Subject: Mathematics
State: Massachusetts
2 What is the value of 2 + 2 + 2 ?0 -1 -2
-6(A)
-5(A)
-(B)
(C)
5(D)
1 5
1 5
3 4
4 Which of the following is equivalent to the expression below? (2
)(2 )5 6
2(A) 11
2(B) 30
4(C) 11
4(D) 30
3 Simplify:
5 In this problem represents a positive number. c
The quotient rule for exponents says that if and are positive
integers with > , then = c m n m n m-n
After explaining to yourself why this is true, complete the
following exploration of the quotient rule when m ≤ :n
a. What expression does the quotient rule provide for when = ?m
n
b. If = , simplify without using the quotient rule.m n
c. What do parts (a) and (b) above suggest is a good definition for
c ?0
d. What expression does the quotient rule provide for ?
e. What expression do we get for if we use the value for found in
part (c)?c0
f. Using parts (d) and (e), propose a definition for the expression
− .c n
cm
cn
cm
cn
cm
cn
c0
cn
c0
cn
6 (a) An ant has a mass of approximately 4 × 10 grams and an
elephant has a mass of approximately 8 −3
metric tons. How many ants does it take to have the same mass as an
elephant?
(b) An ant is 10 cm long. If you put all the ants from your answer
to part (a) in line front to back, how long −1
would the line be?
(c) Find two cities in the United States that are a similar
distance apart to illustrate the length in part (b).
Note: 1 kg = 1000 grams, 1 metric ton = 1000 kg, 1 m = 100 cm, 1 km
= 1000 m.
8 The body of a 154-pound person contains approximately 2 × 10
milligrams of gold and 6 × 10-1 1
milligrams of aluminum. Based on this information, the number of
milligrams of aluminum in the body is how many times the number of
milligrams of gold in the body?
7 A dollar bill is about 3 × 10 inches thick. A penny is
about 6 × 10 inches thick.-4 -3
Which statement correctly compares these measurements?
(A) The thickness of a dollar bill is the thickness of
a penny.
(B) The thickness of a dollar bill is the thickness of
a penny.
(C) The thickness of a dollar bill is 10 times the thickness
of a penny.
(D) The thickness of a dollar bill is 20 times the thickness
of a penny.
1 10
1 20
9 It is said that the average person blinks about 1000 times an
hour. This is an order-of-magnitude estimate, that is, it is an
estimate given as a power of ten. Consider:
100 blinks per hour, which is about two blinks per minute.
10,000 blinks per hour, which is about three blinks per
second.
Neither of these are reasonable estimates for the number of blinks
a person makes in an hour. Make order-of- magnitude estimates for
each of the following:
a. Your age in hours.
b. The number of breaths you take in a year.
c. The number of heart beats in a lifetime.
d. The number of basketballs that would fill your classroom.
Can you think of others questions like these?
10 3908 Nyx is an asteroid between Mars and Jupiter. Let
represent the approximate distance from 3908 d Nyx to the
Sun.
The average distance from Venus to the Sun is about 7 × 10 miles.
The average distance from Jupiter to the 7
Sun is about 5 × 10 miles.8
At a certain time of year, the square of the distance from 3908 Nyx
to the Sun is equal to the product of the average distance from
Venus to the Sun and the average distance from Jupiter to the Sun.
This equation can be used to find the distance from 3908 Nyx to the
Sun, , at this time of year.d
= (7 × 10 )(5 × 10 )d 2 7
8
Which of the following numbers is equal to ?d 2
1.2 × 10(A) 16
3.5 × 10(B) 16
1.2 × 10(C) 57
3.5 × 10(D) 57
11 A company’s profit of $1.71 ×10 will be shared equally by its
3,800 employees. How much money 6
will each employee receive?
$4.5 ×10(A) 2
$4.5 ×10(B) 1
$4.5 ×10(C) 3
$4.5 ×10(D) 5
12 The figure shows a spreadsheet Shona made to record the mass, in
grams, of several samples of cells.
