Algebra I

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Algebra I ...



    Algebra I

    Cecil J. Picard

    State Superintendent of Education

    ? April 2005

    Algebra I

    Table of Contents

Unit 1: Understanding Quantities, Variability, and Change .....................................................1

    Unit 2: Writing and Solving Proportions and Linear Equations .............................................14

    Unit 3: Linear Functions and Their Graphs, Rates of Change, and Applications ..................23

    Unit 4: Linear Equations, Inequalities, and Their Solutions ...................................................34

    Unit 5: Systems of Equations and Inequalities .......................................................................43

    Unit 6: Measurement...............................................................................................................52

    Unit 7: Exponents, Exponential Functions, and Nonlinear Graphs ........................................62

    Unit 8: Data, Chance, and Algebra .........................................................................................72

    Algebra I

    Unit 1: Understanding Quantities, Variability, and Change

Time Frame: Approximately three weeks

Unit Description

This unit examines numbers and number sets including basic operations on rational

    numbers, integer exponents, radicals, and scientific notation. It also includes

    investigations of situations in which quantities change and the study of the relative nature

    of the change through tables, graphs, and numerical relationships. The identification of

    independent and dependent variables is emphasized as well as the comparison of linear

    and non-linear data.

Student Understandings

Students focus on developing the notion of a variable. They begin to understand inputs

    and outputs and how they reflect the nature of a given relationship. Students recognize

    and apply the notions of independent and dependent variables and write expressions

    modeling simple linear relationships. They should also come to understand the difference

    between linear and non-linear relationships.

Guiding Questions

    1. Can students perform basic operations on rational numbers with and without


    2. Can students perform basic operations on radical expressions?

    3. Can students evaluate and write expressions using scientific notation and integer


    4. Can students identify independent and dependent variables?

    5. Can students recognize patterns in and differentiate between linear and non-

    linear sequence data?

Unit 1 Grade-Level Expectations (GLEs)

    GLE # GLE Text and Benchmarks Number and Number Relations 1. Identify and describe differences among natural numbers, whole numbers,

    integers, rational numbers, and irrational numbers (N-1-H) (N-2-H) (N-3-H)

    2. Evaluate and write numerical expressions involving integer exponents (N-2-H)

    3. Apply scientific notation to perform computations, solve problems, and write

    Algebra I?Unit 1? Understanding Quantities, Variability, and Change 1

    representations of numbers (N-2-H)

    4. Distinguish between an exact and an approximate answer, and recognize errors

    introduced by the use of approximate numbers with technology (N-3-H) (N-4-

    H) (N-7-H)

    5. Demonstrate computational fluency with all rational numbers (e.g., estimation,

    mental math, technology, paper/pencil) (N-5-H) 6. Simplify and perform basic operations on numerical expressions involving

    23?53?73radicals (e.g., ) (N-5-H)


    7. Use proportional reasoning to model and solve real-life problems involving

    direct and inverse variation (N-6-H)

    8. Use order of operations to simplify or rewrite variable expressions (A-1-H) (A-


    9. Model real-life situations using linear expressions, equations, and inequalities

    (A-1-H) (D-2-H) (P-5-H)

    10. Identify independent and dependent variables in real-life relationships (A-1-H)

    15. Translate among tabular, graphical, and algebraic representations of functions

    and real-life situations (A-3-H) (P-1-H) (P-2-H) Data Analysis, Probability, and Discrete Math

    28. Identify trends in data and support conclusions by using distribution

    characteristics such as patterns, clusters, and outliers (D-1-H) (D-6-H) (D-7-H)

    29. Create a scatter plot from a set of data and determine if the relationship is linear

    or nonlinear (D-1-H) (D-6-H) (D-7-H)

    34. Follow and interpret processes expressed in flow charts (D-8-H)

    Sample Activities

Activity 1: The Numbers (GLEs: 1, 4, 5)

Use a number line to describe the differences and similarities of whole numbers, integers,

    rational numbers, irrational numbers, and real numbers. Have the students identify types

    of numbers selected by the teacher from the number line. Have the students select

    examples of numbers from the number line that can be classified as particular types.

    9Example questions could include:What kind of number is? What kind of number is 23.6666? Identify a number from the number line that is a rational number.

Discuss the difference between exact and approximate numbers. Have the students use

    Venn diagrams and tree diagrams to display the relationships among the sets of numbers.

