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DOCUMENT RESUME
ED 389 887 CE 070 372
AUTHOR Brodsky, Stanley M.
fITLE Tech-Prep Mathematics in New York State.
INSTITUTION City Univ. of New York, N.Y. Center for AdvancedStudy in Education.
SPONS ACENCY New York State Education Dept., Albany. Bureau ofPostsecondary Grants Administration.
REPORT NO CASE-10-95PUB DATE Sep 95CONTRACT VATEA8080-95-0005NOTE 107p.
PUB TYPE Reports Research/Technical (143)
EDRS PRICE MF01/PC05 Plus Postage.
DESCRIPTORS College Preparation; Integrated Curriculum;*Mathematics Curriculum; *Mathematics Instruction;Secondary Education; State Curriculum Guides; *TechPrep
IDENTIFIERS *New York
ABSTRACTA study sought to determine the extent to which
mathematics content in tech prep programs in New York state matchedor differed from the content of the three Regents college preparatory
courses. Data were gathered through questionnaires received from a
sample of 24 of the state's 30 tech prep consortia providinginformation about 27 curricula. Some of the results included the
following: (1) almost 60 percent of the tech-prep mathematicscurricula reported using only applied courses and no collegepreparatory sequential mathematics courses; (2) only two mathematicscurricula were integrated into other subjects rather than being givenas separate courses; (3) 20 percent of the tech prep programsincluded either 2 or all 3 of the Regents college preparatorymathematics courses; (4) the curricula typically covered just over 40
percent of the topics in the Regents curriculum; (5) agreement with
State Framework Math Statements was substantial; and (6) anticipatedchanges to be brought about by various legislation and technologychanges will have an impact on the curricula of the tech prepschools. (The report includes 31 tables and figures representing thetopics covered in the various mathematics courses in the tech prepschools and in the mathematics framework.) (KC)
Reproductions supplied by EDRS are the best that can be madefrom the original document.
********).......%).***************************************************
TECH-PREP MATHEMATICSIN lsIEW YORK STATE
Stanley M. Brodsky, Ph.D., P.E.Project Director
U S DEPARTMENT OF EDUCATIONry . ,. 1,,I,11,c lir.ea, alEDUCATIONAL RESOURCES INFORMATION
IVCENTER ;ERIC)
This docum eM has been reproduCed asI ec. el ye d ham the poison or organilatirmoriginating it
0 Mins changes have been made tOimprove reproducliOn quality
Paints cif von, or opinions stald In thtsdocument do not necessarily reprer.entofficial OEDI position or policy
"PERMISSION TO REPRODUCE THISMATERIAL HAS BEEN GRANTED BY
V.,/ "'I
TO THE EDUCATiONAL RESOURCESINFORMATION CENTER (ERIC)"
This study was supported as part of the activities of the Tech-Prep TechnicalAssistance Center at the Center for Advanced Study In Education at the City Universityof New York Graduate School by a grant from the Bureau of Post-Secondary GrantsAdministration of the New York State Education Department, VATEA No. 8080-95-0005.
September 1995
CASE #10-95
I3EST COPY AVAILABLE
TABLE OF CONTENTS
Page
Table Of Contents ii
List Of Tables & Figures iii
Acknowledgements
Executive Summary vi
Tech-Prep Mathematics In New York State 1
Background 1
The Study 2
Assumptions 4
Findings 5
Coverage Of Topics From Sequential Mathematics Courses 6
Coverage Of Topics By Subject Area 7
Time Allotment Comparison: Tech-Prep vs. Traditional Courses 8
Curriculum Analysis By Subject Areas & Topics 10
Additional Topics Includei In Tech-Prep Programs 11
Instructional Materials 12
Responses To Framework Mathematics Statements 12
:Iiscussion 14
Tables & Figures 17 -- 62
LIST OF TABLES & FIGURESPage
Table 1 Tech-Prep Mathematics Curriculum Category 17
Table 2A Category A Tech-Prep Curricula With No Sequential Courses: N = 16 17
Topics In Sequential Courses Covered In Tech-Prep Mathematics
Curricula By Sequential Course
Table 28 Category B Tech-Prep Curricula With One Sequential Course: N = 5 18
Topics In Sequential Courses Covered In Tech-Prep Mathematics
Curricula By Sequential Course
Table 2C Category C Tech-Prep Curricula With Two Sequential Courses: N = 2 18
Topics In Sequential Courses Covered In Tech-Prep Mathematics
Curricula By Sequential Course
Figure 1 % Median Number Of Topics From Sequential Math Courses Which
Are Covered In Tech-Prep Curricula By Category & Sequential Course 19
Table 3 Number Of Topics In Sequential Math Courses By Subject Area 20
Table 4 Topics In Sequential Courses Covered In Tech-Prep Mathematics 21
Curricula By Subject Area
Table 5A Category A Tech-Prep Curricula With No Sequential Courses: N = 16 22
Time Devoted To Topics From Sequential Math Courses In Tech-Prep
Mathematics Curricula By Subject Area & Sequential Course
Table 5B Category B -- Tech-Prep Curricula With One Sequential Course: N = 5 23
Time Devoted To Topics From Sequential Math Courses In Tech-Prep
Mathematics Curricula By Subject Area & Sequential Course
Table 5C Category C -- Tech-Prep Curricula With Two Sequential Courses: N = 2 24
Time Devoted To Topics From Sequential Math Courses In Tech-Prep
Mathematics Curricula By Subject Area
Figure 2 % Median Time Used For Topics In Tech-Prep Curricula Compared To 25
Sequential Math Course "Standard" Time By Category & Sequential
Cou rse
Table 6 Logic Topics: Topics In Sequential Math Courses Covered In Tech-Prep 26
Math Curricula
Table 7 Number Sense & Numeration Concepts Topics: Topics In Sequential 27
Math Courses Covered In Tech-Prep Math Curricula
Table 8 Operations On Number Topics: Topics In Sequential Math Courses 28
Covered In Tech-Prep Math Curricula
Table 9 Geometry Topics: Topics In Sequential Math Courses Covered In Tech- 29
Prep Math Curricula(Continued)
LIST OF TABLES & FIGURES (Continued)Page
Table 10 Measurement Topics: Topics In Sequential Math Courses Covered In 33
Tech-Prep Math Curricula
Table 11 Probability & Statistics Topics: Topics In Sequential Math Courses 34
Covered In Tech-Prep Math Curricula
Table 12 Algebra Topics: Topics In Sequential Math Courses Covered In Tech- 36
Prep Math CurriculaTable 13 Trigonometry Topics: Topics In Sequential Math Courses Covered In 40
Tech-Prep Math CurriculaTable 14 Other Topics: Topics In Sequential Math Courses Covered In Tech-Prep 42
Math CurriculaTable 15 Topics From Sequential Math Courses Which Were Most Often And 43
Least Often Covered In 16 Category A Tech-Prep Math Curricula
Table 16 Additional Topks Covered In Tech-Prep Math Curricula As Reported 48
By Respondent;
Table 17 Instructional Materials In Tech-Prep Math Curricula As Reported By 50
Respondents
Table 18 Percent Agreement With New York State Framework Mathematics 51
Statements By Category And By Subject AreaFigure 3 Percentage Of Topics In Sequential Math Courses Which Are Covered 52
In A Tech-Prep Curriculum vs. Percentage Agreement With StateFramework Math Statements For Each Tech-Prep Curriculum
Table 19 Responses To Framework Mathematics Statements On Logic 53
Table 20 Responses To Framework Mathematics Statements On Number Sense & 54
Numeration ConceptsTable 21 Responses To Framework Mathematics Statements On Operations On 55
Number
Table 22 Responses To Framework Mathematics Statements On Geometry 56
Table 23 Responses To Framework Mathematics Statements On Measurement 57
Table 24 Responses To Framework Mathematics Statements On Probability & 58
Stat istics
Table 25 Responses To Framework Mathematics Statements On Algebra 60
Table 26 Responses To Framework Mathematics Statements On Trigonometry 61
Table 27 State Framework Mathematics Statements With Agreement By 62
50 % Or Fewer Of The Respondents
ACKNOWLEDGEMENTS
This study was triggered by an idea which Lee Traver of the New York StateEducation Department suggested and was endorsed by the Tech-Prep Technical Assistance
Center Advisory Committee consisting of Tech-Prep consortium personnel: Anne Gawkins,New York City Technical College; Kevin Hodne, SUNY College of Technology at Delhi;
Marilyn Kelly, Oneida-Madison BOCES; Constance Spohn, Capital District Consortium &Two-Year College Development Center at SUNY Albany; Suzy Tankersley, OnondagaCommunity College; and Kathleen Thompson, Eastern Suffolk BOCES. The plan andinitial draft of the questionnaires were guided by Dr. Bert Flugman, Director of the Centerfor Advanced Study in Education at the CUNY Graduate School and critiqued by BenLindeman of the New York State Education Department. James Donsbach and David
Martire, both of the New York State Education Department were consistently supportive ofthis effort. Two graduate students, Sherri Horner and Frederick Nassauer, Jr. providedexcellent direct assistance in data analysis, table construction and necessary research. Eachof those mentioned above had a substantial and positive effect on this study and I amgrateful to them all.
The Tech-Prep Project Coordinators of the 30 consortia in New York Statefacilitated this project by generally favoring the methodology, seeking and identifying
mathematics professionals from within their consortia to participate and respond to thequestionnaires, and providing follow-up when requested to do so. While not everyconsortium participated, those that did not participate did discuss their situations with me.
These folks are among the most dedicated professionals I have had the pleasure of workingwith. The mathematics faculty respondents provided their professional expertise to thisstudy and I salute them all for a job well done.
The study benefitted greatly from the work of several mathematics experts. I amgrateful to Dr. Arthur Kramer of New York City Technical College and Irvin Barnett ofBushwick High School. Susan Zakaluk, formerly Head of the Central Mathematics Unit atthe New York City Board of Education, helped greatly by providing her technical expertiseand advise as well as a comprehensive review of the draft. Helpful reviews of the draftreport were provided by Bernard McInerney (State Eduration Department), Jack Fabozzi(Two Year College Development Center), Kevin Hodne (SUNY Delhi), Linda Halligan
(Oneida-Madison Tech-Prep Consortium) and Richard Lukaschek (Bayshore High School,Eastern Suffolk Tech-Prep Consortium).
S. M. Brodsky, September 1995
TECH-PREP MATHEMATICS IN NEW YORK STATE
EXECUTIVE SUMMARY
This study was designed to determine the extent to which mathematics content in
Tech-Prep programs matched or differed from the content of the three Regents college
preparatory courses. In addition, the applicability to Tech-Prep mathematics curricula
of statements in "Framework for Mathematics, Science & Technology" was measured.
Secondary school mathematics was studied in a sample of Tech-Prep programs in
New York State. Useful responses were received from 24 of the State's 30 Tech-Prep
consortia providing information about 27 curricula on two extensive questionnaires and
a brief follow-up questionnaire.
One questionnaire listed 317 mathematics topics taken from the Regents college
preparatory curriculum series of three courses, with 89 topics from Sequential Math
Course I, and 114 topics each from Sequential Math Courses II and III. Respondents
were asked to indicate whether a topic was not included in the Tech-Prep program
(NO), was optional or the respondent was uncertain (?), or was included (YES). For
items answered YES, an estimate of the clock hours devoted to that topic was asked for.
A second questionnaire listed 92 statements taken from the "Framework for
Mathematics, Science & Technology" which is in draft form and was distributed for
discussion by the New York State Education Department in mid-1994. These statements
are grouped into 8 broad subject areas. While the State document provides statements
at three levels, elementary, intermediate and commencement, only the latter two were
included in this questionnaire.
The following are among the more interesting results.
Almost 60 percent of Tech-Prep mathematics curricula reported using only
applied courses and no college preparatory Sequential Math Courses.
Typical applied course titles were Tech-Prep Math and Applied Math. Only two
math curricula were integrated into other subjects rather than being given as separate
math courses.
Another one-fifth of the Tech-Prep curricula reported including Sequential Math
vi
Course I along with additional applied mathematics.
The remaining one-fifth included either two or all three of the Regents college
preparatory math courses.
The curricula with no Sequential Math Courses exhibited a wide range in the
percentage of topics covered with an overall median of just over 40 percent. The typical
curriculum included 65 of the 89 topics from Sequential Course I and another 65 out of
228 topics from the other two Sequential Courses, totaling 130 topics. Most of these
applied courses avoided proofs and derivations, but devoted more hours to the topics
than traditionally allotted. In many cases, these curricula covered additional math
topics, not included in the Sequential Courses, as well as other topics relevant to the
technical curriculum.
The curricula which included Sequential Math Course I, also tended to include a
larger number of topics from the other two Sequential Courses, with median coverage of
over half of the Sequential Course II topics and two-fifths of the Sequential III topics.
While these curricula also showed a rather wide range, the typical curriculum of this
type was clearly closer to the Regents course topical series (overall median coverage of
about 60 percent) than those with no Sequential Courses.
Mathematics topics were also grouped into the following subject areas which
mirrored the groupings of the statements in the State Mathematics Frameworks.
Logic topics
Number Sense & Numeration Concepts topics
Operations on Number topics
Geometry topics
Measurement topics
Probability & Statistics topics
Algebra topics
Trigonometry topics
Other topics not included above
For the curricula with no Sequential Courses, the three subject areas with the
lowest median percentage coverage were Logic (25%), Geometry and Trigonometry
(31% each). The three areas with the highest median percentage coverage for these
vii
curricula were Measurement (75%), Numeration Concepts (69%) and Operations with
Number (66%). However, the four subject areas with the largest number of topics
covered in all groups were Algebra, Geometry, Trigonometry and Probability &
Statistics which accounted for almost 80 % of the topics. Similar median percentage
and numerical trends were noted for the curricula with one Sequential Math Course.
One or more curricula with no Sequential Courses included 299 of the 317 topics
listed and required approximately 25 percent more class time than was judged to be
used in Sequential Courses for these same topics. Semester class time for a Sequential
Math Course is traditionally 120 clock-hours, but Tech-Prep applied math courses
ranged from 120 to 200 clock-hours with a median of about 140 hours.
Curriculum developers can use the subject area Tables 6 through 14 to find, for
each of the 317 topics, the number of Tech-Prep curricula with no Sequential Math
Courses which include that topic and the median number of hours reported for each
compared with a set of "standard" hours. In addition, for these same curricula, Table
15 lists by subject area those topics which are included in 75 % or more curricula and
25 % or fewer curricula; i.e. the most often and least often used topics in Tech Prep.
Agreement with State Framework Math Statements was substantial with an
overall median of 87 percent for the curricula with no Sequential Courses and 91
percent for those with one Sequential Course. The three subject areas with the lowest
percentage agreement for those with no Sequential Courses Vs1:1',. Logic (44%),
Probability & Statistics (66%) and Geometry (77%). All other subject areas had median
agreement percentages from 96% to 100%. Evidently, those involved with applied
curricula may agree with many of the broader Framework statements even though
coverage of specific topics may vary.
Readers who are interested in the detailed responses to the State Framework
Math Statements will find that information in subject area TaHes 19 through 26. In
addition, Table 27 lists those Framework Math Statements which received 50 % or less
agreement from respondents.
Several anticipated changes in Tech-Prep curricula are outlined in the Discussion
section. In addition, that section also treats the issue of curriculum decision rules which
may result in diverse mathematical curricular content.
viii
TECH-PREP MATHEMATICS IN NEW YORK STATE
BACKGROUND
Tech-Prep is a national program as part of the Vocational & Applied Technology
Education Act (VATEA) which encourages secondary and post-secondary institutions to
form consortia to create applied curricula which smoothly connect the 11th and 12th
grades through an associate degree in career fields. The legislation also allows
equivalent apprenticeship programs in lieu of associate degrees but the programs in this
state have focused on associate degree goals and beyond. The program fosters close
working relationships between high school and college faculty and counselors including
joint training, planning and curriculum development. Tech-Prep normally includes
development of workplace readiness skills, career education and exploration, initial
stages of workplace learning, educational preparation for rigorous associate degree
technical programs, and often opportunities for advanced standing through attending
college level work prior to completing high school.
