Exploring Pascal's Triangle and Other Recursive Patterns - Carole Bersani
Contents of Curriculum Unit
Intent of Unit
The purpose of this unit is to develop students' critical thinking skills
and problem solving strategies by teaching them to recognize and create
patterns. The patterns will be developmentally introduced starting with
actions, shapes, and pictures, and ending with numbers. Class discussion
will be a large component in order for the students to understand how others
think. Mathematical discussions are beneficial because students see that
there is quite often more than one way to solve a problem. I intend to
discuss why Pascal's Triangle is an effective tool as one of the focuses
for this unit. Additionally, there will be an emphasis on integrating children's
literature within the unit. Lastly, students will be writing in math journals
at the completion of each lesson. There will be three specific lesson plans
designed for a third, fourth, or fifth grade classroom. The time required
for completing this unit is two weeks, depending on the ability level of
In addition to focusing on patterns, I have suggested a few extensions in the area of Number Theory and Discrete Mathematics, specifically combinations of numbers. I recommend having a poster of Pascal's Triangle as well as providing multiple copies of the triangle for each student to use. The extensions are not presented with as much detail as the actual lesson plans for the simple reason that they are not the focus of the unit. At the end of the unit is a comprehensive bibliography, including children's literature, teacher resources, and Internet sites.
Why Patterns are Important
A familiarity with patterns will allow students to gain a deeper understanding of attributes and relationships between numbers. These patterns will build a base for understanding fractions, algebra, geometry, and calculus.
Why Pascal's Triangle?
Pascal's Triangle is an excellent choice for a unit on recursive patterns because there is such a wide variety of patterns including, but not limited to primes, Fibonacci's Sequence, factorials, polygonal numbers, and Serpinski's Right Triangle. A pattern is recursive if the terms in a sequence use the same procedure that the pattern starts with. Most of the sequences found in the triangle are recursive because they copy the same rule of progression throughout the entire sequence. Students will be able to see visual patterns if presented with a copy of the triangle with certain elements shaded. One suggestion is to shade the multiples of any given number and have the students guess the visual pattern. Another option is to use the triangle for divisibility rules by shading in numbers divisible by any given number.
Pascal's Triangle Defined
The triangle continues infinitely. The sum of two adjacent numbers is equal to the number directly below and between them. The array is vertically symmetrical. The numbers in horizontal lines are called rows, which begin with the 0th row. Diagonals consist of numbers in an oblique line. They are also called columns, and begin with the 0th diagonal. Each number is referred to as an element, and the number of elements in a row is always one more than the number of the row. For example, there are four elements in the third row.
At the start of each of the lessons, a book will be shared with the students to motivate them. The primary purpose of reading to the students is to integrate literature with mathematics. Each book has a connection to patterns, and this will be emphasized by the teacher. For example, Sea Squares introduces the concept of square numbers. In the second lesson, I recommend reading this book to the class because it has a direct connection to the focus of the lesson.
Writing in mathematics is valid because it allows the teacher to authentically
assess a student's understanding or misunderstanding of a topic. The teacher
will be able to observe the manner in which the student processes information.
The student will be compelled to think about what s/he has actually learned.
I encourage individual journal writing, however it is possible for students
to respond in groups or with a partner. Specific prompts will be given,
and the students will be encouraged to respond with writing and pictures.
For example, a prompt included in the first lesson asks the student to
describe the pattern that he or she created.
Blaise Pascal was born in 1623 in Clermont-Ferrand, France. His father
educated Blaise and decided that he should not learn mathematics until
the age of 15. Blaise decided to work on geometry himself at the age of
12, and he discovered that the sum of the angles of a triangle is two right
angles. When his father found out that Blaise had a ãgiftä for mathematics,
he allowed Blaise to borrow a copy of Euclid.
Blaise designed a calculator and conducted many experiments on atmospheric pressure. He is credited with inventing the one-wheeled wheelbarrow, he designed a public transportation system for the city of Paris, and he was a large contributor to creating the probability theory. Among his many projects was a triangle of numbers. This triangle of natural numbers was originally discovered by Chinese algebraist, Chu Shu-Kie, in 1303. However, Blaise discovered so many patterns and properties of the triangle that the array became commonly known as "Pascal's Triangle."
