Untitled Lesson
Presentation
•
Mathematics
•
7th Grade
•
Practice Problem
•
Hard
Molly Sorensen
FREE Resource
100 Slides • 0 Questions
1
AlgebraByE ample
{ }
a
=
x³
Analyzing. Explaining. Solving.
±
b
Example-based
Problem Sets
Student Workbook
2
3
It’s easy to make a mistake, but it’s not always so easy to learn from one. This workbook is designed to help you learn from all kinds of math
mistakes. Flip to any assignment in this book and you’ll see something unusual. Every problem you need to solve is paired with an example that
shows you how someone else tried to do a similar problem.
Introduction for students
Step 2
Complete the similar problem on the right.
AlgebraByE ample
Each set has been strategically designed based on classroom and laboratory research to target the kinds of algebra mistakes many students make.
By studying the examples and answering questions about them, you’ll learn from mistakes others have made. Understanding the way other people
have done the math will help you better understand the math yourself.
Pablo didn’t rewrite this expression correctly.
Here is what he wrote:
SET 2: Using the distributive property, rewrite the expression in simplest form.
What did Pablo forget when
distributing the 4w?
What should Pablo’s final
expression be?
✗
Your Turn:
−4w 5+12
()
4w 5+12
()
Step 1
Examine the example problem on the left and
answer the questions about it.
The or
icon lets you know
whether the example
shows correct or
incorrect work.
✔ ✗
4
5
1) P R E - A L G E B R A
Absolute Value
3
Combining Like Terms
5
Decimals
7
Distributive Property
9
Fractions
11
Order of Operations
13
Analyzing Answers to Word
Problems
15
2) G R A P H I N G
Slope
17
Slope Intercept Form
19
Writing Equations in Slope
Intercept Form
21
Graphing Linear Equations
23
3) L I N E A R E Q U AT I O N S
Solving 1 and 2 Step Equations
25
Solving Multi-Step Equations
27
Solving Multi-Step Equations with
Fractions
29
Writing Proportions
31
Solving Proportions
33
Writing Expressions and
Equations from Words
35
4) S O L V I N G S Y S T E M S
O F E Q U AT I O N S
by Graphing
37
by Substitution
39
by Elimination with Addition and
Subtraction
41
by Elimination with Multiplication
43
5) I N E Q U A L I T I E S
Adding and Subtracting Inequalities
45
Multiplying and Dividing Inequalities
47
Graphing Inequalities
49
Compound Inequalities
51
Writing Inequalities from Words
53
Multiplication and Division Properties
of Exponents
55
Power to Power Properties of
Exponents
57
Zero and Negative Properties of
Exponents
59
Multiple Properties of Exponents
61
Graphing and Tables of
Exponents
63
Growth and Growth Rate
65
Decay and Decay Rate
67
Decay and Growth Rate
69
7) E X P O N E N T I A L
A P P L I C AT I O N S
6) E X P O N E N T I A L
P R O P E R T I E S
Adding and Subtracting
Polynomials
71
Multiplying Monomials by
Polynomials
73
Factoring Binomials
75
Multiplying Binomials
77
8) P O L Y N O M I A L S A N D
F A C T O R I N G
9) Q U A D R AT I C S
Quadratic Formula
79
Solving Quadratics by
Factoring
83
Solving Quadratics Using the
Square Root
87
Graphing Quadratic Functions
91
TABLE OF CONTENTS
The development of these materials was supported by the Institute of Education Sciences,
U.S. Department of Education, through Grant R305A100150 to Strategic Education Research
Partnership Institute. The opinions expressed are those of the authors and do not
represent views of the Institute or the U.S. Department of Education.
AlgebraByE ample
6
7
For each set, first examine the problem on the left and answer the question(s) about it. Then complete the similar problem on the right.
Your Turn:
Name: ____________________________________ Date: _________________
✔
SERP Institute, 2014
Your Turn:
✔
Assignment 1.1
absolute value
Chad simplified this expression correctly.
Here is what he wrote:
Yapeng simplified this expression correctly.
Here is what she wrote:
Why is the negative sign not
included in the answer?
Why did Yapeng subtract 7 from 5
as her first step?
SET 1: Write the following expressions in simplest form.
SET 2: Write the following expressions in simplest form.
|–5|
|5 – 7|
|–7|
|2 – 7|
page 3
8
Your Turn:
SERP Institute, 2014
Your Turn:
✗
Assignment 1.1, cont.
absolute value
SET 3: Write the following expressions in simplest form.
SET 4: Solve the equations.
George tried to simplify this expression but didn’t do it correctly.
Here is what he wrote:
What did George do wrong in the
step marked with an arrow?
Jada solved this equation correctly.
Here is what she wrote:
Why doesn’t this equation have a
solution?
✔
|5|–|–7|
–5 =|x|
–|5|–|–7|
|x| = –15
page 4
9
For each set, first examine the problem on the left and answer the question(s) about it. Then complete the similar problem on the right.
Your Turn:
Name: ____________________________________ Date: _________________
SERP Institute, 2014
Your Turn:
Assignment 1.2
combining like terms
What did Helaina do wrong in her
first step?
Would it have been okay to write
5 + 2 – 4x? Explain why or why
not.
SET 1: Write the following expressions in simplest form.
SET 2: Write the following expressions in simplest form.
✗
✗
Joseph tried to simplify this problem but didn’t do it correctly.
Here is his work:
Helaina tried to simplify this expression, but she didn’t do it correctly.
Here is her first step:
5− 4x + 2
12x + 4 −5x
In the step marked with an arrow,
what did Joseph do wrong when
choosing the terms to combine?
If Joseph had combined like terms
correctly, what should he have gotten
as the answer in simplest form?
5 – 4x + 12x
x – 12 + 5
page 5
10
Your Turn:
SERP Institute, 2014
Your Turn:
Assignment 1.2, cont.
combining like terms
SET 3: Write the following expressions in simplest form.
SET 4: Write the following expressions in simplest form.
Ramon simplified this expression correctly.
Here is what he wrote:
Monica simplified this expression correctly.
Here is what she wrote:
Why didn’t Monica combine
–2x and –20x2?
Why didn’t Ramon combine the
12x with the 2 or –5?
✔
✔
12x + 2−5
3x + 2−5x − 20x2
3x + 4 − 2
3x + 4 −6x2− 9
page 6
11
Joey solved this problem correctly.
Here is his work:
For each set, first examine the problem on the left and answer the question(s) about it. Then complete the similar problem on the right.
Your Turn:
SET 1: Solve.
SET 2: Solve.
✗
Name: ____________________________________ Date: _________________
Why did Joey rewrite the problem
so that decimal points are aligned?
✔
SERP Institute, 2014
Your Turn:
When rewriting the problem, Joey
put a decimal point after the 4. If he
put the decimal point after the 2,
would it change the value of the
number? Why or why not?
Assignment 1.3
decimals
378.5− 0.24
0.37
× 0.14
3.785+ 24
3.7
× 1.4
Sally didn’t solve this problem correctly.
Here is her work:
Sally didn’t put the decimal point in
the right place. Where should the
decimal point go?
When multiplying, how do you
determine where the decimal point
goes in the product?
page 7
12
Carlos didn’t convert the fraction to a decimal correctly.
Here is his work:
Your Turn:
SET 3: Convert the fraction into a decimal.
SET 4: Convert the decimal into a fraction in simplest form.
Assignment 1.3, cont.
decimals
How did Kenisha know 0.75 =
?
75
100
What operation should Carlos have
used to convert the fraction into a
decimal?
SERP Institute, 2014
Your Turn:
What part of the fraction tells you to
use that operation?
✗
✔
Kenisha converted the decimal to a fraction correctly.
Here is her work:
Why did Kenisha divide the
numerator and the denominator
by 25?
1
5
0.4
1
8
0.75
page 8
13
Makala rewrote this expression correctly.
Here is what she wrote:
Pablo didn’t rewrite this expression correctly.
Here is what he wrote:
For each set, first examine the problem on the left and answer the question(s) about it. Then complete the similar problem on the right.
Your Turn:
SET 1: Using the distributive property, rewrite the expression in simplest form.
SET 2: Using the distributive property, rewrite the expression in simplest form.
What did Pablo forget when
distributing the 4w?
What should Pablo’s final
expression be?
✗
Name: ____________________________________ Date: _________________
Why was it important for Makala to
multiply the 5 by both the 4 and the
12?
✔
SERP Institute, 2014
Your Turn:
Is 5(4) + 12 the same expression
as 5(4 + 12) ? Explain.
Assignment 1.4
distributive property
−4w 5+12
(
)
4 12−5
(
)
5 4 +12
(
)
4w 5+12
(
)
page 9
14
Destiny didn’t rewrite this expression correctly.
Here is what she wrote:
Your Turn:
SET 3: Using the distributive property, rewrite the expression in simplest form.
SET 4: Using the distributive property, rewrite the expression in simplest form.
Assignment 1.4, cont.
distributive property
Where did the +20 come from in
the step marked with an arrow?
What did Destiny do wrong when
applying the distributive property?
SERP Institute, 2014
Your Turn:
You can change just one small part
in Destiny’s answer and make it
correct. Explain what you can
change.
✗
✔
Paul rewrote this expression correctly.
Here is what he wrote:
Would he have gotten the same
answer if he first subtracted 5
from 12? Explain.
−5 w + 4
(
)
12+5 w − 4
(
)
−5 w − 4
(
)
12−5 w − 4
(
)
page 10
15
For each set, first examine the problem on the left and answer the question(s) about it. Then complete the similar problem on the right.