The New York State Education Department launched Tech-Prep as part of the
newly reauthorized VATEA in 1991 when seven consortia were funded. In subsequent
years, an additional seven, then fourteen and finally two consortia were approved giving
a total of 30 operating consortia, geographically distributed across the state. In
addition, two Tech-Prep Technical Assistance Centers have been supported, one at the
Two-Year College Development Center at the State University of New York (SUNY) in
Albany, and the other at the Center for Advanced Study in Education (CASE) at the
City University of New York (CUNY) Graduate School in New York City.
The New York State Education Department was interested in assessing how
applied curricula in core subject areas compared with content in college preparatory
courses. A Technical Assistance Center Advisory Committee consisting of Tech-Prep
Consortium Project Directors endorsed the idea and recommended starting with
mathematics. The CASE Technical Assistance Center was asked to carry out this study.
1
THE STUDY
Each of the 30 Tech-Prep Consortium Project Directors was asked to select an
experienced mathematics person who was familiar with their Tech-Prep curriculum to
participate in the study. Those identified were sent two extensive questionnaires and
detailed instructions for completing them. In addition, the author spoke to each of the
potential respondents as well as the 30 Project Directors to ensure that the materials
were understood and that the math person would be able to respond in a timely
manner. A total -A 25 respondents from 24 of the state's 30 (80%) Tech-Prep consortia
provided usefv; information about 27 different Tech-Prep curricula. The remaining
consortia did not participate either because there was no available respondent or
because their math curriculum was not sufficiently developed to provide complete
responses to the questionnaires.
One questionnaire consisted of a listing of math topics from the three Regents
college preparatory courses in the state. These courses are officially designated as a
"Three Year Sequence for High School Mathematics, Courses I, II & III". In this
study, these courses will be called "Sequential Math Courses I, II and III" or variations
of that terminology. The topics in the questionnaire were listed by Sequential Course
with 89 topics associated with Sequential Math Course I, and 114 topics for each
Sequential Math Course II and III; a total of 317 items. For each topic, the respondent
was to indicate by check mark if the topic was not included in the Tech-Prep math
curriculum (NO), if the topic was either optional or the respondent was uncertain about
it (?), or if the topic was included in the Tech-Prep curriculum (YES). For each YES
response, the number of clock-hours used for that topic in the curriculum was to be
estimated to the nearest quarter hour.
The list of topics was taken from the 1988 New York City revision of Sequential
Courses 1, 2 and 3, which detailed lesson topics numbering 120, 120 and 130,
respectively. This source was selected over the New York State syllabi for these courses
because the latter were subdivided into many more topical items. Although the state
2
syllabi were more recent, use of the New York City version limited the length of the
content questionnaire while covering essentially the same material.
The topics were listed on the questionnaire in the order they appeared in the
source noted above. After the questionnaires were distributed to the various consortia
in early February 1995, three mathematics experts provided a classification of these
items into 9 subject areas:
Logic
Number Sense & Numeration Concepts
Operations on Number
Geometry
Measurement
Probability & Statistics
Algebra
Trigonometry
Other
Responses to the questionnaire were subdivided and analyzed by Sequential Math
Course from which the item was drawn and by subject area classification.
An expect in secondary mathematics assigned hours normally needed to cover
each of the topics in each Sequential Math Course with instructional hours totaling 120.
These hours represented a math course meeting for 180 instructional days in 40 minute
periods each day. These hours were designated as "standard" hours and were used for
comparison with those reported by the respondents.
The items in the topics questionnaire were arranged in three sections. Items 1
through 89 were from Sequential Math Course I, items 90 through 203 were from
Sequential Math Course II, and the last group from Sequential Math Course III were
items 204 through 317. To simplify the respondent's task, those whose Tech-Prep
curricula included any of the Sequential Math Courses were instructed to skip the
corresponding section of the questionnaire. Thus, the four respondents who reported
that their Tech-Prep curricula included all three Sequential Math Courses did not need
3
to give information for any of the 317 topical items.
The second questionnaire incorporated 92 statements from "Framework for
Mathematics, Science and Technology" issued by the New York State Education
Department as a draft for discussion. The statements, as issued, are given in three
levels; elementary, intermediate and commencement. Statements from the latter two
levels were included in the questionnaire as being appropriate to secondary grades 9
through 12.
The Framework statements were organized, in the publication noted above, into 8
subject areas:
Logic
Number Sense & Numeration Concepts
Operat;ons on Number
Geometry
Measurement
Probability & Statistics
Algebra
Trigonometry.
The items in the topics questionnaire were classified into the same set of subject areas
as the Framework statements after the topics questionnaire was distributed.
Respondents were asked to indicate for each statement if it applied to their
curriculum (YES), if it did not apply to their curriculum (NO), or if they were
uncertain (?).
A brief follow-up questionnaire solicited information about the number of
separate mathematics courses in their curricula, the course titles, the time devoted to
eacii course, and for integrated math, the subjects with which it is included.
ASSUMPTIONS
This study is delimited by the following assumptions.
All respondents were math professionals who were involved in one or more Tech-
4
Prep curricula and completed the questionnaire using their best professional judgement.
The topics listed in the questionnaire accurately describe the essential content of
the Sequential Math Courses I, II and III.
All topics in the questionnaire can be treated as though they are numerically
equal in importance to facilitate quantification. The question of interest here is the
comparison of the extent of coverage of the content of Sequential Mathematics courses.
For this purpose, no consideration of the relative importance was made, and they were
not weighted but rather treated as equal.
All topics listed in the questionnaire for a given Sequential Math Course were
included when such courses were taught in their entirety as part of a Tech-Prep
curriculum.
The "standard" hours assigned to topics in Sequential Math Courses in this study
accurately reflect the time distribution in such courses generally and these hours apply
when such courses are taught in their entirety as part of a Tech-Prep curriculum.
All Framework statements can be treated as though they are numerically equal in
importance to facilitate quantification, rather than being weighted in some way.
Both the intermediate and commencement levels of the State Framework
statements are applicable to secondary mathematics curricula.
FINDINGS
The 27 Tech-Prep curricula reported fall into four different categories as shown
in Table 1. About 60 percent of the curricula (Category A) do not require any
Sequential Math Courses, but include applied mathematics instruction. Another 18.5
percent (Category B) included Sequential Math Course I, usually with additional applied
mathematics instruction, while the remaining curricula included either two (Category C)
or three (Category D) Sequential Math Courses.
The respondents reported that applied mathematics instruction was most often
offered in separate courses with a variety of titles, the most common of which were
Tech-Prep Math and Applied Math. Only two programs had the math integrated into
5
other subjects rather than M separate math courses. Approximately half of the
curricula use materials developed by the Center for Occupational Research &
Development (CORD) of Waco, Texas either as the primary instructional material or as
a supplement to texts or teacher-made material. Most of the others rely on textbooks,
including all of the Category C & D, most of the Category B and several of Category A
curricula. A list of the instructional materials reported, including the titles of the 40
CORD units, is shown in Table 17.
COVERAGE OF TOPICS FROM SEQUENTIAL MATHEMATICS COURSES
Table 2A shows the range and median of the number of topics from the
Sequential Math Courses which are covered in the 16 Category A Tech-Prep
mathematics curricula; those with no Sequential Math Courses. Also given is the
median of the percentages of topics covered which is determined by dividing the median
number of topics covered by the total number of topics in that group. See the example
in the footnote to Table 2A which shows that the median Tech-Prep curriculum in
Category A covered about three-quarters of the Sequential Course I topics. This table
shows the extremely broad range of topical coverage within this category. Note that
while a median of three-quarters of the topics from Sequential Course I are covered in
Tech-Prep curricula, this measure drops to one-third for Sequential Course II topics
and to less than one-quarter for Sequential Course III topics. Overall, a median of
about two-fifths of the topics are covered. Thus, the typical Category A curriculum
includes about 65 topirs from Sequential Course I, and another 65 topics from
Sequential Courses II and III, totaling 130 topics from college preparatory courses.
The corresponding data for Categories B and C are shown in Table 2B and 2C. The
median curriculum with one Sequential Math Course (Category B) appears to
incorporate about three-fifths of the college preparatory topics, while those with two
Sequential Math Courses (Category C) show almost 90 percent coverage. Note that is
was assumed that all topics listed for Sequential Math Courses were covered in those
Tech-Prep curricula which included specific Sequential Courses. These latter two
categories (B and C) include very small numbers of programs, 5 and 2, respectively,
6
which suggests the probability that these data may be unreliable. Category D curricula
were assumed to cover all 317 topics in the three Sequential Math Courses which they
include and therefore no separate table for Category D topics was needed. Variations
from the assumption of full topical coverage when Sequential Math Courses are given as
part of Tech-Prep curricula may occur in some cases but were not measured. A
comparison of the median data from Tables 2A, 2B and 2C is shown graphically in
Figure 1. Category D curricula are represented by tars which reach the 100 percent
level. Category C data start at 100 percent for Courses I and II and then drops to 68.4
percent for Course III. Category B data start at 100 percent for Course I and then
drops to 52.6 and 36.0 percent. While tech-prep math curricula with no Sequential
Courses cover only a fraction of the topics in the college preparatory courses, the tech-
prep math curricula cover additional topics in math and other areas relevant to the
technical curriculum; see Table 16.
Coverage of Topics by Subject Area
The 317 topics from the three Sequential Math Courses have been subdivided
into nine subject area classifications as shown in Table 3. In order of total number of
topics, Algebra and Geometry predominate with more than half of the topics (91 and
87, respectively), followed by Trigonometry (39), Probability & Statistics (34), Logic and
Operations on Number (20 each), Number Sense & Numeration Concepts (11), and
Measurement (7). Eight topics were classified as Other.
The range, median numbers, median percentages and size ranks of subject area
topics covered in Tech-Prep curricula for Categories A, B and C are shown in Table 4.
The rank order of all 317 topics is somewhat different than the rank order of topics
covered in Tech-Prep curricula. While Algebra and Geometry continue to dominate as
1st and 2nd ranks, the remaining size rankings in Category A are different than for
Sequential Courses. Probability & Statistics, Number Sense & Numeration Concepts,
and Measurement all moved up one rank position to 3rd, 6th & 7th,
respectively, while Logic dropped back more than two positions to 8th and
Trigonometry moved one position down to 4th. Rankings for Category A and B were
7
similar except that in Category B Logic was down one-half position to 6th and
Numeration and Measurement were tied for 7th. Category C which included two
Sequential Courses matched the Sequential Course rankings with one small exception.
TIME ALLOTMENT COMPARISON: TECH-PREP VS. TRADITIONAL COURSES
The median number of hours devoted to Tech-Prep topics in each subject area
compared to the "standard" hours for the same topics is shown for Category A curricula
in Table 5A. In this table and the two which follow, the number of topics listed for
each subject area are those which are included in one or more Tech-Prep curricula of
the corresponding category. Thus, if a topic in the questionnaire is not represented in
any Category A curriculum, it is not included in this table and neither are its
"standard" hours or Tech-Prep hours. Furthermore, responses which indicated that a
topic was included in the Tech-Prep curriculum but provided no time estimate were
included in the No. TP TOPICS column but were not involved in either "standard"
hours or medians. These Tech-Prep programs appear to spend more hours than the
"standard" hours for each group of topics. The percentage of increased time in Table
5A for each Sequential Course group of topics is +22.6 % for I, +33.8 % for II and
+16.2 % for III, with an overall increment of +23.7%. This last number can be seen
in the TOTAL line under % TP MEDN, shown as 123.7. The percentage hourly
increments are calculated from the TOTAL line by dividing TP MEDN by STAND.
Certain Tech-Prep subject areas appear to involve from 70 to 100 percent more time
than Sequential Courses; Number Sense & Numeration Concepts (+100 %), and
Operations on Number and Measurement (+70 % each), although taken together these
three subject areas represent relatively few topics. Ten percent fewer median hours
than "standard" are devoted to Probability & Statistics by Category A curricula which
may imply somewhat briefer coverage with less problem solving time than is given
proportionately to other topics. The three mathematics subject :reas to which these
Tech-Prep programs appear to devote the heaviest time commitments are Algebra,
Geometry and Trigonometry. These topics together account for 66 percent of total
Tech-Prep math course time, and exceed Sequential Course "standard" hours by about
8
one-fifth for Algebra and Geometry, and by about one-third for Trigonometry.
Table 5B for Category B curricula, shows the time devoted to Tech-Prep topics
compared to "standard" hours. Note that the assumed time for topics in Sequential
Math Course I, which these Tech-Prep Category B curricula include, is equal to the
"standard" time. Therefore, the hours listed for Sequential Course I under the heading
TP MEDN are the same as those listed under STAND, as are both column totals.
However, the Tech-Prep median time allocations for Sequential Courses II & HI topics
which total 135.9 and 112.6 hours, respectively, exceed the "standard" hours of 109.8
and 100.8 by +23.8 % and + 11.7 %, respectively, with an overall increment of + 11.5
%. This difference is half as much as for Category A programs and may be attributed
to the inclusion of Sequential Math Course I in Category B programs which eliminates
the time differential for the first 89 topics. Note that the largest increment is for the
Sequential Course II topics, which was true of Category A as well. For these five
Category B curricula, the largest subject matter hourly median increment was in Logic
(+31.9 %), followed by Trigonometry (+19.7 %) and Geometry (+17 %). The major
time allocations were for Algebra, Geometry and Probability & Statistics which together
accounted for almost 70 percent of the hours in the median Tech-Prep math curriculum.
Table 5C shows only the overall data for two Category C curricula which include
Sequential Courses I & II in the Tech-Prep math program. The hourly allocations for
the two Sequential Course topics were, once again, assumed to be the "standard" hours.
Therefore, the only effect on the overall increment comes from the Sequential Course
III topical hours. With an overall increment of +11.9 %, topics for III contributed a
surprising +36.5 % more time than Sequential Course III topics. This may be partially
explained because most of the Trigonometry topics appear in the III group of topics
which show a median increment of +58.3 % over "standard" hours. Trigonometry
combined with Algebra and Geometry command almost 70 percent of the median Tech-
Prep hours for this category.
Figure 2 shows graphically the relations between the median percentage time
allocations for topics in all four categories of Tech-Prep curricula. Category A & B
9
curricula appear to follow somewhat similar patterns with higher incremental hours
devoted to Sequential Course II topics. One would have expected the increment for
Category C in group III topics to have been closer to those of the other two categories;
i.e. somewhere near +10 to +15 %. Instead, an increment of almost triple the expected
number was found. While this may be partially explained as noted above, these data
come from just two programs which were the only Category C curricula reported by
respondents in this study. Therefore, this result may not be typical of the group.
CURRICULUM ANALYSIS BY SUBJECT AREAS AND TOPICS
In order to provide Tech-Prep mathematics curriculum developers with a
maximum of useful information, the 317 topics used in this study are arrayed by subject
area ;4,3 Tables 6 through 14. Individuals may differ with the classification of topics into
the subject areas shown, but the data for each topic represent the responses from math
professionals who were unaware of the subject area classification when they responded.
The participants in the study and the 30 Tech-Prep Project Directors in New
York State should have a copy of the topics questionnaive and those who responded
would have copies of their own completed questionnaires. To relate the following nine
tables to a copy of a completed questionnaire, it will be useful to number the items from
1 to 317, consecutively. There is no need to number a blank topics questionnaire nor
does one need a blank questionnaire since all 317 topics are identified in Tables 6 -- 14.
These nine tables reflect the data for the 16 curricula in Category A, that is,
those Tech-Prep math curricula with no Sequential Math Courses. Tech-Prep personnel
whose curricula fall into Categories B or C might simply use the less detailed data in
Tables 4, 58 and 5C, or examine the appropriate group topics from II and or III in
Tables 6 through 14. Each of these nine tables gives for each topic in one subject area,
the number of curricula and the percentage of the 16 programs for which respondents
checked YES, indicating that the topic was being covered in their Tech-Prep math
program. In addition, the assigned "standard" hours are given for each topic, the range
of Tech-Prep hours reported for each topic and the median Tech-Prep hours for each
topic. Since not all respondents who checked YES for a topic provided time estimates,
10
the median hours were determined from those reporting the time, which may have been
fewer curricula than the number of YES responses.
Tech-Prep math study participants and consortium Project Directors who have
their completed questionnaires may want to compare their msponses with these tables
for several possible purposes, for example:
To see how their program compares with the statewide patterns
To examine whether to include or delete a series of topics based on what othershave done
To adjust their time allocations for certain topics or subject areas
To broaden the coverage of their program, where warranted
To include or delete Sequential Math Courses, or
To provide a basis for seeking Regents credit recognition for applied mathcourses via the variance process.
