Mathematics
The Mathematics Section of the HSPA assessment is divided into four content clusters. Each of
these clusters reflects knowledge and skills specified in New Jersey’s Core Curriculum Content
Standards.
High School Proficiency Assessment (HSPA) Mathematics Clusters
I.
Number Sense, Concepts, and Applications
A. Understand types of numbers, our numeration system, and the ways they are used and
applied in real-world situations.
B. Apply ratios, proportions, and percents in a variety of situations.
II.
Spatial Sense and Geometry
A. Recognize, visualize, analyze, and apply geometric properties, relationships, and
patterns in real-world and/or problem-solving contexts using models, manipulatives,
or technology.
B. Use coordinate geometry in problem-solving situations and apply the principles of
congruence, similarity, and transformations.
C. Apply the principles of measurement and geometry to solve problems involving
direct and indirect measurement.
III.
Data Analysis, Probability, Statistics, and Discrete Mathematics
A. Determine, interpret, and use probabilities of simple and compound events.
B. Understand and interpret statistical distributions and apply to real-world situations.
C. Collect, organize, represent, analyze, and interpret data.
D. Apply the concepts and methods of discrete mathematics to model and explore a
variety of practical situations.
E. Use iterative and recursive patterns and processes to model a variety of practical
situations and solve problems.
IV.
Patterns, Functions, and Algebra
A. Recognize, create, and extend a variety of patterns and use inductive reasoning to
understand and represent mathematical and other real-world phenomena.
B. Use various types of functions to represent mathematical or real-world situations.
C. Use algebraic concepts and processes to concisely express, analyze, and model realworld situations.
17
Types of Questions
The multiple-choice (MC) questions on the HSPA Mathematics test assess higher-level cognitive
processes than the questions in traditional multiple-choice tests. It will take you an average of
between one and two minutes to answer each MC question. The answers are computer scored
and have a weight of one point each.
Open-ended (OE) questions require you to construct your own written or graphical responses and
explain your responses. It will take approximately ten minutes to answer each OE question. Your
responses are hand scored on a scale from 0 to 3.
The general scoring guide on page 20 was created to help readers score open-ended questions
consistently within a single test and across different forms of the test. This scoring guide is used
by the trained readers who will score the Mathematics open-ended questions on the HSPA.
The following table shows you how many multiple-choice and open-ended questions to expect.
Question Type
Number of Questions
11th Grade
MC
40
OE
8
You will be provided with a Mathematics Reference Sheet that contains a ruler, geometric
shapes, formulas, and other information you may find useful as you take the test. You will also
be provided with a calculator to help you solve problems.
18
19
Scoring Guide for Mathematics Open-Ended (OE) Questions
(Generic Rubric)
3-Point Response
The response shows complete understanding of the problem’s essential
mathematical concepts. The student executes procedures completely and gives relevant
responses to all parts of the task. The response contains few minor errors, if any. The
response contains a clear, effective explanation detailing how the problem was solved so
that the reader does not need to infer how and why decisions were made.
2-Point Response
The response shows nearly complete understanding of the problem’s essential
mathematical concepts. The student executes nearly all procedures and gives relevant
responses to most parts of the task. The response may have minor errors. The explanation
detailing how the problem was solved may not be clear, causing the reader to make some
inferences.
1-Point Response
The response shows limited understanding of the problem’s essential
mathematical concepts. The response and procedures may be incomplete and/or may
contain major errors. An incomplete explanation of how the problem was solved may
contribute to questions as to how and why decisions were made.
0-Point Response
The response shows insufficient understanding of the problem’s essential
mathematical concepts. The procedures, if any, contain major errors. There may be no
explanation of the solution, or the reader may not be able to understand the explanation.
The reader may not be able to understand how and why decisions were made.
The generic rubric above is used as a guide to develop specific scoring guides or rubrics for each
of the open-ended (OE) questions that appear on the New Jersey fourth-grade (ESPA), eighthgrade (GEPA), and eleventh-grade (HSPA) proficiency assessments in Mathematics. The generic
rubric helps ensure that students are scored in the same way for the same demonstration of
knowledge and skills regardless of the test question.
