Whatcom Math Project
Algebraic Expressions
Course: Algebra
Grade Level: 9 - 12
College Readiness Standard Name(s) and Number(s):
Standard 7 Algebra
The student accurately describes and applies
concepts and procedures from algebra.
Standard 2 Communication
The student can interpret and communicate
mathematical knowledge and relationships in both mathematical and
everyday language.
Standard 3 Connections
The student extends mathematical thinking across mathematical content
areas, and to other disciplines and real life situations.
Student Attribute:
Perseveres when faced with time-consuming or complex tasks by
being willing to work on problems that require time and thought, particularly
problems that cannot be solved by mimicking a previously seen example.
Student Learning Outcomes − College Readiness Standard Component and
Number:
7.1 Recognize and use appropriate concepts, procedures, definitions, and properties to
simplify expressions and solve equations.
2.1 Summarize and interpret mathematical information which may be in oral or written
formats.
2.2 Use symbols, diagrams, graphs, and words to clearly communicate mathematical
ideas, reasoning, and their implications.
3.1 Use mathematical ideas and strategies to analyze relationships within mathematics
and in other disciplines and real life situations.
3.2 Understand the importance of mathematics as a language.
3.4 Abstract mathematical models from word problems, geometric problems, and
applications.
1
Learning objectives:
7.1.a Explain the distinction between factor and term.
2.1.c Create symbolic representations for situations described in everyday language.
2.2.b Identify units associated with these variables and constants.
2.2.c Use correct mathematical symbols, terminology and notation.
3.1.b Recognize patterns and apply mathematical concepts and procedures in other
subject areas and real world situations.
3.2.b Transfer mathematical vocabulary, concepts, and procedures to other disciplinary
contexts and the real world.
3.4.a Recognize and clarify mathematical structures that are embedded in other
contexts.
Prerequisite Skills:
Properties of addition and subtraction
Properties of multiplication and division
Exponents and order of operations
Write word phrases as algebraic expressions
Estimated Time For Completion:
15 minutes for pre-assessment
One 55min period for the lesson
Another 55 min period for the application and assessment
Material for Students:
Lesson worksheets (optional)
Application group worksheets
Teaching Aids:
White or chalk board for lesson
2
I.
PREASSESSMENT:
Students should be able to correctly do all of these questions, with the exception of
number nine. Number nine is leading the students into the lesson. If the students are
unable to do all of the preassessment, then this would not be a suitable lesson for the
class. Below is the teacher’s version, followed by the student version.
Multiple Choice (1 point each)
1. The length of a rectangular pig pen is three feet longer than the width. Let w
represent the width of the pig pen in feet. What is represented by w  3 ?
a.
b.
c.
d.
The length of the pig pen in feet
The number of widths
The number of feet in the width
The perimeter in feet
correct
confusing number with length
confusing width with length
misunderstanding perimeter
2. The length of a rectangular pig pen is three feet longer than the width. Let w
represent the width of the pig pen in feet. What is represented by 2w  2(w  3) ?
a.
b.
c.
d.
Area of the pig pen in feet
Area of the pig pen in square feet
Perimeter of the pig pen in feet
Perimeter of the pig pen in square feet
misunderstanding area and units
incorrect understanding of area
correct
incorrect units
3. The length of a rectangular pig pen is three feet longer than the width. Let w
represent the width of the pig pen in feet. What is represented by w (w  3) ?
a.
b.
c.
d.
Area of the pig pen in feet
Area of the pig pen in square feet
Perimeter of the pig pen in feet
Perimeter of the pig pen in square feet
incorrect units of area
correct
misunderstanding perimeter
misunderstanding perimeter units
4. The expression 7 x  5 x 2  202 may also be written as…
a.
b.
c.
d.
5 x 2  7 x  202
 5 x 2  (7 x)  202
7 x  (5x 2 )  (202)
7 x  (5x ) 2  (202)
misunderstanding commutative property
misunderstanding commutative property
correct
misunderstanding exponent properties
3
5. The expression  5  12x may also be written as…
a.
b.
c.
d.
12x  5
12 x  ( 5)
 5  ( 12 x )
5  ( 12 x )
dropping negative with subtraction
dropping negative with correct conversion to addition
Correct
misunderstanding subtraction of negatives
Short Answer (2 points each)
6. Write an algebraic expression for “22 less than the total T”.
Solution: T-22
2-point response: The student shows understanding of algebraic expressions by
writing T – 22.
1-point response: The student shows some understanding of algebraic expressions
by indicating at least two of the following.


