GCSE Mathematics: Number - TSSmaths



GCSE Mathematics – Aiming for an A or Better

Grade Criteria and exemplar examination questions to get a Grade A or A* in the following topics:

1. Surds

2. Recurring Decimals

3. Limits of Accuracy

4. Indices

5. Proportionality

6. Rearranging Formulae

7. Algebraic Fractions

8. Using Graphs

9. Quadratic Equations

10. Simultaneous Equations

11. Algebraic Proofs

12. Circle Theorems

13. Trigonometry – for triangles which are not right-angled

14. Vectors

15. Similar Triangles

16. Congruent Triangles

17. Scale Factors – for volumes & surface areas of similar shapes (2D or 3D)

18. Stratified Samples

19. Histograms

20. Tree Diagrams

21. Mixed Questions

NC = non-calculator question

GCSE Mathematics: Surds

|Grade Criteria |Can Do |

|To get a grade B you must be able to: | |

|Work out the square roots of some decimal numbers | |

|To get a grade A you must be able to: | |

|Simplify surds by rewriting with the smallest integer inside the root | |

|To get a grade A* you must be able to: | |

|Simplify surds by rationalising denominators | |

|Manipulate & simplify expressions involving surds | |

Question 1

[i]

[pic]

[ii]

[pic]

GCSE Mathematics: Recurring Decimals

|Grade Criteria |Can do |

|To get a grade A you must be able to: | |

|Convert recurring decimals to fractions | |

Question 2

[pic]

[ii]

Convert the recurring decimal [pic]to a fraction.

………………………………

(2)

GCSE Mathematics: Limits of Accuracy

|Grade Criteria |Can Do |

|To get a grade B you must be able to: | |

|Find limits of accuracy for numbers given to whole number accuracy | |

|To get a grade A you must be able to: | |

|Find limits of accuracy for numbers given to d.p. or s.f. accuracy | |

|To get a grade A* you must be able to: | |

|Find limits of accuracy for compound measures (speed, gradient, etc) | |

Question 3

(i)

[pic]

(ii)

[pic]

GCSE Mathematics: Indices

|Grade Criteria |Can Do |

|To get a grade C you must be able to: | |

|Multiply and divide numbers in index form | |

|To get a grade A you must be able to: | |

|Know how to use indices rules for negative & fractional powers | |

Question 4

[pic]

GCSE Mathematics: Proportionality

|Grade Criteria |Can Do |

|To get a grade A you must be able to: | |

|Find formulae that describe direct & inverse proportionality (or variation) and use them to solve problems | |

|To get a grade A* you must be able to: | |

|Solve variation problems involving 3 variables | |

Question 5

[i]

[pic]

[ii]

[pic]

GCSE Mathematics: Rearranging Formulae

|Grade Criteria |Can Do |

|To get a grade B you must be able to: | |

|Rearrange more complicated formulae | |

|To get a grade A you must be able to: | |

|Rearrange formulae where the subject appears twice | |

|To get a grade A* you must be able to: | |

|Rearrange formulae where the subject appears with a power | |

Question 6

[pic]

GCSE Mathematics: Algebraic Fractions

|Grade Criteria |Can Do |

|To get a grade B you must be able to: | |

|Solve linear equations involving algebraic fractions where the subject (x) is part of the numerator | |

|To get a grade A you must be able to: | |

|Combine algebraic fractions using all 4 operations | |

|To get a grade A* you must be able to: | |

|Simplify algebraic fractions by factorising & cancelling | |

|Solve equations containing fractions with algebraic denominators | |

Question 7

(i)

[pic]

(ii)

[pic]

GCSE Mathematics: Using Graphs

|Grade Criteria |Can Do |

|To get a grade B you must be able to: | |

|Work out the coordinates in a table of values and plot graphs of the form y = ax2 + bx + c (quadratic), y = ax3 (cubic), y = | |

|a/x (reciprocal) | |

|To get a grade A you must be able to: | |

|Give the equation of a line parallel or perpendicular to given lines & passing through a given point | |

|Interpret more complex real-life graphs | |

|To get a grade A* you must be able to: | |

|Transform graphs of a given function with the equation either written in f(x) form or fully specified | |

|Identify a function’s equation from its graph which has been transformed from a known function | |

|Solve equations using the intersection of two graphs | |

Question 8

[pic]

[pic]

GCSE Mathematics: Quadratic Equations

|Grade Criteria |Can Do |

|To get a grade B you must be able to: | |

|Factorise a quadratic expression of the form x2 + bx + c | |

|Solve a quadratic equation of the form x2 + bx + c = 0 | |

|To get a grade A you must be able to: | |

|Solve a quadratic equation of the form ax2 + bx + c = 0 by factorising or by using the formula | |

|To get a grade A* you must be able to: | |

|Solve quadratic equations by completing the square | |

|Solve real problems requiring constructing & solving quadratic equations | |

Question 9

(i) [pic]

(ii)[pic] [pic]

GCSE Mathematics: Simultaneous Equations

|Grade Criteria |Can Do |

|To get a grade B you must be able to: | |

|Solve 2 linear simultaneous equations | |

|To get a grade A you must be able to: | |

|Set up & solve 2 linear simultaneous equations from practical problems | |

|To get a grade A* you must be able to: | |

|Solve 1 linear & 1 non-linear (probably quadratic, typically the equation of a circle) simultaneous equations | |

Question 10

(i)

[pic]

[pic]

GCSE Mathematics: Algebraic Proofs

|Grade Criteria |Can Do |

|To get a grade B you must be able to: | |

|Verify results by substituting numbers into them | |

|Understand the proofs of simple theorems (an exterior angle of a triangle is the sum of the two opposite interior angles, etc) | |

|To get a grade A you must be able to: | |

|Show an algebraic statement is true by using both sides to justify your answer | |

|To get a grade A* you must be able to: | |

|Prove algebraic results using rigorous & logical mathematical arguments | |

Question 11

[i]

[pic]

[ii]

[pic]

[iii]

(a) Write down an expression, in terms of n, for the nth multiple of 5.

