July 2006 Public Schools of North Carolina State Board of Education Department of Public Instruction Secondary Education Division Mathematics/Science Section

State Board of Education Howard N. Lee Chairman :: Raleigh

Jane P. Norwood Vice Chair :: Charlotte

Kathy A. Taft Greenville

Michelle Howard-Vital Wilmington

Edgar D. Murphy Durham

Shirley E. Harris Troy

Melissa E. Bartlett Mooresville

Robert “Tom” Speed Boone

Wayne McDevitt Asheville

John A. Tate III Charlotte

Beverly Perdue Lieutenant Governor :: New Bern

Richard Moore State Treasurer :: Kittrell

Patricia Willoughby Raleigh

NC Department of Public Instruction June St. Clair Atkinson, ED.D., State Superintendent

301 North Wilmington Street * Raleigh, North Carolina 27601-2825 * Website: www.ncpublicschools.org

Resources for Algebra

Contents

Introduction

Goals and Objectives (G-1 – G-43) Number and Operations 1

Geometry and Measurement 20

Data Analysis and Probability 22

Algebra 29

Week by Week Essentials (W-1 – W-80) Writing Ideas odd #

Connections to the World odd #

WORDS odd #

Teacher to Teacher odd #

Calculator Tips odd #

Warm Ups even #

Challenges even #

Answer Key 73

Extra Essentials (E-1 – E-47) Curriculum Resource Organizer, Pacing Guides 1

Accommodations, Assessments, and Rubrics 8

K-8 Curriculum and NCTM Standards 14

Game Descriptions 19

More Warm Ups and Challenges 22

Answer Key 45

Activities (A-1 – A-46)

Blackline Masters (B-1 – B-134)

Introduction

Algebra is a universal theme that runs through all of mathematics and is important in nearly

every aspect of the workplace. Algebraic thinking is a necessity for the intelligent consumer of

goods and information.

Algebraic ideas continue to be a strand that runs through the K-8 North Carolina Mathematics

Standard Course of Study (see E-14). In 1991 the North Carolina State Board of Education

designated Algebra 1 as one of three units of mathematics required for high school graduation.

The Resources for Algebra document was originally offered as a culmination to state-wide staff

development in algebra following the release of NCTM’s Principles and Standards for School

Mathematics (2000) and an opportunity for teachers to examine the 1992 Algebra 1 Standard

Course of Study more critically. With the adoption of a new 9-12 North Carolina Mathematics

Standard Course Study for the fall of 2006, it was imperative that the Resources for Algebra

document reflect changes in that curriculum.

Resources for Algebra presents Algebra 1, wherever possible, as calculator-assisted and

activity- or application-centered. Teachers should feel free to modify, reorder, and/or rewrite

any or all of the ideas and activities. This is a document that teachers can build upon and remake

to fit their classroom. It is a good model for a staff development activity within a school or

school district. Using a similar format, teachers can create their own entries and share their

resources in a similar document.

Problems and activities are correlated with the Algebra 1 Standard Course of Study in the

Goals and Objectives section (G-1 – G-43).

Week by Week Essentials (W-1 – W-72) provide seven resources in a weekly format. Although

they are presented in rough chronological order, teachers should review the entire section before

jumping in. The second page of each Week is in larger type so that it can be copied for use as a

transparency.

• Writing Ideas provides an algebra prompt. Writing in the context of algebra requires

students to organize their thoughts and to present them in a coherent fashion. It allows

students the opportunity to reflect, persuade, and/or report mathematics for themselves as

well as for others.

• Connections to the World identifies situations or problems where algebra is an important

tool in their investigation and explanation.

• The Words are… is intended to highlight and encourage clear, specific mathematics

vocabulary in the classroom and among colleagues.

• Teacher to Teacher is ideas from colleagues about organization, management,

assessment, curriculum, standards, projects, and grading.

• Calculator Tips tries to identify the calculator routines most likely to be used in an

Algebra 1 classroom. Since these were written from a TI-83/84 perspective, teachers

who use other models or brands should construct similar routines for their calculators.

• Warm Ups are problems that review a recently covered concept or skill or prior

knowledge/prerequisite skills and can be solved by most students in ten minutes or less.

(See W-73 – W-80 for answers.)

• Challenges are problems for students to do outside of class and extend concepts or skills

students have used or are experienced with.

Extra Essentials (E-1 – E-44) includes blackline masters for teachers (curriculum organizer,

pacing guides, rubrics, etc.), K-8 algebra, NCTM Standards, games, TI-83/84 calculator tips, and

additional Warm Ups and Challenge problems. (See E-45 – E-47 for answers.)

The Activities (A-1 – A-46) are detailed layouts (concepts and competencies, goal, materials,

description, and procedure) for classroom exercises which usually extend for an hour or more.

Although they are identified in the Goals and Objectives by objective, the Activities, like many

other entries, address several objectives. Many of the supporting materials for the Activities,

Goals and Objectives, and Week by Week Essentials are provided in the Blackline Masters

section (B-1 – B-134).

The principle contributors to the writing, editing, and revision of ideas and activities in this third

edition of Resources for Algebra are Carmella Fair, Beverly Ghesquiere, and Debra Kinsey.

Please feel free to forward any comments and/or suggestions to:

Everly Broadway

Mathematics/Science

Section Chief

ebroadway@dpi.state.nc.us

Carmella Fair

Mathematics Consultant

cfair@dpi.state.nc.us

Beverly Ghesquiere

Mathematics Consultant

bghesqui@dpi.state.nc.us

Debra Kinsey

Mathematics Consultant

dkinsey@dpi.state.nc.us

Updates and corrections will be posted on the NCDPI mathematics web page,

http://community.learnnc.org/dpi/math.

G-0

Number and Operations Geometry and Measurement Data Analysis and Probability Algebra

G-1

Number and Operations The learner will perform operations with numbers and expressions to solve problems. 1.01 Write equivalent forms of algebraic expressions to

solve problems. a) Apply the laws of exponents. b) Operate with polynomials. c) Factor polynomials.

A. Play Krypto (B-18) with students. A deck of Krypto cards is described as follows: three of each number 1-10; two of each number 11-17; one of each number 18-25. Randomly select five cards. All five of these numbers will be combined, using operations (including exponents and roots), to make an expression equal to your target number, a sixth card chosen randomly. With multiple sets of cards, groups of students can play. Example: Suppose 23 is the target number and 7,8,4,3, and 16 are the numbers to work with.

)78()3416(23 !•++=

or

)416(3823 ÷!•=7

B. Roll seven dice (five WHITE, one RED, and one GREEN). The outcomes of the WHITE dice must combine correctly, using any operations, to equal 10 • GREEN + RED. Example: GREEN = 4, RED = 6, WHITES = 6, 4, 4, 3, 1 One solution is 14)34(646 •++•=

C. Four in a Row

Students will need game boards (B-22), markers of two different shapes or colors, and two paper clips. Play begins by the first player placing the two paper clips on any pair of factors along the bottom edge of the game board. The player then places a marker on the square which is the product of the two factors. The next player is allowed to move exactly ONE clip and cover the square which is the product of the two indicted factors. (Both clips can be placed on the same factor to square that factor.) Play alternates until someone gets four markers in a row horizontally, vertically, or diagonally. The teacher may want to demonstrate the game on the overhead with the class before students play one another.

G-2

D. In early April, use the Income Tax Rate Schedules (B-69) to have students calculate

taxes.

E. Develop or have the students develop flowcharts to simplify expressions.

F. Investigate advertisements that involve percent and use the information to create

problems. For instance, a car dealer offers two discounting methods. (1) Take $2000 off of the original price, then 10% more, or (2) take 10% off of the original price, then $2000 more. Which method would give the lower price to the buyer? Explain algebraically which method is best.

G-3

G. Collect five or six articles from the newspaper that use percentages. Attach each article inside a file folder and have students write problems related to the articles. Place those in the folder. In cooperative groups, have the students work the file folder problems.

H. Have students in cooperative groups make a list of as many ways as possible to use percentages. Have groups write a problem for each use. Let students report their work to the whole class. Compile the problems and use as homework, starter problems, quiz items, or test items.

I. Determine several ways to raise a number to a power using the calculator.

J. Race to the Top (B-19, 20, 21)

Enter five numbers in the triangles along the base of the triangle. To fill in space in the row above, carry out a teacher-specified operation on the numbers in the two spaces immediately below. See the example below where addition of integers is the operation.

K. Lining Up Dominoes (A-1)

Students will make a train of dominoes by successfully simplifying an expression. Blank domino sheets (B-80) can be made available so that students can create versions of the game that practice various operational and algebraic skills throughout the year, such as evaluating algebraic expressions, simplifying real number expressions, raising a number to a power, simplifying radical expressions, multiplying binomials which contain roots, solving a variety of equations and inequalities, and operating with polynomials.

L. Relays (B-23, 24)

Students will work in teams. Individually team members will complete a problem share the result with the rest of the team so that a team task can be finished. The format can also be used to address evaluating algebraic expressions, simplifying real number expressions, raising a number to a power, simplifying radical expressions, multiplying binomials which contain roots, solving a variety of

equations and inequalities, and operating with polynomials.

M. Use area models to explain binomial multiplication. B-57, B-58, B-59, B-60, B-61, B-62.

N. Have students to compare which is largest? 1004 10003 100002

G-4

O. Patterns with Exponents (B-25, 26) A problem set that explores patterns generated when numbers are exponentially increased.

P. I Have … Who Has …(A-3)

Students will listen, perform operations, and respond when appropriate in a round-robin format. Students will need to be able to complete operations with integers using paper and pencil. Students will use the format to create their own versions of the activity. The format can also be used to address evaluating algebraic expressions, simplifying real number expressions, raising a number to a power, simplifying radical expressions, multiplying binomials that contain roots, solving a variety of equations and inequalities, and operating with polynomials.

Q. Play match game with cards matching the expansion or answer with the number raised to

a power.

R. How Do They Fit? (A-11) Students will assemble a 33! array of puzzle pieces so that adjacent sides match

mathematically. Students will be expected to create their own puzzles and have the teacher share those with the class throughout the remainder of the school year. Well constructed and edited student puzzles will provide the teacher a pool of materials to use thereafter. The format can also be used to address evaluating algebraic expressions, simplifying real number expressions, raising a number to a power, simplifying radical expressions, multiplying binomials which contain roots, solving a variety of equations and inequalities, and operating polynomials.

S. Scientifico (A-5)

Students practice translating scientific notation numbers into standard notation. Students take turns rolling three dice and constructing a number in scientific notation. Ex. 3, 6, 4 can be written 106.3 !

4. After recording this number on the recording chart, the student

places a marker in the proper place on the game board. The student who can make three numbers in a row, column, or diagonal is the winner.

