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Pearson Edexcel Level 1/Level 2 GCSE (9 – 1) in Mathematics (1MA1)

Higher tier diagnostic document

For first teaching from September 2015

Contents

Introduction 5Higher course overview 6Higher units 7

Pearson Edexcel Level 1/Level 2 GCSE (9 – 1) in Mathematics Higher tier diagnostic document – Issue 1 April 2016 © Pearson Education Limited 2016

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Introduction

This Higher tier diagnostic document is intended to support students in accessing the Higher tier of the new GCSE (9–1) Mathematics specification.

This document lists the units in the Higher tier scheme of work, suggests questions to establish whether a student has the required prior knowledge, and provides a mapping of references to the Foundation scheme of work (and occasionally the Access to Foundation tier scheme of work) should the student need to refresh their understanding or develop a particular skill. Teachers can then turn to the relevant unit(s) in the Foundation scheme of work for additional support, including objectives, possible success criteria, opportunities for reasoning and problem-solving, and common misconceptions.

For later Higher tier units, prior knowledge has sometimes not been covered in the Foundation scheme of work. In these instances, a reference to an earlier Higher tier unit is provided, along with diagnostic questions to check that this knowledge has been acquired.

Our free support for the GCSE Mathematics specification (1MA1) can be found on the Edexcel mathematics website (http://qualifications.pearson.com/en/home.html) and on the Emporium (www.edexcelmaths.com).


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Unit Title

1a Calculations, checking and roundingb Indices, roots, reciprocals and hierarchy of operationsc Factors, multiples, primes, standard form and surds

2a Algebra: the basics, setting up, rearranging and solving equationsb Sequences

3 a Averages and rangeb Representing and interpreting data and scatter graphs

4 a Fractions and percentagesb Ratio and proportion

5 a Polygons, angles and parallel linesb Pythagoras’ Theorem and trigonometry

6a Graphs: the basics and real-life graphsb Linear graphs and coordinate geometryc Quadratic, cubic and other graphs

7a Perimeter, area and circlesb 3D forms and volume, cylinders, cones and spheresc Accuracy and bounds

8 a Transformationsb Constructions, loci and bearings

9 a Solving quadratic and simultaneous equationsb Inequalities

10 Probability11 Multiplicative reasoning 12 Similarity and congruence in 2D and 3D

13 a Graphs of trigonometric functionsb Further trigonometry

14 a Collecting datab Cumulative frequency, box plots and histograms

15 Quadratics, expanding more than two brackets, sketching graphs, graphs of circles, cubes and quadratics

16 a Circle theorems b Circle geometry

17Changing the subject of formulae (more complex), algebraic fractions, solving equations arising from algebraic fractions, rationalising surds, proof

18 Vectors and geometric proof

19a Reciprocal and exponential graphs; Gradient and area under

graphsb Direct and inverse proportion


Foundation tier

UNIT 1: Powers, decimals, HCF and LCM, positive and negative, roots, rounding, reciprocals, standard form, indices and surds

Return to OverviewSUB-UNITS

a Calculations, checking and roundingb Indices, roots, reciprocals and hierarchy of operationsc Factors, multiples, primes, standard form and surds

PRIOR KNOWLEDGE

Students will be able to: Possible diagnostic questions

Students will need to work on the objectives covered in:

understand place value, order integers and decimals and use the four operations

Given the digits 2, 5, 7 and 9, make all the possible three-digit number with one decimal place and put them in order.

