Year 9: What's the Difference
A first lesson
In the first lesson introduce a set of number
patterns for students to extend and find rules of, at least one
or two of which should be ones they will not get  to set up a motivation
to work on methods for doing this.
On the board, have written, eg:
a) 4, 6, 8, 10,
b) 2, 6, 10, 14,
c) 16, 19, 22, 25,
d) 3, 6, 11, 18,
e) 0, 4, 10, 18,
f) 0, 20, 80, 198,
~ Find the fifth, sixth, tenth and hundredth terms
of these sequences. I will give you 15 minutes to get as far as
you can  start at sequence (a). Some of these you may be able to
do, some of them you may not, in 15 minutes I'd like to talk about
what methods or strategies you can use to answer this kind of question.
Tip: calculate your sequences and the tenth, hundredth
terms etc on a spreadsheet!
Having given the class around 15 mins feedback,
in a whole class discussion, what answers students have got and,
crucially, what methods they used to find them. Highlight anyone
who found the algebraic rules as a way of getting the hundredth
term. It may be that you can show how a student's method can be
written algebraically.
A very common misconception is to eg double the
fifth term to get the tenth  the first sequence is designed to
be one where students can easily see this doesn't work. It is important
in this discussion to also introduce the term "difference pattern".
If this discussion has been fruitful you may want
to give students another 5 or so minutes to try and get any rules
they didn't get before.
At some point, the students should get stuck 
they can't find the rule even though they know the difference pattern.
This provides a motivation to work on the challenge for this project,
so that hopefully they can come back and solve these question later:
Given any rule what can you tell about the difference
pattern?
Students should choose a rule of their own, find
the first 5 (at least) terms and write down what they notice about
the difference pattern.
At first allow students a free choice of rules
(you may need to get from the class some examples to start off with).
It is likely students will not get anywhere doing this  and will
provide a motivation for a discussion about how to be organised
in picking rules. Write up at least 5 or 6 rules on the board from
suggestions of students (eg 4n+3, n2, 2n9, etc)  you can then
direct students to try these if they can't make up any of their
own.
As you go around supporting students, encourage
them to come up with conjectures eg "The first level difference
is the number infront of the n." With conjectures like this,
write them on the board and also encourage students to express them
using the language: "If the rule is an + b then ...".
Where this can go
At some stage in this series of lessons,
students should start developing conjectures about sets of rules.
These should be discussed as a class, written as generally as possible
(eg "an + b rules have a difference of a") and recorded
on a poster for the remainder of the lessons on this project. Whenever
a conjecture is mentioned get someone else in the class to make
a prediction using that conjecture. Further conjectures can first
be written on the board by students as a way of sharing results.
Encourage students to always have a conjecture
they are testing.
Extension
 If I have a rule an + b, can you prove that
the difference will always be a?
 If I have a rule an2 + bn + c, what will the
first and second differences be eg from looking at the first five
terms  how can you use this to find quadratic rules?
 Extend to a generalised cubic rule and beyond.
Guidance for notes at the end of the topic
Reading algebra: eg give examples of the
difference between 2n and n2 or between (2n) 2 and 2n2
Give notes on how to find the rule of a linear sequence.
? Give notes on how to find the rule of a quadratic sequence (perhaps
top set only).
Typical mathematical content
 Reading algebra
 Substituting into formulae
 Finding rules for linear and quadratic sequences
A homework to be done by everybody
Find the fifth, sixth, tenth, hundredth term
and the rule for these sequences:
(a) 2, 4, 6, 8,
(b) 3, 5, 7, 9,
(c) 1, 5, 9, 13,
(d) 17, 27, 37, 47,
(e) 1, 4, 9, 16,
(f) 0, 2, 6, 12,