Uses the variable T and 22 in an expression
Indicates “less than” with subtraction in an expression
0-point response: The student shows little or no understanding of algebraic
expressions.
7. Write an algebraic expression for “three fourths of the time t”.
Solution:
3
t
4
2-point response: The student shows understanding of algebraic expressions by
3
writing t .
4
1-point response: The student shows some understanding of algebraic expressions
by indicating at least two of the following.
3
 Uses the variable t and
in the expression
4
 Indicates “of” with multiplication in an expression
4
0-point response: The student shows little or no understanding of algebraic
expression.
8. Write an algebraic expression for “the sum of twice a and ten”.
Solution: 2a+10
2-point response: The student shows understanding of algebraic expressions by
writing 2a+10.
1-point response: The student shows some understanding of algebraic expressions
by indicating at least two of the following.
 Uses the variable a and 10 in the expression
 Indicates “sum” with addition in an expression
 Indicates “twice” with multiplication by 2 in an expression
Short Answer (8 points, 1 point for each part)
9. You have some money to invest and decide to put some money in Account A
which earns a simple interest rate of 3%.
a. Complete the following table.
Amount of Money invested in
Interest earned in one year from
Account A
Account A
20,000
17,000
16,500
Solutions: 600, 510, 495
b. Explain in words the process you completed to find the amount of interest
earned in one year in account A.
Solution: multiplied the amount of money invested in account A by
0.03 to find the interest earned in one year from account A.
c. Write an algebraic expression for the amount of interest earned in one
year from account A if you invested x dollars in account A.
Solution: 0.03x
5
d. The total amount of money you have to invest is $30,000. You decide to
split it into two different accounts. Account A is less risky and account B is
more risky but earns a simple annual interest rate of 5%.Complete the
following table.
Amount of money
Amount of money
Interest earned in one
invested in Account A invested in Account B year from Account B
20,000
17,000
16,500
Solutions for money invested: $10,000, $13,000, and $13,500.
Solutions for interest: $500, $650, and $675
e. Explain in words the process you completed to find the amount of money
invested in account B.
Solution: To find the amount of money invested in account B I
subtracted the amount of money invested in account A from the
30,000 dollars.
f. Write an algebraic expression for the amount of money invested in
account B if you invested x dollars in account A.
Solution: (30,000-x)
g. Explain in words the process you completed to find the amount of interest
earned in one year in account B.
Solution: To find the amount of interest earned in one year in
account B, I took the amount of money in account B and multiplied
it by 0.05.
h. Write an algebraic expression for the amount of interest earned in one
year from account B if you invested x dollars in account A.
Solution: (30,000-x)(0.05)
i.
Write an algebraic expression for the total amount of interest earned in
one year if you invested x dollars in account A.
Solution: x(.03)+(30,000-x)(0.05)
6
Pre-Assessment
Name _____________________
Date ________
Period _____
Multiple Choice (1 point each)
1. The length of a rectangular pig pen is three feet longer than the width. Let w
represent the width of the pig pen in feet. What is represented by w  3 ?
a.
b.
c.
d.
The length of the pig pen in feet.
The number of widths
The number of feet in the width
The perimeter in feet
2. The length of a rectangular pig pen is three feet longer than the width. Let w
represent the width of the pig pen in feet. What is represented by 2w  2(w  3) ?
e.
f.
g.
h.
Area of the pig pen in feet
Area of the pig pen in square feet
Perimeter of the pig pen in feet
Perimeter of the pig pen in square feet
3. The length of a rectangular pig pen is three feet longer than the width. Let w
represent the width of the pig pen in feet. What is represented by w (w  3) ?
a.
b.
c.
d.
Area of the pig pen in feet
Area of the pig pen in square feet
Perimeter of the pig pen in feet
Perimeter of the pig pen in square feet
4. The expression 7 x  5 x 2  202 may also be written as…
e.
f.
g.
h.
5 x 2  7 x  202
5x 2  ( 7 x )  202
7 x  (5x 2 )  (202)
7 x  (5x ) 2  (202)
7
5. The expression  5  12x may also be written as…
a.
b.
c.
d.
12x  5
12 x  ( 5)
 5  ( 12 x )
5  ( 12 x )
Short Answer (2 points each)
6. Write an algebraic expression for “22 less than the total T”.
7. Write an algebraic expression for “three fourths of the time t”.
8. Write an algebraic expression for “the sum of twice a and ten”.
8
Short Answer (8 points, 1 point for each part)
9. You have some money to invest and decide to put some money in Account A
which earns a simple interest rate of 3%.
a. Complete the
Amount of Money invested
Interest earned in one
table to the right.
in Account A
year from Account A
20,000
17,000
16,500
b. Explain in words the process you completed to find the amount of interest
earned in one year in account A.
c. Write an algebraic expression for the amount of interest earned in one
year from account A if you invested x dollars in account A.
d. The total amount of money you have to invest is $30,000. You decide to
split it into two different accounts. Account A is less risky and account B is
more risky but earns a simple annual interest rate of 5%.Complete the
following table.
Amount of money
invested in Account A
20,000
17,000
16,500
Amount of money
invested in Account B
Interest earned in one
year from Account B
e. Explain in words the process you completed to find the amount of money
invested in account B.
f. Write an algebraic expression for the amount of money invested in
account B if you invested x dollars in account A.
