Algebra II EOC Success Lesson Plan Summary CCRS: Reporting Category 5 Course: Algebra II Topic : Square Root Functions Content Objectives: TEKS 2A.9F: TSIET analyze situations modeled by square root functions, formulate equations or inequalities, select a method, and solve problems. (Readiness Standard) TEKS 2A.9D: TSIET determine solutions of square root equations using graphs, tables, and algebraic methods. (Supporting Standard) Language Objective: College and Career Readiness Standards: c3E: TSIET share information in cooperative VII.B.1: Understand and analyze features of a learning interactions. function.

Vocabulary:  domain  elapsed time  inclined plane  range  square root function

VIII.C.2: Use a function to model a real world situation. Materials: Student Pages:  Name Tags for Job  Inclined Plane Data Collection Responsibilities  Finding the Solution to an Inclined  Meter sticks Plane Problem  Books  On Your Own: Solving Another Problem Modeled by a Square  Transparent tape Root Function  Graphing calculators RtI:  Marbles  Square Root Functions  Stopwatch Scavenger Hunt Recording Sheet  Highlighters  Square Root Functions  Graph Paper Scavenger Hunt Cards  Markers Enrichment:  Square Root Functions Scavenger Hunt Recording Sheet  Square Root Functions Scavenger Hunt Cards

Prior Knowledge: Students have been introduced to the square root parent function as the inverse of the quadratic parent function. They have investigated transformations to the square root parent function. They have solved radical equations by squaring both sides of the equation. Activities RtI Tier 1 Differentiation Activity Students will work on the Square Root Functions Scavenger Hunt. If necessary, the teacher can work with students who demonstrate gaps by working the Square Root Functions Scavenger Hunt with them.

Instructional Phase Engage Students will watch a video of an airplane evacuation practice. Then students will remember and discuss what they have learned about inclined planes in science class. Explore Students will collect data using a marble on an inclined plane. They will measure the elapsed time for various distances on the inclined plane. Then

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Enrichment Differentiation Activity Students will complete Square Root Functions Scavenger Hunt.

  Models of proficient problem solving: The teacher will demonstrate how to locate key values on the graph and in the table and how to solve algebraically. Guided practice: Working with a partner, the students will complete the remaining parts of the Scavenger Hunt. Corrective feedback: As students work to complete the activity, the teacher will monitor the student work and ask for justification of reasoning noting for the student when the reasoning does not correspond to the recorded work. Corrective feedback should be detailed and specific enough to guide students to make corrections to their thinking and, in some cases, may require reteaching of key concepts and procedures.

students analyze the data by finding a function that models the data. If appropriate for ELL students: Encourage students to work as a cooperative group. Explain Students will use the given function model from a fictional group’s inclined plane experiment. They will use the function model to find the solution from a graph, a table, and algebraically. Formative Assessment Students will find the solutions to three different questions, using a different solution method for each.

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Inclined Plane Data Collection Answer Key The job responsibilities for a group of four are listed. Each person in the group should choose a job responsibility. Each manager should wear a name tag. The equipment manager is in charge of the materials. The data collection manager is in charge of recording the data. The calculator manager is in charge of any calculations required. The time manager is in charge of accurately measuring time with the stopwatch. Instructions for setting up the experiment to collect data for the inclined plane: 1.

The equipment manager should get the materials from the teacher. Necessary materials include two meter sticks, one book, transparent tape, a graphing calculator, and a marble.

2.

Set up your inclined plane by taping two meter sticks together with the metric system markings matching. Use one long piece of tape. Place the piece of tape down the middle of the two meter sticks, holding them together. The smoother the tape the better the data collection results; any wrinkles or double-taped areas will skew your results.

3.

Set-up the inclined plane using the book to create an incline.

4.

Place the 0 cm mark of the meter stick at the lower end of the inclined plane and the 100 cm mark at the higher end.

5.

Use one piece of tape to hold the meter sticks in a V-shape at the top of the ramp as shown in the photograph below.

6.

Practice a few rolls with the marble on the inclined plane to be certain all materials have been set up correctly and the marble rolls freely.

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Inclined Plane Data Collection Answer Key (cont) Instructions for collecting data: 7.

The equipment manager will release the marble from the distance listed in the table. The time manager will use the stopwatch to measure the amount of time required for the ball (elapsed time) to reach the 0 cm mark on the meter stick.

8.

Repeat from the same distance three times and record all three measurements. The calculator manager will find the average of the three time trials and record the average elapsed time. Round all of the averages to hundredths place.

9.

Repeat the measurements again using the other distances listed in the table. Continue to calculate and record the average elapsed times. Leave the column labeled k blank at this time.

