Program Model A Pacing Guide
Applications-Driven Model for High School Mathematics
(For students not intending to take calculus, see Model A')
Every citizen is deluged with numbers: claims and counter-claims, polls and statistics, measures of risk, and promises of certainty. Each student must attain a level of quantitative sophistication sufficient to decide what to believe and what to challenge. The model presented here uses applications, including probability and data, to motivate student learning of algebra and geometry. An approach that combines applications, computation, and theory will engage students throughout their studies and will help prepare them for employment or further education.
This model requires that students have frequent experience with rich problems in order to understand the mathematical topics fully. Students must be challenged throughout the sequence with tasks that require creative problem solving and reasoning skills. They must also learn to communicate mathematical ideas using formal mathematical language.
First Year Course
First Year Course Rationale
Students learn best when they are engaged with interesting and meaningful problem
tasks. A project that involves the analysis of data captures student interest best
when the students themselves generate the data. Tables, lists, graphs and formulas
that grow naturally from the data lead into the full range of algebraic and logical
skills. When students study procedures and algorithms in the context of an application,
they will learn more, retain their knowledge longer, and begin to appreciate the
importance and beauty of mathematics. This approach enables students to make deep
connections between conceptual learning and the procedural learning required by
the mathematical content. Such a program can ensure that the benchmarks of the Mathematical
Processes Standard are met as well as the subject matter content standards.
First Year Course Description
This course is designed to be a first-year algebra course with applications-driven development of the content.
The early emphasis is on linear expressions and relationships. The curriculum begins with the study of bivariate
data that have a linear relationship. Intuition is developed before linear functions and equations are formally
presented. Classical topics from algebra are emphasized, such as solutions and graphs of linear functions and
solutions of linear equations, arithmetic of polynomials, factorization of trinomials, and solving quadratic equations.
Fluency with numerical computation (decimals, fractions, scientific notation, radicals, etc) with and without technology
will be reinforced throughout the curriculum.
Second Year Course
Second Year Course Rationale
Geometry was developed in the ancient world for surveying, architecture, astronomy, and navigation. However,
the main thrust of the second year course is the logical development of geometry and the beginnings of abstract
mathematical thought. For more than 2300 years Euclid’s Elements has served as the model for instruction in
mathematics and logic. The study of Euclidean geometry is necessary for anyone interested in understanding the
foundations of western civilization. This second course moves from concrete applications, through the logical
beauty of Euclidean geometry, to geometric ideas used in contemporary mathematics.
Second Year Course Description
The course uses coordinate geometry to connect the algebra learned in Year 1 to geometric topics learned in
earlier grades and in this course. Geometry is introduced informally, in the context of the coordinate plane.
Subsequently students learn the core ideas of logic and deduction in more formal Euclidean geometry, while also
understanding geometric interpretations of results in the preceding algebra course. Geometry software such as
Geometer’s Sketchpad or Cabri can be used to advantage.
The main part of the course emphasizes logic, proofs, and classical synthetic Euclidean plane geometry. This
section should occupy more than half of the year. The course concludes with sections on right triangle trigonometry,
transformational geometry, and informal solid geometry. Measurement topics of units and scaling should receive
attention throughout the course including units, conversion between units, scale factors.
Third Year Course
Third Year Course Rationale
This course allows for a deeper study of some topics included in previous courses and introduces new topics
necessary for students who will continue their mathematical studies. A variety of teaching strategies should
be used, with the underlying theme of applications-driven, exploratory activities and real-world applications.
Third Year Course Description
Prerequisite to this course is working knowledge of key topics from years one and two, including number line
and interval notation, solving equations and inequalities, and absolute value and distance. The third year course
begins with data analysis, statistics, and probability. These topics are data-driven and can be introduced and
expanded through classroom experiments and observations. By observing different trends in bivariate data,
students are introduced to linear, quadratic, cubic, exponential, and logarithmic functions. Students discuss
various properties of those functions, including their symmetry and inverses. The course also includes a deeper
study of quadratic functions, radicals, and systems of linear equations. Real-world applications and technology
should be used to promote a better understanding of the topics.
