GLEN RIDGE PUBLIC SCHOOLS

Curriculum Guide

 

 

 

 

Course Title:                                                     PRE-ALGEBRA

 

Subject:                                                            Mathematics

 

Grade:                                                              9-10

 

Department/School:                                          Mathematics/Glen Ridge High School

 

Duration:                                                          Full Year

 

Number of Credits:                                           5

 

Prerequisite:                                                      None

 

Elective or Required:                                         Required

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Author:  Sean Fitzpatrick

Date Submitted:  Summer 2007

 

Course Description

                  

The primary aim of this course is to prepare the student for success in the Core-Algebra course; therefore close attention is given to the development of a strong foundation of basic mathematical skills.  Once such a foundation has been established, students begin their study of algebra.  Instruction time will be utilized during the year to prepare for the math section of the H.S.P.A. test given in the junior year.

GLEN RIDGE PUBLIC SCHOOLS

MATHEMATICS MISSION STATEMENT AND GOALS

 

Mathematics and Computer Science are an integral part of our lives.  Students must be actively involved in their mathematics education with problem solving being an essential part of the curriculum.  The mathematics and computer science curricula should emphasize thinking skills through a balance of computation, intuition, common sense, logic, analysis and technology.  Students will be engaged and challenged in a student-centered learning environment that is developmentally appropriate.  Students will communicate mathematical ideas effectively by applying hands-on manipulatives, basic computational skills, mathematical models, and technology in order to solve practical problems.

New Jersey Mathematics Standards

 

The Mathematics Standards consist of five statements, which describe what is essential to excellent mathematics education, and present a view of mathematics teaching and learning that integrates the processes of mathematical activity, the content of mathematics, and the learning environment in the classroom.  The following standards were adopted by the New Jersey State Board of Education.

 

This course will cover the following Core Curriculum Standards:

 


STANDARD 4.1 (NUMBER AND NUMERICAL OPERATIONS) ALL STUDENTS WILL DEVELOP NUMBER SENSE AND WILL PERFORM STANDARD NUMERICAL OPERATIONS AND ESTIMATIONS ON ALL TYPES OF NUMBERS IN A VARIETY OF WAYS.

 

Descriptive Statement:  Numbers and arithmetic operations are what most of the general public think about when they think of mathematics; and, even though other areas like geometry, algebra, and data analysis have become increasingly important in recent years, numbers and operations remain at the heart of mathematical teaching and learning.  Facility with numbers, the ability to choose the appropriate types of numbers and the appropriate operations for a given situation, and the ability to perform those operations as well as to estimate their results, are all skills that are essential for modern day life.

 

Building upon knowledge and skills gained in preceding grades, by the end of Grade 12, students will:

  1. Number Sense
    1. Extend understanding of the number system to all real numbers.
    2. Compare and order rational and irrational numbers.
    3. Develop conjectures and informal proofs of properties of number systems and sets of numbers.
  2. Numerical Operations
    1. Extend understanding and use of operations to real numbers and algebraic procedures.
    2. Develop, apply, and explain methods for solving problems involving rational and negative exponents.
    3. Perform operations on matrices.

§        Addition and subtraction

§        Scalar multiplication

    1. Understand and apply the laws of exponents to simplify expressions involving numbers raised to powers.
  1. Estimation
    1. Recognize the limitations of estimation, assess the amount of error resulting from estimation, and determine whether the error is within acceptable tolerance limits.

 

 


STANDARD 4.2 (GEOMETRY AND MEASUREMENT) ALL STUDENTS WILL DEVELOP SPATIAL SENSE AND THE ABILITY TO USE GEOMETRIC PROPERTIES, RELATIONSHIPS, AND MEASUREMENT TO MODEL, DESCRIBE AND ANALYZE PHENOMENA.

 

Descriptive Statement:  Spatial sense is an intuitive feel for shape and space.  Geometry and measurement both involve describing the shapes we see all around us in art, nature and the things we make.  Spatial sense, geometric modeling, and measurement can help us to describe and interpret our physical environment and to solve problems.

 

Building upon knowledge and skills gained in preceding grades, by the end of Grade 12, students will:

A.  Geometric Properties

1.       Use geometric models to represent real-world situations and objects and to solve problems using those models (e.g., use Pythagorean Theorem to decide whether an object can fit through a doorway).

2.       Draw perspective views of 3D objects on isometric dot paper, given 2D representations (e.g., nets or projective views).

