Understanding by Design
Small Learning Community High Schools, NYC
Algebra Curriculum Unit Plan
Subject Area: Mathematics
Unit Title: Polynomials and Factoring
Course/Grade Level: Integrated Algebra
Number of Days: 16
Designers: Kate Coleridge, Murray Bergtraum
T. Diamont, Queens Vocational and Technical HS
Jessica Ferrara, Queens Vocational and Technical HS
Jim Lavan, Abraham Lincoln HS
Unit Summary: Students will identify and factor polynomial expressions and equations. Students will
be shown four common factoring methods: GCF, difference of 2 squares, trinomials when the coefficient of x2
term is 1, and trinomials when the coefficient of x2 is greater than 1. Key skills will include adding,
subtracting, multiplying and factoring polynomials. Sentence about performance assessment needed
DESIRED RESULTS (STAGE 1)
State Standards and or/ grade level performance indicators addressed:
Standard 1
Analysis, Inquiry, and Design Students will use mathematical analysis, scientific inquiry, and
engineering design, as appropriate, to pose questions, seek answers and develop solutions
Standard 3
Mathematics Students will understand mathematics and become mathematically confident by
communicating and reasoning mathematically, by applying mathematics in real world settings and by solving problems
through the integrated study of number systems, geometry, algebra, data analysis, probability and trigonometry.
Standard 5
Technology Students will apply technological knowledge and skills to design, construct, use and evaluate
products and systems to satisfy human and environmental needs
Standard 6
Interconnectedness – Common Themes Students will understand the relationships and common themes
to connect mathematics, science and technology and apply the themes to these and in other areas of learning
* AA8 – Analyze and solve verbal problems that involve quadratic equations
*AA12 – Multiply and Divide monomial expressions with a common base, using properties of exponents
*AA19 – Identify and factor the difference of two perfect squares
*AA20 – Factor algebraic expressions completely including trinomials with leading coefficients of 1 (after factoring out a
GCF)
*AA28 – Understand the difference and connection between roots of a quadratic equation and factors of quadratic
expressions
Understanding by Design © Jay McTighe and Grant Wiggins
1
Overarching Understanding(s) from Curriculum
Framework Grade or Course Understandings:
Students will understand that…

Utilizing concepts of problem solving to
formulation, implementation, conclusion, and
real life problems?

What do good problem solvers do when they
are stuck on a problem?
Modeling real life situations using algebraic

How do mathematicians use technology to
problems and solutions.
model and analyze real world applications and
Algebra can be represented using a variety of
quantitative relationships?
technologies, including graphing calculators

How do algebraic concepts relate to the world
around us?
and computer programs.

How does organizing our thinking using algebra
allow us to better approach any mathematical or
concepts enables organization and analysis of


address real world scenarios will include a
mathematical reasoning.

Overarching Essential Question(s) from Curriculum
Framework Grade or Course Essential Questions:
Students will need to consider such overarching questions
as…
Algebraic reasoning can be applied to other
areas of learning, not just mathematics.
Topical Understanding(s) Specific to Unit: Students will
understand that…

Polynomials contribute to the construction of
Topical Essential Questions for Unit: To understand,
students will need to consider such unit questions as....

manmade structures
decisions?

There is a difference between x and 2x

Sum and product are not the same
2

What is the relationship between monomials,
binomials, and trinomials?

Understanding by Design © Jay McTighe and Grant Wiggins
How can factoring assist engineers in making
How is factoring related to division?
2
To understand, students will need to know and be able to do the following…
know… Students will know the following in order
be able to… Students will be able to (DO—skills,
to…(e.g., facts, vocabulary, rules, theories, principles)
procedures, processes): Use action verbs
(nouns)
 Identify polynomials
 Distributive Property
 Add and subtract polynomials
 Perfect Squares
 Multiply polynomials
 Rational Number
 Factor polynomials
 FOIL Method
 Understand and apply the distributive property
 Prime Numbers
Essential new vocabulary:
- Polynomial
- Monomial
- Binomial
- Trinomial
- Factor/Factoring
- Degree of Polynomial
- Variable
- Coefficient
Common misunderstanding(s):
 Combining like terms

Possible considerations to differentiate skills, including
advanced skills for more capable learners and more concrete
and scaffolded skills for struggling learners. Essential
Questions and Understandings are not differentiated. (Omit
for initial draft)


Use of more manipulatives
Different questions for different groups of students
Rules of exponents
Possible considerations to differentiate declarative
knowledge, including advanced content and materials for
more capable learners or more appropriately accessible
materials and content for struggling learners. Essential
Questions and Understandings are not differentiated.(Omit
for initial draft)

