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FIRST TERM
SUBJECT: MATHEMATICS CLASS: JSS3
SCHEME OF WORK
WEEKS TOPIC
1 | Whole numbers: Binary number system, translation of word problems |
2 | Whole numbers: Expressions involving brackets, direct and inverse proportion, application, compound interest. |
3 | Addition and subtraction of numbers in base two |
4 | Multiplication and division of numbers in base two |
5 | Rational and non – rational numbers, Variations |
6 | Factorization |
7 | Simple equations involving fractions |
8 | Change of subject of formulae |
9 | Measure of central tendency |
10 | Application of measures of central tendency |
REFERENCE BOOKS
New General Mathematics by M. F Macrae et al bk 3
Essential Maths by AJS OluwasanmiBk 3
WEEK ONE
NUMBER BASE CONVERSIONS
People count in twos, fives, twenties etc. Also the days of the week can be counted in 24 hours. Generally people count in tens. The digits 0,1,2,3,4,5,6,7,8,9 are used to represent numbers. The place value of the digits is shown in the number example: 395:- 3 Hundreds, 9 Tens and 5 Units. i.e. 3X102 + 9 X 101 +5 X 100.
Since the above number is based on the powers of tens it is called the base ten number system i.e. 300 + 90 + 5
Also 4075 = 4 Thousand 0 Hundred 7 Tens 5 Units i.e. 4 x 103 + 0 X 102 + 7 X 101 + 5 X 100 Other Number systems are sometimes used.
For Example: The base 8 system is based on the power of 8. For example: Expand 6478, 265237, 1011012,
(a) 6478 = 6 x 82 + 4 x 81 + 7 X 80 =6 x 64 + 4 x 8 + 7 x 1
(b) 265237 =2 x 74 + 6 x73 + 5 x 72 + 2x 71 + 3 x 70
(c) 1011012= 1 x 25 + 0 x 24 + 1 x 23 + 1 x 22 + 0 x21 + 1 x 20
EVALUATION
Expand The Following
1. 4343
2. 1011112
CONVERSION TO DENARY SCALE (BASE TEN)
When converting from other bases to base ten the number must be raised to the base and added.
Worked Examples:
Convert the following to base 10
(a) 178
(b) 110112
Solutions:
(a) 178 = 1 X 81 + 7 X 80 = 1 X 8 + 7 X 1 = 8 + 7 = 15
(b) 110112 = 1 X 24 + 1 X 23 + 0 X 22 + 1 X 21 + 1 X 20 = 1 X 16 + 1 X 8 + 0 X 4 + 1 X 2 +
1 X 1= 16 + 8 + 0 + 2 + 1 = 27
EVALUATION
Convert The Following To Base Ten:
(a) 101002
(b) 21203
CONVERSION FROM BASE TEN TO OTHER BASES
To change a number from base ten to another base
1. Divide the base ten number by the new base number.
2. Continue dividing until zero is reached
3. Write down the remainder each time
4. Start at the last remainder and read upwards to get the answer.
Worked Examples:
1. Convert 6810 to base 6
2. Covert 12910 to base 2
Solutions:
1. 6 68
6 11 R 2
6 1 R 5
0 R 1
= 1526
2. 2 129
2 64 R 1
2 32 R 0
2 16 R 0
2 8 R 0
2 4 R 0
2 2 R 0
2 1 R 0
2 0 R 1
= 100000012
EVALUATION
1. Convert 56910 to base 8
2. Convert 10010 to base 2
GENERAL EVALUATION
Convert the following to base seven
1. 405ten
2. 876ten
Evaluate the following
3. 5 – 3 + 4
READING ASSIGNMENT
New Gen Math Book 3 pg 15-17
Essential Mathematics for J.S.S.3 Pg 5 -9
WEEKEND ASSIGNMENT
1. Express 3426 as number in base 10 (a) 134 (b) 341 (c) 143
2. Change the number 100102 to base 10 (a) 1001 (b) 40 (c) 18
3. Express in base 2, 10010 (a) 100100 (b) 1100100 (c) 11001
4. Convert 120 base 10 to base 3 (a) 111103 (b) 12103 (c) 121103
5. Convert 25 base 10 to base 2 (a) 110012 (b) 10012 (c) 11002
THEORY
1. Convert 12648 to base 10
2. Convert 10510 to base 2
WEEK TWO
SOLVING EQUATION EXPRESSIONS
WORD PROBLEMS
Worked Examples:
- Find 1/4 of the positive difference between 29 & 11
- The product of a certain number and 5 is equal to twice the number subtracted from 20. Find the number
- The sum of 35 and a certain number is divided by 4 the result is equal to double the number. Find the number.
Solutions:
1. Positive Difference 29 – 11 = 18
1/4 of 18 = 4 2/5
2. Let the number be x
x X 5 = 20 – 2x
5x = 20 – 2x
5x + 2x = 20
7x = 20
x = 20/7 = 2
3. Let the number be n
sum of 35 and n = n + 35
divided by 4 = n + 35
4
result = 2 X n
therefore n + 35 = 2n
4
n + 35 = 8n
8n – n = 35
7n = 35
n = 35/7 = 5
EVALUATION
1. From 50 subtract the sum of 3 & 5 then divide the result by 6
2. The sum of 8 and a certain number is equal to the product of the number and 3 find
the number.
SOLVING EQUATION EXPRESSIONS WITH FRACTION
Always clear fractions before beginning to solve an equation.
To clear fractions, multiply each term in the equation by the LCM of the denominations of the fractions.
Examples:
Solve the following
1. x = 2
9
2. x + 9 + 2 + x = 0
5 2
3. 2x = 5x + 1 + 3x – 5
7 2
Solutions:
1. x = 2
9
Cross multiply
x = 18
2. x + 9 + 2 + x = 0
5 2
Multiply by the LCM (10)
10 X (x + 9) + 10 X (2 + x) = 0 X 10
5 2
2 (x + 9) + 5 (2 + x) = 0
2x + 18 + 10 + 5x = 0
2x + 5x + 28 = 0
7x = -28
x = -28/7 = -4
3. 2x = 5x + 1 + 3x – 5
7 2
Multiply by the LCM (14)
14 X 2x = 14 (5x + 1) + 14 (3x – 5)
7 2
28x = 2 (5x + 1) + 7 (3x – 5)
28x = 10x + 2 + 21x – 35
28x = 31x – 33
28x – 31x = -33
-3x = -33
x = 33/3 = 11
EVALUATION
Solve the following equations.