G-5

T. Use base ten blocks to show the students that the square root is one “side” of a square.

Perfect squares makes a square. You can approximate the square root by finding a “close” square.

U. Getting to the Root of the Number (A-35) Working in pairs, students will use base-10 blocks to build incomplete squares that represent an approximate value for a specified irrational number.

V. Bull’s Eye (B-29) The game is played with two or more people in which each player tries to reach a specified goal number with the least number of rolls of a pair of dice. Decide who goes first. If there are more than two people playing, proceed in a clockwise manner. The teacher assigns the goal number. On each student’s turn the student will: roll the dice and compete the sum (difference, product, quotient, powers, roots) of the two numbers, either add or subtract that result from his/her cumulative total, and record the proceeds on the score sheet. The winner is the person who reaches the goal number with the fewest rolls of the dice. If no one reaches the goal number after 16 rounds, the winner is the student who is closest to the goal number. Calculators may be reserved to use only when powers and roots are the operations of choice.

G-6

W. Use algebra tiles to demonstrate addition and subtraction of like terms in polynomials.

X. Have students write two monomials whose product is 3420 yx! . Record possible

answers on the board/overhead. Have the students make generalizations about possible

answers.

G-7

Y. Look at the following patterns and find the nth terms in each pattern.

Z. After trying Basketball: With the game on the line … (A-25), try Basketball

Extension 3 (B-17). This activity deals with generating polynomials that model free-throw shooting. The Basketball activities generally connect probability and statistics with algebra.

AA. Model multiplying monomials with chips on the overhead. After modeling

several examples, have students discuss rules they could use to multiply monomials. Ask students to discuss how this rule differs from adding two monomials.

BB. Divide students into pairs. Have pairs design a puzzle to match problems with

monomial · monomial and solution. Students can cut up puzzle, place in an envelope to exchange with another to solve.

CC. Have students write two monomials whose product is 3420 yx! . Record

possible answers on the board/overhead. Have the students make generalizations

about possible answers.

G-8

DD. Multiplication With Algebra Tiles (A-29)

Working in pairs, students will use pairs of binomials as the dimensions of a rectangle. The students will use the algebra tiles to build rectangles of given dimensions and find the area of the rectangle, the product of the binomials.

EE. Use the matrix method to multiply binomials (B-50, 52). Make appropriate connections with multiplying using algebra tiles.

Fill in the space at the top of each column with the terms from one of the binomials. Fill in the space a the left side of the matrix for each row with the terms from the other binomial.

G-9

Take the term from the first row and multiply it with each term at the top of the matrix and place the products in the appropriate spaces in the first row.

Do the same with the term from the second row and place the products in the appropriate spaces in the second row.

In the matrix (table) of the products, add along the diagonal from right to left.

G-10

FF. Polynomial Four in a Row (B-55)

Students will need game boards, markers of two different shapes or colors, and two paper clips. Play begins by the first player placing the two paper clips on any pair of factors along the bottom edge of the game board. The player then places a marker on the square which is the product of the two factors. The next player is allowed to move exactly ONE clip and cover the square which is the product of the two indicated factors. (Both clips can be placed on the same factor to square that factor.) Play alternates until someone gets four markers in a row, horizontally, vertically, or diagonally. The teacher may want to demonstrate the game on the overhead with the class before students play one another. A blank game board for Four in a Row is provided (B-56) so that teachers can give students the opportunity to create their own versions and address specifically other objectives in Algebra.

GG. Before using algebra tiles, demonstrate binomial multiplication using an area model. See B-57, 58, 59, 60, 61, 62.

HH. Operating With Binomials (A-33)

Students will fill in the entries for Y1 and Y2 with binomials and , using the calculators, determine and record the graphs of the products of binomials. Students are expected to identify the solutions (x-intercepts) of linear and quadratic equations for each graph in the matrix. Students can use a similar process to explore the sums, differences, products, and quotients of varying degrees of polynomials.

II. Use the matrix (table) method to multiply polynomials. Fill in the space at the top of each column with the terms from one of the polynomials. Fill in the space at the left side of the matrix for each row with the terms from the other polynomial.

Take the term from the first row and multiply it with each term at the top of the matrix and place the products in the appropriate spaces in the first row.

G-11

Do the same with the term from the second row and place the products in the Appropriate spaces in the second row.

In the matrix (table of products), add along the diagonals from right to left.

G-12

JJ. The Month of Algebra (A-27)

In pairs, students will select several sets of dates in the month of Algebra and complete computations according to teacher directions. By replacing the numbers in a set with appropriate variable expressions and repeating the computations, students will be able to justify algebraically the pattern.

KK. Algebra Uno

Use sets of 100 index cards marked like the layout (B-63) of suggested cards (include four wild cards). The game is for three or four players and proceeds much like regular “Uno”. In order to lay down a card, the player’s card must show a term that has a common factor, other than one, card facing up on the discard pile. The player who goes out first wins. A more challenging version could include polynomials on the cards.

LL. Use the matrix (table) method to factor the difference of two squares (B-51, 52). MM. Have students reverse the multiplication process to find two binomials that multiply to

give these answers. (a) x2-25, (b) a2

-16, (c) 4y2-16. What pattern do they notice? Have

the students write their rule for finding two binomials that have a product called the difference f two squares. Discuss how factoring is the process of reversing multiplication.

NN. Demonstrate a2

– b2 = (a + b)(a – b). See B-64

Give directions orally. Cut a square and label the sides a. What is the area of this square? Draw a smaller square in a corner of the first square and label its sides b. What is the area of this second square?

G-13

Cut off the smaller square from the corner of the larger square. What is the area of this figure?

Cut the figure as shown. Rearrange these two new pieces so that they form a rectangle.

What is the area of this new rectangle? Have the students discuss the length and width of the rectangle. Ask the students to write an analysis of how this demonstrates the difference of two squares.

G-14

OO. Use the matrix (table) method for factoring a quadratic trinomial (B-51, 52)

For a quadratic expression of the form ax2 + bx + c (b can equal zero), place the ax

2 term

in the upper location (first row, first column) and the c term in the lower right (second row, second column).

What are two possible factors of ax

2? Place the factors in the locations along the boundary for the first row and the first column.

G-15

What are two possible factors of c? Place the factors in the locations along the boundary for the second row and the second column.

Multiply the binomials that re now in the boundary. Do the two diagonal products add up to box? If not, go back and adjust the factors for c and/or ax

2?

PP. Factoring Trinomials with Algebra Tiles (A-31) Working in pairs, students will select algebra tiles corresponding to the terms of a given quadratic trinomial. The students will create a rectangular arrangement with the tiles and identify the dimensions of the rectangle. Each dimension will be one of the algebraic factors of the original trinomial.

QQ. Have students review types of factoring by writing about the relationship of factoring to

multiplication. Use an example from each type of factoring that we have studied. Explain the relationship between multiplying the factors to obtain a polynomial and factoring the polynomial into the factors.

G-16

RR. Fill in each square with a digit 0 – 9. You may use each digit only once (B-65, 66). Similar puzzles are available commercially.

TT. The difference of two squares can be used with mental math. To multiply 24 and 26, the product can be found using 6241625)125)(125( =!=+! . Have the

students try 1317 • . Using your overhead calculator, enter y = x2. Set your table

generator to begin at ten and increment by five ( or students can do this with individual calculators). Show the table of squares to the class. Have each student write a multiplication problem that would use the difference of two squares and one of the squares in the table. Take up the cards and form two teams. Read a problem and call on a person to give the product using only mental arithmetic. Each correct problem is worth one point. UU. Have the students investigate the relationship of factoring the difference of two

squares and the graph of a quadratic function.

G-17

VV. Make the Message (B-67) Factor quadratic and rational expressions to create a message. Using the same format, create similar messages, copy, and assign for homework, review, or extra credit.

WW. Use the game formats outlined on E-19, E-20, E-21 to practice skills and concepts with real numbers and to practice skills and concepts with polynomials. Encourage students to create their own versions of popular (legal, of course) games. XX. Use the puzzle formats of How Do They Fit? (A-11), Lining Up Dominoes

(A-1), “I Have … Who Has …” (A-3), and Relays (B-23, 24) to create puzzles for student use. Whenever possible, let students create the puzzles. These

formats can also be used to address evaluating algebraic expressions, simplifying real number expressions, raising a number to a power, simplifying radical expressions, multiplying binomials which contain roots, solving a variety of equations and inequalities, and operating with polynomials.

YY. Number Crunching With Ease! (B-27)

ZZ. Calculator Tips (W-1, 3, 5, 9, 11) AAA. Connections(W-1, 5, 7, 9, 19, 33, 35, 41, 53, 57, 63) BBB. Warm Ups (W-2, 4, 6, 8, 10, 12, 14, 16, 22, 26, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72) CCC. Challenges (W-2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 32, 38, 42, 44, 48, 50, 52, 58, 64) DDD. Extra Essentials (E-22, 23, 24, 25, 26, 28, 30, 31, 32, 33, 35, 36, 37, 38, 39, 40, 41, 42, 44)

G-18

1.02 Use formulas and algebraic expressions, including iterative and recursive forms, to model and solve problems.

A. What is the length of a rectangle that has an area of 12.1 cm2 and a width of 5 cm?

B. The drip rate for intravenous fluids can be determined by using the formula T

VD

4=

where D is the drip rate in drops per minute, V is the prescribed volume of fluid in cubic centimeters, and T is the prescribed amount of time for the fluid to be given to the patient in hours. If a doctor orders 1000 cc of fluid be given to a burn victim in 4 hours, what should the hospital staff adjust the IV drip rate to?

C. The following sequences are defined using recursion formulas. Write the first five terms

of each sequence.

a. 51=a and 7

1+= !nn

aa for 3!n

b. 41=a and

13 !=

nnaa for 2!n

c. 61=a and 122

1+= !nn

aa for 1!n

D. A deposit of $10,000 is made in an account that earns 8% interest compounded quarterly.

The balance in the account after n quarters is given by the sequence

,...3,2,1,4

08.01000,10 =!

"

#$%

&+= na

n

n Find the balance in the account after ten years.

Round to the nearest cent.

E. You buy a new car for $28,000. At the end of n years, the value of the car is given by the

sequence .4

3000,28

n

na !

"

#$%

&= Find

3a and write a sentence explaining what this value

represents. Describe the nth term of the sequence in terms of the value of your car at the end of each year.

F. You are considering two job offers. The first company will start you at $22,000 annually

and guarantee a raise of $2800 per year. The second company will start you at a higher salary, $28,000 annually, but will only guarantee a raise of $1100 per year. Find the total salary that each company will pay you over a fifteen-year period. Which company will pay the greater salary?

G. A theater has 20 seats in the first row, 22 seats in the second row, increasing by 2 seats

per row for a total of 30 rows. How many seats are there in the theater?

