Patterns & Relationships Activities
Grade 6 Mathematics
Explore algebraic thinking through pattern recognition, coordinate graphing, and functional relationships. These activities bridge concrete experiences with abstract mathematical concepts.
Learning Outcomes
By the end of these activities, students will be able to:
- Identify and describe patterns in tables of values and graphs involving perimeter, area and volume calculations
- Translate patterns from one representation to another (concrete, pictorial, symbolic)
- Describe pattern rules using symbols and one or more operations
- Write and solve problems with expressions and equations using unknowns
- Determine equality and inequality of quantities using mathematical expressions
- Use coordinate systems to represent and analyze mathematical relationships
Types of Mathematical Patterns
Different pattern types students will explore and analyze
Linear Growing Patterns
Patterns that increase by a constant amount
- • 3, 6, 9, 12... (add 3)
- • 5, 10, 15, 20... (add 5)
an + b
Geometric Growing Patterns
Patterns involving area and perimeter relationships
- • Square perimeters: 4, 8, 12, 16...
- • Rectangle areas: 2, 4, 6, 8...
Based on geometric formulas
Position-Value Patterns
Patterns where position determines value
- • Term 1=2, Term 2=4, Term 3=6...
- • Position × 2
f(n) = expression with n
Coordinate Patterns
Patterns visible on coordinate graphs
- • Points forming lines
- • Geometric shape patterns
y = mx + b relationships
Growing Pattern Investigation
Students investigate growing patterns using concrete materials and develop algebraic thinking through pattern analysis.
Materials Needed
- Pattern blocks or colored tiles
- Graph paper
- Colored pencils
- T-charts for recording
- Calculators
- Coordinate grid paper
Assessment Strategies
Formative Assessment:
- • Observe pattern building and extension accuracy
- • Check table completion and relationship identification
- • Monitor algebraic expression development
Summative Assessment:
- • Accuracy of pattern rule and predictions
- • Quality of graphical representations
- • Mathematical communication in explanations
Step-by-Step Instructions
Step 1: Pattern Creation
Create a growing pattern using blocks (e.g., 3, 6, 9, 12... or L-shapes that grow). Build the first 4-5 terms physically.
Step 2: Data Recording
Record pattern in a table showing term number and value. Look for relationships between term position and value.
Step 3: Graphical Representation
Graph the pattern on coordinate plane with term number on x-axis and pattern value on y-axis.
Step 4: Algebraic Expression
Write the pattern rule algebraically (e.g., 3n for the pattern 3, 6, 9, 12...). Test the rule with known values.
Step 5: Prediction and Verification
Predict the 20th term using the algebraic rule and verify by extending the pattern or using the formula.
Real-World Connections
Extension Activities
- Investigate non-linear growing patterns (quadratic)
- Create patterns for area and perimeter relationships
- Design patterns that decrease or have negative terms
- Connect patterns to real-world growth scenarios
Coordinate Art Gallery
Students create and interpret coordinate art while developing understanding of coordinate systems and algebraic relationships.
Materials Needed
- Graph paper (large grid)
- Colored pencils or markers
- Coordinate pair lists
- Rulers and straightedges
- Digital graphing tools (optional)
- Gallery display materials
Assessment Strategies
Formative Assessment:
- • Check coordinate plotting accuracy
- • Observe understanding of coordinate system
- • Monitor pattern recognition in coordinates
Summative Assessment:
- • Accuracy of original coordinate art instructions
- • Quality and creativity of artistic designs
- • Ability to follow and give coordinate directions
Step-by-Step Instructions
Step 1: Mystery Picture Introduction
Provide coordinate pairs that form a picture when plotted. Start with simple shapes to build confidence.
Step 2: Coordinate Plotting
Students plot points accurately and connect in order to reveal the hidden picture (school mascot, seasonal theme, etc.).
Step 3: Pattern Analysis
Analyze the coordinate relationships. Look for patterns in x and y values, symmetry, and transformations.
Step 4: Original Art Creation
Create their own coordinate picture for classmates. Plan design, determine coordinates, and write clear directions.
Step 5: Gallery Exhibition
Display artwork with coordinate instructions. Students visit gallery and recreate each other's coordinate art.
Real-World Connections
Extension Activities
- Use graphing software like Desmos for digital creation
- Explore coordinate transformations (reflection, rotation)
- Create 3D coordinate art using xyz coordinates
- Design coordinate art animations showing movement
Function Machine Experiments
Students explore functional relationships through hands-on function machine activities and algebraic expression building.
Materials Needed
- Function machine templates or boxes
- Input/output cards
- Recording sheets
- Operation cards (+, -, ×, ÷)
- Number cards
- Algebraic expression builders
Assessment Strategies
Formative Assessment:
- • Observe rule discovery strategies and logical thinking
- • Check algebraic expression accuracy
- • Monitor understanding of input-output relationships
Summative Assessment:
- • Accuracy of function rule identification
- • Quality of algebraic expression writing
- • Creativity and accuracy of student-designed function machines
Step-by-Step Instructions
Step 1: Function Machine Introduction
Introduce the concept of function machines that take inputs and produce outputs according to a rule.
Step 2: Rule Discovery
Give students input-output pairs and challenge them to discover the hidden function rule through testing.
Step 3: Expression Building
Write the discovered rule as an algebraic expression using variables (e.g., n + 5, 2n - 1).
Step 4: Prediction Testing
Use the algebraic expression to predict outputs for new inputs, then verify with the function machine.
Step 5: Machine Design
Design their own function machines with unique rules for other groups to solve and express algebraically.
Real-World Connections
Extension Activities
- Create multi-step function machines
- Explore inverse functions and reverse operations
- Connect to computer programming concepts
- Design function machines for geometry relationships
Table and Graph Connections
Students explore connections between tables, graphs, and algebraic expressions through area and perimeter investigations.
Materials Needed
- Grid paper
- Geometric shapes and manipulatives
- Graphing paper
- Measuring tools
- Table templates
- Calculators
Assessment Strategies
Formative Assessment:
- • Check measurement accuracy and calculations
- • Observe pattern recognition in tables and graphs
- • Monitor algebraic thinking development
Summative Assessment:
- • Accuracy of all three representations (table, graph, equation)
- • Quality of pattern analysis and connections
- • Understanding demonstration across multiple formats
Step-by-Step Instructions
Step 1: Perimeter Investigation Setup
Create rectangles with width 2 and varying lengths (1, 2, 3, 4, 5). Calculate perimeter for each rectangle.
Step 2: Data Table Creation
Record length and perimeter data in table format. Look for patterns in how perimeter changes as length increases.
Step 3: Graphical Analysis
Graph the relationship with length on x-axis and perimeter on y-axis. Analyze the shape and slope of the line.
Step 4: Pattern Rule Development
Develop algebraic expression for the pattern (P = 2l + 4). Test rule with table values.
Step 5: Multiple Representation Comparison
Compare the same relationship shown in table, graph, and equation. Discuss advantages of each representation.
Real-World Connections
Extension Activities
- Investigate area patterns for different shapes
- Explore volume patterns for 3D objects
- Create patterns involving multiple variables
- Connect to real architectural measurement problems
Developing Algebraic Thinking
Key concepts and progressions for building algebraic understanding
Concrete Level
- • Build patterns with physical materials
- • Count and measure concrete objects
- • Use manipulatives to show relationships
- • Connect to real-world situations
Representational Level
- • Create tables and graphs
- • Draw pictures and diagrams
- • Use coordinate systems
- • Make visual connections between formats
Abstract Level
- • Write algebraic expressions
- • Use variables and operations
- • Solve equations with unknowns
- • Generalize mathematical relationships