Sierpinski's Triangle

What is Sierpinski's Triangle?

This activity allows the user to step through the process of building the Sierpinski's Triangle.

This activity is meant to show how changing the shape but using the same idea for a generator in a geometric fractal can yield a predictable final product. This activity should be tried along with the Sierpinski's Carpet Activity for comparison purposes.

To build the Sierpinski's Triangle, start with an equilateral triangle with side length 1 unit, completely shaded. (Iteration 1, the initiator)

Divide each triangle into four equal triangles by finding the midpoint of each side and connecting the midpoints. This generates 4 equal size triangles. Then cut out the middle one.

Repeat this process on all shaded triangles. Stage 3 is shown below.

The limiting figure for this process is called the Sierpinski's Triangle.

How Do I Use This Activity?

This activity allows the user to step through the process of building the Sierpinski's Triangle.

Controls and Output

  • The Next Stage and Previous Stage buttons at the bottom of the activity control the stage of the fractal that is being viewed.
  • The Output Box at the top of the activity lists the current stage's number of triangles and the individual triangle side length.

Description

This activity allows the user to step through the process of building the Sierpinski's Triangle. This activity would work well in same ability groups of three or four for about thirty-five minutes if you use the exploration questions and five minutes otherwise.

Place in Mathematics Curriculum

This activity can be used to:

  • practice students' area and perimeter skills
  • demonstrate fractal objects

Standards Addressed

Grade 6

  • Geometry

    • The student demonstrates conceptual understanding of similarity, congruence, symmetry, or transformations of shapes.

Grade 7

  • Geometry

    • The student demonstrates conceptual understanding of similarity, congruence, symmetry, or transformations of shapes.

Grade 8

  • Geometry

    • The student demonstrates conceptual understanding of similarity, congruence, symmetry, or transformations of shapes.

Grade 9

  • Geometry

    • The student demonstrates an understanding of geometric relationships.
    • The student demonstrates conceptual understanding of similarity, congruence, symmetry, or transformations of shapes.
    • The student demonstrates a conceptual understanding of geometric drawings or constructions.

Grade 10

  • Geometry

    • The student demonstrates an understanding of geometric relationships.
    • The student demonstrates conceptual understanding of similarity, congruence, symmetry, or transformations of shapes.
    • The student demonstrates a conceptual understanding of geometric drawings or constructions.

Fourth Grade

  • Operations and Algebraic Thinking

    • Generate and analyze patterns.

Fifth Grade

  • Operations and Algebraic Thinking

    • Analyze patterns and relationships.

Eighth Grade

  • Geometry

    • Understand congruence and similarity using physical models, trans- parencies, or geometry software.

Geometry

  • Similarity, Right Triangles, and Trigonometry

    • Prove theorems involving similarity

Grades 3-5

  • Algebra

    • Understand patterns, relations, and functions
    • Use mathematical models to represent and understand quantitative relationships

  • Geometry

    • Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships

Grades 6-8

  • Algebra

    • Use mathematical models to represent and understand quantitative relationships

  • Geometry

    • Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships
    • Use visualization, spatial reasoning, and geometric modeling to solve problems

Grades 9-12

  • Algebra

    • Represent and analyze mathematical situations and structures using algebraic symbols
    • Understand patterns, relations, and functions

  • Geometry

    • Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships
    • Use visualization, spatial reasoning, and geometric modeling to solve problems

  • Measurement

    • Apply appropriate techniques, tools, and formulas to determine measurements

Grade 5

  • Number and Operations, Measurement, Geometry, Data Analysis and Probability, Algebra

    • COMPETENCY GOAL 1: The learner will understand and compute with non-negative rational numbers.

Grade 6

  • Number and Operations, Measurement, Geometry, Data Analysis and Probability, Algebra

    • COMPETENCY GOAL 1: The learner will understand and compute with rational numbers.

Grade 7

  • Number and Operations, Measurement, Geometry, Data Analysis and Probability, Algebra

    • COMPETENCY GOAL 1: The learner will understand and compute with rational numbers.

Grade 8

  • Number and Operations, Measurement, Geometry, Data Analysis and Probability, Algebra

    • COMPETENCY GOAL 3: The learner will understand and use properties and relationships in geometry.

Introductory Mathematics

  • Data Analysis and Probability

    • COMPETENCY GOAL 3: The learner will understand and use properties and relationships in geometry.