Others who did not participate in this study might simply work from the dataprovided herein to accomplish some of the same objectives mentioned above.
Table 15 provides a listing of the topics used by 75 percent or more of the
Category A curricula (12 or more of the 16) and the topics used by 25 percent or less ofthe Category A curricula (4 or fewer of the 16) of the 317 used in this study. In
viewing this table, one should keep in mind that more than 100 of the topics used in
Tech-Prep curricula are not shown on Table 15. The largest contribution to the topicsin this table which are used less often comes from topics involving proofs or derivations,
which many Tech-Prep math curricula minimize to allow for emphasis on applications
and additional topics related to specialized mathematics or a technical field.
ADDITIONAL TOPICS INCLUDED IN TECH-PREP PROGRAMS
Table 16 lists some of the additional topics used in Tech-Prep math curricula.Not all respondents provided information about such topics, but one would suspect thatmost Tech-Prep math programs do include some additional instructional topics. Thelargest group of items in this table were in mathematics. Although a few were actually
in the questionnaire list, some represented either advanced or specialized math topics.
11
In the technology group, items were related to manufacturing, drafting, surveying,
electronics, and health careers. A number of items dealt with applications of computers
and calculators, while others focused on measurements and instrumentation.
INSTRUCTIONAL MATERIALS
Table 17 lists some of the instructional materials which were reported. The most
widely used material is from the Center for Occupational Research & Development
(CORD) which has been adopted in whole or part in 13 of the 27 curricula in this study.
This material is organized into 40 units, as shown in Table 17, 36 of which were
available at the beginning of the 1994-1995 academic year. The units come with a
variety of back-up media and lab instruction. There are enough units to support three
applied math courses and several of the curricula have done precisely that.
Eleven of the curricula reported using particular textbooks. Specific texts were
reported used by 4 Category A, 2 Category B, 2 Category C and 3 Category D
curricula. Many of the textbooks listed in Table 17 were either supplemented by other
materials or were themselves supplementary. There does not appear to be very many
texts which would directly support Tech-Prep programs, perhaps because of the variety
of activities which are normally included in such programs as well as the tendency to
avoid single source and print-only instructional materials.
RESPONSES TO FRAMEWORK MATHEMATICS STATEMENTS
Table 18 shows the extent of the reported agreement of these Tech-Prep curricula
with the Framework statements. The median percentage agreement with statements
exceeded 85 percent over all 27 programs. The statements were classified by subject
area in the Framework document and were taken from the intermediate and
commencement levels, as noted earlier. Each statement was treated numerically
equally. The statements in the five subject areas of Algebra, Trigonometry, Number
Sense & Numeration Concepts, Operations on Number, and Measurement produced
strong agreement from the typical (median) respondent with either complete agreement
or nearly so. The statements for the subject area of Logic showed a median agreement
of only 50 percent. Curricula with Sequential Math Courses typically showed
substantial agreement with the Framework statements. In general, the respondents
were much more likely to check YES for a Framework statement, indicating that the
statement applied to their curriculum, than for a Sequential Math Course topic, which
would have indicated that the topic was part of their curriculum. This finding would
imply that the Framework mathematics statements apply broadly to Tech-Prep
programs with notable individual exceptions.
The dispersion of Tech-Prep math curricula reported when measured by both the
percentage of Sequential Math Course topics covered and the percentage agreement with
the Framework statements is shown graphically in Figure 3. Note that those programs
which include all three Sequential Math Courses come close to perfect matches with the
Frameworks while they were assumed to match the college preparatory topics. Many of
the Category A curricula cluster around the median data for both of these variables,
although a few show lower matching percentages on both variables. While there were
only two Category C curricula reported, they both showed somewhat lower Framework
agreements than the Category B programs and many of the Category A programs.
Tables 19 through 26 show the responses to each of the Framework statements,
with each table covering one of the eight subject areas. Within each table, the responses
from Category A curricula (those with no Sequential Math Courses) are given separately
and the other categories (B, C & D) are grouped. In each set of subject area data, the
number of responses and the percentage of the total (for that group) which agreed that
the statement was appropriate to the Tech-Prep math curriculum (YES), the number
that disagreed (NO), and the number of uncertain responses (?) are given for each
statement. The statements shown in these tables have been substantially abbreviated in
order to limit each to one line in the tables. The statements which were given in the
questionnaire were much more elaborate and represented the published Framework
statements closely. If one is interested in the applicability of any of the Framework
statements to Tech-Prep mathematks curricula, these eight tables will provide that
information.
Table 27 shows selected Framework statements which received 50 percent or
13
24
fewer YES responses from either the 16 Category A curricula or the 11 group of
responses from Categories B, C & D combined. The latter grouping represents all those
programs with at least one Sequential Math Course. Of the 12 statements listed which
received 50 percent or fewer YES responses, Logic and Geometry each had 4, although
one of the Probability & Statistics statements was the only entry in the Category BCD
column. Several statements seemed to cause respondents some uncertainty with
numbers of responses (grouped for all 27 ABCD curricula) ranging from 4 to as high as
7. These are listed in the No. ? column in Table 27. It is likely that some statements
may have caused confusion and perhaps need revision, although the uncertainty about
"Validate formulas used to compute measurements" is surprising.
DISCUSSION
A good deal of previously unmeasured information has been gathered from
knowledgeable mathematics professionals regarding the nature and content of Tech-Prep
mathematics curricula in New York State. While these data are believed to be accurate
for the spring 1995 semester, it is clear that several changes are in process giving these
programs a dynamic element. Among these anticipated changes are:
The new School-To-Work Opportunities Act which 'NM impact Tech-Prep
curricula in several predictable and other unpredictable ways. A number of the New
York State Tech-Prep consortia are involved in School-To-Work partnerships which
may change the way they operate. There will undoubtedly be an increase in workplace
learning which might include some elements of the mathematics curricula.
The increasing use of calculators will impact the way in which certain topics are
taught. This year, scientific calculators are to be used in Regents exams. One expects
that there will be changes in the "standard" time distributions and the relative
importance of topics.
In New York City, the Board of Education has approved a change which will
phase in a requirement that all students take the Sequential Math Courses. This corr ing
fall (1995) the requirement for Sequential Math Course I will be implemented and the
14
others will follow in successive years. Currently, two of the five New York City Tech-
Prep consortia include Sequential Math Courses (Category D; all three Sequential
Courses) and shortly they will be joined by the other three.
Approximately half of the respondents reported using math instructional
materials produced by the Center for Occupational Research & Development (CORD)
either solely or as supplements to texts or other materials. Several respondents noted
that they were awaiting one or more of the new units which CORD had under
development at the time they responded to this study. The units mentioned specifically
were Logic, Coordinate Geometry, and Transformations, all of which showed limited
current inclusion in the math curricula, particularly in the Category A (no Sequential
Math Courses) programs.
Several Tech-Prep programs are in their 4th or 5th years of operation and are
planning to institutionalize their programs or have done so in large measure. Some
other consortia may find it imperative to institutionalize their current programs due to
reduced or eliminated external funding even if they may be at early stages. There may
be consortia which are planning a major expansion to meet the mandate in the newer
legislation to serve all students. Still others may terminate their Tech-Prep articulation
agreements and shut down the programs.
Those mathematics faculty in Tech-Prep or similar programs who make use of
the detailed tables of mathematics topics, particularly Tables 6 through 14, as well as
those who re-examine the number and timing of their math courses may decide to
consider changes in their programs as a result of this report. To the extent that this
happens, the report may be thought to have a useful impact.
The wide range of mathematics topics which are revealed in this study may be
attributable to the nature of decision rules related to curriculum planning for career
paths for associate degree graduates in business, allied health, engineering technology
and industrial technology fields. Associate degree programs in some of these fields are
governed by specialized accreditation requirements which influence many curriculum
decisions. Other fields may be shown not to be math dependent and curricula for these
15
areas often place emphasis elsewhere in the curriculum and lessen the mathematics
demand. Some Tech-Prep curriculum planners insist on the strongest mathematics
content for their program espousing the view that everyone should take college
preparatory mathematics, regardless of their occupational goals. It appears that all of
these decision rules have operated to some extent in the Tech-Prep programs sampled in
this study.
The New York State Education Department which promulgated the Framework
statements in draft form for discussion may find the tables which deal with the
Framework statements of some use in considering future editions of the "Framework for
Mathematics, Science and Technology". The pertinent tables are Tables 19 through 26
and Table 27 in particular which shows several of the statements which appear to
deviate from what the respondents felt were valid for their Tech-Prep math curricula.
There were a few statements noted in Table 27 which indicated some uncertainty on the
part of a cluster of respondents which may be sufficient cause to re-examine those
statements.
The cooperative response which this study received from the Project Directors
and the mathematics faculties across the state is an indication that Tech-Prep is
important to these practitioners, and by extension to the students of New York State.
Tech-Prep has typically generated significant enthusiasm from all concerned and has
been responsible for a long list of innovative approaches to secondar:,. and community
college education. While this study has focused narrowly on mathematics courses and
their content, one should be aware of the joint curriculum planning in all subject areas
by secondary and post-secondary faculty working together -- itself an important
innovation. These efforts involve visits by students, faculty and counselors to each
other's institutions as well as a good deal of bridging into college courses by secondary
students prior to their graduation. While the mathematics respondents seemed to
indicate relatively little computer activity in math courses by students, there is extensive
computer use throughout most of the T-ch-Prep programs, most often in the
communications, science and technology components.
16
TABLE 1
TECH-PREP MATHEMATICS CURRICULUM CATEGORY
CURRICULUM NO. OF PERCENT OFCATEGORY CURRICULA CURRICULA
A 16 59.3
B 5 18.5
C 2 7.5
D 4 14.8
TOTAL 27 100
A = Curricula with no Sequential Math Courses; All Math in Applied Courses
B = Curricula with Sequential Math Course I and other Applied Math
C = Curricula with Sequential Math Courses I & II and other Applied Math
D = Curricula with Sequential Math Courses I, II & III
TABLE 2A
CATEGORY A -- TECH-PREP CURRICULA WITH NO SEQUENTIAL COURSES(N 16)
TOPICS IN SEQUENTIAL COURSES COVERED INTECH-PREP MATHEMATICS CURRICULA BY SEQUENTIAL COURSE
SEQUENTIAL No. TOPICS COVEREDCOURSE RANGE MEDIAN % MEDIAN*
I 21 -- 88 67.5 75.8
II 11 -- 76 38.5 33.8
III 2 -- 96 26 22.8
TOTAL 37 -- 251 130 41.0
* % MEDIAN = Median No. of Topics / Total No. of Topics in the Sequential Course
e.g. For Course I, % MEDIAN = 67.5 / 89 = .758 or 75.d%
17
2 I,
TABLE 2B
CATEGORY B TECH-PREP CURRICULA WITH ONE SEQUENTIAL COURSE= 5)
TOPICS IN SEQUENTIAL COURSES COVERED INTECH-PREP MATHEMATICS CURRICULA BY SEQUENTIAL COURSE
SEQUENTIAL No. TOPICS COVEREDCOURSE RANGE MEDIAN % MEDIAN*
I 89 89 100
II 26 -- 88 60 52.6
III 24 -- 87 41 36.0
TOTAL 149 -- 264 190 59.9
* See Footnote to Table 2A
TABLE 2C
CATEGORY C TECH-PREP CURRICULA WITH TWO SEQUENTIAL COURSES(N = 2)
TOPICS IN SEQUENTIAL COURSES COVERED INTECH-PREP MATHEMATICS CURRICULA BY SEQUENTIAL COURSE
SEQUENTIAL No. COVEREDCOURSE MEDIAN % MEDIAN*
I 89 100
II 114 100
III 78 68.4
TOTAL 280.5 88.5
* See Footnote to Table 2A
18
9 "i
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b-3
C-4
TABLE 3
NUMBER OF TOPICS IN SEQUENTIAL MATH COURSESBY SUBJECT AREA
NUMBER OF TOPICSSEQUENTIAL COURSE
SUBJECT 1 II III TOTAL % of TOT RANK
LOG 7 12 1 20 6.3 5.5
NUM 5 ---- 6 11 3.5 7
OPS 13 1 6 20 6.3 5.5
GEO 16 52 19 87 27.4 2
MEA 2 3 2 7 2.2 8
PRB 15 9 10 34 10.7 4
ALG 29 31 31 91 28.7 1
TRG ---- 6 33 39 12.3 3
OTHER 2 6 8 2.5 ----
TOTAL 89 114 114 317 100
LOG = Logic TopicsNUM = Number Sense & Numeration Concepts TopicsOPS = Operations on Number TopicsGEO = Geometry TopicsMEA = Measurement TopicsPRB = Probability & Statistics TopicsALG = Algebra TopicsTRG = Trigonometry TopicsOTHER = Topics Not Included Above
20
RANK = Size Order of Numberof Topics
TA
BL
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TO
PIC
S IN
SE
QU
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TIA
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S C
OV
ER
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IN
TE
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MA
TIC
S C
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UL
A B
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UB
JEC
T A
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A
CA
TE
GO
RY
A (
N =
16)
CA
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GO
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B (
N =
5)
CA
TE
GO
RY
C (
N =
2)
AL
L T
OPI
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TO
PIC
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H-P
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No.
RA
NK
RA
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ED
IAN
RA
NK
% M
DN
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ED
IAN
RA
NK
% M
DN
ME
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N R
AN
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N
LO
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5.5
0 --
16
2.5
812
.57
-- 1
89
645
.019
5.5
95.0
NU
M11
71
-- 1
07
663
.66
-- 1
16
7.5
54.5
107
90.9
OPS
205.
55
-- 2
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555
.013
--
1914
570
.019
5.5
95.0
GE
O87
213
--
6230
.52
35.1
24 -
- 74
502
57.5
75.5
286
.8
ME
A7
82
- 7
67
85.7
4 --
76
7.5
85.7
68
85.7
PRB
344
0 --
34
173
50.0
15 -
32
223
64.7
32.5
395
.6
AL
G91
17
-- 8
239
.51
43.4
49 -
- 80
681
74.7
871
95.6
TR
G39
30
-- 3
613
.54
34.6
0 --
32
154
38.5
29.5
475
.6
OT
HE
R8
----
0 --
71
----
12.5
22
----
25.0
2--
--25
.0
TO
TA
L31
7
LO
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PRB
A L
GT
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R
37 -
- 25
113
041
.014
926
419
0
= L
ogic
Top
ics
= N
umbe
r Se
nse
& N
umer
atio
n C
once
pts
Top
ics
= O
pera
tions
on
Num
ber
Top
ics
= G
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Top
ics
= M
easu
rem
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opic
s=
Pro
babi
lity
& S
tatis
tics
Top
ics
= A
lgeb
ra T
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.;=
Tri
gono
met
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opic
s=
Top
ics
Mt I
nclu
ded
Abo
ve
59.9
280.
588
.5
RA
NG
E=
Min
imum
& M
axim
um N
o. o
f T
opic
s C
over
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Tec
h-Pr
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of
All
Sequ
entia
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ours
e T
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s in
a S
ubje
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rea
% M
DN
= M
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n of
the
% C
over
age
in T
ech-
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Cur
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f A
ll Se
quen
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ours
e T
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sin
a S
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rea
RA
NK
= S
ize
Ord
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n N
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s
TA
BL
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A
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S(N
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SE
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QU
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TIA
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L I
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No.
TP
HO
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TP
No.
TP
TP
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PT
P
TO
TA
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No.
TP
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STA
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G7
8.0
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----
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21.5
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134.
4
NU
M5
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----
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1
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32
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1522
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511
8.6
TR
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----
----
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5.0
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4
OT
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8.3
----
----
----
65.
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4
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100
100.
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113
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741
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OM
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Top
ics
= N
umbe
r Se
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& N
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atio
n C
once
pts
Top
ics
= O
pera
tions
on
Num
ber
Top
ics
= G
eom
etry
Top
ics
= M
easu
rem
ent T
opic
s=
Pro
babi
lity
& S
tatis
tics
Top
ics
= A
lgeb
ra T
opic
s=
Tri
gono
met
ry T
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s=
Top
ics
Not
Inc
lude
d A
bove
No.
