Maaa 028036

102

STRUCTURED CURRICULUM LESSON PLAN

Day: 028 Subject: Advanced Algebra Grade Level: High School

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 group’s 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



Correlations (SG, CAS, CFS): 8C2





Unit Focus


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


Review previous day’s 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.


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


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.


Technology:

Assessment

Evaluate the groups’ paragraphs using the Structured Curriculum Scoring Rubric.

Homework


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 -1∙y = 2

3−x

b)

c) f (-1) = 1f –1 (1) = -2

109








Unit Focus


Instructional Focus

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

Materials

Soak’Em Company ProjectGraph paperClassroom set of graphing calculatorsOverhead graphing calculatorOverhead projectorHandout 3.3


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


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 group’sfindings to the whole class.




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.)


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 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.

Teacher Notes

Remind students they have one week to complete the “Soak ‘Em” activity.

Solutions to Educational Strategies problems:

g x x x

g

g

g

( )

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) ( )

= −

− = − − − = −

= − = −

= − =

3

3

3

3

3

2 2 3 2 2

1 1 3 1 2

0 0 3 0 0

h xx

h

h

h

( )

( )

( )

( )

= +

− = − + = −

= + =

= + =

1

3

22 1

3

1

3

11 1

3

2

3

00 1

3

1

3

Solutions to Explorations 1:

1. f (g (x)) = -0.5x + 1 g (f (x)) = 0.5 (-x + 1)=-0.5x + 0.5

See student response for comparison

2. g (h(x)) =

05 1

2

1

4

0 5 1

2

.

( ( )).

x x

h g xx

−=

−

=−

See student response for comparison

112

3. See student responses for problem #3. Students should come to the conclusion that the functions are not commutative.

Solutions to Exploration 2:

1. f (x) = 3x+ 1; f − = −1 1

3x

xb g h x

xh x x( ) ; ( )= + = −−2

33 21

2. a.

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.

3.

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

113

Solutions to Soak ‘Em Project:

1. y = -2x + 4010 or y = -2x + 30 if you use only the last digit in the year.

2. 3. The percentage in 1992 is 26%

Part II

4. y = 4x-79725. See graph above.

Part III

6. y = 2x + 3 or f (x) = 2x + 3

7. The compostion of y and y −1 ; f (x) = 2x + 3 f −1

(x) = x − 3

2.

Part IV

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.

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Handout 3.3

Given: f(x) = –x + 1; g(x) = 0.5x; h(x) = 2

1−x

Exploration 1:

1) Find the compositions f (g(x)) and g(f(x))Compare these compositions. How are they alike? How are they different?

2) Find the compositions g (h (x)) and h (g (x))Compare these compositions. How are they alike? How are they different?

3) Do you think the composition of functions is commutative? Why or why not?

f x x h xxb g = + = +

3 12

3; ( )

Exploration 2:

1) Find the inverse functions for f and h.

2) Graph f and f -1 on the same coordinate plane. What is the equation of the axis of symmetryof these two graphs?

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?

3) Graph h and h -1 on the same coordinate plane. What is the equation of the axis ofsymmetry of these two graphs?

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?

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?

5) What do you think is the composition of any function and its inverse?

115

Soak ‘Em ProjectUnit 3 Project

The Soak’Em 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 company’s revenues.To do this, you will use various types of functions and their graphs.

Part I:

Overall, the percentage of the entertainment dollar spent on participant sports in the Midwest ison steady decline. Table 1 summarizes this trend.

Table 1Year % of entertainment dollar

spent on participant sports1994 221995 201996 181997 16

1. Express these percentages as a function of time. ______________________________

2. Use the coordinate grid to graph this function. Use 1990 as the 0 point on the x-axis.

3. If the percentage is symmetrical about the year 1994, what was the percentage in 1992? _________________________________________

116

Part II:

Suppose participant sports regain their popularity during the next three years and raise their shareof the entertainment dollar to 28%.

4. Express this percentage as a function of time for the years 1997-2000. ______________________________________________________

5. Graph this function on the same grid you used in Part I. ______________________________________________________

Part III:

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.

6. Express the percentage of young male participants as a function of young female participants. _________________________________________________________

7. What corrective function will equalize these percentages?_________________________________________________________

Part IV:

The Soak’Em Company operates three parks in the Midwest. Table 2 shows the admissions yearby year to each park as a percentage of the company’s total admissions.

