Upper Lower Bounds

Upper and Lower Bounds

When we estimate answers, we will either overestimate or underestimate the correct answers. These estimations can give us a range which the correct answer must be within.

What is 5.43 & times 12?

This multiplication is hard to work out, but we can find a range for the answer by rounding 5.43.

Upper bound

If we round 5.43 up to the nearest whole number, we will overestimate the answer.
- We know that the real answer is less than this overestimate so we can write (5.43 × 12) < 72.

Lower bound

If we round 5.43 down to the nearest whole number we will underestimate the answer.
- We know that the real answer is more than this overestimate so we can write 60 < (5.43 × 12).

Inequality

From these estimations we know that 5.43 &times 12 is more than 60 and less than 72.
- We can write this using the double inequality: 60 < (5.43 × 12) < 72.

1Number

1.1Place Value

1.1.1Ordering Numbers

1.1.2Comparing Numbers

1.1.3Equals Symbols

1.1.4Adding Negative Numbers

1.1.5Subtracting Negative Numbers

1.1.6Multiplying Negative Numbers

1.1.7Dividing Negative Numbers

1.1.8End of Topic Test - Place Value

1.2Factors & Multiples

1.2.1Factors

1.2.2Prime Numbers

1.2.3Multiples

1.2.4Prime Factors

1.2.5Highest Common Factors

1.2.6Highest Common Factor Prime Factors

1.2.7Lowest Common Multiples

1.2.8Lowest Common Multiples Prime Factors

1.2.9End of Topic Test - Factors & Multiples

1.3Operations

1.3.1Addition

1.3.2Subtraction

1.3.3Multiplication 10,100...

