ALGEBRA IN SIMPLEST TERMS – Funded by the Annenberg/CPB Project

In this instructional video series of 26 half-hour programs on high school/college algebra, host Sol Garfunkel clearly explains baffling algebraic concepts while demonstrating how algebra is used for solving real-world problems. Graphic illustrations and on-location examples make concrete connections between mathematics and daily life. The series also has applications to geometry and calculus.

Produced by the Consortium for Mathematics and Its Applications and Chedd-Angier. 1991.

Lesson 2: The Language of Algebra

Exponents and radicals: definitions, rules, and applications to positive numbers. Using Rules for Exponents to simplify expressions. Discussion of the O-ring failure of the Challenger Space Shuttle.

This program defines polynomials and describes how the distributive property is used to multiply common monomial factors with the FOIL method. It covers factoring, the difference of two squares, trinomials as products of two binomials, the sum and difference of two cubes, and regrouping of terms.

This is the first program in which equations are solved. It shows how solutions are obtained, what they mean, and how to check them using one unknown. Concepts are worked out in an application problem involving a modern sewage plant near Los Angeles, where a linear equation is set up and solved to determine how long to keep open an inlet pipe.

To the sets of numbers reviewed in previous lessons, this program adds complex numbers — their definition and their use in basic operations and quadratic equations. Students will learn how to combine like terms, apply the FOIL method, and rationalize the denominator for finding the product or quotient of two complex numbers.

This program reviews the quadratic equation and covers standard form, factoring, checking the solution, the Zero Product Property, and the difference of two squares. Environmental and aviation examples provide realistic problems, and the method of Completing the Square is used to solve them.

This program teaches students the properties and solution of inequalities, linking positive and negative numbers to the direction of the inequality. The program presents three applications of inequalities: modeling problems of the U.S. Postal Service, finding the cheapest way to travel, and conducting market research in the pizza industry.

In this program, the concept of absolute value is defined, enabling students to use it in equations and inequalities. One application example involves systolic blood pressure, using a formula incorporating absolute value to find a person’s “pressure difference from normal.” The recipe for making fireworks offers another example.

This program looks at the linear relationship between two variables, expressed as a set of ordered pairs. Students are shown the use of linear equations to develop and provide information about two quantities, as well as the applications of these equations to the slope of a line.

The circle and parabola are presented as two of the four conic sections explored in this series. The circle, its various measures when graphed on the coordinate plane (distance, radius, etc.), its related equations (e.g., center-radius form), and its relationships with other shapes are covered, as is the parabola with its various measures and characteristics (focus, directrix, vertex, etc.). An earthquake epicenter provides a real-life illustration.

The ellipse and hyperbola, the other two conic sections examined in the series, are introduced. The program defines the two terms, distinguishing between them with different language, equations, and graphic representations. Architecture and surgery provide interesting application examples.

13. Functions
This program defines a function, discusses domain and range, and develops an equation from real situations. The cutting of pizza and encoding of secret messages provide subjects for the demonstration of functions and their usefulness.

Graphics are used to introduce composites and inverses of functions as applied to calculation of the Gross National Product. One-to-one functions and the horizontal line test are introduced, and more encoded messages and the hazards of “the bends” in scuba diving provide instructive applications of the functions discussed.

15. Variation
In this program, students are given examples of special functions in the form of direct variation and inverse variation, with a discussion of combined variation and the constant of proportionality. These are explored in relation to polynomials and assorted equations, with applications from chemistry, physics, astronomy, and the food industry.

This program explains how to identify, graph, and determine all intercepts of a polynomial function. It covers the role of coefficients; real numbers; exponents; and linear, quadratic, and cubic functions. This program touches upon factors, x-intercepts, and zero values. These terms are demonstrated with the baking of pizza.

A rational function is the quotient of two polynomial functions. The properties of these functions are investigated using cases in which each rational function is expressed in its simplified form. The relationship between numerator and denominator is clarified, and sign and other graphs are used to determine intercepts, symmetry, and asymptotes.

Students are taught the exponential function, as illustrated through formulas. The population of Massachusetts, the “learning curve,” bacterial growth, and radioactive decay demonstrate these functions and the concepts of exponential growth and decay.

This program covers the logarithmic relationship, the use of logarithmic properties, and the handling of a scientific calculator. How radioactive dating and the Richter scale depend on the properties of logarithms is explained. Many rules and tests from previous programs are also incorporated into the lesson.

The case of two linear equations in two unknowns is considered throughout this program. Elimination and substitution methods are used to find single solutions to systems of linear and nonlinear equations. Consistent, inconsistent, and dependent systems are also explored through examples from ship navigation and garment production.

