Supporting STAAR™ Achievement: Targeting the … STAAR A2 Sample.pdfReadiness Standards? ..... iii What Is in a Lesson Found in Supporting STAAR Achievement: Targeting the TEKS and

Supporting STAAR™ Achievement:

Targeting the TEKS and Readiness

Standards

Algebra II

Teacher Edition

Product ID: 407-1677

© Region 4 Education Service Center Supporting STAAR™ Achievement: Algebra II All rights reserved. i

Table of Contents Introduction ......................................................................................... iii–x

What Is Supporting STAAR™ Achievement: Targeting the TEKS and Readiness Standards? ......................................................................... iii What Is in a Lesson Found in Supporting STAAR™ Achievement: Targeting the TEKS and Readiness Standards? ...................................................... iv References and Bibliography ................................................................ ix

Domain and Range of Functions .......................................................... 2–13

Lesson Notes ...................................................................................... 2 Answer Keys ....................................................................................... 7 Activity Masters and Student Pages ................................................. on CD

Scatterplots ....................................................................................... 14–33

Lesson Notes .................................................................................... 14 Answer Keys ..................................................................................... 20 Activity Masters and Student Pages ................................................. on CD

Function Transformations ................................................................. 34–51


Systems of Equations: Three Unknowns ............................................ 52–67


Systems of Equations: Substitution ................................................... 68–77


Systems of Linear Inequalities .......................................................... 78–95


Determining Reasonableness .......................................................... 96–109

Lesson Notes .................................................................................... 96 Answer Keys ................................................................................... 102 Activity Masters and Student Pages ................................................. on CD

Supporting STAAR™ Achievement: Algebra II © Region 4 Education Service Center ii All rights reserved.

Quadratic Representations ............................................................ 110–119 Lesson Notes .................................................................................. 110 Answer Keys ................................................................................... 115 Activity Masters and Student Pages ................................................. on CD

Quadratic Situations ...................................................................... 120–131

Lesson Notes .................................................................................. 120 Answer Keys ................................................................................... 125 Activity Masters and Student Pages ................................................. on CD

Quadratic Formula ......................................................................... 132–143


Standard and Vertex Forms ........................................................... 144–157


Square Root Functions .................................................................. 158–171


Rational Functions ........................................................................ 172–187


Inverse of an Exponential Function ............................................... 188–207


Exponential Functions ................................................................... 208–227


© Region 4 Education Service Center Supporting STAARTM Achievement: Algebra II All rights reserved.

What Is Supporting STAARTM Achievement: Targeting the TEKS and Readiness Standards?

A resource that focuses on the TEKS identified as readiness standards while integrating appropriate supporting standards and mathematical processes and skills

A resource that provides opportunities for rigorous mathematical conversations while providing supports for students at varying levels of readiness

A resource that provides support for English language learners and students struggling to learn mathematics through Tier I differentiated activities, preteaching experiences, scaffolds for activities such as hint cards and graphic organizers, and facilitation questions

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2

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4 A resource that supports beginning as well as experienced teachers through clear instructions and facilitation questions that focus on potential stumbling blocks for students in the effort to bridge to formal understanding of mathematics

5 A resource of classroom-ready 5E lessons. The Engage phase of each lesson consists of a student-centered activity that either bridges from students’ prior knowledge or encourages interest in deeper exploration of the concepts in the lesson. The Explore phase of each lesson provides students with an opportunity to “do mathematics” and begin to formulate ideas and conjectures. In the Explain phase of each lesson, students formalize the mathematical ideas from the Explore phase with a focus on academic vocabulary, as well as procedures related to the concepts. The Elaborate phase of each lesson allows students to apply or extend their understanding of the concepts in the lesson. The Evaluate phase consists of four selected-response or griddable items that can be used to assess student understanding.

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Each readiness standard has been rewritten in student-friendly language so that students may gauge their learning.

Each lesson includes an English Language Proficiency Standard rewritten in student-friendly language.

Each lesson includes prerequisite knowledge that may impact student success within the lesson as well as vocabulary that will be used during the lesson.

Additional TEKS that support the conceptual and procedural development of the readiness standard within this lesson are identified.

What Is in a Lesson Found in Supporting STAARTM Achievement: Targeting the TEKS and Readiness Standards?

iv Supporting STAARTM Achievement: Algebra II © Region 4 Education Service Center

All rights reserved.


Materials for each phase are summarized on one page for ease in preparation.

Grouping strategies for each phase are summarized to assist in the arrangement of the classroom.

The Elaborate phase has two concurrent components: a student-facilitated activity and a teacher-facilitated activity that focuses on the needs of students struggling with the content.

