Mathematics · Year 6 · Algebraic Thinking and Patterns · Term 2

Identifying Linear Patterns and Rules

Identifying rules that describe the relationship between two sets of numbers.

• ACARA Content Descriptions
• AC9M6A01

About This Topic

Linear patterns and rules involve identifying the relationship between two sets of numbers, often represented as 'input' and 'output'. This topic, aligned with AC9M6A01, introduces students to algebraic thinking by asking them to find a rule that describes a sequence. Students learn to use tables of values to organize their data and look for constant differences, which indicates a linear relationship.

In Australia, students might investigate patterns in nature, such as the growth of a plant or the arrangement of petals, or patterns in Indigenous art and weaving. This topic is a bridge to formal algebra, moving from 'what comes next' to 'what is the rule for any term'. This topic comes alive when students can physically model the patterns using blocks or counters to see the growth visually.

Key Questions

1. How can we predict the hundredth term in a sequence without calculating every step?
2. What is the difference between an additive pattern and a multiplicative pattern?
3. How can a table of values help us identify a hidden rule?

Learning Objectives

• Identify the constant difference in a linear number sequence.
• Formulate a rule for a linear pattern using algebraic notation.
• Calculate the value of any term in a linear sequence given its rule.
• Compare additive and multiplicative patterns to determine linearity.
• Analyze tables of values to predict future terms in a sequence.

Before You Start

Number Patterns and Sequences

Why: Students need prior experience recognizing and extending simple number patterns, including additive and multiplicative ones.

Basic Operations (Addition, Subtraction, Multiplication)

Why: The ability to perform these operations is fundamental to identifying and applying patterns and rules.

Key Vocabulary

Sequence	A set of numbers that follow a specific order or pattern.
Term	Each individual number within a sequence.
Linear Pattern	A pattern where the difference between consecutive terms is constant, indicating a steady rate of change.
Rule	A mathematical expression or statement that describes the relationship between the position of a term and its value in a sequence.
Position	The place of a term in a sequence, often represented by 'n' or 'x'.

Watch Out for These Misconceptions

Common Misconception

The rule only applies to the next number in the sequence.

What to Teach Instead

Students often find the 'recursive' rule (add 3 each time) but struggle with the 'functional' rule (multiply by 3). Use a table of values to show how the rule must work for the 'term number' to find any position.

Common Misconception

All patterns are linear.

What to Teach Instead

Students may try to find a constant difference in patterns that don't have one. Introduce simple non-linear patterns (like square numbers) to show that rules can be more complex.

Active Learning Ideas

Inquiry Circle: Pattern Detectives

Groups are given a series of 'growing patterns' made of matchsticks. They must create a table of values, identify the rule (e.g., multiply by 2, add 1), and predict the 10th and 100th term.

45 min·Small Groups

Stations Rotation: Function Machines

Students rotate through stations where one student acts as the 'machine' with a secret rule. Others provide an 'input' number, and the machine provides the 'output' until the rule is guessed.

40 min·Small Groups

Gallery Walk: Visualizing Rules

Students create posters showing a pattern, its table of values, and its algebraic rule. They walk around the room and try to match rules to patterns created by their peers.

30 min·Whole Class

Real-World Connections

• City planners use linear patterns to predict population growth or traffic volume over time, helping them decide where to build new roads or public transport.
• Financial analysts model savings or loan repayments using linear rules to forecast account balances at future dates, assisting clients with financial planning.
• Engineers designing simple machines might use linear relationships to describe how the output changes with a consistent input, such as the distance a lever moves based on the force applied.

Assessment Ideas

Exit Ticket

Provide students with the sequence: 3, 7, 11, 15. Ask them to: 1. Identify the type of pattern (additive or multiplicative). 2. State the rule for the sequence. 3. Calculate the 10th term.

Quick Check

Display a table with two columns, 'Input' and 'Output', showing values like (1, 5), (2, 10), (3, 15). Ask students to write down the rule connecting Input to Output and predict the Output for an Input of 7.

Discussion Prompt

Present two sequences: Sequence A (2, 4, 6, 8) and Sequence B (2, 4, 8, 16). Ask students: 'Which sequence has a linear rule? How do you know? What is the rule for Sequence A? Can you find a rule for Sequence B?'

Frequently Asked Questions

How can active learning help students understand linear patterns?

By physically building patterns with blocks, students can see the 'constant' part of a rule and the 'growing' part. For example, in '2n + 1', they see the 1 block that never changes and the 2 blocks added for every new step. This visual and tactile experience makes the abstract algebraic rule much easier to grasp than just looking at a list of numbers.

What is a 'term' in a sequence?

A term is a single number in a sequence. Its 'position' refers to where it sits (1st, 2nd, 3rd, etc.).

Why do we use tables of values?

Tables help organize data so that the relationship between the position and the value becomes clear. It is the first step toward graphing linear equations.

How do I find the rule for a pattern?

Look for the difference between the values. If it increases by 3 each time, the rule likely involves 'multiply by 3'. Then adjust the rule to fit the first term.