H. Teachers’ earnings n years after 1989 can be described by 060,291472 += nan

.

According to this model, what will teachers earn in 2089? Describe two possible situations that would make this predicted salary inaccurate.

G-19

1.03 Model and solve problems using direct variation.

A. Write an equation that expresses each relationship. Use k as the constant of variation. a. c varies directly as p b. r is directly proportional to the square of t c. w is directly proportional to the cube of y d. Charles’s Law states that the volume V of an enclosed ideal gas at a constant

pressure varies directly as the absolute temperature, T.

B. Determine the constant of variation for each stated condition. a. c varies directly as p, and c = 45 when p = 3 b. r varies directly as t2, and r = 80 when t = 4 c. w varies directly as y3, and w = 16 when y = 2

C. y varies directly as x. y = 35 when x = 5. Find y when x = 24. D. Does the following table represent a direct-variation relationship? Why or why not?

E. A person’s wages, w, vary directly as the number of hours worked, h. a. Write an equation that expresses this relationship. b. For a 40-hour work week, Debra earned $1600. Substitute this information

into the equation from part (a) and find k, the constant of variation. Then write the equation that describes Debra’s wages in terms of the number of hours she works.

c. Use the equation from part (b) to find Debra’s wages for 25 hours of work.

F. The cost of an airplane ticket varies directly as the number of miles in the trip. If a 3000-mile trip costs $525, what is the cost of a 650-mile trip?

G. The distance an object falls varies directly as the square of the time it has been

falling. If an object falls 150 feet in 3 seconds, how far will it fall in 5 seconds?

H. The power P (in watts) produced by a windmill is proportional to the cube of the wind speed v (in mph). If a wind of 10 mph generates 25 watts of power, how much power is generated by winds of 25, 35, and 65 mph? Make a table and explain the pattern.

I. In your own words, explain how to solve a direct variation problem?

x 40 20 16 8

y 2 4 5 10

G-20

Geometry and Measurement The learner will describe geometric figures in the coordinate plane algebraically. 2.01 Find the lengths and midpoints of segments to solve

problems.

A. Highways and the mile markers on the sides of highways can represent lines and segments. Therefore, the distance between mile markers is representative of the length of a segment. Using this information, answer the following question: If Mary sees mile marker 135 as she enters the highway and mile marker 98 when she exits, how far did Mary travel?

B. Use the figure at the right to answer the following.

o Find the length of CE .

o Find the midpoint of BE .

o What is the coordinate of the midpoint of AC ?

C. U is the midpoint of ST , V is the midpoint of SU , W is the midpoint of SV , X is the

midpoint of VW , Y is the midpoint of W�

, and Z is the midpoint of TV . If UV =

32, find ZY.

D. S is the midpoint of both �Q and PR . If 14=SP and 287 != xPR , find x , RP ,

and RS .

E. If L is the midpoint of GM and 183 += yGL and 69 != yML , what are the

measures of LG and GM ?

F. If a triangle has vertices (5, -1), (-2, -2), and (3, 8), find the perimeter of the triangle

and the midpoint of each side of the triangle.

G-21

2.02 Use the parallelism or perpendicularity of lines and segments to solve problems.

A. The following picture is a screen from the TI-83/84 showing Quadrants I and II, a

fixed line with two points marked, and line Y1 = AX + B with A = 0 and B = 5.

Look at the screen and answer the following questions:

o How would you change the value of A to make the two lines the same?

o How would you change the value of A to make the two lines parallel?

o How would you change the value of A to make the two lines perpendicular?

B. If Y1 = 5X + 10, write equations for Y2 and Y3 so that their graphs are parallel and

perpendicular, respectively, to the graph of Y1. Graph the lines. C. Write the equations of lines parallel and perpendicular to 3x + 7y = 16. D. Exploring Perpendiculars Draw several segments on graph paper and give each

student a copy (B-43). Have the students find the slope of each segment and record. Have the students fold the graph paper so that the fold is perpendicular to the original line. (This can be done by matching the endpoints and folding.) Have the students find the slopes of the folds and compare the results with the slopes of the corresponding segments.

G-22

Data Analysis and Probability The learner will collect, organize, and interpret data with matrices and linear models to solve problems. 3.01 Use matrices to display and interpret data.

A. National League Matrix (1997 season through July 13, from USA Today)

The rows (top to bottom) and columns (left to right) are arranged alphabetically by teams: Atlanta, Chicago, Cincinnati, Colorado, Florida, Houston, Los Angeles, Montreal, New York, Philadelphia, Pittsburgh, St. Louis, San Diego, San Francisco. A team’s number of victories against each opponent can be found by reading across. The number of losses can be found by reading down. What is Atlanta’s overall record against the other National League teams?

At Ch Ci Co Fl Ho LA Mo NY Ph Pi St SD SF

At 0 5 5 3 1 3 2 7 2 8 4 5 5 5 Ch 0 0 4 0 0 2 2 2 3 5 5 2 4 3 Ci 0 3 0 2 3 4 2 3 2 5 3 3 2 2 Co 3 4 3 0 6 3 1 5 2 3 0 3 4 3 Fl 3 5 3 3 0 4 4 5 3 4 4 2 4 3 Ho 3 6 4 2 3 0 5 1 2 3 3 5 5 1 LA 3 3 3 3 1 3 0 3 3 4 4 3 4 6 Mo 1 6 3 1 3 3 3 0 4 4 3 3 4 4 NY 5 3 5 4 5 2 2 4 0 5 6 5 2 1 Ph 0 3 2 4 2 3 1 2 1 0 3 1 1 1 Pi 3 4 3 4 1 4 1 4 2 4 0 6 3 2 St 1 5 5 3 3 2 6 1 1 3 2 0 4 4 SD 2 1 4 6 1 4 6 1 3 4 2 2 0 2 SF 4 3 3 7 5 4 4 1 5 4 2 1 2 0

B. Use the NHL standings from the newspaper to create a matrix that reports points (based

on wins and ties), goals scored, and goals allowed. Based on the matrix, how do points correlate with goals scored? Goals allowed? If a team had 25 wins, how many goals would it be expected to score? Allow?

C. An almanac is a great source of data to use in matrix formats. For instance, use the

Average Daily Temperatures in Tourist Cities from the Information Please Almanac to create matrices for North America, Europe, and the Southern Hemisphere. Compare temperature variations among the cities. Compare the data between Europe and the Southern Hemisphere; differentiate between the two sets. Compare data between cities close to the equator and those not. Find similar information about major US cities and report in a matrix. Do the same for NC cities.

G-23

D. Use the matrix which accompanies the NC maps published by the NC Department of Transportation to determine distances between NC cities. Use the matrix to estimate driving distance for a trip to the NC mountains during Leaf Season. One such trip may follow this route for a round trip from Raleigh. Raleigh – Greensboro – Charlotte – (via US 74 and I-26) - Asheville – Boone – Winston-Salem – Raleigh.

E. Create and update a matrix for ACC basketball, through the ACC tournament, similar to

the one for major league baseball.

F. Use the tax tables in the current IRS 1040 Forms and Instructions or NC Individual

Income Tax Forms and Instructions (B-69) tax materials to determine taxes owed for varying amounts of taxable income. Describe how the tax tables are organized (income span per row, relationships among filing categories, how does the tax owed change as the income increases, identify similarities and differences between the state and federal tables, etc.). Compare the tables with the actual tax rate schedules for both forms.

G. Have students bring in prices of a large pizza with one topping, large drink, and extra

topping from three or four favorite pizza restaurants. Set up a 3 • 4 matrix. Here is one example:

H. Have students investigate information from two matrices. For example: Pete investigates interest rates at First Bank and Credit Savings for certificates of deposit (CD) and money market accounts. This information is listed in the 2 • 3 matrix.

What would be the interest for one year on a 48 month CD for option I at First Bank? What would be the total interest for option I at Credit Savings? Now compare this to option I at First Bank? Compare the total interest for option II at both banks. Which investment option would be the best for Pete?

I. Using the matrix for In-Line Skating (B-30), add a row that estimates calories consumed for a 130 pound skater. As speed increases how does the number of calories consumed at each weight change? Do the calories increase at the same rate at every weight? Using the matrix for Personal Fitness (B-31), add a row that estimates calories consumed by a person running 13 minute miles. For a person weighing 130 pounds, how many calories does he consume in each activity?

Italian Delight

Dot’s Pizza House

Round Romans

Large Pizza $8.99 $11.40 $12.20 $8.00 Large Drink .99 .85 .90 .89 Extra Topping 1.30 1.25 1.50 1.30

12 CD 48 CD MM 12 CD 48 CD MM

1st Bank 5.8% 6.8% 4.8% Option I $15,000 $30,000 $6,000

CreditSav 6.5% 6.5% 5.4% Option II $10,000 $25,000 $16,000

G-24

J. Determine the top 15 countries that earned gold, silver, and bronze medals at the 2002 Olympics. Organize a matrix to display the information. Order the data so that there are columns for gold, silver, and bronze medals won. Order the rows according to the country that won the most gold medals at the top of the matrix and the country that won the least gold medals at the bottom. If two or more countries had the same number of gold medals, use the most silver medals, and then bronze if needed. Add a fourth column with the medal totals. The matrix can be expanded by adding a column that shows the most recent census or population estimate of each country. In another column record how many medals were won per thousand people in each country. Analyze and discuss any trends or relationships that you notice in the matrix.

K. Have students set up a matrix of sports data for their favorite team (see B-15 for an

example) on a spreadsheet. Have them enter at least three formulas to calculate averages or totals for the players or the team. How could the matrix (B-15) be organized to better investigate correlations between minutes played and points scored or free throw attempts and rebounds? If a player had 1700 minutes of playing time, how many points should he have been expected to score? If a player collected 250 total rebounds, how many free throws would I have expected him to attempt?

L. Have students set up a matrix of data on the planets of our solar system on a spread sheet.

Enter columns for equatorial diameter, mean solar distance, and orbital period. Use the spreadsheet to create new columns in the matrix which compare planetary diameters with Earth’s and mean solar distance in astronomical units. Look for patterns in the data and potential correlations (mean solar distance and orbital period) by reordering rows according to mean solar distance, equatorial diameter, etc.

M. Connections (W-3, 11, 15, 17, 35, 43, 49, 51, 61, 63, 67, 71)

G-25

3.02 Operate (addition, subtraction, scalar multiplication) with matrices to solve problems.

A. Given the following matrices:

[A] =

!!!

"

#

$$$

%

&

493830

342524

463432

[B] =

!!!

"

#

$$$

%

&

252020

202010

302010

[C] =

!!!

