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Exemplar Unit Plan

You have found an exemplar unit plan. Before viewing the plan, there are a couple clarifications to note. First, all unit plans are different. There was not a consistent template used across all plans in the bank. Second, all of the unit plans in the Exemplar Unit Plan Bank are of high quality. While they are different, they each accomplish the goal of planning a unit which weaves together many standards and objectives in a coherent way. The Sanford Inspire Program has collected several different unit plans to demonstrate the variety possible within effective unit planning. Exemplar Unit Plan Uses

Brief Description Uses

Idea Generation

Allow this unit plan and others in the bank to serve as idea generators for yourself and your unique setting.

Pick and choose components of the unit plan which serve you well and use them as you see fit.

Adopt the format of the plan while replacing all of the components with more appropriate material for your specific setting.

Implementation

Take and use the unit plan in your specific setting. Confirm with administrators and instructional support staff that the plan fulfills the needs your classroom has.

Implement unit plan in your classroom as it is planned in the document. Use accompanying materials prescribed and objectives listed.

GENERAL UNIT INFORMATION

Grade/Subject: 10th Grade/ Advanced Algebra 2

Unit Name: Polynomials and Polynomial Functions

Dates of Unit

Implementation: October 22– November 9

UNIT STANDARD(S)

Common Core State Standards

Standard(s)

HSA-SSE.A.1a - Interpret parts of an expression, such as terms, factors, and coefficients.

HSA-SSE.B.3c - Use the properties of exponents to transform expressions for exponential

functions. For example the expression 1.15t can be rewritten as (1.151/12)12t ≈ 1.01212t to reveal

the approximate equivalent monthly interest rate if the annual rate is 15%.

HSA-APR.A.1 - Understand that polynomials form a system analogous to the integers, namely,

they are closed under the operations of addition, subtraction, and multiplication; add, subtract,

and multiply polynomials.

HSA-SSE.A.2 - Use the structure of an expression to identify ways to rewrite it.For example, see

x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 –

y2)(x2 + y2).

HSA-APR.B.3 - Identify zeros of polynomials when suitable factorizations are available, and

use the zeros to construct a rough graph of the function defined by the polynomial.

HSA-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).

Remedial

Standards

HSA-SSE.A.1b - Interpret complicated expressions by viewing one or more of their parts as a

single entity. For example, interpret P(1+r)n as the product of P and a factor not depending on

P.

HSA-SSE.B.3a - Factor a quadratic expression to reveal the zeros of the function it defines.

Note – there are many more remedial standards, however, these would be the most recently

learned and relevant standards

Enrichment

Standards

HAS-BF.B.3 & 4 – building new functions from existing functions could be an enrichment,

however, these standards are also explicitly taught in the next unit

Students can also be given extension problems where they must, without guidance, try to build a

possible function given the graph of a polynomial.

Curricular

Resources

Geogebra.org – free graphing utility that students and/or teacher can use to investigate the

connection between the polynomial expressions and their graphs.

Demonstrations.wolfram.com – site contains visuals that can show the transformation of

polynomials and their graphs. Search for “polynomial”.

‘Big goal’, Assessment and Investment

BIG GOAL

Qualitative:

Students will develop and be able to articulate their deep understanding of the connection between an

algebraically expressed function and its graph including an understanding of roots of an equation and their

representation on a graph. Students will develop the skills necessary to find the zeros of a polynomial

equation and use their knowledge to sketch a graph of a polynomial.

Quantitative:

At least 80% of students will score at least 80% on the unit quizzes and unit test.

All students will earn at least a 90% on the Exponent Qualifier with in the first two attempts.

FORMATIVE & SUMMATIVE ASSESSMENT

How will I measure PROGRESS towards the Unit Goal?

Students will be given daily warm-ups that will be used as a quick informal check for understanding.

Many of these will address the qualitative goals of the unit.

Students’ homework assignments will also be checked during the warm up. The teacher will choose one

problems she anticipates to have been challenging and will check that problem as she checks homework.

