YOUNGSTOWN CITY SCHOOLS
MATH: ALGEBRA 1
UNIT 4A - - EXPRESSIONS AND EQUATIONS – PART I (3 WEEKS) 2013-2014
Synopsis: The major concepts in this unit are polynomials and factoring expressions. Students will work on
interpreting expressions and parts of expressions, and also adding, subtracting, multiplying, and factoring
polynomials.
The star symbol * indicates mathematical modeling
STANDARDS
A.SSE.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. For example, interpret P(1+r)n
as the product of P and a factor not depending on P.
A.APR.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
A.SSE.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).
MATH PRACTICES:
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with Mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning
LITERACY STANDARDS:
L.2 Communicate using correct mathematical terminology
L.4 Listen to and critique peer explanations of reasoning
L.5 Justify orally and in writing mathematical reasoning
MOTIVATION
TEACHER NOTES
1. Discuss with students the real-life uses of polynomial expressions: (A.SSE.1, A.APR.1,MP.4, L.2)
 calculate areas and volumes
 calculate value of a savings account;
 examine the path of a baseball looking at distance and time;
 design roller coasters;
 conduct cost analysts to interpret cost versus sales data to aid companies in decision making
 make calculations with energy and voltage formulas when working with currents
2. Preview expectations for the end of the Unit
3. Have students set both personal academic goals for this Unit.
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YCS ALGEBRA I: Unit 4A: EXPRESSIONS AND EQUATIONS PART I 2013-2014 1
TEACHING-LEARNING
Term
Constant
Coefficient
TEACHER NOTES
VOCABULARY
FOIL
Monomial
Closure
Binomial
Linear term
Trinomial
Polynomial
Leading coefficient
Degree of a polynomial
1. Teacher asks students to give examples and definition of the terms: polynomial, term, constant, and
coefficient, leading coefficient, linear term, degree of a polynomial. Ask students to write a polynomial with
the following qualifications: (e.g., 3 terms, a constant, and a coefficient of 2 - - they might write something
like the following - - 2x2 + 3x – 4); discuss the parts they use, and explain how there can be more than one
answer. Ask if the examples developed are correct, and have students explain why/why not. (A.SSE.1 a,
b, MP.4, MP.7, L-2, L-4, L-5)
2. Have students do polynomial matching activity monomial, binomial, and trinomial; use everyday terms
that have prefixes for mono- (monopoly, monotone, etc.) bi- ( bicycle, biceps, bisect ), and tri- (tripod ,trio,
tricycle, etc.) as introduction. To reinforce monomial, binomial, or trinomial use diagram on Prezzi
developed for this: (A.SSE.1, MP.4, MP.5, MP.7, L.2)
http://prezi.com/udpk9qi7dq_6/untitled-prezi/?kw=view-udpk9qi7dq_6&rc=ref-31300653
#2
4x2 +3x -7 = trinomial
5x3y4z = monomial
3x – 4 = binomial
NOTE to Teacher: you
may want to review laws
of exponents at this time.
3. Have students add and subtract with polynomials by combining like terms. Teacher selects problems
(Kuta, textbook, etc.) (A.APR.1, MP.2, MP.4, L.2)
#4
4. Teacher models for students how to multiply two monomials, a monomial by a binomial or trinomial, a
binomial by a binomial, or a binomial by a trinomial. Teachers: you can use the vertical approach or
horizontal (distribution) approach (FOIL) (A.APR.1, MP.2, MP.4, L.2)
(2x+3)(x-1)
5. After adding, subtracting, and multiplying polynomials, ask students: “If I add 2 polynomials together, what
type of answer do I get?” Students should respond that the answer is a polynomial; same with subtracting
and multiplying; this means that these operations are “closed” (you are always going to get an answer that
is in the particular set) SEE Teacher Notes (A.APR.1, MP2, MP.4, MP.7, L.2)
STOP FOR A TEACHER GENERATED ASSESSMENT
6. Teacher uses “Teach Me How To Factor” (YouTube) (rap about factoring) (A.SSE.2, MP.4, MP.5, MP.7,
L-2)
7. Discuss divisibility rules by asking students to list the rules for 2, 3, 4, 5, 6, 9, and 10. Discuss the answers
with the class and have the class create a list of the rules together similar to ones below and create ways to
remember them: (note; this was in the old 5th grade standards, so students should have a sense of these
rules)
Divisible by 2: All even numbers
Divisible by 3: Add up all the digits in the number. If the sum is divisible by 3, so is the number.