20
HSPA MATHEMATICS SAMPLE QUESTIONS
Cluster I, Macro A
cáåÇ=íÜÉ=äÉåÖíÜI=ïáÇíÜI=~åÇ=~êÉ~=çÑ=É~ÅÜ=çÑ
íÜÉ=R=ëÜ~ÇÉÇ=êÉÅí~åÖäÉëK
tÜ~í=áë=íÜÉ=íçí~ä=~êÉ~=êÉéêÉëÉåíÉÇ=Äó=íÜÉ=R
êÉÅí~åÖäÉë\
eçï=Çç=óçì=íÜáåâ=íÜÉ=~êÉ~=çÑ=íÜÉ=R
êÉÅí~åÖäÉë=Åçãé~êÉë=íç=íÜÉ=~êÉ~=çÑ=íÜÉ
êÉÖáçå=ìåÇÉê=íÜÉ=ÅìêîÉ\=bñéä~áå=óçìê
êÉ~ëçåáåÖK
Rationale:
Since each of the shaded rectangles has the same width of 2 units, you only need to read
the height of each from the graph and then multiply the width and height to obtain the area. The
height of the first rectangle is 100 units, so its area is 100 units * 2 units or 200 sq. units.
The total area of the 5 rectangles is
(100 * 2) + (90 * 2) + (75 * 2) + (50 * 2) + (20 * 2) = 670 sq. units.
The answer for the third bullet may vary, but you must explain your reasoning.
21
Cluster I, Macro B
OK qÜÉ=çêáÖáå~ä=íáÅâÉí=éêáÅÉ=çÑ=~=ëÜáêí=áë=AORKVVK
aìêáåÖ=~=ÅäÉ~ê~åÅÉ=ë~äÉI=íÜáë=ëÜáêí=áë=êÉÇìÅÉÇ=Äó
QMB=çÑ=íÜÉ=íáÅâÉí=éêáÅÉX=íÜÉå=ORB=çÑ=íÜÉ=êÉÇìÅÉÇ
éêáÅÉ=áë=í~âÉå=çÑÑ=~í=íÜÉ=Å~ëÜ=êÉÖáëíÉêK
oçìåÇÉÇ=íç=íÜÉ=åÉ~êÉëí=éÉååóI=ïÜ~í=áë=íÜÉ
éêáÅÉ=é~áÇ=Äó=íÜÉ=ÅìëíçãÉê\
(Answer to first bullet: ANNKSV=or ANNKTMF
tÜ~í=éêáÅÉ=ïçìäÇ=íÜÉ=ÅìëíçãÉê=Ü~îÉ=é~áÇ=áÑ
íÜáë=ëÜáêí=ïÉêÉ=ëçäÇ=~í=~=çåÉJíáãÉ=êÉÇìÅíáçå
çÑ=SRB=Ñêçã=íÜÉ=çêáÖáå~ä=éêáÅÉ\
tÜó=ÇáÇå í=íÜÉ=ëíçêÉ=ëáãéäó=ëÉää=íÜáë=ëÜáêí=~í
SRB=çÑÑ=íÜÉ=çêáÖáå~ä=ëíáÅâÉê=éêáÅÉ\
Rationale:
Amount of 40% discount: $25.99 * 0.40 = $10.396 ≈ $10.40
Price after 40% discount: $25.99 – $10.40 = $15.59
Amount of 25% discount: $15.59 * 0.25 = $3.897 ≈ $3.90
Price after 25% discount: $15.59 – $3.90 = $11.69
($11.70 is acceptable if you round at the end instead of after each step.)
By breaking up the 65% discount into 40% and 25% discounts, the store was able to sell
the shirt at a higher price than it would have if the store sold the shirt at 65% off the original
price.