TP
TO
PCS
STA
ND
TP
ME
DN
% T
P M
ED
N
123.
7
= N
umbe
r of
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uent
ial C
ours
e T
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s W
hich
are
Cov
ered
in O
ne o
r M
ore
Tec
h-Pr
epC
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cula
in a
Sub
ject
Are
a by
Cou
rse
= S
tand
ard
Hou
rs N
eede
d to
Cov
er T
opic
s in
No.
TP
TO
PCS
in a
Sub
ject
Are
a=
Med
ian
Hou
rs in
Tec
h-Pr
ep C
urri
cula
toC
over
Sub
ject
Top
ics
in N
o. T
P T
OPC
S=
Per
cent
age
Whi
ch T
P M
ED
N H
ours
is o
fST
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7
Sent
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s, S
tate
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Ope
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Intr
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to T
ruth
Tab
les:
Neg
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Con
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isju
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Log
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Mat
hem
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ente
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Sets
& L
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Sym
bolic
Log
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Con
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tate
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tate
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onve
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aws
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est V
alid
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39
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204
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223
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pplic
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aws
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l Num
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22L
aws
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s39
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1119
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r &
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r Se
atin
g Pr
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225
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able
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43
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50Id
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icat
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51Sy
mm
etry
in a
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52In
trod
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nsfo
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53Id
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pert
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54Id
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& W
orki
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s &
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port
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57M
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eter
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grue
nt T
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58A
ngle
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uppl
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tary
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plem
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59C
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ifyi
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by
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& A
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sure
s
60T
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t\-)
61C
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62R
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of
Stra
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Lin
es in
a P
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alle
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g
63M
akin
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fere
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iven
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teri
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on
65C
lass
ific
atio
n of
Qua
drila
tera
ls B
ased
on
Para
llelis
m o
f Si
des
66M
akin
g In
fere
nces
Abo
ut S
peci
al P
aral
lelo
gram
s
73-R
evie
w o
f A
naly
tic G
eom
etry
II99
Bas
ic T
erm
inol
ogy
of A
ngle
s &
Ang
le M
easu
rem
ent
100
Dif
fere
nt T
ypes
of
Ang
les
Pair
s
101
Spec
ialii
ne S
egm
ents
Ass
ocia
ted
with
a T
rian
gle
102
Con
stru
ct S
peci
al L
ine
Segm
ents
Ass
ocia
ted
with
a T
rian
gle
106
Con
gnie
nt A
ngle
s
107
l'erp
endi
cula
r L
ines
& A
ltitu
des
108
Bis
ecto
rs &
Mid
poin
ts
(Con
tinue
(1)
45
1168
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500.
60 -
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00
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000.
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000.
60 -
- 2.
001.
25 46
TA
BL
E 9
(C
ontin
ued)
GE
OM
ET
RY
TO
PIC
S
TO
PIC
S IN
SE
QU
EN
TIA
L M
AT
H C
OU
RSE
S C
OV
ER
ED
IN
TE
CH
-PR
EP
MA
TH
CU
RR
ICU
LA
YE
S11
OU
RS
CO
UR
SE &
ITE
M N
O.
TO
PIC
S
II11
0C
ongr
uent
Pol
ygon
s11
1Si
de-A
ngle
-Sid
e Pr
oofs
112
Ang
le-S
ide-
Ang
le &
Sid
e-Si
de-S
ide
113
Mor
e D
iffi
cult
Con
grue
nt T
rian
gles
114
Cor
resp
ondi
ng P
arts
of
Con
grue
nt T
rian
gles
115
Isos
cele
s T
rian
gles
116
Equ
ilate
ral T
rian
gles
117
Prov
e O
verl
appi
ng T
rian
gles
are
Con
grue
nt11
8M
ore
Dif
ficu
lt O
verl
appi
ng T
rian
gle
Proo
fs
011
9U
sing
Tw
o Pa
ir o
f C
ongr
uent
Tri
angl
es in
a P
roof
120
Perp
endi
cula
r L
ines
121
Para
llel L
ines
122
Prov
e T
hat L
ines
are
Par
alle
l12
3Pr
oper
ties
of P
aral
lel L
ines
124
Sum
of
the
Mea
sure
s of
the
Ang
les
ofa
Tri
angl
e12
5Pr
ove
Tri
angl
e C
ongr
uent
126
Con
vers
e of
Iso
scel
es T
rian
gle
The
orem
127
Tro
ve R
ight
Tri
angl
es a
re C
ongr
uent
128
Rel
atin
g E
xter
ior
to N
on-A
djac
ent I
nter
ior
Ang
les
129
Form
ulas
Con
cern
ing
Inte
rior
Ang
les
130
'For
mul
as C
once
rnin
g E
xter
ior
Ang
les
131
Prop
ertie
s of
a P
aral
lelo
gram
47(C
ontin
ued)
NST
AN
DA
RD
RA
NG
EM
ED
IAN
318
.80.
501.
501.
50
16.
31.
501.
501.
50
16.
31.
50--
----
--
00.
01.
00--
---
--
16.
31.
001.
501.
50
1275
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500.
25 -
- 2.
001.
00
1168
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500.
25 -
- 2.
001.
00
00.
02.
00--
----
-0
0.0
1.50
----
----
00.
02.
00--
----
--
1381
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500.
30 -
- 4.
001.
50
1391
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500.
30 -
- 4.
001.
502
12.5
1.00
1.50
1.50
956
.31.
000.
30 -
- 2.
001.
50
1610
01.
000.
25 -
- 3.
001.
002
12.5
0.50
2.00
2.00
318
.80.
501.
50 -
- 2.
001.
75
16.
30.
501.
501.
50
637
.50.
500.
60 -
- 1.
500.
754
25.0
0.50
0.60
--
1.25
0.75
425
.00.
500.
60 -
- 1.
000.
7512
75.0
1.00
0.25
--
2.00
1.00
48
TA
BL
E 9
(C
ontin
ued)
GE
OM
ET
RY
TO
PIC
S
TO
PIC
S IN
SE
QU
EN
TIA
L M
AT
H C
OU
RSE
S C
OV
ER
ED
IN
TE
CII
-PR
EP
MA
TH
CU
RR
ICU
LA
CO
UR
SE &
YE
SH
OU
RS
ITE
M N
O.
TO
PIC
SN
% S
TA
ND
AR
D R
AN
GE
1113
2Pr
ove
Tha
t a Q
uadr
ilate
ral i
s a
Para
llelo
gram
133
Prov
e T
hat a
Qua
drila
tera
l is
a R
ecta
ngle
134
Prov
e T
hat a
Qua
drila
tera
l is
a R
hom
bus
135
Prov
e T
hat a
Qua
drila
tera
l is
a Sq
uare
136
Prov
e T
hat a
Qua
drila
tera
l is
an I
sosc
eles
Tra
pezo
id13
7R
atio
s &
Pro
port
ions
138
Prop
ortio
n A
ssoc
iate
d w
ith a
Tri
angl
e13
9Pr
ove
Tha
t Tri
angl
es a
re S
imila
r14
0Pr
ove
Tha
t Lin
e Se
gmen
ts a
re in
Pro
port
ion
141
Prov
e T
hat P
rodu
cts
of L
ine
Segm
ents
are
Equ
al14
2Sp
ecia
l Pro
port
ions
of
Lin
e Se
gmen
ts in
Sim
ilar
Poly
gons
143
Prop
ortio
ns in
the
Rig
ht T
rian
gle
144
App
ly th
e Py
thag
orea
n T
heor
em14
530
°-60
3-90
° T
rian
gle
146
45°-
45°-
90°
Tri
angl
e
153
Rec
tang
ular
Coo
rdin
ate
Syst
em
155
The
Mid
poin
t For
mul
a
157
Slop
es o
f Pa
ralle
l & P
erpe
ndic
ular
Lin
es16
5Fi
ve B
asic
Loc
i
166
Geo
met
ric
App
licat
ions
of
Loc
us16
7Fi
ndin
g a
Loc
us b
y C
onst
ruct
ion
168
Inte
rsec
tion
of L
oci
171
Loc
us o
f Po
ints
Equ
idis
tant
fro
m T
hree
Non
-col
linea
r Po
ints
49(C
ontim
ied)
ME
DIA
N
16.
31.
500.
750.
75
16.
31.
000.
750.
75
16.
30.
500.
750.
75
16.
30.
500.
750.
75
16.
30.
500.
750.
75
1487
.52.
001.
50 -
10.
005.
00
956
.31.
000.
50 -
- 5.
001.
50
318
.81.
500.
25 -
- 2.
000.
75
16.
31.
500.
250.
25
00.
01.
50--
----
-5
31.3
0.50
0.60
--
1.50
0.88
956
.31.
500.
60 -
- 3.
001.
50
1593
.81.
000.
50 -
- 5.
002.
00
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500.
60 -
- 2.
001.
25
1275
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500.
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- 2.
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13
1168
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- 3.
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1062
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- 3.
001.
00
00.
00.
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----
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01.
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----
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00.
01.
00--
----
--
00.
01.
00--
----
--
16.
30.
50--
----
--
50
TA
BL
E 9
(C
ontin
ued)
GE
OM
ET
RY
TO
PIC
S
TO
PIC
S IN
SE
QU
EN
TIA
L M
AT
H C
OU
RSE
S C
OV
ER
ED
IN
TE
CH
-PR
EP
MA
TH
CU
RR
ICU
LA
CO
UR
SE &
YE
SH
OU
RS
ITE
M N
O.
TO
PIC
SN
% S
TA
ND
AR
DR
AN
GE
III
235
236
237
238
239
240
241
242
243
244
245
246
247
250
306
307
308
309
310 51
Tra
nsfo
rmat
iona
l Geo
met
ry
Prop
ertie
s of
Lin
e R
efle
ctio
ns U
sing
Coo
rdin
ate
Geo
met
ryPo
int R
efle
ctio
n &
Poi
nt S
ymm
etry
Tra
nsla
tion
Rot
atio
n
Dila
tion
Part
s of
a C
ircl
e
Prov
ing
Arc
s &
Cho
rds
Con
grue
nt
Prov
ing
Cho
rds
Con
grue
nt
Mea
suri
ng I
nscr
ibed
Ang
les
Tan
gent
s to
a C
ircl
e
Mea
suri
ng A
ngle
s: T
ange
nt-C
hord
, 2 T
ange
nts,
2 S
ecan
ts...
Mea
suri
ng a
n A
ngle
For
med
by
Tw
o C
hord
sC
onst
ruct
ions
Rel
ated
to th
e C
ircl
e
Com
posi
tion
of T
rans
form
atio
ns
Geo
met
ric
Tra
nsfo
rmat
ions
with
Mat
hem
atic
al S
yste
ms
Glid
e R
efle
ctio
n
Isom
etri
cs
Sim
ilari
ty T
rans
form
atio
ns
TO
TA
LS
ME
DIA
N
425
.01.
001.
501.
50
531
.31.
000.
50 -
1.5
00.
75
531
.31.
000.
50 -
- 1.
500.
75
531
.31.
000.
50 -
1.5
00.
75
531
.31.
000.
50 -
- 1.
500.
75
637
.51.
000.
50 -
- 2.
001.
13
1381
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500.
50 -
- 4.
001.
00
16.
30.
501.
001.
00
00.
00.
50--
----
--
531
.31.
001.
00 -
- 3.
001.
00
531
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000.
60 -
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002.
00
318
.81.
500.
60 -
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000.
75
318
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000.
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750.
60
425
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000.
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2.0
00.
75
318
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501.
501.
50
00.
02.
00--
----
--
16.
31.
000.
750.
75
212
.51.
001.
501.
50
16.
31.
001.
501.
50
557
92.7
592
.98
52
TA
BL
E 1
0
ME
ASU
RE
ME
NT
TO
PIC
S
TO
PIC
S IN
SE
QU
EN
TIA
L M
AT
H C
OU
RSE
S C
OV
ER
ED
IN
TE
CII
-PR
EP
MA
TII
CU
RR
ICU
LA
CO
UR
SE &
I' E
sII
OU
RS
ITE
M N
O.
TO
PIC
SN
STA
ND
AR
D R
AN
GE
I67
Find
ing
Are
as o
f Po
lygo
nal R
egio
ns88
Vol
ume
II15
4T
he D
ista
nce
Form
ula
156
The
Slo
pe F
orm
ula
164
Find
ing
Are
as b
y U
sing
Rec
tang
les
& T
rape
zoid
sII
I24
8Fi
ndin
g th
e L
engt
h of
Lin
e Se
gmen
ts24
9C
ircu
lar
Cir
cum
fere
nce,
Are
a, A
rc L
engt
h, S
ecto
r, S
egm
ent
TO
TA
LS
53
ME
DIA
N
1593
.82.
000.
50 -
- 5.
002.
0015
93.8
0.50
0.50
--
5.00
3.25
1168
.81.
500.
25 -
- 3.
001.
50
1381
.31.
000.
50 -
- 4.
001.
75
1168
.81.
001.
20 -
- 6.
502.
009
56.3
1.00
0.25
--
6.75
2.00
956
.31.
500.
60 -
- 5.
002.
00
838.
50
54
14.5
0
TA
BL
E 1
1
PRO
BA
BIL
ITY
& S
TA
TIS
TIC
S T
OPI
CS
TO
PIC
S IN
SE
QU
EN
TIA
L M
AT
H C
OU
RSE
S C
OV
ER
ED
IN
TE
CH
-PR
EP
MA
TII
CU
RR
ICU
LA
CO
UR
SE &
YE
SH
OU
RS
ITE
M N
O.
TO
PIC
SN
% S
TA
ND
AR
DR
AN
GE
ME
DIA
N
I24
Wha
t is
Prob
abili
ty?
25W
hy d
o w
e In
trod
uce
Sam
ple
Spac
e?
26W
hat a
re th
e R
elat
ions
hips
Bet
wee
n L
ogic
& P
roba
bilit
y?27
How
do
we
App
ly P
roba
bilit
y to
Rea
l Lif
e Si
tuat
ions
?46
The
Fun
dam
enta
l Cou
ntin
g Pr
inci
ple
47Pe
rmut
atio
ns
48Fa
ctor
ials
49M
ultis
tage
Pro
babi
lity
Prob
lem
s
68In
trod
uctio
n to
Sta
tistic
s
69D
ata
Tab
les
70G
raph
s &
His
togr
ams
71M
easu
res
of C
ent.
ai r
ende
ncy
72G
roup
ed D
ata
86C
umul
ativ
e Fr
eque
ncy
His
togr
ams
87M
easu
res
of P
osili
on: M
edia
n, Q
uart
iles,
Per
cent
iles
II18
8-S
impl
e Pr
obab
ility
Pro
blem
s
189
Perm
utat
ions
190
Perm
utat
ions
with
Rep
etiti
ons
191
Usi
ng P
erm
utat
ions
to S
olve
Wor
d &
Num
eric
al P
robl
ems
193
Com
b in
at io
ns
194
App
lyin
g C
ombi
natio
ns to
Cou
ntin
g
195
App
lyin
g C
ombi
natio
ns to
Geo
met
ry
196
Usi
ng C
ombi
natio
ns to
Sol
ve P
roba
bilit
y Pr
oble
ms
197
Mor
e D
iffi
cult
Prob
lem
s in
Cou
ntin
g &
I'ro
babi
lity
55(C
ontin
ued)
1381
.31.
500.
60 -
- 3.
001.
25
1275
.00.
500.
60 -
- 5.
001.
00
425
.00.
500.
25 -
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600.
43
1275
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000.
60 -
- 8.
003.
00
1062
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500.
50 -
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00
850
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500.
25 -
- 4.
001.
00
637
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50 -
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501.
00
637
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801.
00
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00
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60 -
20.
001.
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1487
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001.
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- 7.
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002.
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00
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60 -
- 1.
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,18
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60 -
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501.
05
31.3
2.00
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--
3.00
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531
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000.
60 -
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25
425
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000.
60 -
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00
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000.
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001.
00
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000.
60 -
- 2.
001.
00
56
TA
BL
E 1
1 (C
ontin
ued)
PRO
BA
BIL
ITY
& S
TA
TIS
TIC
S T
OPI
CS
TO
PIC
S IN
SE
QU
EN
TIA
L M
AT
H C
OU
RSE
S C
OV
ER
ED
IN
TE
CH
-PR
EP
MA
TH
CU
RR
ICU
LA
CO
UR
SE &
yEs
ITE
M N
O.