Divisible by 4: If the last two digits are divisible by 4, so is the number.
Divisible by 5: Numbers ending in 5 or 0
Divisible by 6: Numbers divisible by 2 and 3 both
Divisible by 9: Add up the digits in the number. If the sum is divisible by 9, so is the number
Divisible by 10: Numbers ending in 0
Using the divisibility rules, review factor lines and state if numbers are prime or composite using the
examples below. Then play the factor game at http://www.coolmath-games.com/0-math-linesxfactor/index.html (A.SSE.2, MP.2, MP.4, MP.6, L.2)
Factor : 12 = 1 * 12
Factor: 54 = 1*54
Factor: 31 = 1*31
= 1*3*4
= 1*6*9
prime
= 1*3*2*2
= 1*2*3*3*3
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2x +3
x -1
#5The definition of
closure: if you take a set
of numbers (e.g.,
polynomials) and perform
an operation (e.g., addition)
on them and you always
produce a polynomial, then
that set of numbers is
closed under that operation.
We say the set of
polynomials is closed under
addition.
An example of a set of
numbers NOT closed under
an operation would be to
take polynomials as the set
and division as the
operation. The polynomials
are not closed under
division, because
is not a polynomial since
= (2x)(3x+4)-1 and
polynomials cannot have
negative exponents.
YCS ALGEBRA I: Unit 4A: EXPRESSIONS AND EQUATIONS PART I 2013-2014 2
TEACHING-LEARNING
1*22*3
=
composite
TEACHER NOTES
1*2*33
=
composite
8. Review GCF and have student find the GCF of 24 and 81 and write an explanation of the process.
Discuss this process and have students refine it so that everyone understands the process. Then
proceed to problems using variables (note: be sure to explain that the process is the same whether you
are using numbers or variables): Find the GCF for 16xy2z2 and 72xyz3 (A.SSE.2, MP.1, MP.2, MP.4,
MP.6, L-2)
Have students work the following real-life examples pertaining to finding the GCF:
1. A company sells pencils packaged in boxes of 24 or 64. Within the boxes, the pencils are packaged into
smaller boxes. The smaller boxes contain the same number of pencils before they are placed in the larger
boxes. What number of pencils should be placed in the smaller boxes to make the least number of smaller
boxes?
2. Shannon is making identical balloon arrangements for a party. She has 32 maroon balloons, 24 white
balloons, and 16 orange balloons. She wants each arrangement to have the same number of each color.
What is the greatest number of arrangements that she can make if every balloon is used?
9. Have students factor out the GCF using the three examples: factor 5x2 + 10x, 4x5 -6x3 + 14x, and 4x2y –
16xy2 . To reinforce this, play tic tac toe game (attached on page 5 of the unit plan). (A.SSE.2, MP.1,
MP.2, MP.4, MP.6, L.2, L.4)
10. Discuss with students the definition of factoring polynomials. Compare it to prime factorization of numbers,
breaking something up into its factors. Refer back to multiplying two polynomials together and explain
factoring as the reverse operation for multiplying. Discuss factoring the difference of two squares. (Note:
you may want to quickly review the numbers squared from 0 to 15 before you begin). (Worksheet #1 is
provided on page 6 and answers on page 7). (A.SSE.2, MP.2, MP.4, MP.6, L.2)
STOP FOR A TEACHER GENERATED ASSESSMENT
11. After students understand factoring the difference of two squares, work on factoring trinomials whose
leading coefficient is one. At first make sure all the signs are positive, and then work with the linear term
being negative and the constant term positive. Next work with the linear term being positive and the
