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ALGEBRA I HOUSTON ISD PLANNING GUIDE 4TH SIX-WEEKS

- English Language Proficiency Standards (ELPS) - Literacy Leads the Way Best Practices - Aligned to Upcoming State Readiness Standard

- State Process Standard Ⓡ - State Readiness Standard Ⓢ - State Supporting Standard Ⓣ - TAKS Tested Objective (only 11th grade)

© Houston ISD Curriculum

2012 – 2013

of 4

Planning Guide User Information

Unit 9: Patterns in Polynomials and Exponents

Time Allocations

Unit

4 lessons (90-minutes each) or

8 lessons (45-minutes each)

Unit Overview

Patterns in Polynomials and Exponents – Students perform operations with polynomials and exponents and apply these operations to real-world situations.

TEKS/SEs (district clarifications/elaborations in italics)

Ⓢ ALGI.3A Use manipulatives, drawings, verbal descriptions, and symbols to represent unknowns and variables in

real-world situations.

Ⓡ ALGI.4A Find specific function values; add, subtract, multiply, or divide to simplify polynomial expressions;

transform and solve equations including factoring as necessary in problem situations which are expressed in verbal, algebraic, or pictorial (algebra tiles) representations.

Ⓢ ALGI.11A Use patterns to generate properties of exponents and apply these properties in problem-solving situations

when given like numerical or variable bases and integer exponents.

English Language Proficiency Standards

ELPS C.1a Use prior knowledge and experiences to understand meanings in English.

ELPS C.5b Write using newly acquired basic vocabulary and content-based grade-level vocabulary.

College and Career Readiness Standards

CCRS 1.B1 Perform computations with real and complex numbers.

CCRS 2.B1 Recognize and use algebraic field properties, concepts, procedures, and algorithms to combine, transform, and evaluate expressions (e.g., polynomials, radicals, rational expressions).

Key Concepts

exponent

notation

operation

polynomial

property

Academic Vocabulary

pattern rule

Content-Specific Vocabulary

power rational scientific notation

Essential Understandings / Guiding Questions

Patterns in exponents convey properties of real numbers. 1. How are prime and composite numbers different? 2. How may prime and composite numbers have different representations?

Operations on polynomials and exponents demonstrate real-world representations. 1. How are the product, quotient, and “power to a power” rules derived? 2. How do these rules relate to real-world representations?

Scientific notation represents real-world applications of exponents. 1. How are positive and negative exponents used to represent very large and very small numbers? 2. Why does any number (except zero), when raised to the zero power, equal to one? 3. What is the relationship of scientific notation to real-world applications? 4. Why is scientific notation important?

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2012 – 2013

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Assessment Connections

Performance Expectation o Students will use manipulatives, drawing, verbal descriptions, and symbols to perform operations with

polynomials and exponents. o Students will use properties of real numbers to demonstrate real-world representations.

Formative Assessment – The activity Area of Shapes Using Polynomials evaluates students’ ability to use the properties and attributes of polynomials in practical situations. Students may work individually or in pairs.

SpringBoard® Algebra – Embedded Assessment #1: “Decisions, Decisions” – #6

STAAR Sample Item – Item #5 (ALGI.4A) Texas English Language Proficiency Assessment System (TELPAS): End-of-year assessment in listening, speaking, reading, and writing for all students coded as LEP (ELL) and for students who are LEP but have parental denials for Language Support Programming (coded WH). For the Writing TELPAS, teachers provide five writing samples – one narrative about a past event, two academic (from science, social studies, or mathematics), and two others.

Instructional Considerations

Information in this section is provided to assist the teacher with the background knowledge needed to plan instruction that facilitates students to internalize the Key Concepts and Essential Understandings for this unit. It is recommended that teachers thoroughly read this section before implementing the strategies and activities in the Instructional Strategies section. Prerequisites and/or Background Knowledge for Students Students have concretely explored distributive property and combining like terms in Algebra I Unit 3.

Middle school students determined the perimeter and area of plane figures with numeric side lengths. (Ⓡ MATH.6.8B)

In Algebra I Units 2 and 3, students have simplified expressions for the perimeter of plane figures with algebraic side lengths.

In grade 6, students investigated exponents involving prime factorization and order of operations. (Ⓢ MATH.6.1D,

Ⓡ MATH.6.2E)

Background Knowledge for Teacher Critical Content

Add and subtract polynomials in context of concrete examples;

Use laws of exponents;

Justify use of zero and negative exponents;

Translate standard notation of numbers to scientific notation and reverse; and

Evaluate expressions involving rational exponents.

Convert expressions involving rational exponents into their equivalent radical form.

Convert radical expressions into their equivalent exponential form.

Note word variations of “Engage, Explore, Explain, Elaborate, and Evaluate” that imply the 5E Lesson Model.

