Download Unit 1F 2013-14 - Youngstown City Schools

YOUNGSTOWN CITY SCHOOLS
MATH: ALGEBRA II
UNIT 1F: POLYNOMIAL, RATIONAL, AND RADICAL RELATIONSHIPS PART V (3 WEEKS) 2013-2014
Synopsis: It is critical that students learn to simplify rational expressions and solve rational and radical equations if
they wish to continue their study of mathematics. These concepts will be used extensively in their Precalculus and
Calculus courses, so this unit is designed to give them an excellent background in these concepts. The students will
see a definite connection between rational numbers and rational expressions, which will aid them in simplifying the
expressions. Also, they will solve rational and radical equations using algebraic methods and then extend that to
solving them by graphing with technology.
STANDARDS
A.APR.7 (+) Understand that rational expressions form a system analogous to the rational numbers, closed under addition,
subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions.
A.REI.2 Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may
arise.
A.REI.11 Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the
solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables
of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute
value, exponential, and logarithmic functions.
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.1
L.2
L.4
L.5
L.6
Learn to read mathematical text (including textbooks, articles, problems, problem explanations)
Communicate using correct mathematical terminology
Listen to and critique peer explanations of reasoning
Justify orally and in writing mathematical reasoning
Represent and interpret data with and without technology
MOTIVATION
TEACHER NOTES
1. Teacher poses the following problem to the class: You are going to prom, but you need to wash your
car before your date arrives. You have not started washing the car, and your date will show up in 16
minutes. You can wash the car in 40 minutes. Your sister claims that she can do it in 30 minutes. If
you work together, how long will it take to do the job? Will this give you enough time before your date
arrives? (A.REI.2, MP.1, MP.2, MP.4, L.1, L.2, L.5)
Solution: We want to find out how long it will take to finish the job if you and your sister work together. We
don’t know that time, so let’s define that as our variable: Let x = time to finish working together in 1 hour.
You do
of the job. Your sister does
of the job. Together you do
So amount done together = amount you do + amount sister does:
of the job.
3x + 4x = 120
7x = 120
X = 17.14 It will take 17.14 minutes to wash the car, so you won’t make it.
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MOTIVATION
TEACHER NOTES
2. Have students set academic and personal goals for the unit.
3. Preview the authentic assessment for students so they know what will be expected at the end of the
unit.
TEACHING-LEARNING
TEACHER NOTES
Vocabulary:
expressions
closure
polynomial
rational numbers
LCD
exponential
absolute value
irrational numbers
radical
linear
complex fractions extraneous solutions
(General Notes to teachers:
a. This is an excellent resource for quizzes, extra examples and etc.
http://www.glencoe.com/sec/math/algebra/algebra2/algebra2_05/)
rational equations
b. Teachers can post in the room examples of adding, subtracting, multiplying, and dividing
rational expressions at their discretion.
c. Engage students in a quick review on factoring out negatives before starting the unit.
Examples might include:
 Factor a negative one out of 3 – 2x and rearrange the terms. (Ans. – (2x–3))
 Factor a negative one out of 5 – 3x – 4x2 and rearrange the terms.
(Ans. – (4x2 + 3x – 5)
 Factor a negative one out of -2x2 + 8x – 3. (Ans. – (2x2 – 8x +3)
_______________________________________________________________________
1. Discuss closure with respect to addition, subtraction, multiplication, and division of rational
numbers. Review adding, subtracting, multiplying, and dividing rational numbers. Students
need a strong background on adding, subtracting, multiplying, and dividing fractions with numbers
to enhance the understanding when transitioning to rational expressions. Review worksheets are
listed below.
 http://www.mathaids.com/cgi/pdf_viewer_3.cgi?script_name=fractions_adding.pl&difficult=4&probs=15&lang
uage=0&memo=&answer=1&x=110&y=16
 http://www.mathaids.com/cgi/pdf_viewer_3.cgi?script_name=fractions_subtracting.pl&difficult=4&probs=15&
language=0&memo=&answer=1&x=115&y=14
 http://www.mathaids.com/cgi/pdf_viewer_3.cgi?script_name=fractions_multiply.pl&difficult=1&probs=15&lan
guage=0&memo=&answer=1&x=85&y=19
 http://www.mathaids.com/cgi/pdf_viewer_3.cgi?script_name=fractions_divide.pl&difficult=1&probs=15&lang
uage=0&memo=&answer=1&x=164&y=19
(A.APR.7, MP.7, MP.8, L.2)
2. Discuss with students the importance of working with rational expressions such as solving the
work problem from the motivation section; solving rate, distance, time problems; and solving
electrical engineering problems. After the students understand the importance of rational
expressions, work with them on multiplying and dividing rational expressions, make sure to
include working with complex fractions. Use chapter 9 resources for the textbook. Students are
to take notes and practice on dry erase boards. (A.APR.7, MP.1, MP.2, MP.4, MP.7, L.1, L.2)
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TEACHING-LEARNING
TEACHER NOTES
3. Review the process of finding the LCD
a. Factor each denominator and write the factors to the highest power
b. To write the LCD, take each factor once and if it is to a power, take it to the highest power.