Elimination and substitution are used again to solve systems of linear inequalities. Linear programming is shown to solve problems in the Berlin airlift, production of butter and ice cream, school redistricting, and other situations while constraints, corner points, objective functions, the region of feasible solutions, and minimum and maximum values are also explored.

When the growth of a child is regular, it can be described by an arithmetic sequence. This program differentiates between arithmetic and nonarithmetic sequences as it presents the solutions to sequence- and series-related problems. Definitions include sequence, arithmetic sequence, arithmetic series, fixed number, and common difference.

This program provides examples of geometric sequences and series (f-stops on a camera and the bouncing of a ball), explaining the meaning of nonzero constant real number and common ratio. Finite and infinite geometric series and the sequence of partial sums are also defined in the discussion.

Mathematical proofs applied to hypothetical statements shape this discussion on mathematical induction. This segment exhibits special cases, looks at the development of number patterns, relates the patterns to Pascal’s triangle and factorials, and elaborates the general form of the theorem.

How many variations in a license plate number or poker hand are possible? This program answers the question and shows students how it’s done. Techniques for counting the number of ways in which collections of objects can be arranged, ordered, and combined are demonstrated.

26. Probability
In this final program, students see how the various techniques of algebra that they have learned can be applied to the study of probability. The program shows that games of chance, health statistics, and product safety are areas in which decisions must be made according to our understanding of the odds. It also shows how the subject of probability has evolved to support such fields as genetics, social science, and medicine.  Jewels Educational Services for Up-and-coming Scholars
Algebra I  Old Link
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O   O
O                                            ARITHMETIC PROPERTIES
1.0 Students identify and use the arithmetic properties of subsets of integers and rational, irrational, and real numbers, including propeties of equality for addition and multiplication, and closure properties for addition, subtraction, multiplicaion, and division where applicable.

VARIABLE EXPRESSIONS AND EQUATIONS
25.2 Students judge the validity of an argument according to whether the properties of the real number system and the order of operations have been applied correctly at each step.

PROPERTIES OF NUMBERS
1.1 Students use properties of numbers to demonstrate whether assertions are true or false.

EXPONENTS
2.0 Students understand and use such operations as taking the opposite, finding the reciprocal, taking a root, and raising to a fractional power. They understand and use the rules of exponents, including negative exponents and scientific notation.

SOLVING LINEAR EQUATIONS
3.0 Students solve equations and inequalities involving absolute values.

SOLVING LINEAR INEQUALITIES
4.0 Students simplify expressions before solving linear equations and inequalities in one variable, such as 3(2x-5) + 4(x-2) = 12.

INTRODUCTION TO PROBLEM SOLVING
Students use Polya's four step process to solve word problems involving numbers, rectangles, supplementary angles, and complementary angles

SOLVING WORD PROBLEMS
5.0 Students solve multistep problems, including word problems, involving linear equations and linear inequalities in one variable and provide justification for each step.

PERCENT AND PROLEM SOLVING

FURTHER PROBLEM SOLVING

6.0 Students graph a linear equation and compute the x- and y- intercepts (e.g., graph 2x + 6y = 4). They are also able to sketch the region defined by linear inequality (e.g., they sketch the region defined by 2x + 6y < 4).

SLOPE
7.0 Students verify that a point lies on a line, given an equation of the line. Students are able to derive linear equations by using the point-slope formula.

LESSON TWO
8.0 Students understand the concepts of parallel lines and perpendicular lines and how those slopes are related. Students are able to find the equation of a line perpendicular to a given line that passes through a given point.

THE RECTANGULAR COORDINATE SYSTEM

GRAPHING LINEAR EQUATIONS AND INEQUALITIES
9.0 Students solve a system of two linear equations in two variables algebraically and are able to interpret the answer graphically. Students are able to solve a system of two linear inequalities in two variables and to sketch the solution sets.

10.0 Students add, subtract, multiply, and divide monomials and polynomials. Students solve multistep problems, including word problems, by using these techniques.

11.0 Students apply basic factoring techniques to second-and simple third-degree polynomials. These techniques include finding a common factor for all terms in a polynomial, recognizing the difference of two squares, and recognizing perfect squares of binomials.

12.0 Students simplify fractions with polynomials in the numerator and denominator by factoring both and reducing them to the lowest terms.

13.0 Students add, subtract, multiply, and divide rational expressions and functions. Students solve both computationally and conceptually challenging problems by using these techniques.

14.0 Students solve a quadratic equation by factoring or completing the square.

15.0 Students apply algebraic techniques to solve rate problems, work problems, and percent mixture problems.

16.0 Students understand the concepts of a relation and a function, determine whether a given relation defines a function, and give pertinent information about given relations and functio

17.0 Students determine the domain of independent variables and the range of dependent variables defined by a graph, a set of ordered pairs, or a symbolic expression.