Pascal, besides making a large impact on mathematics, put together an introspective collection of personal meditations entitled Pensees ("Thoughts"). One particularly amusing quotation from Pensees attributes all the troubles of man being caused by his inability to sit still.
Pascal never married because of his decision to dedicate his life to physics, philosophy, religion, and mathematics. At one point he chose to forsake his work in science and mathematics and devote himself entirely to religion. It is believed that this decision came about one day while he was out driving his carriage. For some reason his horses became spooked and they bolted over a bridge. Pascal's carriage dangled on a precipice as his horses plunged to their deaths. Pascal interpreted this as a sign to give himself fully to religious studies.
After suffering from many health problems throughout his life, he died in 1662 from a malignant stomach ulcer at the age of 39. One can only imagine what Pascal would have discovered if he had lived longer. He was truly a genius and ahead of his time in many ways.
Pascal was not the only mathematical genius. There are many others that contribute(d) greatly to the advancement of mathematics. Listed in the bibliography is a set of excellent books about historical mathematicians. Students will benefit greatly if encouraged to research and explore these great men and women in history.
Expected Grade Competencies
To present a concentrated effort to teach patterns, we should know what the students are expected to learn at each grade level. The National Council of Teachers of Mathematics recommends that students in grades three through five become proficient at decoding and creating various types of patterns. Patterns, functions, and Algebra, the second NCTM standard, suggests three objectives that should be used as a guideline in planning instructional programs. The first objective states that all students should understand various types of patterns and functional relationships. The second objective states that all students must use symbolic forms to represent and analyze mathematical situations and structures. The final objective states that all students should use mathematical models and analyze change in both real and abstract contexts. This unit will focus on the first objective.
Third-Fifth Grade Competencies
Understand various types of patterns and functional relationships
In grades 3-5, all students should
? identify, describe, and extend geometric and numeric patterns including growing and shrinking patterns;
? identify, express, and verify generalizations and use them to make predictions (e.g., doubling a number then doubling again is the same as multiplying by four).
"Algebra" is not a term that has been commonly heard in upper elementary
classrooms, but students in these grades often rely on algebraic reasoning
to solve problems. These principles of algebraic thinking provide contexts
for advancing understanding in mathematics, and especially aid in the comprehension
of algebra in middle and secondary school classrooms. It is important that
students develop a strong base in algebraic thinking if they are to be
successful in intermediate school.
Algebraic ideas emerge in grades three through five as students study numeric and geometric patterns. For clarification, numeric, or arithmetic, patterns are sequences of numbers in which the difference of two consecutive numbers is the same. The difference of the two consecutive numbers is called the common difference. The following is an arithmetic progression: 3, 6, 9, 12, 15, . . . The common difference is three. In general, a, a+d, a+2d, a+3d, . . . is an arithmetic progression, where a is the first term, and d is the common difference.
A polygonal number sequence is a progression of numbers in which each term is based on the number of vertices in a given shape. For example, the pentagonal number sequence begins with 1 (as do all polygonal sequences). The second term is 5 because there are five vertices on a pentagon. The following terms are found by extending two of the sides of the preceding polygonal numbers, creating larger polygons by placing new vertices, points, and sides where necessary. Then, by adding the vertices and points in the new figure, the third polygonal term is found. This growing pattern may also be formed by adding consecutive numbers to each successive term; 1, 5, 12, 22, 35, . . . The pattern is +4, +7, +10, +13, . . . There is a connection to the arithmetic sequences in that there is a common difference. (See above.)
A geometric sequence is a progression of numbers in which each succeeding term is obtained by multiplying the preceding term by the same number. The following is a geometric progression: 1, 2, 4, 8, 16, 32, . . . Two is the number that is used to multiply each term with, and it is known as the common ratio.
As students investigate and build patterns, they gain confidence and ease in working with number sequences. Working with various patterns aids students in the process of seeing patterns in all areas of mathematics. Students should be encouraged to explain patterns verbally and to predict what will follow in given sequences. This will allow students to identify properties of numbers.
Lesson Plan #1
Objectives: to explore patterns using a variety of manipulatives and actions; to describe non-numeric growing patterns; to create non-numeric growing patterns; to extend non-numeric growing patterns.
