July 7, 2011

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This folder contains 44 lesson plans written by Kathy and Bruce Yoshiwara of Pierce College in Los Angeles. Each lesson covers the topic succinctly, provides clear instruction on the associated procedures, and many also provide contextual situations where key concepts may be explored. Each lesson also includes a homework assignment. The lessons are structured in such a way that there are ample opportunities for students to work in groups (or individually) or the entire lesson could likely be completed in groups with the teacher circulating and providing tips, gauging student understanding, and choosing apt candidates to present to the class or kick off whole class discussions. The 44 units are divided into nine units: 1) Linear Models 2) Applications of Linear Models 3) Quadratic Models 4) Applications of Quadratic Models 5) Functions and Their Graphs 6) Powers and Roots 7) Exponential Functions 8) Polynomials and Rational Functions 9) Equations and Graphs

- Mathematics > General
- Mathematics > Algebra

- Higher Education
- Graduate
- Undergraduate-Upper Division
- Undergraduate-Lower Division

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Table of Contents

- Lesson 1: Linear Models
- Lesson 2: Intercepts (linear models)
- Lesson 3: Graphs and Equations
- Lesson 4: Slope
- Lesson 5: Equations of lines
- Lesson 6: Linear Regression
- Lesson 7: Linear Systems
- Lesson 8: Algebraic Solution of Systems
- Lesson 9: Gaussian Reduction
- Lesson 10: Extraction of Roots
- Lesson 11: Intercepts, Solutions, and Factors
- Lesson 12: Graphing parabolas
- Lesson 13: Completing the Square
- Lesson 14: Quadratic Formula
- Lesson 15: The Vertex
- Lesson 16: Curve Fitting
- Lesson 17: Quadratic Inequalities
- Lesson: 18 Functions
- Lesson 19: Graphs of Functions Reading
- Lesson 20: Some Basic Graphs
- Lesson 21: Variation
- Lesson 22: Functions as Models
- Lesson 23: Integer Exponents
- Lesson 24: Roots and Radicals
- Lesson 25: Rational Exponents
- Lesson 26: The Distance and Midpoint Formulas
- Lesson 27: Working with Radicals
- Lesson 28: Radical Equations
- Lesson 29: Exponential Growth and Decay
- Lesson 30: Exponential Functions
- Lesson 31: Logarithms
- Lesson 32: Properties of Logarithms
- Lesson 33: Applications of Exponential Functions
- Lesson 34: Logarithmic Scales
- Lesson 35: Polynomial Functions
- Lesson 36: Algebraic Fractions
- Lesson 37: Operations on Algebraic Fractions
- Lesson 38: More Operations on Fractions
- Lesson 39: Equations with Algebraic Fractions
- Lesson 40: Properties of Lines
- Lesson 41: Conic Sections: Ellipses
- Lesson 42: Conic Sections: Hyperbolas
- Lesson 43: Linear Inequalities in Two Variables
- Lesson 44: Nonlinear Systems

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This folder contains 44 lesson plans written by Kathy and Bruce Yoshiwara of Pierce College in Los Angeles. Each lesson covers the topic succinctly, provides clear instruction on the associated procedures, and many also provide contextual situations where key concepts may be explored. Each lesson also includes a homework assignment. The lessons are structured in such a way that there are ample opportunities for students to work in groups (or individually) or the entire lesson could likely be completed in groups with the teacher circulating and providing tips, gauging student understanding, and choosing apt candidates to present to the class or kick off whole class discussions. The 44 units are divided into nine units:
1) Linear Models
2) Applications of Linear Models
3) Quadratic Models
4) Applications of Quadratic Models
5) Functions and Their Graphs
6) Powers and Roots
7) Exponential Functions
8) Polynomials and Rational Functions
9) Equations and Graphs

A unit on linear modeling using tables, equations, and graphs to explore different contexts (bike rentals, fuel consumption). A homework assignment is included at the end of the unit.

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Finding the intercepts of a linear model graph. The intercepts are also interpreted in terms of the fuel consumption problem from lesson 1 where intercepts naturally arise as the moments when the tank is full and the tank is empty. The general form of a linear equation and the intercept method of graphing are also introduced in this lesson. Homework assignment follows at end of lesson.

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This lesson begins with linear equations and inequalities in 1 variable and then moves on to linear equations in 2 variables. Graphs of linear equations in 2 variables are introduced as "a picture of all its solutions." Exercises targeting the links between equations, solutions, points, and graphs follows, with the final activities focusing on use of a graphing calculator to graph equations and find coordinates. There aren't any application problems in this lesson.

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Treating slope as a rate of change of two quantities with different units. There are both applications based examples and non contextual examples. Slope is found both from a graph and from the relationship between two quantities given in a context.

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Begins with the slope-intercept equation of a line and then introduces the coordinate formula for slope before the introduction of the point-slope equation of a line. Linear models are then revisited.

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Using real world data, this lesson introduces linear regression using lines of best fit that may calculated by hand by selecting two pints that appear to fall on the line of best fit. The lesson could also be used with a calculator to find the actual regression line. Interpolation and extrapolation are also introduced as well as scatter plots.

