YOUNGSTOWN CITY SCHOOLS
MATH: ALGEBRA 1
UNIT 4B - - EXPRESSIONS AND EQUATIONS – PART II (4 WEEKS) 2013-2014
Synopsis: In this unit, students will work with various forms of quadratic expressions, functions and equations.
They will be factoring, completing the square, and using the quadratic formula to solve equations, graph
functions, find maximums or minimums, zeros, and axis of symmetry. To expand on previous knowledge, they
will be creating quadratic and exponential equations and inequalities and solving them given real-life problems
and the Angry Bird game. In addition, they will encounter solving a simple system of equations consisting of
linear and quadratic equations.
STANDARDS
A.REI.4 Solve quadratic equations in one variable.
a. Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)2 = q
that has the same solutions. Derive the quadratic formula from this form.
b. Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic
formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives
complex solutions and write them as a ± bi for real numbers a and b.
A.REI. 4a, 4b: Students should learn of the existence of the complex number system, but will not solve quadratics with complex
solutions until Algebra II.
A.REI.7 Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and
graphically. For example, find the points of intersection between the line y = –3x and the circle x2 + y2 = 3.
A.REI.7: Include systems consisting of one linear and one quadratic equation. Include systems that lead to work with fractions.
For example, finding the intersections between x2+y2=1 and y = (x+1)/2 leads to the point (3/5, 4/5) on the unit circle,
corresponding to the Pythagorean triple 32+42=52.
A.SSE.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented
by the expression.  The star symbol * indicates mathematical modeling
a. Factor a quadratic expression to reveal the zeros of the
function it defines.
b. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.
c. 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%.
A.SSE.3
It is important to balance conceptual understanding and procedural fluency in work with equivalent expressions. For example,
development of skill in factoring and completing the square goes hand-in-hand with understanding what different forms of a
quadratic expression reveal.
A.CED.1, 2, 4
Extend work on linear and exponential equations in Unit 1 to quadratic equations. Extend A.CED.4 to formulas involving squared
variables.
A.CED.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear
and quadratic functions, and simple rational and exponential functions.
A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate
axes with labels and scales.
A.CED.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example,
rearrange Ohm’s law V = IR to highlight resistance R.
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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 Learn to read mathematical text (including textbooks, articles, problems, problem explanations)
L.2 Communicate using correct mathematical terminology
MOTIVATION
TEACHER NOTES
1. Westerville South High School Video: “Quad-Solve” video:
http://www.youtube.com/watch?v=jGJrH49Z2ZA
2. Use Angry Birds activity to introduce to students what they will be doing at the end of the unit. Show
them the graphs and explain that they will know how to do this problem by the end of the unit, and this
will be the Authentic Assessment; website below has Angry Birds
http://www.teachmaths-inthinking.co.uk/activities/angry-birds.htm
3. Preview expectations for the end of the Unit
4. Have students set both personal academic goals for this Unit.
TEACHING-LEARNING
TEACHER NOTES
NOTE: Teachers, if you are
making up problems in this
section, stay away from
imaginaries
VOCABULARY
foil
factoring
monomial
binomial
trinomial
polynomial
quadratic
root / zero
intersection
quadratic formula
real numbers
complex numbers
completing the square
zero product factor property
principle square root
Solving Quadratic Equations:
1. Factoring: Teacher reviews linear equations and solving for x. Now we will be solving equations for x2.
Introduce with problem they need to factor; use an expression and then throw in that it equals zero and
ask what the answers will be. Discuss how to solve and introduce the zero product factor property (use
Kuta worksheet at the following website:
http://www.kutasoftware.com/FreeWorksheets/Alg1Worksheets/Solving%20Quadratic%20Factoring.pdf).