TO
PIC
S
III
296
Tre
e D
iagr
ams
for
Prob
abili
ties
in 2
+ S
tage
Exp
erim
ents
297
.Sol
ving
Con
ditio
nal P
roba
bilit
y Pr
oble
ms
298
Prob
abili
ty o
f an
Eve
nt in
Seq
uenc
e of
Ind
epen
dent
Tri
als
299
Usi
ng th
e B
inom
ial T
heor
em to
Sol
ve P
roba
bilit
y Pr
oble
ms
300
-Dat
a fr
om E
xper
imen
ts to
Det
erm
ine
& V
erif
y Pr
obab
ilitie
s30
1U
sing
Sum
mat
ion
Not
atio
n
302
Usi
ng F
requ
ency
Tab
les
to O
rgan
ize
Dat
a &
Fin
d th
e M
ean
303
Find
ing
Mea
sure
s of
Cen
tral
Ten
denc
y30
4Fi
ndin
g M
easu
res
of D
ispe
rsio
n
305
Prob
abili
ties
of a
n E
xper
imen
t Ila
ving
a N
orm
al D
istr
ibut
ion
TO
TA
LS
57
% S
TA
ND
AR
DII
OU
RS
RA
NG
EM
ED
IAN
425
.01.
000.
60 -
- 2.
001.
25
425
.02.
000.
60 -
- 0.
750.
75
425
.01.
000.
60 -
- 0.
750.
75
318
.82.
000.
60 -
- 1.
501.
50
425
.02,
500.
60 -
- 3.
001.
13
318
.81.
000.
60 -
- 2.
501.
55
1062
.51.
000.
60 -
12.
001.
50
1381
.31.
500.
25 -
- 8.
001.
38
1275
.01.
000.
50 -
- 6.
001.
50
956
.32.
000.
50 -
- 2.
500.
75
270
48.5
0
58
44.8
3
TA
BL
E 1
2
AL
GE
BR
A T
OPI
CS
TO
PIC
S IN
SE
QU
EN
TIA
L M
AT
H C
OU
RSE
S C
OV
ER
ED
IN
TE
CH
-PR
EP
MA
TH
CU
RR
ICU
LA
YE
SH
OU
RS
CO
UR
SE &
ITE
M N
O.
TO
PIC
S1
6L
ogic
, Gra
phs
& S
olut
ion
Sets
7U
sing
Log
ic to
Hel
p So
lve
Ineq
ualit
ies
19E
xpre
ssin
g R
elat
ions
hips
with
Var
iabl
es &
Ope
n Se
nten
ces
20M
onom
ials
23M
ultip
licat
ion
& D
ivis
ion
of M
onom
ials
28C
ompa
re/C
ontr
ast A
dditi
on &
Mul
tiplic
atio
n of
Mon
omia
ls29
'Mul
tiplic
atio
n &
Div
isio
n of
Mon
omia
l Ter
m A
lgbr
Fra
ctio
ns30
Solu
tion
of L
inea
r E
quat
ions
Inv
olvi
ng O
neO
pera
tion
31So
lutio
n of
Lin
ear
Equ
atio
ns I
nvol
ving
Tw
oO
pera
tions
32So
lvin
g L
inea
r In
equa
litie
s In
volv
ing
One
Ope
ratio
nal
33A
ddin
g &
Sub
trac
ting
Poly
nom
ials
34M
ultip
lyin
g a
Poly
nom
ial b
y a
Mon
omia
l40
Intr
oduc
tion
to Q
uadr
atic
Equ
atio
ns41
Solv
ing
Qua
drat
ic E
quat
ions
in F
orm
of
a Pr
oduc
t = 0
42In
trod
uctio
n to
Fac
tori
ng43
Mul
tiply
ing
Bin
omia
ls44
Solv
ing
Qua
drat
ic E
quat
ions
by
Fact
orin
g45
Ver
bal P
robl
ems
Lea
ding
to Q
uadr
atic
Equ
atio
ns75
Slop
e of
a S
trai
ght L
ine
76E
quat
ion
of a
Str
aigh
t Lin
e77
Gra
phin
g a
Lin
ear
Equ
atio
n78
Gra
phic
Sol
utio
n of
a S
yste
m o
f T
wo
Lin
ear
Equ
atio
ns79
Gra
phin
g a
Lin
ear
Ineq
ualit
y
59(C
ontin
ued)
N%
ST
AN
DA
RD
RA
NG
EM
ED
IAN
425
.00.
500.
50 -
- 2.
001.
25
425
.01.
000.
20 -
- 0.
501.
25
1610
03.
000.
50 -
20.
002.
0016
100
0.50
0.60
- 4
.00
1.25
1381
.30.
500.
50 -
- 5.
001.
50
1168
.80.
500.
25 -
- 3.
001.
50
956
.31.
500.
60 -
3.0
01.
50
1593
.81.
000.
60 -
- 8.
002.
5015
93.8
2.50
1.00
--
8.00
4.25
1275
.00.
500.
50 -
- 6.
002.
0012
75.0
1.25
0.50
- 4
.00
1.50
1275
.01.
000.
25 -
- 4.
001.
25
1487
.50.
500.
50 -
- 3.
001.
00
1275
.00.
500.
50 -
- 3.
001.
50
1275
.01.
000.
50 -
- 4.
002.
0014
87.5
1.50
0.50
--
3.00
1.50
1168
.83.
000.
50 -
- 3.
001.
75
1275
.01.
500.
50 -
- 5.
002.
0015
93.8
1.50
0.60
- 5
.00
1.75
1593
.82.
000.
50 -
- 5.
002.
0014
87.5
2.00
0.60
--
8.00
2.00
1275
.04.
000.
50 -
- 4.
001.
75
1275
.01.
000.
50 -
- 4.
002.
00
60
TA
BL
E 1
2 (C
ontin
ued)
AL
GE
BR
A T
OPI
CS
TO
PIC
S IN
SE
QU
EN
TIA
L M
AT
H C
OU
RSE
S C
OV
ER
ED
IN
TE
CII
-PR
EP
MA
TH
CU
RR
ICU
LA
l'ES
IIO
UR
SN
% S
TA
ND
AR
D R
AN
GE
ME
DIA
NC
OU
RSE
&IT
EM
NO
.T
OPI
CS
I80
Gra
phic
Sol
utio
n of
a S
yste
m o
f L
inea
r In
eq-.
.alit
ies
81A
lgeb
raic
Sol
utio
n of
a S
yste
m o
f L
inea
r E
quat
ions
82R
educ
tion
of A
lgeb
raic
Fra
ctio
ns83
Mul
tiplic
atio
n «
Div
isio
n of
Alg
ebra
ic F
ract
ions
84A
dditi
on &
Sub
trac
tion
of A
lgeb
raic
Fra
ctio
ns85
App
licat
ions
of
Alg
ebra
ic F
ract
ions
, Equ
atio
ns, I
nequ
aliti
es
II10
3E
quiv
alen
ce R
elat
ion
104
Add
ition
& R
efle
xive
Pro
pert
ies
105
Subt
ract
ion
Prop
erty
158
Gra
phin
g a
Lin
ear
Func
tion
Lk)
159
Poin
t-Sl
ope
Form
of
the
Equ
atio
n of
a L
ine
160
Slop
e-In
terc
ept F
orm
of
the
Equ
atio
n of
a L
ine
163
Are
a in
Coo
rdin
ate
Geo
met
ry16
9E
quat
ion
of a
Loc
us
170
Equ
atio
n of
a C
ircl
e
172
Find
ing
Equ
atio
ns o
f L
ocus
in th
e C
oord
inat
e Pl
ane
173
The
Equ
atio
n of
a P
arab
ola
174
Inte
rsec
tion
of L
oci i
n th
e C
oord
inat
e Pl
ane
175
Solv
ing
Qua
drat
ic E
quat
ions
by
Fact
orin
g17
6So
lvin
g Q
uadr
atic
Equ
atio
ns--
Inco
mpl
ete
or P
erfe
ct S
quar
es17
7So
lvin
g Q
uadr
atic
Equ
atio
ns -
- L
eadi
ng C
oeff
icie
nt I
sn't
= I
178
Gra
phin
g Q
uadr
atic
Rel
atio
ns17
9So
lvin
g Q
uadr
atic
Equ
atio
ns b
y G
raph
ing
61(C
ontin
ued)
1381
.32.
000.
50 -
4.0
01.
60
1275
.03.
001.
20 -
- 6.
002.
00
637
.51.
500.
50 -
3.0
00.
68
531
.31.
500.
60 -
3.0
01.
13
531
.32.
000.
60 -
3.0
01.
13
743
.82.
000.
60 -
- 6.
001.
38
743
.81.
000.
50 -
4.0
01.
25
425
.00.
500.
50 -
- 1.
500.
75
425
.00.
500.
75 -
1.5
01.
50
1381
.31.
500.
60 -
- 5.
001.
75
956
.31.
000.
60 -
3.0
01.
50
1381
.31.
000.
60 -
- 5.
001.
75
743
.82.
000.
60 -
2.0
01.
50
16.
31.
003.
003.
00
637
.50.
500.
60 -
1.5
01.
00
318
.81.
001.
20 -
- 2.
001.
60
50.0
1.00
1.20
--
5.00
2.00
16.
31.
002.
502.
50
1062
.52.
001.
20 -
- 5.
001.
50
743
.80.
501.
00 -
- 2.
001.
60
850
.01.
001.
00 -
- 1.
501.
20
50.0
1.00
0.60
--
8.00
1.25
1062
.52.
000.
50 -
1.5
01.
00
62
TA
BL
E 1
2 (C
ontin
ued)
AL
GE
BR
A T
OPI
CS
TO
PIC
S IN
SE
QU
EN
TIA
L M
AT
II C
OU
RSE
S C
OV
ER
ED
IN
TE
CH
-PR
EP
MA
TH
CU
RR
ICU
LA
YE
SH
OU
RS
CO
UR
SE &
rrE
m N
O.
TO
PIC
SII
180
Com
plet
ing
the
Squa
re
181
Der
ivin
g th
e Q
uadr
atic
For
mul
a18
2N
atur
e of
the
Roo
ts o
f a
Qua
drat
ic E
quat
ion
183
Rel
atio
n B
etw
een
Coe
ffic
ient
s &
Roo
ts o
f Q
uadr
atic
Equ
atio
n18
4Fi
ndin
g Po
ints
of
Inte
rsec
tion
of a
Par
abol
a &
a L
ine
185
Solv
ing
Oth
er Q
uadr
atic
-Lin
ear
Syst
ems
Gra
phic
ally
186
Solv
ing
Qua
drat
ic-L
inea
r Sy
stem
s A
lgeb
raic
ally
187
Wor
d Pr
oble
ms
Whi
ch L
ead
to Q
uadr
atic
Equ
atio
ns19
8Id
entit
y, I
nver
se &
Clo
sure
199
Com
mut
ativ
e, A
ssoc
iativ
e &
Dis
trib
utiv
e Pr
oper
ties
(.4
CO
200
Clo
ck A
rith
met
ic20
1D
efin
ition
of
a G
roup
202
Gro
ups
in M
athe
mat
ical
Sys
tem
s20
3D
efin
ition
of
a Fi
eld
III
206
Solv
ing
Firs
t Deg
ree
Equ
atio
ns
207
Solv
ing
Ver
bal P
robl
ems
208
Solv
ing
Lin
ear
Ineq
ualit
ies
209
Solv
ing
Com
poun
d In
equa
litie
s21
0So
lvin
g A
bsol
ute
Val
ue E
quat
ions
211
Solv
ing
Abs
olut
e V
alue
Ine
qual
ities
212
Add
ition
, Sub
trac
tion
& M
ultip
licat
ion
of P
olyn
omia
l Ter
ms
213
Fact
orin
g Po
lyno
mia
ls21
4D
ivis
ion
of P
olyn
omia
ls
63(C
ontin
ued)
NST
AN
DA
RD
RA
NG
EM
ED
IAN
25.0
0.50
0.75
--
1.20
1.00
425
.01.
500.
60 -
- 1.
000.
75
425
.01.
000.
60 -
- 0.
750.
6(25
.00.
750.
60 -
- 0.
75P-
68
318
.81.
001.
501.
50
425
.01.
501.
50 -
- 2.
502.
004
25.0
1.50
1.50
--
2.50
2.00
1381
.31.
500.
50 -
- 5.
002.
152
12.5
1.00
1.00
--
2.00
1.50
637
.51.
000.
60 -
- 3.
002.
000
0.0
1.00
----
----
00.
01.
00--
----
--
00.
00.
75--
----
--
00.
01.
00--
----
--
1381
.30.
500.
60 -
- 5.
002.
0014
87.5
2.00
0.50
--
6.50
2.00
956
.31.
000.
60 -
- 2.
001.
00
637
.50.
500.
50 -
- 2.
501.
13
31.3
1.00
0.60
--
1.00
0.88
425
.01.
000.
50 -
- 2.
000.
68
850
.00.
500.
50 -
- 5.
001.
75
850
.01.
000.
60 -
- 3.
001.
00
43.8
1.00
0.60
--
2.00
0.75
64
TA
BL
E 1
2 (C
ontin
ued)
AL
GE
BR
A T
OPI
CS
TO
PIC
S IN
SE
QU
EN
TIA
L M
AT
II C
OU
RSE
S C
OV
ER
ED
IN
'FI?
,C1I
-PR
EP
MA
TII
CU
RR
ICU
LA
CO
UR
SE &
ITE
M N
O.
TO
PIC
SII
I21
5M
ultip
lyin
g &
Div
idin
g R
atio
nal E
xpre
ssio
ns
216
Add
ing
& S
ubtr
actin
g R
atio
nal E
xpre
ssio
ns21
7Si
mpl
ifyi
ng C
ompl
ex F
ract
ions
218
Solv
ing
Frac
tiona
l Equ
atio
ns21
9So
lvin
g Q
uadr
atic
Ine
qual
ities
by
Fact
orin
g22
0Si
mpl
ifyi
ng R
adic
al E
xpre
ssio
ns22
1M
ultip
lyin
g &
Div
idin
g R
adic
al E
xpre
ssio
ns
222
Solv
ing
Rad
ical
Equ
atio
ns22
6N
atur
e of
the
Roo
ts o
f a
Qua
drat
ic E
quat
ion
227
Sum
& P
rodu
ct o
f th
e R
oots
of
a Q
uadr
atic
Equ
atio
nu.
, w22
8R
elat
ion'
s
229
-Fun
ctio
ns
230
Func
tion
Not
atio
n
231
Gra
phin
g R
elat
ions
& F
unct
ions
232
Com
posi
tion
of F
unct
ions
233
Inve
rse
of a
Rel
atio
n
234
Find
ing
the
Val
ue o
f a
Func
tion
by L
inea
r In
terp
olat
ion
270
Solv
ing
Equ
atio
ns w
ith F
ract
ions
, Var
iabl
es in
Exp
onen
t27
1M
eani
ng o
f th
e E
xpon
entia
l Fun
ctio
n &
Its
Gra
ph27
2In
vers
e of
Exp
onen
tial F
unct
ion:
The
Log
arith
mic
Fun
ctio
n27
8So
lvin
g E
xpon
entia
l & L
ogar
ithm
ic E
quat
ions
279
App
lyin
g L
ogar
ithm
s to
Pro
blem
s In
volv
ing
Gro
wth
65T
OT
AL
S
YE
SN
% S
TA
ND
AR
DII
OU
RS
RA
NG
EM
ED
IAN
850
.00.
500.
60 -
1.5
01.
50
850
.01.
000.
60 -
1.5
01.
50
25.0
1.00
0.60
- 1
.50
0.75
425
.01.
000.
75 -
- 2.
001.
50
637
.51.
000.
50 -
- 2.
000.
75
637
.50.
500.
60 -
3.0
01.
00
637
.51.
000.
50 -
3.0
01.
00
743
.81.
000.
60 -
2.0
01.
00
425
.01.
000.
25 -
1.5
00.
80
318
.81.
000.
60 -
- 1.
501.
00
743
.81.
000.
50 -
1.5
01.
00
956
.31.
000.
75 -
7.0
01.
38
850
.00.
500.
75 -
1.5
01.
00
956
.31.
000.
75 -
8.0
01.
50
425
.01.
000.
75 -
- 1.