"

#

$$$

%

&

444444

505050

606060

Find:

! [A] + [B] ! 2[A]

! [A] + 2

1[C]

! [C] – [B]

B. The Booster Club at a local high school sells food, candy, drinks, megaphones, hats, and rally towels during home sporting events. The following matrices show the total sales, cost of supplies, and operating expenses for the football, basketball, and baseball seasons during 2004 and 2005. Find the total sales, costs of supplies, and operating expenses for the Booster Club.

Football Sales Cost of Supplies Operating Expenses

2004 $19,208 $ 8,211 $2,609

2005 $21,553 $10,420 $2,511

Baseball Sales Cost of Supplies Operating Expenses

2004 $ 9,428 $6,201 $1,807

2005 $11,206 $7,005 $1,775

Basketball Sales Cost of Supplies Operating Expenses

2004 $13,174 $6,426 $2,540

2005 $12,845 $5,315 $2,187

G-26

3.03 Create linear models for sets of data to solve problems. a) Interpret constants and coefficients in the context of the data. b) Check the model for goodness-of-fit and use the model, where appropriate, to draw conclusions or make predictions.

A. Have students graph data from an almanac or other source. Teacher to Teacher (W-51)

identifies several sources for useful data, see B-30 and B-31 for some examples. Write an equation to model the relationship.

B. Take advantage of data that appear in a newspaper or magazine. Here is an example. In

1986, the 20th Super Bowl was played in New Orleans. The price of a 30-second television commercial for the game was $550,000. In 1993, the 27th Super Bowl was played in Pasadena and the price of a 30-second commercial had risen to $850,000. Write a linear equation that models the change in the price of television commercials for this sporting event.

C. Scoring and Winning (A-7) Have students gather data from the NFL (or similar data from the NBA, NHL, MLB, or local minor leagues) to create scatter plots and find lines of best fit. Students will discuss the characteristics of those lines and make predictions.

D. Find the flight times and direct air distances from the larger North Carolina airport

(Charlotte, RDU, Piedmont Triad, etc.) near you. Use airline tables and be aware of time zone considerations as flight times and air distances are compiled. Plot the data, find the line (or other curve) of best fit. Discuss the meaning of slope and x- and y-intercepts in the context of the data.

E. The Good Estimator (A-9) Individual students or groups will estimate ten distances in the classroom or other parts of the school. Then the students will measure the same distances and compare their estimates with the “real” measures.

F. The Wave (A-13) In a whole class setting, increasingly larger groups of students will perform the “wave”. The students will collect and interpret data, determine a linear function of the time to complete the “wave” dependent upon the number of students participating, and use the linear function to make predictions.

G. Patterns in Perimeter (B-32, 33, 34, 35) Students will use perimeter to generate data, find the algebraic expression for the linear pattern, and graph the data that are generated. Although it is expected that students will work individually or in pairs on the pattern sheets, overhead versions of two of the problems are provided for whole class discussion. Using pattern blocks to build the figures is particularly helpful for many students’ understanding.

G-27

H. After trying Basketball: With the game on the line ... (A-25), try Basketball Extension

1 (B-13). The Basketball activities generally connect probability and statistics with algebra.

I. Used Cars Have students ask 10-12 teachers the ages and mileages for their cars. Have the class draw a scatter plot and a line of best-fit. Ask students to estimate a slope and interpret the slope in the context of the situation.

J. Give the equation y = 3.5x + 25, have students create a situation modeled by the

equation. For instance, let the equation describe the relationship between the number of martial arts classes (x) and their cost (y) at the YMCA. What does the y-intercept represent? What does the slope represent?

K. Connecting Units of Measure (A-21) Students will measure several objects in the classroom using both centimeters and inches. Students will plot corresponding pairs of measurements on a graph and interpret the information to determine the relationship between centimeters and inches.

L. Making Sense of Slope The handouts B-44, 45, 46 contain five problems that are related to graphing and slopes of lines. Each problem contains a graph or chart that the students are to analyze and answer questions about. Work problem #1 as a class to give students a good idea of what is expected. It would be appropriate for students to work in groups of three or four to complete the assignment.

M. Patterns in Area and Volume (B-36, 37, 38) Students will use surface area and volume to generate data, find the algebraic expression for the pattern, and graph the data that are generated. Using blocks to build the figures is particularly helpful for many students’ understanding. There are linear, exponential, and quadratic patterns among the sequences. Only after students have described the patterns in arithmetic terms should students use the curving fitting utilities on their calculators. The method of finite differences can be used to identify quadratic patterns. (With Algebra II and Technical Math 2 students, follow up finite differences by setting up the matrix equation and solve it to derive the quadratic expression.)

N. After students have created an equation based on data, have them use it to make

predictions. Here is a continuation of the example from 3.03B. What was the price of a 30-second television commercial in 2000? What was it actually? Explain any differences. Do the same for the first Super Bowl in 1967.

O. According to postal rate information, B-47, what will be the cost of a stamp in the year

2010? In what year do you predict stamps will cost 50¢?

P. It’s All Downhill From Here (A-19) Create a linear model to represent rolling a ball down a ramp. Use the model to predict how far the ball will roll down a ramp of given height.

G-28

Q. Students can use the internet to find data on housing costs to graph. Have students look

up data at www.realtor.com or go through their local chamber of commerce web site. They can make a table for ten houses listing the square footage of a house and the cost of a house. Using a calculator, students can draw a scatter plot of the data and find the line of best fit. Have students define and compare the slope and y-intercept for different cities or different locations in one large city. (Most often the slope would be the average cost per square foot and the y-intercept is the average price of the lot.)

R. Give students a table of data. Ask them to determine the slope and interpret. Example:

Hours Worked Wages Earned

5 28.75

10 57.50

15 86.25

20 115.00

S. How Do You Measure Up? (A-15) Students will measure their forearm and height. The collected data will be organized, interpreted, and used to make predictions.

T. What Shape Are You? (A-17) In groups, students will measure their height and arm

span. They will graph their height vs arm span to investigate data and make predictions.

U. Toothpick Triangles (A-23) This activity will have students building sets of triangles with toothpicks. Students will use the toothpicks to visualize patterns and use the data generated by the pattern to generate a linear relation and make predictions.

V. Creating “Real World” Problems (B-9) Students will create linear equations and make

predictions.

W. Calculator Tips (W-35, 39, 43)

X. Connections (W-49)

Y. Extra Essentials (E-23, 24, 35, 43)

G-29

Algebra The learner will use relations and functions to solve problems. 4.01 Use linear functions or inequalities to model and solve

problems; justify results. a) Solve using tables, graphs, and algebraic properties. b) Interpret constants and coefficients in the context of the problem.

A. Develop an experiment to send a toy car down a ramp on various heights. Clock the time

and determine the height of the ramp. Collect data, make graphs, find the line of best-fit, and interpret the slope and y-intercept in the context of the problem. Make predictions of time traveled based on changing heights of the ramp.

B. The Picture Tells the Linear Story (B-39, 40, 41)

Students investigate families of linear equations. C. Moving on the Graph (B-42)

What do your students know about transforming a graph?

D. Connections problems (W-23, 53) E. Drive Time problem (B-68) F. Used Cars problem

Have students ask 10-12 teachers the age and mileage (odometer reading) for their cars. Have the class determine the line of best-fit, and interpret the slope and y-intercept in the context of the problem.

G. Give the equation y = 3.5x + 25. Have students create a situation modeled by the equation.

For instance, let the equation describe the relationship between the number of martial arts classes (x) and their cost (y) at the recreation center. What does the y-intercept represent? What does the slope represent?

H. Use a relay format to practice solving inequalities. I. Select an inequality (example: x – 5 < 2). Enter each side of the inequality on the calculator

(Y1 = x – 5 and Y2 = 2). Graph, locate the intersection, and identify for which values of x is the left side above (greater) or below (less) the right side.

G-30

J. Placekicker In football, the place kicker can score points two different ways. He can score 3 points with a field goal or 1 point with a point-after-touchdown (PAT). The coach expects his kicker to score at least 50 points during the season. How many field goals and PATs could the kicker score?

K. Select an inequality (example: -2 < x + 7 < 6). Enter each part of the inequality on the

calculator (Y1 = -2, Y2 = x + 7, and Y3 = 6). Graph, locate the intersections, and identify for which values of x is the center part between the other two parts. Solve by hand and compare results with the calculator results.

L. Divide the students into teams. Have the students use any method they wish (including a

calculator) to solve the equation. Keep score. Reward appropriately the team with the most correct solutions. Be sure to divide the teams equitably.

M. Re-solving Equations (B-132, 133)

Students are given several simple linear equations that have been solved incorrectly. Students are expected to identify and explain errors and then solve the equation correctly. Use the blank setup (B-134) to adjust the exercises according to topics being studied.

N. Select a linear inequality. (example: 4x + 2y ! 6) Solve for y in terms of x (y ! -2x + 3).

Enter the corresponding linear equation in the calculator and graph. Identify the region above (greater) or below (less) the line as the solution set. Using the appropriate calculator function, (W-21, W-23) graph and record. Choose several points in the region to check in the original inequality. Have the students use this process with several inequalities.

O. Use algebra tiles to demonstrate solving simple equations using the additive inverse.

Let students use tiles to solve similar problems generated from the teacher or textbook. For example:

G-31

P. Use algebra tiles to demonstrate solving simple equations using the multiplicative inverse.

Let students use tiles to solve similar problems generated from the teacher or textbook. For example:

G-32

G-33

4.02 Graph, factor, and evaluate quadratic functions to solve problems.

A. Have students hand-graph examples of quadratic equations that are in the textbook. Pair

students and ask them to generate a list of characteristics of each equation and quadratics in general.

B. Have students graph 12 != xy and 12!= xy on separate graphs. Ask them to write a

description of how the graphs are alike and how they are different.

C. The Picture Tells the (Quadratic) Story (B-70, 71). Students investigate families of quadratic equations.

D. Have students use and investigate quadratic equations that could represent physical

phenomena. The quadratic equation 5.1309.42

++!= xxy describes in meters the

height of a baseball that is traveling at 30 meters per second and has been hit from a height of 1.5 meters above the ground. (1) Make a table of values, (2) graph the equation, (3) as time increases, describe the height of the ball, and (4) how does the graph differ from a linear equation?

E. Have students graph some quadratic equations which model a situation they may be

familiar with.

F. Max-Min (B-72) Give students three points to graph on graph paper. Have them sketch in a parabola that includes the three points and estimate the maximum or minimum value for the parabola. Have the students share their results. After several examples, give students two points to work with. Again, have students share their results. How many points are needed to determine a parabola?

G. Open Boxes (A-39)

In groups, students will collect and analyze data from the construction of several boxes and estimate the maximum volume. Students will use an algebraic model to determine a maximum volume and compare it with the experimental results.