1. 7/3c = 21/2
2. 6 = 11
y + 3 y – 2
3. 3 – 4 = 0
2b – 5 b – 3
Furthermore, we can consider the word equations or expressions into:
- Sum & Differences
- Products
- Expressions with fractions & equations
SUM & DIFFERENCES
The sum of a set of numbers is a result obtained when the numbers are added together. The difference between two numbers is a result of subtracting one number from the other.
Worked Examples:
1. Find the sum of -2 & -3.4
2. Find the positive difference between 19 & 8
3. The difference between two numbers is 7. If the smaller number is 7 find the other.
4. The difference between -3 and a number is 8, find the two possible values for the number.
5. Find the three consecutive numbers whose sum is 63.
Solutions:
1. -2 + -3.4 = -5.4
2. 19 – 8 = +11
3. let the number be Y i.e. Y -7 = 7
i.e Y = 7 + 7 = 14
4. Let M represent the number
M – (-3) = 8
m + 3 = 8
m = 8 – 3
m = +5
also -3 – m = 8
-m = 8 + 3
-m = 11
m = -11
the possible values are +5 & -11
5. Consecutive numbers are 1,2,3,4,5,6,………….. Consecutive odd numbers are
1,3,5,7,9……….. consecutive even numbers are 2, 4, 6, 8,10……….
Representing in terms of X, we have 2X, 2X + 2, 2X + 4, 2X + 6, 2X + 8, 2X + 10…………
for consecutive even numbers, we have X, X + 2, X + 4, X + 6…….
for consecutive odd numbers, we have X + 1, X + 2, X + 3, X + 4…
for consecutive numbers.
let the first number be x,
let the second number be x + 1
let the third number be x + 2
Therefore x + x + 1 + x + 2 = 63
3x + 3 = 63
3x = 63 – 3
3x = 60
x = 60 /3
= 20
The numbers are 20, 21, and 22.
EVALUATION
1. Find the sum of all odd numbers between 10 and 20
2. The sum of four consecutive odd numbers is 80 find the numbers
3. The difference between 2 numbers is 9, the largest number is 32 find the numbers.
PRODUCTS
The product of two or more numbers is the result obtained when the numbers are multiplied together.
Worked Examples:
1. Find the product of – 6, 0.7, &
2. The product of two numbers is 8 .If one of the numbers is 1/4 find the other.
3. Find the product of the sum of -2 & 9 and the difference between -8 & -5.
Solutions:
1. Products -6 x 0.7 x
-6 x 7/10 x 20/3 = -6 x 7 x 20
10 x 3
= -2 x 7 x 2 = -28
2. Let the number be x
X x = 8
multiply both sides by 4
x = 8 x 4 = 33
3. Sum = -2 + 9 = 7
Difference = -5-(-8) = -5 + 8 = 3
Products= 7 x 3 = 21
EVALUATION
- The product of three numbers is 0.084 if two numbers are 0.7 & 0.2 find the third number.
- Find the product of the difference between 2 & 7 and the sum of 2 & 7.
- From 50 subtract the sum of 3 & 5 then divide the result by 6.
- The sum of 8 and a certain number is equal to the product of the number and 3 find the number.
Reading Assignment
New Gen Maths for J.S.S 3 Pg 20- 24
Essential Mathematics for J.S.S 3 Pg 85-87
PROPORTION
Proportion can be solved either by unitary method or inverse method. When solving by unitary method, always
- Write in sentence the quantity to be found at the end.
- Decide whether the problem is either an example of direct or inverse method
- Find the rate for one unit before answering the problem.
Examples
- A worker gets N 900 for 10 days of work, find the amount for (a) 3 days (b) 24 days (c) x days
Solution
For 1 day =N 900
1 day = 900/10 = N90
a. For 3 days =3 x 90 = 270
b. For 24 days = 24×90 = N 2,160
c. For x days =X x 90 = N 90 x
INVERSE PROPORTION
Example
- Seven workers dig a piece of ground in 10 days. How long will five workers take?
Solution:
For 7 workers =10 days
For 1 worker =7×10=70 days
For 5 workers=70/5 =14 days
- 5 people took 8 days to plant 1,200 trees, How long will it take 10 people to plant the same number of trees
Solution:
For 5 people =8 days
For 1 person =8×5=40 days
For 10 people =40/10 =4 days
CLASS WORK
- A woman is paid N 750 for 5 days, Find her pay for (a) 1 day (b) 22 days
- A piece of land has enough grass to feed 15 cows for x days. How long will it last (a) 1 cow (b) y cows
- A bag of rice feeds 15 students for 7 days .How long would the same bag feed 10 students
Note on direct proportion: this is an example of direct proportion .The less time worked (3 days) the less money paid (#270) the more time worked (24 days) the more money paid (N 2,160)
COMPOUND INTEREST
Interest is a payment given for saving or borrowing money. It can either be simple interest or compound interest. It is simple interest when the interest is calculated on the principal while it is compound interest if interest is calculated on the amount at the end of each period. Amount is the sum of the principal and the interest.
Example:
Find the amount on borrowed for at simple interest.
Solution:
and so that . Substituting the values, we will obtain
Example:
Find the amount that becomes if saved for at per annum simple interest.
Solution:
1st year Principal
interest
2nd year Principal
interest
3rd year Principal
interest
AMOUNT
Alternatively, we can also solve the question with the use of the formula where represent the time or duration.
Then, substituting into the formula, we can have
()
EVALUATION
- What is simple interest?
- Define compound interest.
- Calculate of
WEEKEND ASSIGNMENT
- Esther is 3 times as old as her sister Tolu, if the sum of their ages is 20 years. Find the difference between their ages.(a) 20 years (b) 8 years (C) 10 years
2. 9 was subtracted from a certain number and the result was divided by 4 if the finalanswer is 5 what was the original number? (a) 29 (b) 18 (c) 20
3. A woman is 4 times as old as her son. In five years time she will be 3 times as old as her son. How old is the woman (a) 50 yrs (b) 40 yrs (c) 45 yrs
4. Bayo is 4 times as old as his sister Tolu. If the sum of their ages is 20 years, find the difference between their ages. (a) 12 yrs (b) 15 yrs (c) 18 yrs
5. Subtract the square root of 4 from the square of 4 and divide the result by 2
(a) 2 (b) 4 (c) 7
THEORY
1. Divide 36 by the difference between the product of 3 & 6 and the square root of 36.
2. When I add 45 to a certain number, and divide the sum by 2, the result is the same as five times the number, what is the number?