Sample data: Distance (cm), x

Time Trial 1 (sec)

Time Trial 2 (sec)

Time Trial 3 (sec)

Average Elapsed Time (sec), y

k

90 cm

2.03

2.03

2.00

2.02

22.06

75 cm

1.91

1.93

1.89

1.91

20.56

65 cm

1.72

1.69

1.75

1.72

21.97

50 cm

1.53

1.47

1.50

1.50

22.22

40 cm

1.34

1.25

1.31

1.30

23.67

25 cm

1.00

1.03

1.03

1.02

24.03

0 cm

0.00

0.00

0.00

0.00

?

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Inclined Plane Data Collection Answer Key (cont) 10.

Make a scatterplot of Average Elapsed Time (y) vs. Distance (x) for your data. Label the axes. Using the sample data:

11.

What are the domain and range for this situation? The domain for the situation is {x│0 ≤ x ≤ 100}. The range for the situation is {y│0 ≤ y ≤ 2.1}. If the ramp is longer, the domain and range would both change based on the new ramp.

12.

Why is Average Elapsed Time the dependent variable in this situation? The Average Elapsed Time depends on the distance the marble travels, so the Average Elapsed Time is the dependent variable.

13.

What do you notice about the data? I noticed that as the distance increases, the time increases, but not at a constant rate.

14.

Which parent function do you predict will best model this data? Why? The square root parent function might work since the data starts at (0, 0), continues to increase, but does not appear to be growing exponentially or linearly.

15.

In the early 1600s Galileo began studying the inclined plane and found that the distance a marble rolls varies directly as the square of elapsed time. Every inclined plane has its own constant, k. Use the formula, k  d2 , and your data to find the constant, k, for your inclined

t

plane. Calculate k for each row in the table using the average elapsed times. Record your values in the table. ©2011 Texas Education Agency. All Rights Reserved. 

 

Inclined Plane Data Collection Answer Key (cont) 16.

Find the average of your values for k and record your average here. All of the succeeding answers use the sample data from the table. k = 22.42

17.

Find a function to model the data by solving k  d2 for t and substituting your value for k.

t

(Substitute x for d and y for t after you have solved for t.) t

d 22.42 or

y 

x 22.42

18.

What are the domain and range for the function you found to model the data? The domain for the function is all real numbers greater than or equal to 0. The range is also all real numbers greater than or equal to 0.

19.

Predict the elapsed time, if the distance is 30 cm, using the function model you found.

30 22.42 y  1.16 sec y

20.

Release the ball from 30 cm; use the stopwatch to measure the time. After completing three trials, find the average elapsed time. Record your trials and your average here. Trial times: 1.14 sec, 1.21 sec, and 1.13 sec, so the average is 1.16 seconds.

21.

How accurately did your model predict your measured time? Very closely, the actual measurement and the calculated measurements are approximately the same, 1.16 seconds.

22.

Use your graphing calculator to plot the data values and your function rule. How closely do the data and the function model match? They match very closely.

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Inclined Plane Data Collection Answer Key (cont) 23.

Use three different methods to determine the distance mark upon which to place the marble if you wanted the elapsed time to be approximately 1.8 seconds. a)

Algebraically

y

x 22.42

1.8 

x 22.42

 x  1.8    22.42   3.24  x 22.42 72.64 cm  x

b)

Tabularly

c)

Graphically

2

2

24.

Compare the elapsed time in the function rule table for the two given distances. Compare 20 cm to 80 cm.

Compare 10 cm to 40 cm.

Compare 25 cm to 100 cm.

Compare 40 cm to 160 cm.

What do you notice about elapsed times when distances are quadrupled? Four times the distance results in only twice as much time

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Finding the Solution to an Inclined Plane Problem Answer Key Group E used the function below to model their data based on the inclined plane experiment. They measured distance, x, in centimeters and time, t(x), in seconds.

t(x) 

x 9.82

Use three different methods to find the approximate distance the marble will travel in 3 seconds using Group E’s function model. a)

Use a table

b)

Sketch a graph

c)

Solve algebraically

t(x ) 

x 9.82

3

x 9.82

 x  32    9.82   9 x 9.82 cm 88.38 sec  x

Highlight the solution in each method. Explain which method you prefer for solving and why you prefer that method to your partner.

©2011 Texas Education Agency. All Rights Reserved. 

2

On Your Own: Solving Another Problem Modeled by a Square Root Function Answer Key Find the solutions to the problems below using a different method for each one. You may choose the solution method for each problem; however, you must use each method exactly once. a)

tabularly

b)

graphically

c)

algebraically

The relationship between the distance an object has fallen in feet, h, and the velocity in feet per second, v, of a dropped object can be modeled by the function

v  8 h, where v represents the velocity of a dropped object after it has fallen h feet. 1.