Her spreadsheet automatically converts the masses into scientific
notation.
List the four samples from least mass to greatest mass.
13 The total cost in dollars, , of a membership at each of four
health clubs is represented below in terms y of , the number of
months of the membership.x
Which representation has the greatest rate of change?
Health Club A(A)
Health Club B(B)
Health Club C(C)
Health Club D(D)
14 John and Kim wrote down two different functions that have the
same rate of change.
John’s function is represented by the table shown below.
Graph a function that could be Kim’s function.
15 The students at a middle school want to hire a DJ for an
end-of-the-year dance. The information below can be used to find
the total cost of hiring a DJ at each of four different
companies.
Which company’s cost has the greatest rate of change?
Awesome Entertainers(A)
Cool Beats(B)
Turntable Tunes(C)
Rock-N-Sounds(D)
16 Samir was assigned to write an example of a linear functional
relationship. He wrote this example for the assignment.
(a) Fill in the table above to create an example of the populations
of a certain county that is increasing in population by 10% each
year.
(b) State whether Samir's relationship represents a linear
functional relationship. Explain your reasoning.
17 Classify each equation as defining as a linear or
non-linear function of . Check one cell per column.y x
18 Which of the following graphs shows a linear function?
(A)
(B)
(C)
(D)
19 Which of these equations define as a nonlinear function of
? y x
Select correct answer.each
= + 3(B) y x2
= 12 − (F) y x2
20 A student has had a collection of baseball cards for several
years. Suppose that , the number of B baseball cards in the
collection, can be described as a function of , which is the
time in years since the t collection was started. Explain what each
of the following equations would tell us about the number of cards
in the collection over time.
(a) = 200 + 100 B t
(b) = 100 + 200 B t
(c) = 2000 − 100 B t
(d) = 100 − 200B t
21 A car is traveling down a long, steep hill. The elevation,
, above sea level (in feet) of the car when it is E miles from
the top of the hill is given by the equation = 7500 −
250 , where can be any number from 0 d E d d
to 6.
Find the slope and intercepts of the graph of this function and
explain what they mean in the context of the moving
car.
22 The prices and sizes of several houses for sale in a
neighborhood are shown in the scatter plot below.
Which equation represents the line of best fit?
y = 100x(A)
y = 1,000x + 50,000(D)
23 The graph below shows Carlos’s speed on his trip to
school.
Based on the graph, when is Carlos’s speed decreasing most
rapidly?
for times between B and C(A)
for times between D and E(B)
for times between E and F(C)
for times between H and I(D)
24 The school is 100 meters from Jason's house. The following
describes his most recent trip:
He walked 50 meters toward school in 2 minutes. He realized he left
a book at home.
He turned around and walked home at the same speed.
He spent 1 minute looking for his book before he found it.
He walked all the way to school at twice his original speed.
Finish the graph to accurately represent Jason's trip.
25 Aqua Water Company purchased a new water tank. The tank is a
cylinder with a diameter of 14 feet and a height of 24 feet. What
is the volume of the tank in cubic feet?
168π cubic feet(A)
336π cubic feet(B)
1,176π cubic feet(C)
4,704π cubic feet(D)
26 The sphere below has a 3-inch radius. The right cylinder has a
radius of 3 inches and a height of 4 inches.
(a) What is the volume, in cubic inches, of the sphere?
(b) What is the volume, in cubic inches, of the cylinder?
(c) Lin claims that the volume of any sphere with radius inches is
always to the volume of a cylinder r equal
with a radius of inches and a height of inches,
when = . Show all work necessary to justify Lin's r h h
r
claim.
4 3
28 A square is drawn on a coordinate plane. Which transformation
would change the perimeter of the square?
a translation of 2 units down(A)
a reflection across the x-axis(B)
a dilation with a scale factor of 2(C)
a 90 clockwise rotation about the origin(D)
27 The figure below shows a cylindrical tank that is to be filled
with gasoline. How many cubic feet of gasoline will be required to
fill the tank?