Your Turn:
Name: ____________________________________ Date: _________________
SERP Institute, 2014
Your Turn:
Assignment 1.5
fractions
What is the greatest common
factor of 16 and 24?
Why would knowing the greatest
common factor be helpful when
simplifying a fraction?
What did Yetta do wrong while
adding the fractions?
SET 1: Reduce each fraction to its simplest form.
SET 2: Find the sum or difference for each expression and write the solution in simplest form.
✗
✗
Yetta didn’t find the sum correctly.
Here is her work:
Yoshiaki didn’t reduce this fraction to its simplest form correctly.
Here is his work:
Is Yetta’s answer bigger or
smaller than the correct answer?
How can you tell?
18
24
5
2
−3
4
16
24
2
5
+3
4
page 11
16
Your Turn:
SERP Institute, 2014
Your Turn:
Assignment 1.5, cont.
fractions
SET 3: Find the product for each expression and write the solution in simplest form.
SET 4: Find the quotient for each expression and write the solution in simplest form.
John didn’t solve this problem correctly.
Here is his work:
Tierra solved this problem correctly.
Here is her work:
Why did Tierra change to ?
3
4
4
3
What did John do wrong in the step
marked with an arrow?
✔
✗
−2
5
i −3
4
−2
5
÷4
3
−5
2
i3
4
−2
5
÷ 3
4
page 12
17
Betsy simplified this expression correctly.
Here is what she wrote:
Joan didn’t simplify this expression correctly.
Here is her work:
For each set, first examine the problem on the left and answer the question(s) about it. Then complete the similar problem on the right.
Your Turn:
SET 1: Using the order of operations, write each expression in simplest form.
SET 2: Using the order of operations, write each expression in simplest form.
In the step marked with an
arrow, Morgan subtracted. What
operation should she have done
instead of subtraction?
✗
Name: ____________________________________ Date: _________________
Why did she write +2 in her third
line?
✔
SERP Institute, 2014
Your Turn:
Assignment 1.6
order of operations
8 ÷ 22−6
(
)
52+6 ÷ 2⋅4
52−5÷ 2⋅4
8− 22−6
(
)
page 13
18
Your Turn:
SET 3: Using the order of operations, write each expression in simplest form.
SET 4: Using the order of operations, write each expression in simplest form.
Assignment 1.6, cont.
order of operations
What did Tyrone do in the step
marked with an arrow?
Explain how John used the order of
operations to do the step marked
with an arrow correctly.
SERP Institute, 2014
Your Turn:
Explain why John’s answer is not a
fraction.
✔
✗
John simplified this expression correctly.
Here is his work:
Tyrone didn’t simplify this expression correctly.
Here is his work:
What should Tyrone have done
instead?
2y + 5(y-4)
−22+ 22
5− 3
Evaluate the following expression for y = 8:
−22− 22
5− 3
Evaluate the following expression
for y = 8: 2+5 y − 4
()
page 14
19
Anya gave a good explanation for the
problem. Here is her work:
Preston didn’t give a good explanation. Here
is his work:
For each set, first examine the problem on the left and answer the question(s) about it. Then complete the similar problem on the right.
Your Turn:
SET 1: Explain whether the answer given for the word problem is reasonable.
SET 2: Explain whether the answer given for the word problem is reasonable.
How could Preston have figured out
that the answer didn’t make sense?
Without solving the problem, give a
number that would be a more
reasonable answer to the problem.
Explain.
✗
Name: ____________________________________ Date: _________________
Explain why 100 hours would have
been an unreasonable answer.
✔
SERP Institute, 2014
Your Turn:
Amira is quilting blankets for gifts this year. She finishes ⅓ of
a blanket each day. How many days will it take her to finish 6
blankets?
- Is 2 days a reasonable answer? Explain.
Assignment 1.7
analyzing answers to word problems
Nina hopes to visit the capitals of all 50 U.S. states someday.
After she visits 3 more state capitals, she will have been to
over one quarter of the state capitals. How many capitals has
she visited?
- Is 11 capitals a reasonable answer? Explain.
Smith makes $14 per hour at his job. If he
made $77 on Wednesday, how many hours
did he work that day?
- Is 5 ½ hours a reasonable answer? Explain.
Joel travels 10 miles at 55 miles per
hour. How much time does Joel’s drive
take?
- Is 5.5 hours a reasonable answer?
Explain.
page 15
20
Your Turn:
SET 3: Explain whether the answer given for the word problem is reasonable.
SET 4: Explain whether the answer given for the word problem is reasonable.
✗
SERP Institute, 2014
Your Turn:
Ms. Lina is hosting a pottery-making party for her students.
She needs to pay $12 for pottery clay for each student, but
her own pottery clay will be free. If she had $200 to spend on
the party, how many students can Ms. Lina invite?
- Is 16 students a reasonable answer? Explain.
Joohyung and Rhonda made 32 cupcakes together. They each
ate 3 cupcakes and then packed the rest up for the class bake
sale. How many cupcakes did they take to the bake sale?
- Is 26 cupcakes a reasonable answer? Explain.
Tanya found sneakers online for $24.
She ordered 4 pairs and had to pay an
additional $5.75 for shipping and
handling. How much did the whole
order cost Tanya?
- Is $47 a reasonable answer?
Explain.
Assignment 1.7, cont.
analyzing answers to word problems
Serafina didn’t give a good response for this problem.
Here is what she wrote:
✗
277 students have signed up to go to
the aquarium next week. How many
buses does the trip coordinator need
to order if each bus holds 50
students?
- Is 5.54 a reasonable answer?
Explain.
Serafina did the calculations and got
5.54, but why is 5.54 not a reasonable
answer for this word problem?
Would 5 buses be a reasonable
answer? Why or why not?
If one pair of sneakers costs $24,
does it make sense that four
would cost $47? Explain.
Without solving the problem, give a
number that would be a more
reasonable answer for the problem,
and explain why it is reasonable.
Lakin didn’t give a very good response for this problem.
Here is what he wrote:
page 16
21
Kaemon found the slope correctly.
Here is his work:
Akemi didn’t find the slope correctly.
Here is her work:
For each set, first examine the problem on the left and answer the question(s) about it. Then complete the similar problem on the right.
Your Turn:
SET 1: Find the slope for each line using the slope formula.
SET 2: Find the slope for each line using the slope formula.
What did Akemi do wrong when
plugging the points into the
slope formula?
Using the graph, how could
she have checked her work?
✗
Name: ____________________________________ Date: _________________
If Kaemon had done , would
he still have been correct? Why or
why not?
✔
SERP Institute, 2014
Your Turn:
Why did Kaemon put –6 – 3 in
the numerator rather than in the
denominator?
Assignment 2.1
slope
x–20246
y–6–3036
3− −6
()
4 − −2
()
x–4–1258
y–20246
x
y
0
(1, 5)
(4, 3)
x
y
0
(–1, 2)
(–5,0)
page 17
22
Phil didn’t identify the slope of this line correctly.
Here is his work:
Your Turn:
SET 3: Graph the line and then find the slope for that line using the slope formula.
SET 4: Find the slope for each line using the slope formula and describe the meaning of the slope.
Assignment 2.1, cont.
slope
How did Stephanie know what
the slope represented?
Phil graphed this line correctly,
however the slope is not zero. What
is the correct slope of this line?
SERP Institute, 2014
Your Turn:
How do you know that is not
equal to 0?
✗
✔
Stephanie solved this problem correctly.
Here is her work:
The line contains the points (2, 5) and (2, –3).
8
0
x
y
0
x
y
01
4
20
28
12
23
Hours Worked
x
y
0
5
1
3
5
7
10
15
Pencils
Slope:
Meaning of slope:
The line contains the points (5, 2) and (–3, 2).
page 18
23
Jaslene rewrote this equation correctly.
Here is her work:
Mark didn’t graph this equation correctly.
Here is his work:
For each set, first examine the problem on the left and answer the question(s) about it. Then complete the similar problem on the right.
Your Turn:
SET 1: Rewrite the equation in slope-intercept form. Then identify the slope and y-intercept.
SET 2: Graph each equation.
Mark incorrectly graphed this
slope. What slope did he graph?
How many times up and to the
right from point (0, 3) should
have Mark gone when
graphing this slope?
up _________________
right ________________
✗
Name: ____________________________________ Date: _________________
When an equation is in slope-
intercept form, how can you tell the
difference between the slope and
the y-intercept?
✔
SERP Institute, 2014
Your Turn:
Assignment 2.2
slope-intercept form
3x + y = −14
−2x + y = 12
a. Rewrite the equation.
b. What is the slope?
c. What is the y-intercept?
y =1
2
x + 3
y = 2x − 3
x
y
0
page 19
24
Your Turn:
SET 3: Graph each equation.
SET 4: Answer all questions about each word problem.
Assignment 2.2, cont.
slope-intercept form
What is the correct slope and
y-intercept?
slope ____________________
y-intercept ________________
How did the negative sign in this
equation affect the graph?
SERP Institute, 2014
Your Turn:
How did Daiquan know what the
slope was when there is no written
coefficient in front of x?
To rent a truck, a moving company charges $30 plus $2 per
mile. The equation that represents the total cost is y = 2x + 30.
a.What is the slope and y-intercept?
slope ____________________
y-intercept ________________
b.What does the slope represent in this word problem?
✔
✗
Daiquan graphed this equation correctly.
Here is his work:
Monica didn’t identify the slope and y-intercept correctly.
Here is her work:
y = −x + 3
y = x + 3
x
y
0
A caterer charges a $100 fee plus
$15 per person. The equation that
represents the total cost is
y = 100 + 15x. Identify the slope
and y-intercept.