  • Geometry and Measurement

    • COMPETENCY GOAL 2: The learner will use properties and relationships in geometry and measurement concepts to solve problems.

Geometry

  • Geometry and Measurement

    • Competency Goal 2: The learner will use geometric and algebraic properties of figures to solve problems and write proofs.

Technical Mathematics I

  • Geometry and Measurement

    • Competency Goal 2: The learner will measure and apply geometric concepts to solve problems.

Technical Mathematics II

  • Geometry and Measurement

    • Competency Goal 1: The learner will use properties of geometric figures to solve problems.

Advanced Functions and Modeling

  • Algebra

    • Competency Goal 2: The learner will use functions to solve problems.

Discrete Mathematics

  • Algebra

    • Competency Goal 3: The learner will describe and use recursively-defined relationships to solve problems.

Integrated Mathematics III

  • Geometry and Measurement

    • Competency Goal 2: The learner will use properties of geometric figures to solve problems.

6th Grade

  • Algebra

    • The student will demonstrate through the mathematical processes an understanding of writing, interpreting, and using mathematical expressions, equations, and inequalities.

5th Grade

  • Algebra

    • The student will demonstrate through the mathematical processes an understanding of the use of patterns, relations, functions, models, structures, and algebraic symbols to represent quantitative relationships and will analyze change in various contexts.

7th Grade

  • Algebra

    • The student will demonstrate through the mathematical processes an understanding of proportional relationships.

  • Geometry

    • The student will demonstrate through the mathematical processes an understanding of proportional reasoning, tessellations, the use of geometric properties to make deductive arguments. the results of the intersection of geometric shapes in a plane, and the
    • The student will demonstrate through the mathematical processes an understanding of proportional reasoning, tessellations, the use of geometric properties to make deductive arguments. the results of the intersection of geometric shapes in a plane, and the relationships among angles formed when a transversal intersects two parallel lines.

Intermediate Algebra

  • Algebra

    • The student will demonstrate through the mathematical processes an understanding of sequences and series.

Geometry

  • Geometry

    • Standard G-2: The student will demonstrate through the mathematical processes an understanding of the properties of basic geometric figures and the relationships between and among them.

3rd Grade

  • Patterns, Functions, and Algebra

    • 3.24 The student will recognize and describe a variety of patterns formed using concrete objects, numbers, tables, and pictures, and extend the pattern, using the same or different forms (concrete objects, numbers, tables, and pictures).

7th Grade

  • Patterns, Functions, and Algebra

    • 7.19 The student will represent, analyze, and generalize a variety of patterns, including arithmetic sequences and geometric sequences, with tables, graphs, rules, and words in order to investigate and describe functional relationships.

4th Grade

  • Patterns, Functions, and Algebra

    • 4.21
    • 4.21 The student will recognize, create, and extend numerical and geometric patterns, using concrete materials, number lines, symbols, tables, and words.

5th Grade

  • Geometry

    • 5.15c The student, using two-dimensional (plane) figures (square, rectangle, triangle, parallelogram, rhombus, kite, and trapezoid) will investigate and describe the results of combining and subdividing shapes

Secondary

  • Algebra II

    • AII.16 The student will investigate and apply the properties of arithmetic and geometric sequences and series to solve practical problems, including writing the first n terms, finding the nth term, and evaluating summation formulas. Notation will include Σ and an.
    • AII.16

Textbooks Aligned

Grade Six

  • Bits and Pieces I

    • Investigation One: Fund-Raising Fractions
    • Investigation Two: Comparing Fractions
    • Investigation Three: Cooking with Fractions
    • Investigation Four: From Fractions to Decimals
    • Investigation Five: Moving Between Fractions and Decimals
    • Investigation Six: Out of One Hundred

  • Covering and Surrounding

    • Investigation One: Measuring Perimeter and Area
    • Investigation Two: Measuring Odd Shapes
    • Investigation Six: Measuring Triangles

Grade Seven

  • Comparing and Scaling

    • Investigation Three: Comparing by Using Ratios

Grade Eight

  • Looking for Pythagoras

    • Investigation Two: Finding Areas and Lengths

Book 1

  • Module 2 - Patterns and Designs

    • Section 2: Fractions
    • Section 3: Equivalent Fractions
    • Section 5: Decimal Place Value