H. The Maximum Garden (A-41)

Students will use a table to list possible values for the dimensions and area of a garden space. Students will write an equation to graph the width of the garden verses the area. This will be a quadratic function with a maximum value.

G-34

I. Have students relate the solutions of quadratic equations to the graphs of related functions.

(1) Solve by factoring

a. 062

=!+ xx

b. 01282

=+! xx

(2) Now graph on the calculator the related functions and record the graphs. Where

does each function cross the x-axis?

a. �2!+= xxy

b. 1282

+!= xxy

(3) How can the solutions to quadratic equations be found by graphing the related

function? Why?

(4) Find the solutions to the nearest tenth.

a. 0722

=!+ xx

b. 03822

=+! xx

c. �8�

2+= xx

J. Make an overhead of a picture of a dolphin jumping. Have students graph on a graphing

calculator this equation, xxy 48.135.02+!= where x is the horizontal distance of a

dolphin jump and y is the vertical distance of the jump in meters. Ask students to find

how high the dolphin jumps and after how many meters will the dolphin enter the water.

K. Shuttle Launch (A-43): Working in pairs, students will use a pair of quadratic equations to identify the critical points along the flight of the solid rocket boosters (SRB) that are used to launch the space shuttle. Students will connect algebraic ideas (intersection, vertex, x-intercept, and evaluating expressions) with points along the flight path (engine shutdown, maximum altitude, splashdown, and altitude versus elapsed time).

L. Use quadratic equations to represent a commercial situation. The profit of a business can

be described by the equation P = 1.8T2 – 20T +250, where P is the profit in thousands of

dollars and t is the number of years since 1995 (t =0 corresponds to 1995). Describe the profit trend over the last ten years. Use the model to predict the year in which the profit will be double that of 1995.

M. Have students solve equations in factored form with a product of zero from the textbook.

Have them explain (1) why this method works, (2) how you can tell if a solution will be positive or negative, and (3) how you can tell if a solution will be non-integral.

G-35

N. Have the students reverse the processes. Ask them to write a quadratic equation that has (1) two positive solutions, (2) two negative solutions, (3) two solutions that are fractions, and (4) only one solution.

O. Have students discuss why the following is incorrect.

Tanya solves a quadratic equation this way. (2x – 1)(x + 2) = 3

2x – 1 = 3 or x + 2 = 3

x = 2 or x = 1

Does x = 2 check as a solution to the equation? Does x = 1 check as a solution? What is wrong with Tanya’s method?

P. Operating with Binomials (A-33). Students will fill in the entries for Y1 and Y2 with

binomials and, using calculators, determine and record the graphs of the products of the binomials. Students are expected to identify the solutions (x-intercepts) of linear and quadratic equations for each graph in the matrix. Students can use a similar process to explore the sums, differences, and products of varying degrees of polynomials.

Q. Factoring Trinomials with Algebra Tiles (A-31). Working in pairs, students will select

algebra tiles corresponding to the terms of a given quadratic trinomial. The students will create a rectangular arrangement with the tiles and identify the dimensions of the rectangle. Each dimension will be one of the algebraic factors of the original trinomial.

R. Polynomial Four-in-a-Row (B-55). Students will need game boards, markers of two

different shapes or colors, and two paper clips. Play begins by the first player placing the two paper clips on any pair of factors along the bottom edge of the game board. The player then places a marker on the square which is the product of the two factors. The next player is allowed to move exactly ONE clip and cover the square which is the product of the two indicated factors. (Both clips can be placed on the same factor to square that factor.) Play alternates until someone gets four markers in a row, horizontally, vertically, or diagonally. The teacher may want to demonstrate the game on the overhead with the class before students play one another. A blank game board for Four in a Row is provided (B-56) so that teachers can give students the opportunity to create their own versions and address specifically other objectives in algebra.

S. Use the matrix method for factoring quadratic expressions. See 1.01 OO.

T. Have students solve second-degree equations by factoring using problems from their

textbook. Have students explain why the method does or does not always work. Have them write a second-degree equation that cannot be factored.

G-36

U. Give students a table of values for y = x2 – 2x – 8.

x -3 -2 -1 0 1 2 3 4 5

y 7 0 -5 -8 -9 -8 -5 0 7

(1) From the table, find solutions when y = -5, or x2 – 2x – 8 = -5.

(2) Now use factoring to find the solutions.

(3) Solve each of the following using the table, then factoring.

a. x2 – 2x -8 = 7 b. x2 – 2x -8 = -8 c. x2 – 2x -8 = 0

(4) Develop a table with ten ordered pairs using the function y = x2 + 2x – 15. Write four equations you can solve using your table. Solve the four equations by factoring.

V. Use the cover-up method to have students solve equations such as (x + 2)2 = 49. On the

overhead, write (x + 2)2 = 49 and place a square of paper over the term inside of the

parentheses: ( )2 = 49. So (x + 2) = 7 or (x + 2) = -7 which means x = 5 or x = -9.

W. Use an application. Each side of a square patio was increased in length by 5 feet to give an area of 150 square feet. What was the original length of the patio? [If x is the original length, solve for x when (x + 5)

2 = 150.]

X. When given equations like x2 – 3x + 4 = 7, have students enter each side of the equation

in the calculator as Y1 = x2 – 3x + 4 and Y2 = 7. Graph, trace, and zoom to locate the

intersection.

Y. Using the North Carolina population information (B-47) predict the population in 2010. In what year should the population of North Carolina reach ten million? Try the quadratic curve of best fit with the calculator.

Z. Quadratic Functions (A-37) Students will find solutions to quadratic equations using

two different methods, factoring the equation and graphing the related function to find the x-intercepts.

AA.Connections (W-29)

G-37

4.03 Use systems of linear equations or inequalities in two variables to model and solve problems. Solve using tables, graphs, and algebraic properties; justify results.

A. Olympic Swimming (B-48) With the Olympics occurring every two years (alternating summer and winter games), there are several events in which both men and women compete. For instance, there are winning results for men and women in the 400 meter free-style swimming since 1924. Working in pairs, students will select (or be assigned) an event. Using their calculators, they would determine best-fit linear equations for each of the men’s and women’s data. If appropriate, use the equations to determine men’s and women’s performances for 1940 and 1944. (Why were there no results in those years?) Predict the winning results for the next several Olympics. Ask the students to determine, according to their calculations, if the women’s performance will ever equal or exceed the men’s performance in their event. Research as to whether this is likely to happen. At the 2004 Athens Olympics the winning times in the 400 meter free-style events were 223.10 seconds for the men and 245.34 seconds for the women.

B. Have students research the transportation costs for travel between cities. Assume that the

costs identified represent a linear trend. Here is an example from January 11, 2003:

Distance (round trip) Car Air Rail

Raleigh-Charlotte 300 miles $108 $224 $40

Raleigh-New York 1300 miles $468 $169 $147

Determine the distance at which driving a car is less expensive than riding the train. When does it become cheaper to fly rather than drive? For what distance is the train the most expensive mode of travel? Identify some advantages and disadvantages for each mode of transportation. What other variables affect the cost of travel?

C. Select two linear equations and enter them on the calculator. (Example: Y1 = x – 4 and

Y2 = -x + 3). Graph, trace, and identify the intersection. Verify with substitution. Record the graph and coordinates of the intersection. Have students try other pairs of equations and record results. Use a friendly range for best results.

D. During the 1999 baseball season, each team in Major League Baseball played 162 games.

On April 30, the Boston Red Sox had played 22 games and won 11. Meanwhile the New York Yankees had won 14 of their 21 games. By May 31, Boston had played 50 games and won 31. New York had played 48 games and won 28. Create a linear equation in slope-intercept form that describes the trend for each team. Define the slope of your equations with respect to the quantities being discussed. Explain how you created your equations and use them to predict the number of games each team should win by the end of the season. How do your results compare with the actual results for that season? Explain the difference, if any.

G-38

E. Divide students into pairs. Give each pair two systems to graph. Student A graphs the first system at the same time Student B graphs the second system. Have the pair exchange papers to find the solution to the systems from the graph and check the solutions. Each pair works together if difficulties arise.

F. Divide class into cooperative groups. Give each group three graphs of systems of

equations and the tables from a spreadsheet. Have groups write the equations for each system, find the solutions from the graphs, and check the solutions using the equations.

G. Select two linear inequalities and enter the corresponding linear equations on the

calculator. Using the appropriate calculator function (Calculator Tips W-21, 23) graph and identify the intersection. Record the graph. Have students try other pairs of inequalities and record results.

H. Survey students and create problems similar to the one that follows. Mildred makes $3

an hour babysitting and $6 an hour when she works at Wendy’s. Her parents do not want her to work more than 20 hours per week. Mildred would like to earn at least $70 a week. Write a system of inequalities that show the number of hours she could work at each job. Graph the system. Write at least four possible solutions.

I. Use information form local businesses to create problems. Here is an example. The

Twin Theater charges $7 for adult tickets and $4 for children 12 or under. The theater has 470 seats. The manager wants to have a nightly income (two shows) of at least $5000. Write a system of inequalities for the number of children and adult tickets that can be sold. Write at least four possible solutions.

J. Have students write a system of inequalities whose solution set is (1) a triangle, (2) a

trapezoid, (3) a kite, (4) a hexagon.

K. A business will make money if revenues exceed expenses. Discuss with students the break-even point when revenues equal expenses. Have students consider the following hamburger business. The owner pays $20,000 for the franchise and has expenses of $750 per thousand hamburgers. The price of a hamburger is $2.09. Have students write an equation to represent the cost of the business and the equation to represent the revenue. Graph the equations. Ask the students to find the number of hamburgers the business needs to sell to break even. What would be the profit if 100 thousand burgers are sold? Many franchises work this way. Have students talk with local owners and share the information with the class.

L. Use business-type situations that arise at school to create problems. Here is an example.

The Silver Ratio Band wants to talk to the school principal concerning a contract to play for the Valentine dance. The group is considering three possible rates: (1) the band will charge $3 per person; (2) the school will pay the band $50 plus $2.50 per person; (3) the band will rent the fellowship hall at a local church for $125 and charge $4 per ticket. Which method would be best for the group to use? Have students write a summary of their findings.

G-39

M. Ask students to describe the advantages of each method of solving systems of equations:

graphing a system of equations on a calculator and using a spreadsheet to analyze a system of equations.

N. I’m Thinking of Two Numbers

a. I’m thinking of two numbers. Their sum is 12. The sum of the first with twice the second is 7. What are the numbers? (17, -5)

b. I’m thinking of two numbers. Their difference is 9. The sum of twice the first and three times the second is 63. What are the numbers? (18, 9)

c. I’m thinking of two numbers. The first is three times as large as the second. Their sum is 48. What are the numbers? (36, 12)

d. Ask students to create their own versions of “I’m thinking of two numbers.” Collect those from the students, compile and edit, and redistribute for students to solve.