Give students index cards with x’s, x2’s, etc. and have
them group themselves.
ASSESSMENT EVIDENCE (STAGE 2)
Diagnostic Assessment(s) (To determine students’ readiness (based upon required knowledge and skills), interests,
and learning profiles):
Options for pre-assessment for the unit

Skills Check – combine like terms, use distributive property

Comparison of polynomials and binomials

Prime factorization of integers

Vocabulary

Creating a word problem that can be modeled using an algebraic expression and presenting to a group or class
What instructional adjustments, groupings or options will be made as a result of the diagnostic evidence:
Understanding by Design © Jay McTighe and Grant Wiggins
3

Differentiate based on results

Peer tutoring in pairs/groups

Re-teach material, if necessary
Summative Performance Assessment Task(s) for Understandings Using G.R.A.S.P.S.:
Goals: Redesign a new Albino Tiger Habitat exhibit using only the materials already on hand at the zoo.
Role(s): Engineer
Audience:
 Zoo Administration
 Wildlife Conservation Society
Situation: You may only use a predetermined amount of materials that you will be given. You will be limited in the
amount of wood, concrete, fencing, etc. that you can use to create this habitat.
Product or Performance: Develop an enclosure using the set amount of materials, so that it creates the maximum
roaming area. This can be a drawing or a model and must demonstrate the use of mathematical concepts of polynomials
applying them to area and perimeter.
Standards or Criteria for Evaluation/Traits for Rubrics:
 See rubric
 Judged by zoo administration and the Wildlife Conservation Society
 Meet all standards for safety
How can the product/performance, role or audience be differentiated to provide options for students’ readiness,
interest and/or learning profiles?
Students can be the zoo keeper, an engineer, a visitor, or a Wild Life Conservation Society representative
Student Directions for performance task:
All calculations must be done in class with a calculator.
Drawing and models can be started in class and should be completed at home.
Special Teacher Direction for performance tasks:
Provide all rules, materials, explanations, etc. that students need to complete the task.
Other Evidence (Tests, Quizzes, Academic Prompts):
Possible Differentiation options:

Quiz on area using polynomials
Level 1Students – area of square/rectangle

Quiz on perimeter
Level 2/3 Students – area of triangles and/or trapezoids

Do Now’s and vocabulary assessment
Level 4 Students – area of more complex polygons
Self-Assessment (Including Self-Evaluations Using Rubrics and Checklists, Peer Review, Reflective Journals and
Think Logs):
 Peer Reviews – zoo administrator and Wildlife Conservation Society representative must judge the exhibit, therefore
Understanding by Design © Jay McTighe and Grant Wiggins
4
they must know it too.