(A) 8π ft3
(B) 16π ft3
(C) 18π ft3
(D) 32π ft3
29 Pentagon is shown on the coordinate plane. CDEFG
Pentagon is translated 7 units up and 5 units left, resulting
in pentagon ' ' ' ' ' (not shown).CDEFG C D E F G
Complete the statements below using “greater than”, “less
than” or “equal to”.
The length of is ____________________ to the length of . FG ’
’F C
The perimeter of pentagon is ____________________ the perimeter of
pentagon ’ ’ ’ ’ ’.CDEFG C D E F G
30 Lines and are parallel on a coordinate plane. Lines and are
transformed by the same rotation‚ m n m n resulting in image lines
and . Which statement describes the relationship between lines and
?s t s t
Lines and are parallel.(A) s t
Lines and are perpendicular.(B) s t
Lines and are intersecting but not perpendicular.(C) s t
The relationship between lines and cannot be determined without
knowing the angle of (D) s t the rotation.
31 Parallelogram is shown on the coordinate plane.WXYZ
Parallelogram (not shown) is the reflection of
parallelogram across the -axis. Which W′X′Y′Z′ WXYZ y statement is
true?
Line is parallel to line (A) W’X’ W’Z’
Line is parallel to line (B) W’X’ Z’Y’
Line is perpendicular to line (C) W’X’ W’Z’
Line is perpendicular to line (D) W’X’ Z’Y’
32 Consider the equation 5 − 2 = 3. If possible, write a
second linear equation to create a system of x y equations that
has:
(a) Exactly 1 solution.
(b) Exactly 2 solutions.
(d) Infinitely many solutions.
For each of these cases, how many such equations are
possible?
33 Let (x, y) be the solution of the system of equations
below. 4x + 3y = 17 3x +
2y = 12 What is the value of − x
y?
-11(A)
-1(B)
5(C)
9(D)
35 The equation of line is = −2 + 8 .j y x
The equation of line is y = 3 − 7 .k x
34 The table below shows two systems of linear
equations.
Indicate whether each system of equations has no solution,
one solution or infinitely many solutions by selecting the correct
cell in the table. Select one cell per column.
Graph the system of equations on the coordinate plane
Plot point on the graph. P
36 A transformation of Pentagon 1 resulted in Pentagon 2.
37 Below is a picture of two congruent rectangles:
Which transformation was used?
a translation 10 units to the right(B)
a translation 5 units to the right(C)
a reflection across the x-axis(D)
(a) Show that the rectangles are congruent by finding a translation
followed by a rotation which maps one of the rectangles onto the
other.
(b) Explain why the congruence of the two rectangles can not be
shown by translating Rectangle 1 to Rectangle 2.
(c) Can the congruence of the two rectangles be shown with a single
reflection? Explain.
38 Triangles and are shown below in the coordinate plane:ABC
PQR
(a) Show that Δ is congruent to Δ with a reflection followed by a
translation.ABC PQR
(b) If you reverse the order of your reflection and translation in
part (a) does it still map Δ to Δ ?ABC PQR
(c) Find a second way, different from your work in part (a), to map
Δ to Δ using translations, ABC PQR rotations, and/or
reflections.
39 In a park, the locations of a drinking fountain, a streetlight,
and a flagpole form a right triangle. The drinking fountain is 12
yards north of the streetlight, and the flagpole is 5 yards east of
the drinking fountain, as shown in this diagram.
What is the shortest distance from the streetlight to the
flagpole?
7 yards(A)
11 yards(B)
13 yards(C)
17 yards(D)
40 You are a representative for a cell phone company and it is your
job to promote different cell phone plans.
Your boss asks you to visually display the costs of three plans and
compare them so you can point out the advantages of each plan to
your customers.