1.3.4Grid Multiplication

1.3.5Division by 10s

1.3.6Powers

1.3.7Roots

1.3.8Surds

1.3.9Rationalise denominator

1.3.10Simplify Surds

1.3.11BIDMAS

1.3.12Inverse Ops

1.3.13Standard Form

1.3.14Addition in Standard Form

1.3.15Subtraction in Standard Form

1.3.16End of Topic Test - Standard Form

1.4Fractions, Decimals & Percentages

1.4.1Fractions

1.4.2Mixed & Improper Fractions

1.4.3Write 2 Numbers as Fractions

1.4.4Adding Fractions

1.4.5Subtracting Fractions

1.4.6Multiplying Fractions

1.4.7Dividing Fractions

1.4.8Finding Fractions

1.4.9Improper & Mixed Fractions

1.4.10Decimal Place Value

1.4.11Fraction to Decimal

1.4.12Find Percentage Unitary

1.4.13Fraction to Percentage

1.4.14Percentage to Fraction

1.4.15Percentage to Decimal

1.4.16Decimal to Percentage

1.4.17Percentage using Decimal

1.4.18End of Topic Test - Fractions

1.5Growth & Decay

1.5.1Population Growth

1.5.2Compound Growth & Decay

1.5.3Compound Interest

1.5.4Compund Interest 2

1.5.5Depreciation

1.6Measurements & Units

1.6.1Standard Units 1

1.6.2Standard Units 2

1.6.3Standard Units 3

1.6.4Length

1.6.5Area

1.6.6Volume

1.6.7Capacity

1.6.8Time

1.6.9Money

1.6.10Mass

1.7Accuracy

1.7.1Decimal Rounding

1.7.2SigFig Rounding

1.7.3Estimations

1.7.4Upper Lower Bounds

2Algebra

2.1Introduction to Algebra

2.1.1Algebra Basics

2.1.2Algebraic Concepts

2.1.3Writing Expressions

2.1.4Collecting Terms

2.1.5Common Factors

2.1.6Expanding Brackets

2.1.7FOIL

2.1.8Rearranging Formulae

2.1.9End of Topic Test - Introduction to Algebra

2.2Graphs

2.2.1Coordinates

2.2.2Gradient and Intercept

2.2.3Calculate Gradient &Y-intercept

2.2.4Finding Equation

2.2.5Quadratic Graphs

2.3Solving Equations

2.3.1Solve Linear Equations

2.3.2Words To Equation

2.3.3Solve Linear Equations Graphically

2.3.4Solve Inequality

2.3.5Graphing Inequalities

2.3.6End of Topic Test - Graphs & Equations

2.3.7Simultaneous Equations

2.3.8Simultaneous Equations Graph

2.3.9Quadratic +/-

2.3.10Factorise Quadratics

2.3.11Complete the Square

2.4Sequences

2.4.1Arith & Geo Sequences

2.4.2Key Sequences

2.4.3Nth Term Rules

2.4.4Find NthTerm in Linear Sequence

2.4.5End of Topic Test - Sequences

2.4.6Find Nth Term in Quadratic Sequence

3Ratio, Proportion & Rates of Change

4Geometry & Measures

4.1Introduction to Geometry

4.1.1LinesAngles1

4.1.2LinesAngles2

4.1.3Regular Polygons

4.1.4Triangles and Angles

4.1.5Quadrilaterals

4.1.6Midpoints

4.1.7Bisectors and Perpendiculars

4.1.8Angle Rules

4.1.9Angles in Parallel Lines

4.1.10Angles in a Polygon

4.1.11Angles in a Triangle

4.1.12Bearings

4.1.13End of Topic Test - Geometry

4.2Ruler & Compass Constructions

4.2.1Midpoint

4.2.2Bisectors and Perpendiculars

4.2.3Loci

4.3Perimeter & Area

4.3.1Perimeter

4.3.2Area

4.4Triangles, Quadrilaterals & Circles

4.4.1Types of Triangle

4.4.2Angles & Triangles

4.4.3Congruence

4.4.4Congruent Triangles

4.4.5Similar Triangles

4.4.6Scale Factors & Enlargement

4.4.7Properties of Circle 1

4.4.8Properties of a Circle 2

4.4.9Area of a Circle

4.4.10Circumference of a Circle

4.5Transformations

4.5.1Types of Transformations

4.5.2Effects of Transformations

4.6Triangle Formulae

4.6.1Pythagoras' Theorem

4.6.2Trigonometric Ratios

4.73D Shapes

4.7.1Types of 3D Shape

4.7.2Plans & Elevations

4.7.3Nets

4.7.4Volume of 3D Shapes

4.7.5Volume of a Cuboid

4.7.6Volume of a Cylinder

4.7.7Volume of a Sphere

4.7.8Volume of a Hemisphere

4.7.9Volume of a Cone

4.7.10Volume of a Pyramid

4.7.11Surface Area of a Cuboid

4.7.12Surface Area of a Cylinder

4.7.13Surface Area of a Sphere/Hemisphere

4.7.14Surface Area of a Cone

4.7.15Surface Area of Composite Shapes

4.7.16End of Topic Test - Triangles, Area, & 3D Shapes

4.8Vectors

4.8.1Vector Introduction

4.8.2Vectors

5Probability

5.1Introduction to Probability

5.1.1Probability

5.1.2Probability scale

5.1.3Randomness & Fairness

5.1.4Sum of Probabilities

5.1.5Relative & Expected Frequency

5.1.6Frequency Tables

5.1.7Use Frequency Trees

5.1.8Complete Frequency Trees

5.2Representing Probability

5.2.1Sample Space Diagrams

5.2.2Sets

5.2.3Venn Diagrams

5.3Probability Calculations

5.3.1Theoretical Probability

5.3.2Venn Diagram Probabilities

5.3.3Multiple Events

5.3.4Dependent & Independent Events

5.3.5Calculating Probabilities

5.3.6Complex Tree Diagrams

5.3.7End of Topic Test - Probability

6Statistics

6.1Introduction to Statistics

6.1.1Statistics Basics

6.1.2Statistics Basics 2

6.1.3Comparing Data

6.1.4Comparing Datasets

6.1.5Writing Comparisons

6.1.6Frequency Table

6.2Charts & Graphs

6.2.1Bar Charts

6.2.2Pie Charts

6.2.3Pictograms

6.2.4Scatter Graphs

6.2.5Making Predictions

6.2.6Correlation and Causation

6.2.7Outliers

6.2.8End of Topic Test - Statistics

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