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Chicago Public Schools Advanced Algebra, days 28-36 lesson plans

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<ul><li><p>102</p><p>STRUCTURED CURRICULUM LESSON PLAN</p><p>Day: 028 Subject: Advanced Algebra Grade Level: High School</p><p>Correlations (SG,CAS,CFS): 8C2</p><p>TAP:Analyze and interpret data presented in charts,</p><p>graphs, tables, and other displays</p><p>ISAT:Identify, analyze, and solve problems using</p><p>equations, inequalities, functions, and theirgraphs</p><p>Unit Focus</p><p>Exploring Equations and Functions</p><p>Instructional Focus</p><p>Identify the image and pre-image points and the axis of symmetry of a set of ordered pairs.(Determine the relationship of coordinates of points reflected over the y-axis, the x-axis, and theline y = x.)</p><p>Materials</p><p>Graph paperPins</p><p>Educational Strategies/Instructional Procedures</p><p>Discuss symmetry and how to determine a line of symmetry (line of reflection). Have studentslook at various patterns and determine their line of symmetry. Have students look for otherobjects in the classroom with lines of symmetry. Assign two groups to complete each of threeexplorations (1, 2, and 3) as follows:</p><p>a. Plot 4 points A, B, C, and D on a coordinate plane and use a pin to poke a holethrough these points.</p><p>b. Fold the paper along the y-axis (Exploration 1) / the x-axis (Exploration 2) / theline y = x (Exploration 3) and use the pin to poke the image points through theholes.</p><p>c. Determine the coordinates of the new points A, B, C, and D.</p></li><li><p>103</p><p>Have each group answer the following questions:</p><p>a. How are the coordinates of the points A, B; C; and D; similar to the coordinates of theoriginal pre-image points A, B, C, and D? Which coordinates have the same sign?Which coordinates have opposite signs?</p><p>b. Explain what it means for the image points to be symmetrical to the pre-image pointswith respect to the y-axis / the x-axis / the line y = x.</p><p>c. Let F(a, b) be any point on the coordinate plane. What are the coordinates of the imageof F with respect to the y-axis / the x-axis / the line y = x?</p><p>One student from each group will present the groups findings to the class.</p><p>Discuss the findings with the class.</p><p>Integration with Core Subject(s)</p><p>LA: Understand explicit, factual informationUnderstand the meaning of works in context</p><p>SC: Analyze and interpret data</p><p>Connections</p><p>Enrichment: Have students draw a parallelogram in Quadrant I, reflect that parallelogram overthe line y = x, and determine the coordinates of the vertices of the image parallelogram.Complete a problem write-up to explain observations. (Students should discover that the line ofsymmetry of two points A and A, where A is the image of A, is the perpendicular bisector of</p><p>AA .)</p><p>Fine Arts: Have students find pictures in newspapers and magazines that show symmetry anddescribe the symmetry shown.</p><p>Home: Have students keep a record of all assignments. A parent on a weekly basis will sign thehomework record. Have students list the symmetry they have observed in nature.</p><p>Remediation: Work with students in small groups on their areas of deficiency.</p><p>Technology: Have students use a computer drawing program and explore reflecting polygonsover the x-axis, y-axis and through the origin.</p></li><li><p>104</p><p>Assessment</p><p>Teacher observation</p><p>Homework</p><p>Assign appropriate problems from your text.</p><p>Teacher Notes</p></li><li><p>105</p><p>STRUCTURED CURRICULUM LESSON PLAN</p><p>Day: 029 Subject: Advanced Algebra Grade Level: High School </p><p>Correlations (SG, CAS, CFS): 8C2</p><p>TAP:Analyze and interpret data presented in charts,</p><p>graphs, tables, and other displays</p><p>ISAT:Identify, analyze, and solve problems using</p><p>equations, inequalities, functions, and theirgraphs</p><p>Unit Focus</p><p>Exploring Equations and Functions</p><p>Instructional Focus</p><p>Defining and Determining the Inverse of a Function (Using the Horizontal Line Test toDetermine Whether the Inverse is also a Function)</p><p>Materials</p><p>Graph paperClassroom set of graphing calculatorsBlue and red pensOverhead projectorOverhead graphing calculator</p><p>Educational Strategies/Instructional Procedures</p><p>Review previous days homework and answer students questions.