WEEK THREE
BINARY NUMBERS (BASE 2 NUMBERS)
- Addition in base 2
- Subtraction in base 2
- Multiplication & Division in base 2
ADDITION IN BASE TWO
We can add binary numbers in the same way as we separate with ordinary base 10 numbers.
The identities to remember are:-
0 + 0 = 0, 0 + 1 = 1, 1 + 0 = 1, 1 + 1 = 10, 1 + 1 + 1 = 11, 1 + 1 + 1 + 1 = 100
Worked Examples
Simplify the following
1. 1110 + 1001 2. 1111 + 1101 + 101
Solutions:
1. 1110
+ 1001
10111
2. 1111
+ 1101
101
100001
Note: 11 take 1 carry 1
10 take 0 carry1
100 take 0 carry 10
EVALUATION
1. Simplify the following 101 + 101 +111
2. 10101 + 111
ADDITION IN BICIMALS
In bicimals, the binary point are placed underneath each other exactly the same way like ordinary decimals.
Example:
1. 1.1011two + 10.1001two + 10.01
2. 10.001two+ 101.111
Solution:
1. 1 . 1011
10 . 1001
10 . 0100
110. 1000two
2. 101.111
10.001
1000.000
SUBTRACTION IN BASE TWO
The identities to remember on subtraction are: 0 – 0 = 0, 1 – 0 = 1, 10 – 1 = 1, 11 – 1 = 10, 100 – 1 = 11
Worked Examples
Simplify the following:-
(a) 1110 – 1001 (b) 101010 – 111
Solutions:
(a) 1110
– 1001
101
SUBTRACTION IN BICIMAL
Example
101.101two – 11.011two
101.101
11.011
10.010two
EVALUATION
1. 10111÷110
2. 10001 x 11
READING ASSIGNMENT
New Gen Maths Book 3, chapter 1 Exercise 1e pg 18 Nos 1-12
Essential Mathematics for J.S.S.3 Pg 8-10
WEEKEND ASSIGNMENT
1. Express 3426 as a number in base 10. (a) 342 (b) 3420 (c) 134
2. Change the number 10010 to base 10 (a) 18 (b) 34 (c) 40
3. Express in base two the square of 11 (a) 1001 (b) 1010 (c) 1011
4. Find the value of (101)2 in base two (a) 1010 (b) 1111 (c) 1001
5. Multiply 1000012 by 11 (a) 1001 (b) 1100011 (c) 10111
THEORY
1. Calculate 1102 x (10112 + 10012 – 1012)
2. Convert 110111 to base five
WEEK FOUR
MULTIPLICATION AND DIVISION IN BASE TWO
In multiplication, 0 x 0 = 0, 1 x 0 = 0, 1 x 1 = 1.
When there is long multiplication of binary numbers, the principle of addition can be used to derive the answer. Under division, the principle of subtraction can be used.
Worked Examples:
1. 1110 x 111
Solution:
1. 1110
X 110
0000
1110
1110
1010100
2. Multiply 11.01 by 1.1 in base two
Solution:
1101
x 11
1101
1101
100.111
DIVISION IN BASE TWO
Example:
Divide 110010101 by 1111
Solution:
1100101012 ÷ 11112 = 110112
Example:
Find the square root of 10100012
Solution:
First convert it to base 10
10100012 = 1 x 2 6+ 0 + 1 x 2 4+ 0 + 0 + 0 + 1 x 20
= 64 + 16 + 1 = 81ten
The square root of 81 = 9ten
2 9 1
2 4 0
2 2 0
2 1 1
0 = 10012
EVALUATION
1. 10111÷110
2. 10001 x 11
READING ASSIGNMENT
New Gen Maths Book 3, chapter 1 Exercise 1e pg 18 Nos 1-12
Essential Mathematics for J.S.S.3 Pg 8-10
WEEKEND ASSIGNMENT
1. Express 3426 as a number in base 10. (a) 342 (b) 3420 (c) 134
2. Change the number 10010 to base 10 (a) 18 (b) 34 (c) 40
3. Express in base two the square of 11 (a) 1001 (b) 1010 (c) 1011
4. Find the value of (101)2 in base two (a) 1010 (b) 1111 (c) 1001
5. Multiply 1000012 by 11 (a) 1001 (b) 1100011 (c) 10111
THEORY
1. Calculate 1102 x (10112 + 10012 – 1012)
2. Convert 110111 to base
WEEK FIVE
RATIONAL AND NON-RATIONAL NUMBERS AND COMPOUND INTEREST
RATIONAL AND NON-RATIONAL NUMBERS
Numbers which can be written as exact fractions or ratios in the form are called rational numbers. For example, we can write these numbers as
In addition, rational numbers are also numbers that can be written as recurring decimals, for instance: is equivalent respectively to the following:
WeNumbers which cannot be written as exact fractions or recurring decimals are called non-rational numbers. Examples of non-rational numbers are
SQUARE ROOTS
Since rational numbers are not perfect squares, so their square roots cannot be obtained easily except by trial and error method or by the use of Table of Square Roots in the four-figure table.
Example 1:
Find to three significant figures by the use of tables.
Solution:
gives from the table. Hence, answer is to .
Example2:
Find to the nearest tenth by the use of tables.
Solution:
is equivalent to . This is equal to. We can now look up from the table to give. So that.
Hence, answer is to the nearest tenth.
EVALUATION
- Which of the following is an irrational number?
- Which of the following is a rational number?
- Find the square root of, leaving your answer in one decimal place.
DIRECT AND INVERSE VARIATION
DIRECT VARIATION
This is used to describe quantities which vary in proportions to each other, such that as one increases the other increases, and as one decreases the other decreases. Thus, if P varies directly as R, then the expression symbolically becomes. The expression can now be written in equation form as
Where has been replaced by is a constant of variation. It can also be expressed as
The equation is the equation of variation.
Example 1:
If varies directly as the square of, find the law of variation between given that when Find the value of when and the value of
Solution:
and the law of variation becomes
For substitution gives .