Addition, subtraction, multiplication and division questions with up to three digits and one decimal place

Foundation Unit 1: Number, powers, decimals, HCF and LCM, roots and rounding

find integer complements to 10 and to 100

46 + = 100 Foundation Unit 1a: Integers and place value

See also Access Unit 5: Addition and subtraction 2

recall multiplication facts to 10 × 10

Quick-fire multiplication and division questions. e.g.6 × 7 =8 × 9 =35 ÷ 5 =132 ÷ 12 =

Foundation Unit 1a: Integers and place value

multiply and divide by 10, 100 and 1000

Multiply 24.75 by 10, 100, 1000

Divide 72430 by 10, 100, 1000.


recall and identify squares, square roots, cubes and cube roots

Which of these numbers is a square number? Which is a cube? Explain your answers.2, 5, 8, 12, 16, 20, 28

Foundation Unit 1c: Indices, powers and roots


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Foundation tier

UNIT 2: Expressions, substituting into simple formulae, expanding and factorising, equations, sequences and inequalities, simple proof


a Algebra: the basics, setting up, rearranging and solving equationsb Sequences

PRIOR KNOWLEDGE



use negative numbers with the four operations, recall and use the hierarchy of operations and understand inverse operations

4 – (–6) =

–6 × 3 =

18 ÷ = –3

4 × 7 – 16 ÷ 2 =



deal with decimals and negatives on a calculator

Use a calculator to calculate:

–6.5 × –4.2 =


use index laws numerically

43 × 45 =

67 ÷ 62 =



Foundation tier

UNIT 3: Averages and range, collecting data, representing data


a Averages and rangeb Representing and interpreting data and scatter graphs

PRIOR KNOWLEDGE



read scales on graphs, draw circles, measure angles and plot coordinates in the first quadrant

On cm-squared paper, draw axes for x and y from 0 to 8. Plot these points: (1, 0), (2, 6), (7, 8). Join to make a triangle. Measure the angles.

On the same coordinate grid, use a pair of compasses to draw a circle centre (5, 4), radius 4 cm. What are the coordinates of the point where the circle touches the x-axis?

Foundation Unit 3a: Tables, charts and graphs

Foundation Unit 3b: Pie charts

Foundation Unit 6a: Properties of shapes, parallel lines and angle facts

Foundation Unit 15a: Plans and elevations

use tally charts What number does this represent?

Write 24 in tallies.

Foundation Unit 3: Drawing and interpreting graphs, tables and charts

See also Access Unit 22: Data handling 2

use inequality notation Take a pair of two-digit numbers and use < and > correctly. e.g. 46 and 78 or 62 and 35


find the midpoint of two numbers

What number is in the middle of 3 and 9? 42 and 50?

Foundation Unit 7: Statistics, sampling and the averages

See also Access Unit 22: Data handling 2


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Foundation tier

UNIT 4: Fractions, percentages, ratio and proportion


a Fractions and percentagesb Ratio and proportion

PRIOR KNOWLEDGE



use the four operations of number

See questions for Unit 1 Foundation Unit 1a: Integers and place value

find common factors What factor is common to 8 and 12? To 14 and 35?

Foundation Unit 1d: Factors, multiples and primes

understand fractions as being ‘parts of a whole’

Shade of

Shade of

Foundation Unit 4a: Fractions, decimals and percentages

See also Access Unit 11: Fractions, decimals and percentages 2

understand percentage as ‘number of parts per hundred’ and recognise that percentages are used in everyday life

Shannon got the questions

in a test correct. What is as a percentage?

In a sale, prices are reduced by 10%. What is 10% as a fraction?

Foundation Unit 4b: Percentages


30°x

45°y

Foundation tier

UNIT 5: Angles, polygons, parallel lines; Right-angled triangles: Pythagoras and trigonometry


a Polygons, angles and parallel linesb Pythagoras’ Theorem and trigonometry

PRIOR KNOWLEDGE



rearrange simple formulae and equations

If t = 6h – 3, write an expression for h

Foundation Unit 2: Expressions, substituting into simple formulae, expanding and factorising

recall basic angle facts On squared paper, draw a right-angled triangle with one acute and one obtuse angle.

Find the size of the angles marked x and y.

Foundation Unit 6a: Properties of shapes, parallel lines and angle facts 6b - G3, G6

understand that fractions are more accurate in calculations than rounded percentage or decimal equivalents

≈ 0.3Which of the following give the most accurate answer?