Students will also be given regular exit tickets one 1 – 3 quick problems as a check for understanding.

These tickets also will often include a computational question as well as a question requiring concept

comprehension and a written response. Many of the concept questions will address the qualitative goal

of the unit.

There will be one unit quiz that counts toward students’ grades (see Calendar – Quiz 6A)

How will I measure ACHIEVEMENT of the Unit Goal?

Students must pass a “qualifying” exam for their exponent rules. Students will have two opportunities

to pass this exam with at least a 90% or higher. Students will be given one chance in class and will be

required to attend tutoring and complete additional practice if they do not pass with a 90%. Only after

tutoring and additional practice will they be allowed to re-take the exam with a 10% grade deduction for

each additional try.

Note: Our vertical alignment team has discovered that not knowing exponent rules well enough at this

level will cause enormous challenges later on in this course as well as every subsequent math course. I

have implemented this strict qualifying exam for two years and all students have passed by the second

time. I also send home a letter to parents explaining the severe expectations for this exam. The letter is

included after the exams. Each time a student takes the qualifier he or she receives a different form of

the exam.

Students will also complete a unit exam that will cover all the objectives, except for the exponent rules.

INVESTMENT PLAN SUMMARY (*cut & paste acceptable)

Engagement Students will be given an assignment to investigate their own thoughts about what the graph of a

function truly represents. Although we have discussed this at the beginning of the school year with linear,

absolute value, and linear piecewise functions, they have not investigated this question in detail with curves.

Students will have to answer the following questions over the course of the three state test days and

then we will discuss their ideas when they return at the end of the first week of the unit (see Calendar for more

clarification).

What about an equation causes its graph to be curved? Why do you think this happens?

Why might it be important to know where a function will cross the x-axis? Consider a function that

represents your money in a stock where the x-axis represents time and the y-axis represents the amount

of money you have. What does the x-axis actually represent in this situation?

How do you find the x-intercept of a linear function? How would you find the x-intercept of a non-

linear function? How are these related?

What does the graph of an equation represent?

Complete the table below for the given function and then plot each point on the graph below. What do

you think the graph looks like beyond the points your plotted? Sketch, using a pencil, your prediction.

Students will be asked to share their findings with a partner when they return at the end of the week and

will be given the opportunity to share their thoughts with the class. The teacher will not directly answer any of

their questions at this time, but will announce that they will have concrete answers throughout the unit.

Class Tracking The class averages and mean values of the assessments will be posted when the assessments are returned to

students.

Individual Tracking

Students keep a sheet where they track their assessment scores so that they can see their progress.

Student Reflections On the back or at the end of some of the exit tickets, students will describe how they feel about the unit and what

they are learning. Depending on the comment, the teacher may write a response to be handed back the next day.

UNIT SUMMATIVE ASSESSMENT

ALIGNMENT GUIDE

The assessments are included at the end of the unit plan. T represents the unit test and Q represents the

Exponent Qualifier.

Standard # Standard

Aligned to

item #’s

Points

Possible

HSA-SSE.B.3c

Use the properties of exponents to transform expressions for

exponential functions. For example the expression 1.15t can be

rewritten as (1.151/12)12t ≈ 1.01212t to reveal the approximate

equivalent monthly interest rate if the annual rate is 15%.

Q1-Q30 30

HSA-APR.A.1

Understand that polynomials form a system analogous to the

integers, namely, they are closed under the operations of

addition, subtraction, and multiplication; add, subtract, and

multiply polynomials.

T5, T6, T7 3

HSA-SSE.A.2

Use the structure of an expression to identify ways to rewrite it.

For example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a

difference of squares that can be factored as (x2 – y2)(x2 + y2).

T8 - T11 8

HSA-APR.B.3

Identify zeros of polynomials when suitable factorizations are

available, and use the zeros to construct a rough graph of the

function defined by the polynomial.

T3, T4, T19-T20 4

HSA-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).