  • Module 3 - Statistical Safari

    • Section 2: Fractions, Percents

  • Module 4 - Mind Games

    • Section 3: Fraction Multiplication

  • Module 5 - Creating Things

    • Section 1: Common Denominators

Book 2

  • Module 1 - Making Choices

    • Section 2: Sequences
    • Section 2: Exponents

  • Module 3 - A Universal Language

    • Section 2: Tree Diagrams and Probability

  • Module 4 - The Art of Motion

    • Section 1: Multiplying Fractions, Dividing Fractions

  • Module 5 - Recreation

    • Section 3: Percents

Book 3

  • Module 4 - Patterns and Discoveries

    • Section 1: Sequences
    • Section 1: Fractals
    • Section 1: Constructing Triangles
    • Section 1: Sierpinski Triangle
    • Section 3: Equations with Fractions

6th

  • Module 1 - Patterns and Problem Solving

    • Section 2: Patterns and Sequences

  • Module 3 - Mind Games

    • Section 6: Decimal Multiplication

  • Module 6 - Comparisons and Predictions

    • Section 5: Geometry and Proportions

7th

  • Module 6 - Flights of Fancy

    • Section 5: Triangles and Similarity

8th

  • Module 6 - Visualizing Change

    • Section 3: Modeling Exponential Change

  • Module 8 - MATH-Thematical Mix

    • Section 1: Patterns and Sequences

Book 1

  • From Zero to One and Beyond

    • Lesson 1: Folding Fractions
    • Lesson 4: Out of One Hundred
    • Lesson 5: Percents That Make Sense
    • Lesson 9: All Three at Once

  • Number Powerhouse

    • Lesson 5: Pluses and Minuses
    • Lesson 6: Multiplication Made Easy
    • Lesson 7: The Great Fraction Divide

Book 2

  • Buyer Beware

    • Lesson 6: Which Brand Has the Most Chocolate?

  • Making Mathematical Arguments

    • Lesson 6: Counterexamples and Special Cases

Grade 8

  • Patterns and Figures

    • Patterns
    • Generalities
    • Progressions
    • Rectangular Numbers
    • Triangular Numbers
    • Pascal's Triangle

  • Reflections on Number

    • Divisibility and Prime Factorization
    • Multiplication and Division
    • Operations with Inverses

  • Triangles and Patchwork

    • Similarity
    • Similar Triangles
    • Tessellations

Grade 5

  • Grasping Sizes

    • Ratios
    • Proportional Enlargements and Reductions
    • Scale Lines
    • Calculating Ratios

  • Measure for Measure

    • Equivalent Decimals
    • Fraction/Decimal Equivalence
    • Adding and Subtracting Decimals

  • Patterns and Symbols

    • Variables
    • Patterns
    • Pattern Rule

  • Per Sense

    • Using Percents
    • Fraction/Percent/Ratio Equivalence
    • Estimating Percents

  • Some of the Parts

    • Fractions
    • Relationships between Fractions
    • Operations with Fractions

Grade 6

  • Fraction Times

    • Operations with Fractions
    • Fraction/Percent/Decimal/Ration Relationships

  • Made to Measure

    • Length
    • Volume
    • Surface Area
    • English Units
    • Metric Units

  • More or Less

    • Fraction/Decimal/Percent Relationships
    • Operations with Percents
    • Decimal Multiplication

  • Ratios and Rates

    • Ratio/Fraction/Decimal/Percent Relationships
    • Part-Part Ratios
    • Part-Whole Ratios
    • Scale Factor
    • Linear Functions

  • Reallotment

    • Estimating Area
    • Perimeter
    • Surface Area
    • Volume
    • English and Metric Units

Grade 7

  • Building Functions

    • Algebraic Descriptions
    • Sequences
    • Equivalent Expressions
    • Squares and Square Roots

  • Cereal Numbers

    • Volume
    • Surface Area
    • Relationship between Volume and Surface Area
    • Comparisons with Ratios
    • Fractions
    • Decimals and Percents
    • Multiplying and Dividing Fractions

  • Packages and Polygons

    • Geometric Shapes
    • Models
    • Properties of Regular and Semi-Regular Polyhedra

Be Prepared to

  • give implicit directions on what they are to do. For example, "Today we are going to fill out the work sheet and see if we can find formulas..."
  • answer the question "What does it mean by the n-th case?"
  • discuss how to find patterns in numbers