Materials that are provided as supports for students in need of additional help are labeled with an asterisk.

v © Region 4 Education Service Center Supporting STAARTM Achievement: Algebra II All rights reserved.


Each lesson includes a preteach activity that teachers may use with students who might benefit from exposure to related concepts prior to the lesson.

Each phase includes directions to implement the activity and the identification of additional student supports for the activity.

Each phase includes facilitation questions to help students who may be struggling to interpret or process components of the activity.

vi Supporting STAARTM Achievement: Algebra II © Region 4 Education Service Center

All rights reserved.


Titles of activity masters and student pages are printed in bold for ease of reference.

The Tier I intervention provides instructions on how to make the mathematics content more explicit for students struggling with the concepts within the lesson. The activity is at the same rigor as the activity being completed by the students in a self-directed environment.

vii © Region 4 Education Service Center Supporting STAARTM Achievement: Algebra II All rights reserved.

Each selected-response item is labeled with the STAARTM reporting category and student expectation. Incorrect answer choices are classified according to type.

Exponential Functions

A Exponential Functions Readiness Standard 2A.11F The student is expected to analyze a situation modeled by an exponential

function, formulate an equation or inequality, and solve the problem.

Content Objective I can analyze a situation using an exponential function or equation when appropriate to solve problems.

Additional TEKS

2A.1B The student is expected to collect and organize data, make and interpret scatterplots, fit the graph of a function to the data, interpret the results, and proceed to model, predict, and make decisions and critical judgments.

2A.11C The student is expected to determine the reasonable domain and range values of

exponential and logarithmic functions, as well as interpret and determine the reasonableness of solutions to exponential and logarithmic equations and inequalities.

ELPS c4J The student is expected to demonstrate English comprehension and expand

reading skills by employing inferential skills such as predicting, making connections between ideas, drawing inferences and conclusions from text and graphic sources, and finding supporting text evidence commensurate with content area needs.

Language Objective I can make predictions using information in tables, graphs, and verbal descriptions.

Additional ELPS

c1F The student is expected to use accessible language and learn new and essential language in the process.

Prerequisite Knowledge

Write linear functions Analyze data and represent situations involving exponential growth and decay using

concrete models, tables, graphs, and algebraic methods.

Vocabulary Focus

Exponential Initial Value Successive Quotients

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© Region 4 Education Service Center Supporting STAAR™ Achievement: Algebra II All rights reserved.

LesAson 1: Notes Students may use a graphing calculator as needed throughout the lesson. Read and select facilitation questions as appropriate to meet your students’ needs.

Preteach

Materials Instructional Grouping

Extending Function Values Small group with teacher facilitation

Phase Materials one per student unless otherwise noted

Instructional Grouping

Engage The Great Divide Graphing calculator Pairs of students

Explore

The Sky Is Falling Bean Tray A, B, or C (1 tray per pair of students) Dry beans (about 100 beans per pair of students) Scissors Tape Paper cup (Optional)

Pairs of students

Explain What’s Going On? What’s Going On?*

Whole-group discussion

Elaborate

Chew on This, Part I Chew on This, Part II Individual

Intervention Chew on This, Part I* Chew on This, Part II*

Small group with teacher facilitation

Evaluate Evaluate: Exponential Functions Individual

*for targeted students only

209


Supporting STAAR™ Achievement: Algebra II © Region 4 Education Service Center All rights reserved.

1. Distribute Extending Function Values to each student. 2. Prompt students to look at Graph 1. 3. Ask, “Which points in the table can we complete using the graph?” Listen for student

understanding of how to identify independent and dependent values of a function using a graph.

4. Ask, “What is true about the rate of change for a linear function?” Listen for student answers to include that linear functions have a constant rate of change.

5. Ask, “How could you use slope to identify additional points on the line?” Listen for student

understanding of slope as the ratio

rise yorrun x

.

6. Prompt students to use rate of change to complete the table to extend the graph. 7. Prompt students to look at Graph 2. 8. Ask, “How can you describe the rate of change for a quadratic function?” Listen for student

answers to include that the rate of change for a quadratic function is not constant or that it is different for each pair of points chosen.

9. Prompt students to complete as many values as possible using only the information from the part of the graph that is shown.

10. Ask, “How could you use the symmetry of a quadratic graph to complete the graph?” Listen for student understanding of the axis of symmetry being used as line of reflection for the given points to find other points that lie on the graph of this function.

11. Prompt students to complete the graph.

1. Distribute The Great Divide to each

student. 2. Prompt students to complete the activity,

using the hints as needed. 3. Ask students, “How does the change in

y-values for an exponential function differ from the change in y-values for a linear function?” Listen for student observations that the next consecutive value in the table is found by multiplying by a specific value compared to linear functions where the next consecutive value is found by adding a specific value.