T1, T12-T18 15

HSA-SSE.A.1a Interpret parts of an expression, such as terms, factors, and

coefficients. T2 1

Total points possible: 61

***Assessment outcome data needed for final presentation!

Lesson Objectives

(*A.K.A. : ’Breaking it down, day-by-day’)

Standard

List the main standards

to be taught here.

Note: The following

standard is too complex

to be taught in its

entirety on the same

each day. Therefore, it

will be repeated BUT,

broken down into daily

objectives.

Daily Lesson Objective(s)

SWBAT

Timing

Most lessons will

be taught for 55

min. /day

A total of 10 days

will be used to

teach the daily

lesson objectives

in order to meet

the unit goal.

Summarized (yet specific )Lesson Plan Notes

This area is to include ideas you have for teaching the lesson. Include strategies,

materials and potentially, procedures.

HSA-SSE.B.3c

simplify expressions using the first four

properties of exponents (see Exponent

Qualifier)

Students will already be familiar with many of these rules, so these can be taught

through inquiry. Students can be lead through investigating the basic meaning of

an exponent as a “rapid” form of multiplication and then they can write their own

rules. Students can work in groups to do this and can share their rules on

designated spaces around the room.

Simplify expressions using the last four

properties of exponents (see Exponent

Qualifier)

Students can use what they learned the previous day to expand their rules. The

first three will be straight forward and the students can investigate these by

themselves for a few moments and then they can review them whole group.

The last rule is often difficult for students to think about so this will be more

direct instruction but the teacher should have students guess at what fractional

exponents mean.

HSA-APR.A.1

HSA-SSE.A.1a

Label polynomials based on their degree

and number of terms.

Use arithmetic so simplify polynomial

expressions

Students will probably have questions on the previous lesson after having done

their homework, so almost half of the class time should be devoted to that. The

objectives for this day are very quick and students should be quite familiar with

simplifying polynomial expressions. Students can play a game to help memorize

and recognize the names of the polynomials.

HSA-SSE.A.2

Factor polynomials including the sum and

difference of two cubes and expressions

that are “like quadratics”

Students have already factored quadratics and by grouping in the previous unit,

so most of this standard has already been taught. Students can be presented with

several different types (see below) of polynomials and will be asked to put them

into different groups and to figure out which ones they know how to factor and

which ones they do not know how to factor. As a class, students can make

suggestions for how we could factor the types they are unfamiliar with. (As

foreshadowing, the warm-up for the day should be multiplying a binomial and

trinomial that simplify to the sum or difference of two cubes.)

HSA-APR.B.3

Use synthetic division to divide, factor,

and find the roots of polynomials given a

root or factor.

Ensure that students know the difference between a root and a factor of a

polynomial. Review their initial engagement activity and provide some light

practice in defining roots and factors. Begin the lesson by reviewing long

division and having students write out steps as if they were going to describe

these to someone else. Apply their steps to long division of a polynomial and

then blow their minds by circling the coefficients, then doing the same problem

with synthetic division. (Color coding would probably be very helpful here;

perhaps highlighting the coefficients as you go when doing the synthetic

division.)

Find zeros using the fundamental theorem

of algebra and write polynomial functions

given roots

In between the last objective and this one, students should learn the Rational

Root Theorem (HAS-APR.B.2)

Students should be directed back to their engagement activity and should review

and/or revise their thoughts. This lesson should include a lot of conceptual

checks for understanding and students should be asked to explain their reasoning

and explain the connection between zeros and the equation of a polynomial

function.

Sketch and analyze the graphs of

polynomials given different pieces of

information

55-75 min.

between two days

Students should discuss the accuracy of their sketches and what information they

would need to make their sketches more accurate. As an extension for all,

students should be asked at the end of the lesson to write the equation of a

polynomial given its graph. All students should attempt this.

Write a possible equation of a polynomial

given its graph This is a very short lesson and may not consist of a whole class period.