Instructional Accommodations for Diverse Learners

Use suggested activities to support exploring these critical content topics concretely, pictorially, and symbolically, as stated in ALGI.3A.

For a review of measurement, determine the perimeter and area of figures with side lengths expressed as polynomials. Consider using examples, such as those found in the McDougal-Littell textbook p. 566 (37-42, 49-50), for determining perimeter and area.

http://www.tea.state.tx.us/student.assessment/ell/telpas/





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Instructional Considerations

Connections to Science Money Talks provides a science connection by discussing dimensional analysis, which allows students to solve problems involving rates and relationships between quantities. For instance:

? $ 1 gallon $3.60250 miles

18 miles gallonTrip

Instructional Strategies / Activities

The strategies and activities in this section are designed to assist the teacher to provide learning experiences to ensure that all learners achieve mastery of the TEKS SEs for this unit. It is recommended that the strategies and activities in this section be taught in the order in which they appear.

Nonlinguistic Representations

Allow student pairs to use algebra tiles to create unusual shapes, express the length of each side as an algebraic expression, and determine an expression for the perimeter. (SpringBoard

® Mathematics with Meaning: Algebra 1,

Activity 4.4 “Adding and Subtracting Polynomials”) (ALGI.4A) C.5b

Complete each clarifying activity, Product of Powers and Power of a Power Properties, Quotient of Powers, and Product of Powers and Power of Quotient Properties, Part 2, to prove the rules for operations with exponents. Use examples of perimeter, area, and volume to illustrate real-world examples of operations involving exponents. (ALGI.11A)

In these activities, students first write each exponential expression in expanded form so that they can discover the rules for multiplying (adding the exponents) and dividing (subtracting the exponents). (SpringBoard

® Mathematics

with Meaning: Algebra 1, Activity 4.1 “Exponent Rules”) Instructional Accommodations for Diverse Learners

Students may complete Addition and Subtraction of Polynomials to see examples of using algebraic expressions to determine perimeter. Use Warm-up: Perimeters with Polynomials to create perimeter problems with polynomials – see Resources. C.1a, C.5b (ALGI.3A)

For additional strategies to assist diverse learners, access Recommendations for Accommodating Special Needs Students: Algebra I, Cycle 4, Unit 9. Cues, Questions and Advance Organizers

Frayer Model (Pump Up the Vocab) Students may transition to the properties of exponents by using a Frayer Model for vocabulary. See Unit 3 for an example of how students may apply that graphic organizer. Students may show as non-examples other rules that they have confused in the past. Check that students use appropriate vocabulary in describing the exponential rules. Summarizing and Note-Taking

KWL (Pen/cil To Paper)

Use students’ understanding from science classes to engage them in discussions about how positive and negative powers represent large and small values. Segue to a discussion on scientific notation. In groups, have students "popcorn" what they remember about scientific notation as they take notes on chart paper. Students should recall that scientific notation allows very large or very small numbers to be written in a form that will be easier to manage.

Even graphing calculators do not have the capability to display very large or very small numbers in decimal form. Debrief the activity by having each group make a brief presentation while the audience compares and contrasts the information on each chart.



https://docs.google.com/viewer?a=v&pid=explorer&chrome=true&srcid=0BzaY2soe3hc_YzJiYzM3NjgtZWM2OS00YzMzLTg2NTgtNzg3NGFjYjhkY2Q4&hl=en_US&authkey=CPKiw68F

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http://houstonisdpsd.org/literacy-routines.html

http://houstonisdpsd.org/literacy-routines.html





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Instructional Strategies / Activities

Use Money Talks and Power of Exponents for clarifying activities with real-world examples involving scientific notation. “Use Scientific Notation” on page 519 in the Algebra I textbook gives instructions on using scientific notation with the graphing calculator.

Extensions for Pre-AP As an extension, connect properties of exponents to rational exponents. Discuss the process students will use to evaluate expressions containing fractional exponents. Remind students that the denominator of the exponent tells them what root to take and the numerator tells them what power to use. Consider using examples such as those found on pp. 509 – 510 of the McDougal-Littell textbook.

Resources

Adopted Instructional Materials McDougal-Littell, Algebra 1:

“Add and Subtract Polynomials,” pp. 554 – 559

“Investigations: Products and Powers,” p. 488

“Apply Exponent Properties Involving Products,” pp. 489 – 494

“Apply Exponent Properties Involving Quotients,” pp. 495 – 500

“Investigation: Zero and Negative Exponents,” p. 502

“Define and Use Zero and Negative Exponents,” pp.503 – 508

“Define and Use Fractional Exponents,” pp. 509 – 510

“Use Scientific Notation,” pp. 512 – 517

“Graphing Calculator: Use Scientific Notation,” p. 519

SpringBoard

® Mathematics with

Meaning: Algebra 1

4.1 “Exponent Rules”

4.4 “Adding and Subtracting Polynomials”