</p><p>Have each group answer the following questions:</p><p>Write the following chart on the chalkboard or overhead projector:</p><p>Pre-image Image Symmetric with respect to(4, 3) y-axis</p><p>(-1, 5) x-axis(3, 2) (-3, 2)(1, -3) y = x(5, -7) (5, 7)</p></li><li><p>106</p><p>Have the students complete the chart.</p><p>Have students work in groups of four to complete the following exploration to determine theinverse of a function. Distribute a different set of numbers to each group, as shown below. Thepoints should be plotted in red, the reflection points in blue.</p><p>Group A: f = {(-3, 3), (-2, 2), (1, 1), (-4, -2)}Group B: f = {(5, -2), (-4, 1), (-1, 2), (3, 3)}Group C: f = {(-2, 4), (3, -2), (-1, 1), (3, 2)}Group D: f = {(4, -2), (-3, 2), (-1, -2), (2, 3)}Group E: f = {(-3, 1), (4, -2), (-3, -3), (1, 4)}Group F: f = {(-2, 3), (4, -2), (-2, -1), (3, 1)}</p><p>Have each group complete an x, y table for the given function in blue and a corresponding x, ytable for the set of reflection points in red. (Students should discover that the coordinates of thered points are the reverse of the coordinates of the blue points.)</p><p>Define and discuss the inverse of a function f 1.</p><p>Write the following instructions on the overhead.</p><p>f = {(-2, -4), (-1, 0), (1, 2), (2, 4)}g = {(3, 4), (2, 1), (4, 0), (1, 4)}</p><p>Have the students find the inverse of each function and determine whether the inverse is also afunction.</p><p>Given f(x) = 2x + 3, students will:</p><p>a. Find f 1.b. Graph f and f 1 on the same coordinate plane, as well as the line Y1</p><p> = x.c. Use TRACE to find f (-1) and f 1 (b) = a.</p><p>Integration with Core Subject(s)</p><p>LA: Understand explicit, factual informationUnderstand the meaning of words in context</p><p>SC: Analyze and interpret data</p><p>Connections</p><p>Enrichment: Challenge students to identify the equation of an absolute value function from itsgraph, which is translated.</p></li><li><p>107</p><p>Fine Arts:</p><p>Home: Have students keep a record of all assignments. A parent on a weekly basis will sign thehomework record.</p><p>Remediation: Work with students in small groups on their areas of deficiency.</p><p>Technology:</p><p>Assessment</p><p>Evaluate the groups paragraphs using the Structured Curriculum Scoring Rubric.</p><p>Homework</p><p>Assign appropriate problems from your text.</p><p>Teacher Notes</p><p>Suggested activity: Class discussion about the horizontal line test on graphing calculators.</p><p>Solutions to problems in Educational Strategies:</p><p>Pre-image Image Symmetric with respect to(4, 3) (-4, 3) y-axis</p><p>(-1, -5) (-1, 5) x-axis(3, 2) (-2, 3) origin(1, -3) (3, 1) y = x(5, -7) (5, 7) x-axis</p><p>Reflections:</p><p>Group A: f = {(3, 3), (2, -2), (1, 1), (-2, -4)}Group B: f = {(-2, 5), (1, -4), (2, -1), (3, 3)}Group C: f = {(4, -2), (-2, 3), (1, -1), (2, 3)}Group D: f = {(-2, 4), (2, -3), (-2, -1), (3, 2)}Group E: f = {(1, -3), (-1, 4), (-3, -3), (4, 1)}Group F: f = {(3, -2), (-2, 4), (-1, -2), (1, 3)}</p></li><li><p>108</p><p>f = {(-4, -2), (0, 1), (2, 1), (4, 2)} functiong = {(4, 3), (1, 2), (0, 4), (4, 1)} not a function</p><p>f (x) = 2x+3</p><p>a) f -1y = 2</p><p>3x</p><p>b) </p><p>c) f (-1) = 1f 1 (1) = -2</p></li><li><p>109</p><p>STRUCTURED CURRICULUM LESSON PLAN</p><p>Day: 030 Subject: Advanced Algebra Grade Level: High School</p><p>Correlations (SG,CAS,CFS): 8C1</p><p>TAP:Analyze and interpret data presented in charts,</p><p>graphs, tables, and other displays</p><p>ISAT:Identify, analyze, and solve problems using</p><p>equations, inequalities, functions, and theirgraphs</p><p>Unit Focus</p><p>Exploring Equations and Functions</p><p>Instructional Focus</p><p>Defining and Describing the Relationship Between Dependent and Independent Variables(The Composition of Functions)</p><p>Materials</p><p>SoakEm Company ProjectGraph paperClassroom set of graphing calculatorsOverhead graphing calculatorOverhead projectorHandout 3.3</p><p>Educational Strategies/Instructional Procedures</p><p>Have the students complete the following:</p><p>Evaluate each function for x = 2, x = 1, and x = 0g (x) = x3 3x</p><p>h (x) = 3</p><p>1+x</p><p>Review previous days homework and answer students questions.