Explorations in Pascal's Triangle
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Explorations in Pascal's Triangle Explorations in Pascal's Triangle Educators' Science and Math Institute Series Educators' Science and Mathematics Institute Series About the Series Goals Institutes Lesson Plans Links Photo Albums Explorations in Pascal's Triangle Artistic Expression Lesson Plan Jon Stasiuk P A S C A L S T R I A N G L E Search ESMIS Introduction: As I stated last year, mathematics is a wonderful subject for students interested in solving problems but seems to only be loved by a few. Students seldom get to appreciate the beauty and creativity that mathematics has to offer. This unit makes an attempt to show the student that mathematics isn't a subject to be studied by a few and actually exists in the real world. Students will explore Pascal's Triangle and all of its patterns and extend this pattern into combinatorics, algebra, and artistic expression. Rationale for the Project: Usually in second semester Algebra, when squaring binomials, I bring up Pascal's triangle as a sideline to how the coefficients are formed. To my surprise, most students have never even heard of Pascal's Triangle, or if they have it is not known by name and it's historical significance has been slighted. Overview of the Project: I have included many activities and non-traditional assignments that actively engage the students. I am also using this as a lesson plan for the first 3 - 4 days of class at the beginning of the school year. Introductory Lesson - Discovering Pascal's Triangle, extending the pattern, checking previous background knowledge of applications. Pretest Assignment - Identify other number patterns within Pascal's Triangle Lesson 1 - Historical background of Pascal and the Chinese Triangle Activity - Coin flipping, heads/tails, making a list and noticing how it relates to Pascal's Triangle file:///Users/suehill/Downloads/ExplorationsinPascalsTriangle.html[1/1/13 4:51:36 PM] Explorations in Pascal's Triangle Assignment - What's My Pattern and Missing Numbers handouts Lesson 2 - Further relations to combinatorics and algebra Activity - The Smithville Families Assignment - Binomial Expansion and King Strut's Cube Lesson 3 - Artistic expression Activity - Picture Patterns in coloring in different multiples in pascal's triangle Wrapping it up Posttest Relating art project to Sierpinski's Triangle . Resources Visual Patterns in Pascal's Triangle, Dale Seymour, Copyright 1986 by Dale Seymour Publications http://mathforum.org/workshops/usi/pascal/pascal/sierpinski.html Applications of Pascal's Triangle to Sierpinski's triangle and fractals http://mathforum.org/workshops/usi/pascal/pascal_hsdisc.html Explorations in Pascal's triangle using number patterns http://mathforum.org/workshops/usi/pascal/mid.color_pascal.html Explorations in Pascal's Triangle involving Coloring Multiples http://hsb.iitm.ernet.in/~jm/mar_april02/articles/pascal.htm Background in Pascal's Triangle http://www.roma.unisa.edu.au/07305/pascal.htm Binomial Theorem and the Pascal Triangle http://www.pbs.org/mathline The Smithville Families - applications to probability Standards Addressed: From the Michigan Curriculum Framework (Math) I.1.1, I.1.2, I.1.3, I.1.4, I.1.5, II.3.1, II.3.6, III.3.1, III.3.4, III.3.5, VI.2.1, VI.2.2, VI.2.6 I. Patterns, Relationships and Functions Students recognize similarities and generalize patterns, use patterns to create models and make predictions, describe the nature of patterns and relationships, and construct representations of mathematical relationships. Analyze and generalize mathematical patterns, including sequences, series, and recursive patterns. Analyze, interpret and translate among representations of patterns including tables, charts, graphs, matrices, and vectors. Study and employ mathematical models of patterns to make inferences, predictions, and decisions. Explore patterns and structural patterns within systems of objects, operations, or relations. file:///Users/suehill/Downloads/ExplorationsinPascalsTriangle.html[1/1/13 4:51:36 PM] Explorations in Pascal's Triangle Use patterns and reasoning to solve problems and explore new content. II Geometry and measurement 3 Students compare attributes of two objects, or of one object with a standard (unit), and analyze situations to determine what measurement(s) should be made and to what level of precision. Select and use appropriate tools; make accurate measurements using both metric and common units, and measure angles in degrees and radians. III Data Analysis and Statistics 3 Students collect and explore data, organize data into a useful form, and develop skill in representing and reading data displayed in different formats. 