Table 11994 1995 1996 1997

Park A 50 55 45 40Park B 20 25 30 40Park C 30 20 25 20

8. Express each park’s share of the company’s total admissions as a function of time.a. Park A___________________b. Park B___________________c. Park C___________________

117

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.

Part V:

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.

10. Use the coordinate grid to graph this step function for t = 0 to t = 8, with 1994 as t = 0.

118








Unit Focus


Instructional Focus

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

Materials


Review previous day’s homework and answer the questions.

Present the following problem:

Bargain Town is offering a cash rebate of $15 with a 10% discount of its best-sellingcompact disc player. If you were planning to buy this CD player, which option wouldresult in the lower price, taking the $15 rebate first and then the 10% discount or takingthe 10% discount first and then the $15 rebate?

Let x = price of CD playerLet D(x) = discount function10% discount is the same as paying 90% of the price, so R(x) = 0.9xLet R(x) = rebate function (price less $15), so R(x) = x-15

a. find R(D(x))b. find D(R(x))

Which results in a lower selling price?

119

Discuss the solution with students.

Solution:

R(D(x)) = .9x-15D(R(x)) = .9x-13.5R(D(x)) gives a better price.

Have the students begin their homework and monitor their progress.




Connections

Enrichment: Have students find a sale item from the newspaper and try the class example to seeif it works. If class time permits have students present their findings.

Fine Arts:



Technology:

Assessment

Teacher observation

Homework


Teacher Notes

120



Correlations (SG,CAS,CFS):


graphs tables, and other displays



Unit Focus


Instructional Focus

Defining and Graphing Translations, Reflections, and Scalar Transformations of the AbsoluteValue Function (Solving Equations Involving Absolute Value Symbols by Using Graphing andAlgebraic Methods)

Materials

Graph paperClassroom set of graphing calculatorsOverhead graphing calculatorOverhead projector


Review previous day’s homework.

Instruct the students to graph the function y = |x| by first completing an x, y table.Have students identify characteristics of the graph (Expected responses: Domain = all realnumbers; Range = all real numbers such that y≥0; the graph is that of a function; axis ofsymmetry is the y-axis; minimum point is (0,0), V-shaped).

A: f(x) = |x| + 1 g(x) = |x +1|B: f(x) = 2|x| g(x) = ½|x|C: f(x) = 3|x| g(x) = -3|x|

121

Solutions:

y = |x|x y = |x|-3 3-1 10 01 13 3

A: f(x) = |x| + 1 g(x) = |x + 1|

B: f(x) = 2|x| g(x) = ½|x|

3

2

1

-3- 2 -1 1 2 3 -1

10

8

6

4

2

1 2 3 4 5-5 -4 -3 -2 -1-2

-4

-6

-8

-10

f(x) = |x| + 1

g(x) = |x +1|

-1

-2

-3

-4

-5

-6 -5 -4 -3 -2 -1

1 2 3 4 5 6

7

6

5

4

3

2

1

122

C: f(x) = 3|x| g(x) = -3|x|




Connections

Enrichment:

Fine Arts:

Home: Students will keep a record of all assignments. The homework record will be signed bya parent on a weekly basis.


Technology:

Assessment

Teacher observation

-2

-4

-6

-8

-10

-6 -5 -4 -3 -2 -1

12

10

8

6

4

2

1 2 3 4 5 6

123

Homework


Teacher Notes

Remind students that Chapter Projects are due tomorrow.

124



Correlations (SG,CAS,CFS): 8D2; 8A


graphs, tables and other displays



Unit Focus


Instructional Focus

Defining and Graphing the Greatest Integer Function

Materials

Copies of sales tax table



Distribute copies of a tax table and direct students to create an x, y table and a graph of sales taxon amounts up to and including $2.00.

Discuss a step function.

Present and discuss:

greatest-integer function

Examples: [4]=4 [7.4] = 7 [ 5 ] = 2 [π] = 3

least-integer function

Examples: 4 17. =5 π = 4 − = −2 6 3. 6 7 7. =

125




Connections

Enrichment: What is the cost of a phone call that lasts 4 minutes 37 seconds from Chicago toLos Angeles if the charge is 27 cents per minute?