g. Explain in words the process you completed to find the amount of interest
earned in one year in account B.
h. Write an algebraic expression for the amount of interest earned in one
year from account B if you invested x dollars in account A.
i.
Write an algebraic expression for the total amount of interest earned in
one year if you invested x dollars in account A.
9
II.
INTRODUCTION:
The Magic of Algebra!
“Everyone think of your favorite number and write it down on your paper. Now add five
to your favorite number and write this new number down. Now, multiply your new
number by six. Now subtract 30 from the new number. Lastly, divide this new number
by three. Now, I am going to use my magic abilities so that if you tell me your favorite
number, I will tell you your final result.”
Have a student tell you his or her favorite number and double it in your head and tell
them. For example, if a student says his number is five, then tell him his final number
was ten. Do this for a couple of students then ask the class if they notice a pattern by
show of hands without revealing the secret. If a student thinks they do notice the
pattern have them guess a classmates final number.
Stump your friends with the power of algebra!
Have a student come up and write the expression on the board. Have the rest of the
x  56  30 where x is the
class correct it if something is wrong. Correct solution is
3
student’s favorite number.
(If continuing to go over simplifying expressions you should show how to simplify to get
the result of 2 x . (Words  expression  simplify))
10
III.
LESSON:
Instructor Notes:
This lesson is to be directed by the instructor in front of the entire class, but does
involve student interaction. Further notes within the lesson will be in parenthesis.
There is a worksheet at the end of the lesson to accompany the lesson, so that students
do not need to scramble to keep up with writing down all of the information in the story
problems.
Tell the class that we will be learning about terms and factors of algebraic expressions
and once we are able to distinguish between the two we look at applications (word
problems) where we will create an algebraic expression from a real world situation and
then interpret the terms and factors of each term. So, not only will we be able to identify
the terms and factors, but also be able to correctly explain which each represents in
context of the problem and what units each has.
Math is a language and when we learn a language we must build a vocabulary, so that
we are able to clearly communicate with one another. So to begin today’s lesson, let’s
define some terms.
(Write the following definitions and examples on the board)
Vocabulary
1. Addition separates expressions into parts. These parts are called terms.
(Note: we are only counting nonzero terms, for example 2x only has one term
even though it could be written as 2x + 0)
Examples: Determine the number of terms.
a. x 2  2x  1 (correct answer is 3, discuss that the terms are separated by
addition so x 2 is the first term, 2x is the second term and 1 is the third
term.)
b. 7x  4
(explain that even though in this case, the two terms are
separated by subtraction, that we can rewrite the expression as 7x  ( 4) ,
which clearly shows that there are two terms and the second term is
negative 4)
2. A term that only consists of a number is called a constant term.
Examples: What is the constant term?
11
a. x 2  2x  1
(the third term, 1, is the constant term)
b. 7x  4  7x  ( 4) (negative 4 is the constant term)
3. Factors are numbers or variables which are multiplied together. If a  b  c then a
and b are factors of c.
(Note: When counting factors we are only considering non-one factors similar to
prime factorizations)
Examples: Determine the number of terms, the constant term and the factors of
each term.
a. - 5x 2
(one term, factors are - 5 and x 2 or -1, 5, x and x or
any other combination, no constant term)
b. 6y 4  8xy 3  10
(three terms, factors of term one are 6 and y 4 , or 3 y
and 2y 3 , …, factors of second term are  8x and y 3 , or  4xy 2 and 2 y ,
…, factors of the third term are factors of ten. The third term is the
constant term.
c.  t 
t
 3t  4
10
(four terms, factors of first are  1 and t, possible
1
, possible factors of the third
10
term are -3 and t, and the fourth term is the constant term.)
factors of the second term include t and
d.
ab
b
(one term, possible factors are a and
, or ab and
47 yz
47 yz
1
, or … , no constant term)
47 yz
e. 6a  2(x  y)
(two terms, the first being 6a and the second being
2(x+y). Factors of the first are 6 and a and factors of the second are 2 and
(x+y). no constant term. Discuss that even though the x and y are
separated by addition, they are not separate terms because they are both
multiplied by the 2.)
f.
2(3a  x  y )
(one term, factors are 2 and 3a+x+y. Again even
though the second factor contains three terms, it is part of a factor of the
entire expression so there is only one term)
12
g. 6a  2 x  2y
(3 terms. Note that this is an equivalent expression to
part f, but in this form there are three terms instead of one.)
Now that we have an idea of what our new vocabulary words mean, let’s apply these
terms to expressions created from real world situations. We will create symbolic
representations for situations described in everyday language by recognizing patterns.
Then we will discuss the terms and factors of each of the expressions.
(For each of these be sure you look at patterns by first plugging in numbers and then
looking at patterns to come up with your variables. Be sure you define your variable as
well. Your goal is to lead the students to finding the patterns on their own, not simply
giving them the answer.)
Examples
1. A pool is x feet wide and we want to have 5 lanes each with equal width. We
want to write an expression that represents the width of one lane. So, what if the
pool was 50 ft wide, how wide would each of the lanes be?