This is your LCD. For polynomials, usually one does not multiply these factors together,
so leave the LCD in factored form. Use the following examples. Discuss finding the LCD
and then adding them:
c.
(Ans.
, a, b, c ≠0);
(Ans:
then work with students
,x≠
) and
on adding and subtracting rational expressions. Use chapter 9 resources. Students are
to take notes and practice on dry erase boards. (A.APR.7, MP.2, MP.7, MP.8, L.2)
4. Teacher facilitates students comparing addition, subtraction, multiplication, and division with
respect to rational numbers and rational expressions. Have students do following problems:
i.
compared to
(Ans.
ii.
compared to
(Ans. 8, 4(x+2))
iii.
compared to
iv.
compared to
v.
+
compared to
vi.
+
compared to
vii.
(Ans.
)
)
(Ans. ½ , 3)
+
(Ans.
+
)
(Ans.
(Ans. ½,
compared to
viii.
)
)
(Ans.
)
(A.APR.7, MP.2, MP.7, MP.8, .-2)
5. Discuss solving rational equations. Have students solve simple rational equations by
multiplying both sides by the LCD.
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TEACHING-LEARNING
i.
ii.
+
TEACHER NOTES
= -2 (Ans.
+
=
iii.
)
(Ans.
)
(Ans. -8)
After the students have mastered these, introduce them to more difficult problems - - website:
http://www.kutasoftware.com/FreeWorksheets/Alg2Worksheets/Solving%20Rational%20Equation
s.pdf or skills practice from chapter 9-6, equations only. Make sure to include discussion and
solving problems with extraneous solutions. (A.REI.2, MP.2, MP.7, MP.8, L.2)
DISTRICT ASSESSMENT: Unit 1F part 1
6. Present to students the following questions in order for students to discover rate*time=distance,
distance/rate=time and distance/time=rate:
i.
If you were traveling in a car going 50 MPH and you drove for two hours, how far
did you go?
ii.
If you drove 300 miles in 5 hours, how fast were you driving?
iii.
If you run 5 miles in half an hour, what was your speed in miles/minute?
iv.
The Raging Rapids ride at Kennywood is 2000 feet long and it goes 400 feet per
minute. How long does your friend have to wait for you to come off the ride?
After students have a good grasp of this concept, continue with the following worksheet dealing
with motion problems:
http://images.schoolinsites.com/SiSFiles/Schools/NC/OnslowCounty/NorthwoodsMiddle/Uploads/
DocumentsCategories/Documents/Applied%20(Word)%20Problems%20NoteSheets%20%20Wor
kSheets%20(Distance%20--%20Rate%20--%20Time).pdf
(A.REI.2, MP.1, MP.2, MP.4, MP.7, L.1, L.2)
7. Discuss with students work problems using the following examples:
"Work" problems involve situations such as two people working together to paint a house.
You are usually told how long each person takes to paint a similarly-sized house, and you
are asked how long it will take the two of them to paint the house when they work together.
Many of these problems are not terribly realistic (since when do two laser printers work
together on printing one report?), but it's the technique that they want you to learn, not the
applicability to "real life."
The method of solution for work problems is not obvious, so don't feel bad if you're totally
lost at the moment. There is a "trick" to doing work problems: you have to think of the
problem in terms of how much each person / machine / whatever does in a given unit of
time. For instance:
 Suppose one painter can paint the entire house in twelve hours, and the
second painter takes eight hours. How long would it take the two painters
together to paint the house?
If the first painter can do the entire job in twelve hours and the second painter can
do it in eight hours, then (this here is the trick!) the first guy can do 1/12 of the job
per hour, and the second guy can do 1/8 per hour. How much then can they do per
hour if they work together?