Find the approximate velocity of a dropped object after it has fallen 100 feet.

v 8 x v  8 100 v  80 ft/sec

2.

How far must an object fall before it has a velocity of 32 feet per second?

v 8 x 32  8 x 32  8

x

4

x

42  ( x )2 16 ft  x 3.

If an object falls from the top of a refrigerator that is 6.5 feet tall, how fast is the object moving when it hits the floor?

v 8 x v  8 6.5 v  20.4 ft/sec

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Square Root Scavenger Hunt Recording Sheet Answer Key Students in Group X should have the following cards recorded on their recording sheet in this order but not necessarily with the same starting card: E, S, C, P, K, D Students in Group Y should have the following cards recorded on their recording sheet in this order but not necessarily with the same starting card: J, A, G, M, R, I Students in Group Z should have the following cards recorded on their recording sheet in this order but not necessarily with the same starting card: L, O, B, H, N, F

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Algebra II End of Course Success College & Career Readiness Standards Reporting Category 5

Exploring Reporting Category 5 Demonstrate an understanding of the properties of square root functions.

71

Analyze a Situation M d l db Modeled by a Square Root Function

Content Objective I can investigate a situation modeled by a square root function, write an equation for the situation, and find solutions for problems in the situation.

72

Language Objective I can use technology to collect and share information in a group.

Airplane Evacuation Example: http://www.youtube.com/watch?v=XIaovi1JWyY

73

Airplane Evacuation The evacuation slide on an airplane is an example l off an iinclined li d plane. l What do you remember about inclined planes from your science class?

Inclined Plane Data Collection Assign job responsibilities for your group: • Equipment Manager • Data Collection Manager • Calculator Manager • Time Manager Wear your nametag.

74

Setting Up the Inclined Plane

Inclined Plane Data Collection Distance (cm)

Time Trial 1 (seconds)

Time Trial 2 (seconds)

90 75 65 50 40 25 0

75

Time Trial 3 (seconds)

Average Elapsed Time

k

Finding the Solution to an Inclined Plane Problem

y x

On Your Own: Solving Another Problem Modeled by a Square Root Function

76

Square Root Functions Scavenger Hunt

1. Start at the bottom of one card. 2. Find the answer to the question at the bottom of the card. 3. Look for the answer at the top of another card. 4. Move to the answer card. question at the bottom 5. Find the answer to the q of that card. 6. Continue moving around until you are back to your original card.

Pulling It All Together

77

 

Inclined Plane Data Collection The job responsibilities for a group of four are listed. Each person in the group should choose a job responsibility. Each manager should wear a name tag. The equipment manager is in charge of the materials. The data collection manager is in charge of recording the data. The calculator manager is in charge of any calculations required. The time manager is in charge of accurately measuring time with the stopwatch. Instructions for setting up the experiment to collect data for the inclined plane: 1.

The equipment manager should get the materials from the teacher. Necessary materials include two meter sticks, one book, transparent tape, a graphing calculator, and a marble.

2.

Set up your inclined plane by taping two meter sticks together with the metric system markings matching. Use one long piece of tape. Place the piece of tape down the middle of the two meter sticks, holding them together. The smoother the tape the better the data collection results; any wrinkles or double-taped areas will skew your results.

3.

Set-up the inclined plane using the book to create an incline.

4.

Place the 0 cm mark of the meter stick at the lower end of the inclined plane and the 100 cm mark at the higher end.

5.

Use one piece of tape to hold the meter sticks in a V-shape at the top of the ramp as shown in the photograph below.

6.

Practice a few rolls with the marble on the inclined plane to be certain all materials have been set up correctly and the marble rolls freely.

©2011 Texas Education Agency. All Rights Reserved. 

81

 

Inclined Plane Data Collection (cont) Instructions for collecting data: 7.

The equipment manager will release the marble from the distance listed in the table. The time manager will use the stopwatch to measure the amount of time required for the ball (elapsed time) to reach the 0 cm mark on the meter stick.

8.

Repeat from the same distance three times and record all three measurements. The calculator manager will find the average of the three time trials and record the average elapsed time. Round all of the averages to hundredths place.

9.

Repeat the measurements again using the other distances listed in the table. Continue to calculate and record the average elapsed times. Leave the column labeled k blank at this time. Distance (cm), x

Time Trial 1 (sec)

Time Trial 2 (sec)

Time Trial 3 (sec)

90 cm 75 cm 65 cm 50 cm 40 cm 25 cm 0 cm

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82

Average Elapsed Time (sec), y

k

 

Inclined Plane Data Collection (cont) 10.