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A calculator based introduction to systems of linear equations. Systems are solved using the graphing method: first by estimating the apparent intersection of the two lines and then later by using the intersect function on the calculator to find the exact solution. Inconsistent and consistent solutions are also discussed. There are both applications based problems and non applications based problems.

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An introduction to solving systems of equations using substitution and elimination. There are both applications based and non applications based problems.

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An introduction to solving a 3x3 linear system of equations. Back substitution and triangular form are first introduced, and then a general procedure for solving systems is presented. Application and non application problems.

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This lesson introduces quadratic equations and graphs. Equations of the form ax^2 + c = 0 are solved via extraction of roots. Later application problems involving volume and surface area and compound interest (problems of the form a(x - p)^2 = q ) are presented.

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The lesson begins with a situation where the height of a pop fly in baseball is modeled using a quadratic equation. This motivates an inquiry into finding a method for obtaining solutions beyond the a(x-p)^2 = q case, to the general quadratic ax^2 + bx + c = o. Using factoring and the zero product principle is then presented as a method for solving quadratic equations. Area application problems are covered and at the end of the lesson a method for finding a quadratic equation when given the solutions is presented.

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The lesson begins with an exploration of the family of graphs of y = ax^2, with an emphasis on tracking the changes in the y-values for differing values of the parameter a. The vertical shifts of y = ax^2 + c follow, leading into the graphs of y = ax^2 + bx and the derivation of the formula for the vertex.

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This lesson introduces completing the square as a means of expanding the set of quadratic equations that may be solved beyond the extraction of roots and factoring. Simpler cases are first presented and then towards the end of the lesson a procedure for completing the square of ax^2 + bx + c = 0 is given.

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Completing the square is applied to the general quadratic to derive the quadratic formula. Before an area application example is given there is a quick review of the four methods that have been presented for solving quadratic equations. Complex numbers are introduced before the discriminant is presented.

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This lesson begins with a review of the vertex formula for a quadratic and looks at two quadratics- one a vertical shift of the other- to verify that correct vertex is obtained for any quadratic by substituting x = -b/2a into the equation. Application problems concerned with minima and maxima are covered before the vertex form of the quadratic equation is presented. The lesson concludes with examples where the vertex of a given parabola is given and subsequently used to find the related equation.

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Reminding students how 2 points were used to find the equation of a line, using three points and Gaussian elimination is introduced to find the equation of a parabola. An application relating velocity and gas mileage is worked before using a calculator for quadratic regression and choosing and appropriate model is discussed.

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The lesson begins with using graphs to solve quadratic inequalities. AN equation modeling the height of a rocket is graphed along with a second equation that represents the minimum height at which the rocket can legally and safely be exploded. The intersections of the graphs provide the solution interval. A second method is then presented where the inequality is put into standard form and then solved for its x-intercepts. Interval notation and union of sets is reviewed before a purely algebraic procedure for solving the inequalities is presented. The lesson concludes with an application problem.

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Beginning with linear functions, this lesson looks at functions of real world data defined by tables and graphs before moving into functions defined by equations. Function notation is introduced at the end of the lesson and various examples are provided to get students familiar with the new notation.

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The lesson begins with graphs of the Dow Jones Industrial Average and water levels of Lake Huron where points on the graph are interpreted. Intervals of increase and maxima are introduced before the graph of F(x) = sqrt (x+4) is completed by first generating a table of data. This is followed by the vertical line test and using graphs to solve equations and inequalities.

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A short lesson introducing the cube root and absolute value functions and their graphs.

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The lesson begins with a comparison of data tables and graphs of two functions, one directly proportional (cost of gas) and the other exponential (population), before a definition for direct variation is introduced. Direct variation is then linked to linear function (f(x)= kx)and the scaling property of direct variation is examined (i.e. a multiple of the independent variable will always correspond to that same multiple of the dependent variable). Direct variation with a power of x follows with a test for direct variation before indirect variation and indirect variation with a power of x are introduced.

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This lesson focuses on finding appropriate non linear functions to model real world phenomena. Various cases are examined before the absolute value function and equations and inequalities are introduced.

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To motivate the convention for negative and zero exponents, the lesson begins by observing the halving pattern found in continuously decreasing the exponent of a power of two by 1. After an application looking at the formula for Body Mass Index calculation, power functions of the form f(x) = kx^p are introduced. Radiation Intensity application problems follow before pure algebraic manipulation of exponential expressions are presented. The lesson concludes with a review of scientific notation.

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Exponential notation for Nth roots and radicals is introduced. A short discussion about Nth roots and irrational numbers follows before symbolic manipulation of fractional exponents and solving equations is presented. Power functions and solving radical equations are presented before the lesson concludes with roots of negative numbers.

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Using Kleiber's rule- which linked an animals metabolic rate to its mass- to motivate working with rational exponents, this lesson begins by linking work with Nth roots from the last lesson to working with powers of Nth roots. Rational exponents are linked with radical expressions, and then operations and equations with rational exponents are presented.

