Secondary Mathematics Practices Content Page

Secondary Mathematics Content Page

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General Practice 

Secondary Maths : Daily Short Practice

Practice :

1. 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

2.  y is proportional to x and y = 20 when x = 25. Find y when x = 10

3. y is inversely proportional to x and y = 20 when x = 25. Find y when x = 10

4a.  Write 50 000 cm in kilometres.

       A map is drawn to a scale of 1 : 50 000

b.   The distance between two stations on the map is 12cm

      Find the real distance between the stations in kilometres

c.   The distance between two towns is 35 km. How far is this distance on the map?

5.  Simplify 2x  - 3(x - 2) [12/I/17/2/A]

                   3           5

6.  The nth term of a sequence is given by 11 - 6n. Write down the first 3 terms of the sequence.     

7.               -2 ,1 , 4, 7

a.  Write down the next 2 terms in this sequence

b.   Find an expression for the nth term in this sequence.     [17/I/24a/1,1/A]

<< END>>

Short Practice 1

1.  Simplify 4x – 3(x – 3)

2.  Factorise 2x²+ 5x – 3 

3.  Solve these simultaneous equations.

            4x + 3y = 8

            2x – y = 9

4.  Anna invests $2000 for 3 years. She receives compound interest at 1.65% per year. How much is the investment worth at the end of the three years?

Short Practice 2

1. Simplify (2x – 1)²

2.  Solve 2x= 5

                3

3.  y is proportional to x and y = 20 when x = 25. Find y when x = 10

4. y is inversely proportional to x and y = 20 when x = 25. Find y when x = 10

Short Practice 3

1. Simplify 4a² x  ab

     b       6

2. Rearrange v²= u²+ 2as to make u the subject

3a.  Sally has a map drawn to the scale 1 : 250 000.

       The distance on the map between the airport and the city centre is 7.5cm

       Calculate the actual distance, in kilometres between the airport and the city centre.

3b. The airport covers an area of 13km²

      Calculate the area, in square centimetres, covered by the airport on the map.

Short Practice 4

1. Solve 4x + 5 = 3

2. Simplify 3x – 4(5 – 2x)

3. Solve these simultaneous equations

        3x + 2y = 79

          x + 4y = 68

4. Annie cycles at 20km/h for 2 hours 30 minutes from A to B. She then cycles a further 16 km in 1 hour 15 minutes from B to C. Calculate her average speed, in km/h, for the whole journey from A to C.

Short Practice 5

1.  Solve   9    = 4

              x + 2

2. Rearrange this equation to make h the subject

         g = 3h – 2f

                   5

3. Felice is 10 years old and Lena is 6 years old.

a.   Write the ratio Felice’s age : Lena’s age in the simplest form.

b.   Felice and Lena are given some money to share in the ratio of their ages. Felice’s share is $30. Calculate Lena’s share.

c.   A year later, they are given $54 to share in the ratio of their ages. Calculate Felice’s share.

Short Practice 6

1.  Given that y = 6 – 3x, find the value of y when x = -4

2.  Make d the subject of the formula c = 3d + 2e

3. The scale of a map is given as 1 : 50000

a.  Rewrite the scale in the form of 1 cm to x km

b.  A road on the amp is 18cm long. Find the actual length of the road in kilometres.

c.  A lake has an area of 90km². Find the area of the lake on the map in square centimetres.

Short Practice 7

1a.      Make x the subject of a = bx + c                                        (2,2/10/p1/19/T)

1b.      Write as a single fraction in its simplest form 2   -    4

                                                                                  x     2x - 1

2a.  Write 156 as a product of its prime factors.                                     [1]

2b.  Find the highest common factor of 156 and 390                             [2]

Short Practice 8

1.   A car travels for 1½ hours at 50km/h. It then travels for 2 hours at 65 km/h. Find the average speed for the whole journey.          (3/13/p2/1/A)

2.         Solve the simultaneous equation         (3/12/p2/7b/A)    

                        4x – 3y = 20

                        2x + 5y = -3

3.   Solve (x – 5)(2x + 1) = 0                       (2/12/p1/3/A)

Short Practice 9

1.                     u = 2x + 1     v = x – 3                                              (2,2/15/p1/24/T)

