Unit 3 Foundation


Past Papers







Don't forget to visit the Unit 3 MOCK page to see the Foundation Mock from Dec 2011 and FULL VIDEO SOLUTIONS

A list of topics for Unit 3:
Unit
Module number
Title
Video Links
3
3-1
Percentage problems

3
3-2
Further fractions

3
3-3
Algebraic manipulation

3
3-4
Solving linear equations & inequalities

3
3-5
Substitution & changing the subject

3
3-6
Trial and improvement

3
3-7
Drawing & interpreting graphs

3
3-8
Polygons

3
3-9
Transformations

3
3-10
Constructions, loci, similarity and congruency

3
3-11
Pythagoras’ theorem

3
3-12
3-D Shapes, nets and plans

3
3-13
Circles & cylinders

3
3-14
Further units


A detailed breakdown:
MODULE
CONTENTS
PRIOR KNOWLEDGE
OBJECTIVES
DIFFERENTIATION & EXTENSION
NOTES
3-1
Percentage problems
Use percentages in real-life situations
Solve percentage problems, including increase and decrease in a variety of functional contexts
Four operations of number
An awareness that percentages are used in everyday life
Percentages (Unit 2)
By the end of the module the student should be able to:
• Find a percentage increase/decrease of an amount (2.2)
• Use a multiplier to increase by a given percent, eg 1.1 ´ 64 increases 64 by 10% (2.2)
• Calculate simple interest (2.1)
• Solve a variety of Functional elements questions involving percentages (2.1)
Combine multipliers to simplify a series of percentage changes
Percentages which convert to recurring decimals, eg 33 %, and situations which lead to percentages of more than 100%
Problems which lead to the necessity of rounding to the nearest penny, eg real-life contexts
Comparisons between simple and compound interest calculations (for very brightest)
Formulae in simple interest/compound interest methods
Amounts of money should always be rounded to the nearest penny where necessary, except where such rounding is premature
In preparation for this unit students should be reminded of basic percentages and recognise fraction-and-decimal equivalents from Unit 2
Use percentages in real-life situations:
• VAT
• Simple interest
• Income tax calculations
• Annual rate of inflation
• Find prices after a percentage increase or decrease
• Percentage profit and loss
3-2
Further fractions
Calculate a given fraction of a quantity, expressing the answer as a fraction
Understand ‘reciprocal’ as multiplicative inverse
Multiply and divide a given fraction by an integer, by a unit fraction and by a general fraction
Use a calculator to solve real-life problems involving fractions
Fractions (Units 1 & 2)
By the end of the module the student should be able to:
• Convert a fraction to a decimal, or a decimal to a fraction (2.1)
• Find the reciprocal of whole numbers, fractions, and decimals (1.1)
• Multiply and divide a fraction by an integer, by a unit fraction and by a general fraction (expressing the answer in its simplest form) (1.1)
• Know some common fractions as recurring decimals (and visa versa) (2.1)
• Use fractions in contextualised problems (1.1)
Use a calculator to find fractions of given quantities
Revise addition of fractions and improper/mixed fractions (Unit 1 and 2)
Use combinations of the four operations with fractions (and in real-life problems, eg to find areas using fractional values)
Constant revision of fractions is needed
All work needs to be presented clearly, with the relevant stages of working shown even if a calculator is used
Functional Elements contexts provide a rich source of questions using fractions in a real-life context
3-3
Algebraic manipulation
Simplify terms, products and sums
Multiply a single term over a bracket
Take out common factors
Know that a letter can be used to represent a number
Ability to use negative numbers with the four operations
Experience of using BIDMAS in calculations without a calculator
By the end of this Unit the student should be able to:
• Simplify expressions with like terms, eg x2 + 3x2; 3ab + 5ab + 2c2 (3.1, 3.2)
• Expand and factorise expressions with one pair of brackets,
eg expand x(2x + 3y); factorise 3xy2 - 6x2y (3.3 - 3.6)
• Expand and simplify expressions involving more than one pair of brackets, eg 3(x + 4) – 2(x – 3); (2x + 3)(3x – 4) (3.3 - 3.6)
Expand algebraic expressions like (x – 2) (x + 2) to illustrate the ‘middle’ term cancels
Multiply out two linear brackets (2x – 4) (3x + 2) (for brightest students)
Emphasise the correct use of symbolic notation, eg 3x2 rather than 3 ´ x2
Present all work neatly, writing out the questions with the answers to aid revision at a later stage
3-4
Solving linear equations & inequalities
Solve equations by using inverse operations or by transforming both sides in the same way
