Patterns and Algebraic Thinking
Overview
This unit introduces students to algebraic thinking as a natural extension of recognising and describing patterns. Students will explore number patterns, use variables to represent unknown quantities, write and evaluate expressions, and solve simple equations. Through pattern investigation and problem-solving, students will develop the foundational skills needed for formal algebra.
Key Concept
Relationships — Mathematics is the study of relationships. Algebra provides a language for expressing and working with relationships between quantities.
Related Concepts
- Patterns — Regularities in numbers, shapes, and situations that can be described and extended.
- Generalisation — Creating rules that describe patterns and relationships in a general way.
- Models — Using algebraic expressions and equations to represent real-world situations.
Global Context
Scientific and Technical Innovation — How does algebraic thinking enable scientific discovery and technological innovation?
Statement of Inquiry
Algebraic thinking generalises patterns and relationships, providing powerful tools for problem-solving.
Inquiry Questions
Factual Questions
- What is a variable and how is it used?
- What is the difference between an expression and an equation?
- How do you evaluate an algebraic expression?
Conceptual Questions
- How do patterns help us understand mathematical relationships?
- Why do we use letters to represent numbers?
- How can algebra be used to solve real-world problems?
Debatable Questions
- Would mathematics be easier without variables?
- Is algebra more about finding answers or about understanding relationships?
- Can everything in the universe be described mathematically?
ATL Skills
Thinking Skills
- Identify patterns and describe them using mathematical language.
- Apply generalisation to create rules and formulas.
- Analyse relationships between quantities in real-world contexts.
Communication Skills
- Use algebraic notation correctly.
- Explain the meaning of variables and expressions in context.
- Present solutions with clear logical steps.
Research Skills
- Investigate patterns in nature, art, and daily life.
- Explore historical development of algebraic notation.
- Gather data to identify relationships and patterns.
Self-Management Skills
- Develop organised approaches to pattern investigation.
- Maintain practice records for skill development.
- Set goals for algebraic reasoning growth.
Content
Week 1: Recognising and Extending Patterns
- Visual patterns: shapes, colours, arrangements.
- Number patterns: arithmetic sequences.
- Describing patterns in words.
- Predicting future terms in a pattern.
Week 2: Introduction to Variables
- What is a variable? Using letters for unknown quantities.
- Writing expressions from word phrases.
- Evaluating expressions by substituting values.
- Variables in real-world contexts.
Week 3: Algebraic Expressions
- Building expressions: terms, coefficients, constants.
- Simplifying expressions by combining like terms.
- Writing expressions to represent pattern rules.
- The difference between expressions and equations.
Week 4: Solving Simple Equations
- What is an equation? Balancing both sides.
- Solving one-step equations using inverse operations.
- Checking solutions by substitution.
- Writing equations from word problems.
Week 5: Patterns in Geometry and Measurement
- Perimeter and area formulas as algebraic relationships.
- Patterns in geometric figures.
- Using variables in measurement contexts.
- Creating formulas from geometric patterns.
Week 6: Algebraic Thinking in the Real World
- Using algebra to solve everyday problems.
- Patterns in data and statistics.
- Algebraic thinking in computer programming.
- Unit review and summative assessment.
Summative Assessment
Pattern Investigation Project: Students will investigate a pattern from the real world or from mathematics, describe it in words, create a table of values, write an algebraic rule, and use the rule to make predictions. The project must include a written report and a visual presentation.
Algebraic Problem-Solving Test: Students will complete a test demonstrating their ability to write expressions, evaluate expressions, solve simple equations, and apply algebraic thinking to word problems.
Formative Assessment
- Pattern extension and description exercises.
- Variable and expression identification activities.
- Expression evaluation practice.
- Equation solving with visual models (balance scales).
- Word problem translation to algebra.
- Quizzes on algebraic vocabulary and concepts.
Interdisciplinary Connections
- Science: Using formulas and expressing relationships in science investigations.
- Design: Patterns in design and coding logic.
- Music: Patterns in rhythm and musical notation.
- Physical Education: Patterns in movement and sports statistics.
Service as Action
- Create pattern-based puzzles or games for younger students.
- Design a mathematics trail around the school identifying patterns.
- Develop a resource explaining algebraic thinking to peers.
- Organise a problem-solving workshop for students preparing for algebra.
IB Learner Profile
- Thinkers: Students apply logical reasoning to identify and extend patterns.
- Inquirers: Students explore mathematical relationships with curiosity.
- Communicators: Students express relationships clearly using algebraic language.
- Knowledgeable: Students understand how algebra provides tools for generalisation.
- Reflective: Students reflect on their developing algebraic reasoning skills.
Self-Test
- What is a variable? Give an example of how a variable is used.
- Extend the pattern: 3, 7, 11, 15, __, __. What is the rule?
- Write an expression for "five more than three times a number."
- Evaluate 2x + 5 when x = 3.
- Simplify: 4a + 2b - a + 3b.
- Solve: x + 7 = 15. How did you find the answer?
- Write an equation for: "A number multiplied by 4 equals 20."
- What is the difference between an expression and an equation?
- A rectangle has length L and width W. Write an expression for its perimeter.
- Create a pattern of your own, describe it in words, and write a rule for the nth term.