501.
00
318
.81.
500.
75 -
1.0
00.
881
6.3
0.50
----
---
425
.01.
502.
00 -
- 3.
003.
005
31.3
2.00
0.75
--
4.00
1.00
425
.01.
500.
60 -
- 1.
000.
753
18.8
1.50
0.60
--
2.00
0.75
425
.02.
000.
60 -
- 3.
002.
00
688
110.
2512
6.11
66
TA
BL
E 1
3
TR
IGO
NO
ME
TR
Y T
OPI
CS
TO
PIC
S IN
SE
QU
EN
TIA
L M
AT
H C
OU
RSE
S C
OV
ER
ED
IN
TE
CH
-PR
EP
MA
TH
CU
RR
ICU
LA
CO
UR
SE &
YE
SH
OU
RS
ITE
M N
O.
TO
PIC
SN
% S
TA
ND
AR
D R
AN
GE
ME
DIA
N
II14
7T
he T
ange
nt R
atio
148
Solv
ing
Prob
lem
s U
sing
the
Tan
gent
Rat
io14
9T
he S
ine
& C
osin
e R
atio
s15
0So
lvin
g Pr
oble
ms
Usi
ng th
e Si
ne &
Cos
ine
Rat
ios
151
Find
ing
Len
gths
or
Ang
les
in G
eom
etri
c Fi
gure
s
152
Usi
ng T
rigo
nom
etry
to F
ind
the
Are
a of
a P
olyg
onII
I25
1Fi
ndin
g Si
des
& A
ngle
s of
Rig
ht T
rian
gle
Usi
ng T
rigo
nom
etry
252
Tri
gono
met
ric
Val
ues
of 3
00, 4
50, &
60"
253
Stan
dard
Pos
ition
of
an A
ngle
254
Rad
ian
Mea
sure
cp25
5D
efin
ition
s of
Sin
, Cos
& T
an o
f A
ngle
in S
tand
ard
Posi
tion
256
Find
ing
Tri
g Fu
nctio
ns o
f A
ngle
Giv
en V
alue
of
1 T
rig
Func
tn25
7-G
raph
ing
of y
= s
in x
and
y =
cos
x25
8G
raph
ing
of y
= a
sin
x a
nd y
a co
s x
259
Gra
phin
g of
y =
a s
in b
x an
d y
= a
cos
bx
260
Gra
phin
g of
y =
tan
x26
1Fi
ndin
g C
otan
gent
, Cos
ecan
t & S
ecan
t
262
Find
ing
Quo
tient
& P
ytha
gore
an I
dent
ities
263
Prov
ing
Tri
gono
met
ric
Iden
titie
s26
4So
lvin
g L
inea
r T
rigo
nom
etri
c E
quat
ions
265
Solv
ing
Qua
drat
ic T
rigo
nom
etri
c E
quat
ions
266
Solv
ing
Tri
g E
quat
ions
with
Mor
e T
han
One
Fun
ctio
n26
7So
lvin
g T
rigo
nom
etri
c E
quat
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to th
e N
eare
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inut
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67(C
ont i
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13
(Con
tinue
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TR
IGO
NO
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TR
Y T
OPI
CS
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SE
QU
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TIA
L M
AT
II C
OU
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S C
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IN
TE
CH
-PR
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RR
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LA
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SE &
YE
SH
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ITE
M N
O.
TO
PIC
SN
% S
TA
ND
AR
DR
AN
GE
III
280
Der
ivin
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e L
aw o
f Si
nes
281
Usi
ng th
e L
aw o
f Si
nes
282
Usi
ng th
e L
aw o
f Si
nes
Giv
en S
ide-
Side
-Ang
le
283
Solv
ing
Tri
gono
met
ric
Prob
lem
s U
sing
the
Law
of
Sine
s28
4D
eriv
ing
the
Law
of
Cos
ines
285
Usi
ng th
e L
aw o
f C
osin
es
286
App
lyin
g th
e L
aws
of S
ines
& C
osin
es to
Rig
ht T
rian
gles
287
Solv
ing
Para
llelo
gram
of
Forc
es P
robl
ems
288
Usi
ng S
um &
Dif
fere
nce
Form
ulas
for
Sin
e &
Cos
ine
289
Usi
ng th
e D
oubl
e A
ngle
For
mul
as f
or S
ine
& C
osin
e
290
Usi
ng th
e H
alf-
Aog
le F
orm
ulas
for
Sin
e &
Cos
ine
291
Tan
gent
For
mul
as f
or S
um, D
iffe
renc
e, D
oubl
e &
Hal
f-A
ngle
292
App
lyin
g T
rigo
nom
etri
c Fo
rmul
as in
Pro
ving
Ide
ntiti
es29
3A
pply
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Tri
g Fo
rmul
as in
Sol
ving
Tri
gono
met
ric
Equ
atio
ns29
4E
valu
atin
g In
vers
e T
rigo
nom
etri
c R
elat
ions
& F
unct
ions
295
Usi
ng P
olar
Coo
rdin
ates
TO
TA
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63
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DIA
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0
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Stra
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l'rep
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for
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Exa
min
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I I
311
Sequ
ence
s31
2A
rith
met
ic S
eque
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313
Ari
thm
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Ser
ies
314
Geo
met
ric
Sequ
ence
s31
5G
eom
etri
c Se
ries
316
Sum
of
an I
nfin
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eom
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71
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2512
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8
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S FR
OM
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AT
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S W
HIC
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AT
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CH
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CU
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S IN
12
(75%
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R M
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Subj
TO
KC
S IN
4 (
25%
) O
R F
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CU
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ICU
LA
No.
Top
icN
o.T
opic
LO
G2
Sent
ence
s, S
tate
men
ts &
Ope
n Se
nten
ces
LO
G55
Sym
bolic
Log
ic R
elat
ed to
Con
ditio
nal S
tate
men
tsN
UM
1N
umbe
rs &
The
ir N
ames
56C
ondi
tnal
Sta
tem
nts:
Inv
erse
, Con
vers
e, C
ontr
apos
9In
trod
uctio
n to
Rea
l Num
bers
91L
aws
of L
ogic
35In
trod
uctio
n to
Squ
are
Roo
ts92
App
ly L
aws
of L
ogic
to T
est V
alid
ity o
f A
rgum
ent
74O
rder
ed P
airs
in a
Coo
rdin
ate
Syst
em93
Law
s of
Syl
logi
sm &
Dis
junc
tive
Infe
renc
e26
9U
sing
Sci
entif
ic N
otat
ion
to W
rite
Num
bers
95U
se L
ogic
Law
s to
Wri
te M
ore
Com
plic
atd
Proo
fsO
PS10
Add
ition
of
Rea
l Num
bers
96R
ole
of Q
uant
ifie
rs11
Subt
ract
ion
of R
eal N
umbe
rs97
Foun
datio
ns o
f an
Axi
omat
ic S
yste
m12
Iden
tity
Ele
men
ts &
Inv
erse
s10
9Fo
rmal
Pro
of13
Ass
oc A
dditi
on, D
istr
ib M
ultip
lic, O
rder
of
Ope
rs16
1Pr
ove
Geo
met
ric
Rel
atns
with
Coo
rdin
ate
Geo
met
ry14
Mul
tiplic
atio
n of
Rea
l Num
bers
162
Prov
e G
ener
al T
heor
ems
with
Coo
rdin
ate
Geo
met
ry15
Ass
ocia
tivity
317
Prin
cipl
e of
Mat
hem
atic
al I
nduc
tion
16C
omm
utat
ivity
NU
M22
3Im
agin
ary
Num
bers
17A
pplic
atio
ns o
f th
e L
aws
of A
lgeb
ra22
4Pr
oper
ties
of C
ompl
ex N
umbe
rs21
Div
isio
n of
Rea
l Num
bers
OPS
192
Lin
ear
& C
ircu
lar
Seat
ing
Prob
lem
s22
Law
s of
Exp
onen
ts22
5D
ivid
ing
Com
plex
Num
bers
GE
O58
Ang
les
Rei
s: S
uppl
emen
try,
Com
plem
ntry
, Ver
tical
273
Find
ing
the
Log
arith
m o
f a
Num
ber
59C
lass
ifyi
ng T
rian
gles
by
Side
Lng
th &
Ang
le M
eas
274
Usi
ng C
omm
on L
ogar
ithm
Tab
les
60T
he P
ytha
gore
an T
heor
em27
5Fi
ndin
g th
e L
ogar
ithm
of
a Pr
oduc
t61
Cir
cles
276
Find
ing
the
Log
arith
m o
f a
Quo
tient
62R
eis
of S
trai
ght L
ines
in P
lane
: Par
alle
l & I
nter
sctg
277
Find
ing
the
Log
arith
m o
f a
Pow
er o
r R
oot
63M
akin
g In
fere
nces
Abo
ut G
iven
Par
alle
l Lin
esG
EO
57M
etho
ds o
f D
eter
min
ing
Con
grue
nt T
rian
gles
65C
lass
ific
atio
n of
Qua
drila
tera
ls: S
ide
Para
llelis
m11
0C
ongr
uent
Pol
ygon
s73
Rev
iew
of
Ana
lytic
Geo
met
ry11
1Si
de-A
ngle
-Sid
e Pr
oofs
99B
asic
Ter
min
olog
y of
Ang
les
& A
ngle
Mea
sure
men
t11
2A
ngle
-Sid
e-A
ngle
& S
ide-
Side
-Sid
e
7374
TA
BL
E 1
5 (C
ontin
ued)
Top
icSu
bj.
TO
PIC
S IN
12
(75%
) O
R M
OR
E C
UR
RIC
UL
AT
opic
Suhj
TO
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S IN
4 (
25%
) O
R F
EW
ER
CU
RR
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LA
No.
Top
icN
o.T
opic
GE
O10
0D
iffe
rent
Typ
es o
f A
ngle
Pai
rsG
EO
113
Mor
e D
iffi
cult
Con
grue
nt T
rian
gles
106
Con
grue
nt A
ngle
s11
4C
orre
spon
ding
Par
ts o
f C
ongr
uent
Tri
angl
es10
7Pe
rpen
dicu
lar
Lin
es &
Alti
tude
s11
7Pr
ove
Ove
rlap
ping
Tri
angl
es a
re C
ongr
uent
108
Bis
ecto
rs &
Mid
poin
ts11
8M
ore
Dif
ficu
lt O
verl
appi
ng T
rian
gle
Proo
fs11
5Is
osce
les
Tri
angl
es11
9U
sing
Tw
o Pa
ir o
f C
ongr
uent
Tri
angl
es in
a P
roof
120
Perp
endi
cula
r L
ines
122
Prov
e T
hat L
ines
are
Par
alle
l12
!Pa
ralle
l Lin
es12
5Pr
ove
Tri
angl
es C
ongr
uent
124
Sum
of
the
Mea
sure
s of
the
Ang
les
in a
Tri
angl
e12
6C
onve
rse
of I
sosc
eles
Tri
angl
e T
heor
em13
1Pr
oper
ties
of a
Par
alle
logr
am12
7Pr
ove
Rig
ht T
rian
gles
are
Con
grue
nt13
7R
atio
s &
Pro
port
ions
129
Form
ulas
Con
cern
ing
Inte
rior
Ang
les
144
App
ly th
e Py
thag
orea
n T
heor
em13
0Fo
rmul
as C
once
rnin
g E
xter
ior
Ang
les
145
30°-
60°-
90°
Tri
angl
e13
2Pr
ove
Tha
t a Q
uadr
ilate
ral i
s a
Para
llelo
gram
146
45*-
450-
900
Tri
angl
e13
3Pr
ove
Tha
t a Q
uadr
ilate
ral i
s a
Rec
tang
le15
7Sl
opes
of
Para
llel &
Per
pend
icul
ar L
ines
134
Prov
e T
hat a
Qua
drila
tera
l is
a R
hom
bus
241
Part
s of
a C
ircl
e13
5Pr
ove
Tha
t a Q
uadr
ilate
ral i
s a
Squa
reM
EA
67Fi
ndin
g A
reas
of
Poly
gona
l Reg
ions
136
Prov
e T
hat Q
uadr
ilate
ral i
s an
Iso
scel
es T
rape
zoid
88V
olum
e13
9Pr
ove
Tha
t Tri
angl
es a
re S
imila
r15
6T
he S
lope
For
mul
a14
0Pr
ove
Tha
t Lin
e Se
gmen
ts a
re in
Pro
port
ion
PRB
24W
hat i
s Pr
obab
ility
?14
1Pr
ove
Tha
t Pro
duct
s of
Lin
e Se
gmen
ts a
re E
qual
25W
hy D
o W
e In
trod
uce
Sam
ple
Spac
e?16
5Fi
ve B
asic
Loc
i27
How
Do
We
App
ly P
roba
bilit
y to
Rea
l Lif
e?16
6G
eom
etri
c A
pplic
atio
ns o
f L
ocus
68In
trod
uctio
n to
Sta
tistic
s16
7Fi
ndin
g a
Loc
us b
y C
onst
ruct
ion
69D
ata
Tab
les
168
Inte
rsec
tion
of L
oci
70G
raph
s &
His
togr
ams
171
Loc
us o
f Po
ints
Equ
idis
tant
fro
m 3
Non
-col
linr
Pnts
71M
easu
res
of C
entr
al T
ende
ncy
235
Tra
nsfo
rmat
iona
l Geo
met
ry72
Gro
uped
Dat
a24
2Pr
ovin
g A
rcs
& C
hord
s C
ongr
uent
188
Sim
ple
Prob
abili
ty P
robl
ems
243
Prov
ing
Cho
rds
Con
grue
nt30
3Fi
ndin
g M
easu
res
of C
entr
al T
ende
ncy
246
Mea
suri
ng A
ngle
s: T
an-C
hord
, 2 T
ans,
2 S
ecan
ts...
304
Find
ing
Mea
sure
s of
Dis
pers
ion
247
Mea
suri
ng a
n A
ngle
For
med
by
Tw
o C
hord
sA
LG
19E
xpre
ss R
elat
ions
with
Var
iabl
es &
Ope
n Se
nten
ces
250
Con
stru
ctio
ns R
elat
ed to
the
Cir
cle
20M
onom
ials
306
Com
posi
tion
of T
rans
form
atio
ns
7576
TA
BL
E 1
5 (C
ontin
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Top
icSu
tiA
LG
TR
G
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TO
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S IN
12
(75%
) O
R M
OR
E C
UR
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UL
AT
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Sub'
.
TO
PIC
S IN
4 (
25%
) O
R F
EW
ER
CU
RR
ICU
LA
No.
