Free Mathematics Lesson Note JSS 2

Free Mathematics Lesson Note JSS 2

This MATHEMATICS Lesson Note was pulled from our book (Lesson Note on MATHEMATICS for JSS1 MS-WORD); Compiled to serve as a reference material to help teachers draw out their lesson plan easier, saving you valuable time to focus on the core job of teaching.

The Lesson notes are based on the current NERDC curriculum (UBE compliant)

This MATHEMATICS Lesson Notes CoversThe Following Topics

1. WHOLE NUMBERS
2. LCM AND HCF OF WHOLE NUMBERS
3. FRACTIONS
4. APPROXIMATION
5. ALGEBRAIC EXPRESSIONS – FACTORS AND FACTORIZATION
6. ARITHMETIC IN THE HOME AND OFFICE
7. APPROXIMATION AND ESTIMATION
8. DIRECTED NUMBERS – MULTIPLICATION AND DIVISION
9. EXPANSION OF ALGEBRAIC EXPRESSIONS
10. SIMPLE EQUATIONS
11. LINEAR INEQUALITIES
12. LINEAR INEQUALITY (GRAPHICAL REPRESENTATION)
13. GRAPHS OF LINEAR EQUATIONS
14. STRAIGHT-LINE GRAPHS
15. PLANE FIGURES OR SHAPES
16. SCALE DRAWING
17. ANGLES BETWEEN LINES
18. ANGLES OF ELEVATION AND DEPRESSION
19. STATISTICS 2 – PRESENTATION OF DATA
20. PROBABILITY
21. SOLVING EQUATIONS
22. USING CALCULATORS AND TABLES
23. PYTHAGORAS’ THEOREM
24. TABLES, TIMES TABLES AND CHARTS

Sample note

Topic: WHOLE NUMBERS

Factors and Prime factors (revision)

40 ÷ 8 = 5 and 40 ÷ 5 = 8

8 and 5 divide into 40 without remainder.

8 and 5 are factors of 40.

A prime number has only two factors, itself and 1, 2, 3, 5, 7, 11, 13, … are prime numbers.

1 is not a prime number.

Standard form

Standard form is a way of writing down very large or very small numbers easily. 10³ = 1000, so 4 × 10³ = 4000 . So 4000 can be written as 4 × 10³ . This idea can be used to write even larger numbers down easily in standard form.

Small numbers can also be written in standard form. However, instead of the index being positive (in the above Example, the index was 3), it will be negative.

The rules when writing a number in standard form is that first you write down a number between 1 and 10, then you write × 10(to the power of a number).

Example

Write 81 900 000 000 000 in standard form: 81 900 000 000 000 = 8.19 × 10¹³

It’s 10¹³ because the decimal point has been moved 13 places to the left to get the number to be 8.19

Example

Write 0.000 001 2 in standard form:

0.000 001 2 = 1.2 × 10^-6

It’s 10^-6 because the decimal point has been moved 6 places to the right to get the number to be 1.2

On a calculator, you usually enter a number in standard form as follows: Type in the first number (the one between 1 and 10). Press EXP . Type in the power to which the 10 is risen.

Manipulation in Standard Form

This is best explained with an Example:

Example

The number p written in standard form is 8 × 10⁵

The number q written in standard form is 5 × 10^-2

Calculate p × q. Give your answer in standard form.

Multiply the two first bits of the numbers together and the two second bits together:

8 × 5 × 10⁵ × 10^-2

= 40 × 10³ (Remember 10⁵ × 10^-2 = 10³)

The question asks for the answer in standard form, but this is not standard form because the first part (the 40) should be a number between 1 and 10.

= 4 × 10⁴

Calculate p ÷ q.

Give your answer in standard form.

This time, divide the two first bits of the standard forms. Divide the two second bits. (8 ÷ 5) × (10⁵ ÷ 10^-2) = 1.6 × 10⁷

Express the following fractions in standard form

1. 0.000 07
2. 0.000 000 022
3. 0.075

Whole Numbers in Standard Forms

A number is said to be in standard form if it is re-written as a figure between 1 and 10 and then multiplied by a power of ten without changing its original value. i.e. P × 10^x.