</p><p>Develop the idea of composition of two functions by placing one function into the other. Use thesame f (x) and g (x) for x = 4, 2, 0, 3 to find f (g (x)) and g (f (x)).</p><p>Have the students work in groups of four. Assign Exploration 1 on the commutativity ofcompositions to one-half of the groups; assign Exploration 2 on inverses of functions (Hand-out</p></li><li><p>110</p><p>3-3 to the other half of the groups. Selected students from each group will present the groupsfindings to the whole class.</p><p>Integration with Core Subject(s)</p><p>LA: Understand explicit, factual informationUnderstand the meaning of words in context</p><p>SC: Analyze and interpret data</p><p>Connections</p><p>Enrichment:</p><p>Fine Arts:</p><p>Home: Have students keep a record of all assignments. (The homework record will be signedby a parent on a weekly basis.)</p><p>Remediation: Work with students in small groups on their areas of deficiency.</p><p>Technology:</p><p>Assessment</p><p>Find the inverse of each function:</p><p>1. f(x) = {(2, 2), (1, 4), (2, 18), (3, 2)}</p><p>2. g (x) = 21</p><p>1</p><p>x+</p><p>3. h (x) = )2(4</p><p>3+x</p><p>Solutions:1. f = 1 2 2 4 1 18 2 2 3( ) {( , ), ( , ), ( , ), ( , )}x</p><p>2. g =1 1( )x</p><p>x</p><p>x</p><p>3. h =1 4 6</p><p>3( )x</p><p>x</p></li><li><p>111</p><p>Homework</p><p>Assign from your text appropriate problems on the composition of functions.</p><p>Groups will complete the Soak Em Company project, due the day of the End-of-UnitAssessment (approximately one week). One write-up is required per group. Allow some timeeach day for students to work on this project.</p><p>Teacher Notes</p><p>Remind students they have one week to complete the Soak Em activity.</p><p>Solutions to Educational Strategies problems:</p><p>g x x x</p><p>g</p><p>g</p><p>g</p><p>( )</p><p>( ) ( ) ( )</p><p>( ) ( ) ( )</p><p>( ) ( ) ( )</p><p>= </p><p> = = </p><p>= = </p><p>= =</p><p>3</p><p>3</p><p>3</p><p>3</p><p>3</p><p>2 2 3 2 2</p><p>1 1 3 1 2</p><p>0 0 3 0 0</p><p>h xx</p><p>h</p><p>h</p><p>h</p><p>( )</p><p>( )</p><p>( )</p><p>( )</p><p>=</p><p>+</p><p> =</p><p> += </p><p>=</p><p>+=</p><p>=</p><p>+=</p><p>1</p><p>3</p><p>22 1</p><p>3</p><p>1</p><p>3</p><p>11 1</p><p>3</p><p>2</p><p>3</p><p>00 1</p><p>3</p><p>1</p><p>3</p><p>Solutions to Explorations 1:</p><p>1. f (g (x)) = -0.5x + 1 g (f (x)) = 0.5 (-x + 1)=-0.5x + 0.5</p><p>See student response for comparison</p><p>2. g (h(x)) =</p><p>05 1</p><p>2</p><p>1</p><p>4</p><p>0 5 1</p><p>2</p><p>.</p><p>( ( )).</p><p>x x</p><p>h g xx</p><p>=</p><p>=</p><p>See student response for comparison</p></li><li><p>112</p><p>3. See student responses for problem #3. Students should come to the conclusion that the functions are not commutative.</p><p>Solutions to Exploration 2:</p><p>1. f (x) = 3x+ 1; f =1 1</p><p>3x</p><p>xb g h x</p><p>xh x x( ) ; ( )=</p><p>+= </p><p>2</p><p>33 21</p><p>2. a.</p><p> b. Axis of symmetry is the line y = x. c. f (f 1 (x)) = x ; f 1 (f (x)) =x d. The graphs are the same.</p><p>3.</p><p> b. The equation of the line of symmetry is y = x. c. h (h 1(x)) = x; h 1 (h (x)) = x d. The graphs are the same.4. They are all the same.5. x</p></li><li><p>113</p><p>Solutions to Soak Em Project:</p><p>1. y = -2x + 4010 or y = -2x + 30 if you use only the last digit in the year.</p><p>2. 3. The percentage in 1992 is 26%</p><p>Part II</p><p>4. y = 4x-79725. See graph above.</p><p>Part III</p><p>6. y = 2x + 3 or f (x) = 2x + 3</p><p>7. The compostion of y and y 1 ; f (x) = 2x + 3 f 1 (x) = x 3</p><p>2.</p><p>Part IV</p><p>8. a. Park A y = -4 x+ 8029.5 b. Park B y = 6.5x 12942 c. Park C y = -2.5x + 5012.59. 10.</p></li><li><p>114</p><p>Handout 3.3</p><p>Given: f(x) = x + 1; g(x) = 0.5x; h(x) = 2</p><p>1x</p><p>Exploration 1:</p><p>1) Find the compositions f (g(x)) and g(f(x))Compare these compositions. How are they alike? How are they different?</p><p>2) Find the compositions g (h (x)) and h (g (x))Compare these compositions. How are they alike? How are they different?</p><p>3) Do you think the composition of functions is commutative? Why or why not?