Textbook page 497; real-world problems on 498-500. Discuss with students what to do if the equation is
not factorable, such as x2 = 12. Where x2 – 12 = 0, you cannot factor, so what can you do? (NOTE to
teacher: this is explained in #2, so briefly discuss here). Discuss number of possible solutions (0, 1, 2)
Give students a perfect trinomial square which has one solution such as (x2 +6x + 9) = 0 and an equation
that has no real solutions such as x2 + 16 = 0. At this time, briefly discuss the existence of imaginary
numbers and tell them they will learn to solve these equations in Algebra II. (A.REI.4, A.CED.1,
A.CED.4) A.SSE.3a, MP.1, MP.2, MP.4, MP.7, MP.8, L.1, L.2)
2. NOTE: teacher, do simplifying square roots before you move to the next step. Ask students what they
do if the number is not a perfect square? Ask them to work with numbers and factor with prime factors
and do factor trees to simplify radicals. Keep problems basic such as
,
,
,
,
, and
. Include variables with these - - e.g.,
,
, or
. Discuss the difference between
rational approximation and the exact solution. (Kuta worksheet:
http://www.kutasoftware.com/FreeWorksheets/Alg1Worksheets/Simplifying%20Radicals.pdf
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TEACHING-LEARNING
TEACHER NOTES
3. Take the Square Root of each side: discuss with students if x2 = 16, then what are the values for x?
(Caution: some students will say 8; so review why this is incorrect). Solve simple equations of the type
2m2 +10 = 210. Use other equations that are multi-step and have the squared variable on both sides of
the equation. (Kuta worksheet:
http://www.kutasoftware.com/FreeWorksheets/Alg1Worksheets/Solving%20Quadratic%20Roots.pdf
Solving Quadratic Equations with Square Roots - with calculator approximations and as an exact
solution) Note to teacher: square roots of negative numbers have no solution at this time, however, it
should be discussed that there is a new number system, the complex numbers that will contain solutions
to these equations and will be taught in Algebra II. (A.REI.4b, MP.2, MP.4, MP.5, MP.6, MP.8, L-2)
4. Completing the Square: Discuss the standard quadratic form: ax2 + bx +c = 0. This is the form needed
before starting the process of completing the square. Teachers can use Algebra Tiles (regentsprep.org
has explanation) to illustrate this or Textbook, pages 539 – 543 (old book); New Book: Chapter 9 –
Section 4: p. 574, or the material attached to unit on pages 7-11 (note: this attachment was developed
in a non-English speaking cou8ntry, and the grammar is not correct, so use as a reference, and perhaps
use the problems with some editing. The following link explains the process of completing the square
step by step: http://algebralab.org/lessons/lesson.aspx?file=Algebra_completingthesquare.xml
Use Kuta worksheet; attached on pages 12-15 with answer key (A.SSE.3b, A.REI.4a, MP.1, MP.2,
MP.4, MP.7, MP.8, L.1, L.2)
5. Quadratic Formula: give students the formula and show a video on how to derive the quadratic formula
http://www.youtube.com/watch?v=eYzUXfi8EEw Give students formula and arrive at ax2 + bx + c = 0.
(put standard formula and quadratic formula next to each other and ask students to derive this; see chart
on page 16) Textbook, page 550-551 and skills practice 599-600. Kuta worksheet
http://www.kutasoftware.com/FreeWorksheets/Alg1Worksheets/Quadratic%20Formula.pdf or math.comalgebra I-algebra worksheet generator
(A.REI.4, a & b, A.CED.1, MP.1, MP.2, MP.3, MP.4, MP.7, L.1, L.2)
GIVE STUDENTS A TEACHER GENERATED ASSESSMENT AT THIS TIME
Graphing Quadratic Functions
6. Give students f(x) = 2x2 and make a table using -2, -1, 0, 1, 2 and then graph the function. Discuss the
shape of the curve in relation to the equation: for instance all y values are positive; there are two x values
for each y except for the vertex point (for instance y = 8, x is 2 and -2). Look for axis of symmetry, does it
open up or down, and which point is the minimum or maximum. Find the zeros of the function.
(A.SSE.3a, b, MP.1, MP.2, MP.4, MP.5, MP.8, L.1, L.2)
7. Give students f(x) = -x2 +2x and make a table using -2, -1, 0, 1, 2 and then graph the function. Discuss
the possibility or plotting more points and the shape of the curve in relation to the equation: there are two
x values for each y, except for the vertex point (for instance: y = 0, x is 0 and 2), look for axis of
symmetry, does it open up or down and which point is the minimum or maximum, find the zeros of the
function. (A.SSE.3a, b, MP.1, MP.2, MP.4, MP.5, MP.8, L.1, L.2)
8. Give students f(x) = -x2 +8x -15 and find the zeros by factoring as your first step, discuss symmetrical
property for parabolas and find the third point, maximum or minimum (teacher’s choice - - can then graph
two more points), and then graph the function. Discuss the shape of the curve in relation to the equation - for instance, the only properties visible in this form are the fact that it opens up, which implies that there
is a minimum and the y intercept is 15. (A.SSE.3a, b, MP.1, MP.2, MP.4, MP.5, MP.8, L.1, L.2)
9. Use equation f(x) = -x2 + 8x - 15 and change to vertex form - - f(x) = a(x-h)2 +k by completing the square
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YCS ALGEBRA I: Unit 4B: EXPRESSIONS AND EQUATIONS PART II 2013-2014 3
TEACHING-LEARNING
TEACHER NOTES
as shown below.