Reflective journals – summaries of what students are working on each day

Self evaluation rubrics
SCALE
4
HIGHEST
3
2
RUBRIC FOR PERFORMANCE TASK(S)
EXPLANATION
DIAGRAMS
CLARITY,
(INCLUDES
AND
NEATNESS
WRITTEN
SKETCHES
AND
ORGANIZATION
SUMMARY)
MATHEMATICAL
REASONING
25%
Uses complex
and refined
mathematical
reasoning to build
the maximum
area enclosure
with material
limitations, using
polynomials.
25%
Explanation is
detailed and clear
and shows
understanding of
all math concepts
related to
polynomials.
Uses effective
mathematical
reasoning to build
the maximum
area enclosure
with material
limitations, using
polynomials.
Explanation is
clear and shows
understanding of
math concepts
related to
polynomials.
Some evidence of
mathematical
reasoning to build
the maximum
area enclosure
with material
limitations, using
polynomials.
Explanation is a
little unclear, but
includes critical
components of
math concepts
related to
polynomials.
Understanding by Design © Jay McTighe and Grant Wiggins
STRATEGY/
PROCEDURES
15%
Diagrams and/or
sketches and
greatly add to the
reader’s
understanding of
the mathematical
procedure(s).
10%
The work is
presented in neat,
clear, and
organized fashion
that is easy to
read.
25%
Typically, uses an
efficient and
effective math
strategy to solve
the problem(s).
Diagrams and/or
sketches are clear
and easy to
understand.
The work is
presented in a
neat and
organized fashion
that is generally
easy to read.
Typically, uses an
efective strategy
to solve the
problem(s).
Diagrams and/or
sketches are
difficult to
understand to
some extent.
Work is presented
in an organized
fashion, but may
be difficult to
read at times.
Sometimes uses
an effective
strategy to solve
problems, but
does not do it
consistently.
Correct grammar
is used for the
written summary.
A few (less than
4) grammatical
errors.
5
1
Little evidence of
mathematical
reasoning to build
the maximum
area enclosure
with material
limitations, using
polynomials
Understanding by Design © Jay McTighe and Grant Wiggins
Explanation is
difficult to
understand and is
missing several
components
OR
was not included.
Diagrams and/or
sketches are
difficult to
understand
OR
not used.
The work appears
sloppy,
unorganized, and
inconsistent.
Rarely uses an
effective strategy
to solve
problems.
6
Stage 3: Creating Daily Lessons and Activities
Lessons and Activities should be aligned with Stages 1 and 2 best outlined in the order they are to be
taught. To ensure that lessons are aligned, enter your assessments first (including any lessons in
preparation of or for the assessments. Next, examine Stage 1 for a logical sequence of lessons and activities
which address all components of Stage 1 including knowledge and skills. Each day’s lesson may have
several activities. When you have completed the day by day sequence, then label the activities as A, M, or
T.
***INSTRUCTIONAL NOTES***
The following sequence of teaching activities is simply a standard of structure in which to frame the
learning, and is not intended as prescriptive nor restrictive. Teachers may accommodate student needs
through individualized and small group instructions, as well as provide independent work for more
advanced learners. The Understandings and Essential Questions for this unit should be used to drive
instruction and promote student inquiry.
Day 1
Unit Intro: How can we design an enclosure for the new tiger exhibit at the Bronx Zoo that
maximizes the roaming area? Introduce topical essential questions.
 Considerations for the job
 Practice basic skills – combing like terms, exponents, etc.
 A, (w, h)
Day 2
Adding/Subtracting Polynomials
 Identification/classification of polynomials
 Use candy to add/subtract polynomials
 A, (e1, h)
Day 3
Multiplying a Binomial by a Monomial
 Using the distributive property
 Develop rule for multiplying a binomial by a monomial
 A, M, (r, e1)
Day 4
Multiplying Binomials
 FOIL
 Using the box
 Example of using the box method: (x + 3)(x + 6) Fill in the squares as shown. Important: keep the
sign with the number.
x
+3
x
+6
 In order to fill in the box, multiply each column by each row
x
+3
X
x2
3x
+6
6x
18
 Complete the example by combining like terms.
Understanding by Design © Jay McTighe and Grant Wiggins
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 A, M, (e1, t2)
Day 5
Multiplying a Trinomial by a Binomial
 Use the box method described in Day 4
 Place the trinomial along the top row, and the binomial along the left side.
 Use the box to multiply the terms, then combine all like terms.
 M, (r, t2)
Day 6: Preparation for Performance Task
Finding the Area and Perimeter of Polygons
 Have students calculate area and perimeter of existing enclosures at the zoo
 Give students envelops with different shapes. Their task is to use a ruler to measure the dimensions,
and then use their calculator to calculate the area and perimeter of their shape.
 A, M, T, (e1, t)
Day 7
Assessment
 Quiz on multiplying polynomials
 (e2)
Day 8
Factoring Common Monomial Term
 Example: 3x2 + 12x
 Step One: Break down each term into prime factors (3 * x * x + 2 * 2 * 3 * x)
 Step Two: Circle all common factors (3 and x)
 Step Three: Whatever is circled, multiply together and place outside parentheses. EX: 3x( )
 Step Four: The remaining factors go back inside the parentheses in the same order that it is broken
down. EX: 3x(x + 2 * 2)
 Step Five: Multiply the remaining factors inside the parentheses. EX: 3x(x + 4)
 A, E1, T
Day 9
Factoring Difference of Two Squares
 Example: x2 – 36
 Step 1: Teach how to identify the difference of two squares (There are only 2 terms, both are perfect
squares, and they are connected by subtraction)
 Step 2: Find the square root of each term. Place them into two sets of parentheses. EX: (x 6)(x 6)
 Step 3: Place a plus sign in one group and a minus sign in the other group. EX: (x + 6)(x – 6)
 Think, Pair, Share: Partner the students, and have each student create several of their own examples.
Then have the partners trade their examples, so that each student is factoring their partner’s problem.
Then trade back and have students check their partner’s work.
 A, (e1)
Day 10
Factoring Trinomial with the coefficient of 1
 Use the sum-product method
 Example: x2 + 3x + 2
 Step 1: Find all factors of the last term. EX: 1, 2 and -1, -2
Understanding by Design © Jay McTighe and Grant Wiggins
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

