In this exercise, we used letters to represent numbers and we used the correct order of operations in algebraic expressions.

You can write an expression by using a letter to represent an unknown number.

E.g. 3n + 8
3n in this expression is a term
The letter n is called the variable, because it can have different values.
The co-efficient of n is 3 because it’s the number that multiplies the variable.
The number 8 is called a constant.


2.2 - Using expressions and formulae

In this exercise, we represented situations either in words of as a formula. We also changed the subjects of a formula.

E.g. F=ma
       F/a = m
       m = F/a




2.3 - Expanding brackets

In this exercise, we expanded brackets with expressions.

E.g. 2(3x + 4)
       6x + 8
We basically just multiply the number outside the brackets with the number inside the brackets.


2.4 - Factorising

In this exercise, we factorised expression using HCF, we just did  the opposite thing we did in the last exercise.

E.g. 8x + 10
The HCF of 8x and 10 is 2
2(4x + 5)
        

2.5 - Constructing and solving equations

In this exercise, we wrote and solved equations.

E.g. 3y + 7 = 31
        3y = 31-7
        3y = 24
        y = 24/3
        y = 8
        



2.6 - Inequalities
In this exercise, we used letters and inequalities to represent open and closed intervals.

E.g. x > 5
This means that x can be any number greater than 5.
21 < x < 46
This means that x is greater than 21 and less than 46.
The smallest x can be is 22 and the largest it can be is 45.
E.g. 21 ≤ x ≤ 46
This means x is greater than or equal to 21 and less than or equal to than 46.
The smallest x can be is 21 and the largest it can be is 46.

5.1 - Parallel lines
In this exercise we used geometric vocabulary for equal angles formed when lines intersect.

When you have a pair of parallel lines and there’s a line going through it the lien is called a transversal line.
An angle that is formed on one side is equal to the angle that is formed corresponding to it on the other parallel line.
An angle is also equal to the angle that is formed alternate to the angle which is vertically opposite to the corresponding one.

5.2 - The exterior angle of a triangle
In this exercise, we learnt how to identify the exterior angle of a triangle. 
A way to make things simpler is to know that the exterior angle of a triangle is equal to the sum of the two interior opposite angles.




5.3 - Constructions

There are different types of ways to construct triangles based on the angles and sides given.

Here are the 4 types :

- ASA : This is when you know two angles and the side between them

- SAS : This is when you know two angles and the side between them.

- SSS : This is when you know all three sides, but no angles.

- RHS : This is when you know there is a right angle, you know the length of the hypotenuse, and one other side.

The decimal with the greater digit in the leftmost place value column is the larger number.

If the digits in the leftmost place value column are the same, move to the next place value column and repeat the comparison.

4.2 - Multiplying decimals

For example, if you multiply 3.14 by 2 (which has no decimal places), you would first get 628 (pretending there are no decimals). Since 3.14 has two decimal places, you would move the decimal point in 628 two places to the left to get 6.28.

4.3 - Dividing by decimals

You can divide a number by a decimal by multiplying both the top and bottom numbers by a power of 10. This makes the decimal a whole number, which is easier to divide by.

For example, dividing by 0.7 is the same as dividing by 7 if you multiply the top and bottom numbers by 10.

4.4 - Making decimal calculations easier 
There are two ways to make decimal calculations easier.

Here are some of the things to consider when choosing a sample:

The best way to choose a sample will depend on the specific investigation.

For example, 0.717171... is a recurring decimal.

You can use a bar over the repeating digits to show that a decimal is recurring. For example, 0.71̅ can be used to represent 0.717171...

7.2 - Ordering Fractions 

There are two ways to compare fractions:

7.3 - Subtracting mixed numbers 

7.4 - Multiplying an integer by a mixed number 

You can use partitioning to multiply a mixed number with an integer. For example, 4 x 5 1/2 = 4 x 5 + 4 x 1/2  = 20 + 2 = 22

7.5 - Dividing an integer by a fraction

You can think of dividing by a fraction as multiplying by the reciprocal of the fraction.

The reciprocal of a fraction is the fraction flipped upside down. For example, the reciprocal of 3/2  is 2/3

Here are the steps for dividing an integer by a fraction:

The formula for the circumference is c = 2πr, where c is the circumference, π (pi) is a mathematical constant, and r is the radius of the circle.

The radius is the distance from the center of the circle to any point on the circle.

The diameter is the straight line that goes through the center of the circle and touches two opposite points on the circle. The diameter is twice the radius (d = 2r).

8.3 - 3D Shapes

For 3D shapes, there are different views, such as front elevation, side elevation.

Formula that links vertices, faces and edges,
V+F=E+2

9.2 - Finding rules for sequences

From finding the term-to-term rule, you can find the the position-to-term rule. ( the expression that links the position number to the term.) This way you can find the term easily after knowing the position number.