What does the y-intercept
represent in this word problem?
What does the slope represent in
thisword problem?
page 20
25
Eddie didn’t write this equation correctly.
Here is his work:
Your Turn:
SET 1: Write an equation in slope-intercept form using the information provided.
SET 2: Write an equation in slope-intercept form using the information provided.
How did Sarah know she had to
solve for b first?
Which variable (m or b) does Eddie
think represents the slope in the
equation?
SERP Institute, 2014
Your Turn:
Rewrite the equation correctly.
✗
✔
Sarah wrote this equation correctly.
Here is her work:
Assignment 2.3
writing equations in slope-intercept form
Name: ____________________________________ Date: _________________
For each set, first examine the problem on the left and answer the question(s) about it. Then complete the similar problem on the right.
The line has a slope of and a y-intercept of 3.
−1
5
The line has a slope of 3 and contains the point (2, 2).
The line has a slope of and a
y-intercept of –3.
1
5
The line has a slope of and contains the point
(2, –3).
1
2
page 21
26
Bao wrote this equation correctly.
Here is his work:
Rasheena didn’t write this equation correctly.
Here is her work:
Your Turn:
SET 3: Write an equation in slope-intercept form using the information provided.
SET 4: Write an equation in slope-intercept form using the information provided.
Rasheena substituted correctly for
x and y. Which point did she use to
replace x and y in this equation?
✗
In the step marked with an arrow,
which coordinates did Bao use in
this equation?
✔
SERP Institute, 2014
Your Turn:
If Bao had used the other point,
would he have come up with the
same answer? Explain your
reasoning.
Assignment 2.3, cont.
writing equations in slope-intercept form
Rasheena did not substitute
correctly for b. What value should
she have put in for b?
If Rasheena substituted the point
(3, –2) for x and y, would she have
been correct? Why or why not?
The line contains the points (2, 3) and (6, 4).
The line has an x-intercept of –2 and a y-intercept of 3.
The line contains the points (3, 1) and (–3, –1).
The line has an x-intercept of 3 and a
y-intercept of –2.
page 22
27
For each set, first examine the problem on the left and answer the question(s) about it. Then complete the similar problem on the right.
Your Turn:
Name: ____________________________________ Date: _________________
✔
SERP Institute, 2014
Your Turn:
✔
Assignment 2.4
graphing linear equations
Andrew solved this problem correctly.
Here is his work:
Reza graphed this equation correctly.
Here is her work:
What about the point (0, –3)
indicates that it should be drawn
on the y-axis instead of the x-axis?
How could Andrew have checked
whether he graphed the line
correctly?
Would the point (3, 2) ever be on
the line x = 2? Explain.
The line contains the point
(–2, 5) and has a slope of –4.
a. Write an equation
in slope-intercept
form.
b. Graph the line.
SET 1: Write an equation in slope-intercept form using the information provided. Then graph the line.
SET 2: Graph the line.
The line contains the point (2, 5) and has a slope
of 4.
x
y
0
x = 2
What is the slope of the line?
y = –2
x
y
0
page 23
28
The line has an x-intercept
of 4 and a y-intercept of -3.
a. Use the information
above to write the points
of the x-intercept and
y-intercept.
b. Graph the line.
Your Turn:
SERP Institute, 2014
Your Turn:
✗
✗
Assignment 2.4, cont.
graphing linear equations
SET 3: Graph the line.
SET 4: Write the ordered pair for each intercept. Then graph the line.
Mao didn’t graph this line correctly.
Here is his work:
Callie forgot to write the ordered pairs and didn’t graph this line correctly.
Here is her work:
What intercepts did Callie graph?
Callie might have graphed the
points correctly if she had written
them out first. Write the correct
points of an x-intercept of 3 and a
y-intercept of 4.
What slope did Mao use?
The line contains the point
(–3, 1) and has a slope of .
The line contains the point (2, 3) and
has an undefined slope.
0
1
x
y
0
x
y
0
The line has an x-intercept of 3 and a
y-intercept of 4.
x-intercept ____________
y-intercept ____________
x-intercept
( ___ , ___ )
y-intercept
( ___ , ___ )
x-intercept
( ___ , ___ )
y-intercept
( ___ , ___ )
page 24
29
Marvin didn’t solve this problem correctly.
Here is his work:
Your Turn:
SET 1: Solve.
SET 2: Solve.
Why couldn’t Hannah just
subtract 6 to get the k by itself?
Why is Marvin’s work incorrect?
SERP Institute, 2014
Your Turn:
How could Marvin have checked
whether –3 was the correct answer?
✗
✔
Hannah solved this problem correctly.
Here is her work:
Assignment 3.1
solving 1- and 2-step equations
Name: ____________________________________ Date: _________________
For each set, first examine the problem on the left and answer the question(s) about it. Then complete the similar problem on the right.
k −6 = 3
k +6 = 3
6k = 3
k
6
= 3
page 25
30
SERP Institute, 2014
Assignment 3.1, cont.
solving 1- and 2-step equations
Mackenzie didn’t solve this problem correctly.
Here is her first step:
Your Turn:
SET 3: Solve.
SET 4: Solve.
Why did Eliza subtract 6 from
both sides of the equation?
If Mackenzie wanted to start by
getting rid of the 2 in 2k, what
should she have done differently?
Your Turn:
✗
✔
Eliza solved this problem correctly.
Here is her work:
6 = 3+ 2k
6 = −3+ 2k
6− k = −3
Why did Eliza divide by –1?
−6− k = 3
page 26
31
Ken solved this problem correctly.
Here is his work:
Jackson didn’t solve this problem correctly.
Here is his first step:
For each set, first examine the problem on the left and answer the question(s) about it. Then complete the similar problem on the right.
Your Turn:
SET 1: Solve.
SET 2: Solve.
Which terms did Jackson
incorrectly combine to get 9x?
Give an example of two terms
that would correctly add to 9x.
✗
Name: ____________________________________ Date: _________________
In the first step, Ken combined 6x
and 5x. Why didn’t he also add the
3 to get 14x?
✔
SERP Institute, 2014
Your Turn:
Assignment 3.2
solving multi-step equations
6x +5x + 3 = 11+14
−11= −6x − 3x − 2
3 + 6x = 5x
–6x + 3 = –5x
page 27
32
Umi didn’t solve this problem correctly.
Here is her first step:
Your Turn:
SET 3: Solve.
SET 4: Solve.
Assignment 3.2, cont.
solving multi-step equations
Where did Lupe get 6x + 18 from?
What did Umi do wrong in her first
step?
SERP Institute, 2014
Your Turn:
What should Umi have done on
the right side of the equation in
order to solve it?
✗
✔
Lupe solved this problem correctly.
Here is her work:
3x = 4x −6+5
6x = 3x −5− 4
−3 x +6
()= 4 −5x
6 x + 3
()= 12+ 4x
page 28
33
Pablo didn’t solve this problem correctly.
Here is his work:
Your Turn:
SET 1: Solve.
SET 2: Solve.
Why do you think Inez multiplied
all terms in the equation by 3
instead of subtracting 6 from
both sides?
What did Pablo forget to do in the
step marked with an arrow?
SERP Institute, 2014
Your Turn:
✗
✔
Inez solved this problem correctly.
Here is her work:
Assignment 3.3
solving multi-step equations with fractions
Name: ____________________________________ Date: _________________
For each set, first examine the problem on the left and answer the question(s) about it. Then complete the similar problem on the right.
6− 4x
3
= 5
4x −6
3
= 6
4
3
x +6 =10
3
4
3
x −6 =10
3
page 29
34
Maggie solved this problem correctly.
Here is her work:
Troy didn’t solve this problem correctly.
Here is his work:
Your Turn:
SET 3: Solve.
SET 4: Solve.
In the step marked with an arrow,
what did Troy do wrong?
How can Troy check his answer?
✗
In the step marked with an arrow,
why did Maggie multiply both sides
by (4x – 6)?
✔
SERP Institute, 2014
Your Turn:
Assignment 3.3, cont.
solving multi-step equations with fractions
3
4x −6
= 5
6
6x − 4
= 3
1
3
3x −6
(
)= 4 −5x
−1
3
x −6
()= 4 −5x
page 30
35
For each set, first examine the problem on the left and answer the question(s) about it. Then complete the similar problem on the right.
Your Turn:
Name: ____________________________________ Date: _________________
SERP Institute, 2014
Your Turn:
Assignment 3.4
writing proportions
SET 1: Write a proportion to represent the words. You do not need to solve the proportions.
✗
Vinnie didn’t write this proportion correctly.
Here is what he wrote:
✔
Nancy wrote this proportion correctly.
Here is what she wrote:
Why did Nancy set the two
fractions as equal?
What did Vinnie do wrong when
setting up the proportion?
Show two different ways to write
the proportion correctly.
6 is to 7 as y is to 5.
SET 2: Write a proportion to represent the words. You do not need to solve the proportions.
6 is to y as 7 is to 5.
x is to 24 as 5 is to 6.
Will x be more or less than 24?
How can you tell?
6 is to 24 as 5 is to x.
page 31
36
Your Turn:
SERP Institute, 2014
Your Turn:
Assignment 3.4, cont.
writing proportions
SET 3: Write a proportion to represent the words. You do not need to solve the proportions.
SET 4: Write a proportion to represent the words. You do not need to solve the proportions.
Kim didn’t write this proportion correctly.
Here is what she wrote:
Dan wrote this proportion correctly.
Here is what he wrote:
What does x represent in Dan’s
proportion?
What would Dan have to do to find
the total number of field goals
scored during the entire season?
