049i - Generalising Patterns from Linear Families

Lesson 49 — Generalising Patterns from Linear Function Families

Strand: Algebra | Descriptor: AC9M8A04 | Duration: 45 minutes

Learning Intentions

• To generalise the rule connecting a table of values to the equation of a linear function.
• To describe a family of linear functions using a general algebraic rule.

Success Criteria

I can:

1. Generate a table of values from a linear function and identify the constant rate of change.
2. Generalise the rule connecting and as from a pattern or table.
3. Describe an entire family of lines (e.g. for varying ) in words and algebra.
4. Predict an unknown value using a generalised rule.

Warmup

(5 minutes — pattern-spotting, mini whiteboards)

Find the next two values and describe the rule in words.

Answers: 1. — add each time; 2. — subtract each time; 3. — increases by each time increases by ; 4. — decreases by each time.

Discussion: Ask, “Where have you seen this constant-difference idea before?” (link explicitly to gradient from Lessons 38–40 and 47–48.)

Activities

Activity 1 — From Table to Rule (10 min)

Teacher-led, guided. Build a table of values for :

	0	1	2	3	4
	2	5	8	11	14

Notice: each time increases by , increases by — a constant difference. This constant difference is the gradient, .

Notice: when , — this is the intercept, .

Generalising: the rule is , matching with , .

Guided practice — reverse the process. Given the table below, find the rule.

	0	1	2	3
	-1	2	5	8

Constant difference , so ; value at is , so . Rule: .

Your turn — pairs:

(Answers: 1. ; 2. ; 3. .)

Activity 2 — Investigating a Family with the Digital Tool (14 min)

Pairs, at a device with a graphing/spreadsheet tool.

Task 1. Generate tables of values (for ) for the family with . Record all four tables side by side.

• What is the same about the pattern of differences in every table?
• What changes between the tables?
• Generalise: for the whole family , what is true no matter what is?

Task 2. Generate tables for the family with .

• What is the same about every table (look at )?
• How does the pattern of differences change as changes?
• Generalise: for the whole family , what is true no matter what is?

Socratic scaffolding for the generalising step (Task 1):

Prompt	Purpose
Understand: what varies across your four tables, and what is fixed?	The starting value () varies; the constant difference () is fixed.
Devise a plan: how do you describe “always true” mathematically?	State it as a rule that works for any value of , not a specific number.
Carry out the plan	”For the family , the value of always increases by for every increase of in , regardless of .”
Look back — test it	Check against all four of your tables: does the difference stay in every one?
Look back — express it algebraically	The general rule already is — the algebra itself is the generalisation.

Answers: Task 1 — the constant difference () is identical in every table; only the starting value () changes; generalisation: every member of the family has rate of change , regardless of . Task 2 — every table has at ; the constant difference equals each time, so it changes across tables; generalisation: every member of the family passes through , regardless of .

Activity 3 — Applying Generalisation to a Growing Pattern (10 min)

Pairs.

A matchstick pattern grows as follows: square uses matches, squares in a row use matches, squares use matches, squares use matches.

1. Build a table of values ( = number of squares, = number of matches).
2. Find the constant difference and generalise the rule connecting and .
3. Does this rule belong to the family ? What are and in this context, and what do they mean physically?
4. Predict the number of matches needed for squares, without extending the table by hand.

Answers: 1. , ; 2. difference , rule ; 3. yes — represents the extra matches needed per additional square, represents the “extra” match beyond the repeating pattern (the initial square’s fourth side); 4. matches.

Checks for Understanding

(6 minutes — exit ticket, collected)

1. Find the rule for the table: .
2. A family of lines is . What is true for every member of this family, regardless of ?
3. A family of lines is . What point do all members share?
4. A pattern starts at and increases by each step. Write the general rule connecting the step number (starting at ) and the value .
5. Reasoning. Explain how you can tell, just from a table of values, whether it could represent a linear function.

Answers: 1. ; 2. Every member has gradient — a constant rate of change of ; 3. ; 4. ; 5. Check whether the difference between consecutive -values is constant whenever increases by a constant amount — a constant “second difference” of zero (constant first difference) indicates a linear relationship.

Common Misconceptions

Misconception	How to pre-empt it
Reading the constant difference directly as the -value at , rather than as the rate of change.	Explicitly separate “the value when ” (intercept) from “how much it changes by” (gradient) in every table.
Believing a family “changing ” also changes the rate of growth.	Refer back to Activity 2, Task 1 — the constant difference stayed fixed throughout.
In growing-pattern problems, treating as another “step” rather than a one-off starting adjustment.	Use the matchstick example to show represents a genuine physical difference (the extra side), not a pattern step.
Assuming any table with increasing -values must be linear.	Provide a counterexample table with non-constant differences (e.g. ) and ask students to test it against the constant-difference rule.
Over-generalising from too few data points (e.g. only two rows of a table).	Insist on checking at least three consecutive differences before generalising a rule.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). A table shows and . Find the rule and predict when .

Answer

Difference , intercept : . At : .

E2 (AMC Junior style). A family of functions is . For how many whole-number values of between and inclusive does the line pass through a point with a whole-number -value when ?

Answer

At : . Since is a whole number, is automatically a whole number for every one of the values from to — 11 values.

E3 (Challenge). A growing pattern of tiles has tiles at stage , at stage , at stage . Find the general rule, and determine at which stage the pattern first exceeds tiles.

Answer

Difference ; at stage , gives , so using : , giving . Solve , so stage is the first to exceed tiles ().

E4 (Challenge). Two families of functions, and , share a common member. If that common member has gradient and passes through , find and confirm .

Answer

Since gradient is : found from : . For the second family, also (matching), and check intercept: passes through ✓ matching the family .

Homework

1. Find the rule for each table: (a) (b) (c)
2. A family of lines is . What is true of every member, regardless of ?
3. A family of lines is . What point do all members share?
4. A pattern starts at and decreases by each step ( starting at ). Write the general rule connecting and the value .
5. A tiling pattern has tiles at stage , at stage , at stage . Find the general rule and predict the number of tiles at stage .
6. Reasoning. Explain why a table with a constant difference of still represents a linear function, and describe what its graph would look like.
7. Challenge. A family of functions is graphed for several values of . Find the fixed point shared by every member of the family, and justify algebraically.

Answers: Q1 — (a) (b) (c) . Q2 — every member has gradient , a constant rate of decrease of . Q3 — . Q4 — . Q5 — difference ; at , , so , giving ; at stage : tiles. Q6 — a constant difference of means never changes as increases — this is still “linear” in the sense of constant (zero) rate of change; its graph is a horizontal line. Q7 — substitute : for all ; every line passes through .