Scale Factor & Enlargement

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Scale Factor and Enlargement

When a shape is enlarged, its lengths are all multiplied by S.

Length

Area

When a 2D shape is enlarged by a scale factor S, because all the lengths are multiplied by S, its area will be multiplied by S².

Volume

When a 3D shape is enlarged by a scale factor S all the lengths are multiplied by S.
The areas in the shape will be multiplied by S² and the volumes will be multiplied by S³.

Example

A cube of side length 1cm is enlarged by scale factor 2.
Its initial surface area was 6cm² and its initial volume was 1cm³.
After enlargement each edge is now 2cm long.
Its surface area is now 6 × 4cm² = 24cm² which is 4 times as large as before.
Its volume is now 2³cm³ = 8cm³ which is 8 times as large as before.

Scale Factor Enlargement

If a shape is enlarged by a scale factor, S, all the lengths in that shape will be multiplied by S.

Fractional scale factors

When enlarging something by a fractional scale factor, it will actually get smaller.
- E.g. Enlarging a 10 cm line by scale factor $\frac{1}{2}$ will multiply the length by $\frac{1}{2}$ .
  - This means that the line will be 5 cm long after the enlargement.

Negative scale factors

Negative scale factors still enlarge a shape by multiplying the lengths by scale factor S.
- E.g. Enlarging a 10 cm line by scale factor −2 will multiply the length by −2.
  - But, the line won't be −20 cm long because length is a scalar quantity and, therefore, cannot negative. This means that the line will be 20 cm long after the enlargement.

Scale Factors with Area and Volume

To work out the area and volume of a shape after enlargement, consider that an enlargement occurs in every dimension.

Length

Enlarging a length by scale factor S will multiply the length by S.
- E.g. Enlarge a line with a length of 5 cm by scale factor 2.
  - 5 × 2 = 10 cm
  - The enlarged line will have a length of 10 cm.

Area

Enlarging an area by scale factor S will multiply the area by S².
This is because the lengths of both dimensions of the 2D shape will be enlarged by a scale factor S.
- E.g. Enlarge a square with an area of 5 cm² by scale factor 2.
  - 5 × 2² = 20 cm²
  - The enlarged square will have an area of 20 cm².

Volume

Enlarging a volume by scale factor S will multiply the volume by S³.
This is because the lengths of all dimensions of the 3D shape will be enlarged by a scale factor S.
- E.g. Enlarge a cube with an volume of 5 cm³ by scale factor 2.
  - 5 × 2³ = 40 cm³
  - The enlarged cube will have a volume of 40 cm³.

