Lesson Binomial Theorem
Date: _____________ Teacher(s): ____________________
Course: Common Core Algebra II GT, Unit 4
Start/end times: _________________________
Lesson Standards/Objective(s): What mathematical skill(s) and understanding(s) will be developed? Which
Mathematical Practices do you expect students to engage in during the lesson?
Use polynomial identities to solve problems.
A.APR.C.5 (+) Know and apply the Binomial Theorem for the expansion of (x + y)n 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.
Interpret the structure of expressions.
A.SSE.A.2 Use the structure of an expression to identify ways to rewrite it.
MP2:
MP3:
MP5:
MP7:
MP8:
Reason abstractly and quantitatively.
Construct viable arguments and critique the reasoning of others.
Use appropriate tools strategically.
Look for and make use of structure.
Look for and express regularity in repeated reasoning.
Common Core Algebra II GT, Unit 4
Lesson Launch Notes: Exactly how will you use the first
ten minutes of the lesson?
The triangular array displayed below is called Pascal’s
Triangle. What patterns do you see in the numbers in
this array? Have students discuss in pairs.
Lesson Closure Notes: Exactly what summary activity,
questions, and discussion will close the lesson and
connect big ideas? List the questions. Provide a
foreshadowing of tomorrow.
Write a short paragraph describing how you would use
Pascal’s Triangle and the Binomial Theorem to justify
these simple statements:
(x + y)1 = x + y
(x + y)0 = 1
Randomly select pairs to give one pattern. Many patterns
exist and should be explored. It is critical to highlight the
most basic one that, with ones (1) given on the edges,
every other entry is the sum of the two numbers
diagonally above it.
(Excellent pattern reference: http://ptri1.tripod.com/)
(Look for evidence of MP2 and MP7.)
Lesson Tasks, Problems, and Activities (attach resource sheets): What specific activities, investigations,
problems, questions, or tasks will students be working on during the lesson? Be sure to indicate strategic
connections to appropriate mathematical practices.
HCPSS Secondary Mathematics Office (v2.1); adapted from: Leinwand, S. (2009). Accessible mathematics: 10 instructional shifts that raise student
achievement. Portsmouth, NH: Heinemann.
Lesson Binomial Theorem
Course: Common Core Algebra II GT, Unit 4
Date: _____________ Teacher(s): ____________________
Start/end times: _________________________
1. Assign students to groups of 4. Refer to Activity Sheet 1. Give each group diagram one. Have groups find the
expanded version of this expression (x+y)2. Tell groups to formulate a justification based both on symbolic
manipulation and graphic manipulation based on diagram one. (For the diagram the area of the square is (x+y)2
which, if you take each of the 4 sections separately and sum them, is x2 + xy+ xy + y2 = x2 + 2xy + y2 ). Give
each group diagrams 2 and 3. Have groups find the expanded version of this expression (x+y)3. Tell groups to
formulate a justification based both on symbolic manipulation and graphic manipulation based on diagram two.
(For the diagram the volume of the cube is (x+y)3 which, if you take each of the 8 sections and separately and
sum them, is x2 + x2y+ x2y + x2y +xy2 + xy2+ xy2 +y3 = x3 + 3x2y +3xy2 + y3). Have randomly selected groups
share their methods and observations. Allow for discussion and critique. (Look for evidence of MP2 and MP5.)
2. Highlight any group responses that suggest a relationship between the coefficients of the simplified versions of
the binomial expansions and Pascal’s Triangle. If no groups look at the significance of the lesson launch, then
instruct groups to look back at the lesson launch and compare Pascal’s Triangle to their answers.
3. Ask groups to predict the coefficients of (x+y)4 and of (x+y)5. Ask groups to write notes describing how they
would justify their answers symbolically and using a diagram. ( It is not likely groups will be able to come up
with a diagram that demonstrates this as visual familiarity with dimensions beyond the third dimension is quite
limited! Asking students to try will highlight this.) (It is also a bit tedious to have students multiply these out to
demonstrate. You should have them only describe their method, not necessarily carry it out) Have randomly
selected groups share their methods and observations. Allow for discussion and critique.
4. Ask groups to predict the variable combinations that would accompany the coefficients. Have randomly selected
groups share their methods and observations. Allow for discussion and critique. Through large group discourse,
have the class caucus, with a recorder and moderator, and write a version of the binomial theorem. Ask groups
to use their new theorem to expand (w+z)9. Circulate, check for accuracy, and use as formative assessment. It
may be necessary to ask groups to try a few additional examples. (Look for evidence of MP2, MP7, and MP8.)