Help students understand how approximate values affect the accuracy of answers by

    having them experiment with calculations involving different approximations of a

    number. For example, have the students compute the circumference and area of a circle

    Algebra I?Unit 1? Understanding Quantities, Variability, and Change 2

    ?. Use measurements as examples of approximations using various approximations for

    and show how the precision of tools and accuracy of measurements affect computations of values such as area and volume. Also, use radical numbers that can be written as

    2approximations such as .

Activity 2: Using a Flow Chart to classify real numbers (GLEs: 1, 34)

    A flow chart is a pictorial representation showing all the steps of a process. Guide students to create a flow chart to classify real numbers as rational, irrational, integer, whole and/or natural. A sample flow chart is given at the end of this unit. Tell students that in most flow charts, questions go in diamonds, processes go in rectangles, and yes or

    no answers go on the connectors. Have students come up with the questions that they must ask themselves when they are classifying a real number and what the answers to those questions tell them about the number. Many word processing programs have the capability to construct a flow chart. If technology is available, allow students to construct the flow chart using the computer. After the class has constructed the flow chart, give students different real numbers and have the students use the flow chart to classify the numbers. (Flow charts will be revisited in later units to ensure mastery of GLE 34)

Activity 3: Operations on rational numbers (GLE 5)

    Have students review basic operations with whole numbers, fractions, decimals, and integers. Include application problems of all types so that students must apply their prior knowledge in order to solve the problems. Discuss with students when it is appropriate to use estimation, mental math, paper and pencil, or technology. Divide students into groups and give examples of problems in which each method is more appropriate; then have students decide which method to use. Have the different groups compare their answers and discuss their choices.

Activity 4: Comparing Radicals (GLE 6)

    Have students work with a partner for this activity. Provide the students with centimeter graph paper. Have them draw a right triangle with legs 1 unit long and use the

    2Pythagorean theorem to show that the hypotenuse is units long. Then have them

    repeat with a triangle that has legs that are 2 units long, so they can see that the

    822hypotenuse is or units long. Have them continue with triangles that have legs of

    3 and 4 units long. For each hypotenuse, have them write the length two different ways and notice any patterns that they see. This activity leads to a discussion of simplifying radicals.

Algebra I?Unit 1? Understanding Quantities, Variability, and Change 3

Activity 5: Basic Operations on Radicals (GLEs: 6, 8)

Review the distributive property with students and its relationship to combining like

    terms. (i.e.) Provide students with variable expressions to 35358xxxx??????

    32?52simplify. Give the following radical expression to students: . Guide students

    to the conclusion that the distributive property can also be used on radical expressions,

    32?52?82thus . Provide radical expressions for students to simplify.

Activity 6: Scientific Notation (GLEs: 2, 3, 4)

Have students use a calculator to make a chart with powers of 10 from 5 to 5. Discuss

    the patterns that are observed and the significance of negative exponents. Provide

    students with real-life situations for which scientific notation may be necessary, such as

    the distance from the planets to the sun or the mass of a carbon atom. Have students

    investigate scientific notation using a calculator. Allow students to convert numbers from

    scientific notation to standard notation and vice versa. Relate the importance of scientific

    notation in the areas of physical science and chemistry.

Activity 7: Variation (GLEs: 7, 9, 10, 15, 28, 29)

Part 1: Direct variation

    Have the students collect from classmates real data that might represent a relationship

    between two measures (e.g., foot length in centimeters and shoe size for boys and girls)

    and make charts for boys and girls separately. Discuss independent and dependent

    variables and have students decide which is the independent and which is the dependent

    variable in the activity. Instruct the students to write ordered pairs, graph them, and look

    for relationships from the graphed data. Is there a pattern in the data? (Yes, as the foot

    length increases, so does the shoe size. Does the data appear to be linear? Data should

    appear to be linear.) Help students notice the positive correlation between foot length

    and shoe size. Have students find the average ratio of foot length to shoe size. This is the

    constant of variation. Have students write an equation that models the situation (shoe size

    = ratio x foot length). Following the experiment, discuss direct variation and have the

    students come up with other examples of direct variation in real life.