Then .
For substitution gives
such that
then
GRAPHICAL REPRESENTATION OF DIRECT VARIATION
Data collected from quantities that vary directly can be represented graphically. This will give a straight line graph through the origin as shown below.
Example 2:
Given that distance varies directly with time, consider the table below and plot a graph for such relationship.
Distance | 5 | 10 | 15 | 20 | 25 |
Time | 1 | 2 | 3 | 4 | 5 |
Solution:
EVALUATION
- varies directly as and when Find when
- If increases by from question find the percentage change in .
INVERSE VARIATION
This variation means that related quantities vary inversely or as reciprocal to each other. Hence as one increases the other decreases; and as one decreases, the other increases. Thus if varies inversely as , symbolically this is written as .The expression can now be written in equation form as.
Where has been replaced by is a constant of variation. It can also be expressed as
The equation is the equation of variation.
Example 3:
Given that is inversely proportional to, and that, find the (a) relationship between and (b) value of when
Solution:
such that and
(a) is the required relationship between and (b)
EVALUATION
- Find the percentage change in the current from question if the resistance is decreased by
GRAPHICAL REPRESENTATION OF INVERSE VARIATION
The graph here will not be a straight line from the origin instead it will give us a curve.
Speed | 80 | 40 | 20 | 10 | 5 |
Time | 0.5 | 1 | 2 | 4 | 8 |
Example 4: Given that speed varies inversely to time use the below table to plot a graph of an inverse relationship between
Solution:
GENERAL EVALUATION
- Factorize the expression
- Factorize
- What is the value of the digit in the?
- What is the highest common factor of and ?
- Simplify
READING ASSIGNMENT
Essential Mathematics for J.S.S. 3 by Oluwasanmi A.J.S. 2014 edition; Pages
Essential Mathematics Workbook for J.S.S. 3 by Oluwasanmi A.J.S.; Exercise 7.1, numbers
WEEKEND ASSIGNMENT
- If and when find the value ofwhen. A. B. C.
- and when find the value of when . A. B. C.
- If varies directly as and what is value of when ?
A. B. C.
- and when Find the relationship between m and n.
A. B. C. D. m =
- Find the value of m when. A. 6B. C.
THEORY
- and when find (a) when (b) the percentage change in if increases by .
JOINT AND PARTIAL VARIATION
JOINT VARIATION
Joint variation is obtained when a quantity varies with more than one other quantity either directly and/or inversely. For instance, is jointly proportional to both and as in. Also, is directly proportional to and inversely proportional to as in.
Example 1:
If .When
- Find the relation between
- Find when
Solution:
- and
After substituting, we have
The relation between them is given by
Example 2:
The universal gas law states that the volume of a given mass of an ideal gas varies directly with its absolute temperature and inversely with its pressure A certain mass of gas at an absolute temperature and pressure has a volume. Find:
- the formula that connects .
- the pressure of the gas when its absolute temperature is and its volume is 0.018m3
Solution:
- and , such that
Substituting the values, becomes
and the relationship is
EVALUATION
- Suppose . When Find
- Find the percentage change in when increases by and decreases by
PARTIAL VARIATION
Partial variation problems occur everywhere around us. Some examples are described below:
- When a hairdresser makes hair, the money he/she charges M, is dependent on both the cost of the wool (thread or weavon in some cases) C which is constant, and on the time T, taken to make the hair. The less the weaves, the less the time it will take to complete and the less the charges. We can write a partial equation for this as: , where are constants.
- Domestic electricity prepaid meter bills are prepared on two components which are rental charge (independent of the amount of power consumed) and consumption charges (dependent on the quantity of power consumed). We can also write the total bill T in partial equation as: , where are constants depending on the customer.
Thus, partial variation statements can come in these formats described below:
- is partly constant and partly varies as is interpreted as
- varies partly as and partly inversely as can also be interpreted as
In these cases, and are constants that can be obtained simultaneously.
Example 3:
is partly constant and partly varies as the square of Write an equation connecting and Given that when and when Write down the law of variation. Find when
Solution:
The equation connecting and is , where and are constants.
When, becomes
When , we have becomes
Combining the two equations and solving simultaneously,
Subtracting: and
Substitute for into , so that
And . The law of variation becomes
When becomes
Example 4:
varies as partly as and partly as the cube of . When and when . Write the law connecting and . Find when
Solution:
The equation connecting and is , where and are constants.
when , becomes
when , becomes
Combining the two equations and solving simultaneously to eliminate
Subtracting: and
Alternatively, dividing through by , gives and dividing through by , gives .
Then,
Subtracting:
And as obtained above.
Substitute for into , so that and
So that . The law of variation becomes
When becomes
Example 5:
The cost in naira of making a dress is partly constant and partly varies with the amount of time in hours it takes to make the dress. If the dress takes 3 hours to make, it costs N2700, and if it takes 5 hours to make the dress, it costs N3100. Find the cost if it takes hours to make the dress.
Solution:
Using and to represent the cost and time respectively, we can proceed by writing
From first statement:
From second statement:
Solving the simultaneously,
Subtracting: and
Substitute for into , so that
and.
So that . The law of variation becomes
If it takes to make the dress, the cost becomes
EVALUATION
- varies partly directly with and partly varies inversely with y. When and when. Find when
- An examination fee is partly constant and partly varies with the number of subjects entered. When the examination fee is three subjects are entered. When the fee is , five subjects are entered. Find the number of subjects entered if the fee is
GENERAL EVALUATION
- Express in tonnes.
- Express in fraction in its lowest term.
- What is the sum of and kobo expressed in kobo?
- Factorize
- A trader gives discount on an article in his kiosk marked . How much would a customer pay on such article?
READING ASSIGNMENT
Essential Mathematics for J.S.S. 3 by Oluwasanmi A.J.S. 2014 edition; Pages
Essential Mathematics Workbook for J.S.S. 3 by Oluwasanmi A.J.S.; Exercise, numbers
WEEKEND ASSIGNMENT
- and Find when . A. B. C. D. 5
- In the question above. A. increases by B. decreases by C. increases by D. X decreases by 25%
- is partly constant and partly varies with y. This is statement can be represented as
A. B. C. D. P = a + y
- If and then A. B. C.
- How many constants do we have in partial variation? A. B. C.
