Lesson Title: The Remainder Theorem
Date: _____________ Teacher(s): ____________________
Course: Common Core Algebra II, 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?
Understand the relationship between zeros and factors of polynomials.
A.APR.B.2 Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the
remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x).
Interpret the structure of expressions.
A.SSE.A.1 Interpret expressions that represent a quantity in terms of its context.*
a. Interpret parts of an expression, such as terms, factors, and coefficients.
b. Interpret complicated expressions by viewing one or more of their parts as a single entity.
Interpret the structure of expressions.
A.SSE.A.2 Use the structure of an expression to identify ways to rewrite it.
MP3:
MP5:
MP7:
MP8:
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, Unit 4
Lesson Launch Notes: Exactly how will you use the
first five minutes of the lesson?
Since 3 and 4 are a pair of factors of the number 12,
write a true mathematical statement using 3, 4, and 12.
Ask students to share in table groups. Ask how many
different statements were found in their groups.
Have randomly selected students display their
statements on the document camera
(Be certain that (3)(4)=12 and several forms of
12
12
4
3
3
4
are present.)
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.
Suppose we know that a function can be divided by these
three factors with a remainder of zero:
x + 4, x – 5, x – 5 and when it is divided by x + 1, the
remainder is 5.
In your groups draw a very rough sketch of the graph of
this function. Tell the class answers will be shared and
defended tomorrow.
Ask what happens if we divide by a number that is not
a perfect factor, for example 5. Allow for discussion of
the concept of a remainder. Ask what the remainder is
in the 12, 4, 3 division, where 3 or 4 can be a perfect
factor. (Look for evidence of 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.
Instructional Note: Students will need to access the internet for this part of the lesson.
1. Have students factor the following quadratic function: f(x)= x2 + 7x – 18. Display these questions about the
function:
 What are the zeroes of the function?
HCPSS Secondary Mathematics Office (v2.1); adapted from: Leinwand, S. (2009). Accessible mathematics: 10 instructional shifts that raise student
achievement. Portsmouth, NH: Heinemann.
Lesson Title: The Remainder Theorem
Course: Common Core Algebra II, Unit 4
Date: _____________ Teacher(s): ____________________
Start/end times: _________________________
 How did you find them?
 Why do we call them zeroes?
Allow students to find and compare answers in their table groups. Randomly select students to share their
answers. (Encourage answers to the third question that are both graphical involving x-intercepts, and analytical
concerning making the function value zero.) (Look for evidence of MP2,MP7)
2. As in the lesson launch, ask students to write a true mathematical statement involving x2 + 7x – 18 and the two
factors that were found. (Look for evidence of MP7)
3. Assign students to small groups. Have groups visit one of the following websites to explore how to divide
polynomials:
http://www.webmath.com/polydiv.html
http://www.deadhat.com/poly/index.php
4. Have students divide x2 + 7x – 18 by one of its factors. Have groups note the remainder. Draw a parallel to
the lesson launch. Ask groups to formulate a statement about the connection between a factor, the remainder,
the graph, and the value of the function. Have groups share and allow for critique and discussion. (Highlight
statements that draw the connection between a zero remainder and an value of zero.) (Look for evidence of MP2
and MP7.)
5. Have groups use the polynomial divider to divide x2 + 7x – 18 by another linear binomial of their choice. Ask
groups to write what differences there are in this answer. Now have groups graph f(x)= x2 + 7x – 18 on a
graphing calculator. Have groups, using the differences noted in their writing, write a similar statement about the
non-factor, the remainder, the graph, and the value of the function. Have groups share and allow for critique and
discussion. (Highlight statements that draw the connection between the remainder when divided by x-a and an
value of f(a).) (Look for evidence of MP2 and MP7.)
6. Have the class caucus, with a recorder and moderator, and write a version of the remainder theorem (as stated in
the objective). (It may first be necessary to repeat the process above with another easily factorable quadratic.)
(Look for evidence of MP8.)
7. Using this polynomial function, f(x) = x3 – 13x+12, have groups answer these questions.
 Can you name an x value that is a zero of this function?
 Where is on x intercept of this function?
 What is one factor of this function?
(There are three possible answers to each as there are zeroes at -4, 1, 3. All three answers should focus on one of
these three zeroes, intercepts, factors). Have randomly selected groups give their answers.
8. Ask, ”What is the remainder when x3 – 13x+12 is divided by x-2?” (Answer is -6). Have groups report how they
found their answers. (Highlight the connection between the divided remainder.)
9. Ask groups to find a factor of f(x) = x4 – 2x3-x2+4. (Answer is x – 2 since x = 2 can be found graphically as a
zero ). Circulate around the room to monitor method and understanding. Then ask groups what the remainder
is when x4 – 2x3 – x2+4 is divided by x+2. (Answer is 32). Circulate around the room to monitor method and
understanding. (Look for evidence of MP7 and MP8.)
10. If time permits, have students practice with more polynomial examples.
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 should be able to find the value of a polynomial function f(a) by knowing the remainder when the
polynomial is divided by x-a. Students should be able to find the remainder when the polynomial is divided by x-a
by knowing the value of a polynomial function f(a). Students should be able to identify a zero, a, of a function by
knowing that dividing the polynomial by x-a leaves no remainder.
The lesson closure and the end of lesson examples will provide a sense of the students ability to connect the concept
of the remainder and the value of a function.
HCPSS Secondary Mathematics Office (v2.1); adapted from: Leinwand, S. (2009). Accessible mathematics: 10 instructional shifts that raise student
achievement. Portsmouth, NH: Heinemann.
Lesson Title: The Remainder Theorem
Course: Common Core Algebra II, Unit 4
Date: _____________ Teacher(s): ____________________
Start/end times: _________________________
Notes and Nuances: Vocabulary, connections, anticipated misconceptions (and how they will be addressed), etc.
This lesson does not teach students how to perform long polynomial division or synthetic division. It uses
technology to demonstrate this type of division. An advanced or curious class may create an appropriate setting to
teach or demonstrate this skill. One of the websites suggested shows some of the process.
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?
Computer access and web access
Most textbooks contain numerous problem sets relevant to
this concept. It may be necessary to access a precalculus
book for appropriate work. Access will be needed to the
web for performing divisions.
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?
Did students make sense of the transitions from numerical examples (lesson launch) to quadratic examples to higher
order polynomial examples?
Were all members of the groups participating?
Did the students develop a statement of the remainder theorem of which they had ownership?
Were students able to complete the examples and closure activity using the skill developed in the lesson?
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.