Make a scatterplot of Average Elapsed Time (y) vs. Distance (x) for your data. Label the axes.

11.

What are the domain and range for this situation?

12.

Why is Average Elapsed Time the dependent variable in this situation?

13.

What do you notice about the data?

14.

Which parent function do you predict will best model this data? Why?

15.

In the early 1600s Galileo began studying the inclined plane and found that the distance a marble rolls varies directly as the square of elapsed time. Every inclined plane has its own constant, k. Use the formula, k  d2 , and your data to find the constant, k, for your inclined t plane. Calculate k for each row in the table using the average elapsed times. Record your values in the table.

©2011 Texas Education Agency. All Rights Reserved. 

83

 

Inclined Plane Data Collection (cont) 16.

Find the average of your values for k and record your average here.

17.

Find a function to model the data by solving k  d2 for t and substituting your value for k. t (Substitute x for d and y for t after you have solved for t.)

18.

What are the domain and range for the function you found to model the data?

19.

Predict the elapsed time, if the distance is 30 cm, using the function model you found.

20.

Release the ball from 30 cm; use the stopwatch to measure the time. After completing three trials, find the average elapsed time. Record your trials and your average here.

21.

How accurately did your model predict your measured time?

22.

Use your graphing calculator to plot the data values and your function rule. How closely do the data and the function model match?

©2011 Texas Education Agency. All Rights Reserved. 

84

 

Inclined Plane Data Collection (cont) 23.

Use three different methods to determine the distance mark upon which to place the marble if you wanted the elapsed time to be approximately 1.8 seconds. a)

24.

Algebraically

b)

Tabularly

c)

Graphically

Compare the elapsed time in the function rule table for the two given distances. a)

Compare 20 cm to 80 cm.

b)

Compare 10 cm to 40 cm.

c)

Compare 25 cm to 100 cm.

d)

Compare 40 cm to 160 cm.

What do you notice about elapsed times when distances are quadrupled?

©2011 Texas Education Agency. All Rights Reserved. 

85

 

Finding the Solution to an Inclined Plane Problem Group E used the function below to model their data based on the inclined plane experiment. They measured the distance, x, in centimeters and time, t(x), in seconds. x . 9.82

t(x) 

Use three different methods to find the approximate distance the marble will travel in 3 seconds using Group E’s function model. a)

Use a table

b)

Sketch a graph

c)

Solve algebraically

Highlight the solution in each method. Explain which method of solving you prefer and why you prefer that method to your partner.

©2011 Texas Education Agency. All Rights Reserved. 

86

 

On Your Own: Solving Another Problem Modeled by a Square Root Function Find the solutions to the problems below using a different method for each one. You may choose the solution method for each problem; however, you must use each method exactly once. a)

tabularly

b)

graphically

c)

algebraically

The relationship between the distance an object has fallen in feet, h, and the velocity in feet per second, v, of a dropped object can be modeled by the function

v  8 h, where v represents the velocity of a dropped object after it has fallen h feet. 1.

Find the approximate velocity of a dropped object after it has fallen 100 feet.

2.

How far must an object fall before it has a velocity of 32 feet per second?

3.

If an object falls from the top of a refrigerator that is 6.5 feet tall, how fast is the object moving when it hits the floor?

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Square Root Scavenger Hunt Recording Sheet Start at your assigned card. Solve the problem on the bottom of the card. Find the solution to the problem on the top of another card. Continue to record letters and search for solutions until you return to your starting card. Card Letter

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Process

Solution

 

 

Name Tags for Job Responsibilities Cut along the solid lines. Each person in the group is assigned one job responsibility. They should wear their nametag. There are two sets of nametags on this page.

Data Collection Data Collection Manager Manager Time Manager

Time Manager

Equipment Manager

Equipment Manager

Calculator Manager

Calculator Manager

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Finding the Solution to an Inclined Plane Problem Group E used the function below to model their data based on the inclined plane experiment. They measured the distance, x, in centimeters and time, t(x), in seconds. t(x) 

x . 9.82

Use three different methods to find the approximate distance the marble will travel in 3 seconds using Group E’s function model. a)

Use a table

b)

Sketch a graph

c)

Solve algebraically

Highlight the solution in each method. Explain which method of solving you prefer and why you prefer that method to your partner.

©2011 Texas Education Agency. All Rights Reserved. 

 

Square Root Functions Scavenger Hunt Cards

f ( x )  0.5 x

A

Find an equation for a square root function with domain [ 0.5,  ) . Look for a function on the scavenger hunt cards with the same domain.   