.............................

(1)

(b) Hence or otherwise

(i) prove that the sum of two consecutive multiples of 5 is always an odd number,

(ii) prove that the product of two consecutive multiples of 5 is always an even number.

(5)

(Total 6 marks)

GCSE Mathematics: Circle Theorems

|Grade Criteria |Can Do |

|To get a grade B you must be able to: | |

|Find missing angles in circles | |

|To get a grade A you must be able to: | |

|Use the alternate segment theorem to find missing angles in circles | |

|To get a grade A* you must be able to: | |

|Use circle theorems to prove geometrical results | |

Question 12

[i]

[pic]

[pic]

GCSE Mathematics: Trigonometry

|Grade Criteria |Can Do |

|To get a grade B you must be able to: | |

|Use trigonometry to find the length of any side in right-angled triangles | |

|Use trigonometry to solve problems involving right-angled triangles | |

|To get a grade A* you must be able to: | |

|Use the sine rule & cosine rule to find missing lengths or angles | |

|Find the area of triangles using Area = ½ab sin C | |

|Solve 3D problems using SOH CAH TOA trigonometry | |

|Solve simple equations involving sin, cos or tan, using the shape of the basic graphs or knowledge of CAST diagrams to find all | |

|possible solutions | |

Question 13

[i]

[pic]

[pic]

[pic]

[pic]

[iii]

[pic]

[pic][pic][pic]

GCSE Mathematics: Vectors

|Grade Criteria |Can Do |

|To get a grade A you must be able to: | |

|Solve problems involving addition & subtraction of vectors | |

|To get a grade A* you must be able to: | |

|Solve more complex geometrical problems using vectors | |

Question 14

[i]

[pic]

[ii]

[pic]

GCSE Mathematics: Similar Triangles

|Grade Criteria |Can Do |

|To get a grade B you must be able to: | |

|Set up equations to find missing sides in similar triangles | |

|To get a grade A* you must be able to: | |

|Solve practical problems involving similar triangles | |

Question 15

[pic]

GCSE Mathematics: Congruent Triangles

|Grade Criteria |Can Do |

|To get a grade B you must be able to: | |

|Know the conditions to show that two triangles are congruent | |

|To get a grade A you must be able to: | |

|Prove that two triangles are congruent | |

Question 16

[pic]

GCSE Mathematics: Scale Factors

|Grade Criteria |Can Do |

|To get a grade A you must be able to: | |

|Solve problems using area or volume scale factors for similar shapes | |

|To get a grade A* you must be able to: | |

|Solve related problems e.g. involving mass or capacity | |

Question 17

[pic]

[pic]

GCSE Mathematics: Stratified Samples

|Grade Criteria |Can Do |

|To get a grade A you must be able to: | |

|Calculate the numbers required for a stratified sample | |

Question 18

[i]

[pic]

GCSE Mathematics: Histograms

|Grade Criteria |Can Do |

|To get a grade A you must be able to: | |

|Draw histograms given data in a frequency table | |

|Complete a frequency table given a histogram | |

Question 19

[i]

[pic]

GCSE Mathematics: Tree Diagrams

|Grade Criteria |Can Do |

|To get a grade B you must be able to: | |

|Draw & use a tree diagram to work out the probability of combined events | |

|To get a grade A you must be able to: | |

|Use “and” / “or” knowledge to find probabilities of specific combinations | |

|To get a grade A* you must be able to: | |

|Draw & use a tree diagram to work out dependent probabilities | |

Question 20

[i]

Joan has two boxes of chocolates.

The boxes are labelled A and B.

Box A contains 15 chocolates. There are 6 plain, 4 milk and 5 white chocolates.

Box B contains 12 chocolates. There are 4 plain, 3 milk and 5 white chocolates.

Joan takes one chocolate at random from each box.

Work out the probability that the two chocolates Joan takes are not of the same type.

………………………………

(4)

[pic][pic]

GCSE Mathematics: Mixed Questions

Question 21

The depth, D metres, of the water at the end of a jetty in the afternoon can be modelled by this formula

D = 5.5 + A sin 30(t – k)(

where

t hours is the number of hours after midday,

A and k are constants.

Yesterday the low tide was at 3 p.m.

The depth of water at low tide was 3.5 m.

Find the value of A and k.

A = ………………………

k = ………………………

(4)

[pic][pic][pic]

| | |

|y is directly proportional to x |......................... |

|y is inversely proportional to x |......................... |

|y is proportional to the square of x |......................... |

|y is inversely proportional to the square of x |......................... |

(Total 2 marks)

-----------------------

x

y

Graph A

x

y

–12

–8

O

12

4

6

12

–4

–8

–12

–4

4

8

x

y

Graph D

x

y

Graph C

x

y

Graph B

................
................

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