9.3 - Using the nth term

You can also write the position-to-term rule as an nth term expression.

To do this, you replace the words 'position number' with the letter n.

So, as I stated above, instead of writing:

term = 2 x position number + 3

you would write: nth term = 2 xn + 3

or, more simply: nth term = 2n + 3

9.4 - Representing simple functions

There are different ways to represent a function, including function machines, mapping diagrams, and algebraic equations.

Here are the steps for finding the percentage change:

For example, if the price of a jacket increases from $75 to $105, the price increase is $30 The percentage increase can be found as follows:

• Percent increase = (30 / $75) × 100% = 40%

10.2 - Multiplier

 You can use a multiplier to find a percentage increase or decrease.

For example, this is how to find a 65% increase in a price of $275.

The total cost of hiring the hall for 3 hours is $40 × 3 + $15 = $135

Suppose the hall is hired for n hours and the cost is $c.

Then c=40n + 15

This function shows how to work out the cost for any number of hours.

The charge per hour is multiplied by the number of hours, and this is added to the booking fee.

If you want to hire the hall for 3 hours, then n= 3 and c=40 × 3 + 15 = 135

The cost is $135.

11.2 - Plotting graphs

Here is a function: y = 2x-1

You can substitute different values of x into the function to find the y-value, for example:

if x=3 then y=2x3-1=5

You plot a table stating x and the corresponding y below.

With this you get a set of coordinates, and when you plot the coordinates on a graph, it’s a straight line.

11.3 - Gradient and y-intercept 

The gradient and intercept of a linear function is based on the following :

In this pastry recipe, the ratio of flour to butter is 0.5 kg: 250g.

Before you simplify a ratio, you must write all quantities in the same units.

Pastry recipe

0.5 kg flour

250g butter water to mix

0.5 kg: 250g is the same as 500g: 250g, which you write as 500:250.

You can now simplify this ratio by dividing both numbers by the highest common factor. In this case the highest common factor is 250.

Divide both numbers by 250 to simplify the ratio to 2:1.

If you cannot work out the highest common factor of the numbers in a ratio, you can simplify the ratio in stages. Divide the numbers in the ratio by common factors

until you cannot divide any more.

12.2 - Sharing a ratio

Here are the steps for sharing a quantity in a given ratio:

For example, Zara, Sofia and Marcus buy a painting for $600. Zara pays $200, Sofia pays $300 and Marcus pays $100. The ratio of their contributions is 2:3:1

12.3 - Ratio and direct proportion

You can see ratios in a variety of situations, such as mixing ingredients in a recipe or sharing an amount among several people.

Ratios can also be used to make comparisons.

For example, suppose you wanted to compare two mixes of paint.

Pink paint is made from red and white paint in a certain ratio (red: white).

The shade which is darker is the shade with the greater proportion of red paint. You can change the ratios into fractions, decimals or percentages to compare the proportions of red paint in each shade.

It is an angle measured from north in a clockwise direction.

A bearing can have any value from 0° to 360°. It is always written with three figures.

14.2 - The midpoint of a line segment

The midpoint of a line segment is a point that lies exactly halfway between two points. It is the same distance from each endpoint of the straight line segment.

The formula for midpoint is (x1 + x2 / 2 ) , (y1 + y2 / 2).

14.3 - Translating 2D shapes
You can describe this movement with a column vector.

translate

This is an example of a column vector:

The top number tells you how many units to move the shape right (positive number) or left (negative number).

The bottom number tells you how many units to move the shape up (positive number) or down (negative number).

For example: 
(2)
(5)

means move the shape 2 units right and 5 units up.

If the scale on the grid is one square to one unit, the numbers tell you how many squares to move the object up/down and across.

When a shape is translated, only its position changes. Its shape and size stay the same. This means that the object and its image are always congruent.

14.4 - Reflecting shapes

To reflect a shape, you use the x-axis or y-axis as the mirror line.

You must also be able to reflect a shape on a coordinate grid in other mirror lines.

To do this, you need to know the equation of the mirror line.

Some examples are shown on the grid on the right.

All vertical lines are parallel to the y-axis and have the equation x=‘any number'.

All horizontal lines are parallel to the x-axis and have the equation y = ‘any number’

14.5 - Rotating shapes

When you carry out a rotation, or describe a rotation, you need three pieces of information:

14.6 - Enlarging shapes

An enlargement of a shape is a copy of the shape that changes the lengths but keeps the same proportions. 

In an enlargement, all angles stay the same size. 

The scale factor is what the 1 unit of the shape before enlargement, is after the enlargement.

The scale factor is 2.

The centre of enlargement tells you where to draw the image on a grid. In this case, as the scale factor is 2, not only must the image be twice the size of the object, it must also be twice the distance from centre of enlargement.

You can check you have drawn an enlargement correctly by drawing lines through the corresponding vertices of the object and image.

All the lines should meet at the centre of the enlargement.