh 300.Make a
table showing the of mangoes (from 1 to
10)and their cost. State whether the
number of mangoes in directly
proportional to the cost.
4. In kilombere junior ,the ratio of
learning to text boom is 2:1,find the
number of textbooks needed for 60
learners in the class.
5. If 80l of diesel cost Ksh 8000.Find the
cost of 50l of diesel from the same petrol

2. 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.

EXAMPLE
Ten men take 12 days to complete a task. How long
will 8 men take to complete the task?
Solution
Number of men decrease in the ratio 8:10
Therefore, the number of days taken increases in
the ratio 10:8
The number of days taken
=12 x 10/8
=15 days

ASSESSMENT
1. Three tractors can plough a field in 6 days. Find the number
of days it will take 9 similar tractors to plough the field.
2. A school bus travelling at a speed of 80km/h takes 6 hours
to reach its destination. Find out the time the bus travelling
at a speed of 60km/h takes to reach the same destination.
3. 4 taps twelve minute to fill a tank. Find the time it will take
to fill the same tank.
4. Two farmers working in the same rate can take 8 days to
complete a task. Find out how long 4 farmers working at the
same rate will take to complete the same task.
5. Eight constructors can build a wall in 12 days. Find how long
it will take six constructors working on the same rate to

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