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LearnAQA8300First teaching 2015Papers 1–3 · Higher tierAlgebra

Papers 1–3 · Module

Algebra

Higher tier: completing the square

11 subtopics≈ 350 min in total

Start the module: Expanding and factorising Open graphing calculator

Expanding and factorising
Expanding removes brackets; factorising puts them back. They are the same skill in reverse: every term in one bracket multiplies every term in the other, and factorising finds the brackets that would multiply out to give the expression.
≈ 30 min
Algebraic fractions and formulae
Algebraic fractions follow exactly the same rules as number fractions: factorise, cancel whole brackets, and use a common denominator to add. Changing the subject of a formula means undoing operations in reverse order, doing the same to both sides.
≈ 35 min
Linear and simultaneous equations
An equation is a balance: whatever you do to one side, do to the other. Linear equations take a few balancing steps. Simultaneous equations have two unknowns, so you eliminate one of them first, then substitute back.
≈ 35 min
Quadratic equations
Rearrange to $ax^2+bx+c=0$ . If it factorises, each bracket gives a solution. Otherwise use the quadratic formula, or complete the square, which also gives the turning point of the graph. In context, reject solutions that make no sense.
≈ 35 min
Inequalities and regions
Inequalities are solved almost like equations, with one trap: multiplying or dividing by a negative flips the sign. On a graph, an inequality in $x$ and $y$ is a region on one side of a line: solid line if included, dashed if not.
≈ 30 min
Sequences
A sequence follows a rule. Linear sequences go up by a constant difference, so their nth term is $dn+c$ . Quadratic sequences have a constant second difference; geometric sequences multiply by the same number each time; Fibonacci-type sequences add the previous two terms.
≈ 30 min
Functions
A function $f(x)$ is a rule turning an input into an output. A composite $fg(x)$ means "do $g$ , then $f$ ". The inverse $f^{-1}$ undoes $f$ : swap $x$ and $y$ and rearrange. Some inputs can't be used, such as values that make you divide by zero.
≈ 30 min
Gradients and straight lines
Two points fix a straight line. From them you can find how steep it is (gradient), how long the segment is (Pythagoras), its middle (average the coordinates), and its equation $y=mx+c$ .
≈ 30 min
Parallel and perpendicular lines
Parallel lines have equal gradients. Perpendicular lines have gradients that multiply to $-1$ : flip the fraction and change the sign. With these two facts, plus one point, you can write down any related line.
≈ 30 min
Travel graphs and curves
On a distance–time graph the gradient is the speed. On a speed–time graph the gradient is the acceleration and the area underneath is the distance. Learn the shapes of quadratic, cubic, reciprocal and exponential graphs so you can recognise them at a glance.
≈ 35 min
Gradients of curves and graphical solutions
A curve's steepness changes, so its gradient at a point is the gradient of the tangent there: draw it with a ruler, pick two far-apart points on it, and find rise over run. Where two graphs cross, both equations are true at once, so the crossing points solve the equation formed by setting them equal.
≈ 30 min