× 50 = 16

0.3 × 50 = 15



10

12 cm

5 cm x cm

3 cm

6 cm

7 cm

Foundation tier

UNIT 6: Real-life and algebraic linear graphs, quadratic and cubic graphs, the equation of a circle, plus rates of change and area under graphs made from straight lines


a Graphs: the basics and real-life graphsb Linear graphs and coordinate geometryc Quadratic, cubic and other graphs

PRIOR KNOWLEDGE



identify coordinates of given points in the first quadrant or all four quadrants

Draw axes for values of x and y from –5 to +5.

Plot the points (2, 3), (–3, 2) and (–2, –3), which form three corners of a square. What are the coordinates of the fourth corner?

Foundation Unit 9a: Real-life graphs

use Pythagoras’ Theorem Find the length of the unknown side.

Foundation Unit 12: Right-angled triangles: Pythagoras and trigonometry

calculate the area of compound shapes

Find the area of this shape. Foundation Unit 8: Perimeter, area and volume

use and draw conversion graphs for common units

5 miles ≈ 8 kilometresDraw axes with scales from 0 to 80 km on the horizontal axis and 0 to 50 miles on the vertical axis. Plot a line to show the relationship between miles and kilometres.

Estimate 20 km in miles.Estimate 40 m in kilometres.


use function machines Find y when x = 3. Foundation Unit 1a: Integers and Pearson Edexcel Level 1/Level 2 GCSE (9 – 1) in Mathematics Higher tier diagnostic document – Issue 1 April 2016 © Pearson Education Limited 2016

×2 +511

Foundation tier



and inverse operations x → → = y

Find x when y = 11.x → → = y

place value

Foundation Unit 5a: Equations and inequalities


×3 –4

12

1.7 cm

Foundation tier

UNIT 7: Perimeter, area and volume, plane shapes and prisms, circles, cylinders, spheres, cones; Accuracy and bounds


a Perimeter, area and circlesb 3D forms and volume, cylinders, cones and spheresc Accuracy and bounds

PRIOR KNOWLEDGE



name and identify the properties of 3D forms

Sketch a cuboid, a cylinder and a square-based pyramid.

How many faces does each shape have? How many vertices? How many edges?

Foundation Unit 15a: Plans and elevations

find perimeter and area by measuring lengths of sides

Measure the sides of this rectangle. Find its perimeter and area.

Foundation Unit 8: Perimeter, area and volume

substitute numbers into an equation and give answers to an appropriate degree of accuracy

Use the formula A = πr2 to find the area of this circle. Give your answer to an appropriate degree of accuracy.


Foundation Unit 1b: Decimals


Foundation tier



understand the various metric units

Match each item to the most appropriate unit you could use to measure it.

mm capacity of an egg cupcm capacity of a bathm length of a pencil km diameter of a coin g mass of a horsekg journey from London

to Edinburghml mass of a mousel length of a room



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Foundation tier

UNIT 8: Transformations; Constructions: triangles, nets, plan and elevation, loci, scale drawings and bearings


a Transformationsb Constructions, loci and bearings

PRIOR KNOWLEDGE



recognise 2D shapes Make different shapes using two congruent right-angled triangles by matching equal sides, and name the shapes produced. (There are six: rectangle, kite, two parallelograms, two isosceles triangles.)

Foundation Unit 6: Angles, polygons and parallel lines

plot coordinates in four quadrants

See questions for Unit 6. Foundation Unit 9a: Real-life graphs

plot linear equations parallel to the coordinate axes

On cm-squared paper, draw axes for x and y from 0 to 8. Plot the lines x = 4 andy = –2.

Foundation Unit 9b: Straight-line graphs


Foundation tier

UNIT 9: Algebra: Solving quadratic equations and inequalities, solving simultaneous equations algebraically


a Solving quadratic and simultaneous equationsb Inequalities

PRIOR KNOWLEDGE



understand the ≥ and ≤ symbols

List the positive integers that satisfy the inequality 10 > x ≥ 6.