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Square Root Functions Scavenger Hunt Cards

f (x)  x  5  2

B

Find the function that is 5 units to the left and 2 units below the square root parent function. 

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Square Root Functions Scavenger Hunt Cards

f ( x )  12 x  3

C

Find the function that is 3 units below the parent function, f(x)= x.  

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Square Root Functions Scavenger Hunt Cards

18

D

A soda can contains 355 cm3 of soda. If x represents the height of the soda in the can in cm, the radius, r, of the can may be found by the function r  355 . Find the radius of the can if  x the height of the can is 12 cm. 

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Square Root Functions Scavenger Hunt Cards

3.07

E

Write an equation for a square root function with endpoint (0, 0) passing through the point (16, 12). 

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Square Root Functions Scavenger Hunt Cards

10.42

F

A soda can contains 355 cm3 of soda. If x represents the height of the soda in the can in cm, the radius, r, of the can may be found by the function r  355 . Find the height of the can if  x the radius of the can is 3.5 cm.   

©2011 Texas Education Agency. All Rights Reserved. 

 

 

Square Root Functions Scavenger Hunt Cards

f ( x )  9 x  0.5

G

Change the square root parent function, f ( x )  x , to a function that is translated down vertically 0.5 units from the parent function.   

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Square Root Functions Scavenger Hunt Cards

f (x)  x  5  2

H

The time for one complete swing of a pendulum can be modeled by the formula T  2 L where L represents the 9.8 length of the pendulum in meters and T represents the time for one complete swing in seconds. Approximate the length of the pendulum if one complete swing takes 3 seconds.

©2011 Texas Education Agency. All Rights Reserved. 

 

 

Square Root Functions Scavenger Hunt Cards

27.71

I

A soda can contains 355 cm3 of soda. If x represents the height of the soda can in cm, the radius, r, of the can is described by the function r  355 .  x Find the radius of the can if the height of the can is 11.8 cm. 

©2011 Texas Education Agency. All Rights Reserved. 

 

 

Square Root Functions Scavenger Hunt Cards

3.09

J

Write an equation for a square root function with endpoint (0, 0) passing through the point (9, –1.5).

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Square Root Functions Scavenger Hunt Cards

2.84

K

The area of the shaded region in the graph below can be described by the function f ( x )  2 x x . 3 Find the area of the shaded region if x = 9.

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Square Root Functions Scavenger Hunt Cards

9.22

L

Write an equation for a square root function with endpoint (0, 2) passing through the point (25, 27).

©2011 Texas Education Agency. All Rights Reserved. 

 

 

Square Root Functions Scavenger Hunt Cards

f ( x )  x  0.5

M

The time for one complete swing of a pendulum can be modeled by the formula T  2 L where L represents the 9.8 length of the pendulum in meters and T represents the time for one complete swing in seconds. Approximate the time it takes for one complete swing if the length of the pendulum is 2.6 meters.

©2011 Texas Education Agency. All Rights Reserved. 

 

 

Square Root Functions Scavenger Hunt Cards

2.23

N

The area of the shaded region in the graph below can be described by the function f ( x )  2 x x . 3 Find the area of the shaded region if x = 6.25.

©2011 Texas Education Agency. All Rights Reserved. 

 

 

Square Root Functions Scavenger Hunt Cards

f (x)  5 x  2

O

Find an equation for a square root function with domain [5,  ). Look for a function on the scavenger hunt cards with the same domain. 

©2011 Texas Education Agency. All Rights Reserved. 

 

 

Square Root Functions Scavenger Hunt Cards

f (x)  x  3

P

The time for one complete swing of a pendulum can be modeled by the formula T  2 L where L represents the 9.8 length of the pendulum in meters and T represents the time for one complete swing in seconds. Approximate the time it takes for one complete swing if the length of the pendulum is 2 meters.

©2011 Texas Education Agency. All Rights Reserved. 

 

 

Square Root Functions Scavenger Hunt Cards

3.24

R

The area of the shaded region in the graph below can be described by the function f ( x )  2 x x . 3 Find the area of the shaded region if x = 12.

©2011 Texas Education Agency. All Rights Reserved. 

 

 

Square Root Functions Scavenger Hunt Cards

f (x)  3 x

S

Find an equation for a square root function with domain [3, ) . Look for a function on the scavenger hunt cards with the same domain. 

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EOC Square Root Functions Lesson_RepCat5.pdf

students analyze the data by finding a function that. models the data. If appropriate for ELL students: Encourage students. to work as a cooperative group.

690KB Sizes 0 Downloads 205 Views

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