Ask the students “how are you arriving at your answer?”
Have a student explain to the class that 50 divided by 5 is ten so the width
of each lane should be 10ft.
“What if it is 25ft wide?”
Have another student describe that 25/5 is 5 so each lane should be 5 ft.
“What about 10ft?”
Have another student explain that 10 divided by 5 is 2 ft.
“Do you notice a pattern?”
The class should notice they are always dividing the width by five.
“So what if x feet?”
x/5 which follows the pattern.
Discuss that this expression contains one term and the factors of
this term, namely 1/5 and x.
2. You are given $900 and want to put some in savings and the remainder towards
your car loan. Choose a variable to represent the amount you put in one of the
two options and write an expression for the amount put in the other.
(Again, for students who do not just get it. Run through this scenario with
numbers.)


Start a table on the board.
Pick a number for the amount in the savings account and ask how much
would go towards the loan until there is enough information on the table to
see the pattern.
13





Solutions: If s = amount in savings account, then 900-s is the amount in
checking. If c= amount in checking, then 900-c is the amount in savings.
“How many terms does expression have?” (Solution: 2)
“Does it include a constant term?” (Yes, 900)
“What are the factors of each term?” (any two numbers that multiply
together to get 900, and -1and c)
“What are the units of each of the factors?” (both $)
3. This fall math enrollment in Math 99 was 19 more than twice that of last spring
quarter. Choose a variable to represent the enrollment in one quarter and write
an expression for the enrollment in the other.
(Again, for students who do not just get it, run through this scenario with
numbers.)








Start a table on the board comparing enrollment in fall to enrollment in
spring.
Pick a number for the enrollment in spring and ask the class how large
was the enrollment in fall.
Repeat this until there is enough information on the table to see the
pattern.
Solutions: If s = enrollment in spring, then 2s+19 is fall enrollment. If f =
fall enrollment, then (f-19)/2 represents spring enrollment.
How many terms does expression have? (2 or one depending on how you
defined your variables)
“Does it include a constant term?” (Yes if expression is 2s+19, no if
expression is (f-19)/2)
“What are the factors of each term?”
“What are the units of each of the factors?” (all are number of students.)
4. Sally loves the bulk candy section of the grocery store. On Tuesday she bought
5 scoops of her favorite candy and 3 scoops of another candy. Her favorite
candy costs $2.95 per pound and the other candy costs $2.50 per pound. The
total cost of all of Sally’s candy on Tuesday can be represented by the
expression 2.50(3w)+2.95(5w).
(Assume each scoops weighs the same amount.)



“How many terms does expression have?” (2)
“Does it include a constant term?” (no.)
“What are the factors of each term?” (2.50 and 3w for the first and 2.95
and 5w for the second.)
14


“What are the units of each of the factors?” (2.50 is in dollars per pound
and 3w is pounds, 2.95 is dollars per pound and 5w is in pounds.)
“What does w represent?” (w represents the number of pounds of candy
per scoop.)
5. A farmer grows and sells berries and finds that blueberries are more profitable
than raspberries so he decides to rip some of the rows of raspberries bushes out
and replace them with blueberries. He knows that if he plants too many bushes
in a row, that the yield of blueberries is less per bush than if he gives each bush
enough room in its row. He also knows that for each additional five bushes per
row, then each bush produces ½ quart fewer berries. In the past he had planted
30 bushes per row which produced 8 quarts per bush.
a. Let n represent the number of additional groups of 5 bushes and write an
expression for the number of quarts harvested per row.
(Use numbers and tables to discover the pattern to lead to the expression.
An example table is shown below.)