THEORY
- If and when calculate
(a)when (b) when
- The charge for a pair of shoe is partly constant and partly varies as the number of pair of shoes. If pairs cost and the cost of pairs is Find the charges for pairs.
GENERAL EVALUATION
- Make L the subject of the formula
- Change 84 in base ten to a number in base two.
- Represent on the number line.
- Convert in a decimal number to an octal number.
- A car travelled in minutes. What is the speed of the car in ?
READING ASSIGNMENT
Essential Mathematics for J.S.S. 3 by Oluwasanmi A.J.S. 2014 edition; Pages
Essential Mathematics Workbook for J.S.S. 3 by Oluwasanmi A.J.S.; Exer., numbersExercise, numbers.
WEEKEND ASSIGNMENT
- The simple interest on for years at is _________.
A. B. C. D. #3,000
- The amount on the above question is A. B. C.
- can also be written as A. B. C. D. 0.405
- The square root of is A. B. C. D. 2695
- Which of the following is a rational number?
THEORY
- What is the compound interest on borrowed for years at rate?
- Find the square root of , leaving your answer in one decimal place.
WEEK SIX
FACTORISATION
CONTENT
- Factorization of simple expression
- Difference of two squares
- Factorization of quadratic expression
FACTORISATION OF SIMPLE EXPRESSION
To factorize an expression completely, take the HCF outside the bracket and then divide each term with the HCF.
Example:
Factorize the following completely.
1. 8xy + 4x2y
2. 6ab – 8a2b + 12ab
Solution:
1. 8xy + 4x2y
8xy = 2 X 2 X 2 X x X y
4x2y = 2 X 2 X xXxX y
HCF = 4xy
8xy + 4x2y = 4xy ( + )
= 4xy ( 2 + x)
2. 9a2bc3 – 12ab2c2
9a2bc3 = 3 X 3 X a X a X c X c X c
12ab2c2 = 2 X 2 X 3 X b X b X c X c
HCF = 3abc2
= 3abc2 (3ac – 4b)
EVALUATION
factorize the following expression
1. 9x2yz2 – 12x3z3
2. 14cd + 35cd2f
3. 20m2n – 15mn2
FACTORISATION BY GROUPING
To factorize an expression containing four terms, you need to group the terms into pairs. Then factorize each pair of terns.
Example:
factorize ab – 2cb + 2cf – af
Solution:
Group ab and af together and 2cb and 2cf together
i.e. ab – 2cb + 2cf –af = ab – af – 2cb + 2cf
= a( b – f ) -2c( b – f )
= (a – 2c)( b – f)
EVALUATION
factorize these expressions;
- 16uv – 12vt + 20mu – 15mt
- ap +aq +bq + bp
- mn – pq-pn +mq
FACTORISATION OF QUADRATIC EXPRESSIONS
A quadratic expression has two (2) as its highest power hence this at times is called a polynomial of the second order. The general representation of quadratic expression is ax 2 + bx + c where a ≠ 0. From above expression, a, b, and c stands for a number.
NOTE
- if ax 2 +bx + c= 0, this is known as quadratic equation
- a is coefficient of x2, b is coefficient of x and c is a constant term.
- When an expression contains three terms, it is known as trinomial.
- To be able to factorize trinomial, we need to convert it to contain four terms.
Examples: factorization of trinomial of the form x2 +bx + c.
- Factorise x2 +7x +6
Steps:
- Multiply the 1st and the last term (3rd term) of the expression.
- Find two factors of the above multiple such that if added gives the second term (middle) and when multiplied gives the result in step 1.
- Replace the middle term with these two numbers and factorize by grouping.
Solution to example:
X2 x 6 = 6x2
Factors: 6 and 1
X2 + 6x + x + 6
X(x+6) +1(x+6)
(x+6)(x+1)
EVALUATION
- z2 – 2z + 1
- x2 +10x – 24
FACTORISATION OF QUADRATIC EQUATIONS OF THE FORM ax 2 +bx +c
Example: 5x 2 -9x +4
Solution:
Product: 5x 2 x 4 = 20x 2
Factors: -5 and -4
Sum: -5-4 = -9
Hence, 5x 2 – 9x + 4
5x2 -5x -4x +4
5x(x-1)-4(x-1)
(5x-4)(x-1)
EVALUATION
- 2x 2 +13x +6
- 13d 2 – 11d – 2
FACTORISATION OF TWO SQUARES
To factorise two squares with difference, we need to remember the law guiding difference of two squares i.e. x 2 – y2 = (x + y) (x- y).
Examples:
- P 2 – Q2 = (P+Q) (P-Q)
- 36y 2 – 1= 6 2 y 2 – 1 2
= (6y)2 – 1 2 = ( 6y+1) (6y-1).
EVALUATION
- 121- y 2
- x2y2 – 42
READING ASSIGNMENT
Essential Mathematics for J.S.S.3 Pg29-36
Exam focus for J.S.S CE Pg101-105-
WEEKEND ASSIGNMENT
- The coefficient of x 2 in x 2 + 3x -5 is A. 3 B. 1 C. -5 D. 2
- Simplify e 2 – f 2 A. (e+f)(e-f) B. (e+f)(f+e) C. (e-f)(f-e) D. e+f
- Factorize x 2 +x -6 A. (x+3)(x+2) B. (x-2)(x+3) C. (x+1)(x+5) D. x + 2
- Solve by grouping 5h 2 -20h + h – 4 A. (h-4)(5h+1) B. (h+4)(5h-1) C. (h+2)(h-5) D. h – 4
- 49m 2 – 64n 2 when factorized will be A. (7m+8n)(8m+7n) B. (8m-7n)(8m+7n)
C. (7m-8n)(7m+8n) D. 7m – 8n
THEORY
Factorise the following
- 4p2 – 12p +9q2
- f 2 – 2f + 1
FACTORISATION OF QUADRATIC EXPRESSIONS
A quadratic expression has two (2) as its highest power hence this at times is called a polynomial of the second order. The general representation of quadratic expression is ax 2 + bx + c where a ≠ 0. From above expression, a, b, and c stands for a number.
NOTE
- if ax 2 +bx + c= 0, this is known as quadratic equation
- a is coefficient of x2, b is coefficient of x and c is a constant term.
- When an expression contains three terms, it is known as trinomial.
- To be able to factorize trinomial, we need to convert it to contain four terms.