Strategies: Informally pre-assess students on their knowledge of patterns by teacher observation. Motivate by integrating literature. Modify amount of time spent with manipulatives according to students' ability to describe, create, and extend patterns using manipulatives. Assess students by having them respond to journal prompts.
Classroom activities: Create patterns by arranging students according to their attributes: glasses, no glasses, glasses, no glasses; short sleeves, long sleeves, short sleeves, long sleeves. Allow students to try a few sequences. Provide students with pattern blocks or other manipulative(s), allowing ample time for exploration. Model a variety of patterns to copy and complete using their manipulatives. Encourage them to verbalize what the patterns are: orange square, blue rhombus, orange square, blue rhombus, . . . Instruct them to create and verbally describe patterns to a fellow student. Share a portion of students' patterns with the class. Discuss similarities and differences of patterns. Provide sequential visual representations of shapes or animals and have students verbally describe and then extend the patterns on paper by cutting and pasting. Have students create their own sequences of shapes or animals. Allow time for students to respond to the prompts in their journals.
Journal prompts: What is a pattern? How did you guess what shape came next in the patterns your teacher made? Describe the pattern you made out of animal shapes. Are you good at solving patterns? Why or why not?
Resources: The Skip Count Song
Materials: pattern blocks, attribute blocks, coins, pencils, crayons,
pictures of shapes and animals to cut and paste, construction paper and
scissors, math journals.
Lesson Plan #2
Objectives: to identify growing numeric patterns; to describe growing numeric patterns; to extend growing numeric patterns.
Strategies: Pre-assess the students on their knowledge of patterns and modify lesson accordingly. Motivate by integrating literature. Post-assess with journal responses and completion of a pattern worksheet.
Classroom activities: Read aloud Sea Squares after telling students that they should look for a numeric pattern throughout the story. Discuss how multiplication is used in the story, and allow students to experiment with square tiles to create the square numbers. They will see a pattern, but they may not know how to verbalize it. Provide graph paper and scissors for the students to cut out squares, beginning with a 1-by-1 square, then a 2-by-2 square, and so forth. They can experiment with different sizes and patterns. Additionally, have them draw the square numbers on dot paper to re-emphasize the square pattern that occurs. Have them display their patterns on construction paper that should include a written explanation of their observations. Introduce the definition of numeric sequences (see above). Provide a worksheet with a variety of numeric sequences for the students to extend. Once they have successfully extended a few numeric patterns, have them explain in their math journals. Challenge the students to find a similar pattern in Pascal's Triangle. It is possible that your students will successfully locate these difficult patterns, but do not be discouraged if they have trouble. The triangular numbers are found in the diagonal that starts in the third row. Square numbers are found by finding the sum of any two consecutive numbers in the same diagonal as the triangular numbers.
Journal prompts: What is a pattern? What is an example of a pattern found in the book Sea Squares? How do you know what number comes next in a pattern? Describe a pattern in Pascal's Triangle that your class did not talk about.
Resources: Sea Squares
Materials: square tiles, graph paper, dot paper, scissors, glue, construction
paper, numeric patterns worksheet, journal prompts, math journals, and
Lesson Plan #3
Objectives: to identify growing geometric patterns; to describe growing geometric patterns; to extend growing geometric patterns.
Strategies: Motivate by integrating literature. Build on knowledge of numeric patterns from the previous lesson. Introduce the constant feature on the calculator. Post-assess with journal responses and completion of a pattern worksheet.
Classroom activities: Read aloud The King's Chessboard after telling students that they should look for a pattern throughout the story. Discuss how multiplication is used in the story, and allow students to experiment with calculators to extend the pattern in the story. Briefly, the story involves a king that wanted to show his appreciation to a servant that had invented the game of chess. The servant proposed that the king place grains of wheat on the chessboard and double the amount for every square. There are 64 squares, the request seemed simple enough, and so the king agreed. There are quite a few variations of this story available. The students will discover that the pattern increases by multiplying each term by two. After sufficient exploration time, introduce the constant feature on the calculator by instructing students to enter the following: C 2 X 1 = = = =. . . This will produce the powers of two, which is of course the pattern in The King's Chessboard. Encourage students to try other numbers by substituting the 2 with any other number. The number of equal signs is the same as the power of the number, also called the exponent. Draw the students' attention to Pascal's Triangle and challenge them to find the pattern. This is very difficult to locate as it involves adding the numbers going across each horizontal row. However, some students may see it if given enough time to explore. Giving hints is a good idea. The first term is 1. The second term is found by adding 1 + 1 to get 2, the third term is 4 (1 + 2 + 1), and so on.
