Welcome to Maths

Making generalisations about the number system

Making generalisations about the number system

Knowledge Content: Number systems, axioms and properties

Threshold Concepts: Humans have not always used base 10 and in specific situations other bases are preferential; a prime number has exactly two factors 

Knowledge Content: Positive and negative numbers; basic algebra

Threshold Concepts: Negative values do exist in the real world in various contexts; it is convention to put the number ahead of the unknown e.g. 2y instead of y2 

Knowledge Content: Angles, classifying 2-D shapes and constructions

Threshold Concepts: The rules for angles in parallel lines can be derived from the basic angle facts; Several shapes can share the same property e.g. having 4 right angles 

Knowledge Content:  Coordinates, area of 2-D shapes and transformations

Threshold Concepts: Co-ordinates require two pieces of information to fully map their location; The formula for the area of a triangle is directly linked to that of a rectangle; There can be several ways of describing a shape’s transformation. 

Knowledge Content: Number properties and fractions

Threshold Concepts: Every positive integer (greater than 1) has a unique prime factor decomposition; A fraction is a convenient way of representing one value being divided by another. 

Knowledge Content: Ratio and percentages

Threshold Concepts; Ratios are inherently linked to fractions; A percentage can easily be converted to a fraction with a denominator of 100. 

Knowledge Content: Equations and inequalities

Threshold Concepts: To solve an equation, the same operation must be done to both sides of the equation. 

Knowledge Content: Linear graphs, accuracy and estimation

Threshold Concepts: A minimum of two points are required to draw a straight line; It is convention to round up with a ‘5’ as the decisive digit. There is no mathematically sound reason as to why – it is purely convention. 

Knowledge Content: Real-life graphs and proportion

Threshold Concepts: If two of speed, distance or time are known, the third can be found; Two values that are directly proportional have a fixed ratio. 

Knowledge Content: Univariate and bivariate data

Threshold Concepts: The mean, mode and median are three different ways of finding and average. In real life, each has its own benefits depending on the situation; Using a line if best fit is a form of estimation. It is not exact but will give a fairly accurate answer. 

Knowledge Content: Angles in parallel lines and in polygons; Bearings

Threshold Concepts: The interior and exterior angle of a polygon will sum to 180; Bearings are always taken from North and in the clockwise direction. 

Knowledge Content: Circles, volume and surface area of prism

Threshold Concepts: Pi (𝜋𝜋) is a fixed number that is the constant ratio between the circumference and diameter of any circle; The volume of a prism is the area of the cross section multiplied by the length. V = A x L; The surface area is the area of the net of the 3-D shape. 

Knowledge Content: Probability and sample spaces

Threshold Concepts: Probably can be represented a fraction, decimal or percentage; Samples spaces represent different ways of interpreting probability problems. 

Knowledge Content: Quadratics

Threshold Concepts: A quadratic expression is a polynomial of degree 2. 

Knowledge Content: Constructions and Pythagoras’ theorem

Threshold Concepts: The loci is the set of all points, whose location satisfies the given conditions; If two side lengths of a right-angled triangle are known, the third can always be found. 

Knowledge Content: Similar shapes, surds and trigonometry

Threshold Concepts: When comparing 3-D objects there are three scale factors to consider: length; area; volume; A surd is a radical (square root) that cannot be reduced to an integer. 

Knowledge Content: Solving graphically and algebraically

Threshold Concepts: The intersection point of two straight lines is the solution to the simultaneous equations; To solve algebraically, students can choose to use the elimination method or the substitution method. 

Knowledge Content: Indices, growth and decay

Threshold Concepts: The laws of indices apply to numerical indices and also algebraic indices; A multiplier greater than 1 represents a percentage increase. A multiplier less than 1 represents a percentage decrease. 

Knowledge Content: Factors, multiples and primes; powers and roots; indices; standard form; sequences

Threshold Concepts: 

• Every positive integer (greater than 1) has a unique prime factor decomposition.
• Standard form is useful in comparing numbers of varying magnitudes.  

Link to Prior Learning: Builds on knowledge of prime factorisation and index laws 

Knowledge Content: Algebra KS3 review; quadratics; quadratic graphs; simultaneous equations  

Threshold Concepts:

• A quadratic expression is a polynomial of degree 2.
• Simultaneous equations can be solved algebraically or graphically.  

Link to Prior Learning: Builds on knowledge of quadratics and equation solving 

Knowledge Content: FDP KS3 review; percentages; probability, sets, venn and sample space diagrams  

Threshold Concepts:

• Probably can be represented a fraction, decimal or percentage.
• Samples spaces represent different ways of interpreting probability problems.