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TEACHING-LEARNING
TEACHER NOTES
To find out how much they can do together per hour, I add together what they can
do individually per hour: 1/12 + 1/8 = 5/24. They can do 5/24 of the job per hour. Now
I'll let "t" stand for how long they take to do the job together. Then they can do 1/t
per hour, so 5/24 = 1/t. Flip the equation, and you get that t = 24/5 = 4.8 hours. That
is:
hours to complete job: completed per hour: adding their labor: t © Elizabeth l 19992011 All Rights Reserved
first painter: 12
first painter: 1/12
1
/12 + 1/8 = 1/t
1
second painter: 8
second painter: /8
5
/24 = 1/t
together: t
together: 1/t
24
/5 = t
They can complete the job together in just under five hours.
As you can see in the above example, "work" problems commonly create
rational equations. But the equations themselves are usually pretty simple.
 Two mechanics were working on your car. One can complete the given job
in six hours, but the new guy takes eight hours. They worked together for the
first two hours, but then the first guy left to help another mechanic on a
different job. How long will it take the new guy to finish your car?
The first guy can do 1/6 per hour. The new guy can do 1/8 per hour. Together, they
can do 1/6 + 1/8 = 7/24 per hour. That is:
hours to
complete job:
first guy: 6
new guy: 8
together: t
completed per
hour:
first guy: 1/6
new guy: 1/8
together: 1/t
adding their labor:
1
7
/6 + 1/8 = 1/t
/24 = 1/t
They worked for two hours, so they got 2( 7/24 ) = 7/12 of the job done. That leaves
5
/12 of the job to do.
(I calculated this last bit by using the fact that (fraction of work) = (rate per
unit) × (number of units). That is, I multiplied how much they can do per hour
by the number of hours they worked, to find the fraction of the entire job that
they had completed.)
The new guy can do 1/8 of the job per hour. So the real problem here is figuring out
how long the new guy will take to do 5/12 of the job. Let "h" indicate the number of
hours he needs. Then the equation is ( 1/8 of the job per hour) times (however
many hours) equals ( 5/12 of the job done). That is, I need to set up the equation so
the units cancel:
( 1/8 job / hour) × (h hours) = 5/12 job
h
/8 job = 5/12 job
h
/8 = 5/12
h = ( 5/12) × ( 8/1) = 10/3 = 3 1/3
It takes the new guy another 3 hours and twenty minutes to finish fixing your
car.
By the way, this means that they took a total of [two hours with two guys] plus
[three hours and twenty minutes with one guy] equals [seven-and-a-third manhours] to fix your car.
Reinforce with additional problems on the following worksheet: attached on pages 9-10
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TEACHING-LEARNING
(A.REI.2, MP.1, MP.2, MP.4, MP.7, L.1, L.2)
TEACHER NOTES
8. Have students create their own rational story problems and present them to the class with
solutions for student critiquing. (A.REI.2, MP.1, MP.2, MP.4, L.1, L.2, L.4)
9. Discuss solving radical equations. Have students solve simple radical equations.
i.
(Ans. x = 49)
ii.
(Ans. x = 125)
iii.
(Ans. No solution)
iv.
(Ans. x = 9)
After this, introduce students to more difficult problems; use skills practice section 5-8, omitting
the inequalities. (Note to teachers: Make sure to include cube roots, fourth roots, etc.) (A.REI.2,
MP.2, MP.4, MP.7, L.2)
10. Discuss extraneous solutions using “Reading to Learn Mathematics” section 5-8 in Algebra
II resources. (A.REI.2, MP.7, L.1, L.2, L.4)
11. Another method of solving equations is by setting each side of the equation equal to y or f(x),
graphing and finding the intersection. Use the worksheet attached on pages 11-14 at the end of
the unit to solve problems such as these. If you do not have graphing calculators use the
following web site that finds intersections of two graphs:
http://go.hrw.com/math/midma/gradecontent/manipulatives/GraphCalc/graphCalc.html
(A.REI.11, MP.2, MP.4, MP.5, MP.6, L.2, L.6)
TRADITIONAL ASSESSMENT
1. Paper-pencil test with M-C questions
TEACHER NOTES
TEACHER CLASSROOM ASSESSMENT
1. 2-and 4-point questions
2. Class assignments
TEACHER NOTES
AUTHENTIC ASSESSMENT
TEACHER NOTES
1. Students will create two functions and find their intersection; they explain the significance
of the point of intersection in relation to the solution found when setting the two functions
equal to each other. (attached on pages 7-8) (A.REI.11, MP.1, MP.2, MP.4, MP.5, MP.7,
L-2, L-5)
2. Evaluate goals for the unit.
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AUTHENTIC ASSESSMENT
STANDARDS: A. REI.11
Create two equations (y=f(x) and y = g(x)) using a combination of linear, polynomial, rational, radical, absolute value,
and exponential functions for each equation. Graph them and find the intersection of the equations. Explain what
this intersection point means when considering the solution for the equation f(x) = g(x).