22
Cluster II, Macro A
PK cçê=~=ëÉïáåÖ=éêçàÉÅíI=q~åó~=Åìí=áëçëÅÉäÉë
íêá~åÖäÉë=Ñêçã=~=ëíêáéÉÇ=éáÉÅÉ=çÑ=ã~íÉêá~ä
ïÜÉêÉ=íÜÉ=ëíêáéÉë=~êÉ=é~ê~ääÉäK=qÜÉ=îÉêíÉñ
~åÖäÉ=çÑ=íÜÉ=áëçëÅÉäÉë=íêá~åÖäÉ=ï~ë=RMŒ=~åÇ
BC =áë=é~ê~ääÉä=íç=íÜÉ=Ä~ëÉK
cáåÇ=íÜÉ=ãÉ~ëìêÉ=çÑ=∠_`b=~ë=ëÜçïå=áå=íÜÉ
Çá~Öê~ãK
^K
==RMÿ
_K
==SRÿ
j `K
NNRÿ
aK
NPMÿ
Rationale:
Since the triangle is isosceles and the vertex angle is given to be 50°, the two remaining
angles must be 65°. 180° = 50° + x + x; x = 65°
The measure of ∠AGE is 180° since that is the measure of a straight line. Therefore,
m∠ACB + m∠BCE = 180.
Since BC and FG are parallel and AG intersects both BC and FG , m∠ACB =
m∠AGF = 65.
To solve for m∠BCE, use the following:
180 – m∠ACB = 180 – 65 = 115.
23
Cluster II, Macro B
QK ^=Äç~í=ëí~êíë=~í=mçáåí=^I=íê~îÉäë=S=ãáäÉë=ÇìÉ
É~ëí=íç=mçáåí=_I=~åÇ=íÜÉå=íìêåë=~åÇ=íê~îÉäë
U=ãáäÉë=ÇìÉ=ëçìíÜ=íç=mçáåí=`=çå=íÜÉ=ëÜçêÉK
•
•
•
•
lå=íÜÉ=ÖêáÇ=éêçîáÇÉÇ=áå=óçìê=~åëïÉê
ÑçäÇÉêI=ÅçåëíêìÅí=~=ëÅ~äÉ=Çê~ïáåÖ=ìëáåÖ
îÉÅíçêë=íç=ëÜçï=íÜÉ=Äç~í ë=ãçîÉãÉåíI
ëí~êíáåÖ=Ñêçã=éçáåí=^K
aê~ï=~=îÉÅíçê=íÜ~í=ïçìäÇ=ëÜçï=íÜÉ
ÇáêÉÅí=é~íÜ=Ñêçã=éçáåí=^=íç=éçáåí=_K
tÜ~í=ïçìäÇ=ÄÉ=íÜÉ=~ééêçñáã~íÉ=åìãÄÉê
çÑ=ãáäÉë=íÜÉ=Äç~í=ÅçìäÇ=Ü~îÉ=íê~îÉäÉÇ
~äçåÖ=íÜáë=é~íÜ\
^ééêçñáã~íÉäó=Üçï=ã~åó=ÇÉÖêÉÉë=Ñêçã
kçêíÜ=ïçìäÇ=íÜáë=é~íÜ=ÄÉ\=bñéä~áå=Üçï
óçì=~êêáîÉÇ=~í=óçìê=~åëïÉêK
Rationale:
If the boat travels 6 miles due east from Point A to Point B and then turns and travels
8 miles due south from Point B to Point C, the boat’s path forms a right angle. If you use straight
lines to connect Point A to Point B, Point B to Point C, and Point C to Point A, the result is a
right triangle. To find the length of AC , you apply the Pythagorean Theorem, 62 + 82 = c2.
Solving for c yields c = 10. The measure of ∠C is approximately 37° since tan 37° ≈ 36.869 ≈
24
6
.