Note

When expressing numbers in standard form, point are either carried from the left hand side (LHS) or right hand side (RHS) of it. While the point carried from the left hand side turns negative the point from right hand side turns positive.

Another very important thing to note is that when expressing either decimal number or whole number in standard form, points are carried until they are between the 1^st and 2^nd value.
Example 1; Express 263,000,000 in standard form.

Solution

The value above is a whole number so you carry point (imaginary) from (RHS) towards (LHS). Let’s do it!
= 2.63 × 10⁸. Answer

Example 2; Express 0.0006927 in standard form.

Solution

The value above is a decimal number so points are carried from the left hand side (LHS) – (RHS).

= 6.927 × 10^-4.

Example 3; Express 34.694 in standard form.

Solution

Even though the value above is also a decimal number, points here will be carried from (RHS) – (LHS).

The result will be; 3.4694 × 10¹.

Ordinary Form

Ordinary form is the opposite of standard form. When you are expressing numbers in ordinary form it means going the other way round to get your answer.

For Example; Express 3.4694 × 10¹ in ordinary form.

Solution

You are going to carry the point once from (LHS) – (RHS). Why? Because 10 is raised to power of 1.

3.4694 × 10¹ = 34.694. (Answer)

Assessment

1. Express the following in standard form;

(a) 54000

(b) 0.0003164

(c) 263.478

(d) 0.00000364

(e) 600.84

2. Find the value of A if 0.000046 = A × 10^-5
3. What is the value of n if 0.0000094 = 9.4 × 10ⁿ?
4. Express the following in ordinary form;

(a) 2.83 × 10⁸

(b) 4.765 × 10^-3

(c) 1.278 × 10²

(d) 9.87 × 10^-9

Rounding off numbers

You have learnt how to round off numbers to the nearest thousand, hundred, tens, etc.

Remember that the digits 1, 2, 3, 4 are rounded down and the digits 5, 6, 7, 8, 9 are rounded up.

Round off the following to the nearest

1. thousand ii. hundred iii. ten
2. 12 835
3. 46 926
4. 28 006

Significant figures

Significant figures begin from the first non-zero digit at the left of a number. As before, the digits 5, 6, 7, 8, 9 are rounded up and 1, 2, 3, 4 are rounded down. Digits should be written with their correct place value.

Read the following Examples carefully.

546. 546.53 = 500 to 1 significant figure (s.f.)

543.52 = 550 to 2 s.f.

543.52 = 547 to 3 s.f.

546.52 = 546.5 to 4 s.f.

8. 8.0296 = 8 to 1 s.f.

8.0296 = 8.0 to 2 s.f.

In this case the zero must be given after the decimal point. It is important.

8.0296 = 8.03 to 3 s.f.

8.0296 = 8.030 to 4 s.f.

Notice that the fourth digit is zero. It is significant and must be written

Decimal Places

Decimal places are counted from the decimal point. Zeros after the point are significant and are also counted. Digits are rounded up or down as before. Place value must be kept.

Read the following Examples carefully.

14. 14.902 8= 14.9 to 1 decimal place (d.p.)
15. 14.902 8 = 14.90 to 2 d.p.
16. 14.902 8 = 14.903 to 3 d.p.

Prime Factors

Prime Factors

Let’s look at the number 32. We can multiply 4 times 8 to get 32, so 4 and 8 are factors of 32. But 4 and 8 are like the frosting and the cream in the donut – they are parts, but they are not the smallest possible parts. The numbers 4 and 8 can each be divided evenly by another number – the number 2. The 2 is a prime number – a number divisible only by 1 and itself. That means 2 is a prime factor of 32.

A prime factor is a factor that is also a prime number. In other words, it is one of the smallest components of the number, and it can only be divided by 1 and by itself.

A factor that is a prime number: one of the prime numbers that, when multiplied, give the original number.

Example: The prime factors of 15 are 3 and 5 (3×5=15, and 3 and 5 are prime numbers). In number theory, the prime factors of a positive integer are the prime numbers that divide that integer exactly. The prime factorization of a positive integer is a list of the integer’s prime factors, together with their multiplicities; the process of determining these factors is called integer factorization. A Prime Number can be divided evenly only by 1 or itself. And it must be a whole number greater than 1.