Examples: factorization of trinomial of the form x2 +bx + c.
- Factorise x2 +7x +6
Steps:
- Multiply the 1st and the last term (3rd term) of the expression.
- Find two factors of the above multiple such that if added gives the second term (middle) and when multiplied gives the result in step 1.
Solution to example:
X2 x 6 = 6x2
Factors: 6 and 1
X2 + 6x + x + 6
X(x+6) +1(x+6)
(x+6)(x+1)
EVALUATION
- z2 – 2z + 1
- x2 +10x – 24
Factorization of quadratic equations of the form ax 2 +bx +c
Example: 5x 2 -9x +4
Solution:
Product: 5x 2 x 4 = 20x 2
Factors: -5 and -4
Sum: -5-4 = -9
Hence, 5x 2 – 9x + 4
5x2 – 5x -4x +4
5x(x-1)-4(x-1)
(5x-4)(x-1)
EVALUATION
- 2x 2 +13x +6
- 13d 2 – 11d – 2
FACTORISATION OF TWO SQUARES
To factorize two squares with difference, we need to remember the law guiding difference of two squares i.e. x 2 – y2 = (x + y) (x- y).
Examples:
- P 2 – Q2 = (P+Q) (P-Q)
- 36y2 – 1= 6 2 y 2 – 1 2= (6y) 2 – 1 2 = ( 6y+1) (6y-1).
EVALUATION
- 121- y 2
- x2y2 – 42
READING ASSIGNMENT
Essential Mathematics for J.S.S.3 Pg29-36
Exam focus for J.S.S CE Pg101-105-
WEEKEND ASSIGNMENT
- The coefficient of x 2 in x 2 + 3x -5 is (a) 3 (b) 1 (c) -5
- Simplify e 2 – f 2 (a) (e+f)(e-f) (b) (e+f)(f+e) (c) (e-f)(f-e)
- Factorize x 2 +x -6 (a) (x+3)(x+2) (b) (x-2)(x+3) (c) (x+1)(x+5)
- Solve by grouping 5h 2 -20h + h – 4 (a) (h-4)(5h+1) (b) (h+4)(5h-1) (c) (h+2)(h-5)
- 49m 2 – 64n 2 when factorised will be (a) (7m+8n)(8m+7n) (b) (8m-7n)(8m+7n)
(c) (7m-8n)(7m+8n)
THEORY
Factorise the following
- 4p2 – 12p +9q2
- f 2 – 2f + 1
FACTORISATION OF QUADRATIC EXPRESSIONS
A quadratic expression has two (2) as its highest power hence this at times is called a polynomial of the second order. The general representation of quadratic expression is ax 2 + bx + c where a ≠ 0. From above expression, a, b, and c stands for a number.
NB:
- if ax 2 +bx + c= 0, this is known as quadratic equation
- a is coefficient of x2, b is coefficient of x and c is a constant term.
- When an expression contains three terms, it is known as trinomial.
- To be able to factorize trinomial, we need to convert it to contain four terms.
Examples: factorization of trinomial of the form x2 +bx + c.
- Factorise x2 +7x +6
Steps:
- Multiply the 1st and the last term (3rd term) of the expression.
- Find two factors of the above multiple such that if added gives the second term (middle) and when multiplied gives the result in step 1.
- Replace the middle term with these two numbers and factorize by grouping.
Solution to example:
X2 x 6 = 6x2
Factors: 6 and 1
X2 + 6x + x + 6
X(x+6) +1(x+6)
(x+6)(x+1)
Evaluation: 1. z2 – 2z + 1
2. x2 +10x – 24
Factorization of quadratic equations of the form ax 2 +bx +c
Example: 5x 2 -9x +4
Solution:
Product: 5x 2 x 4 = 20x 2
Factors: -5 and -4
Sum: -5-4 = -9
Hence, 5x 2 – 9x + 4
5x2 -5x -4x +4
5x(x-1)-4(x-1)
(5x-4)(x-1)
Evaluation:
- 2x 2 +13x +6
- 13d 2 – 11d – 2
FACTORISATION OF TWO SQUARES
To factorize two squares with difference, we need to remember the law guiding difference of two squares i.e. x 2 – y2 = (x + y) (x- y).
Examples:
- P 2 – Q2 = (P+Q) (P-Q)
- 36y 2 – 1= 6 2 y 2 – 1 2
= (6y)2 – 1 2 = ( 6y+1) (6y-1).
Evaluation:
- 121- y 2
- x2y2 – 42
READING ASSIGNMENT
Essential Mathematics for J.S.S.3 Pg29-36
Exam focus for J.S.S CE Pg101-105
WEEKEND ASSIGNMENT
- The coefficient of x 2 in x 2 + 3x -5 is (a) 3 (b) 1 (c) -5
- Simplify e 2 – f 2 (a) (e+f)(e-f) (b) (e+f)(f+e) (c) (e-f)(f-e)
- Factorize x 2 +x -6 (a) (x+3)(x+2) (b) (x-2)(x+3) (c) (x+1)(x+5)
- Solve by grouping 5h 2 -20h + h – 4 (a) (h-4)(5h+1) (b) (h+4)(5h-1) (c) (h+2)(h-5)
- 49m 2 – 64n 2 when factorized will be (a) (7m+8n)(8m+7n) (b) (8m-7n)(8m+7n)
(c) (7m-8n)(7m+8n)
THEORY
Factorise the following
- 4p2 – 12p +9q2
- f 2 – 2f + 1
WEEK SEVEN
SOLVING EQUATION EXPRESSIONS
WORD PROBLEMS
Worked Examples:
- Find 1/4 of the positive difference between 29 & 11
- The product of a certain number and 5 is equal to twice the number subtracted from 20. Find the number
- The sum of 35 and a certain number is divided by 4 the result is equal to double the number. Find the number.
Solutions:
- Positive Difference 29 – 11 = 18
1/4 of 18 = 4 2/5
- Let the number be x
x X 5 = 20 – 2x
5x = 20 – 2x
5x + 2x = 20
7x = 20
x = 20/7 = 2
- Let the number be n
sum of 35 and n = n + 35
divided by 4 = n + 35
4
result = 2 X n
therefore n + 35 = 2n
4
n + 35 = 8n
8n – n = 35
7n = 35
n = 35/7 = 5
EVALUATION
- From 50 subtract the sum of 3 & 5 then divide the result by 6
- The sum of 8 and a certain number is equal to the product of the number and 3 find the number
SOLVING EQUATION EXPRESSIONS WITH FRACTION
Always clear fractions before beginning to solve an equations: –
To clear fractions, multiply each term in the equation by the LCM of the denominations of the fractions.