1Number

1.1Using Numbers

1.2Fractions, Decimals & Percentages

1.2.1Adding Fractions

1.2.2Subtracting Fractions

1.2.3Multiplying Fractions

1.2.4Dividing Fractions

1.2.5Mixed Fractions

1.2.6Calculating Percentages of Numbers

1.2.7Calculating Fractions of Numbers

1.2.8Multiply Decimals

1.2.9Place Value

1.2.10Recurring Decimals

1.2.11Exam-Style Questions - Recurring Decimals

1.2.12Convert Fractions & Terminating Decimals

1.2.13Convert Recurring Decimals

1.2.14Convert Decimals & Percentages

1.2.15Compare Decimals

1.2.16Compare Fractions

1.2.17Exam-Style Questions - Percentages

1.2.18End of Topic Test - Fractions & Decimals

1.3Powers & Roots

1.3.1Powers

1.3.2Roots

1.3.3Indices Advanced

1.3.4Estimating Powers

1.3.5Estimating Roots

1.3.6Multiplying Powers

1.3.7Dividing Powers

1.3.8Power Raised to Power

1.3.9Fractions Raised to a Power

1.3.10Numbers Raised to Negative Powers

1.3.11Roots of Numbers

1.3.12Numbers to the Power of 1 & 0

1.3.13Surds

1.3.14Simplify Surds

1.3.15Rationalise Denominator

1.3.16Exam-Style Questions - Indices

1.3.17End of Topic Test - Powers & Roots

1.4Accuracy

1.4.1Decimal Rounding

1.4.2Significant Figure Rounding

1.4.3Estimation

1.4.4Truncation

1.4.5Limits of Accuracy

1.4.6Bounds Addition

1.4.7Bounds Subtraction

1.4.8Bounds Multiplication

1.4.9Bounds Division

1.4.10End of Topic Test - Rounding & Estimation

2Algebra

2.1Introduction to Algebra

2.1.1Algebra Basics

2.1.2Substituting Values

2.1.3Algebraic Vocabulary

2.1.4Collecting Terms

2.1.5End of Topic Test - Algebra Basics & Power Rules

2.1.6Expanding Brackets I

2.1.7Common Factors

2.1.8Expanding Brackets II

2.1.9Rearrange For x

2.1.10Rearrange with Powers

2.1.11Exam-Style Questions - Basics of Algebra

2.1.12End of Topic Test - Expanding & Rearranging

2.2Manipulating Algebra

2.2.1How to Factorise

2.2.2Factorising Quadratics

2.2.3Algebraic Fractions - Addition & Subtraction

2.2.4Algebraic Fractions - Multiplication & Division

2.3Proofs & Functions

2.3.1Identities & Proofs

2.3.2Exam-Style Questions - Algebraic Proof

2.3.3End of Topic Test - Algebra & Proofs

2.3.4Functions

2.3.5Combine Functions

2.3.6Composite Functions

2.3.7Inverse Functions

2.4Straight Line Graphs

2.4.1Straight Line Graphs

2.4.2y = mx + c

2.4.3Grad + YIntercept

2.4.4Exam-Style Questions - Straight Line Graphs

2.4.5End of Topic Test - Functions & Straight Lines

2.5Common Graphs

2.5.1Shape of Quadratic Graphs

2.5.2Roots of a Quadratic Graph

2.5.3Completing Squares

2.5.4Turning Points

2.5.5Common Functions 1

2.5.6Common Functions 2

2.5.7Trigonometric functions

2.5.8End of Topic Test - Common Graphs

2.6Transformations & Tangents

2.6.1Transformations

2.6.2Tangent on a Circle

2.7Properties of Graphs

2.7.1Estimate Gradient

2.7.2Estimate Area

2.7.3Represent Gradient & Area

2.7.4Distance-Time Graph

2.7.5Velocity-Time Graph

2.7.6End of Topic Test - Transformations & Graphs

2.8Solving Equations

2.8.1Solve Linear Equations

2.8.2Linear Equations with Fractions

2.8.3Solve Linear Equations Graphically

2.8.4Quadratic ±

2.8.5Factorise Quadratics

2.8.6Quadratic Formula

2.8.7Complete the Square

2.8.8Solve Quadratics Graphically

2.8.9End of Topic Test - Solving Equations

2.8.10Simultaneous Equations

2.8.11Quadratic Simultaneous Equations

2.8.12Simultaneous Equations - Graphs

2.8.13Simultaneous Equations - Word Problems

2.8.14Approximate Solutions - Number of Iterations

2.8.15Approximate Solutions - Iterations to 1 DP

2.8.16Approximate Solutions - Iteration Formula

2.8.17Exam-Style Questions - Simultaneous Equations

2.8.18End of Topic Test - Simultaneous Equations

2.9Inequalities

2.9.1Solving Inequalities

2.9.2Solving Double Inequalities

2.9.3Solving Quadratic Inequalities

2.9.4Representing Inequalities

2.9.5Exam-Style Questions - Inequalities

2.9.6End of Topic Test - Inequalities

2.10Sequences

2.10.1Nth Term Rules

2.10.2Recurrence Relationships

2.10.3Find Nth Term in Linear Sequence

2.10.4Check If a Number is In a Sequence

2.10.5Exam-Style Questions - Sequences

2.10.6Arithmetic & Geometric Progressions

2.10.7Key Sequences

2.10.8Quadratic Sequences

2.10.9Find Nth Term in Quadratic Sequence

2.10.10Exam-Style Questions - Geometric Series

2.10.11End of Topic Test - Sequences

3Ratio

3.1Ratios in Practice

3.1.1Converting Between Units

3.1.2Formula Triangles

3.1.3Scale Factors

3.1.4Ratios in Shapes

3.2Manipulating Ratios

3.2.1Simplifying Ratios

3.2.2Turn Ratios into Equations

3.2.3Ratio Notation & Reduction

3.2.4End of Topic Test - Units & Ratios

3.2.5Part : Part Ratio

3.2.6Part : Whole Ratio

3.2.7Two Ratios

3.2.8Differences in Ratios

3.2.9Fractions in Ratios

3.2.10Numbers as Fractions

3.2.11Ratios to Fractions

3.2.12Exam-Style Questions - Ratios

3.2.13End of Topic Test - Fractions & Ratios

3.3Percentage & Interest

3.3.1Convert Percentages to Decimals

3.3.2Convert Percentages to Fractions

3.3.3Find the Percentage

3.3.4More/Less Percentage

3.3.5Price from Percentage Change

3.3.6Calculate Percentage Change from Price

3.3.7End of Topic Test - Percentages

3.3.8Profit & Interest & Discount

3.3.9Original Price after Discounts

3.3.10Simple Interest

3.3.11Compound Growth & Decay

3.3.12Compound Interest

3.3.13Compound Interest - After How Many Days?