 Write an expression it its simple form, in term of x for

          a.   3(v + 2) – 2v

          b.   u/2 + v/5

2a.  Calculate 25 + 144 – 2 x 5 x 12                                             (1,2/5/p1/3/T)

2b.  6% of $450

3. Write the following in order of size, smallest first                     (2/14/p1/2/Y)

            0.03                4%                  1/30                Sin 2^o

Short Practice 10

1.  Round                                                                                          (1/13/1/3/T)

a.  25.6389 correct to 2 decimal places

b.  506423 correct to 3 significant figures

c.   0.00678472 correct to 4 significant figures

2.  The scale of a map is given as 1 : 200 000                           (112/5/p2/5/T)

a.  Complete the following

                        Scale of map is 1cm to _____ km

b.  Find the distance on the map between two towns that are actually 30 km apart.

c.  On the map, a forest has an area of 5 cm². Find the actual area of the forest in square kilometres.

Short Practice 11

1.  Solve the simultaneous equations                                         (3/14/2/7/T)

                        2x + 3y = 16

                        4x + 5y = 27

2. 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

Short Practice 12

1.  Simplify  6x ÷  4                                                                            (2/13/1/3/T)

                    7      7

2.  The ratio of the distance Mary runs to the distance she walks is 5:2.         (2,2/14/2/6/T)                                           
a.   When she runs 4000m, how far does she walk

b.   When she covers a total distance of 3500m during exercise, how far does she run?

3.  Solve the equation                                                                     (3/12/1/9/T)

                        3(2x + 50) = 12                                                         

Short Practice 13

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

2.  Solve the equation 4x + 2= 14                                                   (5/13/1/5/T)

                                           5

3a.  Factorise a²+ 3a                                                                      (12/13/1/6/T)    

3b.  Expand and simplify (x + 3)(x – 2)     

Short Practice 14

1.    Evaluate                                                                    [1]

                             76.501

                        4.92 x 3.503

2a.  Express 180 as the product of its prime factors             (1/10/p1/3/A)

2b.  Write down the smallest positive integers, k, such that 180k is a perfect square.          

3.   Factorise 3ax -6ay + bx – 2by        (3/10/p2/8a)

Short Practice 15

1. By rounding each number to 1 significant figure, estimate the value of           (2/10/p2/4a)

                        62.89 x 8.93

                            3.12

You must show your working.       

2.  A map is drawn to a scale of 1: 40 000. Find the actual length, in kilometer, of a road which is 6cm long on the map.    (2/09/P1/7/A)                                    

3.         Factorise                                           (1/10/p1/7/A)

                   (a)               4y²- 9

                   (b)           x²– 6x - 9

Short Practice 16

1.  Find the fraction exactly halfway between ¾ and 3/5. Give your answer in its simplest form.                        (2/11/p1/8/A)                                                 

2.  Which of these ratio are equivalent to the ratio a : b?   (2/12/p2/5/A)

            a²: b²                          3a:3b              1/b : 1/a                      a+1 : b+ 1

3.   Solve the simultaneous equations         (3/14/p1/12)

                        x + 4y = 1

                        3x + 2y = 13

Short Practice 17

1.   Evaluate 78.6 ÷ 0.02                           (1/12/p1/1a)                                         

                       2.4 – 0.9

2.   The distance between London and Tokyo is 9567 km. Write the distance to the nearest hundred.               (1/12/p1/1b)                              

3.         Solve                                            (12/15/p1/14/A)

a.         4x + 5 = 3

b.         x =    6   

                  x + 5

Short Practice 18

1.   Express 1/25 as a decimal.                 (1/13/p1/1b)             

2.   Express 30% as a fraction in its lowest terms.        (1/13/p1/1a/A)

3.         Solve the inequality 4x ≥ 15         (1,2)

            Solve  6  = 4

                    y - 3

S1T1 - 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

Example: 

Give the value of the digit 6 in the following numbers.

a.   234.60

Digit 6 is in the tenth place. The value is 0.6

b.  620912

The digit 6 is in the hundred thousands place. Its value is 600000

 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)

S1T1 Number Line, Ordering , Simple Inequalities

The number line

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

The numbers are placed at their correct positions, equal distance apart.

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.

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

 

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    ____