MATHEMATICS JSS TWO

MATHEMATICS     JSS TWO

WEEK ONE

DAY ONE

STANDARD FORM

The numbers 2x10³ , 8x10^-2  are all in the form of Ax10ⁿ and ‘n’ is an

Integer [positive and negative whole number]  it is in standard form.

For 54321, the first digit is 5 and can be expressed as 5.4321x10⁴  

Example 1:

            [a]10=10x1=10. Writes as 10¹

            [b] 1000=10x10x10=10³

Example 2:

            [a] 20=2x10¹

            [b]180=18x10¹

Example 3:

            [a]Express 543.2 in standard form

            Solution

543.2 = 5.432x100

           =5.432x10x10

           =5.432x10²

ASSIGNMENT

1:  Express the following in power of 10

[a] 10000 [b] 100 [c] 100000 [d] 10000000002

2:  Express the following in standard form

[a] 6789.54 [b]32456.891 [c]61572 [d]70000

DAY TWO

STANDARD FORM CONTINUES

Example 1:

[b]Express 0.008 in standard form

            Solution

0.008 = 8 1000

              =8/1000

            =8/10x10x10

            =8/10³

                        =8x10^-3

Example 2:

Express in full figure

            8.979x10⁵

            Solution

8.979x10⁵ = 8.979x10x10x10x10x10

                  =8.979x100000                   =897900

ASSIGNMENT

1:  Express the following in standard form

[a] 0.00055 [b] 0.003 [c] 0,0000009 [d] 0.00023

2:  Express the following in full figure

[a] 4.321x10⁴  [b] 5.14x10³  [c] 3.5276x10⁴  [d] 6.5434x10²

                                                DAY THREE

 LARGE AND SMALL NUMBERS

In 1 234, the small gap helps us to read and group numbers easily e.g 123456 is a large number because it has more than three digit and to read it easily, we group the digits into three starting from the right.

Example 1

            Divide 1,000,000 by 10

            Solution

1,000,000= 1,000,000 ÷ 10

                 = 1,000,000/10

                 =100,000

Note: by dividing a number by 10, we are reducing the number of digits by one from the right.

Example 2

            Multiply 0.00000 by 10

            Solution

            0.00000x10

            =1x10/1000000

            =1/100000

            =0.00001

ASSIGNMENT

1.      Divide the following by 10

[a]1,000,000,000 [b]2,000,000 [c]100 [d]10,000

       2.  Multiply the following by 10

            [a]0.1  [b]0.00001 [c]0.0000000001 [d]0.00000002

                        DAY FOUR

 ROUNDING OFF NUMBERS

Digits are either rounded up or down. Digits from 1 to 4 are rounded down while digits from 5 to 9 are rounded up.

Example

            Round off 6325 to the nearest [a] ten [b] hundred [c] thousand

            Solution

6325 = 6330   [nearest ten]

6325 = 6300   [nearest hundred]

6325 = 6000   [nearest thousand]

ASSIGNMENT

Round off the following to the nearest [a ]ten  [b]hundred  [c]thousand

[1] 250107  [2] 65466 [3]123456 [4] 9574

DAY FIVE

 SIGNIFICANT FIGURE

The first non-zero digit on the left of a number is the first significant figure (s. f).

 E.g 3457 = 3460 [to 3 s. f]

Example

            Correct 5260457 to [a] 1 s. f [b] 2 s. f [c] 4 s. f

            Solution

a.      5260457 = 5,000,000  [1 s. f ]

b.      5260457 = 5,300,000  [2 s. f ]

c.       5260457 = 5,260,000  [4 s. f ]

ASSIGNMENT

Correct the following to [a] 1 s. f [b] 2 s. f [c] 3 s. f [d]4 s. f

[1] 0.00063479 [2] 81.2066 [3] 0.004567[4]9.9999

WEEK TWO

DAY ONE

DECIMAL PLACES (d. p)

Example 1:

            Correct 25.0327 to [a] 1 d. p [b] 2 d. p [c]3 d. p

            Solution

a.      25.0327 = 25.0   [1 d. p]

b.      25.0327 =25.03  [2 d. p]

c.       25.0327 =25.033 [3 d. p]

ASSIGNMENT

Correct the following to [a]1 d. p [b]2 d. p [c] 3 d. p [d] 4 d. p

[1]125.20362  [2] 9.00123 [3] 62.13475 [4] 33.33453

           

DAY TWO

LAWS OF INDICES

In a standard form notation, 100= 10x10=10²  and read as ‘’ten raised to the power of two’’

Similarly, aⁿ means any number say ’’a’’ multiply by itself up to n times, such as a³= a x a x a

First law of indices is called the law of addition

Example 1

Simplify    2² x 2³   using the two methods

                        Solution

 Method 1: addition of power

            2² x 2³ = 2 ²⁺³  = 2⁵    [law of addition]

Method 2: expansion

            2² x 2³ = [2x2] x [2x2x2]

                        = 2 ²⁺³  = 2⁵

                       

ASSIGNMENT

Simplify the following using the two methods

[a] y⁴ x y¹²  [b] d⁸ x d⁵  [c] p⁷ x p¹³ [d] 5⁵ x 5³

                                               

DAY THREE

LAWS OF INDICES CONTINUES

First law of indices called law of addition

Example 2

            Simplify 5a³   x 2a²     using the two methods

            Solution

Method 1: subtraction of power

5a³   x 2a²    

= 5 x 2a³⁺²  =10a⁵    [law of addition]

Method 2: Expansion

 5a³   x 2a²   = 5 x a x a x a x 2 x a x a

                  =5 x 2 x a x a x a x a x a

                   =10a⁵  

Generally,  a^m x aⁿ  =a ^m+n         law of addition

This means whenever two or more numbers having the same base are multiplied, we simply add the powers and the result is the base number raised to the sum of the power. This is the first law of indices and it’s sometimes called law of addition 

ASSIGNMENT

Simplify the following using the two methods

[a] 5p⁵ x 4p² [b] 3p⁶ x 8p⁴  [c] 6b⁵ x 4b⁷ [d] 3k⁴ x 2k

DAY FOUR

LAWS OF INDICES

Second law of indices called law of subtraction

Example 1

            Using the two method, simplify a⁵÷ a³

                        Solution

Method 1: subtraction of power

a⁵ ÷a³   = a ^5-3  = a²  [law of subtraction]

method 2: Expansion

 a⁵ ÷ a³ = [a x a x a x a x a] ÷ [a x a x a]

[a x a x a x a x a]

                 [a x a x a]

            = a²

 Generally, a^m ÷aⁿ  = a ^m-n

This means that the index of the quotient is the difference between the given indices of the divisor and dividend provided they have the same base. This is known as the law of subtraction.

                        ASSIGNMENT

Using the two methods, simplify the following

[a] b⁷ ÷ b⁴ [b] k⁶ ÷ k³ [c] 7⁹÷ 7⁸ [d] m¹⁵ ÷m¹⁰

                        DAY FIVE

INDICES CONTINUES

Third law of indices

Example 1:

            Using the three methods, simplify  [a³]²

            Solution

Method 1:  [a³]²  = a ^2x3  =a⁶

Method 2: [a³]²    = [a x a x a]² 

                                                     =[a x a x a] x [a x a x a]

                             =a ^{3 + 3} = a⁶

Method 3:  [a³]²   = [a³][a³]

                            = a ^3x2     = a⁶

Generally, [a^m ]ⁿ   =a ^mxn  = a^mn

ASSIGNMENT

Using the three methods, simplify the following

[a] (9³)²   [b](h⁵)⁵   [c] (7³)²   [d] (f^x)^y