The information on the plans is shown in the boxes below.
(a) Use this information to write equations for each plan, and then
graph the equations and estimate their intersections.
(b) A customer wants to know how to decide which plan will save her
the most money. Describe which plan has the lowest cost for each
possible range of numbers of text messages a customer might
send.
41 The local swim center is making a special offer. They usually
charge $7 per day to swim at the pool. This month swimmers can pay
an enrollment fee of $30 and then the daily pass will only be $4
per day.
(a) Suppose you do not take the special offer. Write an equation
that represents the amount of money you would spend based on how
many days you go to the pool if the passes were bought at full
price.
(b) Write a second equation that represents the amount of money you
would spend if you decided to take the special offer.
(c) Graph your two equations from parts (a) and (b).
(d) After how many days of visiting the pool will the special offer
be a better deal? How can you tell algebraically? How can you see
this graphically?
(e) You only have $60 to spend for the summer on visiting the pool.
Which offer would you take? Explain.
42 The coordinate plane shown below has a figure in the third
quadrant.
Which of the following shows the same figure after it has been
reflected across the y-axis and then reflected across the
x-axis?
(A)
(B)
(C)
(D)
43 Triangle is shown on this coordinate grid.ABC
Triangle is dilated with the origin as the center of the
dilation. Which ordered pair could represent the ABC image of
point (5, 2) after the dilation?C
(2.5, 1)(A)
(5, −2)(B)
(7.5, 4.5)(C)
(−1, −4)(D)
ii. Center and scale factor 3.B
iii. Center and scale factor .C
b. For each dilation, answer the following questions:
i. By what factor do the base and height of the triangle change?
Explain.
ii. By what factor does the area of the triangle change?
Explain.
iii. How do the angles of the scaled triangle compare to the
original? Explain.
1 2
45 Seven line segments are shown on the coordinate
plane.
Which of these segments could be the image of segment after a
sequence of reflections, rotations, and/or AB translations?
Select correct answer. each
line segment (A) CD
line segment (B) EF
line segment (C) GH
line segment (D) JK
line segment (E) LM
line segment (F) NP
46 For each linear equation in this table, indicate whether the
equation has no solution, one solution, or infinitely many
solutions.
47 Consider the equation 3(2 + 5) = + .x ax b
(a) Find one value for and one value for so that there
is exactly one value of that makes the equation a b x true.
Explain your reasoning.
(b) Find one value for and one value for so that there
are infinitely many values of that make the a b x equation
true. Explain your reasoning.
48 Which ordered pair is the solution of the system of equations
below?
x + 2 = 6 y
3 + 8 = 4x y
(2, 2)(A)
(4, 10)(B)
(10, −2)(C)
(20, −7)(D)
49 Ted has some red blocks and some green blocks. • Each red block
weighs the same number of ounces.
• Each green block weighs the same number of ounces.
• The total weight of 2 red blocks and 6 green blocks is 23
ounces.
• The total weight of 3 red blocks and 4 green blocks is 22
ounces
What is the total weight of 1 red block 1 green
block?and
3 ounces (A)
6 ounces (B)
6.5 ounces (C)
13.5 ounces (D)
50 What is the coordinate of the solution to this system of
equations?y
−2 + 3 = −6x y
5 − 6 = 15x y
Write your answer below.
51 Three students solved the equation 3(5 − 14) = 18 in
different ways, but each student arrived at the x correct answer.
Circle YES or NO to indicate whether each solution shows a correct
method for solving the equation.
52 Consider the equation 3(2 + 5) = + .x ax b
(a) Find one value for and one value for so that there
is exactly one value of that makes the equation a b x true.
Explain your reasoning.
(b) Find one value for and one value for so that there
are infinitely many values of that make the a b x equation
true. Explain your reasoning.
53 A history club sold rolls of wrapping paper as a fundraiser. The
wrapping paper was sold in small rolls and large rolls.
The club earned $3.00 for every small roll sold.