5. Ask groups to extend the use of their binomial expansion theorem to expand (x+2y)3. Ask them to write their
answers down and note justifications. Have a randomly selected group share their answer and justification.
Allow for discussion, critique, and revision until all are convinced the answer is correct.
6. Ask groups to consider what would change about the answer if the 2y was subtracted in this way (x - y)3. Ask
them to write their answers down and note justifications. Have a randomly selected group share their answer
and justification. Allow for discussion, critique, and adjustment until all are convinced the answer is correct.
(Look for evidence of MP2 and MP7.)
7. Refer to Activity Sheet 2. Give each group a different binomial to expand. Each expansion is written twice.
Have them write the answer two times and cut the paper to separate them. Have all groups pass one of the two,
in a rotation set up, to the next group. Each group should check and, if they think it necessary, correct the
answer. Repeat this process until all groups have evaluated all the problems and the groups have their original
problem back. Ask all groups to look at the answer and compare it to their original. Ask groups to evaluate if
either of the answers the now correct. Bring groups with discrepancies to the board or a document camera to
argue their case. Allow for discussion and critique. Have the class caucus to a correct answer. (Look for
evidence of MP2 and MP7.)
8. Extention: Classes or students who show curiosity about why the coefficients are correct can be shown the
notation nCr, with n = row number and r = entry number. The first row is row 0 and the one(1) as the first entry
in each row is the 0th entry. (This counts the number of ways to choose x and y. For example, in expanding
(x+y)5 and finding the coefficient of x2y3 there are 5C2 = 10 or 5C3 = 10 ways to choose from the 5 factors which
will contribute the y and which two are left to contribute the x.) This computation can be done on many
calculators. This topic can be addressed in class or offered as investigation project work. (Look for evidence of
MP5 and MP8.)
Evidence of Success: What exactly do I expect students to be able to do by the end of the lesson, and how will I
measure student success? That is, deliberate consideration of what performances will convince you (and any outside
observer) that your students have developed a deepened and conceptual understanding.
Students will be able to expand a binomial raised to a positive integer power.
The concept attainment stage of the lesson as well as the group work and end discussion in Activity Two will
provide ample opportunity to assess understanding.
HCPSS Secondary Mathematics Office (v2.1); adapted from: Leinwand, S. (2009). Accessible mathematics: 10 instructional shifts that raise student
achievement. Portsmouth, NH: Heinemann.
Lesson Binomial Theorem
Course: Common Core Algebra II GT, Unit 4
Date: _____________ Teacher(s): ____________________
Start/end times: _________________________
Performance on homework will provide assessment of understanding as well.
Notes and Nuances: Vocabulary, connections, anticipated misconceptions (and how they will be addressed), etc.
Pascal’s Triangle
Students often have trouble when one or more of the terms of the binomial are negative or subtracted. The
subtraction, in particular, does not lend itself as well to using the negative of the term when raising it to a power.
This may need some special attention in the part of the lesson where this is introduced.
If addressing the use of counting to find entries on Pascal’s Triangle, it is important to remember that the students
have limited exposure to the nCr, notation and its uses.
Resources: What materials or resources are essential
for students to successfully complete the lesson tasks or
activities?
Homework: Exactly what follow-up homework tasks,
problems, and/or exercises will be assigned upon the
completion of the lesson?
Activity Sheet 1
Activity Sheet 2
Image of Pascal’s Triangle (can be copied from the
lesson launch)
Graphing Calculators (for the extension)
Most textbooks contain numerous problem sets relevant
to this concept. It may be necessary to access a
precalculus book for appropriate work.
Lesson Reflections: How do you know that you were effective? What questions, connected to the lesson
standards/objectives and evidence of success, will you use to reflect on the effectiveness of this lesson?
Were all students engaged in the group process?
Were students willing to critique work and have their work critiqued?
Did the class arrive at the Binomial Theorem and the answers to subsequent problems with minimal guidance from
the instructor(s)?
Did student show curiosity about Pascal’s Triangle and how it works?
Howard County Public Schools Office of Secondary Mathematics Curricular Projects has licensed this
product under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.
HCPSS Secondary Mathematics Office (v2.1); adapted from: Leinwand, S. (2009). Accessible mathematics: 10 instructional shifts that raise student
achievement. Portsmouth, NH: Heinemann.