Primary Type: Lesson Plan
Status: Published
This is a resource from CPALMS (www.cpalms.org) where all educators go for bright ideas!
Resource ID#: 47838
Binomial Expansion with Pascal's Triangle
Pascal's Triangle will help students multiply (expand) binomials without having to use the FOIL method or the Distributive Property.
Subject(s): Mathematics
Grade Level(s): 9, 10, 11, 12
Intended Audience: Educators
Instructional Time: 1 Hour(s)
Resource supports reading in content area: Yes
Freely Available: Yes
Keywords: Pascal's Triangle, Binomial Expansion
Resource Collection: CPALMS Lesson Plan Development Initiative
LESSON CONTENT
Lesson Plan Template: General Lesson Plan
Learning Objectives: What should students know and be able to do as a result of this lesson?
Students will be able to use Pascal's Triangle to expand binomial relationships with 100% accuracy.
Prior Knowledge: What prior knowledge should students have for this lesson?
Students must know the definition of coefficient and term. A coefficient is the numerical factor in a term. A term is a number, a variable, or the product of a number
and one or more variables.
Students must know how to multiply two or three binomials together to see the relationship between the coefficients of their answer and the coefficients in Pascal's
Triangle.
Example #1
=
.
The coefficients of the answer are 1, 2, 1 which are the coefficients of row 2 of Pascal's Triangle.
Example #2:
=
.
The coefficients are 1, 3, 3, 1, which are the coefficients of row 3 of Pascal's Triangle. The row will match up with the exponent.
Guiding Questions: What are the guiding questions for this lesson?
1. What does it mean to expand a binomial? (To multiply the binomial factors.)
2. What is the connection between Pascal's Triangle and Binomial Expansion? (The n+1 row of Pascal's triangle gives us the coefficients of the expanded form of the
binomial.)
Teaching Phase: How will the teacher present the concept or skill to students?
page 1 of 4 1. What is Pascal's Triangle? Pascal's Triangle is a triangular array of numbers formed by first lining the border with 1's and then placing the sum of the two
adjacent numbers within a row between and underneath the two original numbers.
FilledInPascalsTriangle.docx
2. When was Pascal's Triangle developed?
The earliest know version of Pascal's Triangle was developed between 300 and 200 B.C. by the Indian mathematician Halayudha. Although other cultures were
aware of the triangle, it has been named for Blaise Pascal (1623–1662), a French mathematician. (Prentice Hall ­ Algebra 2 textbook)
Around 1654, Blaise Pascal began to investigate the triangle. The two major areas where Pascal's triangle is used today are in algebra and probability. Pascal is
credited because he took the information and organized it to where it made sense and was useful. (Leavitt, "The Simple Complexity of Pascal"s Triangle")
3. How is Pascal's Triangle used? Each row of Pascal's Triangle contains coefficients for the expansion of in the row that begins 1, 6, 15 ….
4. How can you find patterns within Pascal's Triangle? Look for multiples of 2, 3, 4, 5 etc. in the triangle to learn about the different types of numbers such as
counting, triangular, hexagonal and tetrahedral.
Students will complete fill in BlankPascalsTriangle.docx. Teacher can fill in rows 0 to 4 with students and then students can continue to the pattern to fill in the rest of
the triangle.
Use Pascal's Triangle to expand
. First you would use the row that has a 6 as its second number. The exponents for
begin with 6 and decrease:
. The exponents for b begins with 0 and increases.
Use Pascal's Triangle to expand
=
. First you would write the pattern for raising a binomial to the third power. These numbers are 1,3,3,1. Since
, substitute x for a and -2 for b.
. Note the pattern
of the signs. If the sign is negative inside the binomial then the answer will be in the pattern of -, +, -, + ....
Guided Practice: What activities or exercises will the students complete with teacher guidance?
Suggest putting students in groups of two for the following two Guided Practice activities:
Activity 1: Use Pascal’s Triangle to answer the following questions:
1. If the row number is odd, how many times does the largest number of the row appear in the row?
2. If the row number is even, how many times does the largest number of the row appear in the row?
3. Do even rows have an even or odd number of terms?
4. Do odd rows have an even or odd number of terms?
5. The number in rows 0 and 1 are odd. List the next three rows which have only odd numbers.
6. Which rows have only even numbers except for the outer 1's?
7. Except for the outer 1's, does it appear that row 16 will also have only even numbers? Why? Predict the next row which will have all even numbers except for the
out 1's. Explain.
