This unit starts from the idea that a pattern can be described by how it changes. A sequence that adds the same amount each time has a rule of the form dn + c, and the same rule reaches the 50th term as quickly as the 5th. When the differences themselves are not constant, the differences of the differences are: a constant second difference means an n² term, and half of it is the number in front.
The next two lessons follow a sequence that multiplies. A common ratio gives a power, and a percentage rise or fall is a multiplier, so interest and depreciation are geometric sequences. The unit then turns from numbers to letters: making a different letter the subject of a formula by undoing steps in reverse order, and, when the letter appears twice, by collecting and factorizing.
The last lesson brings the threads together in direct and inverse proportion. Each has a constant that is found from one pair of values, and powers and roots change which combination of the two quantities stays fixed.
A graphic display calculator (GDC) is optional all the way through. Lessons 1 and 8 have a closed “With a GDC” section to open if you have one; everything else is done by hand or with a scientific calculator.
The lessons, in order
- Lesson 1 · Arithmetic sequences and the nth term Common difference, the position-to-term rule dn + c, finding a term, deciding which term has a value, and recovering a rule from two terms. Change the first term and the difference, and watch six terms line up on one straight line. GDC optional
- Lesson 2 · Second differences and quadratic sequences First and second differences, recognizing a quadratic sequence, and linking the second difference to the number in front of n squared. Change a, b and c and see which row of differences never changes. No calculator
- Lesson 3 · The nth term of a quadratic sequence Finding an squared plus bn plus c: a from the second difference, subtract an squared, then the arithmetic rule for the remainder. Guess the number in front of n squared and see which guess leaves an arithmetic remainder. No calculator
- Lesson 4 · Geometric sequences Common ratio, the nth term a r to the power n minus 1, negative and fractional ratios, and finding terms, positions and ratios. Set the first term and the ratio, and see the bars grow by multiplication, then read the same terms backwards. No calculator
- Lesson 5 · Growth and decay by a percentage Multiplier for a percentage rise or fall, repeated change as a power, compound interest, depreciation, and finding a rate or a number of years. Set a yearly percentage and a number of years, and see how the value curves away from a straight-line rule. Scientific calculator
- Lesson 6 · Rearranging formulae Making a letter the subject by inverse operations in reverse order, including formulas with brackets, fractions, squares and square roots. Run a two-step formula forwards and then undo it step by step to get back to the start. No calculator
- Lesson 7 · When the subject appears twice Collecting terms in the subject, factorizing, and dividing, with formulas where the subject is on both sides, in a bracket or in a fraction. Join two rectangles of the same height and see why ax + bx is (a + b) times x. No calculator
- Lesson 8 · Direct and inverse proportion Direct and inverse proportion, finding k, powers and roots, and recognizing the constant product or quotient and the shape of the graph. Switch between four kinds of proportion and see which combination of x and y never changes. GDC optional
Every lesson has a teacher page with the answers, the misconceptions each wrong answer reveals, discussion questions, differentiation and support for students learning in English, and extra questions in three tiers.