constant term negative. Lastly, work with both the linear term and the constant term being negative. The
worksheet at the website below can be used as a cumulative review of factoring trinomials with leading
coefficient of one. (A.SSE.2, MP.2, MP.4, MP.6, L-2)
http://www.kutasoftware.com/FreeWorksheets/Alg1Worksheets/Factoring%201.pdf
12. Teach factoring trinomials whose leading coefficient is some number other than 1, using the
following: http://alex.state.al.us/lesson_view.php?id=4152 This web-based lesson is comprised of 6 parts
that teach factoring a trinomial in a manner other than trial and error. The first part reviews finding GCF
by playing a game (the attachments on this website illustrate the factoring method and the attachment
labeled “factoring fanatic.ppt” is a power point showing all 6 attachments. In attachment #2 the pattern is
take the two numbers on the right and multiply them to get the number in quadrant II and add them to get
the number in quadrant III. Note: the quadrants are numbered in the same manner as the coordinate
system. (A.SSE.2, MP.2, MP.4, MP.5, MP.6, L-2)
13. Have students multiply the following polynomials together; have students discuss the pattern that is
present. (x+1)(x2 –x +1), (x+3)(x2 - 3x + 9), (x+5)(x2 - 5x + 25), (a-2)(a2 + 2a + 4), (a-6)(a2 + 6a +36), (y4)(y2 + 4y + 16). They should deduce the formulas for the sum and difference of two cubes. Add more
examples if needed. After they understand the formulas, reinforce with the worksheet at the Kuta website
below. (A.APR.1, A.SSE.2, MP.1, MP.2, MP.3, MP.7, L-2, L-4)
http://www.kutasoftware.com/FreeWorksheets/Alg2Worksheets/Factoring%20A%20Sum+Difference%20of%20Cube
s.pdf
14. Discuss rules for factoring attached to unit on page 8. (A.SSE.2)
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YCS ALGEBRA I: Unit 4A: EXPRESSIONS AND EQUATIONS PART I 2013-2014 3
TEACHING-LEARNING
TEACHER NOTES
15. Factor By Grouping. Begin by discussing the importance of factoring out the GCF. Then have students
hypothesize on how they would factor 4(x-3) + y(x-3), x(x+2) + 5(x+2), x2 (x-6) – 3(x-6). After they
understand this part of factoring by grouping, extend to the following problems: factor: 2x – 2y + ax – ay,
x3 – 4x2 + 2x – 8, 3x3 – 3x2 + 5x – 5, x3 – 8x2 + x – 8, 6x – 6y – 5ax + 5ay, 12x3 – 16x2 – 3x + 4, ax – bx –
4a + 4b, xy + 10x – 4y – 40. (A.SSE.2, MP.1, MP.2, MP.7, MP.8, L-2)
16. Review factoring with the game on page 9 of unit plan. Cut out the squares, mix them up and have
students match the sides of the pieces. (A.SSE.2, MP.1, MP.2, MP.5, MP.7, MP.8, L.2, L.4)
TRADITIONAL ASSESSMENT
TEACHER NOTES
1. Unit Test with Multiple Choice Questions
TEACHER CLASSROOM ASSESSMENT
TEACHER NOTES
1. 2- and 4-point questions
2. Other in class assessments
AUTHENTIC ASSESSMENT
TEACHER NOTES
1. Students evaluate goals for the unit
2. Error analysis with 4-5 problems (problems and answer key / rubric attached to unit on pages 10 -12)
(A.APR.1, A.SSE.2, MP.1, MP.2, MP.4, MP.7, L-2, L-4, L-5)
3. Students are given certain qualifications and write 2 equations and explain or they write one equation and
explain why there cannot be 2 equations; show work as to how it factors (attached on page 13) (A.SSE.1
a, b, MP.2, MP.4, L-2)
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YCS ALGEBRA I: Unit 4A: EXPRESSIONS AND EQUATIONS PART I 2013-2014 4
T-L #9
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YCS ALGEBRA I: Unit 4A: EXPRESSIONS AND EQUATIONS PART I 2013-2014 5
T-L # 10
Worksheet 1
Difference of Two Squares Homework
NAME_______________________
Date________
Period____
Factor each completely; remember to factor out GCF first if there is one.