1.)
Solutions:
2.)
Solutions:
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AUTHENTIC ASSESSMENT RUBRIC
ELEMENTS OF
THE PROJECT
Create two
equations
0
1
Did not
attempt
Both equations
contained only one type
of function.
Graph the
equations
Find the
intersection of the
equations
Did not
attempt
Did not
attempt
Explanation of the
point of
intersection
Did not
attempt
Graphed only one
equation correctly
Found the intersection
incorrectly (both the x
and y coordinates are
incorrect)
Vague explanation with
no mention that the x
coordinate is the
solution to f(x) = g(x)
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2
One equation
contained only one
type of function, the
second equation
contained more than
one type of function.
NA
Found the intersection
incorrectly ( one
coordinate is correct,
the is incorrect)
3
Both equations
contained more than
one type of function.
Graphed both
equations correctly
Found the
intersection correctly
Good explanation
which includes the
fact that the x
coordinate is the
solution to f(x) = g(x)
YCS ALGEBRA II: Unit 1F: Polynomial, Rational, and Radical Relationships Part V 2013-2014
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T-L #7
Kuta Software -
Name___________________________________
Period____
Date________________
WORK WORD PROBLEMS
Solve each question. Round your answer to the nearest hundredth.
1) Working alone, Ryan can dig a 10 ft by 10 ft hole in five hours. Castel can dig the same hole in six hours. How
long would it take them if they worked together?
2) Shawna can pour a large concrete driveway in six hours. Dan can pour the same driveway in seven hours. Find
how long it would take them if they worked together.
3) It takes Trevon ten hours to clean an attic. Cody can clean the same attic in seven hours. Find how long it would
take them if they worked together.
4) Working alone, Carlos can oil the lanes in a bowling alley in five hours. Jenny can oil the same lanes in nine hours.
If they worked together how long would it take them?
5) Working together, Paul and Daniel can pick forty bushels of apples in 4.95 hours. Had he done it alone it would
have taken Daniel 9 hours. Find how long it would take Paul to do it alone.
6) Working together, Jenny and Natalie can mop a warehouse in 5.14 hours. Had she done it alone it would have
taken Natalie 12 hours. How long would it take Jenny to do it alone?
7) Rob can tar a roof in nine hours. One day his friend Kayla helped him and it only took 4.74 hours. How long would
it take Kayla to do it alone?
8) Working alone, it takes Kristin 11 hours to harvest a field. Kayla can harvest the same field in 16 hours. Find how
long it would take them if they worked together.
9) Krystal can wax a floor in 16 minutes. One day her friend Perry helped her and it only took 5.76 minutes. How long
would it take Perry to do it alone?
10) Working alone, Dan can sweep a porch in 15 minutes. Alberto can sweep the same porch in 11 minutes. If they
worked together how long would it take them?
11) Ryan can paint a fence in ten hours. Asanji can paint the same fence in eight hours. If they worked together how
long would it take them?
12) Working alone, it takes Asanji eight hours to dig a 10 ft by 10 ft hole. Brenda can dig the same hole in nine hours.
How long would it take them if they worked together?
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Answers: Work Word Problems
1) 2.73 hours
2) 3.23 hours
3) 4.12 hours
4) 3.21 hours
5) 11 hours
6) 8.99 hours
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T-L #11 WORKSHEET: Solving Equations by Graphing
Solve each equation by graphing the function on the left and the function on the right, then find the
intersections. Use some type of technology to graph these.
http://go.hrw.com/math/midma/gradecontent/manipulatives/GraphCalc/graphCalc.html
1.)
2.)
3.)
4.)
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6.)
7.)
8.)
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9.)
10.)
11.)
12.)
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Answers:
1. x = -6
2. x = 9.6
3. x = -3, 4
4. x = 0, 0.6458
5. x = -0.9585, -9.0415
6. x = 1, 2
7. x = 1
8. x = -0.26
9. x = -1.5, 3.5
10. x = 0, 4
11. x = 0
12. x = 0, 2.5813
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