O. Give each row of students one of the following sets of equations to solve using the

addition method: 1) �2 =+ yx , 654 =+ yx

2) ��2 =+ yx , 7�5 =+ yx

3) 543 =+ yx , 876 =+ yx

4) 654 =+ yx , 987 =+ yx

5) 765 =+ yx , 1098 =+ yx

Call on one person on each row to give the solution. Ask students if they notice a pattern to the equations. Why would they have the same solutions? If all systems were graphs on the same axis, what would they look like?

P. Calculcator Wars

(1) Each player enters the equations for two lines into Y1 and Y2. Wise players choose equations that do not intersect on the screen.

(2) After they exchange calculators, the students must manipulate the window so that the crossing lines appear. They must then use the 2nd

CALC INTERSECTION to find the solution to the system of equations. Since the calculation will only find the intersection of the functions displayed on the screen, the students may not skip the step of manipulating windows.

(3) Encourage the players to use the table function to get initial estimates. (4) The first player to show his opponent the correct solution wins the round. As

always, play continues until the teacher calls time.

Q. First Wheels (B-49) Construct a system of equations to make a thoughtful automobile purchase.

G-40

R. Revisit absolute value equations using the substitution method and calculators. Have

students graph on the calculator, 2!= xy and 4=y . Use the trace to find the ordered

pairs for the intersection. Next have students solve the system using substitution. (i.e.

42 =!x ). Ask students to explain a method for solving an absolute value equation, 3 =!x using a calculator and solving by hand.

S. Challenges (W-10, 40, 46, 54, 56, 60)

G-41

4.04 Graph and evaluate exponential functions to solve problems.

A. The Dinosaurs Bite the Dust (B-76)

Simulate the extinction of the dinosaurs using dice to create an exponential model. Point out to students the random nature of the dinosaurs’ demise, yet we are able to discover a mathematical model for the event. The two theories which try to explain the dinosaurs’ disappearance are provided only to the teacher. The teacher can share this information with the students or expect the students to research the topic. Sixty-five million years ago, Earth experienced a global extinction event so severe that it defines the boundary between the Cretaceous (K) and Tertiary (T) geological periods. Causing extinctions on both the lands and in the oceans, that event is referred to as the K-T extinctions. The dinosaurs became extinct during the K-T extinctions. Of all the theories ever devised for cause of the K-T extinctions, only two remain and they are the focus of intense scientific debate. One, the asteroid-impact winter theory created by Nobel laureate Luis Alvarez, states that a giant asteroid struck Earth 65 million years ago. It blasted dust into the stratosphere that blocked out sunlight and plunged Earth into a dark, frozen winter. The other, the volcano-greenhouse theory originated by Dewey M. McLean, relates the K-T extinctions to a major perturbation of earth’s carbon cycle caused by the Deccan Traps Mantle Plume Volcanism in India. This was one of the greatest volcanic events in Earth history and its main eruptions began 65 million years ago. The Deccan Traps released vast quantities of the greenhouse gas, carbon dioxide (CO2), onto Earth’s surface, trapping heat form the sun, and turning Earth’s surface into a hot, sterilizing “greenhouse.” It has been estimated that the dinosaurs disappeared in 100-300 years. Using the

function ( )xGy •= 500000 , where x is the number of years since the extinction event,

y is the remaining dinosaurs, and G ( .01. !!G ) is the rate of population decline,

have students determine a rate of decline when there are 100 dinosaurs left after 300 years.

B. Problems of an Exponential Nature (B-77)

Many problems like these are available in textbooks. Expect students to investigate the situation in a table, using their calculators for computation. Then students can discover and discuss patterns in a graph of their data. Students can explore curve-fitting with their calculators and/or discuss how to use the y = a•b

x form to model the problem.

G-42

C. Patterns with Exponential Equations (A-45)

Students will graph equations in which the base b is a positive number greater than 1. They will investigate what happens as b increases and describe the pattern. Next, students will graph equations in which the base b is between 0 and 1 and describe this pattern.

D. Rolling Dice: An Exponential Experience (B-73)

o Begin with a large number of dice (30+). Place the dice in a cup and roll them. Remove all the dice that show 3. Roll the remaining dice and again remove the 3s. Continue the process until there are only one or two dice remaining. Keep a record of the results for each roll and graph those results. Use the calculator to determine a best-fit exponential function. How does the best-fit function compare with the

expected function

��y !

"

#$%

&•=6

5, where N is the number of dice you begin with,

x is the number of rolls, and y is the remaining dice?

o With a large number of dice (30+) handy, begin with two dice. Place the dice in a

cup and roll them. For every die that shows a 3, add another die. Roll the dice and again add a die for each 3 that appears. Continue the process until all of the dice are used. Keep a record of the results for each roll and graph those results. Use the calculator to determine a best-fit exponential function. How does the best-fit

function compare with the expected function

x�y !

"

#$%

&•=6

7, where N is the number

of dice you begin with, x is the number of rolls, and y is the new dice total?

E. Begin with a large number of coins (30+). Place the coins in a cup, shake, and dump on the

desk top or floor. Remove all the coins that show HEADs. Shake and dump the remaining coins and again remove the HEADs. Continue the process until there are no coins remaining. Keep a record of the results for each turn and graph those results. Use the calculator to determine a best-fit exponential function. How does the best-fit function compare with the

expected function

x

Ny !"

#$%

&•=2

1, where N is the number of coins you begin with, x is the

number of rolls, and y is remaining coins.

With a large number of coins (30+) handy, begin with two coins. Place the coins in a cup, shake, and dump on the desk top or floor. For every coin that shows HEADs, add another coin. Shake and dump the coins and again add a coin for each HEADs that appears. Continue the process until all of the coins are used. Keep a record of the results for each turn and graph those results. Use the calculator to determine a best-fit exponential function. How does the

best-fit function compare with the expected function

x�y !

"

#$%

&•=2

3, where N is the number of

coins you begin with, x is the number of turns, and y is the new coin total?

G-43

F. Use Your Imagination (B-75)

Three problems are presented that are mental experiments involving “folding” a single sheet of paper in half many times. These are good illustrations for the power of exponential growth. (The solutions are: the Sears Tower in 23 folds; Mount Everest in 27 folds; and the Moon in 42 folds.)

G. Have students use a calculator and a table to calculate the cost of jeans at 5% inflation over a period of 5 years. Then, have them compare their yearly results to the compound interest formula. (The example shown is for a $30 pair of jeans.)

Write an expression for the cost of jeans in the year 2015. Use your calculator to simplify.

H. Have students draw a graph to compare the difference in simple interest and compound interest for $2,000 invested at 6%.

I. Connections (W-69) J. Challenges (W-72) K. Extra Essentials (E-22, 25) L. Warm Ups (W-70)

W-0

Writing Ideas

Connections to the World

WORDS

Teacher to Teacher

Calculator Tips

Warm Ups

Challenges

Answer Key

W-1

Calculator Tips: Evaluating an Expression

! Key the value for the variable; STO>; ALPHA; key the variable; ENTER.

! Key in the expression; ENTER.

Curriculum Resource Organizer: When trying to correlate the textbook with the Algebra 1

Standard Course of Study, the format on E-1 – E-4 may be helpful. Beside each objective the

pages in the textbook can be listed and one can quickly identify the parts of the curriculum which

are not addressed in the textbook. Additional resources and materials can be listed with the

appropriate objective in the third column. Activities from other sections of Resources for Algebra

can also be matched with the textbook being used. Working with another teacher (or several other

teachers in your district) will provide an opportunity to share ideas and resources.

W-2

W-3

Pacing Guides: No tow teachers are alike and no two teachers necessarily teach the topics in

Algebra 1 in the same sequence. In order to assist teachers in the implementation of the revised

Algebra 1 curriculum, two suggestions for pacing guides are laid out in six-week and quarterly

formats. See E-5, 6, 7.

W-4

W-5

Calculator Tips: Scientific Notation

When working with numbers expressed in scientific notation, there are several ways in which

one can enter those numbers. Suppose the number to be used is 81074.5 • , you can do one of

the following:

! key 5.74 x 10 ^ 8; ENTER

! key 5.74 x 2ND; LOG; 8

! key 5.74 2ND; , ; 8

Accommodations for Individual Students: Knowing how knowledge is constructed increases

the importance of how to use thinking or cognitive abilities. Students and teachers become

aware of the power of cognitive strategies such as information gathering, organizing, analyzing,

integrating, generating, and evaluating in learning new concepts. Teachers help students to

develop strategies which enable students to understand mathematical ideas. For example,

information gathering requires a learner to listen, observe, use visual aids, and ask questions.

Teachers should take every opportunity to incorporate the teaching of cognitive strategies within

the content of a lesson. E-8 lists ten cognitive strategies identifying behaviors which indicate

deficiencies in those and accommodations that encourage growth in developing those strategies.

Although the chart and list of accommodations on E-8 are usually identified with learning-

disabled students, classroom teachers find that these are appropriate for many students who are

not identified LD. The Essentials for Instruction which appear throughout the

Goals/Objectives pages are accommodations that are important for all students.

W-6

W-7

Maximizing Class Time: Some teachers have found it useful to start the class with two or three

problems to be done as the students come into class. The problems may be a review of

yesterday’s lesson or from a lesson six weeks ago. As a student finishes, give the student a card

with the answers. Pass the answer card around the room as the students finish. Get those who

finish early to help those who are having difficulty. For a wider range of mathematics, the

calendar section of each issue of NCTM’s Mathematics Teacher is an excellent source of starter

and challenge problems. See Warm Ups, Challenges, and Extra Essentials (E-22 – E-44).

W-8

W-9

Calculator Tips: Absolute Value

To find the absolute value of an expression, key MATH; select NUM; select 1:abs;

ENTER; key the expression; ENTER. Absolute value expressions can be entered on the Y=

screen to graph equations.

W-10

W-11

Assessment: Consider these questions to help with your reflections about assessment:

! Am I teaching and assessing important mathematics?

! Do my assessment practices promote further learning?

! Are my students involved in self-assessment and self-direction?

! Do my assessment practices afford all students the opportunity to demonstrate what they

know and understand?

! Do students clearly understand what my expectations are?

! Am I providing appropriate examples and models of good work?

! Am I gathering appropriate evidence and keeping sufficient documentation for

evaluations I make?

! Do my assessments match my goals?

Several assessment methods (selected response, guided constructed response, performances and

presentations, products, observations, and questioning and interviews) are described on E-9.