Teacher Note The Great Divide is intended to review finding successive quotients and writing exponential functions.

Facilitation Questions Does this graph represent a

constant rate of change? Does it appear quadratic? How does the table verify your observations?

What is the y-intercept for this graph? Where is it recorded on the table?

What is the ratio between any two successive given y-values?

Is the successive quotient for each pair of y-values constant?

Listen for . . .

Use of vocabulary: exponential, linear, rates, y-intercepts

Engage

Preteach

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1. Distribute The Sky Is Falling to each

pair of students. 2. Distribute a bag of dry beans and one

Bean Tray (A, B, or C) to each pair of students. Prompt students to cut out and build their tray.

3. Prompt students to complete The Sky Is Falling. Provide students with a paper cup to facilitate pouring the dry beans if needed.

4. After the completion of the experiment, prompt students to prepare to share their results with another group.

5. Regroup students so that each student is paired with a student who had a different tray. Instruct students to share their experiments and their results.

6. After the small groups share their data with each other, debrief The Sky Is Falling. When might we misinterpret an

exponential context as a linear context?

How did your exponential model compare to the other group’s model?

How do the b values in the exponential equation for each tray compare?

What factors in the tray affected the results of the experiment?

What do the a values in the exponential equation for The Sky Is Falling have in common?

What are the factors in the experiment that support using an exponential model?

How would the number of beans have to change with each trial for the linear model to be a better fit?

Facilitation Questions Is there a constant rate of

change? Is there a constant successive

quotient? How do you determine the initial

value from the table? How could you determine which

model is a better fit? Listen for . . .

Justification of selected function model

Use of academic vocabulary such as exponential, growth, decay, initial values, successive quotients, increasing, decreasing.

Supports Provide Tray A to students who are struggling as the larger shaded area provides data that is growing at a faster rate. The exponential relationship can be difficult to see if the window settings are zoomed out. Prompt students who are struggling to adjust the window on their calculators to zoom in closer in order to better see the relationship of the data.

Explore

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1

1. During this phase of the lesson, students should complete What’s Going On? with the teacher through a whole-group discussion. Some students may benefit from a table prompt for considering rates of change and successive quotients. Provide What’s Going On?* to these students.

2. Ask, “What is occurring mathematically in order for a situation to be modeled with a linear function? Exponential function? Neither linear nor exponential function?” Listen for student understanding that the rate of change will be constant for a linear function and that an exponential function will have constant successive quotients.

3. Prompt students to read the first scenario. Use a teacher think-aloud process to explain the reasoning for the answer to the first problem. Allow students time to complete the table of values on What’s Going On?* if necessary.

4. Ask, “What values are necessary to write the equation? How could you determine these values from the situation?” Listen for student understanding that since the first scenario is linear, slope and the y-intercept would be needed to write an equation.

5. Prompt students to read the second scenario and determine what type of function is being described. Allow students time to complete the table of values on What’s Going On?* if necessary.

6. Ask, “When writing an exponential function model, how are you able to determine the b value? a value?” Listen for student understanding of initial values, base, and successive quotients.

7. Guide students to understand how to calculate the b value for an exponential model when given a rate, r%, of the growth or decay (1 r ). Ask, “Are the dependent values in the situation increasing or decreasing each time? How would you determine if the rate should be added to or subtracted from 1 to find the b value?” Listen for students to connect values increasing each time with growth and values decreasing each time with decay. Listen also for student understanding that they are adding to 1 when the initial value is smaller than the final value and vice versa.

8. If students demonstrate a continued need for support, repeat the questioning process for additional questions. If students appear ready to work independently, allow them to do so.

1. Prompt students to complete Chew on

This, Part I and Part II. 2. If a student appears to be struggling with

Chew on This, the student may join the teacher-led intervention group.

Intervention 1. Prompt students to complete Chew on

This, Part I* and Part II*. 2. Pose the following questions:

How do the y-values change when the relationship is linear? Exponential?

How could you determine the next value in the table?

How could you find the initial value from the table?

3. Work with individuals to provide immediate feedback as they complete Chew on This, Part I* and Part II*. Allow students to use a graphing calculator for verification of their results before completing the questions.

Elaborate

Explain

212



Question Number

Correct Answer

Reporting Category TEKS Conceptual

Error Procedural

Error Guess

1 C 7 2A.11F A B D

2 A 7 2A.11F B D C

3 D 7 2A.11F A B C

4 C 7 2A.11F A B D

Evaluate

213



The Sky Is Falling Tray—A (Answer Key)

1. Use the template provided to build your tray. Record which tray you received in the table. You will be dropping dry beans into the tray.