N
0
1
2
…
N
When doing the table, stress the importance of keeping track of
where the numbers came from.
Ask the students “how did you arrive at the number of quarts?” and
record their logic in the table.
Do as many as it takes for the students to realize the pattern.
Then write the expression using the variable to represent the
quantity that is changing and constants for numbers that are not
changing.
In this example the thing that is changing is the number of times
you add five or subtract ½.
Number of quarts harvested
(30bushes)(8quarts/bush per row)=240quarts/row
(five more bushes, so 30+5)(0.5 less so 8-.5)=262.5
(30+5+5)(8-.5-.5)=(30+5(2))(8-.5(2))
…
(30+5n)(8-.5n)
b. What do the following expressions represent?
15
(Look back at the expression and ask what each of the following
represent. Include units in each explanation. Also include number of
terms and a discussion of factors and constant terms.)
i. (30+5n)(8-0.5n)
(This expression represents the number of
quarts produced in each row. This expression has one term and
factors include (30+5n) and (8-.5n).)
ii. 30+5n
(This expression represents the number of
bushes per row. There are two terms.)
iii. 8-0.5n
(This represent the number of quarts per bush
depending on the number of 5 bush increases per row. Two
terms.)
iv. 30
(30 bushes per row. Constant term.)
v. 5n
(number of additional bushes per row. Units
are still bushes per row. One term. Factors are 5 and n.)
vi. 8
(8 quarts per bush when there are 30 bushes
per row. This is one term and it is a constant term.)
vii. 0.5n
(The amount of quarts each bush would
decrease in production by. One term Factors are 0.5 and n.)
16
Algebraic Expressions
Name _____________________
Date ________
Period _____
Vocabulary
1. Addition separates expressions into parts. These parts are called terms.
Examples: Determine the number of terms.
a. x 2  2x  1
b. 7x  4
2. A term that only consists of a number is called a constant term.
Examples: What is the constant term?
a. x 2  2x  1
b. 7x  4
3. Factors are numbers or variables which are multiplied together. If a  b  c then a
and b are factors of c.
Examples: Determine the number of terms, the constant term and the factors of
each term.
a. - 5x 2
b. 6y 4  8xy 3  10
c.  t 
d.
t
 3t  4
10
ab
47 yz
e. 6a  2(x  y)
f.
2(3a  x  y )
g. 6a  2 x  2y
17
Applications
1. A pool is x feet wide and we want to have 5 lanes each with equal width. We
want to write an expression that represents the width of one lane. So, what if the
pool was 50 ft wide, how wide would each of the lanes be?
2. You are given $900 and want to put some in savings and the remainder towards
your car loan. Choose a variable to represent the amount you put in one of the
two options and write an expression for the amount put in the other.
3. This fall math enrollment in Math 99 was 19 more than twice that of last spring
quarter. Choose a variable to represent the enrollment in one quarter and write
an expression for the enrollment in the other.
18
4. Sally loves the bulk candy section of the grocery store. On Tuesday she bought
5 scoops of her favorite candy and 3 scoops of another candy. Her favorite
candy costs $2.95 per pound and the other candy costs $2.50 per pound. The
total cost of all of Sally’s candy on Tuesday can be represented by the
expression 2.50(3w)+2.95(5w).
5. A farmer grows and sells berries and finds that blueberries are more profitable
than raspberries so he decides to rip some of the rows of raspberries bushes out
and replace them with blueberries. He knows that if he plants too many bushes
in a row, that the yield of blueberries is less per bush than if he gives each bush
enough room in its row. He also knows that for each additional five bushes per
row, then each bush produces ½ quart fewer berries. In the past he had planted
30 bushes per row which produced 8 quarts per bush.
19
IV.
APPLICATION:
The students will be put into groups of three and each group will be given one of the
following three problems. Then, the students will be given time to work on their
problems in their groups. Once the groups are finished, form new groups where each
new group has an “expert” from each problem to help the group solve all the problems.
The “experts” are told not to just tell the answer but to guide the others through the
process, similarly to how you guided the class through the examples in the lesson.
The three problems are formatted so that they can be printed out as worksheets on the
following three pages.
The solutions of the three problems follow the worksheets.
20
1. Sally is selling couches for $899. On average she sells 14 couches a month.
She decided to have a holiday sale and during the month of December she
lowered the price per couch to $799 and sold an additional 3 couches.
a. If this trend continues, write an expression that represents the revenue
earned, using x to represent the number of $100 drops in the price of a
couch.
b. How many factors does your expression in part a contain?
c. What are the factors?
d. How many terms does your expression in part a contain?
e. The cost involved in making each couch is $350. Write an expression that
represents the profit in dollars earned.
f. How many terms does your expression in part e contain?
21
2. The longer you have a cow the more weight it will gain, but the price per pound of
cows is decreasing. You want to sell the cow for as much as possible and cows
are bought and sold per pound. Currently the cow weighs 750lb and is gaining
22 lbs/week. At the last auction cows were sold for 17 cents/lb and that price is
decreasing by 0.3 cents per week.
a. If this trend continues, write an expression that represents the price of the
cow using x to represent the number of weeks you have owned the cow.
b. How many factors does your expression in part a contain?
c. What are the factors?
d. How many terms does your expression in part a contain?
22
3. Bill and Sue love to go kayaking. Since Sue is in better shape, their dog Benny
sits in her kayak. This slows Sue down considerably, so that Bill travels 2 miles
per hour faster than Sue. It takes Sue an hour longer to kayak 17 miles than it
takes Bill to go the same 17 miles. Let k represent the speed at which Sue
kayaks in miles per hour.
a. Write an expression that represents the speed at which Bill kayaks.
b. How many terms does the expression from part a contain? What are the
units of each term?
c. Write an expression for the total amount of time that it takes Bill to kayak
17 miles if Sue’s speed is k miles per hour.
d. How many terms does the expression from part c contain? What are the
units of each term?
e. Write a new expression that includes the expression from part c which
represents the total amount of time it takes Sue to kayak the 17 miles.
23
Worksheet solutions
1. Sally is selling couches for $899. On average she sells 14 couches a month.
She decided to have a holiday sale and during the month of December she
lowered the price per couch to $799 and sold an additional 3 couches.
a. If this trend continues, write an expression that represents the revenue
earned, using x to represent the number of $100 drops in the price of a
couch.
(899-100x)(14+3x)
b. How many factors does your expression in part a contain? (2)
c. What are the factors?
(899-100x and 14+3x)
d. How many terms does your expression in part a contain?
(1)
e. The cost involved in making each couch is $350. Write an expression that
represents the profit in dollars earned. (899-100x)(14+3x)-350x
f. How many terms does your expression in part e contain?
(2)
2. The longer you have a cow the more weight it will gain, but the price per pound of
cow is decreasing. You want to sell the cow for as much as possible and cows
are bought and sold per pound. Currently the cow weighs 750lb and is gaining
22 lbs/week. At the last auction cows were sold for 17 cents/lb and that price is
decreasing by 0.3 cents per week.
a. If this trend continues, write an expression that represents the price of the
cow using x to represent the number of weeks you have owned the cow.
(750+22x)(0.17-0.003x)
b. How many factors does your expression in part a contain? (2)
What are the factors?
750+22x and 0.17-0.003x
c. How many terms does your expression in part a contain?
(1)
3. Bill and Sue love to go kayaking. Since Sue is in better shape, their dog Benny
sits in her kayak. This slows Sue down considerably, so that Bill travels 2 miles
per hour faster than Sue. It takes Sue an hour longer to kayak 17 miles than it
takes Bill to go the same 17 miles. Let k represent the speed at which Sue
kayaks in miles per hour.
a. Write an expression that represents the speed at which Bill kayaks. (k+2)
b. How many terms does the expression from part a contain? What are the
units of each term? (2 terms, both terms have units of miles per hour)
c. Write an expression for the total amount of time that it takes Bill to kayak
 17 
17 miles if Sue’s speed is k miles per hour. 

k 2
d. How many terms does the expression from part c contain? What are the
units of each term? (1 term, the units of the term are hours)
e. Write a new expression that includes the expression from part c which
represents the total amount of time it takes Sue to kayak the 17 miles.
 17