8
Cluster II, Macro C
RK
aÉëÅêáÄÉ=áå=ÇÉí~áä=Üçï=óçì=ÅçìäÇ=ìëÉ=~
Å~äÅìä~íçê=ïáíÜ=íêáÖçåçãÉíêáÅ=ÑìåÅíáçåë=íç
ÜÉäé=ÑáåÇ=íÜÉ=ÜÉáÖÜí=çÑ=óçìê=ëÅÜççä=ÄìáäÇáåÖ
áÑ=óçì=âåÉï=íÜÉ=Çáëí~åÅÉ=Ñêçã=íÜÉ=Ä~ëÉ=çÑ
íÜÉ=ÄìáäÇáåÖ=íç=íÜÉ=éçáåí=ïÜÉêÉ=óçì=ïÉêÉ
ëí~åÇáåÖ=~åÇ=íÜÉ=~åÖäÉ=Ñêçã=íÜÉ=ÖêçìåÇ=~í
óçìê=ÑÉÉí=íç=íÜÉ=íçé=çÑ=íÜÉ=ÄìáäÇáåÖK
Rationale:
If you know the distance d from the base of the building to Point A where you are
standing and you know angle θ, which the ground makes with the hypothetical line from Point A
to the top of the building, you can apply the formula tan θ =
building. Solving for h results in h = d (tan θ).
25
h
, where h is the height of the
d
Cluster III, Macro A
SK tÉ~íÜÉêéÉêëçåë=éêÉÇáÅí=íçãçêêçï ë=ïÉ~íÜÉê
Ä~ëÉÇ=çå=ïÜ~í=Ü~ë=Ü~ééÉåÉÇ=áå=íÜÉ=é~ëí=çå
íÜÉ=Ç~óë=ÑçääçïáåÖ=Ç~óë=àìëí=äáâÉ=íçÇ~óK=aìêáåÖ
íÜÉ=é~ëí=RM=óÉ~êëI=íÜÉêÉ=Ü~îÉ=ÄÉÉå=PUM=Ç~óë
íÜ~í=Ü~îÉ=ÄÉÉå=àìëí=äáâÉ=íçÇ~óI=~åÇ=çÑ=íÜçëÉI
OMM=Ü~îÉ=ÄÉÉå=ÑçääçïÉÇ=Äó=~=ÅäÉ~ê=Ç~óK=tÜáÅÜ
çÑ=íÜÉ=ÑçääçïáåÖ=áë=íÜÉ=~ééêçñáã~íÉ=éêçÄ~Äáäáíó
çÑ=~=ÅäÉ~ê=Ç~ó=íçãçêêçï=íÜ~í=ïçìäÇ=ÄÉ=ÖáîÉå=Äó
~=ïÉ~íÜÉêéÉêëçå=ìëáåÖ=íÜÉ=éêÉÇáÅíáçå=êìäÉ
ÇÉëÅêáÄÉÇ=áå=íÜáë=éêçÄäÉã\
^K NPB
_K PQB
j `K RPB
aK SSB
Rationale:
Since there have been 200 days out of 380 days which were followed by a clear day, the
200
experimental probability of tomorrow being a clear day is
≈ 0.5263 ≈ 53%.
380
26
Cluster III, Macro B
TK qÜÉ=Ç~í~=éêçîáÇÉÇ=ëÜçï=íÉëí=ëÅçêÉë=Ñçê=íïÉäîÉ=ëíìÇÉåíë=~åÇ=íÜÉ=åìãÄÉê=çÑ=Üçìêë=íÜÉó=ëíìÇáÉÇ=Ñçê=íÜÉ
íÉëí=ÇìêáåÖ=íÜÉ=íÜêÉÉ=Ç~óë=éêáçê=íç=í~âáåÖ=áíK
•
lå=íÜÉ=ÖêáÇ=éêçîáÇÉÇ=áå=óçìê=~åëïÉê=ÑçäÇÉêI=ÅçåëíêìÅí=~=ëÅ~ííÉê=éäçí=çÑ=íÜáë=Ç~í~K
•
açÉë=íÜÉêÉ=~ééÉ~ê=íç=ÄÉ=~=êÉä~íáçåëÜáé=ÄÉíïÉÉå=~=ëíìÇÉåí ë=íÉëí=ëÅçêÉ=~åÇ=íÜÉ=íáãÉ=ëéÉåí
ëíìÇóáåÖ\=rëÉ=íÜÉ=ëÅ~ííÉê=éäçí=íç=ëìééçêí=óçìê=~åëïÉêK
•
aç=~åó=çÑ=íÜÉ=éçáåíë=~ééÉ~ê=íç=ÄÉ=çìíäáÉêë\=bñéä~áåK
Rationale:
Student must draw a correct scatter plot of the data.
