Patterns and generalisations Notes

Patterns Turn Specific Examples Into Mathematical Structure

1. Mathematics often begins with noticing something repeats or grows in a regular way.
2. A pattern is an observable regularity (in numbers, shapes, or relationships).
3. When you describe that regularity in a way that works beyond the examples you started with, you are making a generalization.

1. In the IB MYP, generalizing is not just about "spotting the next term".
2. It is about using logic to create and validate rules that can become part of a mathematical model.
3. Done well, generalizations help you solve complicated problems by considering a more general case first.

Conjectures Are Testable Claims Suggested By Patterns

When you think you have found a rule, you usually express it first as a conjecture.

Conjectures are essential because they turn noticing into doing:

1. you predict what should happen for the next case,
2. you test the prediction on new cases,
3. you revise the conjecture or work toward a proof.

Representations Help You See And Communicate Relationships

1. In MYP mathematics, a key idea is that relationships are connections and associations between properties, objects, people, and ideas.
2. A pattern is one kind of relationship, and representation is how you show it clearly.

Different representations reveal different features:

1. A table is good for spotting changes from one step to the next.
2. A diagram can reveal geometric structure.
3. A graph highlights trends and rates of change.
4. An algebraic rule is compact and works for any input.

A Reliable Process For Generalizing Patterns

A disciplined approach reduces the risk of incorrect generalizations.

1. Generate examples: compute or draw the first few cases accurately.
2. Organize: make a table (input $n$, output value) and look for structure.
3. Describe the pattern: use words first (what changes each step?).
4. Conjecture a rule: write an algebraic expression in terms of $n$.
5. Test on new cases: check values you did not use to create the rule.
6. Justify: explain why the rule must work (a proof or a convincing argument).

Common Pattern Types You Should Recognise Quickly

Linear Patterns Have A Constant First Difference

1. A linear pattern changes by the same amount each step.
2. If the sequence is $3, 7, 11, 15, \dots$, the differences are $+4, +4, +4, \dots$ so a rule has the form $an+b$.
3. To find $a$ and $b$:
   1. the common difference is $a$,
   2. substitute any term to find $b$.

Quadratic Patterns Have Constant Second Differences

If first differences are not constant, check second differences.

Geometric Patterns Multiply By A Constant Factor

If each term is obtained by multiplying by the same number, the pattern is geometric.

A Famous Cautionary Example: Regions Formed By Joining Dots On A Circle

A classic exploration shows why it is risky to generalize from a small number of cases.

1. You place $n$ dots on the circumference of a circle and connect every pair of dots with a straight line segment (a chord).
2. The chords divide the circle into regions.
3. Let $r$ be the number of regions.

From careful drawings:

1. $n=1 \Rightarrow r=1$
2. $n=2 \Rightarrow r=2$
3. $n=3 \Rightarrow r=4$
4. $n=4 \Rightarrow r=8$
5. $n=5 \Rightarrow r=16$

A table makes the pattern tempting:

It looks like the number of regions doubles each time, so you might conjecture: $$r = 2^{n-1}$$

Testing The Conjecture Matters

1. If the conjecture were correct, then for $n=6$ you would predict $r=32$.
2. However, when you actually draw the $n=6$ case carefully (ensuring that no three chords cross at the same interior point), you get: $$r = 31$$
3. So the simple doubling rule is false, even though it fits perfectly for $n=1$ to $5$.

What Went Wrong With "Doubling"?

1. The early drawings create a doubling illusion because each new dot creates many new intersections, and intersections control how regions increase.
2. The number of intersections does not grow in a way that keeps exact doubling forever.
3. There is a correct general formula for this problem, but it is not obvious.
4. It comes from graph theory and connects vertices, edges, and regions (related to Euler's formula).

Generalizing Relationships Beyond Mathematics

1. Mathematical generalization also supports modeling in other subjects because it clarifies trends among individuals and systems.
2. For example, in biology, organisms coexist through ecological relationships that can be:
   1. oppositional (such as predation or competition),
   2. symbiotic (one organism lives in or on another), sometimes mutualistic (both benefit).
3. You might observe several examples of mutualism and generalize "both organisms benefit".
4. But biology also includes cases where one benefits and the other is harmed, so generalizations must be made with care, clear definitions, and awareness of exceptions.