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, 2n-9, 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,