Solve linear equations with integer or fractional coefficients, in which the unknown appears on either side or on both sides of the equation
Solve linear equations that require prior simplification of brackets, including those that have negative signs occurring anywhere in the equation, and those with a negative solution
Solve simple linear inequalities in one variable, and represent the solution set on a number line
Experience of finding missing numbers in calculations
The idea that some operations are ‘opposite’ to each other
An understanding of balancing
Experience of using letters to represent quantities
Be able to draw a number line
By the end of the module the student should be able to:
• Solve linear equations with one, or more, operations (including fractional coefficients) (3.3 – 3.6)
• Solve linear equations involving brackets and/or variables on both sides
(3.3 – 3.6)
• Solve linear inequalities in one variable and present the solution set on a number line (4)
• Form linear equations from word problems in a variety of contexts and relate the answer back to the original problem (4)
Use of inverse operations and rounding to 1 significant figure could be applied to more complex calculations
Derive equations from practical situations (such as finding unknown angles in polygons or perimeter problems)
Solve equations where manipulation of fractions (including negative fractions) is required
Solve linear inequalities where manipulation of fractions is required (top students)
Students need to realise that not all linear equations can easily be solved by either observation or trial and improvement, so the use of a formal method is important
Students can leave their answers in fractional form where appropriate
Students should use the correct notation when showing inequalities on a number line, eg a filled in circle to show inclusion of a point, an empty circle to show exclusion of a point
3-5
Substitution & changing the subject
Substitute numbers into formulae
Use formulae from mathematics and other subjects that require prior simplification of brackets, including those that have negative signs occurring anywhere in the equation, and those with a negative solution
Change the subject of a formula including where the subject occurs once
Generate a formula in algebra or words
Understand of the mathematical meaning of the words expression, simplifying, formulae and equation
Experience of using letters to represent quantities
Substitute into simple expressions using words
Use brackets in numerical calculations and removing brackets in simple algebraic expressions
Solve Linear Equations
By the end of the module the student should be able to:
• Use letters or words to state the relationship between different quantities (6.1)
• Substitute positive and negative numbers into simple algebraic formulae (6.1, 6.2)
• Substitute positive and negative numbers into algebraic formulae involving powers (6.1)
• Find the solution to a problem by writing an equation and solving it (6.1, 6.2)
• Simple change of subject of a formula, eg convert the formula for changing Centigrade into Fahrenheit into a formula that converts Fahrenheit into Centigrade
• Generate a formula from given information, eg find the formula for the perimeter of a rectangle given its area A and the length of one side
Use negative numbers in formulae involving indices
Use various investigations leading to generalisations
Use further problems in generating formulae form given information
Use the equation of a straight line to solve problems, eg what is the gradient of the
line 4x + 2y = 12?
Apply changing the subject of the formula to physics formulae, ie speed, density, equations of motion etc
Emphasise good use of notation, eg 3ab means 3 ´ a ´ b
Students need to be clear on the meanings of the words expression, equation, and formula
Show a linear equation first and follow the same steps for the similarly structured formula to be rearranged
Functional elements material is a rich source of formulae given in everyday contexts
Link with formulae for area and volume
3-6
Trial and improvement
Use systematic trial and improvement to find approximate solutions of equations where there is no simple analytical method of solving them
Substitute numbers into algebraic expressions
Deal with decimals on a calculator
Order decimals and decimal place approximations.
• Solve quadratic & cubic functions by successive substitution of values of x (3.8)
Solve functions of the form (link with ‘changing the subject’)