1 Design investigations to model and solve problems 4 Collect and explore data through observation, measurement, surveys, sampling techniques and simulations. Employ investigations, mathematical models, and simulations to make inferences and predictions to answer questions and solve problems. VI Probability and Discrete Mathematics 2 Students investigate practical situations such as scheduling, routing, sequencing, networking, organizing and classifying, and analyze ideas like recurrence relations, induction, iteration, and algorithm design.. Derive and use formulas for calculating permutations and combinations. Use sets and set relationships to represent algebraic and geometric concepts. Use discrete mathematics concepts as described above to model situations and solve problems; and look for whether or not there is a solution, determine how many solutions there are and decide upon a best solution. Introductory Lesson I used this on the first day of class. Our school traditionally has a half-day of school and we run 20 - 25 minute classes. So I decided to use the time for educational purposes rather than the traditional approach of going over classroom rules and course objectives. Given the beginning of the pattern, I instructed the students as follows: Introduce the pattern Name the pattern Extend the pattern See how many other patterns you could find in Pascal's Triangle Assignment : Identify other number patterns within Pascal's Triangle. . Square numbers 1, 4, 6, 25, 36 Fibonacci sequence 1, 1, 2, 3, 5, 8, 13, 21 Counting numbers 1, 2, 3, 4, 5, Triangular numbers 1, 3, 6, 10, 15 Powers of two 1, 2, 4, 8, 6 file:///Users/suehill/Downloads/ExplorationsinPascalsTriangle.html[1/1/13 4:51:36 PM] Explorations in Pascal's Triangle Lesson 1 A Brief Historical background of Pascal and the Chinese Triangle Blaise Pascal French Mathematician born in 1623 Fascinated by mathematics At age 19, invented one of the first calculating machines which actually worked. It was called the Pascaline. His dad was a tax collector and used it in his work. Interested in solving problems of chance ( He was a gambler ) and other algebraic problems. Pascal's Triangle Known in China as early as 1261 Also known as the Chinese Triangle. They ( Chinese ) used it for solving algebraic problems. Activity: Coin flipping and relating it to Pascal's Triangle Students given one coin and the list all the outcomes of what could happen. 1 tail 1 head student is given two coins and list all the outcomes It is helpful to give the student two different coins ( penny and nickel ) so that they can see that there are actually 2 different ways of getting a head and a tail 1 both tails 2 ways of getting a head and a tail 1 both heads three coins ( use three different coins ) 1 all tails 3 - 2 tails 1 head 3 - 1 tail 2 heads 1 all heads Students will record all the different outcomes and should discover that when there are two choices, H or T, Pascal's triangle is involved in predicting the amounts. Assignment: What's My Pattern and Missing Numbers handouts From: Visual Patterns in Pascal's Triangle, Dale Seymour. The Missing Numbers handout has portions of Pascal's Triangle and students need to fill in the missing parts by using the property as to how the triangle is extended: Ex: ___ _15 35 35 ____ ___ 56 The What's My Pattern handout has triangular arrays of numbers similar to Pascal's Triangle by have been created using different number patterns. The students are directed to discover each pattern and fill in the missing numbers. file:///Users/suehill/Downloads/ExplorationsinPascalsTriangle.html[1/1/13 4:51:36 PM] Explorations in Pascal's Triangle Lesson 2 Further relations to probability and algebra The previous days activity dealing with combinations of different coin flipping introduced the student to another aspect of Pascal's triangle. Activity: The Smithville Families This activity is designed to determine the total number of possible girl/boy combinations in a five-child family. This is accomplished by