8. If the row number is prime, what is true about all the numbers in the row?
9. Describe any symmetry in the triangle.
Activity 2: Binomial expansion problems (May use GeoGebra to assist in these problems)
1. The teacher will pass out two index cards to each student in the class.
2. Students will write their name on both cards.
3. On one index card the student will make up their own binomial expansion problem.
4. On the other index card, the student will write the answer to their problem.
5. The teacher will collect all index cards and keep the two piles separate.
6. Each student will receive two index cards.
7. One index card will contain a problem and the other card will contain an answer.
8. Students will work on the problem they received.
9. When all students are done, they will interact with each other by walking around the classroom to find the person who has the answer to their new problem.
Rules:
Students cannot match up by name. Students must complete work before they can look for answer.
Independent Practice: What activities or exercises will students complete to reinforce the concepts and skills developed in the
lesson?
Using paper and pencil and then Geogebra - www.geogebratube.org/student/m26261, students can work on the following three problems:
Directions: Use Pascal's Triangle to expand each binomial
Closure: How will the teacher assist students in organizing the knowledge gained in the lesson?
Students can answer the following questions at the end of the class:
1. After you expand an expression, what do you notice about the sum of the exponents?
2.Expand each expression. Write the terms of each expanded expression so that the powers of "a" decrease.
page 2 of 4 3. Describe the relationship between the coefficients in parts (a), (b), and (c) and the rows of Pascal's Triangle.
4. Describe any patterns in the exponents of "a" and the exponents of "b"
Summative Assessment
Students will take a test on Binomial Expansion using Pascal's Triangle. Teacher will grade the test and provide feedback to each student. If students need more
practice then additional homework problems can be given.
summative assessment - binomial expansion
summative assessment - answer key
Formative Assessment
This formative assessment is to review the FOIL method and Distributive property method when multiplying binomials. Students should be able to master this skill to
then compare it to the binomial expansion method using Pascal's Triangle.
Formative Assessment
Formative Assessment - pascal - answer key.docx
Feedback to Students
Teacher will circulate around all areas of the classroom to help each group that needs assistance. Teacher will guide students to reach the correct answer. Teacher
will use the formative assessment to make sure students have the necessary skills to move forward.
ACCOMMODATIONS & RECOMMENDATIONS
Accommodations:
Teacher can give students who need extra help filling in the numbers in Pascal's Triangle a buddy to work with. Students can also use their filled-in Pascal Triangle as
a tool to use when taking summative assessment.
Extensions:
Students can also expand binomials using the binomial theorem.
The Binomial Theorem states:
Here is an example of the binomial theorem:
(3x – 3)10 = 10C0 (3x)10–0 (–3)0 + 10C1 (3x)10–1 (–3)1 + 10C2 (3x)10–2 (–3)2
+ 10C3 (3x)10–3 (–3)3 + 10C4 (3x)10–4 (–3)4 + 10C5 (3x)10–5 (–3)5
+ 10C6 (3x)10–6 (–3)6 + 10C7 (3x)10–7 (–3)7 + 10C8 (3x)10–8 (–3)8
+ 10C9 (3x)10–9 (–3)9 + 10C1 0 (3x)10–1 0(–3)1 0
Special Materials Needed:
Paper
Pencil
Calculator
Blank Pascal Triangle Worksheet
Computer
Geogebra Software
Further Recommendations:
Teacher can explain to students that this lesson about Binomial Expansion using Pascal's Triangle is an introductory lesson. Students should be encouraged to do
research about Blaise Pascal and go back even further to possibly explore hieroglyphics and find the connection between that and Pascal's Triangle.
page 3 of 4 Additional Information/Instructions
By Author/Submitter
Use of the following GeoGebraTube resource is acknowledged: "Binomial Expansion", by Goldenj, mathguru, accessed from http://www.geogebratube.org/student/m26261,
used under a Creative Commons Attribution-Share Alike license.
Lesson may align with the following standards of math practice:
MAFS.K12.MP.4.1 - Model with mathematics.
MAFS.K12.MP.5.1 - Use appropriate tools strategically.
SOURCE AND ACCESS INFORMATION
Contributed by: Wendy Moskowitz
Name of Author/Source: Wendy Moskowitz
District/Organization of Contributor(s): Broward
Is this Resource freely Available? Yes
Access Privileges: Public
License: CPALMS License - no distribution - non commercial
Related Standards
Name
MAFS.912.A-APR.3.5:
Description
Know and apply the Binomial Theorem for the expansion of (x
in powers of x and y for a positive integer n, where
x and y are any numbers, with coefficients determined for example by Pascal’s Triangle.
page 4 of 4