 1

k  2 
24
V.
ASSESSMENT:
Below is the teacher’s version, followed by the student version.
Multiple Choice (1 point each)
CRS Target 7.1a: Explain (or indicate understanding of) the distinction between factor
and term.
1. How many terms are there in the expression ( x  3)(5 x  7)  10 x ?
a.
b.
c.
d.
2
3
4
5
correct
total number of factors in first term
incorrect number of terms in first term
incorrect number of terms in first term plus second term
2. The second term in the expression 3x  5( x  2)  30 x is
a.
b.
c.
d.
5
x
5x
5( x  2)
a factor of the second term
second factor in the expression
dropping -2
correct
3. How many factors are there in the expression 11( x 2  5x  7) ?
a.
b.
c.
d.
1
2
3
4
number of terms
correct
number of terms in second factor
sum of all the terms in both of the factors
4. How many terms are there in the second factor of the expression
2(3x  5y  10)(13xy  1) ?
a.
b.
c.
d.
1
2
3
6
number of terms in the first factor
number of terms in the third factor
correct
sum of all the terms in each of the factors
25
CRS Target 2.1c: Create symbolic representations for situations described in everyday
language.
5. A carpenter is cutting a piece of wood that is eight feet long. He cuts off a piece
that is three feet then he cuts off five more small pieces, each of length x.
Which of the following is the expression which represents the number of feet of
wood that is left over?
a.
b.
c.
d.
8  15x
8  3  5x
8  3  5x
8  (5 x  3)
incorrect interpretation of 3 and five more
correct
not an expression
incorrect use of parenthesis
CRS Target 2.2b: Identify units associated with these variables and constants.
6. A bag of coins contains both quarters and dimes. Let 0.25q  0.1d represent the
total amount of money in the bag in dollars. What does q represent?
a.
b.
c.
d.
25 cents
Number of quarters
Quarters
Value of quarters
incorrect units
correct
not specific enough
incorrect units
7. A bag of coins contains both quarters and dimes. Let 0.25q  0.1d represent the
total amount of money in the bag in dollars. What should 0.25q represent?
a.
b.
c.
d.
Number of quarters
Percent of coins that are quarters
Value of quarters in cents
Value of quarters in dollars
CRS Target 2.2.c
incorrect units
incorrect interpretation of 0.25
incorrect monetary units
correct
Use correct mathematical symbols, terminology and notation.
8. Which expression represents “the difference of 9 and the quantity 5x times
negative four”?
a.
b.
c.
d.
9  5 x ( 4)
9  (5 x  4)
9  (5 x  4 )
(9  5 x )( 4)
correct
dropped parenthesis around -4
not multiplying by negative four
incorrect use of parenthesis
26
CRS Target 3.1.b Recognize patterns and apply mathematical concepts and
procedures in other subject areas and real world situations.
9. A flag is made so that the width is three feet longer than twice the height. Which
of the following mathematical expressions best represents the area of the flag if
the height is h meters?
a.
b.
c.
d.
2(h  3)h
( 2h  3)( h )
2(h  3)h
2( 2h  3)  2h
cannot interpret width
correct
incorrect width and three feet longer than
confusing perimeter with area
Short Answer (2 points each)
CRS Target 7.1.a
Explain the distinction between factor and term.
10. Consider the expression 4( x  y )  3 .



Determine the number of terms of the expression
Determine the constant term of the expression
Determine the factors of each term in the expression
2-point response: The student shows understanding of the distinction between
factor and term by indicating:




There are 2 terms
The constant term is 3
The factors of the first term are 4 and (x+y)
The factor of the second term is 3
1-point response: The student shows some understanding of the distinction
between factor and term by indicating two of above.
0-point response: The student shows little or no understanding of the distinction
between factor and term.
11. Describe the difference between a factor and a term.
27
Solution: Terms are separated by addition and factors are separated by
multiplication.
2-point response: The student shows understanding of the distinction between
factor and term by indicating:



Connection between terms and addition.
Connection between factors and multiplication.
Mentions how each are “separated” from other terms or factors.
1-point response: The student shows some understanding of the distinction
between factor and term by indicating at least one of above.
0-point response: The student shows little or no understanding of the distinction
between factor and term.
CRS Target 2.2b: Identify units associated with these variables and constants;
2.2c: Use correct mathematical symbols, terminology and notation.
12. A company decides to make and sell rubber ducks. The fixed start-up cost to
produce the ducks is $2,000 plus an additional 75 cents per rubber duck made.
Let n represent the number of rubber ducks produced.
Interpret and describe in words what 0.75n represents in this situation. Be sure
to indicate the appropriate units in your description.
2-point response: The student shows understanding of units associated with the
variables and correct mathematical notation by indicating:




The units of 0.75n are dollars
Includes the word “cost”
The expression represents the additional cost, not “total cost” or just “cost”.
Includes that the expression relates to “n rubber ducks”.
Example: 0.75n represents the cost, in dollars, to make n rubber ducks that the
company must pay on top of the $2000 start up costs.
1-point response: The student shows some understanding of units associated with
the variables and correct mathematical notation by including at least two out of the
four bullets.
0-point response: The student shows little or no understanding of units associated
with the variables and correct mathematical notation.
28
CRS Target 2.2b: Identify units associated with these variables and constants;
2.2c: Use correct mathematical symbols, terminology and notation.
3.4.a Recognize and clarify mathematical structures that are embedded in other
contexts.
13. It took Tessa 45 minutes to travel from her house to the pet store. Let r
represent Tessa’s speed in miles per hour during her trip.
Interpret and describe in words what 0.75r represents in regards to this
application. Be sure to indicate the appropriate units in your description.
2-point response: The student shows understanding of units associated with the
variables and correct mathematical notation by indicating:




The units are miles
Includes the word “distance”
Includes the units of r, which are miles per hour.
Includes that the expression relates to “r, Tessa’s speed”.
Example: 0.75r represents the distance in miles that Tessa traveled if her speed
in miles per hour was r.
1-point response: The student shows some understanding of units associated with
the variables and correct mathematical notation by including at least two out of the
four bullets.
0-point response: The student shows little or no understanding of units associated
with the variables and correct mathematical notation.
14. Kat’s Trail mix is made of 75% peanuts and 25% chocolate candies and is sold
by the pound. Let p represent the number of pounds of trail mix.
Interpret and describe in words what 0.75p represents in regards to this
application. Be sure to indicate the appropriate units in your description.
2-point response: The student shows understanding of units associated with the
variables and correct mathematical notation by indicating:



The units are pounds
The expression represents pounds of peanuts
Includes the units of p, which are pounds.
29

Includes that the expression relates to “p, the number of pounds of trail mix”.
Example: 0.75p represents the number of pounds of peanuts in the p pound
mixture of trail mix.
1-point response: The student shows some understanding of units associated with
the variables and correct mathematical notation by including at least two out of the
four bullets.
0-point response: The student shows little or no understanding of units associated
with the variables and correct mathematical notation.
3.1.b Recognize patterns and apply mathematical concepts and procedures in other
subject areas and real world situations.
15. John needs to build a new fence for his goats. He lives along a steep cliff so he
only needs to fence three sides of the rectangular pasture, but he also need to
separate the female from the male goats so he will need a strong partition in the
middle of the pasture as shown. The strong partition costs $3 per linear foot and
the rest of the fencing costs $1.50 per linear foot.
 Give an expression for the total linear feet of fencing, including the
partition.
 How many terms does your expression for the total linear feet of the
fencing contain?
 What are the units of the first term listed in your expression?
y
x
Solution: Total feet of fencing: 3x  y , 2 x  x  y , or x  x  x  y . Number
of terms: 2, 3, or 4, respectively. Units of first term: feet.
2-point response: The student shows understanding by including the following:


The units of the first term are feet.
The expression correctly represents total linear feet of fencing.
30

The stated number of terms matches the stated expression.
1-point response: The student shows some understanding by getting at least two
out of the three bullets correct.
0-point response: The student shows little or no understanding.
3.1.b Recognize patterns and apply mathematical concepts and procedures in other
subject areas and real world situations.
16. John needs to build a new fence for his goats. He lives along a steep cliff so he
only needs to fence three sides of the rectangular pasture, but he also need to
separate the female from the male goats so he will need a strong partition in the
middle of the pasture as shown. The strong partition costs $3 per linear foot and
the rest of the fencing costs $1.50 per linear foot.
 Give an expression for the cost of the center fence.
 Give an expression for the cost of the rest of the fencing.
 Give an expression for the total cost of the fencing.
 How many terms does your expression for the total cost of the fencing
contain?
y
x
Solution: Cost of center: 3x. Cost of rest: 1.50( 2 x  y ) , 1.50( 2)x  1.50y , or
3x  1.50y . Total Cost: 3x  1.50( 2 x  y ) , 3x  1.50( 2)x  1.50y ,
3x  3x  1.50y , or 6 x  1.5y . Number of terms: 2, 3, 3, or 2 respectively.
2-point response: The student shows understanding by including the following:




The expression for the cost of the center fence is correct.
The expression for the cost of the rest of the fence is correct.
The expression for the total cost is the sum of the previous two expressions.
The number of terms matches the student’s expression.
1-point response: The student shows some understanding by getting at least two
out of the four bullets correct.
0-point response: The student shows little or no understanding.
31
Extended response (4 points each)
3.2.b Transfer mathematical vocabulary, concepts, and procedures to other disciplinary
contexts and the real world.
17. A photographer needs to mix a 12% acid solution with a 7% acid solution to get a
10% acid solution. He wants a total of 2 liters of the 10% solution. Let 0.12x
represent the total amount of acid, in liters, in the solution that has a 12%
concentration of acid.



What does x represent?
What would 0.10(2)  0.12x represent?
Include units with both answers.
Solution: x represents the total amount of liters of the 12% concentration of
acid that is used in the mixture. 0.10(2)  0.12x represents the total amount
of acid, in liters in the solution that has a 7% concentration of acid.
4-point response: The student shows understanding of transferring mathematical
vocabulary and concepts to the real world by indicating:



Includes units of x, which are liters.
Specifies x is the liters of “the 12% concentration that is used in the mixture,” or
similar meaning.
Includes the units of 0.10(2)  0.12x , which are liters.