Supporting Resources

Addition and Subtraction of Polynomials

Warm-up: Perimeters with Polynomials

Product of Powers and Power of a Power Properties

Quotient of Powers

Product of Powers and Power of Quotient Properties, Part 2

Money Talks

Power of Exponents

Recommendations for Accommodating Special Needs Students: Algebra I, Cycle 4, Unit 9

Online Resources

BrainPop: Multiplying and Dividing Exponents www.brainpop.com

http://www.brainpop.com/

Materials Transparency of Blackline

Master B12 One copy of Blackline Master

B12 for each student Overhead algebra modeling tiles One set of algebra modeling

tiles for each pair of students

A12 Addition and Subtraction of Polynomials Students have used algebra modeling tiles to combine like terms in Unit 3, Lesson 3, so they should be familiar with the names of the tiles and the concept of zero pairs. (A positive and a negative tile of the same size cancel each other out.) Review these concepts if necessary. Then, use overhead tiles to model the expression 2x2 + 3x + 4 as shown below:

Another name for an algebraic expression whose terms have whole number exponents is polynomial. If the polynomial has one term, it is called a monomial; two terms, it is called a binomial, three terms, it is called a trinomial. Therefore we would call the expression modeled above a trinomial. Demonstrate the addition of polynomials with algebra tiles as shown in example 1 on p. 424, having the students model the polynomials with their own tiles and record the process of eliminating zero pairs and combining like terms on their activity sheet. Do additional examples as necessary. Next, demonstrate subtraction of polynomials as shown below. Model each step concretely, then record symbolically what is happening. Students should also model and record on their activity sheets. Model original problem:

(2x2 – 3x + 2) – (3x2 + 2x – 1) Apply the Definition of Subtraction (add the opposite)

(2x2 – 3x + 2) + (-3x2 -2x + 1) Combine like terms (2x2 + -3x2) + (-3x + -2x) + (2 + 1)

Notes to the Teacher Encourage students to refer to algebraic expressions as monomials, binomials, trinomials, and polynomials because this vocabulary will be helpful later in the unit. For example, you will want to be able to point out that we have a special technique for multiplying two binomials together and that identifying a common monomial factor is a first step in factoring a particularly important to drill students on more specific names such as a cubic trinomial.

–

+

Notes to the Teacher

Simplify -x2 + -5x + 3 Monitor students as they complete the rest of the activity sheet. Solutions:

1. 3x2 – x + 1 2. –x2 + 6 3. 13x – 12 4. (x2 + 7x + 6) – (x2 + 5) = 7x + 1 5. (x2 + 2x + 4) + (x2 + 3x) = 2x2 + 5x + 4 6. 110.5 − 25π cm2

B12 Addition and Subtraction of Polynomials

Pictorial Representation Symbolic Representation (3x2 + 2x + 3) + (2x2 – x – 2)

(2x2 – 3x + 2) – (3x2 + 2x – 1)

(x2 + x – 3) – (2x2 – 2x – 2)

Solve the following problems. You may continue to model with the algebra modeling tiles or work symbolically only. 1. (2x2 + 4x – 1) + (x2 – 5x + 2) 2. (x2 – 3x + 7) – (2x2 – 3x + 1) 3. Find the perimeter. 2x 3x − 7 3x − 7

5x + 2

4. You were working with one big square, seven rectangles, and six small

squares when the teacher walked by and knocked the big square on the floor and the wind blew through and scattered five of the small squares all over the room. Write an equation to represent the tiles you had, what you lost, and the tiles that remained.

5. Write an equation to represent the model below.

+ =

6. Find the area of the shaded region.

5 cm

13 cm

17 cm


Materials One copy of Blackline Master

B5 for each student Calculators (preferably graphing

calculators) Be certain that students understand that when

simplifying b m

b n , if the larger power is in the numerator, the result is a positive power of b. If the larger power is in the denominator, then the result is a negative power of b.

A5 Quotient of Powers As in the last activity, students work in groups with their calculators to generate properties of exponents. Answers:

Problem Value 28 26

4

210 ÷ 28 4 25 ÷ 23 4 29 27

4

1. The answer was always 4. 2. The base of the numerator and the denominator was always

the same. 3. The difference between the exponents was always the same. 4. 4

Problem Value 39 36

27

310 ÷ 37 27 35 ÷ 32 27 39 36

27

5. The answer was always 27. 6. The base of the numerator and the denominator was always

the same. 7. The difference between the exponents was always the same. 8. 27

Problem Value 43 42

41

58 ÷ 56 52 77 ÷ 72 75 73 ÷ 76 7-3 43 48

4-5

x8 x3

X5

am ÷ an am-n The Quotient of Powers Property provides a shortcut when dividing powers—just leave the base the same and subtract the exponents. After debriefing the activity, check for understanding by doing a few problems similar to those that will be on the homework assignment on the board as a large group activity.