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The lesson begins by associating the distance between two points with the right triangle that may be formed by joining the points and extending horizontal and vertical lines through the points. This linking is generalized to derive the distance formula for any two points in the plane. The midpoint formula is then derived by taking the average of the coordinates of the two points. Using the distance formula, the equation for circle is derived and then examples follow for finding the equation of a given circle.

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The lesson begins with the product and quotient rules for radicals, highlighting the frequently made mistakes of students by overgeneralizing the rules. Simplification examples follow and then sums and differences of radicals are presented. The lesson concludes with rationalizing denominators containing both one and two terms.

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The lesson begins with an emphasis on isolating the radical expression in a radical equation and then highlights the importance of checking for extraneous solutions that may be generated when the equation is solved by applying even powers. Equations containing two radical expressions and then presented, followed by coverage of taking the nth root of a^n.

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The lesson begins with a discussion about growth factors and percent increase, leading to the presentation of the compound interest formula. Following this focus on growth, exponential decay is introduced. The lesson concludes with a comparison between exponential and linear growth, highlighting the difference in the additive and multiplicative patterns in their growth patterns.

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Beginning with a formal definition of an exponential function, the lesson then compares the graphs of increasing and decreasing exponential functions. A comparison between exponential and power functions follows, which leads to methods for determining the h value in the power function h(x) = kx^p and the value of the base b in the exponential function f(x) = ab^x. A procedure for solving exponential equations is presented before a population application problem is solved. The lesson concludes with a discussion about using graphs to find approximate solutions to exponential equations.

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The lesson begins with an application problem to motivate the necessity and use of a logarithm. The formal definition linking logs and exponents is then introduced. Exercises in writing exponential equations as logarithms follows before a calculator based method for approximating logarithmic values is discussed. The common log, i.e. logs of base 10, is introduced and a procedure for solving common log equations with a calculator is presented, along with various caveats about proper syntax for the calculator. The lesson concludes with exponential modeling problems where logs may be employed to find the desired exponent.

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The properties of logarithms are presented after the corresponding laws for exponents are reviewed. Various exercises in simplifying logarithmic expressions are provided before solving logarithmic and exponential equations is introduced. The lesson ends with an application problem and a few exercises in solving formulas.

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The lesson begins with population application problems looking at doubling time, the constant time that it takes for an exponentially modeled population to double. Application problems concerning half life are then discussed.

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This lesson introduces logarithmic scales as a means for plotting data that might otherwise be difficult plot due to their widely varying values. Instead of plotting the actual values, the values of their logs are plotted on a scale of powers of ten. A discussion follows that highlights an unfamiliar feature for students: equal intervals on the power of ten scale do not correspond to equal values. Various application problems about acidity and ph, decibels, and the Richter Scale follow.

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Beginning with the definition of a polynomial, polynomial multiplication and degree of polynomial products are introduced. Special products and factoring cubics are presented before modeling with polynomials is discussed.

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The lesson begins with the definition of an algebraic fraction and then a quick review of the fundamental principle of fractions. Exercises in reducing fractions follow before a brief procedure for reducing algebraic fractions is provided. Opposites of binomials are reviewed before rational functions are defined and a motion application problem is discussed.

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The four arithmetic operations on fractions are given here, but in each case examples of using the operation on ordinary fractions is first reviewed, helping to ground the new material in the understanding of the familiar older content. Finding the least common denominator and rewriting the new equivalent fraction is also presented. The lesson concludes with a motion application problem.

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The lesson begins with a definition of a "complex" fraction and then presents a procedure for simplifying them (multiplying the numerator and the denominator by the lcd of all fractions contained in the complex fraction). Exercises with negative exponents are covered before applications are presented. The lesson concludes with division of polynomials.

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The lesson begins with using the lcd to clear the fractions in the equation, and then a review of proportions follows to motivate the "shortcut" of cross multiplication for equations containing only a fraction on each side of the equation. The possible existence of extraneous solutions and solving the equations graphically is discussed before application problems are presented. The lesson concludes with exercises in manipulating formulas.

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The lesson begins with horizontal and vertical lines, first looking at the corresponding sets of points that comprise each, then using the points to find the slope of each type of line. Parallel and perpendicular lines are covered next, ending with applications in geometry.

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Beginning with a general introduction to conics and how they are formed, circles are first presented and then ellipses are motivated by looking at the general equation of a circle centered at the origin. After central ellipses, translated ellipses are discussed, followed by a procedure for writing the equation of the ellipse in standard form. The lesson concludes with a procedure for finding the equation of an ellipse given its vertices.

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This lesson begins with centralized hyperbolas and then moves into translated hyperbolas. A review of conic sections is presented before exercises in determining the type of conic section from its equation are given.

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The lesson begins with a definition of a linear inequality and then looks at individual points that satisfy the inequality to motivate the existence of a larger set of points that satisfy the inequality. The point test is then presented and a general procedure for graphing inequalities. The lesson concludes with systems of inequalities and application problems.

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Using a cost/revenue application problem, the lesson begins with systems involving quadratic equations. Systems with conics are introduced next along with the elimination method for solving these systems.

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