Students should be encouraged to use their calculators efficiently - by using the "replay" or ANS/EXE functions
The square/cube function on a calculator may not be in the same place for different makes of calculator
Take care when entering negative values to be squared (always use brackets)
Students should always write down all the digits on their calculator display, and then round the final answer, given to the required degree of accuracy
3-7
Drawing & interpreting graphs
Interpret information presented in a range of linear and non-linear graphs modelling real situations
Generate points and plotting graphs of quadratic functions
Find approximate solutions of a quadratic equation from the graph of the corresponding quadratic function
Experience at plotting points in all quadrants
Linear sequences and straight line graphs
Substitution
By the end of the module the student should be able to:
• Draw linear graphs from tabulated data, including real-world examples (5)
• Interpret linear graphs, including conversion graphs and distance-time graphs (5.1)
• Draw and interpret graphs in the form y = mx + c (when values for m and c are given) (5.2)
• Understand that lines are parallel when they have the same value of m (5)
• Understand when two straight lines intersect this is the solution to the two simultaneous equations (5.3)
• Generate points and plotting graphs of quadratic functions (5.2)
• Plot the graphs of quadratic functions for positive and negative values of x (5.2)
• Solve a quadratic equation by reading the roots off the x-axis from the graph of that equation (5.3)
Plot graphs of the form y = mx + c where students have to generate their own tables and draw their own axes
Use a spreadsheet to generate straight-line graphs, posing questions about the gradient of lines
Use a graphical calculator or graphical ICT package to draw straight-line and quadratic graphs
Discuss the shape of quadratic curves and introduce the word ‘parabola’
Clear presentation with axes labelled correctly is vital
Recognise linear graphs and hence when data may be incorrect
Link to graphs and relationships in other subject areas, ie science, geography etc
Interpret straight line graphs for Functional Elements and link gradient to cost per unit or rate of change and intercept to initial cost
– Ready reckoner graphs
– Conversion graphs
– Fuel bills
– Fixed charge (standing charge) and cost per unit
3-8
Polygons
Use definitions and names of polygons
Calculate and use the sums of the interior angles of quadrilaterals, pentagons and hexagons
Calculate and use the angles of regular polygons
Understand that inscribed regular polygons can be constructed by equal divisions of a circle
Angle facts for triangles and quadrilaterals (Unit 2)
Experience in drawing triangles, quadrilaterals and circles
By the end of the module the student should be able to:
• Name a polygon with 3, 4, ..., 10 sides (7.1)
• Prove the exterior angle of a triangle is equal to the sum of the two opposite interior angles (Unit 2 revision) (7.1)
• Calculate and use the sums of the interior angles of convex polygons of sides 3, 4, 5, 6, 8, 10 (7.1)
• Know, or work out, the relationship between the number of sides of a polygon and the sum of its interior angles (7.1)
• Know that the sum of the exterior angles of any polygon is 360° degrees (7.2)
• Find the size of each exterior/interior angle of a regular polygon (7.1)
Use triangles to find the angle sums of polygons as well as relying on formulae
Use the angle properties of triangles to find missing angles in combinations of triangles
Use harder problems involving multi-step calculations
Link with symmetry and tessellations
Use plenty of practical drawing examples to help illustrate properties of various shapes – group work/displays
Diagrams used in examinations are seldom drawn accurately
Encourage students to always put the reasons and ‘quote’ the angle fact/theorem used
3-9
Transformations
Transform triangles and other 2-D shapes by translation, rotation and reflection and combinations of these transformations
Understand that translations are specified by a distance and direction (or a column vector), and enlarging by a centre and a scale factor
Rotate a shape about the origin, or any other point
Measure the angle of rotation using right angles, simple fractions of a turn or degrees
Understand that rotations are specified by a centre and an (anticlockwise) angle
Understand that reflections are specified by a mirror line, at first using a line parallel to any axis, then a mirror line such as y = x or y = –x
Recognise, visualising and constructing enlargements of objects using a positive integer scale factor