For the most part, it looks as though a student’s test score improves with more time spent
studying. Answers as to which points are outliers may vary, as long as your response shows a
clear understanding of the definition of outliers and you support your answers.
27
Cluster III, Macro D
Cluster III, Macro C
UK qÜÉ=ÅÜ~êí=ÄÉäçï=ëÜçïë=íÜÉ=åìãÄÉêë=çÑ=ëíìÇÉåíë
VK
áå=~=ÜçãÉêççã=íÜ~í=îçíÉÇ=Ñçê=É~ÅÜ=ëíìÇÉåí
ÅçìåÅáä=Å~åÇáÇ~íÉK
qÜÉ=Çá~Öê~ã=~ÄçîÉ=êÉéêÉëÉåíë=~=åÉíïçêâ=çÑ
ÇáêÉÅí=~áê=êçìíÉë=ÄÉíïÉÉå=ÅáíáÉëK=tÜáÅÜ=çÑ=íÜÉ
ÑçääçïáåÖ=ã~íêáÅÉë=Å~å=êÉéêÉëÉåí=íÜÉ=åÉíïçêâ
fÑ=íÜáë=Ç~í~=ïÉêÉ=Çê~ïå=áå=~=ÅáêÅäÉ=Öê~éÜI
~ÄçîÉ\=^=òÉêç=áåÇáÅ~íÉë=åç=ÇáêÉÅí=~áê=êçìíÉK=^=çåÉ
~ééêçñáã~íÉäó=Üçï=ã~åó=ÇÉÖêÉÉë=áå=íÜÉ=ÅÉåíê~ä
êÉéêÉëÉåíë=íÜ~í=~=ÇáêÉÅí=~áê=êçìíÉ=ÉñáëíëK
~åÖäÉ=çÑ=íÜÉ=ëÉÅíçê=çÑ=íÜ~í=ÅáêÅäÉ=ïçìäÇ=êÉéêÉëÉåí
qáâç ë=îçíÉë\
^K OO
_K SU
j `K TV
aK VM
Rationale:
Tiko received 7 votes out of the total 32
7
students who voted.
≈ 0.218 ≈ 0.22 = 22%.
32
Since there are 360° in a circle, 22% of
360° would be 79.2° or 79°.
Rationale:
Since there are line segments connecting A to
B and B to C, there are 1’s in Row A for Column
B and C corresponding to AB & BC . The rest
of the matrix follows accordingly.
28
Since the number in Column C is always
divisible by 3 and 32 is 1 less than 33
(which is divisible by 3), the number 32
appears in Column B. To find the row
number, substitute the number 32 into the
equation for Column B.
3n – 1 = 32
3n = 32 + 1
3n = 33 and n = 11 (the 11th row)
Cluster IV, Macro A
NNK fã~ÖáåÉ=íÜ~í=íÜÉ=í~ÄäÉ=ÄÉäçï=ÅçåíáåìÉëI=êçï
~ÑíÉê=êçïI=ÑçääçïáåÖ=íÜÉ=ë~ãÉ=é~ííÉêå=ÑçêÉîÉêK
To find the column for 1783, divide by 3.
