GFS Maths Year 9 Revision Booklet

Student Name: Student Tutor Group: Maths Class: Maths teacher:

How to use this revision booklet Repeat these steps till you are 100% with EVERY topic. Step 1 Look at the scheme of work – this lists all the topics you are learning this year, and those in previous years. Highlight or note down the topics you are not 100% confident with. Step 2 Revise the topics you need to go over, using a revision guide or any of the following resources, ranked from 1st to 5th in how useful:  Hegartymaths.com (videos and practice questions for EVERY TOPIC UP TO GCSE) Choose any lesson from each of the 6 strands or search for the lesson in the search bar at the top

 corbettmaths.com (really good resource, with videos and lots of practice)  khanacademy.com (helpful videos walked through clearly)  mathsisfun (good examples to work through and understand)  GCSE bbc bitesize (clear examples in sections)

Step 3 Practise questions on that topic using this revision booklet, and the other resources. Step 4: Ask your maths teacher for extra help if you are still stuck!

In this booklet there will be information and questions on the topics covered in Year 9. Therefore if you see any topics in the Year 7 or year 8 schemes of work that you think you need to revise, visit the websites mentioned above or speak to your teacher. In Year 7 you should have studied the following topics:

In Year 8 you should have studied the following topics:

Note, some pupils may have covered less material dependent on their starting point.

In Year 9 you are studying the following topics:

This booklet contains questions on all the topics above. The expectation is that you use this booklet to practise questions, and you revise using the websites/resources listed earlier. Remember, your maths teacher can help you with things you are stuck on. Remember: The best way to learn in maths is to try a question, check your answer, find your mistake, learn from that mistake, revise the topic and then attempt another question without making that mistake again.

Good luck!

Autumn 1 - Graphs and proportion Cartesian coordinates

HINT: Finding the distance – use Pythagoras theorem. (using the distance along (the difference in x co-ordinates) and the distance up (the difference in y-coordinates)). Remember pythagoras: a2 + b2 = c2 (the squares of the two shorter sides added together are equal to the longest side squared).

Linear graphs

HINT: The general equation of a line is y=mx+c – m is the gradient, c is the y-intercept. So a graph of y=3x-4 has a gradient of 3 (goes up 3, for every 1 it goes along), and crosses the y-axis at -4.

Slight difference: finding the equation of a line after re-arranging – you need to make the line in the form y=mx+c. So you need to make y the subject of the formula. Make y the subject, and then state the gradient (m) and the y-intercept (c): a) b) c) d) e) f)

y-1 = x y+5=x y–x=1 2y = 4x + 2 y +3 = 2x + 4 3y – 2 = 6x + 1

Direct proportion Direct proportion refers to two variables relating to each other at a constant rate. It does not mean both variables increase/decrease by the same amount each time. But as one variable increases or decreases, the other variable increases or decreases accordingly. E.g. as variable a increases by 1 each time, variable b might increase by 2 each time. The formula for this would be b = 2a.

We can use y = kx (where k is the constant of proportionality) to solve proportion problems.

Inverse proportion Inverse proportion is when one value increases as the other value decreases. A simple example of inversely proportional quantities is the lengths and widths of rectangles with the same area. As the length of one side doubles, the width has to be halved for the area to stay the same.

Here the formula will be

as it is inverse proportion.

Standard form Standard form is a method of expressing very large or very small numbers more easily. The general format is ‘a x 10b’ where a must be between 1 and 10 including 1 but not including 10 (1 < a < 10) and b can be any value. For instance instead of 3000, we would write 3 x 103. Instead of 0.0025 we would write 2.5 x 10-3. You need to make sure the power used will move the digits the correct number of places.

When adding/subtracting with standard form, you need to evaluate the number first and then use the column method to add them. (e.g. 3.25 x 103 – 5 x 102 = 3250-500 = 2750). When multiplying/dividing, you can treat the terms separately. (e.g. (5 x 102) x (3 x 103) = 5 x 3 x 102 x 103 = 15 x 105 = 1.5 x 106 (in standard form). This uses indices rules. Be careful to ensure you leave your answer in standard form if it asks for this.

Autumn 2 – Algebraic expressions Arithmetic sequences Arithmetic sequences increase or decrease at a fixed rate. They are like the times tables, but shifted up or down. E.g. the sequence 4, 8, 12, 16, 20 would be 4n. But 5, 9, 13, 17, 21 would be 4n + 1 (the 4 times table shifted up 1).

Geometric sequences Geometric sequences (e.g. 1, 2, 4, 8, 16, 32 …) are sequences that do not increase/decrease at a fixed rate. You need to be able to spot these.