Use congruence to show that translations, rotations and reflections preserve length and angle, so that any figure is congruent to its image under any of these transformations
Recognition of basic shapes
Line and rotational symmetry (Unit 2)
An understanding of the concept of rotation & enlargement
Coordinates in four quadrants
Linear equations parallel to the coordinate axes and the equation of (45°) diagonal
lines y = x, y = –x
By the end of the module the student should be able to:
• Understand translation as a combination of a horizontal and vertical shift including signs for directions (11.2 - 11.6)
• Understand rotation as a (clockwise) turn about a given origin (11.3)
• Reflect shapes in a given mirror line; parallel to the coordinate axes and then y = x or y = –x (11.4)
• Enlarge shapes by a given scale factor from a given point; using a positive integer scale factors (11.5)
• Understand that shapes produced by translation, rotation and reflection
are congruent to its image, hence define congruency (11.2 – 11.4)
The tasks set should be extended to include combinations of transformations
Emphasis needs to be placed on ensuring that students describe the given transformation fully
Diagrams should be drawn carefully in pencil
The use of tracing paper is allowed in the examination
3-10
Constructions, loci, similarity and congruency
Draw accurate constructions of triangles and other 2-D shapes, using a ruler and protractor, given information about side lengths and angles
Understand from their experience of constructing them, that triangles satisfying SSS, SAS, ASA and RHS are unique, but SSA are not
Use straight edge and compasses to do standard constructions
Find loci
Understand similarity of triangles and of other plane figures.
Recognise that enlargements preserve angle but not length
Understand and using SSS, SAS, ASA and RHS conditions to prove the congruence of triangles using formal arguments, and to verify standard ruler and compass constructions
Understand tessellations
Use and interpret maps and scale drawings
An ability to use a pair of compasses
The special names of triangles (and angles)
Understanding of the terms perpendicular, parallel and arc
Transformations (particularly enlargements)
Congruency
By the end of the module the student should be able to:
• Do a range of standard constructions including:
- An equilateral triangle with a given side (10.1)
- The midpoint and perpendicular bisector of a line segment (10.1)
- The perpendicular from a point on a line (10.1)
- The bisector of an angle (10.1)
- The angles 60°, 30° and 45° (10.1)
- A regular hexagon inside a circle, etc (link with polygons) (10.1)
- A region bounded by a circle and an intersecting line (10.3)
- A path equidistant from two points or two line segments (10.2)
• Make clearly the link between constructions and the definition of loci (10.2, 10.3)
• Understand tessellations and explain why some shapes tessellate and why other shapes do not (7.4)• Work out the real distance from a map, eg find the real distance represented by 4 cm on a map with scale 1:25 000
• Work out the distance on a map for a given real distance and scale (link with ratio
from Unit 1 & 2)
Solve loci problems that require a combination of loci
Construct combinations of 2-D shapes to make nets
All working should be presented clearly, and accurately
A sturdy pair of compasses is essential
Construction lines should not be erased as they carry valuable method marks
Use of tracing paper can help immensely with tessellations
3-11
Pythagoras’ theorem
Understand, recall and use Pythagoras’ theorem in 2-D
Calculate the length of a line segment between two coordinates
Use and understand 3-figure bearings
Solve problems involving geometric figures (including triangles within circles) in which a right-angle triangle has to be extracted in order to solve it by Pythagoras
Names of triangles & quadrilaterals and their symmetries
Knowledge of the properties of rectangles, parallelograms and triangles
Indices and roots, substitution, equations and changing the subject
By the end of the module the student should be able to:
• Find missing sides of right-angle triangles by using Pythagoras (12.1, 12.2)
• Find the distance between two coordinates using Pythagoras (link with
drawing and interpreting graphs) (12)
• Give answers as decimals for Pythagoras problems (12)
• Understand how bearings work and solve problems involving bearings using Pythagoras (12)
• Solve problems involving geometric figures (including triangles within circles) in which a right-angle triangle has to be extracted in order to solve it by Pythagoras