List the integers that satisfy the inequality 10 < y ≤ 14.


substitute into, solve and rearrange linear equations

What is the value of h in this formula, if C = 10?C = 5h + 20

Foundation Unit 2b: Expressions and substitution into formulae

factorise simple quadratic expressions

Factorise x2 – x – 6, Foundation Unit 16a: Quadratic equations: expanding and factorising

recognise the equation of a circle

Which of these equations is the equation of a circle?y = x2 + 16x2 + y2 = 52

x + y = 25

Higher Unit 6c: Quadratic, cubic and other graphs


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Foundation tier

UNIT 10: Probability

Return to OverviewPRIOR KNOWLEDGE



distinguish between events which are impossible, unlikely, even chance, likely, and certain to occur

Match to events to how likely they are to occur.

1 Christmas will fall on 25 December this year.

2 The sun will rise at midnight tonight.

3 You will score an even number if you roll an ordinary, fair dice.

4 The next person you meet likes chocolate.

5 If you buy a lottery ticket, you will win the jackpot.

A Impossible B UnlikelyC Even chance D LikelyE Certain

Foundation Unit 13: Probability

understand that a probability is a number between 0 and 1 and mark events and/or probabilities on a probability scale of 0 to 1

A bag contains 20 marbles. Tessa picks a marble at random.

Mark these probabilities on the number line.

P(blue) = P(red) =

P(green) = P(pink) = P(black) = 0 P(marble) = 1

0 1

Foundation Unit 13: Probability

add and multiply fractions and decimals

+ = × =

0.35 + 1.7 = 0.2 × 0.6 =


express one number as a fraction of another number

What is 15 as a fraction of 25?



Foundation tier


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Foundation tier

UNIT 11: Multiplicative reasoning: direct and inverse proportion, relating to graph form for direct, compound measures, repeated proportional change




find a percentage of an amount and relate percentages to decimals

What is 45% of 300?

What is the decimal equivalent of 6%?

Foundation Unit 4b: Fractions and percentages

rearrange equations and use these to solve problems

A square has sides of d + 3. A rectangle has sides of 3d + 1 and d – 3. They have the same length perimeter. Find d.


understand speed = distance/time, density = mass/volume

A car travels 70 miles in 2 hours. What is its average speed?

Cobalt has a density of 8.9 gm/cm3. What is the mass of a cm cube of cobalt?

Foundation Unit 14: Multiplicative reasoning


6 cm40° 35°

x ya

Foundation tier

UNIT 12: Similarity and congruence in 2D and 3D




recognise and enlarge shapes and calculate scale factors

Enlarge this triangle by a scale factor of 2.

Shape B is an enlargement of shape A. What is the scale factor?

B

A

Foundation Unit 10: Transformations

calculate area and volume in various metric measures

What is the area of a rectangle that measures 4.5

m by 6 m?

What is the volume of a cuboid that measures 2 mm by 5 mm by 7 mm?


measure lines and angles and use compasses, ruler and protractor to construct standard constructions

Use compasses and a ruler to construct this triangle accurately.

Measure the length of sides x and y and the size of angle a.

Foundation Unit 3b: Pie charts

Foundation Unit 6a: Properties of shapes, parallel lines and angle facts


Foundation Unit 15b: Constructions, loci and bearings


20

a cm32°

11 cm

Foundation tier

UNIT 13: Sine and cosine rules, ab sin C, trigonometry and Pythagoras’ Theorem in 3D, trigonometric graphs, and accuracy and bounds


a Graphs of trigonometric functionsb Further trigonometry

PRIOR KNOWLEDGE



use axes and coordinates to specify points in all four quadrants

See questions for Unit 6. Foundation Unit 9a: Real-life graphs

recall and apply Pythagoras’ Theorem and trigonometric ratios

See questions for Unit 6.

Use the cosine rule to find the value of a.

Foundation Unit 12: Right-angled triangles: Pythagoras and trigonometry

substitute into formulae See questions for Unit 9. Foundation Unit 5a: Equations and inequalities


Foundation tier

UNIT 14: Statistics and sampling, cumulative frequency and histograms


a Collecting datab Cumulative frequency, box plots and histograms

PRIOR KNOWLEDGE



understand the different types of data: discrete/continuous

Sort the following data into two groups: discrete and continuous.