1) k 2 – 36
2) x 2 − 4
3) n 2 – 49
4) 2x 2 − 18
5) 9n 2 – 4
6) 4n 2 + 1
7) 1 − 16 p 2
8) 9a 2 − 16
9) 49b 2 – 81
10) 5n 2 − 405
11) 81 x 2 – 25
12) 64 p 2 − 9
13) 100m 2 – 81
14) k 2 + 4
15) 36x 2 − 4y 2
16) 2u 2 − 50v 2
17) 16 x 2 − y 2
18) 9u 2 − 16v 2
©R CKguRtka4 kSCoef3tKwiaSr2eq tLkLOC8.D P CANlUlM NrVigg4hCtWsf nr4e5s8ezrRvCeEdS.5 V
kM9aWdsec xw7iptphy pIDnXf3i4naiAtneH zAIl5gvewbFrNag L1J.t
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YCS ALGEBRA I: Unit 4A: EXPRESSIONS AND EQUATIONS PART I 2013-2014 6
Answers to worksheet #1:
1. (k-6)(k+6)
2. (x-2)(x+2)
3. (n-7)(n+7)
4. 2(x-9)(x+9)
5. (3n-2)(3n+2)
6. Not factorable
7. (1-4p)(1+4p)
8. (3a-4)(3a+4)
9. (7b-9)(7b+9)
10. 5(n-9)(n+9)
11. (9x-5)(9x+5)
12. (8p-3)(8p+3)
13. (10m-9)(10m+9)
14. Not factorable
15. 4(3x-y)(3x+y)
16. 2(u-5v)(u+5v)
17. (4x-y)(4x+y)
18. (3u-4v)(3u+4v)
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YCS ALGEBRA I: Unit 4A: EXPRESSIONS AND EQUATIONS PART I 2013-2014 7
T-L #14 Rules for Factoring
Take out the greatest common factor
Look for the difference of two squares a2 – b2 = (a - b)(a + b)
Factoring trinomials: Put polynomials in descending order
x2 + bx + c the choices are:
( + )( + ) or ( - )( - )
x2 + bx – c the choices are
( + )( - )
2
x – bx – c the choices are
( - )( + )
Factoring the sum and difference of two cubes:
(x3 – y3) = (x – y)(x2 + xy + y2)
(x3 + y3)= (x + y)(x2 - xy + y2)
Factor by grouping:
Group two terms together and take out GCF and then take out GCF of those two terms.
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YCS ALGEBRA I: Unit 4A: EXPRESSIONS AND EQUATIONS PART I 2013-2014 8
T-L #16: FACTORING GAME
Cut out the squares, mix them up, and have the students match the polynomial with its
factored form.
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YCS ALGEBRA I: Unit 4A: EXPRESSIONS AND EQUATIONS PART I 2013-2014 9
AUTHENTIC ASSESSMENT: ERROR ANALYSIS
FORM 1: Find the errors, if any, in the following problems; explain why it is incorrect:
1. Find the GCF for 108x3y2 and 56x2y4.
108 = 2*54 = 2*2*27 = 2*2*3*9 = 2*2*3*3*3 = 22*33 so 108x3y2 = 22*33 x3 y2
56 = 2*28 = 2*2*14 = 2*2*2*7 = 23 * 7 so 56x2y4 = 23 * 7 x2 y4
The GCF = 23 * 33 * 7 x3 * y4 = 504 x3 * y4
2. Multiply: (2x – 5) (3x2 – 4x + 1) = 6x3 – 8x2 + 2x – 15x2 – 20x – 5 = 6x3 – 23x2 – 18x – 5
3. Factor: 16x2 – y2 = (4x – y)2
4. Factor: 8ax – 4a – 6x + 3 = 4a(2x – 1) + 3(2x – 1) = (4a + 3)(2x – 1)
FORM 2: Find the errors if any in the following problems and explain why it is incorrect:
1. Find the GCF for 80x4y3 and 170x3y4.
80 = 2*40 = 2*2*20 = 2*2*4*5 = 2*2*2*2*5 = 24*5 so 80x4y3 = 24*5 x4y3
170 = 2*85 = 2*5*17 so 170x3y4 = 2*5*17 x3y4
The GCF = 2 * 5 * x4 * y4 = 10 x4 * y4
2. Multiply: (2x – 5)2 = 4x2 – 25
3. Factor: 8x3 – y3 = (2x – y)3
4. Factor: 12x2 + 13x – 4 = (2x – 1)(6x + 4)
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FORM 3: Find the errors if any in the following problems and explain why it is incorrect:
1. Find the GCF for 72x4y5 and 48x3y.
72 = 2*36 = 2*2*18 = 2*2*2*9 = 2*2*2*3*3 = 23*32 so 72x4y5 = 23*32 x4y5
48 = 2*24 = 2*2*12 = 2*2*2*6 = 2*2*2*2*3 so 48x3y4 = 24*3 x3y
The GCF = 24 * 32 * x3 * y = 144 x3 * y
2. Multiply: (3x – 2)3 = 27x3 – 8
3. Factor: 25x2 – 30x + 9y2 = (5x – 3y)(5x + 3y)
4. Factor: 6x2 – 7x – 20 = (2x + 5)(3x – 4)
FORM 4: Find the errors if any in the following problems and explain why it is incorrect:
1. Find the GCF for 66xy5 and 90x3y3.
66 = 2*33 = 2*3*11 so 66xy5 = 2*3*11 xy5
90 = 2*45 = 2*5*9 = 2*3*3*5 = 2*32*5 so 90x3y3 = 2*32*5 x3y3
The GCF = 2 * 32 * 5 * 11 x3 * y5 = 990 x3 * y5
2. Multiply: (3x – 2)(9x2 + 6x + 4) = 27x3 – 8
3. Factor: 9x3 – 30x2 + 9x = 3(3x + 1)( x + 3)
4. Factor: 3ab – b – 4 + 12a = (3a – 1)(b – 4)
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Answers and points for Authentic Assessment Form 1:
Answer
1. The LCM was found. The GCF is 22 x2 y2. Another explanation might be
to take what is common to both monomials which would be 22 x2 y2.
2. The error is in the -20x. The –20x should be +20x making the linear
term +22x
3. The error was in the method of factoring. The difference of two squares
does not factor into a quantity squared. 16x2 – y2 = (4x – y)(4x + y)
4. The error is in the positive sign in front of the 3 in the first factoring step.
When factoring -6x +3, it is -3(2x – 1) so the answer is (4a - 3)( 2x-1)
Answers and points for Authentic Assessment Form 2:
Answer
1. The variable part of the GCF is incorrect. Should be 10 x3 y3.
Points
Give one point for finding error and
one point for the explanation
Give one point for finding error and
one point for the explanation
Give one point for finding error and
one point for the explanation
Give one point for finding error and
one point for the explanation
2. 2x – 5 was squared incorrectly. When squaring a quantity, you multiply it
by itself and FOIL. So (2x- 5 )2 = 4x2- 20x + 25
3. The error was in the method of factoring. The difference of two cubes is
not a quantity cubed. It is (2x – y)(4x4 + 2xy + y2).
4. The error is in the factors of 12. When factoring 12x2 +13x – 4, the
factors are 4x – 1 and 3x + 4
Points
Give one point for finding error and
one point for the explanation
Give one point for finding error and
one point for the explanation
Give one point for finding error and
one point for the explanation
Give one point for finding error and
one point for the explanation
Answers and points for Authentic Assessment Form 3:
Answer
1. The number part of the GCF is incorrect. The LCM of the numbers was
found. The GCF is 23 * 3 = 24x3y
2. When cubing a quantity, you do not cube each term. You multiply the
binomial by itself 3 times giving 27x3 – 54x2 + 36x - 8
3. The error was in the sign in front of the 3 when factoring. Both signs in
front of the 3’s are negative to give a positive 9x2
4. The error is in the signs of the factors. The signs are reversed,
(2x – 5)(3x + 4)
Points
Give one point for finding error and
one point for the explanation
Give one point for finding error and
one point for the explanation
Give one point for finding error and
one point for the explanation
Give one point for finding error and
one point for the explanation
Answers and points for Authentic Assessment Form 4:
Answer
1. The LCM was found. The GCF is 2 * 3 *x y3. Another explanation might
be to take what is common to both monomials which would be 2 * 3 x y3.
2. The is no error. It is multiplied correctly.
3. There are two errors. There is a common factor of x to factor out with the
3 and the signs in both factors are negative to give a negative 30 x2 as
the middle term.
4. The error is in the negative sign in the second factor. When factoring,
rearrange the terms so that it is 3ab – b + 12a – 4. The 12a is positive
and the 4 is negative so when factoring a positive 4 out you get 3a – 1
whichis common and what is left is b + 4.
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Points
Give one point for finding error and one
point for the explanation
Give two points for stating it is correct
Give one point for finding error and one
point for the explanation
Give one point for finding error and one
point for the explanation
YCS ALGEBRA I: Unit 4A: EXPRESSIONS AND EQUATIONS PART I 2013-2014
12
AUTHENTIC ASSESSMENT #3: (A.SSE.1 a, b, MP.2, MP.4, L-2)
Students are given certain qualifications and are to write 2 equations and explain.
Have students write two expressions that meet the following criteria:
There are two factors. One factor is a trinomial with degree 2 and has a constant term of 10. The other
factor is a binomial with a term that has a coefficient of 4 and a negative constant term.
Note to teacher: to grade this, for each answer give one point if there are two factors in parentheses, one point for a
factor that is a trinomial, one point if the variable is squared in the trinomial factor, and one point if the constant term
is 10 in the trinomial factor. Give one point if the second factor is a binomial, one point if 4 is in front of the variable in
the binomial factor and one point if the constant term is negative in the binomial factor.
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