Kim forgot to include the 45 minutes
in the proportion. What number
should be replaced with the 45?
What would the question need to
have asked for Kim’s proportion to
be correct?
Giselle bought 4 pineapples for $9. How many pineapples
can she buy if she has $45?
Joe found out that after working for 10 months at his new
job he earned 7 days off. How many days off will he have
earned after working for 12 months?
✔
✗
When James read 1 magazine, he
read 16 pages in 25 minutes. At this
rate, how many pages can he read in
45 minutes?
There are 25 games in a football season. A
football team scores 6 field goals in the first 5
games. At this rate, how many more field goals
can you expect them to score in the remaining
20 games?
page 32
37
Roxanne didn’t solve this proportion correctly.
Here is her work:
Your Turn:
SET 1: Solve.
SET 2: Solve.
Look at Jamila’s work in the step
marked with an arrow. Why did she
multiply 5 by both m and 3?
In the step marked with an arrow,
what did Roxanne forget when she
multiplied?
SERP Institute, 2014
Your Turn:
In the original problem, what should
Roxanne have multiplied by to get m
by itself?
✗
✔
Jamila solved this proportion correctly.
Here is her work:
Assignment 3.5
solving proportions
Name: ____________________________________ Date: _________________
For each set, first examine the problem on the left and answer the question(s) about it. Then complete the similar problem on the right.
5
6
=m
2
2
6
=5
m
5
6
=
2
m+ 3
6
5
=
3
m+ 2
page 33
38
Steve didn’t solve this problem correctly.
Here is his work:
Your Turn:
SET 3: Write a proportion for each problem then solve the proportion. Write your answer in sentence form.
SET 4: Write a proportion for each problem then solve the proportion. Write your answer in sentence form.
Steve didn’t set up his proportion
correctly. Explain which part of the
proportion he did not set up
correctly.
Why should Steve know that his
answer should be larger than 10?
✗
SERP Institute, 2014
Your Turn:
It took 10 minutes to download 30 applications to a phone. At
this rate, how long will it take to download 42 applications?
Scott spends 20 hours in a 4-week period practicing his
juggling skills. If he continues at the same rate, how many
hours will he practice in 5 weeks?
Assignment 3.5, cont.
solving proportions
✗
Alejandro didn’t complete this problem correctly.
Here is his work:
Alejandro is incorrect because he
didn’t write his answer in a sentence
and didn’t include units in his
answer. Does his answer refer to the
number of miles or hours? Explain
your reasoning.
Write the answer in sentence form.
A bus is traveling at 20 miles per hour.
At this rate, how many hours will it take
to reach a destination that is 200 miles
away?
10 pounds of soil are needed to plant
6 flower beds. A family wants to plant
8 flower beds. Assuming all flower
beds are the same size, how many
pounds of soil do they need?
page 34
39
Cedric wrote this equation correctly.
Here is what he wrote:
Gabriel didn’t write this expression correctly.
Here is what he wrote:
For each set, first examine the problem on the left and answer the question(s) about it. Then complete the similar problem on the right.
Your Turn:
SET 1: Write an expression or equation to represent the situation. You do not need to solve any equations.
SET 2: Write an expression or equation to represent the situation. You do not need to solve any equations.
To see what Gabriel did wrong,
consider that c = 40 inches. Then,
how tall is Zach?
Is 6 – 80 the same as 80 – 6? Explain
why or why not.
✗
Name: ____________________________________ Date: _________________
How did Cedric know that this was
an equation and not an expression?
✔
SERP Institute, 2014
Your Turn:
Why subtract m from 40 rather
than add m to 40?
The Potts family is moving today. It took 30 boxes to pack all of
their children’s possessions. Mrs. Potts put 3 boxes in the
truck. How many boxes, b, still need to be put into the truck?
There are 6 members in Sonia's chorus. They made 95 t-shirts
to raise money and each member sold y t-shirts. Assuming
every member sold the same number of t-shirts, how many t-
shirts does the chorus have left?
Assignment 3.6
writing expressions and equations from words
Myron got his paycheck of $40 this week. He
had $11 left after he spent m dollars on a new
video game. How much did the video game
cost?
Zach is 6 inches shorter than two times
his cousin’s height, c. How tall is Zach?
page 35
40
Olga didn’t write this equation correctly.
Here is what she wrote:
Your Turn:
SET 3: Write an expression or equation to represent the situation. You do not need to solve any equations.
SET 4: Write an expression or equation to represent the situation. You do not need to solve any equations.
Assignment 3.6, cont.
writing expressions and equations from words
What does the x stand for in
Melinda’s equation?
What information from the word
problem did Olga forget to include in
her equation?
SERP Institute, 2014
Your Turn:
What should the equation look like?
✗
The scarf team at the factory made $500 this week. All 10
members each received $40 and put the rest in the holiday
dinner party fund. How much total money did they put in
the holiday dinner party fund?
✔
Melinda wrote this equation correctly.
Here is what she wrote:
Eva planted 22 vegetables in her garden. She planted an
equal number of three types of tomatoes, as well as 7 bean
plants. How many of each type of tomatoes did Eva plant?
Three siblings baked b brownies
together this weekend. They threw
away the 24 burned brownies and split
the rest evenly, each taking home 10
brownies. How many brownies did they
bake?
Guillaume works as a parking attendant and made
$30 in tips each night he worked this week. He
gave $15 of his week’s tips to the parking assistant
and had $135 left for himself. How many days did
he work this week?
How did she know to subtract 15
rather than add 15?
page 36
41
For each set, first examine the problem on the left and answer the question(s) about it. Then complete the similar problem on the right.
Your Turn:
Name: ____________________________________ Date: _________________
✔
SERP Institute, 2014
Your Turn:
✔
Assignment 4.1
solving systems of equations by graphing
Juan has evaluated this solution correctly.
Here is his work:
Erin has evaluated this solution correctly.
Here is her work:
How does Juan prove that (5, –1) is
a solution for this system?
If Juan had graphed the two lines,
how could he tell if (5, –1) was the
solution?
Why aren’t (0, 2) and (–3, 0) also
solutions?
SET 1: Determine whether the ordered pair is a solution to the given system.
SET 2: Use the graph to determine the solution to the system of equations.
5,−1
()
x + 4 y = 1
2x −6y = 16
⎧
⎨⎪
⎩⎪
6,5
()
3x + y = 23
−x + 3y = −21
⎧
⎨⎪
⎩⎪
x
y
0
x
y
0
page 37
42
Your Turn:
SERP Institute, 2014
Your Turn:
✗
Assignment 4.1, cont.
solving systems of equations by graphing
SET 3: Solve each system of equations by graphing.
SET 4: Solve each system of equations by graphing.
Salim didn’t solve this system correctly.
Here is his work:
Salim graphed one of the lines
incorrectly. By looking at the system
of equations and graphs, how can
you tell which line it is?
What is another way Salim could
have checked his answer?
Francesca solved this system correctly.
Here is her work:
How could Francesca have
known that the origin (0, 0) was
the solution without graphing
the lines?
✔
x = 2
y = x +1
⎧
⎨⎪
⎩⎪
y = −x −1
y = 2
⎧
⎨⎪
⎩⎪
x
y
0
x
y
0
y =1
3
x +1
y = 3x +1
⎧
⎨⎪
⎩⎪
y =1
2
x
y = 4x
⎧
⎨⎪
⎩⎪
page 38
43
Domingo didn’t solve this system correctly.
Here is his work:
Your Turn:
SET 1: Solve the system of equations using the substitution method.
SET 2: Solve the system of equations using the substitution method.
Sasha’s first step was to solve
x + y =1 for x. Her friend Jonah’s first
step was to solve x + y =1 for y. Do
you think that Jonah got the same
answers as Sasha? Explain.
What else must Domingo do to solve
the system of equations?
SERP Institute, 2014
Your Turn:
✗
✔
Sasha solved this system correctly.
Here is her work:
Assignment 4.2
solving systems of equations by substitution
Name: ____________________________________ Date: _________________
For each set, first examine the problem on the left and answer the question(s) about it. Then complete the similar problem on the right.
y = x −1
2x + y = 5
⎧
⎨⎪
⎩⎪
x = 1− y
2x − y = 5
⎧
⎨⎪
⎩⎪
x + y = 1
2x + 3y = 5
⎧
⎨⎪
⎩⎪
x +5y = 1
2x + 3y = 9
⎧
⎨⎪
⎩⎪
page 39
44
Mark solved this system correctly.
Here is his work:
Zalika didn’t solve this system correctly.
Here is her work:
Your Turn:
SET 3: Solve the system of equations using the substitution method.
SET 4: Write and solve the system of equations using the substitution method. Write your answer in sentence form.
Does Zalika’s answer
make sense? Why or
why not?
How should she have
written the equations?
✗
Would Mark have found the same
answer if he had solved both
equations for y, and then set them
equal to each other? Explain.
✔
SERP Institute, 2014
Your Turn:
Augie’s chess team raised money for charity by organizing a
car wash. They washed a total of 80 vehicles and raised a total
of $486. If they charged $5 to wash a car and $7 to wash a
truck, how many of each type of vehicle did they wash?
Assignment 4.2, cont.
solving systems of equations by substitution
3x = y +11
5y − 7x = 1
⎧
⎨⎪
⎩⎪
3x = y +11
4 y −6x = −2
⎧
⎨⎪
⎩⎪
The museum charges $8 for each adult ticket and $6
for each child ticket. This morning, 200 tickets were
sold and the museum collected $1,340. How many of
each type of ticket were sold?
page 40
45
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������������������
�������didn’t ��������������������������������������������������������������������������
������������������
For each set, first examine the problem on the left and answer the question(s) about it. Then complete the similar problem on the right.