Top
icN
o.T
opic
23M
ultip
licat
ion
of M
onom
ials
30So
lutio
n of
Lin
ear
Equ
ats
Invo
lvin
g 1
Ope
ratn
31So
lutio
n of
Lin
ear
Equ
ats
Invo
lvin
g 2
Ope
ratn
s32
Solv
ing
Lin
ear
Ineq
ualit
ies
InvI
vng
1 O
pera
tion
33A
ddin
g &
Sub
trac
ting
Poly
nom
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34M
ultip
lyin
g a
Poly
nom
ial b
y a
Mon
omia
l40
Intr
oduc
tion
to Q
uadr
atic
Equ
atio
ns41
Solv
ing
Qua
drat
ic E
quat
s in
For
m: P
rodu
ct =
042
Intr
oduc
tion
to F
acto
ring
43M
ultip
lyin
g B
inom
ials
45V
erba
l Pro
blem
s L
eadi
ng to
Qua
drat
ic E
quat
s75
Slop
e of
a S
trai
ght L
ine
76E
quat
ion
of a
Str
aigh
t Lin
e77
Gra
phin
g a
Une
ar E
quat
ion
78G
raph
ic S
olui
;on
of S
yste
m o
f 2
Lin
ear
Equ
ats
79G
raph
ing
a L
inea
r In
equa
lity
80G
raph
ic S
olut
n of
Sys
tm o
f L
inea
r In
equa
litie
s81
A:g
ebra
ic S
olut
ion
of S
yste
m o
f L
inea
r E
quat
s15
8G
i7ap
hing
Lin
ear
Func
tion
160
Slop
e-In
terc
ept F
orm
of
the
Equ
atio
n of
a L
ine
187
Wor
d Pr
obs
Whi
ch L
ead
to Q
uadr
atic
Equ
ats
206
Solv
ing
Firs
t Deg
ree
Equ
atio
ns20
7So
lvin
g V
erba
l Pro
blem
s14
7T
he T
ange
nt R
atio
148
Solv
ing
Prob
lem
s U
sing
the
Tan
gent
Rat
io14
9T
he S
ine
& C
osin
e R
atio
s15
0So
lvin
g Pr
oble
ms
Usi
ng S
ine
& C
osin
e R
atio
s25
1Fi
ndin
g Si
des
& A
ngle
s of
Rig
ht T
rian
gle
with
Tri
g18
Exp
lora
tion
& I
nsig
ht in
to P
robl
em S
trat
egie
s
GE
O 3
07G
eom
etri
c T
rans
form
atio
ns w
ith M
athe
mtc
l Sys
tms
308
Glid
e R
efle
ctio
n30
9Is
omet
rics
310
Sim
ilari
ty T
rans
form
atio
nsPR
B26
Wha
t are
Rel
atio
nshi
ps B
etw
n L
ogic
& P
roba
bilit
y?19
0Pe
rmut
atio
ns w
ith R
epet
ition
s19
4A
pply
ing
Com
bina
tions
to C
ount
ing
195
App
lyin
g C
ombi
natio
ns to
Geo
met
ry19
6U
sing
Com
bina
tions
to S
olve
Pro
babi
lity
Prob
lem
s19
7M
ore
Dif
ficu
lt Pr
oble
ms
in C
ount
ing
& P
roba
bilit
y29
6T
ree
Dia
gram
s fo
r Pr
obab
ilitie
s in
2+
Sta
ge E
xper
s29
7So
lvin
g C
ondi
tiona
l Pro
babi
lity
Prob
lem
s29
8Pr
obab
ility
of
an E
vent
in S
eque
nce
of I
ndep
Tri
als
299
Use
Bin
omia
l The
orem
to S
olve
Pro
babi
lity
Prob
lms
300
Exp
erim
ent!
Dat
a to
Det
erm
ine/
Ver
ify
Prob
abili
ties
301
Usi
ng S
umm
atio
n N
otat
ion
AL
G6
Log
ic, G
raph
s &
Sol
utio
n Se
ts7
Usi
ng L
ogic
to H
elp
Solv
e In
equa
litie
s10
4A
dditi
on &
Ref
lexi
ve P
rope
rtie
s10
5Su
btra
ctio
n Pr
oper
ty16
9E
quat
ion
of a
Loc
us17
2Fi
ndin
g E
quat
ions
of
Loc
us in
the
Coo
rdin
ate
Plan
e17
4In
ters
ectio
n of
Loc
i in
the
Coo
rdin
ate
Plan
e18
0C
ompl
etin
g th
e Sq
uare
181
Der
ivin
g th
e Q
uadr
atic
For
mul
a18
2N
atur
e of
the
Roo
ts o
f a
Qua
drat
ic E
quat
ion
183
Rel
Bet
wn
Coe
ffic
ient
& R
oots
of
Qua
drat
ic E
quat
184
Find
ing
Poin
ts o
f In
ters
ectio
n of
a P
arab
ola
& L
ine
185
Solv
e O
ther
Qua
drat
ic-L
inea
r Sy
stem
s G
raph
ical
ly18
6So
lvin
g Q
uadr
atic
-Lin
ear
Syst
ems
Alg
ebra
ical
ly19
8Id
entit
y, I
nver
se &
Clo
sure
7778
TA
BL
E 1
5 (C
ontin
ued) TO
PIC
S IN
4 (
25%
) O
R F
EW
ER
CU
RR
ICU
LA
Top
icSu
bi N
o.T
opic
AL
G20
0C
lock
Ari
thm
etic
201
Def
initi
on o
f a
Gro
up20
2G
roup
s in
Mat
hem
atic
al S
yste
ms
203
Def
initi
on o
f a
Fiel
d21
1So
lvin
g A
bsol
ute
Val
ue I
nequ
aliti
es21
7Si
mpl
ifyi
ng C
ompl
ex F
ract
ions
218
Solv
ing
Frac
tiona
l Equ
atio
ns22
6N
atur
e of
the
Roo
ts o
f a
Qua
drat
ic E
quat
ion
227
Sum
& P
rodu
ct o
f R
oots
of
Qua
drat
ic E
quat
232
Com
posi
tion
of F
unct
ions
233
Inve
rse
of a
Rel
atio
n23
4Fi
nd V
alue
of
Func
tion
by L
inea
:. In
terp
olat
ion
crN
270
Solv
e E
quat
s w
ith F
ract
ions
/Var
iabl
es in
Exp
nt27
2In
vers
e of
Exp
onen
tl Fu
nctn
: Log
rthm
c Fu
nctn
278
Solv
ing
Exp
onen
tial &
Log
aritm
ic E
quat
ions
279
App
lyin
g L
ogar
ithm
s to
Pro
blem
s in
Gro
wth
TR
G 2
61Fi
ndin
g C
otan
gent
, Cos
ecan
t, &
Fec
ant
262
Find
ing
Quo
tient
& P
ytha
gore
an I
dent
ities
263
Prov
ing
Tri
gono
met
ric
Iden
titie
s26
4So
lvin
g L
inea
r T
rigo
nom
etri
c E
quat
ions
265
Solv
ing
Qua
drat
ic T
rigo
nom
etri
c E
quat
ions
266
Solv
ing
Tri
g E
quat
s w
ith M
ore
Tha
n 1
Func
tn26
7So
lvin
g T
rig
Equ
atio
ns to
the
Nea
rest
Min
ute
280
Der
ivin
g th
e L
aw o
f Si
nes
282
Usi
ng th
e L
aw o
f Si
nes
Giv
en S
ide-
Side
.Ang
le28
4D
eriv
ing
the
Law
of
Cos
ines
287
Solv
ing
Para
llelo
gram
of
Forc
es P
robl
ems
288
Use
Sum
& D
iffr
nce
Form
ulas
for
Sin
e &
Cos
73'0
TA
BL
E 1
5 (C
ontin
ued)
LO
GN
UM
OPS
GE
OM
EA
PRB
AL
GT
RG
OT
H
= = = = = = = = =
Log
ic T
opic
sN
umbe
r Se
nse
& N
umer
atio
n C
once
pts
Top
ics
Ope
ratio
ns o
n N
umbe
r T
opic
sG
eom
etry
Top
ics
Mea
sure
men
t Top
ics
Prob
abili
ty &
Sta
tistic
s T
opic
sA
lgeb
ra T
opic
sT
rigo
nom
etry
Top
ics
Oth
er T
or: ;
Not
Inc
lude
d A
bove
Top
icSu
bt
TO
PIC
S IN
4 (
25%
) O
R F
EW
ER
CU
RR
ICU
LA
No.
Top
icU
se D
oubl
e A
ngle
For
mul
as f
or S
ine
& C
osin
eU
se H
alf-
Ang
le F
orm
ulas
for
Sin
e &
Cos
ine
Tan
gent
For
mul
as: S
um, D
ifT
, Dou
ble,
1/2
Ang
lA
pply
Tri
g Fo
rmul
as in
Pro
ving
Ide
ntiti
esA
pply
Tri
g Fo
rmul
as in
Sol
ving
Tri
g E
quat
ions
Eva
luat
ing
Inve
rse
Tri
g R
elat
ions
& F
unct
ions
Usi
ng P
olar
Coo
rdin
ates
Prep
arat
ion
for
Reg
ents
Exa
min
atio
nSe
quen
ces
Ari
thm
etic
Seq
uenc
esA
rith
met
ic S
erie
sG
eom
etri
c Se
quen
ces
Geo
met
ric
Seri
esSu
m o
f an
Inf
inite
Geo
met
ric
Seri
es
TR
G28
929
029
129
229
329
429
5O
TH
89 311
312
313
314
315
316
8 2
TABLE 16
ADDITIONAL TOPICS COVERED IN TECH-PREP MATH CURRICULAAS REPORTED BY RESPONDENTS
SUBJECT TOPIC
MATHEMATICS 3 Equations, 3 Unknowns (Listed by 1 A, 1 B)Matrices (1 A)Surface Area (1 A)3 Dimensional Shapes (1 A, 1 B)Application of Statistics (1 A)Designing a Statistical Study (1 A, 1 B)Applicatiors of Formulae (1 A)Inverse Proportions (1 A)*Ratios & Proportions (1 A)[Topic 137]*2 Equations, 2 Unknowns (1 B)[Topics 78, 81]Topics in Jr. High School Math (1 A)Binary Numbers (1 A)*Systems of Inequality -- Graphically (1 B)[Topic 801Exponents (1 B)Constructing Models in Geometry & Trigonometry (1 B)*Area of Any Triangle (1 C)[Topic 67]Piece-wise Function (1 C)
COMPUTERS &CALCULATORS
Computer Spreadsheets for Statistics (1 A)Computer Spreadsheets -- General (1 A)Using Scientific Calculators (1 A, 1 B, 1 C)Computer Graphics (1 A)Using Computer Graphics to Solve Practical Problems (1 A)Using Scientific Calculator for Linear Regression, Linear
Programming, Quadratic Progression (1 C)
MEASUREMENT Converting Measurements (1 A)SYSTEMS & Using Vernier & Micrometer Calipers (1 A)INSTRUMENTATION Using Stopwatches, Rulers & Line Levels (1 A)
Metric & English Measurement Systems (1 A)English -- Metric Conversion (1 B)
COMMUNICATION &CAREER EDUCATION
Oral & Written Presentations (1 A)Communication (1 B)Career Exploration (1 A, 1 B)
(Continued)
48
83
TABLE 16 (Continued)
ADDITIONAL TOPICS COVERED IN TECHPREP MATH CURRICULAAS REPORTED BY RESPONDENTS
SUBJECT TOPIC
TECHNOLOGY Precision, Accuracy and Tolerances (1 A, 1 B)Drawing Detailed Scale Drawings (1 A, 1 B)Quality Assurance & Process Control (1 A)Outdoor Surveying (1 A)Outdoor Design Mapping (1 A)Using Drawing Instruments (1 A)AND/OR Gates in Electronics (1 A)Euler Circuits (1 A)Survey Taking & Data Analysis in Health Careers (1 A)Measurements in Health Careers (I A)
MISCELLANEOUS Preparation for RCT Examination (1 A)Managing Personal Finances (1 A)Map Coloring (1 A)Graph Coloring (1 A)
* Denotes Topics Reported as Additional Which Were Listed in Questionnaire
49
84
TABLE 17
INSTRUCTIONAL MATERIALS IN TECH-PREP MATH CURRICULAAS REPORTED BY RESPONDENTS
TEXTBOOKSAlgebra 2. Larson, R., Kano ld, T & Stiff, L. Heath. Lexington, MA. 1993.
Curriculum k Evaluation Standards for School Mathematics. Addenda Series. Grades 9-12.
National Council of Teachers of Mathematics. Reston, VA. 1995.
Elementary Algebra for College Students. Angel, A. Prentice-Hall. 1991.
Integrated Math Course I. Dressler, I. & Keenan, E. Amsco School Publishing, Inc. New
York. 1980.
Integrated Math Course II. Keenan, E. & Dressler, L Amsco. New York. 1980.
Integrated Math Course HI. Keenan, E. & Ganter, A. Amsco. New York. 1980
Integrated Mathematics Course I. Kelly, D., Atkinson, P., & Alexander, B. McDougal, Littell
& Co. Evanston, IL. 1991.
Integrated Mathematics Course II. Kelly, B., Atkinson, P., & Alexander, B. McDougal, Linen
& Co. Evanston, IL. 1991.
Math Connections. Gardena, F. Houghton-Mifflin, 1992.
Mathematics: A Topical Approach. Bumby, D. & Klutch, R. C. E. Merrill Publishing Co.
Columbus, OH. 1986.
Mathmatters. Lynch, C. & Olmstead, E. South Western Publishing Co. Cincinnati, OH. 1993.
Practical Problems in Mathematics for Electronics Technicians. Herman, S. & Sullivan, R.
Delmar Publishing Co. Albany. 1995.
CENTER FOR OCCUPATIONAL RESEARCH & DEVELOPMENT MATERIALS
Dividing the 40 Units of CORD Applied Mathematics into One-Year Courses.
9th Grade (15 units)UnitA. Getting to Know Your CalculatorB. Naming Numbers in Different
WaysC. Fmding Answers with Your
Calculator1. Learning Problem-solv-ing
Techniques2. Estimating Answers3. Measurint in English and Metric
Units4. Using Graphs. Clans, and Tables5. Dealing with Data6. Working with Lines and Angles7. Working with Shapes in Two
Dimensions8. Working with Shapes in Three
Dimensions -
9. Using Ratios and Proportions10. Working with Scale Drawings11. Using Signsd Numbezs and .
Vectors12. Using Scientific flotation
106 Grade (13 units)Unit
13. Precision, Accuracy, and Tolerance14. Solving Problems with Powers and
Roots15. Using Formulas to Solve Pioblems16. Solving Problems That Involve -
Linear Equations17. Graphing Data18. Solving Problems That Involve
Nonlinear Fquations19. Working with Statistics"c0. Working with Probabilides23. Factoring24. Patterns and Functions25. Quadratics26. Systems of Equauons27. Inequalities
lltn Grade (12 units)Unit
21. Using Right-triangle Relationships22- Using Trigonometric Functions28. Geometry in the Workplace 129. Geometry in the Workplace 230. Solving Problems with Computer
Spreadsheets31. Solving Problems with Computer
Graphics32. Quality Assurance and Process
Control I ,
33. Quality Assurance and ProcessControl 2
34. Logic (new unit)35. Spatial Viivalization (new unit)36. Coordinate Geometry (new unit)37. Transformations (new unit)
-ED
Applied EDEBvt
matics,,A CCM/DIU 1/4 AVP0A04 TO MOAT= Pmall.MA ICS
e50
TABLE 18
PERCENT AGREEMENT WITHNEW YORK STATE FRAMEWORK MATHEMATICS STATEMENTS
BY CATEGORY AND BY SUBJECT AREA
No. OFFRAMEWK
SUBJECT STATEMNTS % RANGE % MEDN % RANGE % MEDN % MEDN
PERCENT AGREEMENTA
(N = 16) (N = 5)ABCD
(N = 27)
LOG 8 0 -- 100 43.8 0 -- 100 75.0 50.0
NUM 8 62.5 -- 100 100 75.0 -- 100 100 100
OPS 11 63.6 -- 100 100 81.8 -- 100 100 100
GEO 13 23.1 -- 100 76.9 30.8 -- 100 92.3
MEA 12 25.0 -- 100 95.8 25.0 -- 100 91.7
PRB 22 0.0 -- 100 65.9 0.0 -- 95.5 86.4 86.4
ALG 13 46.2 -- 100 96.1 92.3 -- 100 92.3
TRG 5 0.0 -- 100 100 0.0 -- 100 100 100
TOTAL 92 37.0 -- 98.9 86.5 38.0 -- 95.7 90.6 85.9
Category A = Tech-Prep curricula with no standard sequential coursesCategory B = Tech-Prep curricula which include Sequential Course ICategory C = Tech-Prep curricula which include Sequential Courses I & IICategory D = Tech-Prep curricula which include Sequential Courses I, II & HI
LOG = Logic StatementsNUM = Number Sense & Numeral ion Concepts StatementsOPS = Operations on Number StatementsGEO = Geometry StatementsMEA = Measurement StatementsPRB = Probability & Statistics StatementsALG = Algebra StatementsTRG = Trigonometry Statements
51
86
FIGURE 3
PERCENTAGE OF TOPICS IN SEQUENTIAL MATH COURSES
WHICH ARE COVERED IN A TECH-PREP CURRICULUM
VS.
PERCENTAGE AGREEMENT WITH STATE FRAMEWORK MATH STATEMENTS
FOR EACH TECH-PREP CURRICULUM
lod
90
60
E 70-
1-1
Z 50cz?.