Introduce 3-D Pythagoras or surds (not on the Foundation tier course, but a good extension for the brightest)
Students should be encouraged to become familiar with one make of calculator
Emphasise that scale drawings will score no marks for this type of question
A useful mnemonic for remembering Pythagoras is ‘Square it, Square it, Add/Subtract it, Square root it’ (this eliminates the need for formal algebra for the weaker students)
Students should not forget to state the units for the answers
3-12
3-D Shapes, nets and plans
Identify and name common solids
Explore the geometry of cuboids (including cubes), and shapes from cuboids
Names of common 2-D shapes
By the end of the module the student should be able to:
• Identify and name common solids: cube, cuboid, cylinder, prism, pyramid, sphere and cone (9.1)
• Draw nets of solids and recognise solids from their nets (isometric drawing) (9.1)
• Draw and interpret plans and elevations (9.2)
• Recognise, for similar shapes, that all corresponding angles are equal in size when the lengths of the sides are not
• Recognise and name examples of solids, including prisms, in the real world
Make solids using equipment such as clixi or multi-link
Draw shapes made from multi-link on isometric paper
Build shapes from cubes that are represented in 2-D
Work out how many small boxes can be packed into a larger box (Functional Elements example)
Accurate drawing skills need to be reinforced
Some students find visualising 3-D objects difficult, so simple models can assist
Use of tracing paper/mirrors can help
3-13
Circles & cylinders
Find circumferences of circles and areas enclosed by circles, recalling relevant formulae
Solve problems involving surface areas and volumes of prisms and cylinders
Find the surface areas and volumes of compound solids constructed from; cubes, cuboids, cylinders and solve problems including examples of solids in everyday use
Perimeter and area (Unit 2)
Surface area and volume (Unit 2)
Knowledge of names and properties of 3D solids (Unit 2)
Formulae and substitution
By the end of the module the student should be able to:
• Learn that the ratio between the circumference and diameter is always
constant for a circle (ie π) (8.1, 8.2)
• Learn the formula for circumference and area of a circle (8.1, 8.2)
• Solve problems involving the circumference and area of a circle (and simple fractional parts of a circle) (8.1, 8.2)
• Solve problems involving the volume and surface area of a cylinder (9.3)
• Find the surface area and the volume of more complex shapes, eg find the volume of an equilateral triangular prism (9.3)
Extend to volume of a cone (not on course)
‘Now! I Know Pi’ is a good way to learn the approx value (The number of letters of each word and the ! is the decimal point)
Locate π button on a calculator
Also ‘Cherry Pie Delicious’ is C = πD and ‘Apple Pies are too’ is A = πr2
Use a tower of coins to model the volume of a cylinder
Final answers should be rounded to the required degree of accuracy
Need to constantly revise the expressions for area/volume of shapes
Students should be aware of which formulae are on the relevant page on the exam paper and which they need to learn
3-14
Further units
Convert measurements from one unit to another (for area and volume)
Understand the effect of enlargement for perimeter, area and volume of shapes and solids
Knowledge of metric units of length, area, volume and their conversions eg 1m = 100 cm, etc
Experience of multiply by powers of 10, eg 100 ´ 100 = 10 000
Enlargement
By the end of the module the student should be able to:
• Convert between area measures, eg 1 m2 = 100 00 cm2 (9.5)
• Convert between volume measures (Using Metric units) (9.5)
• Convert between volume and capacity measures, eg 1ml = 1 cm³
(1 c.c.) (9.5)
• Understand the effect of enlargement on perimeter, area and volume of
shapes and solids (9.4)
• Use simple examples of the relationship between enlargement and areas and volumes of simple shapes and solids (9.4)
Mention other units (not on course), eg hectares
Draw a large grid made up of 10 by 10 mm squares to show what 1 square cm looks like
Similarly, a grid of 100 by 100 cm squares to show how many cm squares fit into 1 square metre
A good starter is to bring in a small bottle of water and a larger bottle (preferably the larger bottle will be twice the length). Show, by pouring, that eight small bottles will fill the larger bottle
[This can also be done with one small cube and a larger box with lengths twice as long] Initially the class will say two bottles will fill the larger double size bottle