Your Turn:
SET 1: �����������������������������������������elimination with addition and subtraction method�
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������������������������������
����������������������������������
�����������
✗
�������������������������������������������������������������������
����������������������������������
��������������������������������
�������������������������
✔
��������������������
Your Turn:
���������������������x�����������
��������������������������������
�����
���������������
solving systems of
equations by elimination
�����������������������������
�x + �y = �
x − �y = �
⎧
⎨⎪
⎩⎪
�x + y = �
�x − y = �
⎧
⎨⎪
⎩⎪
�x + y = �
−x + y = �
⎧
⎨⎪
⎩⎪
�x + �y = ��
�x + �y = ��
⎧
⎨⎪
⎩⎪
�������
46
Your Turn:
SET 3: Solve each system of equations using the elimination with addition and subtraction method.
SET 4: Write and solve each system using the elimination with addition and subtraction method. Write your answer in sentence form.
Assignment 4.3, cont.
solving systems of
equations by elimination
with addition and subtraction
What other cost from the
problem is missing from
Terrance’s system of equations?
Why did Felipe reorder the equation
before solving?
SERP Institute, 2014
Your Turn:
✔
✗
Felipe solved this system correctly.
Here is his work:
Terrance didn’t solve this system correctly.
Here is his work:
4x = 2y + 4
2x + 2y = 14
⎧
⎨⎪
⎩⎪
3x = y +10
3x + 4 y = −20
⎧
⎨⎪
⎩⎪
Sammy and Cam went to a concert and
each bought one ticket and some
snacks. Sammy bought 3 snacks and
spent $22, while Cam bought 5 snacks
and spent $29.50. Assuming all snacks
cost the same price, how much did the
concert ticket and each snack cost?
Write the system of equations by
including the missing variable.
Tiffany and Brianna each bought a pair of headphones and
some new audiobooks for their upcoming trip. Tiffany bought
6 audiobooks and spent $70.70, while Brianna bought 9
audiobooks and spent $88.55. Assuming all audiobooks cost
the same amount, how much did the headphones and each
audiobook cost?
page 42
47
For each set, first examine the problem on the left and answer the question(s) about it. Then complete the similar problem on the right.
Your Turn:
Name: ____________________________________ Date: _________________
✔
SERP Institute, 2014
Your Turn:
✔
Assignment 4.4
solving systems of equations by elimination
with multiplication
Scott solved this system correctly.
Here is his work:
Danielle solved this system correctly.
Here is her work:
What would Scott have had to do if
he wanted to solve for x first?
Why did Danielle multiply the
first equation by 3 and the
second equation by 2 in the step
marked with an arrow?
SET 1: Solve each system of equations using the elimination with multiplication method.
SET 2: Solve each system of equations using the elimination with multiplication method.
5x + y = 9
10x − 7y = −18
⎧
⎨⎪
⎩⎪
2y − 3x = −4
6y + x = 8
⎧
⎨⎪
⎩⎪
2x + 4 y = 2
3x +5y = 1
⎧
⎨⎪
⎩⎪
3x + 4 y = −1
4x − 3y = 7
⎧
⎨⎪
⎩⎪
page 43
48
Your Turn:
SERP Institute, 2014
Your Turn:
✗
✗
SET 3: Solve each system of equations using the elimination with multiplication method.
SET 4: Write and solve the system of equations using the elimination with multiplication method. Write your answer in sentence form.
Does Natasha’s price for a pencil
seem reasonable? Why or why not?
In the steps marked with an arrow,
Natasha multiplied the left side of
the equation by –4. What should
she have done to keep the
equation equivalent?
What did Claudio do wrong when he
combined the equations in the step
marked with an arrow?
The Williams family has only one oven. It takes 310 minutes to
bake 4 pies and 3 loaves of bread. It takes 400 minutes to bake
5 pies and 4 loaves of bread. Assuming they can only cook 1
item at a time, how long does each item take to cook?
3x −5y = −8
2x = 3y − 2
⎧
⎨⎪
⎩⎪
3x − 9y = 15
2x = 4 y +12
⎧
⎨⎪
⎩⎪
Javier paid $16.30 for 4 binders and 6
pencils. Coco paid $26.10 for 6 binders
and 12 pencils. How much did each
item cost?
Claudio didn’t solve this system correctly. He got stuck and
couldn’t finish the problem. Here is his work:
Natasha didn’t solve this system correctly. She got stuck and
couldn’t finish the problem. Here is her work:
Assignment 4.4, cont.
solving systems of equations by elimination
with multiplication
page 44
49
For each set, first examine the problem on the left and answer the question(s) about it. Then complete the similar problem on the right.
Your Turn:
Name: ____________________________________ Date: _________________
SERP Institute, 2014
Your Turn:
Assignment 5.1
adding and subtracting inequalities
Dylan should not have used an
equals sign in his solution. What
symbol should he have used?
–13 is a possible solution for this
inequality. List another possible
correct solution.
Maria wanted to write the x first
in her solution. What did she
forget to change in order to keep
the answer correct?
SET 1: Solve each inequality.
SET 2: Solve each inequality.
✗
✗
Maria didn’t solve this inequality correctly.
Here is her work:
Dylan didn’t solve this inequality correctly.
Here is his work:
9+ x < −3
−9+ x ≥ −3
12 ≥ x and x ≥ 12 mean two
different things. Explain the
meaning of both.
−9 < x − 3
9 ≥ x − 3
page 45
50
Your Turn:
SERP Institute, 2014
Your Turn:
Assignment 5.1, cont.
adding and subtracting inequalities
SET 3: Solve each inequality.
SET 4: Solve each inequality.
Tyrese solved this inequality correctly.
Here is his work:
Walt solved this inequality correctly.
Here is his work:
Would Walt’s answer have been
the same if he had first subtracted
3x from both sides? Explain.
Would Walt’s answer have been the
same if he had first subtracted 9
from both sides? Explain.
Why did Tyrese combine –3 and +1
before adding something to both
sides?
✔
✔
x − 3+1≥ 9
x − 3−1> 9
3x + 9 ≥ 2x − 4
3x − 9 ≥ 2x − 4
page 46
51
Jaren solved this inequality correctly.
Here is his work:
Chantal didn’t solve this inequality correctly.
Here is her work:
For each set, first examine the problem on the left and answer the question(s) about it. Then complete the similar problem on the right.
Your Turn:
SET 1: Solve each inequality.
SET 2: Solve each inequality.
Chantal incorrectly switched the
symbol. When should you switch
the symbol?
How could Chantal have checked to
see if her answer was right or
wrong?
✗
Name: ____________________________________ Date: _________________
Why did Jaren multiply both sides
by 3 in the step marked with an
arrow?
✔
SERP Institute, 2014
Your Turn:
Assignment 5.2
multiplying and dividing inequalities
x
3
< 6
x
2
≥ 9
3x ≥ −9
−3x ≥ −9
page 47
52
Your Turn:
SET 3: Solve each inequality.
SET 4: Solve each inequality.
Assignment 5.2, cont.
multiplying and dividing inequalities
Explain why it’s not okay to just
cancel out the negative signs in the
step marked with an arrow.
Why didn’t Sohrob switch the
symbol after he multiplied 3 by –9?
SERP Institute, 2014
Your Turn:
✔
✗
Sohrob solved this inequality correctly.
Here is his work:
Yasir didn’t solve this inequality correctly.
Here is his work:
x
−9
≤ 3
−9 <x
3
−2
6
<x
−9
What is the correct answer?
−2
4
≤x
−6
page 48
53
Lina graphed this inequality correctly.
Here is her work:
Terrance didn’t graph this inequality correctly.
Here is his work:
For each set, first examine the problem on the left and answer the question(s) about it. Then complete the similar problem on the right.
Your Turn:
SET 1: Graph each inequality on the number line.
SET 2: Graph each inequality on the number line.
How do you know that Terrance’s
arrow is pointed in the wrong
direction?
✗
Name: ____________________________________ Date: _________________
How did Lina know which way to
draw her arrow?
✔
SERP Institute, 2014
Your Turn:
Why did Lina make the circle around
the 3 open instead of closed?
Assignment 5.3
graphing inequalities
x < 3
x ≥ −3
−2 ≥ x
2 < x
0
4321
-1
-4
-2-3
0
4321
-1
-4
-2-3
page 49
54
Nevaeh didn’t write the inequality correctly.
Here is her work:
Your Turn:
SET 3: Write an inequality represented by the graph.
SET 4: Write an inequality represented by the graph.
Assignment 5.3, cont.
graphing inequalities
How did Franco know that the
inequality symbol should open
towards the 10?
What part of the inequality that
Nevaeh wrote is incorrect?
SERP Institute, 2014
Your Turn:
✗
✔
Franco wrote this inequality correctly.
Here is his work:
If Franco had written 10 ≥ x, would
his answer still be correct? Explain.
0
8642
-2
-8
-4-6
0
8642
-2
-8
-4-6
0
40302010
-10
-40
-20-30
0
40302010
-10
-40
-20-30
page 50
55
Jasmine wrote this compound inequality
correctly. Here is her work:
Sung didn’t write this compound inequality
correctly. Here is his work:
For each set, first examine the problem on the left and answer the question(s) about it. Then complete the similar problem on the right.
Your Turn:
SET 1: Write a compound inequality for each graph.
SET 2: Write a compound inequality for each graph.
Sung used the right numbers and
symbols, so why is his answer
wrong?