Examples:
Solve the following
1. x = 2
9
2. x + 9 + 2 + x = 0
5 2
3. 2x = 5x + 1 + 3x – 5
7 2
Solutions:
1. x = 2
9
Cross multiply
x = 18
2. x + 9 + 2 + x = 0
5 2
Multiply by the LCM (10)
10 X (x + 9) + 10 X ( 2 + x) = 0 X 10
5 2
2 (x + 9) + 5 (2 + x) = 0
2x + 18 + 10 + 5x = 0
2x + 5x + 28 = 0
7x = -28
x = -28/7 = -4
3. 2x = 5x + 1 + 3x – 5
7 2
Multiply by the LCM (14)
14 X 2x = 14 (5x + 1) + 14 ( 3x – 5)
7 2
28x = 2 (5x + 1) + 7 (3x – 5)
28x = 10x + 2 + 21x – 35
28x = 31x – 33
28x – 31x = -33
-3x = -33
x = 33/3 = 11
EVALUATION
Solve the following equations.
1. 7/3c = 21/2
2. 6 = 11
y + 3 y – 2
3. 3 – 4 = 0
2b – 5 b – 3
Furthermore, we can consider the word equations or expressions into:
- Sum & Differences
- Products
- Expressions with fractions & equations
SUM & DIFFERENCES
The sum of a set of numbers is a result obtained when the numbers are added together. The difference between two numbers is a result of subtracting one number from the other.
Worked Examples:
1. Find the sum of -2 & -3.4
2. Find the positive difference between 19 & 8
3. The difference between two numbers is 7. If the smaller number is 7 find the other.
4. The difference between -3 and a number is 8, find the two possible values for the number.
5. Find the three consecutive numbers whose sum is 63.
Solutions:
1. -2 + -3.4 = -5.4
2. 19 – 8 = +11
3. let the number be Y i.e. Y -7 = 7
i.e. Y = 7 + 7 = 14
4. Let M represent the number
M – (-3) = 8
m + 3 = 8
m = 8 – 3
m = +5
also -3 – m = 8
-m = 8 + 3
-m = 11
m = -11
the possible values are +5 & -11
5. Consecutive numbers are 1, 2, 3,4,5,6, Consecutive odd numbers are
1, 3,5,7,9……….. consecutive even numbers are 2, 4, 6, 8,10……….
Representing in terms of X, we have 2X, 2X + 2, 2X + 4, 2X + 6, 2X + 8, 2X + 10…………
for consecutive even numbers, we have X, X + 2, X + 4, X + 6…….
for consecutive odd numbers, we have X + 1, X + 2, X + 3, X + 4…
for consecutive numbers.
let the first number be x,
let the second number be x + 1
let the third number be x + 2
Therefore x + x + 1 + x + 2 = 63
3x + 3 = 63
3x = 63 – 3
3x = 60
x = 60 /3
= 20
The numbers are 20, 21, and 22.
EVALUATION
1. Find the sum of all odd numbers between 10 and 20
2. The sum of four consecutive odd numbers is 80 find the numbers
3. The difference between 2 numbers is 9, the largest number is 32 find the numbers.
PRODUCTS
The products of two or more numbers is the result obtained when the numbers are multiplied together.
Worked Examples:
1. Find the product of – 6, 0.7, &
2. The product of two numbers is 8 .If one of the numbers is 1/4 find the other.
3. Find the product of the sum of -2 & 9 and the difference between -8 & -5.
Solutions
1. Products -6 x 0.7 x
-6 x 7/10 x 20/3 = -6 x 7 x 20
10 x 3
= -2 x 7 x 2 = -28
2. Let the number be x
X x = 8
multiply both sides by 4
x = 8 x 4 = 33
3. Sum = -2 + 9 = 7
Difference = -5-(-8) = -5 + 8 = 3
Products= 7 x 3 = 21
EVALUATION
- The product of three numbers is 0.084 if two numbers are 0.7 & 0.2 find the third number
- Find the product of the difference between 2 & 7 and the sum of 2 & 7
- From 50 subtract the sum of 3 & 5 then divide the result by 6
- The sum of 8 and a certain number is equal to the product of the number and 3 find the number
Reading Assignment
New Gen Maths for J.S.S 3 Pg 20- 24
Essential Mathematics for J.S.S 3 Pg 85-87
WEEKEND ASSIGNMENT
- Esther is 3 times as old as her sister Tolu, if the sum of their ages is 20 years. Find the difference between their ages.
(a) 20 years (b) 8 years (C) 10 years
2. 9 was subtracted from a certain number and the result was divided by 4 if the final
answer is 5 what was the original number?
(a) 29 (b) 18 (c) 20
3. A woman is 4 times as old as her son. In five years time she will be 3 times as old as her son. How old is the woman(a) 50 yrs (b) 40 yrs (c) 45 yrs
4. Bayo is 4 times as old as his sister Tolu. If the sum of their ages is 20 years, find the difference between their ages.
(a) 12 yrs (b) 15 yrs (c) 18 yrs
5. Subtract the square root of 4 from the square of 4 and divide the result by 2
(a) 2 (b) 4 (c) 7
THEORY
1.Divide 36 by the difference between the product of 3 & 6 and the square root of 36.
2. When I add 45 to a certain number, and divide the sum by 2, the result is the same as five times the number, what is the number?
WEEK EIGHT
CHANGE OF SUBJECT OF FORMULA
A formula is a general equation involving two or more unknowns. An example is the formula a = ¶r2 which gives the area of the circle in terms of its radius r. In this formula, a is called the subject of the formula.
Simplifying a formula by substitution
Example:
Given mx+c=y, express x in terms of m, c, and y. Find the value of x if y=10, c=2 and m is 4
Solution: mx+c= y, mx=y-c
X= y-c
M
To find the value of X, X= 10-2
4
X= 8/4 =2
Evaluation: If I= PRT
- Make p the subject and find the value of p if I = 10, R= 4, and T= 5.
CHANGING THE SUBJECT OF A FORMULA
When a variable which forms a part of the formula is made subject, we say we have changed the subject of the formula.