</p><p>f x x h xxb g = + = +3 1 2</p><p>3; ( )</p><p>Exploration 2:</p><p>1) Find the inverse functions for f and h.</p><p>2) Graph f and f -1 on the same coordinate plane. What is the equation of the axis of symmetryof these two graphs?</p><p>Find f (f -1(x)) and f -1(f (x).How do f (f -1(x)) and f -1(f (x)) compare with the equation of their axis of symmetry?</p><p>3) Graph h and h -1 on the same coordinate plane. What is the equation of the axis ofsymmetry of these two graphs?</p><p>Find h (h -1(x)) and h 1 (h (x))How do h (h -1(x)) and h -1(h (x)) compare with the equation of their axis of symmetry?</p><p>4) Graph the line y = x. How does this graph compare with the graphs of the axes ofsymmetry you found in Steps 2 and 3?</p><p>5) What do you think is the composition of any function and its inverse?</p></li><li><p>115</p><p>Soak Em ProjectUnit 3 Project</p><p>The SoakEm Company operates three water parks in the Midwest. Over the last three years, thecompany has noticed a number of trends concerning its business. You have been hired as aconsultant to analyze these trends and to devise the best way to increase the companys revenues.To do this, you will use various types of functions and their graphs.</p><p>Part I:</p><p>Overall, the percentage of the entertainment dollar spent on participant sports in the Midwest ison steady decline. Table 1 summarizes this trend.</p><p>Table 1Year % of entertainment dollar</p><p>spent on participant sports1994 221995 201996 181997 16</p><p>1. Express these percentages as a function of time. ______________________________</p><p>2. Use the coordinate grid to graph this function. Use 1990 as the 0 point on the x-axis.</p><p>3. If the percentage is symmetrical about the year 1994, what was the percentage in 1992? _________________________________________</p></li><li><p>116</p><p>Part II:</p><p>Suppose participant sports regain their popularity during the next three years and raise their shareof the entertainment dollar to 28%.</p><p>4. Express this percentage as a function of time for the years 1997-2000. ______________________________________________________</p><p>5. Graph this function on the same grid you used in Part I. ______________________________________________________</p><p>Part III:</p><p>Another trend in the participant sports industry is the loss of percentage of young femaleparticipants. At this time, the percentage of young male participants is three more than twice thepercentage of young female participants.</p><p>6. Express the percentage of young male participants as a function of young female participants. _________________________________________________________</p><p>7. What corrective function will equalize these percentages?_________________________________________________________</p><p>Part IV:</p><p>The SoakEm Company operates three parks in the Midwest. Table 2 shows the admissions yearby year to each park as a percentage of the companys total admissions.</p><p>Table 11994 1995 1996 1997</p><p>Park A 50 55 45 40Park B 20 25 30 40Park C 30 20 25 20</p><p>8. Express each parks share of the companys total admissions as a function of time.a. Park A___________________b. Park B___________________c. Park C___________________</p></li><li><p>117</p><p>9. Use the coordinate grid to graph each function, each in a different color. Again, use 1990 as the 0 point on the x-axis.</p><p>Part V:</p><p>Soak Em charged $15 per person for admission in 1994. Every year after that, through 1997, theprice increased by $3. From 1998 through 2002, the price will remain constant.</p><p>10. Use the coordinate grid to graph this step function for t = 0 to t = 8, with 1994 as t = 0.</p></li><li><p>118</p><p>STRUCTURED CURRICULUM LESSON PLAN</p><p>Day: 031 Subject: Advanced Algebra Grade Level: High School </p><p>Correlations (SG,CAS,CFS): 8C1</p><p>TAP:Analyze and interpret data presented in charts,</p><p>graphs, tables, and other displays</p><p>ISAT:Identify, analyze, and solve problems using</p><p>equations, inequalities, functions, and theirgraphs</p><p>Unit Focus</p><p>Exploring Equations and Functions</p><p>Instructional Focus</p><p>Defining and Describing the Relationship Between Dependen...</p></li></ul>

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