✗
Name: ____________________________________ Date: _________________
Describe the solution set in words.
✔
SERP Institute, 2014
Your Turn:
Assignment 5.4
compound inequalities
0
456
321-1
-4-5-6
-2-3
0
456
321
-1
-4-5-6
-2-3
0
456
321-1
-4-5-6
-2-3
0
456
321
-1
-4-5-6
-2-3
page 51
56
Your Turn:
SET 3: Solve each inequality and graph the solution. Label your graph.
SET 4: Solve each inequality and graph the solution. Label your graph.
Assignment 5.4, cont.
compound inequalities
Is p = –1 a possible correct solution
for this inequality? Explain.
Rosario should have also subtracted
2 from –1. Why?
SERP Institute, 2014
Your Turn:
✗
✔
Malik solved and graphed this inequality
correctly. Here is his work:
Rosario’s graph correctly represents her
answer. However, she didn’t solve it
correctly. Here is her work:
−1≤ p+ 2 < 8
−2 < p+1≤ 6
−3p+ 2 ≤ −1
2p− 4 < −8
or
Why did Malik have to switch the
symbol in the first inequality, but not
in the second?
3p+ 2 > 8
−3p− 2 ≥1or
page 52
57
Dave wrote this inequality correctly.
Here is his work:
Rita didn’t write this inequality correctly.
Here is her work:
For each set, first examine the problem on the left and answer the question(s) about it. Then complete the similar problem on the right.
Your Turn:
SET 1: Write an inequality that describes each situation.
SET 2: Write an inequality that describes each situation.
Even though the problem says
“least,” why shouldn’t Rita have
used ≤ in her answer?
✗
Name: ____________________________________ Date: _________________
What does the symbol that Dave
used mean?
✔
SERP Institute, 2014
Your Turn:
t is positive.
s is at most 4.
Assignment 5.5
writing inequalities from words
x is negative.
r is at least 2.
page 53
58
Lance didn’t write this inequality correctly.
Here is his work:
Your Turn:
SET 3: Write an inequality that describes each situation.
SET 4: Write an inequality that describes each situation.
Assignment 5.5, cont.
writing inequalities from words
Why did Ann-Elise need to use two
inequality symbols?
Lance should have written ≥ . Which
words in the problem show that
Lance’s answer is wrong? Explain.
SERP Institute, 2014
Your Turn:
✗
The candy store has no more than 24 flavors of jellybeans.
Let f represent the number of flavors.
✔
Ann-Elise wrote this inequality correctly.
Here is her work:
Gerard has less than 16 trains in his box, but the box is not
empty. Let t represent the number of trains.
At least 100 parents attended the
lacrosse game. Let p represent the
number of parents.
The elevator can hold 20 people and
there are more than 5 people in the
elevator at the moment. Let p represent
the number of people.
page 54
59
Brock didn’t simplify this expression correctly.
Here is his work:
Your Turn:
SET 1: Write the following expressions in simplest form.
SET 2: Write the following expressions in simplest form.
Explain why Karin was able to add
the exponents together in order to
get the correct answer.
What is wrong with the placement of
the 3 and the 2 in Brock’s answer?
SERP Institute, 2014
Your Turn:
✗
✔
Karin simplified this expression correctly.
Here is her work:
Assignment 6.1
multiplication and division
properties of exponents
Name: ____________________________________ Date: _________________
For each set, first examine the problem on the left and answer the question(s) about it. Then complete the similar problem on the right.
c ⋅c ⋅c ⋅k ⋅k
j ⋅ j ⋅ j ⋅ j ⋅ g ⋅ g ⋅ g
a5⋅a2
b4⋅b4
page 55
60
Joshua simplified this expression correctly.
Here is his work:
Lila didn’t simplify this expression correctly.
Here is her work:
Your Turn:
SET 3: Write the following expressions in simplest form.
SET 4: Write the following expressions in simplest form.
Lila divided her exponents to get
her answer. What operation should
Lila have used to get the correct
answer?
✗
Explain why equals .
✔
SERP Institute, 2014
Your Turn:
Assignment 6.1, cont.
multiplication and division
properties of exponents
47
42
45
47i1
42
25i1
23
c6
c2
b6
b3
page 56
61
Min didn’t simplify this expression correctly.
Here is her work:
Your Turn:
SET 1: Write the following expressions in simplest form.
SET 2: Write the following expressions in simplest form.
In the step marked with an arrow,
why did Liz add instead of multiply?
Min used the correct exponent for b,
so why is her final expression
wrong?
SERP Institute, 2014
Your Turn:
✗
✔
Liz simplified this expression correctly.
Here is her work:
Assignment 6.2
power to power
properties of exponents
Name: ____________________________________ Date: _________________
For each set, first examine the problem on the left and answer the question(s) about it. Then complete the similar problem on the right.
(ab5)2
(cj3)5
b3(b2)4
(r5)2r3
page 57
62
Jamal simplified this expression correctly.
Here is his work:
James didn’t simplify this expression correctly.
Here is his work:
Your Turn:
SET 3: Write the following expressions in simplest form.
SET 4: Write the following expressions in simplest form.
There is something wrong with the 5
in James’ answer. What should there
be instead?
✗
Does Jamal really need parentheses
for (–10)2 ? Why or why not?
✔
SERP Institute, 2014
Your Turn:
Why does (a6)2 = a12, while a12a2 = a14 ?
(Why do you multiply exponents in
one case and add exponents in the
other?)
Assignment 6.2, cont.
power to power
properties of exponents
(−10a6)2⋅a2
(−2b2)2⋅b4
3a
b
⎛
⎝⎜
⎞
⎠⎟
4
5x
y
⎛
⎝⎜
⎞
⎠⎟
2
page 58
63
For each set, first examine the problem on the left and answer the question(s) about it. Then complete the similar problem on the right.
Your Turn:
Name: ____________________________________ Date: _________________
SERP Institute, 2014
Your Turn:
Assignment 6.3
zero and negative
properties of exponents
SET 1: Write the following expressions in simplest form.
✗
Ashley didn’t simplify this expression correctly.
Here is her work:
✔
Andy simplified this expression correctly.
Here is his work:
Why did the negative sign disappear
when Andy wrote his solution?
How does a negative exponent
affect its base?
What is the correct answer to this
problem?
SET 2: Write the following expressions in simplest form.
4−2
5−3
1
3b−2
2
x−3
Why did the 3 stay in the
denominator?
page 59
64
Your Turn:
SERP Institute, 2014
Your Turn:
Assignment 6.3, cont.
zero and negative
properties of exponents
SET 3: Write the following expressions in simplest form.
SET 4: Write the following expressions in simplest form.
Carlos didn’t simplify this expression correctly.
Here is his work:
Danice simplified this expression correctly.
Here is her work:
Why was x not included in Danice’s
answer?
What does anything to the power of
zero always equal?
What is the correct answer to this
problem?
✔
✗
5⋅a0
(−4)0⋅b
a−4b0
x0y−2
page 60
65
For each set, first examine the problem on the left and answer the question(s) about it. Then complete the similar problem on the right.
Your Turn:
Name: ____________________________________ Date: _________________
✔
SERP Institute, 2014
Your Turn:
✔
Assignment 6.4
multiple properties of exponents
Maya simplified this expression correctly.
Here is her work:
Riyo simplified this expression correctly.
Here is his work:
Where did the 9 come from in Maya’s
expression?
The original expression did not
contain a fraction. Why did Maya’s
answer contain a fraction?
Could Riyo have multiplied the base
numbers first and then simplified?
Why or why not?
SET 1: Write the following expressions in simplest form.
SET 2: Write the following expressions in simplest form.
(3ab−5)2
(5a−3b)2
23⋅3
2⋅33
22⋅32
35⋅2
page 61
66
Your Turn:
SERP Institute, 2014
Your Turn:
✗
✗
Assignment 6.4, cont.
multiple properties of exponents
SET 3: Write the following expressions in simplest form.
SET 4: Write the following expressions in simplest form.
Julian didn’t simplify this expression correctly.
Here is his work:
Kelly didn’t simplify this expression correctly.
Here is her work:
The –3 should have stayed in the
numerator in order for Kelly’s final
answer to be correct. Why does the
–3 belong in the numerator?
Julian made a mistake in the step
that is marked with an arrow. What
operation should he have used for
the exponents instead?
Julian could have figured out the
correct answer without showing any
work. What rule about the power of
zero did Julian forget?
(a3b2)0
(c0b−3)2
−3x−2
(−2c)−4
page 62
67
Lorena evaluated this function correctly for n = 1,
but she wasn’t successful for n = 2 and n = 3.
Here is her work:
Your Turn:
SET 1: Evaluate each exponential function by completing the tables for the given domains.
SET 2: Evaluate each exponential function by completing the tables for the given domains.
How would the second table have
differed if the equation given were
y = (–2)x ?
Why are f(n) = 4·3n and f(n) = (4·3)n
the same when n = 1?
SERP Institute, 2014
Your Turn:
Why are the solutions different for
f(n) = 4·3n and f(n) = (4·3)n when
n ≠ 1?
✗
f(n) = 2(3)nfor the domain {1, 2, 3}
✔
Marcus evaluated this function correctly.
Here is his work:
y = (5)xand y = (–5)x for the domain {0, 1, 2}
Assignment 7.1
graphing and tables of exponents
Name: ____________________________________ Date: _________________
For each set, first examine the problem on the left and answer the question(s) about it. Then complete the similar problem on the right.
f(n) = 4·3n
for the domain {1, 2, 3}
n
4 ⋅ 3n
f(n)
1
2
3
n
2(3)n
f(n)
1
2
3
y = (2)xand y = –(2)x
for the domain {0,1,2}
x
(2)x
y
0
1
2
x
–(2)x
y
0
1
2
x
(5)x
y
0
1
2
x
(–5)x
y
0
1
2
page 63
68
Dmitri completed this graph correctly.