Sample problems: Solutions: (x2)(x7) x9 (x3)12 x36

(y12)(y8) y20 (a13)(a8)(a2) a23 (2x4)(3x5) 6x9 (-7t10)(3t9) -21t19

3

12x

x

x-9

(p)(p3) p4 (q12) (q5) q17

10

3x

x x7

(x3)5 x15

13

6r

r

r7

6x5 + 2x5 8x5 10p4 – 12p4 -2p4 x5 + x3 does not simplify p4 – p4 0


B5 Quotient of Powers You have seen that the patterns that occur when you multiply powers with the same base can lead you to shortcuts for computing problems such as 23 · 27 and (23)5. Now see if you can use patterns to find a shortcut for dividing powers with the same base. Complete the table below. Problem Value

28

26

210 ÷ 28 25 ÷ 23

29

27

1. What did you notice about your answers to each problem?

______________

2. What did you notice about the base used in every problem? _____________

3. What did you notice about the exponents in every problem?

______________

4. Based on what you have seen, what do you think 27 ÷ 25 will equal? _______ Check your answer on your calculator.

Problem Value

39

36

310 ÷ 37 35 ÷ 32

39

36

5. What did you notice about your answers to each problem?

______________

6. What did you notice about the base used in every problem? ______________

7. What did you notice about the exponents in every problem?

______________

8. Based on what you have seen, what do you think 37 ÷ 34 will equal? _______ Check your answer on your calculator.

Keeping the patterns you have observed in the preceding tables in mind, complete the table below. Notice that your answer will be in exponent form. If you follow the pattern established above, you won’t need a calculator. Problem Value

43

42 4?

58 ÷ 56 5? 77 ÷ 72 7? 73 ÷ 76 7?

43

48 4?

x 8

x 3 x?

am ÷ an a? State the Quotient of Powers Property in words: ________________________________________________________________ ________________________________________________________________


B8 for each student. Calculators By now, these property names may all start running together for the students. From this point on, the important thing is not to be able to state the name of the property being used at a particular time, but to be able to visualize the rule symbolically or state it in words so that it can be applied to the appropriate problem.

A8 Power of Products and Power of Quotients Properties

These properties are easier to see, so students should not need too much time to complete this activity either independently or in pairs. Answers:

Problem Expand Group like factors

Rewrite in exponential form

Evaluate coefficient if possible

(3x)4 3x · 3x ·3x ·3x

3·3·3·3·x· x· x· x

34 x4 81x4

(5p)2 5p·5p 5·5·p·p 52p2 25p2 (ab)5 ab·ab·a

b·ab·ab a·a·a·a·a·b·b·b·b·b

a5b5

(2xy)3 2xy·2xy·2xy

2·2·2·x·x·x·y·y·y

23x3y3 8x3y3

1. The Power of a Product Property states that if you have a

product raised to a power, you need to raise both factors of the product to the power.

2. Sample answer: When we use the distributive property we multiply both parts of an expression by the same number. Exponentiation distributes over multiplication because you have to raise both factors to the exponent.

Problem Expand Simplify

numerator and

denominator


( 23

33 23 )3 2

3 · 23 · 2

3 827

x x x x x x 5

45 x

45

x 5

1024 4 · 4 · 4 · 4 · 4 2

3ab 9a 2

b2 3a

b · 3a

b 32a 2

b2 3. an

bn 4. The Power of a Quotient Property states that if you have a

quotient (fraction) raised to a power, you need to raise the numerator and the denominator to the power

5. Sample answer: Exponentiation distributes over quotients because you have to multiply both parts of the quotient—the numerator and the denominator—by the exponent.


B8 Product of Powers and Power of Quotient Properties Complete the tables below to generate more properties of powers that will provide you shortcuts when working with exponents. Problem Expand Group like factors Rewrite in

exponential form

Evaluate coefficient if possible

(3x)4 3x · 3x ·3x ·3x 3·3·3·3·x· x· x· x 34 x4 81x4

(5p)2 (ab)5 (2xy)3 1. The Power of a Product Property states that for all nonzero a and b,

(ab)n = an · bn. Restate this property in your own words. _____________________________________________________ ________________________________________________________________ 2. Mathematicians say that exponentiation distributes over multiplication. What

do you think they mean? ________________________________________________________________ ________________________________________________________________ The Power of a Quotient Property is very similar to the Power of a Product Property. It enables you to find powers of fractions. Complete the table below. Problem Expand Simplify

numerator and denominator


23

3

23 ·

23 ·

23

827

23

33 x

45

x4 ·

x4 ·

x4 ·

x4 ·

x4

3ab

2

3. In general, we can say that for all nonzero a and b, a

bn

= ________. 4. State the Power of a Quotient Property in your own words. ______________ ________________________________________________________________ 5. Could you also say that exponentiation distributes over division? Explain your

answer. ______________________________________________________


B4 for each student. Calculators (preferably graphing

calculators) It is important to allow time for students to generate the properties of exponents using patterns rather than simply asking them to memorize rules that have little meaning to them. .