Journal prompts: What is a pattern? Describe the pattern found in the book The King's Chessboard. How do you know what number comes next in the pattern found in the story? How could Pascal's Triangle have helped the king? How do you use the constant feature on a calculator?
Resources: The King's Chessboard
Materials: calculators, paper and pencil, numeric patterns worksheet,
journal prompts, math journals, and Pascal's Triangle.
Pascal's Triangle can be used to find number combinations. If there
are three flavors of ice cream to choose from and you want to order a double
dip cone, Pascal's Triangle can help you figure out the different combinations
Look at row three of the triangle (representing the number of flavors) in place two (representing the number of scoops of ice cream). The answer is there; there are three possible combinations. What if you had five flavors of ice cream to choose from for your double-dip cone? Find place two (two scoops) in row five (five flavors) and you will see that there are ten possible combinations.
Hockey Stick Pattern
Start at any of the 1's along the perimeter of the triangle and follow the diagonal toward the center of the triangle. Stop at any number inside the triangle. The sum of the numbers inside the selected diagonal is the same as the number just below the end of the diagonal. For example, add 1, 3, 6, 10 and 15 together and you get the sum of 35, which is just below this series of diagonal numbers.
Section off a consecutive series of mini triangles and find the sums
of the elements inside each triangle. Start off using one row; the sum
is 1. Using two rows, the sum will be 3; with three rows the sum is 7;
with four rows, the sum is 15. The formula for this involves exponents,
which is a good extension for Lesson #3. For the nth number of rows in
the triangle, 2 to the nth power minus 1 will be equal to the sum of the
elements inside the triangle. For row eight, 2 to the eighth power subtract
one will give you 255, which is the sum of the elements in the triangle.
Anno, Mitsumasa. Anno's Magic Seeds. Philomel Books, 1995.
Birch, David. The King's Chessboard. Puffin Pied Piper Books, 1988.
Hulme, Joy N. Sea Squares. NY: Hyperion Books, Walt Disney Book Publishing
Williams, Rozanne Lanczak. The Skip Count Song. Creative Teaching
AIMS. Historical Connections in Mathematics (volumes I, II, and III).
Education Foundation, 1992-1995.
Bresser, Rusty. Math and Literature (grades 4-6). Math Solutions Publications, 1995.
Corwin, Rebecca. Talking Mathematics. Heinemann Publishers, 1996.
Creative Publications. Smart Strands. Creative Publications, 1999.
Kaye, Peggy. Afterwards (grades 3-4). Cuisenaire, Inc., 1997.
Newman, Vicki. Math Journals. Teaching Resource Center, 1994.
Pagni, David. Camp-LA Book 3. Cal State Fullerton Press, 1991.
Seymour, Dale. Critical Thinking Activities (grades 4-6). Dale
1988. (also available from this company is a poster of Pascal's Triangle)
Number Patterns Worksheet
Complete the following geometric patterns:
Write the common ratios.
1, 10, 100, --______, ________, _______, . . . _______
5, 10, 20, _____, _____, _____, _____, . . . _______
1, 7, 49, _____, _____, _____, _____, . . . _______
Complete the following numeric patterns:
Write the common differences.
37, 41, 45, _____, 53, _____, 61, _____, 69, . . . . _______
1976, 1980, 1984, _____, _____, _____, 2000, . . . _______
7, 16, _____, 34, 43, _____, _____, _____, 79, . . . _______
11, _____, 33, _____, 55, _____, _____, _____, 88, . . . _______
12, 23, 34, _____, _____, _____, _____, 89, _____, . . . _______
Complete the following polygonal patterns:
Write the shape names.
1, 4, 9, 16, _____, _____, _____, _____, . . . ________________
1, 8, 27, _____, _____, _____, . . . ________________
1, 3, 6, 10, 15, _____, _____, _____, . . . _______________
Challenge: Finish the patterns in a different way.