The first few prime numbers are: 2, 3, 5, 7, 11, 13, and 17 …, and we have a prime number chart if you need more.

Factors

“Factors” are the numbers you multiply together to get another number:

Factor Trees

When trying to determine the basic ingredients of a donut, we look at the recipe. When trying to determine the basic ingredients of a number, the prime factors, we can make a factor tree. Look at this picture of the factor tree for 32.

• The first branch shows that 32 is equal to 4 times 8.
• The next branch shows that 4 is equal to 2 times 2. Both these numbers are prime numbers, so this branch is finished.
• The 8 is broken into 2 times 4. Since 2 is a prime number, its branch is done.
• The 4 is broken into 2 times 2. Now all the numbers on the ends are prime.
• When we look at the circled numbers, we see all the prime factors of 32.
• The prime factorization of 32 is 2 x 2 x 2 x 2 x2.

Prime Factorization

“Prime Factorization” is finding which prime numbers multiply together to make the original number.

Here are some Examples:

Example 1: What are the prime factors of 12 ?

It is best to start working from the smallest prime number, which is 2, so let’s check: 12 ÷ 2 = 6

Yes, it divided evenly by 2. We have taken the first step!

But 6 is not a prime number, so we need to go further. Let’s try 2 again: 6 ÷ 2 = 3

Yes, that worked also. And 3 is a prime number, so we have the answer: 12 = 2 × 2 × 3

As you can see, every factor is a prime number, so the answer must be right.

Note: 12 = 2 × 2 × 3 can also be written using exponents as 12 = 2² × 3

Example 2: What is the prime factorization of 147 ?

Can we divide 147 evenly by 2?

147 ÷ 2 = 73½

No it can’t. The answer should be a whole number, and 73½ is not.

Let’s try the next prime number, 3:

147 ÷ 3 = 49

That worked, now we try factoring 49, and find that 7 is the smallest prime number that works:

49 ÷ 7 = 7

And that is as far as we need to go, because all the factors are prime numbers.

147 = 3 × 7 × 7

(or 147 = 3 × 7² using exponents)

Example 3: What is the prime factorization of 17 ?

Hang on … 17 is a Prime Number.

So that is as far as we can go. 17 = 17

Another Method

We showed you how to do the factorization by starting at the smallest prime and working upwards.

But sometimes it is easier to break a number down into any factors you can … then work those factor down to primes.

Example: What are the prime factors of 90 ?

Break 90 into 9 × 10

• The prime factors of 9 are3 and 3
• The prime factors of 10 are2 and 5

So the prime factors of 90 are 3, 3, 2 and 5

Perfect Squares

In mathematics, a square number or perfect square is an integer that is the square of an integer; in other words, it is the product of some integer with itself. For Example, 9 is a squarenumber, since it can be written as 3 × 3. Square numbers are non-negative.

Examples of Perfect Square

16 can be expressed as a perfect square as 4 x 4 (the product of 2 equal integers)

36 can be expressed as a perfect square as 6 x 6

81 can be expressed as a perfect square as 9 x 9

169 can be expressed as a perfect square as 13 x 13

Squares and square roots

Square roots

7² = 7 x 7 = 49.

In words ‘ the square of 7 is 49’. We can turn this statement round and say. ‘the square root of 49 is 7’.

In symbols, √49 = 7. The symbol √ means the square root of .

To find the square root of a number, first find its factors.

Example

Find √11 025.

Method: Try the prime numbers 2, 3, 5. 7, …

Working:

11 025 = 3² x 5² x 7²

= (3 x 5 x7) x (3 x 5 x 7)

= 105 x 105

Thus √11 025 = 105

It is not always necessary to write a number in its prime factors.

Example

√6 400

6400 = 64 x 100

= 8² x 10²

Thus √6 400 = 8 x 10 = 80

The rules for divisibility can be useful when finding square root.

Assessment

Find by factors the square roots of the following:

1. 225
2. 194
3. 342
4. 484

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