a) F(x) = x2 + 8x - 15
b) –F(x) = x2 – 8x + 15
c) -F(x) -15 = x2 − 8x
d) -F(x) – 15 + 16 = x2 − 8x + 16
e) -F(x) + 1 = (x − 4)2
f) F(x) = -(x − 4)2 + 1 so vertex is (4, 1), axis of symmetry is x = 4 and it opens down
Discuss the ease at finding the vertex, axis of symmetry, opening up or down when the equation is in this
form. Website helpful for teacher: http://www.illustrativemathemataics.org/illustrations/434 There are
word problems here as well. (A.SSE.3a, b, A.CED.1, A.CED.2, MP.1, MP.2, MP.4, MP.5, MP.8, L.1, L.2)
10. Reinforce the importance of the three forms of a quadratic equation, the properties of each of them
reveal, and when it is advantageous to use each (Attached is a worksheet on page 18-19) Reference
them back to the Angry Birds in the motivation that they will be doing for Authentic Assessment. Open
the link to the Angry Birds game, print out the quadratic links activities 1-3 and have students work in
groups to match the cards with the graphs. Leave the link and give students some samples to work
backwards to find the equations, such as:
a. write the equation of the parabola that passes through (0,0) and (6,0) and has a maximum height of
18 units.
b. write the equation of the parabola that passes through (0,0), (10,0) and (9,9). After the students are
able to write the equations of the parabola, go back to the Angry Birds link and have them play
levels one, two and three. Students should be using the given information to calculate the equation
and then check it by entering it into the game. They should not be guessing.
http://www.mathsisfun.com/algebra/quadratic-equation.html review of quadratic functions
http://www.mathsisfun.com/algebra/quadratic-equation-graph.html applet that lets students change a,
b, and c and observe the effects of each (A.SSE.3a, b, A.CED.1, A.CED.2, MP.1, MP.2, MP.4,
MP.5, MP.8, L.1, L.2)
GIVE STUDENTS A TEACHER GENERATED ASSESSMENT
11. After completing graphing and solving quadratic equations, discuss with students the procedure used to
solve quadratic inequalities in one variable. The following links may be helpful:
http://www.youtube.com/watch?v=U8QRzWUloeU and
http://www.regentsprep.org/Regents/math/algtrig/ATE6/Quadinequal.htm omitting example #2
Attached on page 20 is a worksheet on solving quadratic inequalities (A.CED.1, A.REI.4b, MP.2,
MP.4, MP.5, MP.6, MP.7, MP.8, L.2)
12. In order to examine and transform exponential functions to reveal certain information, review the
formulas for exponential growth: y(t) = C(1+r) t, exponential decay: y(t) = C(1−r) t, and compound
interest: A(t) =
. Then have students work in groups to answer the following questions:
a) When $2500 is invested in an account that is compounded quarterly. The value of the account
can be represented as A(t) =
. Find the interest rate by rewriting the function.
(Answer. 6%)
b) You purchased a car and the minute you drive it off the lot, it depreciates in value. The value of
the car after t years can be represented as V(t) = 18,500(0.89) t, Find the initial value of the car
and the rate of depreciation by rewriting the function. (Answer: initial value $18,500, rate of
depreciation 11%)
c) In 2000 the number of women participating in sports is represented as W(t) = 3,400,000(1.085)t.
How many women participated in 2000 and what is the rate of increase? (Answer: 3,400,000
women participated in 2000 and the rate is 8.5%.)