A4 Product of Powers and Power of a Power Properties Students should work in groups of two or three to complete this activity. Monitor carefully to be certain that they correctly state the property that is being generated. Answers:

Problem Value 22·28 1024 23·27 1024 24·26 1024 25·25 1024 210 1024

1. They are all 1024 2. The base is 2 3. The exponents add up to 10 in every problem. 4. 1024

Problem Value 32·37 19683 33·36 19683 34·35 19683 39 19683

5. They are all 19683. 6. The base is 3. 7. The exponents add up to 9 in every problem. 8. 19683

Problem Value 22·28 210 32·312 314 50·54 54

x5 · x3 x8 am · an am+n

The Product of Powers Property provides a shortcut for multiplying two powers with like bases—just keep the base and add the exponents.

Problem Expanded Form Value (exponential form)(35)4 35 · 35 · 35 · 35 320

(x3)2 x3 · x3 x6 (26)5 26 · 26 · 26 · 26 · 26 230

(85)2 85 · 85 810 (f4)3 f4 · f4 · f4 f12

9. If you multiply the exponents in the problem it equals the

Notes to the Teacher The properties could be generated by looking at expanded products and expanded quotients

exponent in the answer. 10. (am)n = am•n 11. The Power of a Power Property provides a shortcut for raising

a power to a power—just keep the base the same and multiply the exponents together.

12. All of the statements are false. 13. No 14. No

B4 Product of Powers and Power of a Power Properties

Product of Powers Property Some interesting things happen with exponents when you multiply or divide two powers. Use your calculator to complete the tables below. Be looking for patterns! Problem Value 22·28 23·27 24·26 25·25 210 1. What did you notice about your answers to each problem? ______________ 2. What did you notice about the base used in every problem? _____________ 3. What did you notice about the exponents in every problem? _____________ 4. Based on what you have seen above, what do you think 21·29 will equal?

Check your answer on your calculator. Problem Value 32·37 33·36 34·35 39 5. What did you notice about your answers to each problem? ______________ 6. What did you notice about the base used in every problem? _____________ 7. What did you notice about the exponents in every problem? _____________ 8. Based on what you have seen above, what do you think 31·38 will equal?

Check your answer on your calculator.

Keeping the patterns you saw in the preceding tables in mind, complete the table below. Notice that your answer will be in exponent form. You shouldn’t need a calculator for this table, just follow the pattern established above. Problem Value 22·28 2? 32·312 3? 50·54 5?

x5 · x3 x? am · an a? State the Product of Powers Property in words: ________________________________________________________________ ________________________________________________________________ Power of a Power Property You can use the shortcut you learned above for multiply powers with the same base to find a shortcut for simplifying expressions such as (56)3. Complete the table below. Problem Expanded Form Value (exponential form) (35)4 35 · 35 · 35 · 35 320

(x3)2 x3 · x3

(26)5 (85)2 (f4)3 9. What do you notice about the exponent in each problem and the exponents in

the answer? ____________________________________. 10. Fill in the blank to write the Power of a Power Property symbolically:

(am)n = _______ 11. State the Power of a Power Property in words: _______________________ ____________________________________________________________

Is there a Sum of Powers Property? 12. Use your calculator to decide if the following are true statements? 25 + 22 = 27 T or F 35 + 31 = 36 T or F 43 + 42 = 45 T or F

13. Choose a value for x and compute x2 + x3 and x5. Are the results the same? ____________ 14. Based on your results in problems 12 and 13, do you think there is a shortcut

for adding powers with the same base—that is, does xm + xn = xm + n ? ______________

ALGEBRA I HOUSTON ISD PLANNING GUIDE

4th SIX-WEEKS

Recommendations for Instructional Enhancements for Students with Special Needs

Unit 9: Patterns in Polynomials and Exponents

Content-specific Accommodations for this Unit

Teach students how to use a box to assist them with multiplying polynomials. 1. Construct a box by counting the number of terms in the first polynomial to

determine the number of rows needed. 2. Count the number of terms in the second polynomial to determine the number of

columns needed. 3. Draw a box similar to the one below. In this example, the polynomials (x +

1)(x+2) are placed in the rows and columns. 4. Columns and rows are multiplied together to fill in the blanks. 5. Combine like terms and write the answer in standard polynomial form.

Polynomial Multiplication Box

Problem: (x+1)(x+2) First term of

second polynomial x

Second term of second polynomial 2

First term of first polynomial x x2 2x

Second term of first polynomial 1 1x 2

Solution: (x+1)(x+2) = x2+1x+2x+2 = x2 + 3x +2

Encourage students to clearly scribe exponents as superscripts so they do not confuse the exponent with the base.