<z zc.)
w 30
C 20
10
0
A = Curriculum with No. Sequential Courses
B = Curriculum with OneSequential Course
C = Curriculum with TwoSequential Courses
D = Curriculum with ThreeSequential Courses
A
A
r-/ 0 20 30 4:0 io 610 710 8'0 90 /00
e
A
A AA B
AA
A A A
A
A
% AGREEMENT WITH STATE FRAMEWORK STATEMENTSIN A TECH-PREP CURRICULUM
52
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1487
.51
111
100
00
1610
00
010
90.9
01
1610
00
011
100
00
1487
.50
29
81.8
11
1381
.31
211
100
00
732
552
12
162
59
114
21
5
93
'FA
BL
E 2
2
RE
SPO
NSE
S T
O F
RA
ME
WO
RK
MA
TH
EM
AT
ICS
STA
TE
ME
NT
S
ON
GE
OM
ET
RY
LE
VE
LFR
AM
EW
OR
K S
TA
TE
ME
NT
INT
ER
ME
D E
xplo
re w
ays
in w
hich
geo
met
ry is
use
d in
the
real
wor
ld
Use
tran
sfor
mat
ions
to s
ee p
rese
rvat
ion
of s
ize
&/o
r sh
ape
Cla
ssif
y 2-
& 3
-dim
ensi
onal
fig
ures
on
part
icul
ar a
ttrib
utes
Con
stru
ct g
eom
etri
c co
nclu
sion
s us
ing
logi
cal r
easo
ning
Use
gem
idea
s to
ana
lyze
pro
bs in
volv
ing
geom
con
cept
scr
iSu
b-T
otal
CO
NIM
EN
Rel
ate
geom
etri
c pr
inci
ples
to r
eal-
wor
ld p
heno
men
a
Show
und
rstn
dg o
f tr
ansf
orm
atio
ns:
cong
ruen
ce, s
imila
rity
Ana
lyze
pat
tern
s in
2 &
3 d
imen
sion
s
Con
stru
ct c
onvi
ncin
g ar
gum
ents
usi
ng g
eom
etri
cco
ncep
ts
App
ly g
eom
etri
c id
eas
in th
e so
lutio
n of
pro
blem
s
From
ass
umpt
ns, d
educ
epr
ops
of &
rel
tn b
etw
n ge
om f
igs
Mak
e co
ord
repr
esen
tatn
s of
geo
met
ric
figu
res
&co
ncep
ts
Ded
uce
prop
s of
fig
ures
usi
ng tr
ansf
rmat
ns &
coo
rdin
ates
Sub-
Tot
al
TO
TA
L
94
CA
TE
GO
RY
AY
ES
NO
?N
o.%
No.
No.
CA
TE
GO
RIE
S B
CD
YE
SN
ON
o.%
No.
No.
1610
00
010
90.9
10
956
.36
29
81.8
20
1593
.81
08
72.7
12
425
.010
28
72.7
12
1062
.54
210
90.9
10
5421
645
64
1487
.52
010
90.9
10
956
.35
29
81.8
11
1275
.03
17
63.6
13
850
.08
J7
63.6
13
1381
.33
09
81.8
11
1275
.04
07
63.6
22
1275
.04
07
63.6
40
425
.011
17
63.6
40
8440
471
1510
138
6110
116
2114
TA
BL
E 2
3
RE
SPO
NSE
S T
O F
RA
ME
WO
RK
MA
TH
EM
AT
ICS
STA
TE
ME
NT
S
ON
ME
ASU
RE
ME
NT
LE
VE
LFR
AM
EW
OR
K S
TA
TE
ME
NT
INT
ER
ME
D K
now
mea
sure
men
ts c
an b
e to
spe
cifi
c de
gree
of
accu
racy
Sele
ct a
ppro
pria
te to
ols
to m
easu
re d
egre
e ac
cura
cy d
esir
ed
Mea
sure
& c
ompu
te in
bot
h E
nglis
h &
met
ric
syst
ems
Info
rmal
ly d
eriv
e an
d us
e fo
rmul
as in
mea
sure
men
t act
ivs
Solv
e w
ide
arra
y of
pro
blem
s us
ing
mea
sure
men
t con
cept
s
Com
pute
qua
ntiti
es: a
rea
& v
olum
e us
ing
stan
dard
uni
ts
Sub-
Tot
al
CO
MM
EN
Rel
mea
srm
t pre
cisi
on w
ith c
alc
accu
racy
usi
ng m
easu
rem
ts
Use
inst
rum
nts
& p
roce
dure
s to
mak
e in
dire
ct m
easu
rem
ts
Use
dim
ensi
onal
ana
lysi
s in
pro
blem
s in
volv
ing
mea
sure
s
Val
idat
e fo
rmul
as u
sed
to c
ompu
te m
easu
rem
ents
Solv
e w
ide
arra
y of
pro
bs in
mat
h, s
cien
ce &
tech
cur
ric
Com
pare
rel
s of
per
imtr
, are
a, v
ol o
f va
riab
le s
ize
figu
res
Sub-
Tot
al
TO
TA
L
96
CA
TE
GO
RY
AY
ES
NO
?N
o.%
No.
No.
CA
TE
GO
RIE
S B
CD
YE
SN
O?
No.
%N
o.N
o,
1593
.80
110
90.9
10
1593
.81
09
81.8
20
1593
.80
110
90.9
10
1593
.81
09
81.8
20
1593
.81
010
90.9
1
1610
00
011
100
00
913
259
7
1381
.32
18
72.7
30
1275
.03
18
72.7
21
1381
.32
18
72.7
21
850
.04
47
63.6
22
1593
.81
010
90.9
01
1487
.51
110
90.9
10
7513
851
105
166
16I
1011
01
117
I5
9 7
TA
BL
E 2
4
RE
SPO
NSE
S T
O F
RA
ME
WO
RK
MA
TH
EM
AT
ICS
STA
TE
ME
NT
S
ON
PR
OB
AB
ILIT
Y &
ST
AT
IST
ICS
LE
VE
LFR
AM
EW
OR
K S
TA
TE
ME
NT
INT
ER
ME
D D
istin
guis
h be
twee
n em
piri
cal &
theo
retic
al p
roba
bilit
ies
Dev
ise/
cond
uct e
xper
imen
ts/s
imul
atns
to f
ind
prob
abili
ties
Use
pro
bab
mod
el to
com
pare
res
ults
with
mat
h ex
pect
atns
Mak
e pr
edic
tions
App
reci
ate
the
perv
asiv
e us
e of
pro
babi
lity
in r
eal w
orld
Col
lect
, org
aniz
e, d
escr
ibe,
inte
rpre
t gro
uped
& in
div
data
Use
sam
plin
g in
sta
tistic
al s
tudi
es
Use
mea
sure
s of
cen
tral
tend
enci
es to
ana
lyze
dat
a
Ext
rapo
late
info
fro
m d
ata
set i
n nu
mer
icor
gra
phic
for
m
Sub-
Tot
al
(Con
tinue
d)
98
CA
TE
GO
RY
AY
ES
NO
?N
o.%
No.
No.
CA
TE
GO
RIE
S B
CD
YE
SN
O?
No.
%N
o.N
o.
1062
.56
09
81.8
11
956
.37
010
90.9
10
1062
.56
08
72.7
12
1168
.85
09
81.8
11
1062
.56
09
81.8
11
1381
.33
010
90.9
10
1381
.33
010
90.9
10
1487
.52
010
90.9
10
1487
.51
19
81.8
11
1104
113
9184
99
TA
BL
E 2
4 (C
ontin
ued)
RE
SPO
NSE
S T
O F
RA
ME
WO
RK
MA
TIW
MA
TIC
S ST
AT
EM
EN
TS
ON
PR
OB
AB
ILIT
Y &
ST
AT
IST
ICS
LE
VE
LFR
AM
EW
OR
K S
TA
TE
ME
NT
CO
MM
EN
Use
pro
babi
l;ty
to r
epre
sent
& s
olve
cha
nce
prob
lem
s
Cre
ate
& in
terp
ret d
iscr
ete
prob
abili
ty d
istr
ibut
ions
Und
erst
and
the
conc
ept &
use
of
rand
omne
ss
Use
com
pute
r &
oth
er s
imul
atio
ns to
est
imat
e pr
obab
ilitie
s
Und
erst
and
a no
rmal
dis
trib
utio
n
Solv
e pr
oble
ms
invo
lvin
g pr
obab
ility
con
cept
s
Kno
w th
at a
s ex
peri
mnt
l n in
crse
s pr
obab
nea
rs th
eore
tic!
Des
ign/
cond
uct s
tatis
tcl s
tudi
es: i
nter
pret
/com
mun
ic r
esul
ts
Dra
w in
fere
nces
fro
m d
ata
in c
hart
s, ta
bles
and
gra
phs
Dif
fere
ntia
te b
etw
een
norm
ally
dis
trib
uted
& s
kew
ed d
ata
App
reci
ate
stat
istic
al m
etho
ds a
s m
eans
of
deci
sion
mak
ing
Det
erm
ine
mea
sure
s of
dis
pers
ion
for
a gi
ven
set o
f da
ta
Use
com
pute
r so
ftw
are
to m
odel
dat
a fr
om th
e re
al w
orld
Sub-
Tot
al
TO
TA
L
160
CA
TE
GO
RY
AC
AT
EG
OR
IES
BC
DY
ES
NO
?Y
ES
NO
?N
o.%
No.
No.
No.
%N
o.N
o.
1062
.56
09
81.8
11
956
.37
07
63.6
31
956
.37
010
90.9
10
531
.39
24
36.4
15
1275
.04
010
90.9
10
1062
.56
08
72.7
12
850
.07
17
63.6
22
1168
.85
09
81.8
11
1487
.52
010
90.9
10
1062
.56
08
72.7
21
1168
.83
210
90.9
10
1168
.85
010
90.9
10
850
.07
16
54.5
32
128
746
108
19IS
232
113
719
228
21
101.
TA
BL
E 2
5
RE
SPO
NSE
S T
O F
RA
ME
WO
RK
MA
TH
EM
AT
ICA
L S
TA
TE
ME
NT
S
ON
AL
GE
BR
A
LE
VE
LFR
AM
EW
OR
K S
TA
TE
ME
NT
INT
ER
ME
D M
odel
sol
utio
n of
sim
ple
equa
tions
usi
ng c
oncr
ete
mat
eria
ls
Gra
ph li
near
rel
atio
nshi
ps
Dev
elop
pro
cedu
res
for
com
putin
g w
ith in
tege
rs
Use
var
iabl
es
Wri
te &
sol
ve e
quat
ions
Com
pare
dir
ect &
indi
rect
var
iatio
n
Sele
ct/u
se a
ppro
p te
chno
logi
es to
inve
stig
alg
ebra
ic c
once
pts
Sub-
Tot
al
CO
MM
EN
Use
alg
ebra
ic &
gra
phic
tech
niqu
es in
sol
ving
equ
atio
ns
Com
pare
/con
tras
t dir
/indi
r va
riat
ion,
incl
udg
use
of g
raph
s
Exp
lore
fun
ctio
ns th
at d
epic
t rea
l wor
ld p
heno
men
a
Use
alg
ebra
ic te
chni
ques
in th
e so
lutio
n of
pro
blem
s
Rel
ate
alge
brai
c id
eas
to c
oord
inat
e ge
omet
ry
Sele
ct/u
se a
ppro
p te
chno
logi
es to
sol
ve a
lgeb
raic
pro
blem
Sub-
Tot
al
TO
TA
L
lo
CA
TE
GO
RY
AC
AT
EG
OR
IES
BC
DY
ES
NO
L.
YFS
NO
?N
o.%
No.
No.
No.
foN
o,N
o.
1381
.31
29
81.8
11
1487
.52
011
100
00
1593
.80
110
90.9
10
1610
00
011
100
00
1593
.81
011
100
00
1168
.83
211
100
00
1487
.51
110
90.9
01
988
673
22
1593
.81
011
100
00
1062
.53
311
100
00
1593
.81
010
90.9
0
1593
.81
011
100
00
1381
.32
19
81.8
1
1487
.52
010
90.9
01
8210
462
1
180
1810
135 1
03
TA
BL
E 2
6
RE
SPO
NSE
S '1
'0 F
RA
ME
WO
RK
MA
TH
EM
AT
ICS
STA
TE
ME
NT
S
ON
TR
IGO
NO
ME
TR
Y
LE
VE
LFR
AM
EW
OR
K S
TA
TE
ME
NT
1NT
ER
ME
D I
nves
tigat
e re
latn
ship
s am
ong
the
side
s of
sim
ilar
tria
ngle
s
Exp
lore
rel
atns
hps
with
sim
ilar
tria
ngle
s in
pro
blem
sol
ving
Sub-
Tot
al
C(M
IME
NR
el c
onsi
stnt
rat
ios
in s
imila
r ri
ght t
rian
gles
to tr
ig f
untio
ns
App
ly tr
igon
omet
ry to
pro
blem
s w
ith r
ight
tria
ngle
s
Use
cal
cula
tors
to g
et tr
igm
mm
etri
c fu
nctio
ns o
f gi
ven
angl
e
Sub-
Tot
al
TO
TA
L
104
CA
TE
GO
RY
AY
ES
NO
?N
o.%
No.
No.
CA
TE
GO
RIE
S 11
CD
YE
SN
O?
No.
%N
o.N
o.
1487
.52
010
90.9
10
1381
.33
010
90.9
10
275
020
20
1381
.33
010
90.9
.1
0
1381
.33
010
90.9
10
1381
.33
010
90.9
10
399
030
30
6614
050
50
105
SIJB
JEC
T/
LO
G--
CO
MM
C
GE
OIN
TM
D
GE
O--
CO
MN
IC
LO
(---
CO
MM
C
PRII
--C
OM
MC
IA)G
--C
OM
MC
!M.I
NT
(M)
GE
O-:
CO
NIM
C
NlE
A--
CO
NIN
IC
PRB
--C
OM
MC
PRB
--C
OM
MC
OPS
--IN
TN
ID
GE
O--
CO
NIM
C
TA
BL
E 2
7
STA
TE
FR
AM
EW
OR
K M
AT
IIE
MA
TIC
S ST
AT
EM
EN
TS
WIT
HA
GR
EE
ME
NT
BY
50
% O
R F
EW
ER
OF
TH
E R
ESP
ON
DE
NT
S
FRA
ME
WO
RK
ST
AT
EM
EN
T
Con
stru
ct s
impl
e ar
gum
ents
usi
ng la
ws
of lo
gic
Use
geo
m id
eas
to a
naly
ze p
robs
invo
lvin
g ge
om c
once
pts
Ded
uce
prop
s of
fig
ures
usi
ng tr
ansf
rmat
ns &
coo
rdin
ates
& ju
dge
valid
ity o
f lo
gica
l arg
umen
ts
Flse
com
pute
r &
oth
er s
imul
atio
ns to
est
imat
e pr
obab
ilitie
s
Rec
ogni
ze &
app
ly in
duct
ive
reas
onin
g
Rec
ogni
ze &
app
ly d
educ
tive
reas
onin
g
Nla
ke &
eva
luat
e m
ath
conj
ectu
res
& a
rgum
ents
Con
stru
ct c
onvi
ncin
g ar
gum
ents
usi
ng g
e(-
netr
ic c
once
pts
Val
idat
e fo
rmul
as u
sed
to c
ompu
te m
easu
rem
ents
Kno
w th
at a
s ex
peri
mnt
l n in
crse
s pr
obab
nea
rs th
eore
ticl
Ilse
com
pute
r so
ftw
are
to m
odel
dat
a fr
om th
e re
al w
orld
Ilse
con
cret
e m
ater
ials
for
ops
with
fra
ctns
, dec
hns,
inte
gs
Ana
lyze
pat
tern
s in
2 &
3 d
imen
sion
s
CA
TE
GO
RIE
SA
% Y
ES
BC
D%
YE
SA
BC
DN
o. ?
18.8
25.0
4
25.0
31.3
31.3
36.4
7 k
37.5
.
43.8
50.0
50.0
.
50.0
6
50.0
50.0
5 4
LO
GL
ogic
lera
mew
ork
Stat
emen
tsIN
Tm
l)=
Int
erm
edia
te L
evel
OPS
= O
pera
tions
on
Num
ber
Fram
ewor
k St
atem
ents
GE
O=
Geo
met
ry le
ratn
ewor
k St
atem
ents
CO
NIN
IC=
Com
men
cem
ent L
evel
NIE
A=
Nle
asur
emen
t Fra
mew
ork
Stat
emen
tsPR
B=
Pro
babi
lity
& S
tatis
tics
Fram
ewor
k St
atem
ents
No.
?=
Num
ber
of U
ncer
tain
Res
pons
es
106
107