3.3.14End of Topic Test - Percentages in Finance

3.4Proportion

3.4.1Proportion

3.4.2Direct Proportion

3.4.3Inverse Proportion

3.4.4Exam-Style Questions - Proportionality

3.5Gradient

3.5.1Gradient on a Straight Line

3.5.2Direct & Inverse Proportions as Graphs

3.5.3Gradient of a Curve

3.5.4Positive & Negative Gradients

3.5.5End of Topic Test - Proportion & Gradient

4Geometry

4.1Introduction to Geometry

4.1.1Geometry Basics

4.1.2Geometry Basics 2

4.1.3Regular Polygons

4.1.4Triangles & Angles

4.1.5Midpoint

4.1.6Bisectors & Perpendiculars

4.1.7Locus

4.1.8Angle rules

4.1.9Parallel Angle Rules

4.1.10Angles in a Polygon

4.1.11Angles in a Triangle

4.1.12Exam-Style Questions - Angles in Shapes

4.1.13End of Topic Test - Geometry

4.2Triangles & Quadrilaterals

4.2.1Types of Triangle

4.2.2Angles & Triangles

4.2.3Congruence

4.2.4Similarity

4.2.5Quadrilaterals

4.3Transformations

4.3.1Types of Transformations

4.3.2Invariant Points

4.3.3Effects of Transformations

4.3.4Exam-Style Questions - Reflection

4.4Circle Basics

4.4.1Properties of a Circle 1

4.4.2Chords

4.4.3Properties of a Circle 2

4.4.4Area of a Circle

4.4.5Circumference of a Circle

4.4.6Fractions of Circles

4.4.7Exam-Style Questions - Circle Formulae

4.4.8End of Topic Test - Triangles & Circles

4.5Circle Theorems

4.5.1Perpendicular Bisector of a Chord

4.5.2Angle at the Centre

4.5.3Angle of a Semicircle

4.5.4Opposite Angles in Cyclic Quadrilaterals

4.5.5Tangent Lengths

4.5.6Angle in the Same Segment

4.5.7Alternate Segment Theorem

4.5.8Exam-Style Questions - Circle Theorems

4.6Measurements & Units

4.6.1Length

4.6.2Area

4.6.3Volume

4.6.4Capacity

4.6.5Time

4.6.6Money

4.6.7Mass

4.6.8Bearings

4.6.9Scale Factor & Enlargement

4.6.10Exam-Style Questions - Enlargements

4.7Calculating Area

4.7.1Area of 2D Shapes

4.7.2Exam-Style Questions - Area

4.7.3End of Topic Test - Circle Theorems, Units & Area

4.8Triangle Formulae

4.8.1Pythagoras' Theorem

4.8.2Pythagoras in 3D

4.8.3Trigonometry

4.8.4Exact Values

4.8.5Sine Rule

4.8.6Cosine Rule

4.8.7Area of a Triangle

4.8.8Exam-Style Questions - Trigonometry & Area

4.93D Shapes

4.9.1Types of 3D Shape

4.9.2Plans & Elevations

4.9.3Nets

4.9.4Volume of 3D Shapes

4.9.5Volume of a Cuboid

4.9.6Volume of a Cylinder

4.9.7Volume of a Sphere

4.9.8Volume of a Hemisphere

4.9.9Volume of a Cone

4.9.10Volume of a Pyramid

4.9.11Surface Area & Composite Shapes

4.9.12Surface Area of a Cuboid

4.9.13Surface Area of a Cylinder

4.9.14Surface Area of a Sphere/Hemisphere

4.9.15Surface Area of a Cone

4.9.16Exam-Style Questions - Volume

4.10Vectors

4.10.1Vectors Introduction

4.10.2Vectors

4.10.3Translation

4.10.4Scalar

4.10.5Exam-Style Questions - Vectors

4.10.6End of Topic Test - Trigonometry, Shapes & Vectors

5Probability

6Statistics

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