HAS-APR.B.2 Find rational zeros by using the rational

root theorem.

Students can be given a set of polynomials and their roots and asked to see if they

can find a pattern between the roots and any set of the coefficients of the

polynomials. The teacher should ask if the students can find a short cut or “way

to cheat” to find the roots simply by looking at the equation.

Total days/minutes taught during the entire unit of instruction. 495 - 550 min. days 9 - 10

Real-Time Calendar

Date YWBAT:

Mon. 10/22 (6.1A) simplify expressions using properties of exponents (rules 1 – 4)

Tue. 10/23 State Testing in the morning, then classes:

Tue.: Per. 1 & 2, Wed.: Per. 3 & 4, Thur.: Per. 5 & 6

(6.1B) simplify expressions using properties of exponents (rules 5 – 8) Wed. 10/24

Thur. 10/25

Fri. 10/26 (6.3) name and simplify polynomial expressions using arithmetic

Mon. 10/29 (6.4) factor polynomial expressions including the sum and difference of two cubes

Tue. 10/30 (6.5) use synthetic division to divide, factor, and find the roots of polynomials, given a root or factor.

Wed. 10/31 (6.6) find rational zeros by using the rational root theorem

Thur. 11/1 (6.7) find zeros using the fundamental theorem of algebra and write polynomial functions given roots

Fri. 11/2 Quiz 6A (6.1 – 6.6)

(6.8) sketch the graphs of polynomials (if time permits)

Mon. 11/5 (6.8) sketch and analyze the graphs of polynomials

Tue. 11/6 Chapter 6 Review (focus on 6.1 – 6.4)

Wed. 11/7

Early Release (38 min.)

Exponent Qualifier: You must earn a 90% or higher. If you do not, you have one chance to re-take it and

again must earn at least a 90% but will get a 10% reduction.

Thur. 11/8 Chapter 6 Review (focus on 6.5 – 6.8)

Fri. 11/9 CHAPTER 6 TEST

Chapter 6 Test

Lawson Risoldi, Adv. Alg. 2 (2011), Pg. 1

Chapter 6 Test Name ____________________________________

QUADRATIC FUNCTIONS Date __________ Period ______

DIRECTIONS: Write neatly, use pencil, blue, or black ink. Show your work and simplify all answers.

PART A: Find the best answer. Write your answer in the space provided. (1 point each)

_____ 1. Evaluate 3 2( ) 2 4 5 8f x x x x at 2x .

(A) 2 (B) 18 (C) 8 (D) 34 (E) 50

_____ 2. Which of the following expressions is a quadratic binomial?

(A) 4 2x x (B)

3x x (C) 24x (D)

2 5x x (E) None of these

_____ 3. Which of the following is a possible root of3 2( ) 4 5 9f x x x x ?

(A) 1

3 (B)

1

9 (C) 18 (D) 9 (E) None of these

_____ 4. Which function below could have the graph shown in Figure A?

(A) 2x (B) 31

2x (C)

4x

(D) 32x (E) None of these

_____ 5. Simplify 3 24 4 3 9x x x x .

(A) 3 24 2 13x x x (B)

3 24 4 5x x x (C) 3 24 2 13x x x

(D) 4 3 28 13 3 36x x x x (E) None of these

_____ 6. Simplify 24 4 9x x x .

(A) 9

4x

x

(B)

2 8 7 36x x x (C) 3 28 7 36x x x

(D) 4 316 7 36x x x (E) None of these

_____ 7. Simplify 5 6 2x x x .

(A) 3 29 8 60x x x (B)

3 29 8 28x x x (C) 3 60x

(D) 3 60

9 85

x xx

(E) None of these

Figure A

Chapter 6 Test

Lawson Risoldi, Adv. Alg. 2 (2011), Pg. 2

PART B: Factor the following polynomials. Write your answers in the answer box. (2 points each)

8. 4 16x 9.

3 25 9 45x x x

10. 327 1x 11.