Part 2: Inverse variation

    Have students work with a partner. Provide each pair with 36 algebra unit tiles. Have

    students arrange the tiles in a rectangle and record the height and width. Discuss

    independent and dependent variables. Does it matter in this situation which variable is

    independent and dependent? (No, but the class should probably decide together which to use.) Have students form as many different sized rectangles as possible and record the

    dimensions. Instruct the students to write ordered pairs, graph them, and look for

    relationships in the graphed data. Help students understand that the constant of variation

    in this experiment is a constant product. Have them write an equation to model the

    situation (height (or dependent) = 36/width (or independent))

    Algebra I?Unit 1? Understanding Quantities, Variability, and Change 4

Provide students with other data sets that will give them examples of direct variation,

     equations that can be inverse variation, and constant of variation. Ask students to write

    used to find one variable in a relationship when given a second variable from the


    Activity 8: Exponential Growth (GLEs: 2, 9, 10, 15, 29)

    18Give each student a sheet of ‖ by 11‖ paper. Have them fold the paper in half as many 2

    times as they can. After one fold, there will be two regions, after two folds four regions,

    etc. Ask the students how the area of the new region compares to the area of the original

    sheet of paper after each fold. Have the students complete a table like the one below and

    provide the data and variable expressions.

    Number of Folds Number of Regions Area of Smallest Region

    0 1 1

    ?11or 21 2 2

    ?21or 22 4 4

    ?31or 23 8 8

    . . . . . . . . .

    ?n1nor 2N n 22

Have the students complete a graph of the number of folds and the number of regions.

    Have them identify the independent and dependent variables. Is the graph linear? This is

    called an exponential growth pattern. Have the students also graph the number of folds

    and the area of the smallest region. This is called an exponential decay pattern. Include the significance of integer exponents as exponential decay is discussed.

Activity 9: Pay Day! (GLEs: 9, 10, 15, 29)

Which of the following jobs would you choose?

    ? Job A: Salary of $1 for the first year, $2 for the second year, $4 for the third

    year, continuing for 25 years

    ? Job B: Salary of $1 million a year for 25 years

At the end of 25 years, which job would produce the largest amount in total salary?

After some initial discussion of the two options, have the students work to explore the

    answer. They should organize their thinking using tables and graphs. Have the students

    represent the yearly salary and the total salary for both job options using algebraic

    expressions. Have them predict when the salaries would be equal. Return to this problem

    later in the year and have the students use technology to answer that question. Discuss

    whether the salaries represent linear or exponential growth.

    Algebra I?Unit 1? Understanding Quantities, Variability, and Change 5

Activity 10: Linear or Non-linear? (GLEs: 10, 15, 29)

Divide students into groups. Give each group a different set of the sample data at the end

    of this unit. Have each group identify the independent and dependent variables of the data

    and graph on a poster board. Let each group investigate their data and decide if it is linear

    or non-linear and present their findings to the class, displaying each poster in the front of

    the class. After all posters are displayed, conduct a whole-class discussion on the findings.

    As an extension, regression equations of the data could be put on cards and have the class

    try to match the data to the equation. Sample data sets are provided at the end of this unit.

Activity 11: Using Technology (GLEs: 10, 15, 29)

Have students enter data sets used in Activity 5 into lists in a graphing calculator and

    generate the scatter plots using the calculator.

Activity 12: Understanding Data (GLEs: 5, 10, 28, 29)

The table below gives the box score for game three of the 2003 NBA Championship series. SAN ANTONIO SPURS REBOUNDS PLAYER POS MIN FGM-A 3GM-A FTM-A OFF DEF TOT AST PF PTS TONY PARKER G 43 9-21 4-6 4-8 1 2 3 6 0 26 STEPHEN JACKSON G 36 2-7 1-2 2-4 0 6 6 2 3 7 TIM DUNCAN F 45 6-13 0-0 9-12 3 13 16 7 3 21 BRUCE BOWEN F 32 0-5 0-2 0-0 1 3 4 0 3 0 DAVID ROBINSON C 26 1-5 0-0 6-8 1 2 3 0 2 8 Emanuel Ginobili 28 3-6 0-0 2-3 2 0 2 4 2 8 Malik Rose 22 4-7 0-0 0-0 0 2 2 0 2 8 Speedy Claxton 5 2-2 0-0 0-0 0 1 1 0 1 4 Kevin Willis 3 1-1 0-0 0-0 1 0 1 0 1 2 Steve Kerr Danny Ferry Steve Smith TOTAL 240 28-67 5-10 23-35 9 29 38 19 17 84 41.8% 50.0% 65.7% Team Rebs: 15 NEW JERSEY NETS REBOUNDS PLAYER POS MIN FGM-A 3GM-A FTM-A OFF DEF TOT AST PF PTS JASON KIDD G 42 6-19 0-5 0-0 2 1 3 11 3 12 KERRY KITTLES G 43 8-16 3-5 2-3 1 3 4 1 2 21 KENYON MARTIN F 42 8-18 0-1 7-8 2 9 11 0 5 23 RICHARD JEFFERSON F 36 3-11 0-0 0-0 2 7 9 0 2 6 JASON COLLINS C 25 0-3 0-0 0-0 4 1 5 1 6 0 Lucious Harris 22 1-6 1-2 4-4 1 0 1 3 2 7 Dikembe Mutombo 18 1-1 0-0 0-0 1 2 3 0 3 2 Rodney Rogers 11 0-3 0-0 2-2 0 2 2 0 2 2 Anthony Johnson 6 2-2 0-0 0-0 0 1 1 0 0 4 Aaron Williams 4 1-2 0-0 0-0 1 1 2 1 1 2 Tamar Slay Brian Scalabrine TOTAL 240 30-81 4-13 15-17 14 27 41 17 26 79 37.0% 30.8% 88.2% Team Rebs: 10

