Ohio Resource Center

## 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 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.

 First Year Chapter List for Model A Instructional Days(suggested) 1.1 Data Analysis and Introduction to Linear Relations 10 - 13 1.2 Variables and Algebraic Expressions 16 - 21 1.3 Equations and Inequalities in One Variable 19 - 24 1.4 Linear Relationships in Two Variables 20 - 24 1.5 Systems of Linear Equations and Inequalities 15 - 18 1.6 Polynomial Algebra 20 - 24 1.7 Linear and Non-linear Functions 15 - 19 1.8 Introduction to Quadratic Polynomials 22 - 27 1.9 Counting Techniques and Elementary Probability Theory 17 - 22

### 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.

 Second Year Chapter List for Model A Instructional Days(suggested) 2.1 Informal Geometric Ideas in the Coordinate Plane 26 - 35 2.2 Classical Euclidean Geometry 80 - 91 2.3 Right Triangle Trigonometry 15 - 22 2.4 Transformational Geometry 14 - 20 2.5 Informal Solid Geometry 20 - 23 2.6 Non-Euclidean Geometries (Optional) 0 - 5

### 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.

 Third Year Chapter List for Model A Instructional Days(suggested) 3.1 Data Analysis, Statistics, and Probability 15 - 20 3.2 Functions 23 - 28 3.3 Quadratic Algebra 37 - 43 3.4 Polynomial and Rational Functions 30 - 39 3.5 Exponential and Logarithmic Functions 30 - 37 3.6 Matrices 20 - 26

### 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, 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, 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 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.