3 24 6 10x x x

PART C: Simplify the following expressions. Write your answers in the answer boxes. (2 points each)

12.

3 23 4 5

2

x x

x

13. 4 3 22 5 1 1x x x x x

PART D: Factor the following polynomials with the given information. Place your answers in the boxes below.

(2 points each)

14. 3 23 13 15x x x , given ( 5)x is a factor. 15.

3 25 2 24x x x , given ( 2)x is a factor.

ANSWER BOX

8.

9.

10.

11.

Chapter 6 Test

Lawson Risoldi, Adv. Alg. 2 (2011), Pg. 3

PART E: Given a zero of the polynomial, find the remaining zeros. Write your answers in the box below.

(2 points each)

16. 3 23 4 12x x x , given 3 is a zero.

PART F: Find all the zeros of the function. Write your answer in the boxes below. (2 points each)

17. 3 2( ) 9 10 17 2f x x x x 18.

4 3 2( ) 7 13 3 18f x x x x x

Chapter 6 Test

Lawson Risoldi, Adv. Alg. 2 (2011), Pg. 4

PART G: Follow the directions for each. (1 point each)

19. Sketch the graph of ( ) 2 2 3f x x x x 20. Write a possible function in factored form of the

function graphed below.

CHALLENGE: You must complete all previous problems in order to receive any extra credit. Box your answers on

this page.

21. Write the equation of the function shown in the graph below. (Hint: Use a piecewise function) (1%)

Exponent Qualifier

Lawson Risoldi, Adv. Alg. 2 (2011) , Pg. 1

Exponent Qualifier

Name ____________________________________

FORM A Date __________ Period ______

PART A: Complete the following expressions. Write neatly. (2 points each – however, you must get all of these

correct in order to “pass”.)

PART B: Simplify the following. Write your answers with positive exponents and in radical notation where

applicable. Write your answers neatly in the answer box. You do not have to rationalize the denominator.

(1 point each)

9. 3 5x x 10.

42 3x y 11.

27 12.

34x

13.

5

3 2

4

16

xy

x y 14.

23

10

5x

y

15. 0 2 3 53 x y x

16. 3/25 17.

61/3 5/6x y 18.

22

5 4

7xy

x y

1. x ya a 3. x

ab 5. xa 7. y

xa

2. x

y

a

a 4.

xa

b

06. a /8. x ya

Answer Box

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

Exponent Qualifier

Lawson Risoldi, Adv. Alg. 2 (2011) , Pg. 2

19.

20

2

x y

xy

20. 3/2x

21. 3

2x

22. 3/2 5/2x x 23.

27

3

24. 0 2

2 7 345 3x y z x y

25. 3

1/5 9/5x x 26.

1/3

1/6

y

y 27.

3/22

28.

2 1

3 0

2

4

x y

x y

29.

12 2 23x y z

30.

31

x

Answer Box

19.

20.

21.

22.

23.

24.

25.

26.

27.

28.

29.

30.

Dear Parent,

I am writing to inform you of a very important test your child will take in this unit of Advanced

Algebra 2. This unit covers a skill that is critical to students’ success later on in this course and nearly

every other math course they will take. Because of this, I am requiring that students pass a test on this

topic with no lower than a 90% because that is the level of proficiency they will need to be successful

later on.

Students will have two chances to take this exam and will receive a 10% score deduction for

taking it more than once. I employ this 10% penalty to encourage students to study well and get the

help they need to do well the first time.

If students do not pass with 90% the first time, they will receive no score in the gradebook and

will be required to attend tutoring and complete additional practice before re-taking the exam. I am

available before and after school for tutoring, but students should check the tutoring schedule to before

attending.

I have required this exam for the last two years and every student has passed within two tries. I

am confident that all students will pass again this year. I have also seen significant improvement in

students’ work in later chapters and courses after implementing this qualifying exam. This year, all of

the Advanced Algebra 2 teachers are giving this exam.

Please reach out to me with any questions or concerns you might have.

Sincerely,