     Source: Key for Table

    Pos Position 3GM-A 3 point goals made 3 point goals attempted

    Min. Minutes Played AST Assists FGMA Field goals madefield goals attempted PF Personal Fouls FTMA Free throws madefree throws attempted PTS Total Points Scored R Rebounds Algebra I?Unit 1? Understanding Quantities, Variability, and Change 6

    Ask detailed questions about the information in the table, such as: Who played the most minutes, who had the most assists, or which team made a larger percentage of free throws? Have students calculate the percentage of field goals made/attempted and the percentage of free throws made/attempted for each player. Which player(s) has the highest percentages? Why do you think this is so?

    Ask the students if they think that the players who attempt the most field goals are generally the players who make the most field goals. Is this a linear relationship? Have the students identify the independent and dependent variables and make a scatter plot showing field goals made and field goals attempted. Designate the players from the different teams using team colors or ―S‖ for each Spur player and ―N‖ for each New Jersey Net.

    The plot shows a positive correlation. Note that a player who makes every basket will be represented by a point on the line containing the points (0,0), (1,1), (2,2), etc. Have the students identify the points representing the players who were the four perfect shooters.

    Have the students write a brief description of their interpretations of the scatter plot, noticing that points seem to cluster into two groups. The cluster in the upper right represents players who played more than forty minutes and the cluster in the lower left represents players who played less time.

    Ask, Do you think that players who get a lot of rebounds also make a lot of assists (i.e. does the number of rebounds depend on the number of assists)? Have the students construct a scatter plot of rebounds (R) and assists (A). This scatter plot will show that there is no relationship. Have students identify other possible relationships of two-variable data and to investigate whether there is a positive correlation, negative correlation, or if no correlation exists.

Algebra I?Unit 1? Understanding Quantities, Variability, and Change 7

    Sample Assessments

    General Assessments

    ? The students will explore patterns in the perimeters and areas of figures such

    as the ―trains‖ described below.

    Train 1

    Train number 1 2 3 4 5 …

     n 1 2 3 4 5

    Area 1 4 9 16 25

     Perimeter 4 8 12 16 20

    Describe the shape of each train. (square)

    What is the length of a side of each square? (n)

    Compare the lengths of the trains with their areas and perimeters. (length-n, area-

    2, perimeter-4n) n

    Train 2

    Train Number 1 2 3 4 5 …

     n 1 2 3 4 5

    Area 1 3 6 10 15

    Perimeter 4 8 12 16 20

    nn?1??Formulas: area - , perimeter 4n 2

    ? The students will solve constructed response items, such as:

    1. Cary’s Candy Store sells giant lollipops for $1.00 each. This price is no

    longer high enough to create a profit, so Cary decides to raise the price. He

    doesn’t want to shock his customers by raising the price too suddenly or

    too dramatically. So, he considers these three plans,

    ? Plan 1: Raise the price by $0.05 each week until the price reaches


    ? Plan 2: Raise the price by 5% each week until the price reaches


    ? Plan 3: Raise the price by the same amount each week for 8 weeks,

    so that in the eighth week the price reaches $1.80.

    Algebra I?Unit 1? Understanding Quantities, Variability, and Change 8

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