having students investigate the number and makeup of different five child families that could be born in a mythical town called Smithville. Coins are used to simulate the births of different children. If the coin shows a head, the child is a girl and if it shows a tail, the child is a boy. The different combinations are presented in an organized manner so that students can discover patterns that will enable them to identify all possibilities. Finally, students work collaboratively to address and analyze questions regarding the theoretical probabilities of other multiple-child families using Pascal's Triangle. Assignment: Binomial Expansion The general form of a binomial is (a + b). (a + b) 2 means (a + b)(a + b) and by using the FOIL method, (a + b) 2 = a2 + 2ab + b2. Two is the numerical coefficient of ab. The numerical coefficients of a2 and b2 is one. Students are directed to expand the following binomials: (a + b) 0 = 1 (a + b) 1 = 1a + 1 b (a + b) 2 = 1a2 + 2ab + 1b2 (a + b) 3 = 1a3 + 3a2b + 3ab2 + 1b3 (a + b) 4 = ( a + b) n Upon doing these, students discover that there is a relationship between Pascal's Triangle and binomial expansion. Lesson 3 Artistic Expression Mathematics is often described as the study of patterns. Learning to look for patterns in all aspects of mathematics will help make this work more entertaining and understandable. The purpose of this project is to discover some of the striking visual patterns in Pascal's Triangle. Activity: Picture patterns in coloring in different multiples in Pascal's Triangle. Students are to choose a whole number between 2 and 9 inclusive. Color in all of the elements in Pascal's Triangle that are divisible by that number. Do as many rows as necessary to determine the geometrical pattern. file:///Users/suehill/Downloads/ExplorationsinPascalsTriangle.html[1/1/13 4:51:36 PM] Explorations in Pascal's Triangle These produce very interesting and beautiful patterns. When all the odd numbers are shaded in, it produces a pattern called the Sierpinski Triangle. This is discussed in the next lesson. Many triangles of different sizes with either numbers filled in or not are available through Visual Patterns in Pascal's Triangle, Dale Seymour. Another option is to use multiple colors For four colors, use one color to color in the multiples of 4, another color to color in one less than the multiples of 4, a third color to color in two less than the multiples of 4, and a fourth color to color in three less than the multiples of 4 A similar strategy using three colors or five colors can be used to create other interesting patterns Wrap up relating art project to Sierpinski's Triangle Activity: Sierpinski Triangle via the Chaos Game In the previous lesson, the Sierpinski Triangle was mentioned as a byproduct of the coloring in of numbers in Pascal's Triangle. I have the students play the Chaos game goes something like this Students use a straightedge and draw a triangle, labeling the vertices A, B, C Randomly place a point in the triangle Roll a die and remember the number Assign the vertices A = 1 or 2 and B = 3 or 4 and C = 5 or 6 Make a dot halfway between the point and the randomly generated vertex. Repeat by rolling the die and making a dot halfway between the previous dot and the newly randomly generated vertex. Repeat this an infinite number of times. Or you could do the following: To speed things up, there are several computer programs and graphing calculator programs that similate the construction Pre and Post Assessment Questions These questions were graded on a ten point basis with all questions receiving one point except for questions 4 and 8 each receiving two points. 1) What is Pascal's Triangle? 2) Describe the pattern of numbers 3) What is the Chinese Triangle? file:///Users/suehill/Downloads/ExplorationsinPascalsTriangle.html[1/1/13 4:51:36 PM] Explorations in Pascal's Triangle 4) Give two other patterns of numbers in Pascal's Triangle 5) What are its applications to probability? 6) What are its applications to algebra? 7) What are its applications to art? 8) Describe the relationship between the Fibonacci sequence, Pascal's Triangle and Sierpinski's Triangle? file:///Users/suehill/Downloads/ExplorationsinPascalsTriangle.html[1/1/13 4:51:36 PM] Explorations in Pascal's Triangle file:///Users/suehill/Downloads/ExplorationsinPascalsTriangle.html[1/1/13 4:51:36 PM]
