Secondary Mathematics Practices Content Page

Secondary Mathematics Content Page

Algebra 
Topical Practice 
General Practice 

ST1 - Numbers, Power, Place Value

NUMBERS                                                                              

Integers

Whole numbers including negative whole number

            Example:          …,-4, -3, -2, -1, 0, 1, 2, 3, 4,…

A positive number is any number greater than zero. 

            Example: 1, 2, ½, 6.4

Negative Numbers

We read -1 as negative one.

A negative number is any number less than zero. 

            Example: -1, -2.5, -4/7

Large Numbers

One Million : 1 000 000 [6 zeros, 1000 times of thousands]

One Billion: 1000 000 000 [9 zeros, 1000 times of millions] 

Place Values

Digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 are used to form numbers.

The value the digit is the position of the digit in the number

            487

          7 è Value of the digit 7 is 7

        80 è Value of 8 is 80

      400 è Value of 8 is 400

 Index/Power

The small, raised number next to a normal letter or number, to be multiplied by itself

Example

b²= b x b

4³ = 4 x 4 x 4

 

Square power 2 : ²

- multiplying by itself

Example

(1)        Square of 4 = 4² = 4 x 4 = 16                    

(2)        3² = 3 x 3 = 9

(3)        Square of (-5)² = (-5) x (-5) = 25

Perfect Square are the squares of whole numbers: 

             (2 x 2) 4, (3 x 3) 9, (4 x 4) 16, …

Square Root - the Opposite of Square

            A number that when multiplied by itself, gives the number

                                    Symbol: √ 

        

5 -->         square 5²       --> 25

                        5 <--    Square root √25  <-- 25

Example

(1) 16 = 4 x 4

     √16 = 4

(2) a² = 25        

      a = + √5 x 5 =    = +5             

Why +5 when a number is square-root?

   (-5) x (-5) = 25 ( -ve x -ve = + ve)

    5 x 5 = 25

  => 25 = (-5)² = 5² = 25

       a2 =  + √a x a = -a or a

Cube - Power of 3 : ³

- Multiplying the number by 3 times

Example:

(1)        Cube of 3 = 33 = 3 x 3 x 3 = 27                                

(2)        4³ = 4 x 4 x 4 = 64

(3)        cube of (-5) ³ = (-5) x (-5) x (-5) = = -125                            

Cube Root

            A value that when ‘cubed’ gives the original number

                        Symbol: ³√

4 -->        cube 4³         --> 64

                4 <--  cube root ³√64    <-- 64

Example: 

                  3√8 = 3√2 x 2 x 2 = 2

             ³√216 = ³√6 x 6 x 6 = 6

Prime Number

 Greater than 1 that cannot be formed by multiplying 2 smaller natural number. 

            Example: 2, 3, 5, 7, …

            2 = 1 x 2, 13 = 1 x 13

* 1 is not a prime number because prime numbers are greater than 1

* All even numbers are not prime number except 2.

Example

List all the prime number that are greater than 10 and less than 20

Step 1: List all the numbers 

 11, 12 , 13 , 14, 15, 16, 17, 18, 19

Step 2: Strike out the non-prime number

            11, 12 , 13 , 14, 15, 16, 17, 18, 19

Step 3: List out the prime number

  11, 13, 17, 19

Practice:

1. Write the following in numerals

(a) 7.18 billion.   (b) 19.05 million

Refresh and Revise

Conversion of Unit of Measurement

Length

(mm = millimeter cm = centimeter   m = metre)

1 cm = 10 mm

1 m = 100 cm                           

1km = 1000 m  

1km = 1000m = 100000cm (1000 x 100)             

Area

1 cm² = 1cm x 1cm = 10mm x 10mm = 100mm²

1 m² = 1m x 1m = 100 cm x 100 cm = 10000 cm²

1km² = 1000m x 1000m = 1 000 000 m²

Mass

1 g = 1000mg

1kg = 1000g

1 ton = 1000kg

TIME

1 min = 60 seconds (s)                         

1 hour (h) = 60 min (m)                         

1h = 60 x 60 = 3600s

1 s = 1/60   x 1/60   = 1/3600 h

            1 s = 1/3600 h

            1 h = 3600 s

            0.5 or ½ hr = 30 mins

Volume

1 litre = 1000ml = 1000cm³

            1cm³ = 1ml

Practice:

1.  Convert 125 km to m

2.  Convert 299 m to km

3.  Convert 700 g to kg

4.  Convert 2 ton to g

5.  Convert 2 hrs to seconds

6.  Convert 1260 seconds to minutes 

7.  Convert 100 kilometres per hour into metres per second.  (2/13/1/2/T)

N1-G123 Number Line, Ordering , Simple Inequalities

The number line

A number line is a straight horizontal line representing numbers in sequential order equal distance apart and extending indefinitely in both directions. 