Algebraic manipulation Always remember to collect like terms and simplify answers fully. This does not mean putting everything together. Remember 2x + 5g cannot be simplified. The classic mistake students make is saying a + b + c = abc (it does not!). But 2x + 3x can and should be simplified to 5x. Remember a + b is different to ab (which means a x b). Algebra: Collecting Like Terms Collecting Like Terms Simplify (HINT: think about fruit in a fruit bowl it cannot be mixed, so we cannot mix different letters) a) 3g + 5g = b) e + f + e + f + e = c) 5p + 2q – 3p – 3q = d) 2xy + 3xy – xy = e) 5x² + 2x – 3x² = (Total 5 Marks) Simplify a) p x p x p x p = b) 2r x 5p = c) 4p x 2q = d) 3b x 4b² x 2c² = (Total 5 Marks)

Algebra: Expanding and Factorising Brackets.

Expanding Brackets (HINT: all terms in the bracket are multiplied by the term outside the bracket) a) Expand 3(x + 2) b)

Expand 3(5p – 2)

c)

Expand 4x(1 + 3x)

d)

Expand 7a(2a – 3)

e)

Expand and Simplify 4b(3 +2b) - b²

f)

Expand and Simplify 2(3x + 1) + 5(3x – 1)

g)

Expand and Simplify 2(r + 3) + 3(2r + 1)

(Total 10 Marks) Factorising Brackets (HINT: The HCF goes outside of the bracket) a) Factorise 5t + 20 b)

Factorise 8p – 6

c)

Factorise 5x + 15

d)

Factorise 16y – 4y

e)

Factorise 48f + 6f

f)

Factorise y³ - y²

g) Factorise 24xy + 6x² (Total 10 Marks)

Algebra: Substitution. Substitution Positive (HINT: think in sport you take a player off and replace him/her with another player – in algebra we replace letters with numbers) If x = 6 and y = 2, calculate the following:

a) x² b) 5x + y c) X + y² d) y + 16 x

(Total 5 Marks)

Substitution Negative

a) Work out the value of 2a + ay, when a = -5 and y = -3

b) Work out the value of 5t² - 7, when t = -4

(Total 5 Marks)

Substitution into a Formula

V = 3b + 2b²

a) Find the value of V, when b = 4.

h = 5t² + 2

b) Work out the value of h, when t = -2

V = u + at

c) Work out the value of V, when a = 4, t = 3, u = 23

d) Work out the value of U, when v = 30, a = 2, t = 8

Change the subject of a formula This is where we re-arrange an equation to make a variable the subject. The key thing to remember is that whatever you do to one side, you must do to the other side. E.g. if you subtract 3 from the left-hand side, you must subtract 3 on the right-hand side.

Expansion Expansion means multiplying out. Everything outside, should be multiplied by everything inside. Remember to collect like terms and simplify after.

Factorisation Factorisation is the reverse of expansion. Instead you are now finding the biggest factor (including algebraic terms) to put outside the brackets. E.g. 2x2 + 4x factorises to 2x(x+2). You can check your factorisation by expanding and you should get back to the original answer.

For quadratics (ax2 + bx + c), you need two numbers that add to b, and multiply to c. e.g. x2 + 7x + 12 factorises to (x+3)(x+4). (3 + 4 = 7 and 3 x 4 = 12).

Spring 1 – 2-D geometry Construction and loci This relies on a pair of compasses, and accurate drawings, forming arcs from the points required (see Corbett/hegarty for videos).

Triangles and quadrilaterals

Types of triangles and angles in triangles

Types of quadrilaterals and angles in quadrilaterals Name each of the following shapes:

Congruence Congruence means identical. The shapes must be exactly the same in all dimensions. They can be rotated/reflected/translated, but they are still the exact same shape.

Similarity Similar shapes, in maths, mean the dimensions of the two shapes are still in proportion, but they are not congruent i.e. the image is an enlargement of the object (either bigger or smaller but proportionally). E.g. all the lengths might have been multiplied by 2.

Angles in polygons

Spring 2 – Equations and inequalities Construct and solve equations and inequalities Solving equations means finding the unknown. To do this, you must isolate it, remembering to do exactly the same operations to both sides to keep it equal.

Bronze

Silver

Gold

Inequalities are just like equations, except instead of the equals sign, you have one of four signs:

You solve them in exactly the same way. They can be represented on a number line with an arrow. The circle should be coloured in if it can be equal to that number or hollow if it cannot e.g. x