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15.1 - Converting between miles and kilometres

One kilometre is about 1.6 miles, which is 5/8 of a mile.

To convert a distance in kilometres to a distance in miles, divide the distance by 8 and then multiply by 5.To convert a distance in miles to a distance in kilometres, divide the distance by 5 and multiply by 8

15.2 - The area of a parallelogram and a trapezium

The formula for the area of a parallelogram is base x height otherwise, b x h.

The formula for the area of trapezium is 1/2(a +b) x l. “a” being the top part of the trapezium, and b being the bottom part of the trapezium.

15.3 - Calculating the volume of triangular prisms

A cuboid is a rectangular prism. The formula for volume of this is l x b x h.

In order to calculate the volume of a triangular prism, or in fact any prism that has a cross section(the shape that is the side elevation) other than a rectangle or a square, is area of cross section x height.

15.4 - Calculating the surface area of triangular prisms and pyramids

The best way to calculate surface area of 3D shapes is to look into their 3D nets.

After calculating the area of one of the shapes that’s part of the 3D net, and multiplying it correspondingly, and adding up all of the values, we can get the total surface area.

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14.1 - Bearings

A bearing describes the direction of one object from another.

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:

the angle of the rotation

the direction of the rotation (clockwise or anticlockwise)

the coordinates of the centre of rotation.

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.

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12.1 - Simplifying ratios

A ratio is a way of comparing two or more quantities.

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:

1.  Add the parts of the ratio together. This will give you the total number of parts to divide the quantity into.

2.  Divide the quantity by the sum of the parts. This will give you the value of each part.

3.  Multiply the value of each part by the corresponding number in the ratio. This will give you the share for each person or thing.

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

So each part is worth $600 / 6 = $100.

Zara paid 2 parts, so she gets 2 x $100 = $200.

Sofia paid 3 parts, so she gets 3 x $100 = $300.

Marcus paid 1 part, so he gets 1 x $100 = $100.

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.

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11.1 - Functions

Different situations can be represented as a function.

This sort of function has a value per variable and a fixed change.

For example, The cost of hiring a hall is in two parts.
There is :
a booking fee of $15
a charge of $40 per hour

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 :

The gradient of a linear function is the slope of the line that represents the function. It describes how steep the line is.

The y-intercept of a linear function is the point where the line crosses the y-axis.

The equation for a linear function is written in the form y = mx + c, where:

m is the gradient of the line

c is the y-intercept

x is the input value

y is the output value

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10.1 - Percentage increases and decreases

To understand percentage increase and decrease, you need to know absolute change.

Absolute change is the difference between a set of values.

A percent increase or decrease can be found using a formula.

Here are the steps for finding the percentage change:

1.  Find the difference between the two numbers. This is called the absolute change.

2.  Divide the absolute change by the original number.

3.  Multiply the answer by 100% to express the answer as a percentage.

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.

To find the multiplier for an increase, add the percentage increase to 100%.

To find the multiplier for a decrease, subtract the percentage decrease from 100%.

Once you have the multiplier, multiply the original value by the multiplier to find the new value.

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

The multiplier for a 65% increase is 165% (100% + 65%).

Multiplying the original price of $275 by 1.65 gives you $453.75, which is the new price after the increase.

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9.1 - Generating sequences

You can find out then next number(term) in a sequence by finding out the term-to-term rule, the change between two consecutive terms.

The rule is to add 1 to get the next number in the sequence.

So the first number in the sequence is any number you   choose, and the next number in the sequence is 1 more than the first number.

For example, if you start with 3, then the next number in the sequence is 4, then 5, and so on.

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.

A function is a relationship between two sets of numbers, called the input and the output.

You can represent a function as a function machine, where the input goes in, and the output comes out.

You can also represent a function as a mapping diagram, which shows how the input numbers map to the output numbers.

You can also represent a function algebraically as an equation, using letters to represent the input and output numbers.

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8.1 - Quadrilaterals and polygons

A quadrilateral is a flat, four-sided shape with straight sides.
Each quadrilateral possesses certain properties.

The main quadrilaterals that are important are :

- Kite
- Rhombus
- Parallelogram
- Rectangle
- Square
- Trapezium
- Isosceles Trapezium

8.2 - The circumference of a circle

The circumference is the distance around the circle.

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

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7.1 - Fractions and recurring decimals

A recurring decimal is a decimal that never ends and repeats a block of digits forever.

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:

1. Write them as fractions with the same denominator (common denominator).

2. Write them as decimals.

7.3 - Subtracting mixed numbers

1. Convert each mixed number to an improper fraction. A mixed number is a whole number plus a fraction. An improper fraction has a numerator (the top number) that is bigger than the denominator (the bottom number).