Fourth Year Course
Fourth Year Course Rationale
With the advent of the new core requirements for Ohio, all students must take mathematics in their senior year. Two options are offered as possible courses
following the three-year sequence above: Pre-Calculus or the Modeling and Quantitative
Reasoning course.
Fourth Year Course, Option 1
Pre-Calculus
Fourth Year, Option 1, Course Rationale
Topics covered in a fourth year course can have many applications to a variety of post-high school pathways.
In order to enable all students to be successful in such topics, a variety of teaching styles is encouraged,
with the depth of theory and application fitted to student needs.
Fourth Year, Option 1, Course Description
As presented here, the fourth course is primarily a course in trigonometry and its geometric applications,
together with discussion of series and applications to finance. The analysis of periodic data in 4.2 can be
expanded if more applications are desired.
| Pre-Calculus Chapter List for Model A | Instructional Days
(suggested) |
| 4.1 Trigonometry | 26 - 33 |
| 4.2 Analysis of Periodic Data | 13 - 19 |
| 4.3 Polar Coordinates | 18 - 23 |
| 4.4 Conic Sections | 20 - 25 |
| 4.5 Vectors | 23 - 28 |
| 4.6 Sequences, Series, and Mathematical Induction | 21 - 28 |
| 4.7 Personal Finance | 16 - 21 |
Fourth Year Course, Option 2
Modeling and Quantitative Reasoning
Fourth Year, Option 2, Course Rationale
One purpose of secondary education in the United States has always been preparing students for their roles
as citizens, as well as preparing them for future study and the workplace. Today numbers and data are
critical parts of public and private decision making. Decisions about health care, finances, science policy,
and the environment are decisions that require citizens to understand information presented in numerical form,
in tables, diagrams, and graphs. Students must develop skills to analyze complex issues using quantitative tools.
In addition to a textbook, teachers will want to use on-line resources, newspapers, and magazines to identify
problems that are appropriate for the course. Students should be encouraged to find issues that can be
represented in a quantitative way and shape them for investigation. Appropriate use of available technology is
essential as students explore quantitative ways of representing and presenting the results of their investigations.
Fourth Year, Option 2, Course Description
This course prepares students to investigate contemporary issues mathematically and to apply the mathematics
learned in earlier courses to answer questions that are relevant to their civic and personal lives. The course
reinforces student understanding of
- percent
- functions and their graphs
- probability and statistics
- multiple representations of data and data analysis
This course also introduces functions of two variables and graphs in three dimensions. The applications in all
sections should provide an opportunity for deeper understanding and extension of the material from earlier courses.
This course should also show the connections between different mathematics topics and between the mathematics and
the areas in which applied.
| Modeling and Quantitative Reasoning Chapter List for Model A | Instructional Days
(suggested) |
| 4M.1 Use of Percent | 15 - 18 |
| 4M.2 Statistics and Probability | 29 - 32 |
| 4M.3 Functions and Their Graphs | 54 - 65 |
| 4M.4 Functions of More Than One Variable | 10 - 15 |
| 4M.5 Geometry | 40 - 48 |
| 4M.6 Exploration of Data (integrated throughout the course) | |
Fifth Year Course
Fifth Year Course Rationale
Students in a fifth year high school mathematics course have been accelerated at some point in their study.
This might involve starting with the first year high school course in eighth grade, doubling up on courses at
some point, or another form of acceleration. Any student who has been successful in the pre-calculus course
is prepared for college-level calculus or statistics courses, and students who have been successful in either
of the other year 4 courses will be prepared for college-level statistics.
Fifth Year Course Description
The fifth year of high school mathematics will be a calculus course for most accelerated students. When a
calculus course is offered in the high school curriculum, the course should be taught at the college level and
students should expect it to replace a first year calculus course in college. This can be assured by using one
of the College Board’s Advanced Placement calculus courses and requiring students to take the AP exam at the end
of the course. In some locations, accelerated students are able to enroll in a mathematics course at an area
college or to take a college level course through distance education, concurrent with their high school studies.
The Program Models also prepare accelerated students to take the College Board’s Advanced Placement statistics
course. For many accelerated students, AP Statistics can be an exciting and appropriate option.
Syllabi for AP Calculus and AP Statistics are provided by the College Board.