W-12

W-13

Calculator Tips: The Best Window for the TI-83

The window on the TI-83 is a 95 by 63 pixel grid. Think of numbering the columns of pixels

from left to right 1-95 and the rows of pixels from bottom to top 1-63. By choosing values

for Xmin and Xmax to have a difference of 94 (95 - 1) and Ymin and Ymax to have a

difference of 62 (63 – 1), you will have a proportionally correct window. That is, the vertical

units will be exactly the same length as the horizontal units. By choosing multiples of 94 and

62 that are derived using the same factor (use a factor of 0.1 to get 9.4 and 6.2), you will have

a proportionally correct window.

North Carolina Standard Course of Study (2003), K-8 Algebra: In case you did not know,

students entering Algebra 1 should have had many algebraic experiences in the earlier grades. See E-14, 15.

W-14

W-15

Calculator Tips: Zooming for a Proportionally Correct Coordinate Plane (TI-83/84)

W-16

W-17

Calculator Tips: Friendly Range on the TI-83/84

W-18

W-19

W-20

W-21

Calculator Tips: Inequalities on the TI-83/84

W-22

W-23

Calculator Tips: Inequalities on the TI-83/84

W-24

W-25

W-26

W-27

Calculator Tips: Entering and Displaying Data (TI-83/84)

! Check MODE to make sure all settings are appropriate.

! Y=; clear all equations; QUIT.

! STAT; 4:ClrList; key the lists you want cleared, separated by commas; ENTER.

! STAT; 1:Edit; enter data (x values in L1 and y values in L2 or other appropriate

lists): QUIT.

! WINDOW; set values for Xmin, Xmax, Xscl, Ymin, Ymax, Yscl; QUIT.

! 2nd Y= (STAT PLOT); 1:Plot1: highlight On by ENTERing; highlight the Type of

graph you want; Select L1 for Xlist and L2 for Ylist (or other appropriate lists);

highlight a Mark; QUIT; GRAPH.

W-28

W-29

Calculator Tips: Entering and Displaying Data (TI-83/84)

W-30

W-31

Calculator Tips: Generating Random Numbers on the TI-83/84

The TI-83/84 will generate random numbers between 0 and 1.

W-32

W-33

Calculator Tips: Generating Sets of Random Numbers on the TI-83/84

The TI-83/84 will generate sets of random numbers so that you do not have to continue to press

ENTER.

To generate a list of random numbers between 0 and 1:

! Key MATH; go to PRB menu; select 1:rand; place the number of values desired in

parentheses; ENTER; use the right and left arrow keys to scroll to the other values

! Repeating ENTER will generate another list of random values.

To generate random integers:

! Key MATH; go to PRB menu; select 5:randInt; key the smallest value, comma,

largest value, comma, and number of values; ENTER ! Repeating ENTER will generate additional sets of values.

Algebra in the NCTM Principles and Standards for School Mathematics (2000): The NCTM

Principles and Standards articulate algebraic expectations from prekindergarten to grade twelve.

Understand patterns, relations, and functions; represent and analyze mathematical situations and

structures using algebraic symbols; use mathematical models to represent and understand quantitative relationships; analyze change in various contexts. See E-16, 17, 18.

W-34

W-35

Calculator Tips: The Best-Fit Line (TI-83/83+)

! Enter data.

! STAT; go to CALC menu; 4:LinReg(ax+b); L1 (or other appropriate list),

comma, L2 (or other appropriate list), comma, VARS, select Y-Vars, 1:Function,

Y1 (or other appropriate equation); ENTER. The calculator will display values

for a and b and also for r2 and r if diagnostics are on.

! The calculator uses the general form of a linear equation, y = ax + b.

! The correlation coefficient r (-1"r"1) provides information about the quality of

the linear fit.

! GRAPH.

! When finished, go to STAT PLOT and set plots to Off. QUIT.

W-36

W-37

W-38

W-39

Calculator Tips: The Best-Fit Line (TI-83/84)

! Enter data.

! STAT; go to CALC menu; 4:LinReg(ax + b); ENTER.

! The calculator will display values for a, b, r and r2 if diagnostics are on.

! The calculator uses the general form of a linear equation, y = ax + b.

! The correlation coefficient r (-1 " r " 1) provides information about the quality of

the linear fit.

! Y=; VARS; 5:Statistics; EQ; 1:RegEQ; GRAPH

! When finished, go to STAT PLOT and set plots to Off. QUIT.

W-40

W-41

W-42

W-43

Calculator Tips: Dealing with r and r2 on the TI-83/84

If r and r2 are not displayed after completing a linear regression, follow these steps:

! Enter 2nd 0 (CATALOG)

! Using the down arrow, move down the list until the arrow is pointed to DiagnosticOn

and press ENTER twice. The calculator should respond with Done.

! Now redo the regression steps.

The Diagnostics are turned off when memory is reset. If students are required to reset memory

before standardized tests, they need to be sure to turn them back on before doing a data analysis

problem.

W-44

W-45

(Science) Do you know why you see the flash of lightning before hearing the roll of thunder?

Light travels at 1.86 x 105 miles per second while sound travels through the air (at 32° F) in about

742 miles per hour. Compare the two velocities. Investigate how the velocities of the two

phenomena are affected by factors such as altitude, temperature, and media.

Calculator Tips: Tables for Your Function on the TI-83/84

Once can view a function evaluated for values of x by using the TABLE function.

! Y=; key in the expression to be evaluated in Y1; 2nd WINDOW (TBLSET); key in the

values for TblStart and !Tbl (increment); 2nd GRAPH (TABLE).

! Scroll in the table using the up and down arrows. Notice values for x and the

corresponding values of Y1 are listed. The table can be expanded to include

expressions for Y2, Y3, …

W-46

W-47

W-48

W-49

Calculator Tips: Setting Zoom Factors (TI-83/84)

ZOOM; go to the MEMORY menu; 4:SetFactors; enter values for XFact and YFact; QUIT.

W-50

W-51

Resources for Data: Almanacs and the Statistical Abstract of the United States are excellent

sources for numerical data covering a wide range of topics. The reference section of your

library should have current editions of these publications. Newspapers generally are another

good source for data and mathematics. The World Wide Web can put you in touch with many

sources for data as well as the most recent discoveries in science and mathematics. Once

students are connected, the information is almost unlimited: The NC Department of Public

Instruction’s web page is at http://www.ncpublicschools.org. For the most complete coverage

of K-12 mathematics in North Carolina check out the mathematics page for the NC Department

of Public Instruction at http://community.learnnc.org/dpi/math.

W-52

W-53

Calculator Tips: Graphing a Family of Equations on the TI-83/84

W-54

W-55

W-56

W-57

W-58

W-59

W-60

W-61

Calculator Tips: The Best-Fit Exponential Equation (TI-83/84)

! Enter data.

! STAT; go to CALC menu; select 0:ExpReg; ENTER

! The calculator uses the general form of an exponential equation, y = a • bx

! Y=; VARS; 5:Statistics; go to EQ menu; 1:RegEQ

! GRAPH

W-62

W-63

Calculator Tips: The Best-Fit Exponential Equation (TI-83/84)

W-64

W-65

W-66

W-67

Calculator Tips: The Best-Fit Quadratic Equations (TI-83/84)

! Enter data.

! STAT; go to CALC menu; 5:QuadReg; ENTER

! The calculator uses the general form of a quadratic equation, y = ax2 + bx + c

! Y=; VARS; 5:Statistics; go to EQ menu; 1:RegEQ

! GRAPH

W-68

W-69

W-70

W-71

Calculator Tips: The Best-Fit Quadratic Equation (TI-83/84)

Scoring Open-ended Problems: When scoring open-ended problems, activities, or projects a

rubric or scoring guide is an appropriate instrument to evaluate a student’s performance. The

Extra Essentials section has a rubric for students’ use (E-10), a comprehensive rubric (E-12), a

general scoring guide (E-13), and scoring information for students (E-11) whose work is to be scored with a rubric.

W-72

W-73

W-74

W-75

W-76

W-77

W-78

W-79

W-80

E-0

Curriculum Resource Organizer E-1

Pacing Guides E-5

Accommodations E-8

Assessment Methods E-9

Rubrics E-10

K-8 Curriculum E-14

NCTM Principles and Standards E-16

Games, Warm-Ups/Challenges E-19

Answers E-45

E-1

Curriculum Resource Organizer

Goals & Objectives Textbook Other Resources

The learner will perform operations

with numbers and expressions to

solve problems.

1.01 Write equivalent forms of

algebraic expressions to solve

problems.

a) Apply the laws of exponents.

b) Operate with polynomials.

c) Factor polynomials.

1.02 Use formulas and algebraic

expressions, including iterative

and recursive forms, to model and

solve problems.

1.03 Model and solve problems

using direct variation.

E-2

Goals & Objectives Textbook Other Resources

The learner will describe geometric

figures in the coordinate plane

algebraically.

2.01 Find the lengths and

midpoints of segments to solve

problems.

2.02 Use the parallelism or

perpendicularity of lines and

segments to solve problems.

E-3

Goals & Objectives Textbook Other Resources

The learner will collect, organize,

and interpret data with matrices

and linear models to solve

problems.

3.01 Use matrices to display and

interpret data.

3.02 Operate (addition,

subtraction, scalar multiplication)

with matrices to solve problems.

3.03 Create linear models for sets

of data to solve problems.

a) Interpret constants and

coefficients in the context of

the data.

b) Check the model for goodness-

of-fit and use the model,

where appropriate, to draw

conclusions or make

predictions.

E-4

Goals & Objectives Textbook Other Resources

The learner will use relations and

functions to solve problems.

4.01 Use linear functions or

inequalities to model and solve

problems; justify results.

a) Solve using tables, graphs, and

algebraic properties.

b) Interpret constants and

coefficients in the context of

the problem.

4.02 Graph, factor, and evaluate

quadratic functions to solve

problems.

4.03 Use systems of linear

equations or inequalities in two

variables to model and solve

problems. Solve using tables,

graphs, and algebraic properties;

justify results.

4.04 Graph and evaluate

exponential functions to solve

problems.

E-5

Algebra 1: A Semester at a Glance

Goals 1st Quarter 2nd Quarter 3rd Quarter 4th Quarter

Number and

Operations The learner will perform

operations with numbers and

expressions to solve

problems.

1.01a

1.02

1.03

1.01b 1.01b

1.01c

Geometry and

Measurement The learner will describe

geometric figures in the

coordinate plane

algebraically.

2.01

2.02

Data Analysis and

Probability The learner will collect,

organize, and interpret data

with matrices and linear

models to solve problems.

3.01 3.02 3.03a

3.03b

Algebra The learner will use relations

and functions to solve

problems.

4.01a

4.01b

4.03

4.02

4.04

E-6

Algebra 1: A Year at a Glance

Goals 1st Quarter 2nd Quarter 3rd Quarter 4

th

Quarter

5th

Quarter

6th

Quarter

Number and

Operations The learner will perform

operations with numbers

and

expressions to solve

problems.