Examples:
- In the formula S=2¶r(r+h), make h the subject.
Make m the subject if F= mv-mu
T
Solution: S=2¶r(r+h)
S=2¶r 2 +2¶rh
S- 2¶r 2 =2¶rh
S-2¶r2 =h
2¶r
2. F =mv-mu
T
Cross multiply: FT =mv-mu
FT =m(v-u)
FT
v-u = m
EVALUATION
- Make r the subject if V =¶r 2h
- Make u the subject if 1/f =1/v + 1/u
READING ASSIGNMENT
- Exam focus for J.S.C.E Pg. 212 -216
- Essential Maths for J.S.S.3 Pg. 47-54
WEEKEND ASSIGNMENT
- Express a in terms of u, v, and t in v= u+at
-
a = vu-t (b) a= v-u (c) a = v+u
tt
-
2. If Z=2p +3, find the value of Z when p=1. (a) Z=2, (b) Z= 5, (c) Z =7
3. In the relation X = m – 6y , how would you write m in terms of x and y?
6
(a) m=x-6 (b) m = 6x -6y (c) m =6x +6y
4. Express n in terms of s, a, and L if S=n/2(a+L)
(a) n=2(a+L) (b) n= 2S (c) 2S-(a-L)
Sa+L a+L
5. Make U the subject if V 2 = U2 +2as.
(a) U= V 2 -2as (b) U = (V 2 -2as) 2 (c) V2 – 2a
THEORY
1. Given L = arn-1, make a the subject of the formula.
2. If S + 2t , make t the subject of the formula. Find the value of t if S=1, d= 2.
d
WEEK NINE AND TEN
TOPIC: MEASURE OF CENTRAL TENDENCY
CALCULATION OF RANGE, MEAN, MEDIAN AND MODE OF UNGROUPED DATA
RANGE
The range of a set of numbers is the difference between the largest and the smallest numbers.
Example: Find the range of the following set of scores: 79, 60, 52, 34, 58, 60.
Solution
Arrange the set in rank order: 79, 60, 60, 58, 52, 34
The range is 79 – 34 = 45
THE MEAN
There are many kinds of average. The mean or arithmetic mean, is the most common kind. If there are n numbers in a set, then
Mean = sum of the numbers in the set/ n
Examples
1) Calculate the mean of the following set of numbers.
176 174 178 181 174
175 179 180 177 182
Solution
Mean = 176 + 174 + 178 +…. + 182/10
= 1776/10
= 177.6
2) Five children have an average age of 7 years 11 months. If the youngest child is not included, the average increases to 8 years 4 months. Find the age of the youngest child.
Solution
Total age of all five children
= 5 x 7 yr 11 mo
= 35 yr 55 mo
= 35 yr + 4 yr 7 mo
= 39 yr 7 mo
Total age of the four older children
= 4 x 8 yr 4 mo
= 32 yr 16 mo
= 32yr + 1 yr 4 mo
= 33 yr 4 mo
Age of youngest child
= 39yr 7 mo – 33 yr 4 mo
= 6 yr 3 mo
EVALUATION
1) Find x if the mean of the numbers 13, 2x, 0, 5x and 11 is 9. Also find the range of the set of numbers.
2) A mother has seven children. The mean age of the children is 13 years 2 months. If the mother’s age is included, the mean age rises to 17 years 7 months. Calculate the age of mother.
MEDIAN AND MODE
MEDIAN: If a set of numbers is arranged in order of size, the middle term is called the median. If there is an even number of terms, the median is the arithmetic mean of the two middle terms.
Examples
Find the median of a) 15, 11, 8, 21, 17, 3, 8 b) 3.8, 2.1, 4.4, 8.3, 9.2, 5.0.
Solution
a) Arrange the numbers in rank order (i.e. from highest to lowest).
21, 17, 15, 11, 8, 8, 3
There are seven numbers. The median is the 4th number, 11.
b) Arrange the numbers from the lowest to highest.
2.1, 3.8, 4.4, 5.0, 8.3, 9.2
There are six numbers. The median is the mean of the 3rd and 4th terms.
Median = (4.4 + 5.0) /2
= 4.7
MODE: The mode of a set of numbers is the number which appears most often, i.e. the number with the greatest frequency.
Example: Twenty-one students did an experiment to find the melting point of naphthalene. The table below shows their results. What was a) the modal temperature b) the median temperature?
temperature (oC) 78 79 80 81 82 83 90
frequency 1 2 7 5 3 2 1
a) Seven students recorded a temperature of 80oC. This was the most frequent result.
Mode = 80oC
b) There were 21 students. The median is the 11th temperature. If the temperatures were written down in order, there would be one of 78oC, two of 79oC, seven of 80oC, and so on. Since 1 +2 + 7 = 10, the 11th temperature is one of the five 81oCs.
Median = 81o C.
Evaluation
For the following set of numbers:
13, 14, 14, 15, 18, 18, 19, 19, 19, 21
a) state the median, b) state the mode, c) calcilate the mean.
WEEKEND ASSIGNMENT
1) The number of goals scored by a team in nine handball matches are as follows: 3, 5, 7, 7, 8, 8, 8, 11, 15
Which of the following statements are true of these scores?
a) The mean is greater than the mode.
b) The mode and the median are equal.
c) The mean, median, and mode are all equal.
Use the table below to question 2-5
The table below shows the number of pupils (f) scoring a given mark (x) in attest.
X 2 3 4 5 6 7 8 9 10 11 12
f 3 8 7 10 13 16 15 15 6 2 5
2) Find the mode.
a) 7 b) 8 c) 9 d) 10
3) Find the median.
a) 6 b) 7 c) 8 d) 9
4) Calculate the mean.
a) 6.7 b) 6.8 c) 6.9 d) 6.95
5) Find the range.
a) 10 b) 11 c) 9 d) 12
THEORY
1)x, x, x, y represent four numbers. The mean of the numbers is 9, their median is 11. Find y
2) Students at a teacher training college are grouped by age as given in table below.
Age (years) 20 21 22 23 24 25
Frequency 4 5 10 16 12 3
a) Find the modal age.
b) Find the median age.
c) Calculate the mean age of the students.
READING ASSIGNMENT
ESSENTIAL MATHS BK 3 PG 201 – 205 Ex 22.5nos 17 – 20