Specifies that 0.10(2)  0.12x is the total amount of acid in the solution that has a
7% concentration.
3-point response: The student shows some understanding of transferring
mathematical vocabulary and concepts to the real world by including at least three
out of the four bullets.
2-point response: The student shows some understanding of transferring
mathematical vocabulary and concepts to the real world by including at least two out
of the four bullets.
32
1-point response: The student shows little understanding of transferring
mathematical vocabulary and concepts to the real world by including at least one out
of the four bullets.
0-point response: The student shows no understanding of transferring
mathematical vocabulary and concepts to the real world.
33
Assessment
Name _____________________
Date ________
Period _____
Multiple Choice (1 point each)
1. How many terms are there in the expression ( x  3)(5 x  7)  10 x ?
a.
b.
c.
d.
2
3
4
5
2. The second term in the expression 3x  5( x  2)  30 x is
a.
b.
c.
d.
5
x
5x
5( x  2)
3. How many factors are there in the expression 11( x 2  5x  7) ?
a.
b.
c.
d.
1
2
3
4
4. How many terms are there in the second factor of the expression
2(3x  5y  10)(13xy  1) ?
a.
b.
c.
d.
1
2
3
6
5. A carpenter is cutting a piece of wood that is eight feet long. He cuts off a piece
that is three feet then he cuts off five more small pieces, each of length x.
Which of the following is the expression which represents the number of feet of
wood that is left over?
a.
b.
c.
d.
8  15x
8  3  5x
8  3  5x
8  (5 x  3)
34
6. A bag of coins contains both quarters and dimes. Let 0.25q  0.1d represent the
total amount of money in the bag in dollars. What does q represent?
a.
b.
c.
d.
25 cents
Number of quarters
Quarters
Value of quarters
7. A bag of coins contains both quarters and dimes. Let 0.25q  0.1d represent the
total amount of money in the bag in dollars. What should 0.25q represent?
a.
b.
c.
d.
Number of quarters
Percent of coins that are quarters
Value of quarters in cents
Value of quarters in dollars
8. Which expression represents “the difference of 9 and the quantity 5x times
negative four”?
a.
b.
c.
d.
9  5 x ( 4)
9  (5 x  4)
9  (5 x  4 )
(9  5 x )( 4)
9. A flag is made so that the width is three feet longer than twice the height. Which
of the following mathematical expressions best represents the area of the flag if
the height is h meters?
a.
b.
c.
d.
2(h  3)h
( 2h  3)( h )
2(h  3)h
2( 2h  3)  2h
35
Short Answer (2 points each)
10. Consider the expression 4( x  y )  3 .



Determine the number of terms of the expression
Determine the constant term of the expression
Determine the factors of each term in the expression
11. Describe the difference between a factor and a term.
12. A company decides to make and sell rubber ducks. The fixed start-up cost to
produce the ducks is $2,000 plus an additional 75 cents per rubber duck made.
Let n represent the number of rubber ducks produced.
Interpret and describe in words what 0.75n represents in this situation. Be sure
to indicate the appropriate units in your description.
13. It took Tessa 45 minutes to travel from her house to the pet store. Let r
represent Tessa’s speed in miles per hour during her trip.
Interpret and describe in words what 0.75r represents in regards to this
application. Be sure to indicate the appropriate units in your description.
36
14. Kat’s Trail mix is made of 75% peanuts and 25% chocolate candies and is sold
by the pound. Let p represent the number of pounds of trail mix.
Interpret and describe in words what 0.75p represents in regards to this
application. Be sure to indicate the appropriate units in your description.
15. John needs to build a new fence for his goats. He lives along a steep cliff so he
only needs to fence three sides of the rectangular pasture, but he also need to
separate the female from the male goats so he will need a strong partition in the
middle of the pasture as shown. The strong partition costs $3 per linear foot and
the rest of the fencing costs $1.50 per linear foot.
 Give an expression for the total linear feet of fencing, including the
partition.
 How many terms does your expression for the total linear feet of the
fencing contain?
 What are the units of the first term listed in your expression?
y
x
37
16. John needs to build a new fence for his goats. He lives along a steep cliff so he
only needs to fence three sides of the rectangular pasture, but he also need to
separate the female from the male goats so he will need a strong partition in the
middle of the pasture as shown. The strong partition costs $3 per linear foot and
the rest of the fencing costs $1.50 per linear foot.
 Give an expression for the cost of the center fence.
 Give an expression for the cost of the rest of the fencing.
 Give an expression for the total cost of the fencing.
 How many terms does your expression for the total cost of the fencing
contain?
y
x
Extended response (4 points each)
17. A photographer needs to mix a 12% acid solution with a 7% acid solution to get a
10% acid solution. He wants a total of 2 liters of the 10% solution. Let 0.12x
represent the total amount of acid, in liters, in the solution that has a 12%
concentration of acid.



What does x represent?
What would 0.10(2)  0.12x represent?
Include units with both answers.
38
VI.
EXTENSIONS:
Work Problems
1. Henry and Hannah decided they needed to repaint their barn. Without consulting
Hannah, Henry painted the barn yellow in 3 days. Hannah did not like the yellow
so she decided to repaint the barn lilac. It took her 2.5 days to paint the barn.
Henry was not pleased when he came home after a weekend away to see a lilac
barn, so Henry and Hannah agreed that they would repaint the barn red together.
Let t represent the number of days that Hannah and Henry paint the barn
together.
Extension for the Couch problem
1. Can plot this relationship on calculator and find max profit [occurs at (.41166,
7736)].
2. What does .41166 represent and what does that tell us about the price Sally
should sell a couch for?
3. What does 7336 represent?
Could have them write their own number “magic” problem.
39