2. Begin with 10 beans. Place the tray on a flat surface and drop the beans into the tray.

3. For each bean that lands in a shaded area, add one additional bean to your total number of beans. When a bean lands on the edge of a shaded region, if the majority of the bean is in the shaded area, consider it as landing in the shaded area.

4. Record the total number of beans in the table below. Repeat the process until you reach a total

of at least 100 beans. Extend the table if needed.

Sample answers:

5. Determine an appropriate window for your data and enter the data into the graphing calculator.

6. Record your window and sketch a graph of your scatterplot below.

Window: Sketch: XMin: 0 XMax: 10 YMin: 0 YMax: 125

Tray ___A___

Number of Trials, x

Number of Beans, y

0 10

1 14

2 20

3 30

4 45

5 67

6 92

7 123

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7. Use the table below to write a linear and an exponential model for your data. Round answers to the nearest thousandth.

Linear Model m by x Exponential Model abxy

m is the rate of change (slope)

b is the initial value

(when 0x )

a is the initial value

(when 0x )

b is the base or the factor of growth (or decay)

Calculate the rate of change for consecutive pairs of points in your table. Then find the average of those values.

Rate of Change

2 1

2 1

y yyx x x

4

6

10

15

22

25

31

Average of rates of

change:

m =____16.143____

b=___10____ a=___10____

Find successive quotients by dividing each y-value by the previous y-value in the table. Then find the average of those values.

Average of successive

quotients:

b =____1.432____

Successive Quotient 1n

n

yy

1.4

1.429

1.5

1.5

1.489

1.373

1.336

Linear Model

m by x

y x 16.143 10

Exponential Model

abxy

xy 10(1.433)

8. On your graphing calculator, graph both the linear and the exponential models you created. 9. Which one of the above models is a better fit for the data? What is happening in the experiment

to support your decision?

The exponential model is a better fit for the data. A percentage of the tray is shaded, and as the trials occur, that percentage of the total number of beans is added each time.

217

Name: _________________________________ Exponential Functions

©Region 4 Education Service Center Supporting STAAR™ Achievement: Algebra II All rights reserved.

Chew on This, Part I The popularity of Genius Gum has been on the decline since the year 2000. Alexis and Mario decided to investigate this trend. They researched the gum company’s data bank and found that in the year 2001, there were 90,000 people who had purchased Genius Gum. The next year there were 81,000 people who purchased the gum. Both Alexis and Mario decided to find a model to predict the popularity of Genius Gum. Let x be the number of years since 2000 and let y be the number of people who purchased the gum, in thousands. Alexis predicts that the relationship will be linear. Write a linear equation to model the situation.

Mario predicts that the relationship will be exponential. Write an exponential equation to model the situation.

Linear Model:

______________________

Exponential Model:

______________________

Name: _________________________________ Exponential Functions


Chew on This, Part I* The popularity of Genius Gum has been on the decline since the year 2000. Alexis and Mario decided to investigate this trend. They researched the gum company’s data bank and found that in the year 2001, there were 90,000 people who had purchased Genius Gum. The next year there were 81,000 people who purchased the gum. Both Alexis and Mario decided to find a model to predict the popularity of Genius Gum. Let x be the number of years since 2000 and let y be the number of people who purchased the gum, in thousands.

Alexis’s Linear Model Alexis predicts that the relationship will be linear. Complete the table, graph the relationship, and determine a linear model for the situation.

Number of years since

2000, x

People, y, in thousands

0

1 90

2 81

3

4

5

1. Calculate the slope for the given points using the formula:

2 1

2 1

y yyx x x

2. Use the value you found to fill in the missing values in the table.

3. Circle the value of y when 0x .

1. Graph each point from the table.


y mx b

m ______________

b ______________

Linear Model:

______________________

Use your calculator to verify that your equation matches the values you found in your table. Sketch the model on your graph.



Mario’s Exponential Model Mario predicts that the relationship will be exponential. Complete the table, graph the relationship, and determine an exponential model for the situation.

Number of years since

2000, x

People, y, in thousands

0

1 90

2 81

3

4

5

1. Calculate the successive quotient for the given points.

2. Use the value you found to fill in the missing values in the table.


1. Graph each point from the table.


xy a b

a ______________

b ______________

Exponential Model:

______________________

Use your calculator to verify that your equation matches the values you found in your table. Sketch the model on your graph.

Documents

Supporting STAAR™ Achievement: Targeting the … STAAR A2 Sample.pdfReadiness Standards? ..... iii What Is in a Lesson Found in Supporting STAAR Achievement: Targeting the TEKS and