Here is his work:
Kyle didn’t complete this problem correctly.
Here is his work:
Your Turn:
SET 3: Graph each exponential function.
SET 4: Evaluate each exponential function by completing the tables. Then graph the functions.
Which of the points is incorrect?
✗
Does this function represent
growth or decay? How can you tell
by looking at the graph?
✔
SERP Institute, 2014
Your Turn:
Assignment 7.1, cont.
graphing and tables of exponents
x2x – 4y
–12-1 – 4–3.5
020 – 4–3
121 – 4–2
222 – 40
323 – 44
y = 2x – 4
y = 2x – 3
x2x – 3
y
–12-1– 3–2.5
020– 3–2
121– 3–1
222– 3
1
323– 3
5
x
y
0
x–2xf(x)
–1
0
1
2
x
y
0
f(x) = –2x
Use x = {-1, 0, 1, 2} in the table to help.
x(2)xf(x)
–1
0
1
2
f(x) = 2x
Use x = {-1, 0, 1, 2} in the table to help.
How do you know f(x) = 2x will
never reach or go below the x-axis?
page 64
69
Gio completed this problem correctly.
Here is his work:
Julia didn’t complete this problem correctly.
Here is her work:
For each set, first examine the problem on the left and answer the question(s) about it. Then complete the similar problem on the right.
Your Turn:
SET 1: Determine if the given table represents an exponential function. Justify your reasoning.
SET 2: Identify the following from the given equation: the initial amount (a), the growth rate (r), and time (t).
Julia did not identify the growth rate
correctly. What did she do wrong?
What is the actual growth rate (as a
percent)?
✗
Name: ____________________________________ Date: _________________
✔
SERP Institute, 2014
Your Turn:
Assignment 7.2
growth and growth rate
x
y
–1
–1
0
1
1
3
2
5
x
y
0
1
1
2
2
4
3
8
y = 12 1+.07
(
)
2
y = 4 1+.90
(
)
3
Gio is correct that the function isn’t
exponential. What pattern would
he see in an exponential function?
page 65
70
Your Turn:
SET 3: Complete the following problems using the growth rate formula. Write your answer in a complete sentence.
SET 4: Complete the following problems using what you know about growth rate. Write your answer in a complete sentence.
Assignment 7.2, cont.
growth and growth rate
Why is there a 1 in the formula?
SERP Institute, 2014
Your Turn:
If you put $600 in a savings account at a 3% interest rate
compounded annually, how much money will be in the savings
account after 4 years?
The price of an item doubles each year. What is the growth rate?
✔
Mateo completed this problem correctly.
Here is his work:
Why isn’t the growth rate 300% per
year?
✔
Francine completed this problem correctly.
Here is her work:
If you put $400 in the bank at 3% interest
compounded annually, how much money
will be in the account after 5 years?
If the population triples each year, what is
the growth rate?
page 66
71
Michelle completed this problem correctly.
Here is her work:
Enid didn’t complete this problem correctly.
Here is her work:
For each set, first examine the problem on the left and answer the question(s) about it. Then complete the similar problem on the right.
Your Turn:
SET 1: Write an equation that represents the data in the table using the decay formula.
SET 2: Find the decay rate. Justify your reasoning.
✗
Name: ____________________________________ Date: _________________
Do you always need to use the
y-intercept when solving for a?
Explain why or why not.
✔
SERP Institute, 2014
Your Turn:
Assignment 7.3
decay and decay rate
x
y
-1
4
0
1
1
2
1
4
1
16
x
y
-1
4
0
2
1
1
2
1
2
y = 4(0.97)x
y = −2(0.30)x
Enid has identified the decay factor,
instead of the decay rate. What is
the decay rate?
page 67
72
Sam didn’t complete this problem correctly.
Here is his work:
Your Turn:
SET 3: Identify the decay factor in each given situation.
SET 4: Identify the original amount, decay rate, and decay factor for each situation. Then write a function for the rate of decay.
Assignment 7.3, cont.
decay and decay rate
Why did Micah say the decay rate
was .07 and not .93?
SERP Institute, 2014
Your Turn:
✗
A 53% decrease
✔
Micah completed this problem correctly.
Here is his work:
A 6% decrease
A population of 300,000 decreases by
7% each year.
Original amount: _____________________
Decay Rate: ________________________
Decay Factor: ______________________
Function:
A computer’s value decreases by 20% each year.
The initial cost was $2,000.
Original amount: _____________________
Decay Rate: ________________________
Decay Factor: ______________________
Function:
What is wrong with Sam’s answer?
page 68
73
Janae completed this problem correctly.
Here is her work:
Ari didn’t complete this problem correctly.
Here is his work:
For each set, first examine the problem on the left and answer the question(s) about it. Then complete the similar problem on the right.
Your Turn:
SET 1: Identify each function as exponential growth or exponential decay. Justify your reasoning.
SET 2: Identify each function as exponential growth or exponential decay. Justify your reasoning.
What is the base?
What does the represent?
✗
Name: ____________________________________ Date: _________________
What is the growth factor?
✔
SERP Institute, 2014
Your Turn:
Why is it important that the base is
greater than 1?
Assignment 7.4
decay and growth rate
y = 0.45 3( )
x
y = 3 0.45
()
x
y =1
16
(4)x
1
16
y = 3
1
15
⎛
⎝⎜
⎞
⎠⎟
x
page 69
74
Nicki didn’t complete this problem correctly.
Here is her work:
Your Turn:
SET 3: Solve the problems using the exponential growth OR the exponential decay formula. Write your answer in sentence form.
SET 4: Solve the problems using the exponential growth OR the exponential decay formula. Write your answer in sentence form.
Assignment 7.4, cont.
decay and growth rate
Why did Juan say that the boat
should not be sold?
Nicki’s calculations are correct, but
she made a mistake when
substituting for the formula. What
was it?
SERP Institute, 2014
Your Turn:
How could Nicki have known that
her answer was not reasonable?
✗
A subway pass was $50 per month in 2001. If the price increased
by 6% each year, how much does a monthly pass cost in 2013?
Round to the nearest cent.
✔
Juan completed this problem correctly.
Here is his work:
Cars lose value over time. If you bought a used car 4 years ago
for $3,000, how much is the car worth today? Assume it loses
15% of its value each year. Round to the nearest cent.
If a ticket to a theme park cost $70 in
2008 and has increased each year by
4%, how much did it cost in 2012?
Round to the nearest cent.
Three years ago, a person bought a used boat for
$5,000. Today, someone else offered to buy it for
$3,500. If the value of the boat decreased by 6%
each year, should the person sell the boat for
$3,500? Round to the nearest cent.
How did Juan know to substitute
3 for t?
page 70
75
Kassandra found the sum correctly.
Here is her work:
Alta tried to subtract these polynomials but
didn’t do it correctly. Here is her work:
For each set, first examine the problem on the left and answer the question(s) about it. Then complete the similar problem on the right.
Your Turn:
SET 1: Find the sum for each of the polynomials.
SET 2: Find the difference for each of the polynomials.
In the step marked with an arrow,
Alta did not change the signs
correctly. Which term does not
have the correct sign attached to it?
Why should the sign be positive?
✗
Name: ____________________________________ Date: _________________
Where did the +2x come from in
Kassandra’s answer?
✔
SERP Institute, 2014
Your Turn:
Assignment 8.1
adding and subtracting polynomials
4x2− x + 3
(
)+ x2 + 3x −1
(
)
4x2+ 3x −6
(
)+ 8x2 + 3x + 9
(
)
3x2− 4x + 8
(
)− x2 − 4
(
)
3x2+ 4x − 8
(
)− x2 − 4x
(
)
page 71
76
Eric tried to add these polynomials but didn’t
do it correctly. Here is his work:
Your Turn:
SET 3: Find the sum for each of the polynomials.
SET 4: Find the difference for each of the polynomials.
Assignment 8.1, cont.
adding and subtracting polynomials
Where did the –3x come from in the
step marked with an arrow?
Eric made a mistake when adding
the terms. What did he add
incorrectly?
SERP Institute, 2014
Your Turn:
What could Eric have done to help
him figure out what terms to
combine?
✗
✔
Jean-Paul subtracted these polynomials
correctly. Here is his work:
−3x3+6x2+ 4
(
)+ 8x3 + 3x − 9
(
)
−3x3+6x2+ 4x
(
)+ 8x3 − 3x + 9
(
)
x3+6x2− x + 2x
(
)− 9x3 − 8x2 + 3x
(
)
x3+6x2− x2+ 2x
(
)− 9x3 − 8x2 + 3x
(
)
page 72
77
Michael found the product correctly.
Here is his work:
Hugo didn’t find the product correctly.
Here is his work:
For each set, first examine the problem on the left and answer the question(s) about it. Then complete the similar problem on the right.
Your Turn:
SET 1: Find the product.
SET 2: Find the product.
The correct answer is –18x2 + 12x.
What multiplication rule did Hugo
forget?
✗
Name: ____________________________________ Date: _________________
Where did the x2 come from in
Michael’s answer?
✔
SERP Institute, 2014
Your Turn:
Assignment 8.2
multiplying monomials by polynomials
x x − 4
()
3x x − 4
()
−3x 6x − 4
(
)
−3x 6x2− 4x
(
)
page 73
78
Your Turn:
SET 3: Find the product.