2. Subtract the improper fractions.

3. Simplify the answer. This means writing the fraction in its lowest terms.

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:

1. Write the integer as a fraction with a denominator of 1.

2. Flip the fraction you are dividing by (find the reciprocal).

3. Multiply the two fractions.

4. Simplify the answer as much as possible.

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6.1 - There are 3 different types of data

Discrete data : Data that can only take certain values is called discrete data or discrete values. This is data that can be counted and has a limited number of values. For example, the attendance at a football game is discrete data.

Continuous data : Continuous data refers to data that can be measured. This data has values that are not fixed and have an infinite number of possible values. For example, height and weight is continuous data.

Categorical data : Categorical variables represent types of data which may be divided into groups. For example, gender is categorical data.

There are also ways to collect data, such as, making observations, interviewing or questionnaire. A good sample is also necessary for accuracy.

6.2 - Sampling

Sampling is the process of choosing a small group of individuals to represent a larger population. The text says that you should choose the best way to choose your sample based on your investigation.

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

The size of the sample: The size of your sample will depend on your investigation. A larger sample will be more accurate, but it can also be more expensive and time-consuming to collect data from a larger sample.

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

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4.1 - Ordering decimals

Decimals are ordered by comparing their digits one place value at a time, starting from the leftmost (whole numbers) and moving to the right (tenths, hundredths, thousandths, etc.).

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

1. Ignore the decimals for now. Multiply the numbers as if they were whole numbers.

2. Count the total number of decimal places in the two numbers you originally multiplied.

3. Place the decimal point in the product: Count the same number of digits to the right in the product as the total number of decimal places you counted in step 2.

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.

5.67 divided by 0.7 is the same as

(5.67 x 10) divided by (0.7 x 10) which is

56.7 divided by 7.

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

Place value of decimals: You can use the place value of decimals to understand how much each digit is worth. For example, in the number 3.14, the 3 is in the ones place and is worth 3. The 1 is in the tenths place and is worth one-tenth (0.1). This can help you line up the numbers correctly when you add or subtract decimals.

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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.

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Unit 2 - Expressions, formulae and equations

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Unit 2 - Expressions, formulae and equations

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2.1 - Constructing expressions

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.

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3.1 - Multiplying and dividing by 0.1 and 0.01

In this exercise, we learnt how to multiply and divide by 0.1 and 0.01. To multiply by 0.1, we simply reverse the operation and make it 10 and we get the result. Same applies for 0.01 except we change it to 100.

E.g.  5 x 0.1
5 / 10 = 0.5

E.g.   87 x 0.01
87 / 100 = 0.87

3.2 - Rounding

In this exercise, we learnt rounding to different decimal places and significant figures. If we were to round off to the 2nd decimal place in a number, we would check if the next number is above or below than 5 and round off accordingly. Significant figures however, if it’s rounded off to the 2nd significant figure, we find the second number (a recurring 0 doesn’t count) and we round it off the same way.

E.g.   54.678 (1 d.p.)
54.7

E.g.   54.678 (1 s.f.)
50

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Integers 1.1 - Factors, multiples and primes

In this exercise we learnt how to do prime factorisation and how with any number, we can find their prime factors. We use this technique to also find HCF and LCM

Integers 1.2 - Multiplying and dividing integers

In this exercise we learnt the rules about multiplying and dividing negative and positive integers. We also learnt how to also deal with exponents and brackets.

Rules
+ x + = +
- x - =  +
+ x - = -
- x + = -

E.g.
3 x 3  = 9
-3 x -3 = 9
3 x -3 = -9
-3 x 3 = -9

Integers 1.3 - Square roots and cube roots

In this exercise we learnt about finding the squares and cubes and square and cube roots of positive and negative and positive numbers.

Note - Square roots of negative numbers are not defined.
Note - Cube or cube root of any negative number will be negative.
Note - Square root of any positive number is defined as that square root negative and positive.
Note - A negative number needs a bracket to be squared or cubed into a positive number. Otherwise it will be a negative number.

E.g. 3³ = 27
-(3)³ = 27
- 3³ = -27

1.4 -  Indices

In this exercise we used positive and zero indices to represent numbers and in multiplication and division.

For multiplication, we add the indices together to get the result

E.g. 3^3 x 3^5 = 3^8

For division however, we subtract the indices and get the the result.

E.g.  3^8/3^5 = 3^3

We also learnt about indices in brackets in which we multiply the two indices to get the result.

E.g. (3^3)^3 = 3^9

Grade 8 -

Keywords :
SURDS - sq. root of non-perfect squares and cube root of non-perfect cubes