18.0 Students determine whether a relation defined by a graph, a set of ordered pairs, or a symbolic expression is a function and justify the conclusion.

19.0 Students know the quadratic formula and are familiar with its proof by completing the square.

20.0 Students use the quadratic formula to find the roots of a second-degree polynomial and to solve quadratic equations.

21.0 Students graph quadratic functions and know that their roots are the x- intercepts.

22.0 Students use the quadratic formula or factoring techniques or both to determine whether the graph of a quadratic function will intersect the x-axis in zero, one, or two points.

23.0 Students apply quadratic equations to physical problems, such as the motion of an object under the force of gravity.

24.0 Students use and know simple aspects of a logical argument:

24.1 Students explain the difference between inductive and deductive reasoning and identify and provide examples of each.

24.2 Students identify the hypothesis and conclusion in logical deduction. 24.3 Students use counterexamples to show that an assertion is false and recognize that a single counterexample is sufficient to refute an assertion.

25.0 Students use properties of the number system to judge the validity of results, to justify each step of a procedure, and to prove or disprove statements:

25.1 Students use properties of numbers to construct simple, valid arguments (direct and indirect) for, or formulate counterexamples to, claimed assertions.

25.3 Given a specific algebraic statement involving linear, quadratic, or absolute value expressions or equations or inequalities, students determine whether the statement is true sometimes, always, or never.

MISCELLANEOUS
Fractions, Reading Graphs, Formulas, Basic Geometry, Central Tendencies, R
PRE-ALGEBRA
The videos linked to below are availabe from Prentice Hall at . . .
http://www.phschool.com/webcodes10
/index.cfm?fuseaction=home.gotoWeb
Code&wcprefix=aze&wcsuffix=07

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Chapter 1: Algebraic Expressions and Integers
Lesson 1: Variables and Expressions
Lesson 2: Order of Operations
Lesson 3: Evaluating Expressions
Lesson 4: Integers and Absolute Value
Lesson 5: Adding Integers
Lesson 6: Subtracting Integers
Lesson 7: Inductive Reasoning
Lesson 8: Reasoning Strategy: Look for a Pattern
Lesson 9: Multiplying and Dividing Integers
Lesson 10: The Coordinate Plane

Chapter 2: Solving One-Step Equations and Inequalities
Lesson 1: Properties of Numbers
Lesson 2: The Distributive Property
Lesson 3: Simplifying Variable Expressions
Lesson 4: Variables and Equations
Lesson 5: Using Models with Equations
Lesson 6: Solving Equations by Multiplying/Dividing
Lesson 7: Reasoning Strategy: Try, Check, and Revise
Lesson 8: Data and Graphs
Inequalities and Their Graphs
Lesson 9: Solving One-Step Inequalities by Adding or Subtracting
Lesson 10: Solving One-Step Inequalities by Multiplying or Dividing

Chapter 3: Decimals and Equations
Lesson 1: Rounding and Estimating
Lesson 2: Estimating Decimal Products and Quotients
Lesson 3: Mean, Median, and Mode
Lesson 3A: Mean and Median on a Graphing Calculator
Lesson 4: Using Formulas
Lesson 4A: Formulas in a Spreadsheet
Lesson 5: Solving Equations by Adding or Subtracting Decimals
Lesson 6: Solving Equations by Multiplying or Dividing Decimals
Lesson 7: Using the Metric System
Lesson 7A: Precision and Significant Digits
Lesson 8: Reasoning Strategy: Simplify a Problem

Chapter 4: Factors, Fractions, and Exponents
Lesson 1: Divisibility and Factors
Lesson 2: Exponents
Lesson 3: Prime Factorization and Greatest Common Factor
Lesson 3A: Venn Diagrams
Lesson 4: Simplifying Fractions
Lesson 5: Reasoning Strategy: Account for All Possibilities
Lesson 6: Rational Numbers
Lesson 7: Exponents and Multiplication
Lesson 8: Exponents and Division
Lesson 9: Scientific Notation
Lesson 9A: Scientific Notation with Calculators

Chapter 5: Operations With Fractions
Lesson 1: Comparing and Ordering Fractions
Lesson 2: Fractions and Decimals
Lesson 3A: Estimating with Fractions and Mixed Numbers
Lesson 3: Adding and Subtracting Fractions
Lesson 4: Multiplying and Dividing Fractions
Lesson 5: Using Customary Units of Measurement
Lesson 5A: Greatest Possible Error
Lesson 6: Reasoning Strategy: Work Backward
Lesson 7: Solving Equations by Adding or Subtracting Fractions
Lesson 8: Solving Equations by Multiplying Fractions
Lesson 9: Powers of Products and Quotients