(A.SSE.3c, MP.1, MP.2, MP.4, MP.6, MP.7, MP.8, L-1, L-2)
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TEACHING-LEARNING
TEACHER NOTES
13. Have students research formulas from science, social studies, geometry that involve one of the
parameters being squared, such as A=
, collect the formulas and create a worksheet from
them asking students to solve for one of the other parameters. For instance in the area of a sector of a
circle formula: A=
L.7)
, have them solve for “r”. (A.CED.4, A.REI.4 a & b, MP.2, MP.4, L.1, L.2,
14. Linear-quadratic systems are to be solved graphically and algebraically. To solve graphically with the TI
calculator see page 553 in the textbook. Before solving graphically, discuss the number of possible
solutions to the system (none, one, two). The following link does a nice job explaining number of
possible solutions: http://www.regentsprep.org/Regents/math/ALGEBRA/AE6/LLinQuadT.htm
To solve graphically by hand and algebraically, refer to the worksheets on pages 21-22 of this unit.
The following link presents several problems to solve graphically and algebraically:
http://www.regentsprep.org/Regents/math/ALGEBRA/AE6/PLinQuad.htm
(A.REI.7, A.CED.2, MP.1, MP.2, MP.4, MP.5, MP.6, MP.7, L.1, L.2)
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. ANGRY BIRDS AUTHENTIC ASSESSMENT: (A.SSE.3.B, A.CED.4, A.CED.4, MP.1, MP.2, MP.4,
MP.5, MP.6, MP.7, MP.8, L.2)
a. Choose one of the following links:
http://geogebratube.org/material/show/id/39916
http://geogebratube.org/material/show/id/39917
http://geogebratube.org/material/show/id/39918
http://geogebratube.org/material/show/id/39919
b. Plot the three given points on graph paper (the statrting point which is on the y axis, a point X on the
graph, and the pig).
c. Write your equation in the form f(x) = ax2 + bx + c showing all work (solving 2 equations with 2
variables)
d. Put equation in f(x) = a(x-h)2 + k showing work
e. State the vertex
f. State the axis of symmetry
Rubric on page 6
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YCS ALGEBRA I: Unit 4B: EXPRESSIONS AND EQUATIONS PART II 2013-2014 5
PROJECT CRITERIA
Plot the three points on
graph paper
Write equation in the
form f(x) = ax2 + bx + c
showing work
Equation in the form
f(x) = a(x-h)2 + k
showing work
List vertex
State the axis of
symmetry
AUTHENTIC ASSESSMENT RUBRIC
0
1
2
Did not
Plotted one point
Plotted two points
attempt
correctly
correctly
Did not
Wrote equation
Wrote equation
attempt
without work
containing errors
with work shown
Did not
Wrote equation
Wrote equation
attempt
without work
containing errors
with work shown
Did not
Neither coordinate One coordinate of
attempt
of the vertex is
the vertex is correct
correct
Did not
Stated axis of
Stated axis of
attempt
symmetry
symmetry as a
incorrectly
number instead of
as an equation
3
Plotted three
points correctly
Wrote equation
correctly with
work
Wrote equation
correctly with
work
Both coordinates
of the vertex is
correct
Stated axis of
symmetry
correctly
Answers to the authentic assessment:
39916: b.
c. f(x) =
d. vertex: (7,
39917: b.
c.
d. vertex: (4, 9) e. x = 4
39918: b.
c.
d. vertex: (3, 4) e. x = 3
39919: b.
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c.
d. vertex: (
e. x = 7
,
YCS ALGEBRA I: Unit 4B: EXPRESSIONS AND EQUATIONS PART II 2013-2014 6
Completing the Square
Year 10 Set 1: First Lesson on Completing the Square
Objectives:
1. Square a bracket
2. Complete the square for x2 + 10x
3. Complete the square for x2 + 4x +1
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Completing the Square support materials attached to email:
PDF entitled: Completing_the_square_presentation.pdf (43 pages)
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T-L #4 Completing the Square
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ANSWER KEY
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T-L #5
DERIVATION #1: WORKING BACKWARDS:
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DERIVATION #2: COMPLETING THE SQUARE
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WORKSHEET: T- L #10
1. Tonia, standing on the ground, throws a ball up to her brother, who is standing on a balcony 38 feet
high. If Tonia throws the ball with an initial velocity of 40 feet per second, the equation h(x) = -16t2 +
40t + 5 gives the height h(x) of the ball after t seconds.
a. Using the quadratic formula, did Tonia throw the ball with enough velocity to reach her
brother.
b. What is the maximum height of the ball and how many seconds did it take to reach that
height?
c. How tall in Tonia?