Fine Arts:

Home: Have students keep a record of all assignments. (The homework record will be signedby a parent on a weekly basis.) Calculate the per minute cost of any long distance calls made bya family member.


Technology: Have students explore least-integer and greatest-integer functions on a graphingcalculator.

Assessment

Teacher observation

Homework


Teacher Notes

126








Unit Focus


Instructional Focus

Reviewing Equations and Functions

Materials



Review Unit Three using a review format from Appendix C.




Connections

Enrichment:

Fine Arts:


Remediation: Assign appropriate review problems from Unit Three

Technology:

127

Assessment

Teacher observation

Homework

Review class notes and homework.

Teacher Notes

Remind students that chapter projects are due tomorrow.

Prepare copies of the Unit Three Assessment.

128








Unit Focus


Instructional Focus

Assessing Equations and Functions

Materials

Copies of the Unit Three Assessment


Administer the Unit Three Assessment.

Collect chapter project (Soak ‘Em Company).




Connections

Enrichment:

Fine Arts:

Home:

Remediation:

129

Technology:

Assessment

Evaluate the Unit Three Assessment and Soak ‘Em Company activity using the StructuredCurriculum Scoring Rubric.

Homework

Teacher Notes

Solutions to Unit 3 Assessment:

52

162

1)3(2))((.1

2

2

2

−=+−=

+−=

x

x

xxgf

144

3)12())((.22

2

++=−+=

xx

xxfg

(No; f(g(x)) and g(f(x)) do not have the samesolutions.)

3. y = 2 x 4. x −1

5. See students’ solutions. 6. See students’ solutions.

0)1((

363

3)1(6)1(31

363

)12(3

)1(3))((.7

2

2

2

=+−=

+−⇒=+−=+−=

−=

gf

x

xx

xx

xxgf

88

896

8)4(24))4((

8244

)26(4))((.8

=−=

−=−==−=

gf

xx

xxfg

9. (1,0) 2,3) (5,4) (9,6) (10,-1)function 10. (3,2) (-3,-3) (3,4) (9,8) (

2

1,

7

6)

not a function11. yes 12. yes

13. xy2

3−= 14. y = x – 1

15. y = -2x + 6 slope = -2

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Unit 3 Assessment

1. Find f (g(x) ) when f (x) =2x +1 and g(x) = x2 3− .

2. Is the solution from problem #1 the same as g (f (x))? What is the solution ofg (f (x))?

Write an equation to match the following graphs:

3. 4.

5. Graph each pair of equations on the same coordinate plane. Discuss the similarities in thetwo graphs:

[ ]xy 2−=[ ]xy 2=

6. Give a real - world example where the step function would apply.

7. If 23)( xxf = and 1)( −= xxg , find the value of )).1((gf

8. If 26)( −= xxf and xxg 4)( = , find the value of )).4(( fg

131

Find the inverse of each relation and determine if it is a function.

9. (0,1) (3,2) (4,5) (6,9) (-1,10)

10. (2,3) (-3,3) (4,3) (8,9) (7

6,

2

1)

Determine whether the functions are inverses:

11.

2

1)(

12)(

+=

+=x

xg

xxf12.

6

14)(

16)(

−=

+=x

xg

xxf

Find the inverse of each function.

13. xxf3

2)( −= 14. y=x+1

15. Graph the equation 2x+y=6. What is the slope of this graph?

132



Correlations (SG,CAS,CFS): 6A1; 8C1





Unit Focus


Instructional Focus

Assessing Equations and Functions

Materials

Copies of the Unit Three Assessment


Return the Unit Three Assessment and review the solution with the students.

Have the students write an essay in which they reflect upon Unit Three. Invite the students to usethe following questions as guidelines for writing their essays:

1. Did I do my best?2. What gave me the most difficulty?3. Where do I need the most help?4. What methods or recourse did I use to improve myself?5. What would I do differently next time?6. What activities did I enjoy the most?

Remind students that an essay has a minimum of three paragraphs: introduction, body, andconclusion.

133




Connections

Enrichment:

Fine Arts:

Home:

Remediation:

Technology:

Assessment

Teacher observation

Homework

Have students select three samples of their work to place in their portfolio, then write anexplanation as to why each piece was selected.

Teacher Notes

Read student essays.

Documents

Maaa 028036