Whole numbers, fractions and decimals can be represented on the number line.

Zero is usually placed in the centre as the starting point. The positive number is to the right of the zero 0, and negative number to the left.

Uses:
Comparing: Numbers further to the right are larger.
Addition: Move to the right
Subtraction: Move to the left

Negative Numbers

We read -1 as negative one.

A negative number is any number less than zero. 

            Example: -1, -2.5, -4/7

Any pair of numbers  eg: 3 and -3 are same distance from the origin.

Ascending and Descending Orders

Ascending : Increasing in value; Moving higher/becoming bigger 

[Tip to remember : A - side of A is going up, from small to big ]

Descending : Moving down in value/becoming smaller

[D => Down , from big to small]

Example

Arrange the following in ascending order

      5.3, 5.25, 5.205       

          5.3                                        Step1: Arrange by the decimal point

          5.25

          5.205

<<Ascending => small to big>>

    5.3 the largest tenth value         Step 2 Compare number from left to right

    5.205 has the smallest hundredth value.  

    5.25 has the smaller tenth value

          5.205, 5.25, 5.3 Step3: Arrange the numbers

Using Number Line

Example 

What is the value of 3 - 5?

      3 - 5 = -2

Ascending and Descending Orders

Ascending : Increasing in value; Moving higher/becoming bigger 

[Tip to remember : A - side of A is going up, from small to big ]

Descending : Moving down in value/becoming smaller

[D => Down , from big to small]

Example

Arrange the following in ascending order

      5.3, 5.25, 5.205       

Step1: Arrange by the decimal point

          5.3

          5.25

          5.205

Step2: Ascending => small to big. 

           Compare the number values from left to right

          5.205 has the smallest hundredth value, 

          5.25 has the smaller tenth value

Step3: Arrange the numbers

          5.205, 5.25, 5.3

Using number line:

 

    5.205, 5.25, 5.3 ( Ascending -> from left to right )

Comparing and Ordering Numbers

Symbols

                        >    greater than

                        <    less than

                        ≥    greater than or equal to

                        ≤    less than or equal to

                        =    equal

                        ≠  not equal to

Tip to remember

                                                   4  >  3

        greater = "open mouth"       >      “point” = less than

  

Similarly,

                                                   3  <  4

                  less than = “point”     <       greater = "open mouth"            

Using number line and Inequalities

Using the dot : o excluding the number  => < or > symbol

                        •  including the number   =>  ≤ or ≥ symbol

 

Using the symbol : > => right arrow  ------>

                                < => left arrow    <-------

Example ( > symbol)

Draw x > 2 on the number line.

Step1 : "stand" at 2, draw o (> symbol)

Step2 : > (greater than) symbol => Right arrow

[To check if arrow direction is correct, use a bigger number (eg 3), and move the arrow in that direction]

Example ( < symbol )

Draw x < 1 on the number line  

Steps : "stand" at 1, draw o (< symbol); then left arrow (<)

 

Example (≥ symbol )

     Draw x ≥ 2 on the number line.

                        

(Draw a solid circle at 2 and then right arrow direction) 

Example ( ≤ )

    Draw x ≤1 on the number line.

    

 (Draw a solid circle at 1 and left arrow direction)

  

Practice

1.  Arrange the following numbers in ascending order.

(a)  1.14, 1.106, 1.13, 1.1    

(b).  0.52, 0.543, 0.5, 0.506

2.  Arrange the following numbers in descending order.

(a)  6.07, 6.32,  6.3, 6.301

(b). 9.123, 9.09, 9.3, 9.15

3.  Round 965.27 km to 

a. 1 decimal place

b. the nearest 10 km

c. 3 significant figures

4.  The following are temperatures, in C, over 6 days in Iwate.

       -1.4, -2, 0, -3, -0.5

(a) List all the temperatures that are >= -1 C

(b) List all the temperatures that are < -0.5 C

5. Use <, > or = to complete each of these statements.            (1/14/2/4/T)

a.         1/3 ___ 0.3

b.         12 ½% ______ 1/8

c.         7/12 _______ 5/9

6. Complete the number line.

        <——|——|——|——|——|——|——|——|——|

               -6     ___ ___    3.      6.    ___  12    15    ____