3.       Apply the properties of geometric shapes.

§        Parallel lines – transversal, alternate interior angles, corresponding angles

§        Triangles

a.       Conditions for congruence

b.       Segment joining midpoints of two sides is parallel to and half the length of the third side

c.       Triangle Inequality

§        Minimal conditions for a shape to be a special quadrilateral

§        Circles – arcs, central and inscribed angles, chords, tangents

§        Self-similarity

4.       Use reasoning and some form of proof to verify or refute conjectures and theorems.

§        Verification or refutation of proposed proofs

§        Simple proofs involving congruent triangles

§        Counterexamples to incorrect conjectures

B.  Transforming Shapes

1.       Determine, describe, and draw the effect of a transformation, or a sequence of transformations, on a geometric or algebraic object, and, conversely, determine whether and how one object can be transformed to another by a transformation or a sequence of transformations.

2.       Recognize three-dimensional figures obtained through transformations of two-dimensional figures (e.g., cone as rotating an isosceles triangle about an altitude), using software as an aid to visualization.

3.       Determine whether two or more given shapes can be used to generate a tessellation.

4.       Generate and analyze iterative geometric patterns.

§         Fractals (e.g., Sierpinski’s Triangle)

§         Patterns in areas and perimeters of self-similar figures

§         Outcome of extending iterative process indefinitely

C.  Coordinate Geometry

1.       Use coordinate geometry to represent and verify properties of lines.

§         Distance between two points

§         Midpoint and slope of a line segment

§         Finding the intersection of two lines

§         Lines with the same slope are parallel

§         Lines that are perpendicular have slopes whose product is –1

2.       Show position and represent motion in the coordinate plane using vectors.

§         Addition and subtraction of vectors

D.  Units of Measurement

1.       Understand and use the concept of significant digits.

2.       Choose appropriate tools and techniques to achieve the specified degree of precision and error needed in a situation.

3.       Degree of accuracy of a given measurement tool

4.       Finding the interval in which a computed measure (e.g., area or volume) lies, given the degree of precision of linear measurements

E.   Measuring Geometric Objects

1.       Use techniques of indirect measurement to represent and solve problems.

§         Similar triangles

§         Pythagorean theorem

§         Right triangle trigonometry (sine, cosine, tangent)

2.       Use a variety of strategies to determine perimeter and area of plane figures and surface area and volume of 3D figures.

§         Approximation of area using grids of different sizes

§         Finding which shape has minimal (or maximal) area, perimeter, volume, or surface area under given conditions using graphing calculators, dynamic geometric software, and/or spreadsheets

§         Estimation of area, perimeter, volume, and surface area

 

 


STANDARD 4.3 (PATTERNS AND ALGEBRA) ALL STUDENTS WILL REPRESENT AND ANALYZE RELATIONSHIPS AMONG VARIABLE QUANTITIES AND SOLVE PROBLEMS INVOLVING PATTERNS, FUNCTIONS, AND ALGEBRAIC CONCEPTS AND PROCESSES.

 

Descriptive Statement:  Algebra is a symbolic language used to express mathematical relationships.  Students need to understand how quantities are related to one another, and how algebra can be used to concisely express and analyze those relationships.  Modern technology provides tools for supplementing the traditional focus on algebraic procedures, such as solving equations, with a more visual perspective, with graphs of equations displayed on a screen.  Students can then focus on understanding the relationship between the equation and the graph, and on what the graph represents in a real-life situation.

 

Building upon knowledge and skills gained in preceding grades, by the end of Grade 12, students will:

A.  Patterns

1.       Use models and algebraic formulas to represent and analyze sequences and series.

§         Explicit formulas for nth terms

§         Sums of finite arithmetic series

§         Sums of finite and infinite geometric series

2.       Develop an informal notion of limit.

3.       Use inductive reasoning to form generalizations.

B. Functions and Relationships

1.       Understand relations and functions and select, convert flexibly among, and use various representations for them, including equations or inequalities, tables, and graphs.

2.       Analyze and explain the general properties and behavior of functions of one variable, using appropriate graphing technologies.

§         Slope of a line or curve

§         Domain and range

§         Intercepts

§         Continuity

§         Maximum/minimum

§         Estimating roots of equations

§         Intersecting points as solutions of systems of equations

§         Rates of change

3.       Understand and perform transformations on commonly-used functions.

§         Translations, reflections, dilations

§         Effects on linear and quadratic graphs of parameter changes in equations

§         Using graphing calculators or computers for more complex functions

4.       Understand and compare the properties of classes of functions, including exponential, polynomial, rational, and trigonometric functions.