Provide students with a skeletal place value chart and assist them with completing the chart. Then allow them to use the place value chart when working with scientific notation. Instruct them to store their charts in their binders for future reference.

When converting very large or very small values to scientific notation, teach students to highlight or mark the places as they count them to determine the correct exponent for 10.

General Accommodations for this Unit

Follow a task on which students perform below expectations with one they are likely to complete proficiently in order to engage reluctant learners and to encourage them to take a risk. End each day of instruction on a positive note.

A9 Money Talks Work through problem 1 with the whole class, demonstrating how to use scientific notation and exponents to simplify their work. Then allow time for students to work the remaining problems. No new concepts are introduced, so it is not important for every student to get to every problem. Answers: 1. Step 1: 1,000,000,000 = 1 x 109

Step 2: = = =−

$1 $60 $3600 $28,800sec min 8hr hour day

Step 3: ×= ×

×

94

4

1 10 3.472 102.88 10

Step 4: 34,720365

= 95 years

Materials Transparency of Blackline Master B9 Calculators Students use dimensional analysis when working problems with different units in their science classes. You may want to be familiar with and/or show a solution such as the following: Step 1

• • • •

=

$1 60sec 60min 81sec 1min 1 1365 $10,512,000

1

hrhr workday

daysyr yr

Step 2

×=

×

9

7

1 10 951.0512 10

yrs

yr

Notes to the Teacher 2. ×

=×

9

2

$1,000,000,000 1 10454 4.54 10

= 2,202,643 lbs

3. ×= = ×

×

96

3

1,000,000,000 1 10 1 101,000 1 10

= 1,000,000 days

≈1,000,000

3652,740 years

4. ×× = = ×

109 5

5

1.56 1015.6 10 1.56 1010

cm kilometers = 156,000 kilometers

B9 Money Talks Exponents are most useful when working problems that involve very large or very small numbers. You should recall that scientific notation allows us to rewrite numbers in a form that will be easier to manage—not only for people, but also for calculators. Even graphing calculators do not have the capability to display very large or very small numbers in decimal form. Use your previous knowledge of scientific notation and your new knowledge of exponents as you consider the following problems.

1. You have won the lottery! The net cash prize is 1 billion dollars. But there’s a catch. The money is being given to you all in $1 bills and in order to keep the money, you must count it for eight hours a day at the rate of $1 per second. Nothing is yours until you have completed the count. How long will it take you to count the money and start your spending spree?

2. 454 one-dollar bills weigh about one pound. How much would your prize

weigh in pounds?

3. If you plan to spend your money at the rate of $1,000 per day, how long would it take you to spend it?

4. The length of a $1 bill is 15.6 cm. How long, in kilometers, would your

money be if you laid the bills side by side length-wise.


B10 for each student. Calculators Chart paper Markers A “gallery walk” is an opportunity for students to walk around the room and view work done by other groups. You may want group members to walk around together. Depending on time, you might want to give them a rubric to “judge” the work they see.

A10 The Power of Exponents Students should work in groups of three or four to solve one or more of the following problems. Each group should write their solution, showing all work, on chart paper and post. Have a “gallery walk” or let one member of each group present their solution to the class. Answers: 1.

5.5 x 10 9 bills

2.74 x 10 8 people 2.007 x 101 = about 20 bills per person 2. 3.2 x 104 x 110 x 5 x 106 1760 x 1010 1.760 x 1013 3. SA = 4π(6.4 x 106)2 SA = 5.1 x 1014 4. a. SA = 24s2; 96s2

b. 8s3 : 64s3 c. length of larger cube is twice the length of smaller cube d. surface area of larger cube is 4 times larger e. 8 times larger

5. 5.075 x 109

B10 The Power of Exponents

1. In 1999, there were approximately 274 million people and 5.5 billion one-dollar bills in circulation. How many dollar bills was this per person?

2. Find the approximate numbers of red blood cells in the body of a 110-lb

person if there are 32000 microliters of blood for each pound of body weight and each microliter of blood contains 5 million red blood cells.

3. The radius of the Earth is about 6.4 million meters. Using the formula for

the surface area of sphere, S = 4πr2, approximate the surface area of the Earth.

2s4s

4. Compute the surface area of each cube shown above.

• Compute the volume of each cube shown above. • What is the relationship between the length of a side of the smaller

cube and the length of a side of the larger cube? • What is the relationship between the surface area of the smaller

cube and the surface area of the larger cube? • What is the relationship between the volume of the smaller cube

and the volume of the larger cube?

5. There are about 290 million people in the United States. If each of us buys five 6-packs of soft drinks a year for $3.50 each, how much money could the soft drink industry expect to take in a year?

Area of shapes using polynomials Write an expression for the shaded area of each figure. Simplify the expression.

