Mathematics · Year 7 · Patterns and Variable Thinking · Term 1

Identifying and Describing Patterns

Students will identify visual and numerical sequences and describe them using words.

• ACARA Content Descriptions
• AC9M7A01

About This Topic

Generalising patterns is the bridge between arithmetic and algebra. In Year 7, students move from simply identifying the 'next number' to describing the underlying rule of a sequence using words and algebraic symbols (AC9M7A01). They explore visual patterns, such as matchstick shapes or tile designs, and translate these into tables of values and linear expressions. This skill is vital for developing abstract reasoning and the ability to make predictions based on data.

By learning to use variables, students gain a powerful tool for describing universal truths rather than specific instances. This topic is most effective when students can interact with physical patterns and collaborate to 'crack the code' of a sequence. Students grasp this concept faster through structured discussion and peer explanation, where they must justify why their rule works for any term in the sequence.

Key Questions

1. Analyze how different patterns can be represented visually and numerically.
2. Construct a rule in words for a given sequence of numbers.
3. Differentiate between increasing and decreasing patterns.

Learning Objectives

• Identify visual and numerical patterns in sequences.
• Describe identified patterns using clear, concise language.
• Construct a rule in words to represent a given numerical sequence.
• Differentiate between increasing and decreasing numerical patterns.
• Analyze how visual patterns can be translated into numerical sequences.

Before You Start

Number Properties and Operations

Why: Students need a solid understanding of addition, subtraction, multiplication, and division to identify and describe numerical patterns.

Basic Shape Recognition

Why: Familiarity with common geometric shapes is helpful for identifying and describing visual patterns.

Key Vocabulary

Pattern	A sequence of numbers or shapes that repeat or follow a specific rule.
Sequence	An ordered set of numbers or shapes that follow a specific rule or pattern.
Rule	The specific instruction or relationship that determines how each term in a sequence is generated from the previous term.
Term	Each individual number or shape within a sequence.

Watch Out for These Misconceptions

Common Misconception

Students often use the 'recursive' rule (e.g., 'add 3 each time') instead of the 'functional' rule (e.g., 3n + 1).

What to Teach Instead

Ask students to find the 100th term. They quickly realise that adding 3 one hundred times is inefficient, which motivates the need for a rule that relates the term number directly to the value. Peer discussion helps highlight this efficiency.

Common Misconception

Thinking that a variable like 'n' must always stand for a specific, hidden number.

What to Teach Instead

Use 'input-output' machines where students see that 'n' can be any number they choose. Hands-on activities where different students 'be' different values of 'n' help show that the variable represents a position, not a fixed constant.

Active Learning Ideas

Inquiry Circle: Matchstick Patterns

Students use matchsticks (or toothpicks) to build a growing geometric pattern. They record the number of sticks for each step in a table and work together to find a rule that predicts the number of sticks needed for the 100th step.

45 min·Small Groups

Think-Pair-Share: Pattern Detectives

Provide students with a sequence of numbers. Individually, they write a rule in words, then pair up to translate that rule into an algebraic expression (e.g., 'double the number and add one' becomes 2n + 1).

20 min·Pairs

Gallery Walk: Visual to Variable

Groups create a visual pattern on a poster but leave the algebraic rule hidden under a flap. Other groups rotate through, try to determine the rule, and write their guess on a sticky note before checking the answer.

35 min·Small Groups

Real-World Connections

• Architects use patterns to design repeating elements in buildings and structures, like the arrangement of windows on a facade or the spacing of support beams.
• Graphic designers create visual patterns for logos, websites, and advertisements, ensuring consistency and aesthetic appeal through repeating motifs or color schemes.
• Musicians develop melodic and rhythmic patterns that form the basis of songs, allowing listeners to recognize and anticipate musical phrases.

Assessment Ideas

Quick Check

Present students with a visual pattern (e.g., growing squares made of dots). Ask them to draw the next two stages and write a sentence describing how the pattern grows. Then, provide a numerical sequence like 3, 6, 9, 12 and ask for the next two numbers and the rule in words.

Exit Ticket

Give each student a card with a different numerical sequence (e.g., 10, 8, 6, 4 or 5, 10, 15, 20). Ask them to write the next two terms and the rule in words. Also, ask them to identify if the pattern is increasing or decreasing.

Discussion Prompt

Display two sequences: one increasing (e.g., 2, 4, 6, 8) and one decreasing (e.g., 15, 12, 9, 6). Ask students: 'How are these patterns different? What words can we use to describe the rule for each sequence? Can you create your own increasing and decreasing pattern?'

Frequently Asked Questions

How can active learning help students understand generalising patterns?

Active learning allows students to physically build and manipulate patterns, making the relationship between the 'step number' and the 'total' visible. When students work together to describe a pattern, they are forced to move from intuitive 'seeing' to formal 'describing,' which is the essence of algebraic thinking. Collaborative problem-solving helps them test and refine their rules in real time.

What is the difference between a sequence and a rule?

A sequence is the list of numbers or shapes themselves (e.g., 2, 4, 6, 8). The rule is the mathematical instruction that tells you how to get those numbers (e.g., multiply the position by 2).

Why do we use letters in math patterns?

Letters, or variables, act as placeholders for any number. They allow us to write a single, short rule that covers every possible step in a pattern, which is much faster than writing out long sentences.

How can I help my child find the rule for a pattern?

Encourage them to look at the 'change' between terms. If the pattern goes up by 5 each time, the rule likely involves '5n'. Then, they can check what they need to add or subtract to get the first term.