Note that the answer is 594 with a remainder
of 1. The remainder indicates that the
number occurs in the first column (Column
A) of the next row, which is Row 595. You
can confirm your answer by using the
formula for Column A:
•
•
•
•
•
•
3n - 2 = 1783
3n = 1783 + 2
3n = 1785
and n = 595
`çãéäÉíÉ=íÜÉ=SíÜ=~åÇ=TíÜ=êçïë=áå=óçìê
~åëïÉê=ÑçäÇÉêK
tÜ~í=åìãÄÉêë=~êÉ=áå=íÜÉ=NMMíÜ=êçï\
têáíÉ=ÉñéêÉëëáçåë=Ñçê=íÜÉ=åìãÄÉêë=áå=íÜÉ
n íÜ=êçïK
få=ïÜáÅÜ=êçï=ïáää=íÜÉ=åìãÄÉê=PO=ÄÉ
ÑçìåÇ\
få=ïÜáÅÜ=Åçäìãå=ïáää=íÜÉ=åìãÄÉê=PO=ÄÉ
ÑçìåÇ\=bñéä~áå=óçìê=~åëïÉêK
få=ïÜáÅÜ=Åçäìãå=ïáää=íÜÉ=åìãÄÉê=NTUP=be
found? Explain your answer.
Rationale:
Column C is always 3 times the row number
(that is, 3n)
Column B is always 3 times the row number
less 1 (that is, 3n – 1)
Column A is always 3 times the row number
less 2 (that is, 3n – 2)
Therefore,
Row 6:
Row 7:
Row 100
16, 17, 18
19, 20, 21
298, 299, 300
30
Cluster IV, Macro B
NOK qÜÉ=Öê~éÜ=çÑ=~=ÑìåÅíáçåI=ÑExFI=áë=ÖáîÉå=ÄÉäçïK
tÜáÅÜ=Öê~éÜ=ïçìäÇ=êÉéêÉëÉåí=ÑExF= =O\
Rationale:
The graph of the function, f(x), is shown.
So, f(x) – 2 is just the graph of f(x) translated
2 units in the negative y direction.
31
Cluster IV, Macro C
NPK qÜÉ=éä~óÉêë=çå=~=Ä~ëâÉíÄ~ää=íÉ~ã=ëÅçêÉÇ
TR=éçáåíë=áå=íÜÉ=Ñáå~ä=Ö~ãÉ=çÑ=íÜÉ=ëÉ~ëçåK
aìêáåÖ=íÜ~í=Ö~ãÉI=íÜÉó=ã~ÇÉ=íïáÅÉ=~ë=ã~åó
ÑáÉäÇ=Öç~äë=~ë=íÜÉó=ÇáÇ=ÑêÉÉ=íÜêçïëK=Eb~ÅÜ
ÑáÉäÇ=Öç~ä=áë=ïçêíÜ=íïç=éçáåíëI=~åÇ=É~ÅÜ=ÑêÉÉ
íÜêçï=áë=ïçêíÜ=çåÉ=éçáåíKF=eçï=ã~åó=éçáåíë
ÇáÇ=íÜÉ=éä~óÉêë=çå=íÜ~í=íÉ~ã=ã~âÉ=çå=ÑêÉÉ
íÜêçïë=ÇìêáåÖ=íÜÉ=Ö~ãÉ\
tÜáÅÜ=çÑ=íÜÉ=ÑçääçïáåÖ=Éèì~íáçåë=Å~ååçí=ÄÉ
ìëÉÇ=íç=ëçäîÉ=íÜÉ=éêçÄäÉã=ÖáîÉå=~ÄçîÉ\
j ^K 2x H=x Z=TR
_K OE2xF=H=x Z=TR
`K 4x H=x Z=TR
aK 5x Z=TR
Rationale:
75 = 2x + x only accounts for the fact
that the team made twice as many field goals
as they did free throws. It doesn’t contain
the additional stipulation that each field goal
is worth two points while each free throw is
worth only one point. All three of the other
distractors simplify to 5x = 75, which is the
correct equation.
32