Name: ________________________________________

Date:_______________________

Period: ________

Formula and expression for area of triangle: Formula and expression for area of rectangle:

Expression for shaded area:

Simplified:

Formula and expression for area of trapezoid: Formula and expression for area of rectangle: Expression for shaded area:

Simplified:

Formula and expression for area of triangle: Formula and expression for area of rectangle: Expression for shaded area:

Simplified:

4x

3x

+ 9

Formula and expression for area of rectangle: Formula and expression for area of circle:

Use = 22

7

Expression for shaded area: Simplified:

3x

8x

– 5

7x

3x + 1

6x

3x

– 1

2x

Height of

trapezoid =

3n 3n n

+ 4

6n + 3

8n – 3

State of Texas Assessments of

Academic Readiness

STAARTM

Algebra I2011 Released Test Questions

These released questions represent selected TEKS student expectations for each reporting category. These questions are samples only and do not represent all the student expectations eligible for assessment.

Copyright © 2011, Texas Education Agency. All rights reserved. Reproduction of all or portions of this work is prohibited without express written permission from the Texas Education Agency.

STAAR Algebra I 2011 ReleaseReleased Test Questions

Page 2

1 The sales tax rate at a clothing store is 8.75%. Sales tax on an item is a function of its price. Which of the following is the dependent quantity in this function?

A The sales tax rate on the item

B The item’s price

C The amount of sales tax on the item

D The item’s size

2 Which of the following relations is a function?

I. {(0, 0), (0, 1), (0, 2)} II. {(0, 0), (1, 1), (2, 4)} III. {(0, 0), (1, 2), (2, 2)} IV. {(0, 0), (1, 2), (1, 3)}

A I, II, and III only

B I and II only

C II and III only

D III and IV only


Page 3

3 Southern Phone Company is promoting a new cell phone service plan: a customer can make up to 500 minutes of calls each month for $39.99. If the number of minutes used in a month exceeds 500, then the function

c І 0.40(m − 500) + 39.99

describes the monthly charge, c, in dollars in terms of m, the total number of minutes used. Which of the following statements best describes this function?

A If the total number of minutes used is more than 500, then every minute beyond 500 costs 40 cents.

B Every minute used costs 40 cents, regardless of the total number of minutes used.

C The first 500 minutes used costs 40 cents each, after which there is an additional charge of $39.99.

D If the total number of minutes used is more than 500, then every minute used costs 40 cents.

4 What is the domain of the function graphed below?

−5

−4

−6

−7

−8

−9

−10

−3

−2

−1

1

2

3

4

5

6

7

8

9

10

−1 1−2−3−4−5−6−7−8−9−10 2 3 4 5 6 7 8 9 10

y

x

A 0 < x ≤ 5

B 2 < x ≤ 5

C 0 < x ≤ 4

D 0 < x < 2


Page 4

5 In the quadratic equation xầ − x + c І 0, c represents an unknown constant. If x І −3 is one of the solutions to this equation, what is the value of c?

Record your answer and fill in the bubbles on your answer document.

6 Which of the following is not a correct description of the graph of the function y І −2x − 7?

A The graph of the function contains the point (−2, −3), and when the value of x increases by 1 unit, the value of y decreases by 2 units.

B The graph of the function contains the points (−1, −5), (2, −11), and (4, −15).

C The graph of the function is a line that passes through the point (0, −7) with a slope of −2.

D The graph of the function contains the points (0, −7), (1, −9), and (3, −1).

7 If (2k, k) and (3k, 4k) are two points on the graph of a line and k is not equal to 0, what is the slope of the line?

A 3

B 3k

C 1

3

D Not here


Page 5

8 The amount an appliance repairman charges for each job is represented by the function t І 50h + 35, where h represents the number of hours he spent on the job and t represents the total amount he charges in dollars for the job. The repairman plans to change the amount he charges for each job. The amount he plans to charge is represented by the function t І 50h + 45. What will be the effect of this change on the amount he charges for each job?

A The total amount he charges for each job will increase by $10.

B The total amount he charges for each job will decrease by $10.

C The amount he charges per hour will increase by $10.

D The amount he charges per hour will decrease by $10.

9 The sum of the perimeters of two different squares is 32 centimeters, and the difference between their perimeters is 8 centimeters. If x represents the side length of the larger square and y represents the side length of the smaller square, which of the following systems of equations could be used to find the dimensions of the squares?

A x + y І 32 x − y І 8

B 4x + 4y І 32 4x − 4y І 8

C 2x + 2y І 32 2y − 2x І 8

D 4x + 2y І 32 4x − 2y І 8


Page 6

10 Some values for two linear equations are shown in the tables below.

x

Equation 1

y

5−711

2−4

5−1 −1

x

Equation 2

y

5−3

01

11−13−4−1

What is the solution to the system of equations represented by these tables?