Link to Prior Learning: Builds on knowledge of FDP and probability

Knowledge Content: Transformations; 2D shapes and circle geometry; 3D shapes; volume and surface area  

Threshold Concepts:

• There can be several ways of describing a shapes transformation.
• Pi (𝜋𝜋) is a fixed number that is the constant ratio between the circumference and diameter of any circle.

Link to Prior Learning: Builds on knowledge of 2D/3D shape properties (area, volume etc.) 

Knowledge Content: Ratio review; compound measures and direct and inverse proportion; Pythagoras’ theorem review; similarity and trigonometry  

Threshold Concepts:

• Proportion problems can be represented algebraically, graphically or in a table.
• Pythagoras’ theorem states that that for a rightangled triangle a2 + b2 = c2 where c is the hypotenuse  

Link to Prior Learning: Builds on knowledge of ratio and trigonometry 

Knowledge Content: Averages and range; data collection and sampling; presenting data including scatter diagrams  

Threshold Concepts:

• Scatter diagrams are used to show trends in a data set.  

Link to Prior Learning: Builds on knowledge of averages and data presentation 

Knowledge Content: Powers and roots; surds and irrational numbers; indices; standard form; sequences

Threshold Concepts:

• Surds can be calculated with but won’t always result in ‘nice-looking’ answers.
• Find the nth term of linear, geometric and quadratic sequences.

Link to Prior Learning: Builds on knowledge of prime factorisation and index laws 

Knowledge Content: Quadratics; quadratic graphs algebraic fractions; simultaneous equations  

Threshold Concepts:

• Simultaneous equations with a quadratic involved can result in two solutions.
• Algebraic fractions can be simplified by dividing the numerator and denominator by a common factor.

Link to Prior Learning: Builds on knowledge of quadratics and equation solving 

Knowledge Content: KS3 FDP review; percentages; probability sets, venn and sample space diagrams (including conditional probability)  

Threshold Concepts:

• Probability can be represented a fraction, decimal or percentage.
• Know the difference between simple and compound interest.
• Problems with conditional probability require more careful consideration.  

Link to Prior Learning: Builds on knowledge of FDP and probability 

Knowledge Content: Transformations; upper and lower bounds; 2D shapes and circle geometry; 3D shapes; volume and surface area  

Threshold Concepts:

• There are 4 main types of transformation.
• Upper and lower bounds show the margin of error as a result of rounding.
• A sector is a fraction of a circle.  

Link to Prior Learning: Builds on knowledge of 2D/3D shape properties (area, volume etc.)

Knowledge Content: Ratio review; compound measures and proportion; Pythagoras review; similarity and trigonometry; 3D triangles  

Threshold Concepts:

• Speed, density and pressure are all examples of compound measures.
• Problems involving rightangled triangles can be solved using Pythagoras or the sin/cos/tan ratios.
• For non-right angled triangles the following can be used: sine rule; cosine rule; area of a triangle using sine rule

Link to Prior Learning: Builds on knowledge of ratio and trigonometry 

Knowledge Content: Averages and range review; data collection and sampling; presenting data including scatter diagrams; further statistical diagrams  

Threshold Concepts:

• Averages can be calculated from a frequency table and estimated from a grouped-data table
• Boxplots can be used to compare the distribution of two data sets.  

Link to Prior Learning: Builds on knowledge of averages and data presentation 

Knowledge Content: Vectors; geometric reasoning; bearings; congruence; construction and loci  

Threshold Concepts:

• Vectors represent a translation and can be written in column vector form.  

Link to Prior Learning: Builds on knowledge of angles and constructions

Knowledge Content: Linear inequalities; linear graphs, non-linear graphs  

Threshold Concepts:

• The method for solving a linear inequality is almost identical to that of a linear equation.
• Solid and dotted lines represent different things on an inequality graph  

Link to Prior Learning: Builds on knowledge of inequalities and graphs 

Knowledge Content: Vectors; geometric reasoning review; circle theorems; bearings; congruence; construction and loci  

Threshold Concepts:

• Parallel vectors are scalar multiples of each other.
• Angles made in circles using the points on the centre and/or circumference can be found using the circle theorems.  

Link to Prior Learning: Builds on knowledge of angles and constructions 

Knowledge Content: Inequalities; linear graphs, non-linear graphs; trig graphs  

Threshold Concepts:

• The graphs of sin(x) and cos(x) are identical but out of phase by 90o.  

Link to Prior Learning: Builds on knowledge of inequalities and graphs 

Knowledge Content: Algebraic proof and reasoning; recurrence relations; functions; transformation of graphs; further graphs  

Threshold Concepts:

• Algebra can be used to construct proofs
• The area under a curve can be estimated using basic triangles and quadrilateral.  

Link to Prior Learning: Builds on knowledge of nonlinear graphs