The club earned $4.50 for every large roll sold.
The club sold 10 more large rolls than small rolls.
How many small rolls did the history club sell?
54 A company uses a new machine and an old machine to print
booklets. Each machine prints booklets at a constant rate. The
graph and the equation represent the relationships between , the
number of minutes the x machines print, and , the number of
booklets printed.y
The company uses both machines to print a total of 1,250 booklets.
Both machines start printing at the same time. During printing, the
old machine breaks down and stops printing. The new machine
continues printing for an additional 14 minutes and completes the
order.
What is the total number of minutes the new machine prints? Show or
explain all your work.
Write your answer and your work below.
55 An equation is incorrectly solved below.
Equation: 2 + 3 = −4x
Step 2: 2 = −1x
Step 3: = − 1 2
Step 4: = −x 1 2
What is the first step that shows an error in the solution to the
equation?
Step 1(A)
Step 2(B)
Step 3(C)
Step 4(D)
2x 2
1 2
1 2
(a) What is the cost per pound of this coffee?
(b) How many pounds of coffee could Lena buy for $1.00?
(c) Draw a graph in the coordinate plane of the relationship
between the number of pounds of coffee and the total cost. If
you can, use technology to do this.
(d) In this situation, what is the meaning of the slope of the line
you drew in part (3)?
57 Amanda graphed line and line on a coordinate plane, as shown
below.m n
Which of the following statements is true?
The slope of line is greater than the slope of line .(A) m n
The slope of line is greater than the slope of line .(B) n m
The -intercept of line is greater than the -intercept of line .(C)
x m x n
The -intercept of line is greater than the -intercept of line .(D)
y n y m
58 The graph below represents , the cost in dollars of pounds of
salad at a salad bar.y x
What is the unit rate for the cost of a salad at the salad
bar?
$0.50 per pound(A)
$1.00 per pound(B)
$2.00 per pound(C)
$4.00 per pound(D)
59 Mindy is saving money She started with $0. After 6 weeks, she
had $90 saved. Mindy is not sure exactly how much money she
saved each week. She assumes that she saved money at a
constant rate from when she started saving money through week
6.
Part A: Create a graph that can be used to model the number of
dollars, , Mindy saves in weeks.y x
Part B: Explain what the slope of the line you drew
represents.
Part C: Explain how the line you drew can be used to predict the
number of weeks it will take Mindy to save $150. Include in your
explanation any assumptions that must be made in order to make the
prediction.
60 The table below shows two coordinate pairs that satisfy
the equation for some (x, y) y = mx + b numbers
and .m b
(a) If = 7, determine possible values for and .
Explain your choices.m y 1 y 2
(b) Find another pair of -values that could work for = 7.
Explain why they would work. How do these -y m y values
compare to the first pair you found for = 7?m
(c) Use the same -values in the table and find possible
values for and if = 3. Explain your
choices.x y 1 y 2 m
(d) Find another pair of -values that could work
for = 3. Explain why they would work. How do these -y m
y values compare to the first pair you found for = 3?m
61 John and Kim wrote down two different functions that have the
same rate of change. John's function is shown in the table
below.
Graph a function on the grid that could be Kim's function.
62 Use the graph shown below to answer the question.
Which statement is true?not
(A) The slope of the line is .
(B) The slope of the line is - .
(C) The smaller triangle and the larger triangle shown are
similar.
(D) The simplified ratio of the vertical side length to the
horizontal side length of each triangle
is .
3 5
3 5
3 5
63 A function machine takes an input, does something to this input
according to a rule, and returns a unique output.
Given below are tables of input-output pairs for different function
machines. Fill in the remaining table entries and describe each
function rule in words.
64 The table below shows a relationship between x and y that is a
function.not
Write one ordered pair that can be removed from the table to make
the relationship between and a x y function.
65 Which of the following tables shows a relationship that is
a function?not
(A)
(B)
(C)
(D)