A (2, 3)

B (3, 5)

C (−1, 1)

D (5, 11)


Page 7

11 The graph of a quadratic function is shown below.

−5

−4

−6

−7

−8

−9

−3

−2

−1

1

2

3

4

5

6

7

8

9

−1 1−2−3−4−5−6−7−8−9 2 3 4 5 6 7 8 9

y

x

Which statement about this graph is not true?

A The graph has a y-intercept at (0, 8).

B The graph has a maximum point at (Ѝ1, 9).

C The graph has an x-intercept at (2, 0).

D The graph has the y-axis as a line of symmetry.


Page 8

12 The graph of a quadratic function is shown below.

−5

−4

−6

−7

−8

−9

−10

−3

−2

−1

1

2

3

4

5

6

7

8

9

10

−1 1−2−3−4−5−6−7−8−9−10 2 3 4 5 6 7 8 9 10

y

x

What is the best estimate of the positive value of x for which this function equals 8?

A 2

B 4

C 13

D 7

13 A population of 1500 deer decreases by 1.5% per year. At the end of 10 years, there will be approximately 1290 deer in the population. Which function can be used to determine the number of deer, y, in this population at the end of t years?

A y t= −1500 1 0 015( . )

B y t= 1500 0 015( . )

C y t= +1500 1 0 015( . )

D y t= 1500 1 5( . )

Item Number

Reporting Category

Readiness or Supporting

Content Student Expectation

Correct Answer

1 1 Supporting A.1(A) C

2 1 Supporting A.1(B) C

3 1 Readiness A.1(E) A

4 2 Readiness A.2(B) A

5 2 Readiness A.4(A) –12

6 3 Readiness A.5(C) D

7 3 Supporting A.6(A) A

8 3 Readiness A.6(F) A

9 4 Supporting A.8(A) B

10 4 Readiness A.8(B) D

11 5 Readiness A.9(D) D

12 5 Readiness A.10(A) D

13 5 Supporting A.11(C) A

For more information about the new STAAR assessments, go to www.tea.state.tx.us/student.assessment/staar/.

STAAR Algebra I 2011 ReleaseAnswer Key

Page 9

The 5 E Learning Cycle Model

Engage Objects, events, or questions are used to engage students. Connections are

made between what students know and can do.

Explore Objects and phenomena are explored through hands-on activities, with

guidance.

Explain

Students explain their understanding of concepts and processes. New

concepts and skills are introduced as conceptual clarity and cohesion are

sought.

Elaboration Activities allow students to apply concepts in contexts, and build on or

extend understanding and skill.

Evaluation Students assess their knowledge, skills, and abilities. Activities permit

evaluation of student development and lesson effectiveness.

Engage:

Learner Teacher

calls up prior knowledge poses problems

has an interest asks questions

experiences doubt or disequilibrium reveals discrepancies

has a question(s) causes disequilibrium or doubt

identifies problems to solve, decisions to be

made, conflicts to be resolved

assess prior knowledge

writes questions, problems, etc.

develops a need to know

self reflects and evaluates

Explore:

Learner Teacher

hypothesizes and predicts questions and probes

explores resources and materials models when needed

designs and plans makes open suggestions

collects data provides resources

builds models provides feedback

seeks possibilities assesses understandings and processes

self reflects and evaluates

Explain:

Learner Teacher

clarifies understandings provides feedback

shares understandings for feedback asks questions, poses new problems and

issues

forms generalizations models or suggests possible modes

reflects on plausibility offers alternative explanations

seeks new explanations enhances or clarifies explanations

employs various modes for explanation

(writing, art, etc)

evaluates explanations

Elaborate:

Learner Teacher

applies new knowledge asks questions

solves problems provides feedback

makes decisions provides resources

performs new related tasks makes open suggestions

resolves conflicts models when necessary

plans and carries out new project

asks new questions

seeks further clarification

Evaluate:

Learner Teacher

self-assess their own learning and

understanding of new concepts

evaluates effectiveness of the instruction

provide feedback to the teacher on lesson

effectiveness

assesses student learning and understanding

reflect with adults and their peers uses information about student learning to

guide subsequent instruction

communicate, in a variety of ways (e.g.

journals, reporting drawing, graphing,

charting) their level of understanding of

concepts that t hey have developed to date

asks open-ended questions to examine

students’ thinking

create and use quality indicators to assess

their own work

employs a rubric on which to give students

feedback on their learning

report and celebrate their strengths and

identify what they'd like to improve upon

Source: http://faculty.mwsu.edu/west/maryann.coe/coe/inquire/inquiry.htm

http://faculty.mwsu.edu/west/maryann.coe/coe/inquire/inquiry.htm

B15 Warm-up: Perimeters with Polynomials 1. Find the perimeter. 5c + 2 9c − 10 2. The perimeter is 13x + 20. Find the missing side.

4x

2x − 1

x + 3

x

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