2. The science teacher has divided your class into groups with the task of making a rocket that will
remain in the air the longest. The flight of your rocket is modeled by the equation h(t) = -16t2 + 240t
launched from the ground.
a. Using factoring, solve for the time the rocket was in the air.
b. What is the maximum height and when did it reach it?
3. Joe and Sam are building a rectangular fire pit 5 feet by 4 feet with a brick border around it. They
want the area of the pit and the border to be 110 sq. ft.
a. Use factoring to find the width of the border.
4. In the early 1900’s, the average American ate 300 pounds of bread and cereal every year. By the
1960’s, Americans were eating half that amount. However, eating cereal and bread is on the rise
again. The comsumption of there types of foods can be modeled by the function y = 0.059x2 – 7.423x
+ 362.1, where y represents the bread and cereal consumption in pounds and x represents the
number of years since 1900. (page 543, #50 in the textbook)
a. If this trend continues, what year will Americans consume 300 pounds of bread and cereal?
Solve by using the quadratic formula.
b. When did Americans eat the least amount of bread and cereal and what was the amount?
c. Explain why this model is not very realistic.
5. From the top of a hill 256 feet high, Zoe launches a rocket out over a river. The height of the rocket t
seconds after it is launched is represented by h(t) = -16t2 + 32t + 256.
a. Using completing the square, how long after the rocket is launched does it take to reach the
river?
b. The beginning height is 256 ft. How is this represented on the graph of the function?
In problems 1-5 have students examine the method used to solve each problem (completing the square,
factor, quadratic formula) and then decide if that was the best method or should it have been solved by a
different method. Explain the reasoning for the choice.
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T-L #10 Answers:
1
a. No, negative number under the radical
b. 30 ft. is the maximum height and it took 1.25 seconds
c. 5 ft.
2. a. 15 sec.
b. 7.5 seconds and 900 feet
3.
a.3 ft.
4. a. 2017
b. 1962 and the minimum amount was 128 pounds
5. a. 5.123 seconds
b. It is the y intercept
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T- L #11: Solving quadratic inequalities in one variable
Solve the following inequalities:
1. 2x2 < 0
2. 4x2−1 < 0
3. -x2 < 0
4. 2(x-1)2 - 5 > 0 find the two decimal approximation
5. -(x+4)2 + 3 > 0 find the exact value
6. x2−8x+12 > 0
7. -2x2+11x−9 < 0
8. -x2−6x−12 < 0 find the exact value
9. x2−6x−13 < 0 find the two decimal approximation
10. –(x+2)(3x−1) > 0
Answers:
1. 0
2. -½ < x <½
4. x < -0.58 or x > 2.58
5.
7. x < 1 or x > 4.5
8.
3. All reals except 0
6. X<2 or x>6
9.
10.
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T - L#14 Solving a linear-quadratic system by graphing by hand.
Solve the following systems by graphing by hand:
1. f(x) = -2x2+ 2x + 4 and g(x) = 2x – 4
2. f(x) = (x+1)2 – 1 and g(x) = x + 2
3. f(x) = -x2 + 5 and g(x) = -2x+2
4. f(x) = (x+2)(x−6) and g(x) = -3x + 8
5. f(x) = -½(x-3)2 + 4 and x – 2y = 1
Answers:
1. (2,0) and (-2, -8)
4. (5, -7) and (-4, 20)
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2. (1, 3) and (-2, 0)
5. (0, -½) and (5, 2)
3. (-1, 4) and (3, -4)
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T- L#14 Solve a linear-quadratic system algebraically:
Solve the following systems algebraically:
1. You are to build a rectangular patio with area 900 sq. ft. The perimeter is to be 260 ft.
What is the length and width of the patio?
2. In playing a game, you state to your friend, I am thinking of two numbers whose sum is
20 and the sum of the squares is 298. What are the numbers?
3. Having just purchased a new dog, you are going to fence in your backyard. You can
afford 240 feet of fencing and the area of your backyard is 3375 sq. ft. What are the
dimensions of your yard?
4. The product of two numbers is 8 and twice the first minus the second is 6. Find the two
numbers.
Answers:
1. 122.7 ft. and 7.3 ft.
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2. 3 and 17
3. 45 ft. x 75 ft.
4. 4 and 2; -1 and -8
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