# maaa 028036

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

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• 102

STRUCTURED CURRICULUM LESSON PLAN

Correlations (SG,CAS,CFS): 8C2

TAP:Analyze and interpret data presented in charts,

graphs, tables, and other displays

ISAT:Identify, analyze, and solve problems using

equations, inequalities, functions, and theirgraphs

Unit Focus

Exploring Equations and Functions

Instructional Focus

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.)

Materials

Graph paperPins

Educational Strategies/Instructional Procedures

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:

a. Plot 4 points A, B, C, and D on a coordinate plane and use a pin to poke a holethrough these points.

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.

c. Determine the coordinates of the new points A, B, C, and D.

• 103

Have each group answer the following questions:

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?

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.

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?

One student from each group will present the groups findings to the class.

Discuss the findings with the class.

Integration with Core Subject(s)

LA: Understand explicit, factual informationUnderstand the meaning of works in context

SC: Analyze and interpret data

Connections

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

AA .)

Fine Arts: Have students find pictures in newspapers and magazines that show symmetry anddescribe the symmetry shown.

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.

Remediation: Work with students in small groups on their areas of deficiency.

Technology: Have students use a computer drawing program and explore reflecting polygonsover the x-axis, y-axis and through the origin.

• 104

Assessment

Teacher observation

Homework

Assign appropriate problems from your text.

Teacher Notes

• 105

STRUCTURED CURRICULUM LESSON PLAN

Correlations (SG, CAS, CFS): 8C2

TAP:Analyze and interpret data presented in charts,

graphs, tables, and other displays

ISAT:Identify, analyze, and solve problems using

equations, inequalities, functions, and theirgraphs

Unit Focus

Exploring Equations and Functions

Instructional Focus

Defining and Determining the Inverse of a Function (Using the Horizontal Line Test toDetermine Whether the Inverse is also a Function)

Materials

Graph paperClassroom set of graphing calculatorsBlue and red pensOverhead projectorOverhead graphing calculator

Educational Strategies/Instructional Procedures

Review previous days homework and answer students questions.

Have each group answer the following questions:

Write the following chart on the chalkboard or overhead projector:

Pre-image Image Symmetric with respect to(4, 3) y-axis

(-1, 5) x-axis(3, 2) (-3, 2)(1, -3) y = x(5, -7) (5, 7)

• 106

Have the students complete the chart.

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.

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)}

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.)

Define and discuss the inverse of a function f 1.

Write the following instructions on the overhead.

f = {(-2, -4), (-1, 0), (1, 2), (2, 4)}g = {(3, 4), (2, 1), (4, 0), (1, 4)}

Have the students find the inverse of each function and determine whether the inverse is also afunction.

Given f(x) = 2x + 3, students will:

a. Find f 1.b. Graph f and f 1 on the same coordinate plane, as well as the line Y1

= x.c. Use TRACE to find f (-1) and f 1 (b) = a.

Integration with Core Subject(s)

LA: Understand explicit, factual informationUnderstand the meaning of words in context

SC: Analyze and interpret data

Connections

Enrichment: Challenge students to identify the equation of an absolute value function from itsgraph, which is translated.

• 107

Fine Arts:

Home: Have students keep a record of all assignments. A parent on a weekly basis will sign thehomework record.

Remediation: Work with students in small groups on their areas of deficiency.

Technology:

Assessment

Evaluate the groups paragraphs using the Structured Curriculum Scoring Rubric.

Homework

Assign appropriate problems from your text.

Teacher Notes

Suggested activity: Class discussion about the horizontal line test on graphing calculators.

Solutions to problems in Educational Strategies:

Pre-image Image Symmetric with respect to(4, 3) (-4, 3) y-axis

(-1, -5) (-1, 5) x-axis(3, 2) (-2, 3) origin(1, -3) (3, 1) y = x(5, -7) (5, 7) x-axis

Reflections:

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)}

• 108

f = {(-4, -2), (0, 1), (2, 1), (4, 2)} functiong = {(4, 3), (1, 2), (0, 4), (4, 1)} not a function

f (x) = 2x+3

a) f -1y = 2

3x

b)

c) f (-1) = 1f 1 (1) = -2

• 109

STRUCTURED CURRICULUM LESSON PLAN

Correlations (SG,CAS,CFS): 8C1

TAP:Analyze and interpret data presented in charts,

graphs, tables, and other displays

ISAT:Identify, analyze, and solve problems using

equations, inequalities, functions, and theirgraphs

Unit Focus

Exploring Equations and Functions

Instructional Focus

Defining and Describing the Relationship Between Dependent and Independent Variables(The Composition of Functions)

Materials

SoakEm Company ProjectGraph paperClassroom set of graphing calculatorsOverhead graphing calculatorOverhead projectorHandout 3.3

Educational Strategies/Instructional Procedures

Have the students complete the following:

Evaluate each function for x = 2, x = 1, and x = 0g (x) = x3 3x

h (x) = 3

1+x

Review previous days homework and answer students questions.

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)).

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

• 110

3-3 to the other half of the groups. Selected students from each group will present the groupsfindings to the whole class.

Integration with Core Subject(s)

LA: Understand explicit, factual informationUnderstand the meaning of words in context

SC: Analyze and interpret data

Connections

Enrichment:

Fine Arts:

Home: Have students keep a record of all assignments. (The homework record will be signedby a parent on a weekly basis.)

Remediation: Work with students in small groups on their areas of deficiency.

Technology:

Assessment

Find the inverse of each function:

1. f(x) = {(2, 2), (1, 4), (2, 18), (3, 2)}

2. g (x) = 21

1

x+

3. h (x) = )2(4

3+x

Solutions:1. f = 1 2 2 4 1 18 2 2 3( ) {( , ), ( , ), ( , ), ( , )}x

2. g =1 1( )x

x

x

3. h =1 4 6

3( )x

x

• 111

Homework

Assign from your text appropriate problems on the composition of functions.

Groups will complete the Soak Em Company pr

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