Step 2: Find the sum of each set of factors. EX: 1+2 = 3 and -1+-2=-3
Step 3: The set of factors that add up to the middle term are the second terms in the parentheses. EX:
( + 1)( + 2)
 Step 4: x is the first term in the parentheses. EX: (x + 1)(x + 2)
 Factor Matching Game: Pair the students. Give each pair an envelope of questions and answers on
strips of paper. Put questions and answers face down in two separate piles. The first student should
randomly select a question and both students should work to solve the problem. The first student
then gets to randomly pick an answer from the answer pile. If it matches their answer, they get a
point. Then the second student goes and follows the same steps as the first. Play continues until all
the questions are used up.
 A, (e1, o)
Day 11
Factoring Trinomial with a coefficient greater than 1
 Use payback method
 Example: 2x2 + 5x + 2
 Step 1: Borrow the first coefficient by multiplying the first coefficient by the last term. EX: 2 * 2 =
4
 Step 2: Re-write the trinomial with the new coefficients. EX: x2 + 5x +4
 Step 3: Factor the resulting trinomial using the previous methods described. EX: (x + 1)(x + 4)
 Step 4: Payback the coefficient you borrowed to each variable of the factors. EX: (2x + 1)(2x + 4)
 Step 5: Simplify one or both of the parentheses by finding a common monomial term. EX: We
cannot reduce the first parentheses. We can reduce the second parentheses by dividing each term by
two. So the new factors will be (2x + 1)(x + 2).
 Step 6: Check the factors by multiplying the two terms to get the original trinomial.
 Timed Gallery Walk: Choose several hard polynomials to factor and place each on separate sheets
of chart paper, which you will hang around the room. Students will work in small groups at each
station for a short timed period. Each group will get a different color marker, and will have a set
amount of time to begin working on the problem. When time is up, each group should move to the
next station and continue working on the problem where the other group left off.
 A, M, (e1, r, t, o)
Day 12
Assessment
 Quiz on factoring
 (e1)
Day 13
Zoo Project – tying it up
 Students play role as engineer to design new exhibit at Bronx zoo.
 Give toothpicks, ice pop sticks or other manipulative that are set measures. Students can draw or
construct their exhibit.
 Is differentiated through different shapes that are created.
 M, T, (h, t, r, o)
Day 14
Zoo Project
 Students present as engineers, the rest of the class is zoo administration or the Wildlife Conservation
Society representatives to get approval or denial of the proposal. Students as society representatives
Understanding by Design © Jay McTighe and Grant Wiggins
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will use the rubric to evaluate each student’s design.
 Denial = need for revision
 T, (r, e2)
Day 15
Revision Day
 Students whose proposal was accepted will help students whose proposals were denied.
 M, T, (r)
Day 16
Assessment
 Unit Test
 End of unit self assessment through journal entry, paragraph on authentic applications that can be
transferred or list of things I learned in this unit.
 (e1)
Labeling Key:
A represents learning experiences which optimize students acquisition of knowledge and skills and will
include a number of equipping and exploring activities
M represents meaning making activities represents learning experiences that increase students’
understanding of knowledge and skills
T represents activities that will ask students to apply their understanding in tasks and procedures that are
authentic and realistic.
Another way of considering the purpose of activities:
Teaching-Learning Activities Based upon W.H.E.R.E.T.O.: These are embedded in the ATM for Lesson Design Document
(WHERETO represents the purpose of lessons, not the sequence) For more detailed information see pages 212-226 of the UbD
Professional Development Workbook.
Where are we (student’s point of view) headed? How will the unit be introduced including the tasks, goals, essential questions?
How will I hook students to engage their interests?
Equip and Explore: What lessons and activities will provide the knowledge, skills, processes, and procedures needed for the unit? How
will these address the needs of all learners?
Revise/Rethink/ Reflect/ Revisit: What opportunities (activities, experiences) will be provided to help students revise/rethink/reflect/ and
revisit?
Evaluation/self/evaluation/: How will we engage students in self-evaluation, goal setting, and self-reflection?
Tailoring: How will we tailor or differentiate the unit and lessons to differentiate for different learning needs and interests? (Materials,
strategies, groupings, mini-lessons, etc.)
Organized: What sequence of lessons or activities will we use to organize the unit in a way that is coherent and makes sense to students?
Understanding by Design © Jay McTighe and Grant Wiggins
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Materials and Resources for Teaching the Unit:
 Calculators
 Rulers
 Envelopes with several different shapes of different sizes
 Graph Paper
 Bronx Zoo Maps
Understanding by Design © Jay McTighe and Grant Wiggins
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