§         Linear vs. non-linear

§         Symmetry

§         Increasing/decreasing on an interval

C. Modeling

1.       Use functions to model real-world phenomena and solve problems that involve varying quantities.

§         Linear, quadratic, exponential, periodic (sine and cosine), and step functions (e.g., price of mailing a first-class letter over the past 200 years)

§         Direct and inverse variation

§         Absolute value

§         Expressions, equations and inequalities

§         Same function can model variety of phenomena

§         Growth/decay and change in the natural world

§         Applications in mathematics, biology, and economics (including compound interest)

2.       Analyze and describe how a change in an independent variable leads to change in a dependent one.

3.       Convert recursive formulas to linear or exponential functions (e.g., Tower of Hanoi and doubling).

D. Procedures

1.       Evaluate and simplify expressions.

§         Add and subtract polynomials

§         Multiply a polynomial by a monomial or binomial

§         Divide a polynomial by a monomial

2.       Select and use appropriate methods to solve equations and inequalities.

§         Linear equations – algebraically

§         Quadratic equations – factoring (when the coefficient of x2 is 1) and using the quadratic formula

§         All types of equations using graphing, computer, and graphing calculator techniques

3.       Judge the meaning, utility, and reasonableness of the results of symbol manipulations, including those carried out by technology.

 

STANDARD 4.4 (DATA ANALYSIS, PROBABILITY, AND DISCRETE MATHEMATICS) ALL STUDENTS WILL DEVELOP AN UNDERSTANDING OF THE CONCEPTS AND TECHNIQUES OF DATA ANALYSIS, PROBABILITY, AND DISCRETE MATHEMATICS, AND WILL USE THEM TO MODEL SITUATIONS, SOLVE PROBLEMS AND ANALYZE AND DRAW APPROPRIATE INFERENCES FROM DATA.

 

Descriptive Statement:  Data analysis, probability, and discrete mathematics are important interrelated areas of applied mathematics.  Each provides students with powerful mathematical perspectives on everyday phenomena and with important examples of how mathematics is used in the modern world.  Two important areas of discrete mathematics are addressed in this standard; a third area, iterations and recursion, is addressed in Standard 4.3 (Patterns and Algebra).

 

Building upon knowledge and skills gained in preceding grades, by the end of Grade 12, students will:

A. Data Analysis

1.       Use surveys and sampling techniques to generate data and draw conclusions about large groups.

§         Advantages/disadvantages of sample selection methods (e.g., convenience sampling, responses to survey, random sampling)

2.       Evaluate the use of data in real-world contexts.

§         Accuracy and reasonableness of conclusions drawn

§         Bias in conclusions drawn (e.g., influence of how data is displayed)

§         Statistical claims based on sampling

3.       Design a statistical experiment, conduct the experiment, and interpret and communicate the outcome.

4.       Estimate or determine lines of best fit (or curves of best fit if appropriate) with technology, and use them to interpolate within the range of the data.

5.       Analyze data using technology, and use statistical terminology to describe conclusions.

§         Measures of dispersion: variance, standard deviation, outliers

§         Correlation coefficient

§         Normal distribution (e.g., approximately 95% of the sample lies between two standard deviations on either side of the mean)

B. Probability

1.       Calculate the expected value of a probability-based game, given the probabilities and payoffs of the various outcomes, and determine whether the game is fair.

2.       Use concepts and formulas of area to calculate geometric probabilities.

3.       Model situations involving probability with simulations (using spinners, dice, calculators and computers) and theoretical models, and solve problems using these models.

4.       Determine probabilities in complex situations.

§         Conditional events

§         Complementary events

§         Dependent and independent events

5.       Estimate probabilities and make predictions based on experimental and theoretical probabilities.

6.       Understand and use the "law of large numbers" (that experimental results tend to approach theoretical probabilities after a large number of trials).

C. Discrete Mathematics—Systematic Listing and Counting

1.       Calculate combinations with replacement (e.g., the number of possible ways of tossing a coin 5 times and getting 3 heads) and without replacement (e.g., number of possible delegations of 3 out of 23 students).

2.       Apply the multiplication rule of counting in complex situations, recognize the difference between situations with replacement and without replacement, and recognize the difference between ordered and unordered counting situations.

3.       Justify solutions to counting problems.

4.       Recognize and explain relationships involving combinations and Pascal’s Triangle, and apply those methods to situations involving probability.

D. Discrete Mathematics—Vertex-Edge Graphs and Algorithms

1.       Use vertex-edge graphs and algorithmic thinking to represent and solve practical problems.

§         Circuits that include every edge in a graph

§         Circuits that include every vertex in a graph

§         Scheduling problems (e.g., when project meetings should be scheduled to avoid conflicts) using graph coloring

§         Applications to science (e.g., who-eats-whom graphs, genetic trees, molecular structures)

2.       Explore strategies for making fair decisions.

§         Combining individual preferences into a group decision (e.g., determining winner of an election or selection process)

§         Determining how many Student Council representatives each class (9th, 10th, 11th, and 12th grade) gets when the classes have unequal sizes (apportionment)