A Heights of 10 studentsB Number of pets owned by

30 studentsC Favourite colours of 15

studentsD Mass of 20 apples

Foundation Unit 3a: Tables, charts and graphs

use inequality notation See questions for Unit 3. Foundation Unit 1a: Integers and place value

multiply a fraction by a number

What is of 48?


understand the data handling cycle

Put these four steps in the correct order.

A Analyse the data.B Draw conclusions.C Collect data.D Specify the problem and

plan an investigation.

Foundation Unit 7: Statistics, sampling and the averages


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Foundation tier

UNIT 15: Quadratics, expanding more than two brackets, sketching graphs, graphs of circles, cubes and quadratics




solve quadratics and linear equations

Solve these equations.3(x – 6) = 6x2 – 3x – 28 = 0


Foundation Unit 16: Algebra: quadratic equations and graphs

solve simultaneous equations algebraically

Solve these simultaneous equations:3x – y = 232x + y = 7

Foundation Unit 20: Rearranging equations, graphs of cubic and reciprocal functions and simultaneous equations


Foundation tier

UNIT 16: Circle theorems and circle geometry


a Circle theorems b Circle geometry

PRIOR KNOWLEDGE



draw circles with compasses

See questions for Unit 3. Foundation Unit 15a: Plans and elevations

recall the words, centre, radius, diameter and circumference

Use the following words to fill in the gaps.

centre circumference diameter radius

The ___ of a circle is a straight line from the ___ to the ___. It is half the length of the ___.

Foundation Unit 17: Circles, cylinders, cones and spheres

recall the relationship of the gradient between two perpendicular lines

Line A has gradient 2.Line B is perpendicular to Line A.Write down the gradient of Line B.

Higher Unit 6b: Linear graphs and coordinate geometry

find the equation of the straight line, given a gradient and a coordinate

Find the equation of the line with gradient 3 that passes through the point (2, 4).

Foundation Unit 20: Rearranging equations, graphs of cubic and reciprocal functions and simultaneous equations


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Foundation tier

UNIT 17: Changing the subject of formulae (more complex), algebraic fractions, solving equations arising from algebraic fractions, rationalising surds, proof




simplify surds Simplify .

Higher Unit 1c: Factors, multiples, primes, standard form and surds

use negative numbers with all four operations

See questions for Unit 2. Foundation Unit 1a: Integers and place value

recall and use the hierarchy of operations

See questions for Unit 2. Foundation Unit 1a: Integers and place value



Foundation tier

UNIT 18: Vectors and geometric proof




use vectors to describe translations

Write as a column vector the transformation that maps shape A onto shape B.

Foundation Unit 10: Transformations

use Pythagoras’ Theorem See questions for Unit 6. Foundation Unit 12: Right-angled triangles: Pythagoras and trigonometry

identify properties of triangles and quadrilaterals

See questions for Unit 8. Foundation Unit 6a: Properties of shapes, parallel lines and angle facts


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Foundation tier

UNIT 19: Direct and indirect proportion: using statements of proportionality, reciprocal and exponential graphs, rates of change in graphs, functions, transformations of graphs


a Reciprocal and exponential graphs; Gradient and area under graphsb Direct and inverse proportion

PRIOR KNOWLEDGE



draw linear and quadratic graphs

Sketch the following graphs.y = 2x – 3y = x2

Foundation Unit 9: Real-life and algebraic linear graphs

Foundation Unit 16b: Quadratic equations: graphs

calculate the gradient of a linear function between two points

A line passes through the points (1, 2) and (7, 5).Find the gradient of the line.


recall transformations of trigonometric functions

Sketch the graph of y = sin x.On the same axes, sketch the graph of y = 2 sin x.

Higher Unit 13a: Graphs of trigonometric functions

write statements of direct proportion and form an equation to find values

a is directly proportional to b.a = 18 when b = 1.5.Form an equation involving a and b and solve it to find the value of a when b = 7.

Foundation Unit 11b: Proportion


Issue 1 – April 2016

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