1.01a

1.02 1.03 1.01b

1.01b

1.01c

1.01b

1.01c

Geometry and

Measurement The learner will describe

geometric figures in the

coordinate plane

algebraically.

2.01

2.02

Data Analysis and

Probability The learner will collect,

organize, and interpret

data with matrices and

linear models to solve

problems.

3.01 3.02 3.02 3.03a

3.03b

Algebra The learner will use

relations and functions to

solve

problems.

4.01a

4.01b

4.01a

4.01b 4.03 4.03

4.02

4.04

E-7

Algebra 1: Quarterly Pacing Guide

First Quarter

1.01a Apply the laws of exponents.

1.02 Use formulas and algebraic expressions, including iterative and recursive forms, to model

and solve problems.

1.03 Model and solve problems using direct variation

3.01 Use matrices to display and interpret data.

Second Quarter

2.01 Find the lengths and midpoints of segments to solve problems.

2.02 Use the parallelism or perpendicularity of lines and segments to solve problems.

3.02 Operate (addition, subtraction, scalar multiplication) with matrices to solve problems.

4.01a Use linear functions or inequalities to model and solve problems using tables, graphs, and

algebraic properties; justify results.

4.01b Interpret constants and coefficients in the context of the problem.

Third Quarter

1.01b Operate with polynomials.

3.03a Create linear models for sets of data to solve problems: Interpret constants and

coefficients in the context of the data.

3.03b Check the model for goodness-of-fit and use the model, where appropriate, to draw

conclusions or make predictions.

4.03 Use systems of linear equations or inequalities in two variables to model and solve

problems. Solve using tables, graphs, and algebraic properties; justify results.

Fourth Quarter

1.01b Operate with polynomials.

1.01c Factor polynomials.

4.02 Graph, factor, and evaluate quadratic functions to solve problems.

4.04 Graph and evaluate exponential functions to solve problems.

E-8

E-9

E-10

E-11

E-12

E-13

E-14

E-15

E-16

E-17

E-18

E-19

E-20

E-21

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E-25

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E-34

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E-36

E-37

E-38

E-39

E-40

E-41

E-42

E-43

E-44

E-45

E-22

E-23

E-24

E-25

E-26

E-27

E-28

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E-46

E-30

E-31

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E-34

E-35

E-36

E-37

E-47

E-38

E-39

E-40

E-41

E-42

E-43

E-44

A - 0

A - 1

Materials One set of dominoes (B-78, 79) per pair (or team) of students, blank domino

sheets (B-80).

Goal Simplify real number expressions and apply the distributive property in an interesting format.

Description Students will make a train of dominoes by successfully simplifying an expression

or applying the distributive property. Blank domino sheets (B-80) can be made

available so that students can create versions of the game that practice various

algebraic skills throughout the year. Well-constructed and edited student

versions of dominoes will provide the teacher a pool of materials to use

thereafter.

A - 2

Copy the blank sheet (B-80) for students to create other versions of dominoes.

A - 3

Materials Two copies of the blackline masters (B-81, 82, 83, 84), one separated.

A - 4

Using the blank page (B-85), students can create their own versions of the activity as a homework

assignment or for extra credit. Well-constructed and edited student versions of the activity

will provide the teacher a pool of materials to use thereafter.

A - 5

Materials Gameboard (B-86), 6 markers of one color for each student

A - 6

Give each pair of students a gameboard (B-86), three dice, and 6 colored markers for each student. (Or the teacher can lead the game by rolling the dice for the class.)

A - 7

Materials Calculators, NFL scoring statistics (B-87).

A - 8

A - 9

Materials Meter sticks, measuring tape, graph paper, calculator, recording sheets (B-88).

A - 10

Individually or in groups, give the students ten distances to estimate from the classroom and around the school grounds. Students will record their estimates on the sheet provided (B-88).

A - 11

Materials Copies of sample puzzle (B-89, 90, 91, 92, 93, 94, 95, 96, 97, or 98) separated,

transparency of sample puzzle (separated), puzzlemaster sheets (B-99).

A - 12

As an assignment, give each student two blank puzzle forms (B-99) so that they can create their own puzzle.

A - 13

Materials Stopwatch, graph paper (B-1 or B-2)

A - 14

A - 15

Materials Measuring tape, calculators, graph paper (B-1 or B-2)

A - 16

A - 33

Materials Calculators, reporting sheets (B-113, 114, 115)

A - 34

Give each student reporting sheets (B-113, 114).

A - 35

Materials Base-10 blocks, transparency of square numbers (B-116), transparency of

example (B-117), recording sheets (B-118), calculators

A - 36

Use the transparency of the squares (B-116).

Use transparency B-117 ( 135 )

Use the blocks to determine rational approximations for the square roots of other numbers (B-118).

A - 37

Materials Calculators, recording sheets B-119, 121.

A - 38

Have students solve the equation using factoring. Students should record the solutions on their recording sheet (B-119).

Part II Have students write explanations as they answer the discussion questions (B-121).

A - 39

Materials Scissors, rulers, tape, grid paper (B-1), recording sheets (B-122), calculators

A - 40

Discuss with students how to determine the algebraic expression that represents the volume of all the boxes. Use B-123 and B-124 when appropriate.

A - 41

Materials Recording sheets (B-125, 126, 127), computer spreadsheet or calculator

Description In this activity, students will use a table to list possible values for the

dimensions and area of a garden space. Students will write an equation to graph

the width of the garden versus the area. This will be a quadratic function with a

maximum value. See B-128 for extensions.

A - 42

To complete using the TI-83/84 table generator, enter the following into the table setup:

2nd TBLSET TABLE SETUP

TblStart = 0

! Tbl = 1.2

Indpnt: Auto Ask

Depend: Auto Ask

A - 43

Materials Calculators, recording sheets (B-129)

A - 44

Pass out to each student a recording sheet (B-129). Ask the students to draw a graph illustrating

the length of time the SRBs are in the air versus their altitude.

A - 45

Materials Recording sheets (B-130, 131), calculators

A - 46

Part II: Student can investigate patterns in exponential equations by graphing several using a calculator

and then sketching the graph on the recording sheet (B-130). Students also record the y-intercept

and whether the graph is increasing or decreasing. Ask students to compare the first three graphs

by graphing them on the same axis. Next, have them compare the last three graphs on the same

axis.

A - 17

Materials Graph paper (B-2), a clear transparency for each group, measuring tapes, B-102

transparency

A - 18

Show the overhead of the graph labeled arm span and height (use B-2). Discuss with the class

where data would lie for people who are square. Have students write an equation to represent the

set of data that are squares. (Let x = arm span, y = height). Graph this line on the transparency of

B-2. Ask student if they are above or below the line y = x.

A - 19

Materials Measuring tape, golf balls (or some other object to roll down the ramp), ramp

construction materials (paper towel tubes, meter sticks, boards, etc. for the balls to

roll down and something that will uniformly raise the ramp – books, bricks, etc.),

calculators, recording sheets (B-103)

A - 20

A - 21

Materials Measuring tapes (or rulers), calculators, recording sheets (B-104)

A - 22

Measure the dimensions of several objects in the classroom using both centimeters and inches.

Record the measures on the data sheet (B-104).

A - 23

Materials Toothpicks, graph paper (B-1 or B-2), calculators, recording sheets (B-105)

A - 24

Have students work in pairs and complete the worksheet (B-105) using toothpicks to create the necessary models.

A - 25

Materials Recording sheets (B-12), calculators, telephone book (optional)

Description Students, working in pairs, are going to simulate the end-of-game situation 25

times and keep a record of their results. Students will use a random number

generator (telephone book, calculator, etc.) to simulate shooting free throws.

Students will use their experimental results to predict the outcome of the game

(win, lose, or overtime). There are three extensions to this activity on B-13, B-16,

and B-17.

A - 26

A - 27

Materials Calculators, calendar transparency (B-106) and student copies of the calendar

(B-107)

A - 28

A - 29

Materials Algebra tiles (although a blackline master, B-109, is provided for you to create

your own tiles, there are several “brands” of tiles commercially available with student and overhead versions), setup sheets (B-108), recording sheets (B-110)

A - 30

Complete the rest of the recording sheet (B-110).

A - 31

Materials Algebra tiles (although a blackline master, B-109, is provided for you to create

your own tiles, there are several “brands” of tiles commercially available with

student and overhead versions), setup sheets (B-108), recording sheets (B-111 and

B-112)

A - 32

Complete the rest of the recording sheet (B-111 and B-112).

B-0

B-1

B-2

B-3

B-4

B-5

B-6

B-7

B-8

B-9

B-10

B-11

B-12

B-13

B-14

B-15

B-16

B-17

B-18

B-19

B-20

B-21

B-22

B-23

B-24

B-25

B-26

B-27

B-28

B-29

B-30

B-31

B-32

B-33

B-34

B-35

B-36

B-37

B-38

B-39

B-40

B-41

B-42

B-43

B-44

B-45

B-46

B-47

B-48

B-49

B-50

B-51

B-52

B-53

B-54

B-55

B-56

B-57

B-58

B-59

B-60

B-61

B-62

B-63

B-64

B-65

B-66

B-67

B-68

B-69

2005 Tax Rate Schedules for Single Tax Payers

2005 Federal Tax Rate Schedule

Taxable Income Is more than But not over The tax is

$ 0 $ 7,300 10% of the taxable income $ 7,300 $ 29,700 $730 + 15% of the amount over $7,300

$ 29,700 $ 71,950 $4,090.00 + 25% of the amount over $29,700 $ 71,950 $ 150,150 $14,652.50 + 28% of the amount over $71,950

$ 150,150 $ 326,450 $36,548.50 + 33% of the amount over $150,150 $ 326,450 ------------ $94,727.50 + 35% of the amount over $326,450

2005 NC Tax Rate Schedule

Taxable Income

Is more than But not over The Tax is $ 0

$ 12,750 $ 60,000

$ 120,000

$ 12,750

$ 60,000 $ 120,000

----------

6% of the taxable income

$765 + 7% of the amount over $12,750 $4,072.50 + 7.75% of the amount over %60,000

$8,722.50 + 8.25% of the amount over $120,000

B-70

B-71

B-72

B-73

B-74

B-75

B-76

B-77

B-78

B-79

B-80

B-81

B-82

B-83

B-84

B-85

B-86

B-87

B-88

B-89

B-90

B-91

B-92

B-93

B-94

B-95

B-96

B-97

B-98

B-99

B-100

B-101

B-102

B-103

B-104

B-105

B-106

B-107

B-108

B-109

B-110

B-111

B-112

B-113

B-114

B-115

B-116

B-117

B-118

B-119

B-120

B-121

B-122

B-123

B-124

B-125

B-126

B-127

B-128

B-129

B-130

B-131

B-132

B-133

B-134