SET 4: Find the product.
Assignment 8.2, cont.
multiplying monomials by polynomials
SERP Institute, 2014
Your Turn:
✔
Alicia found the product correctly.
Here is her work:
Why did Mora add the exponents
instead of multiply them?
✔
Mora found the product correctly.
Here is her work:
−3x 6x2− 4x +1
(
)
−3x 6− 4x + x2
(
)
4x23x2+6x +1
(
)
−4x23x2+6x +1
(
)
Why did Alicia write –18x3 instead of
–18x2?
page 74
79
Fei factored this polynomial correctly.
Here is her work:
Caroline tried to factor this polynomial but she
didn’t do it completely. Here is her work:
For each set, first examine the problem on the left and answer the question(s) about it. Then complete the similar problem on the right.
Your Turn:
SET 1: Factor each binomial completely.
SET 2: Factor each binomial completely.
Why is Caroline’s polynomial not
factored completely?
✗
Name: ____________________________________ Date: _________________
Where did the +1 come from in Fei’s
factored expression?
✔
SERP Institute, 2014
Your Turn:
Assignment 8.3
factoring binomials
−4x − 2
−9x2− 3x
12x2+ 8x
12x2− 8x
page 75
80
Tristan tried to factor this polynomial but
didn’t do it completely. Here is his work:
Your Turn:
SET 3: Factor each binomial completely.
SET 4: Factor each binomial completely.
Assignment 8.3, cont.
factoring binomials
Why can’t Demarcus factor out
an x3 ?
Tristan did not factor this out
completely. What else can be
factored out?
SERP Institute, 2014
Your Turn:
✗
✔
Demarcus factored out this polynomial
correctly. Here is his work:
−4x2− 2x
−3x3−12x2
16x4+ 20x2
20x4− 4x
page 76
81
For each set, first examine the problem on the left and answer the question(s) about it. Then complete the similar problem on the right.
Your Turn:
Name: ____________________________________ Date: _________________
SERP Institute, 2014
Your Turn:
Assignment 8.4
multiplying binomials
SET 1: Multiply the binomials.
✗
Ebony tried to find the product but she didn’t
do it correctly. Here is her work:
✔
Letizia multiplied correctly.
Here is her work:
In the step marked with an arrow,
where did the + sign come from?
In the step marked with an arrow,
what mistake did Ebony make?
SET 2: Multiply the binomials.
x + 4
()x + 3
()
x + 4
()x − 3
()
x − 3
()x − 4
()
x − 3
()x −6
()
page 77
82
Your Turn:
SERP Institute, 2014
Your Turn:
Assignment 8.4, cont.
multiplying binomials
SET 3: Multiply the binomials.
SET 4: Multiply the binomials.
Chuck tried to find the product but didn’t do
it correctly. Here is his work:
Shu-Ju multiplied correctly to find the product.
Here is her work:
Where did the –14x come from in
Shu-Ju’s answer?
What is another way that Chuck
could have written (x – 4)2?
✔
✗
x − 4
()
2
x + 4
()
2
6x + 4
(
)x − 3
()
6x2+ 4x
(
)x − 3
()
Would the same answer be correct if
the problem was ?
Explain why or why not.
How would that have helped him
get the correct answer?
x − 3
()6x + 4
(
)
page 78
83
For each set, first examine the problem on the left and answer the question(s) about it. Then complete the similar problem on the right.
Your Turn:
Name: ____________________________________ Date: _________________
SERP Institute, 2014
Assignment 9.1
quadratic formula
How did Denzel know to substitute +1 for a?
Denzel forgot that there was a ± in the formula and therefore only found one
solution. What is the other solution for w and how do you find it?
SET 1: Solve each equation using the quadratic formula.
✗
Denzel didn’t solve the equation correctly.
Here is his work:
w2+6w + 8 = 0
w2+ 2w − 8 = 0
page 79
84
Your Turn:
SERP Institute, 2014
Assignment 9.1, cont.
quadratic formula
SET 2: Solve each equation using the quadratic formula.
Abdalla solved this equation correctly.
Here is his work.
Why did Abdalla add +1 to both sides before applying the quadratic formula?
Why is there only one solution to this equation?
✔
4w2− 4w = −1
9w2+12w = −4
page 80
85
Your Turn:
SERP Institute, 2014
Assignment 9.1, cont.
quadratic formula
SET 3: Solve each equation using the quadratic formula.
Maya solved this equation correctly.
Here is her work:
Would Maya have gotten the same answer if she had moved x2 and 5x to the left
hand side in the first step instead of moving –5 to the right hand side? Explain
why or why not.
Why are –1.38 and –3.62 approximate solutions (≈) ?
✔
−5 = x2+5x
1+ 3x2= −5x
page 81
86
page 82
87
For each set, first examine the problem on the left and answer the question(s) about it. Then complete the similar problem on the right.
Your Turn:
Name: ____________________________________ Date: _________________
SERP Institute, 2014
Assignment 9.2
solving quadratics by factoring
x = 3 is one of the answers, but Bethanne did not finish solving the problem. What
is the other answer? Explain your reasoning.
SET 1: Solve each equation by factoring.
✗
Bethanne didn’t solve this equation correctly.
Here is her work:
x2− 3x = 0
x2+ 25x = 0
page 83
88
Your Turn:
SERP Institute, 2014
Assignment 9.2, cont.
solving quadratics by factoring
SET 2: Solve each equation by factoring.
Himanshu didn’t solve this equation correctly.
Here is his work:
Himanshu found factors of –48 (24 and –2), but they weren’t the right ones for
this problem. Why not?
✗
x2+ 8x − 48 = 0
x2+ x −12 = 0
page 84
89
Your Turn:
SERP Institute, 2014
Assignment 9.2, cont.
solving quadratics by factoring
SET 3: Solve each equation by factoring.
Mark solved this equation correctly.
Here is his work:
Why can you set both factors equal to zero in the step marked with an arrow?
✔
x2+ 9x + 8 = 0
x2+ 3x − 28 = 0
page 85
90
page 86
91
For each set, first examine the problem on the left and answer the question(s) about it. Then complete the similar problem on the right.
Your Turn:
Name: ____________________________________ Date: _________________
SERP Institute, 2014
Assignment 9.3
solving quadratics using the square root
Osvaldo did not completely solve the problem. He found the first answer
correctly. What did he need to do to find the second answer?
SET 1: Solve each equation by using the square root.
✗
Osvaldo didn’t solve the equation correctly.
Here is his work:
n+ 2
()
2
= 9
n−5
()
2
= 100
page 87
92
Your Turn:
SERP Institute, 2014
Assignment 9.3, cont.
solving quadratics using the square root
SET 2: Solve each equation by using the square root.
Jasmine solved this equation correctly.
Here is her work:
Why does this problem have one solution when the problems in the first set have
two solutions?
✔
n−6
()
2
= 0
n+ 25
(
)
2
= 0
page 88
93
Your Turn:
SERP Institute, 2014
Assignment 9.3, cont.
solving quadratics using the square root
SET 3: Solve each equation by using the square root.
Lacey solved this equation correctly.
Here is her work:
Why is there no solution to this equation?
Without plotting points, draw a quick sketch
of what the graph of this equation might
look like on the graph to the right.
✔
n+ 3
()
2
= −7
n− 3
()
2
= −4
x
y
0
5
5
-5
-5
page 89
94
page 90
95
For each set, first examine the problem on the left and answer the question(s) about it. Then complete the similar problem on the right.
Your Turn:
Name: ____________________________________ Date: _________________
SERP Institute, 2014
Assignment 9.4
graphing quadratic functions
Why is the axis of symmetry always the same as the x value of the vertex?
SET 1: Identify the axis of symmetry, the vertex, and whether the vertex is a minimum or maximum value.
Imani found the characteristics of the function correctly.
Here is her work:
✔
x
y
0
5
5
-5
-5
x
y
0
5
5
-5
-5
Axis of
Symmetry
Vertex
Min or Max?
Axis of
Symmetry
Vertex
Min or Max?
page 91
96
Axis of
Symmetry
Vertex
Min or Max?
Your Turn:
SERP Institute, 2014
Assignment 9.4, cont.
graphing quadratic functions
SET 2: Identify the axis of symmetry, the vertex, and whether the vertex is a minimum or maximum value. Then graph the function.
Bernardo found the characteristics of the function correctly,
but he didn’t finish his graph correctly. Here is his work:
Bernardo found the axis of symmetry correctly. What equation did he use?
✗
By looking at the graph, how could Bernardo have figured out that it was incorrect?
y = −x2+ 4x +1
y = x2− 4x +1
Axis of
Symmetry
Vertex
Min or Max?
x
y
0
5
5
-5
-5
page 92
97
Your Turn:
SERP Institute, 2014
Assignment 9.4, cont.
graphing quadratic functions
SET 3: Identify the axis of symmetry, the vertex, and whether the vertex is a minimum or maximum value. Then graph the function.
Lourdes didn’t find the characteristics of the function correctly.
Here is her work:
Lourdes left out a part of the quadratic function when solving the problem. What
did she forget?
What part of the quadratic function will always affect the direction of the graph?
✗
y = x2− 2x −1
Axis of
Symmetry
Vertex
Min or Max?
y = −x2− 2x −1
Axis of
Symmetry
Vertex
Min or Max?
x
y
0
5
5
-5
-5
page 93
98
page 94
99
100
serpinstitute.org/algebra-by-example
Strategic Education Research Partnership
info@serpinstitute.org • (202) 223-8555
serpinstitute.org
AlgebraByE ample
AlgebraByE ample
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