Chapter 6: Ratios, Proportions, and Percents
Lesson 1: Ratios and Unit Rates
Lesson 1A: Converting Between Measurement Systems
Lesson 2: Proportions
Lesson 3: Similar Figures and Scale Drawings
Lesson 3A: Dilations
Lesson 4: Probability
Lesson 5: Fractions, Decimals, and Percents
Lesson 6: Proportions and Percents
Lesson 7: Percents and Equations
Lesson 8: Percent of Change
Lesson 9: Markup and Discount
Lesson 10: Reasoning Strategy: Make a Table

Chapter 7: Solving Equations and Inequalities
Lesson 1: Solving Two-Step Equations
Lesson 2: Solving Multi-Step Equations
Lesson 3: Multi-Step Equations with Fractions and Decimals
Lesson 4: Reasoning Strategy: Write an Equation
Lesson 5: Solving Equations With Variables on Both Sides
Lesson 6: Solving Two-Step Inequalities
Lesson 6A: Compound Inequalities
Lesson 7: Transforming Formulas
Lesson 8: Simple and Compound Interest

Chapter 8: Linear Functions and Graphing
Lesson 1A: Relating Graphs to Events
Lesson 1: Relations and Functions
Lesson 2: Equations With Two Variables
Lesson 2A: Direct Variation
Lesson 3: Slope and y-intercept
Lesson 3A: Graphing Lines
Lesson 4: Writing Rules for Linear Functions
Lesson 5: Scatter Plots
Lesson 6: Reasoning Strategy: Solve by Graphing
Lesson 7: Solving Systems of Linear Equations
Lesson 8: Graphing Linear Inequalities
Lesson 8A: Graphing Inequalities

Chapter 9: Spatial Thinking
Lesson 1: Points, Lines, and Planes
Lesson 2A: Drawing and Measuring Angles
Lesson 2: Angle Relationships and Parallel Lines
Lesson 3: Classifying Polygons
Lesson 3A: Angles of a Polygon
Lesson 4: Reasoning Strategy: Draw a Diagram
Lesson 5: Congruence
Lesson 6: Circles
Lesson 7: Constructions
Lesson 8: Translations
Lesson 8A: Matrices and Translations
Lesson 9: Symmetry and Reflections
Lesson 10: Rotations
Lesson 10A: Tessellations

Chapter 10: Area and Volume
Lesson 1: Area: Parallelograms
Lesson 2: Area: Triangles and Trapezoids
Lesson 3: Area: Circles
Lesson 4A: Three Views of an Object
Lesson 4: Space Figures
Lesson 5A: Cross Sections of Space Figures
Lesson 5: Surface Area: Prisms and Cylinders
Lesson 6: Surface Area: Pyramids, Cones, and Spheres
Lesson 7: Volume: Prisms and Cylinders
Lesson 8: Reasoning Strategy: Make a Model
Lesson 9: Volume: Pyramids, Cones, and Spheres

Chapter 11: Right Triangles in Algebra
Lesson 1: Square Roots and Irrational Numbers
Lesson 2: The Pythagorean Theorem
Lesson 2A: The Pythagorean Theorem and Circles
Lesson 3: Distance and Midpoint Formulas
Lesson 4: Reasoning Strategy: Write a Proportion
Lesson 5: Special Right Triangles
Lesson 5A: Square Roots of Expressions with Variables
Lesson 6: Sine, Cosine, and Tangent Ratios
Lesson 6A: Finding the Angles of a Right Triangle
Lesson 7: Angles of Elevation and Depression

Chapter 12: Data Analysis and Probability
Lesson 1: Frequency Tables and Line Plots
Lesson 1A: Making Histograms
Lesson 2: Box-and-Whisker Plots
Lesson 2A: Stem-and-Leaf Plots
Lesson 3: Using Graphs to Persuade
Lesson 4: Counting Outcomes and Theoretical Probability
Lesson 5: Independent and Dependent Events
Lesson 6: Permutations and Combinations
Lesson 7: Experimental Probability
Lesson 8: Random Samples and Surveys
Lesson 9: Reasoning Strategy: Simulate the Problem
Lesson 9A: Using Random Numbers

Chapter 13: Nonlinear Functions and Polynomials
Lesson 1: Patterns and Sequences
Lesson 2: Graphing Nonlinear Functions
Lesson 3: Experimental Growth and Decay
Lesson 3A: Non-Linear Functions and Graphing Calculators
Lesson 4: Polynomials
Lesson 4A: Degree of a Polynomial
Lesson 5: Adding and Subtracting Polynomials
Lesson 6: Multiplying a Polynomial by a Monomial
Lesson 7: Multiplying Binomials
Lesson 7A: Binomial Factors of a Trinomial
Lesson 8: Reasoning Strategy: Use Multiple Strategies

Worksheets