Algebra 1 > Notes > Yorkcounty Final > Yorkcounty > Algebra I A & B

Curriculum Guide

Algebra IA

7/04

Algebra IA Content Outline Topic

SOLs

Suggested Time Frame

I.

PATTERNS/FUNCTIONS A. Analyzing Data for Patterns B. Determining Functional Relationships

A.5

5 blocks/ 10 single periods

II.

STATISTICAL METHODS A. Problem Solving B. Box-and-Whisker Graphs C. Measures of Central Tendency D. Range

A.17

5 blocks/ 10 single periods

III.

VARIABLES AND REAL NUMBERS A. Variables and Expressions B. Exponents and Powers C. Properties of Real Numbers D. Operations with Real Numbers E. Justification of Steps in Simplifying Expressions

A.2, A.3

4 blocks/ 8 single periods

IV.

SOLVING LINEAR EQUATIONS/INEQUALITIES A. Solving Multistep Linear Equations with One or More Transformations B. Literal Equations and Formulas C. Problem Solving

A.1

11 blocks/ 22 single periods

V.

MATRICES A. Addition and Subtraction B. Scalar Multiplication

A.4

4 blocks/ 8 single periods

VI.

LINEAR FUNCTIONS AND GRAPHS A. Graphing Linear Functions B. Slope-Intercept C. X- and Y-Intercepts D. Transformations

A.6, A.7

6 blocks/ 12 single periods

VII.

EQUATIONS OF LINES A. Writing Equations of Lines B. Standard Form C. Problem Solving

A.8, A.16

10 blocks/ 20 single periods

Algebra I: Blueprint Summary Table No. of Items

Reporting Categories Expressions and Operations

12

Relations and Functions

12

Equations and Inequalities

18

Statistics

8

Total Number of Operational Items Field-Test Items** Total Number of Items

SOLs A.2; A-10; A.11; A.12; A.13 A.5; A.15; A.18 A.1; A.3; A.6; A.7; A.8; A.9; A.14 A.4; A.16; A.17

50 10 60

* These field-test items will not be used to compute students’ scores on the test.

Algebra IA Virginia Standards of Learning A.1

The student will solve multistep linear equations and inequalities in one variable, solve literal equations (formulas) for a given variable, and apply these skills to solve practical problems. Graphing calculators will be used to confirm algebraic solutions.

A.2

The student will represent verbal quantitative situations algebraically and evaluate these expressions for given replacement values of the variables. Students will choose an appropriate computational technique, such as mental mathematics, calculator, or paper and pencil.

A.3

A.4

A.5

A.6

A.7

The student will justify steps used in simplifying expressions and solving equations and inequalities. Justifications will include the use of concrete objects; pictorial representations; and the properties of real numbers, equality, and inequality. The student will use matrices to organize and manipulate data, including matrix addition, subtraction, and scalar multiplication. Data will arise from business, industrial, and consumer situations. The student will create and use tabular, symbolic, graphical, verbal, and physical representations to analyze a given set of data for the existence of a pattern, determine the domain and range of relations, and identify the relations that are functions. The student will select, justify, and apply an appropriate technique to graph linear functions and linear inequalities in two variables. Techniques will include slope-intercept, x- and y-intercepts, graphing by transformation, and the use of the graphing calculator. The student will determine the slope of a line when given an equation of the line, the

graph of the line, or two points on the line. Slope will be described as rate of change and will be positive, negative, zero, or undefined. The graphing calculator will be used to investigate the effect of changes in the slope on the graph of the line. A.8

The student will write an equation of a line when given the graph of the line, two points on the line, or the slope and a point on the line.

A.9

The student will solve systems of two linear equations in two variables both algebraically and graphically and apply these techniques to solve practical problems. Graphing calculators will be used both as a primary tool for solution and to confirm an algebraic solution.

A.10 The student will apply the laws of exponents to perform operations on expressions with integral exponents, using scientific notation when appropriate. A.11 The student will add, subtract, and multiply polynomials and divide polynomials with monomial divisors, using concrete objects, pictorial and area representations, and algebraic manipulations. A.12 The student will factor completely first- and second-degree binomials and trinomials in one or two variables. The graphing calculator will be used as a tool for factoring and for confirming algebraic factorizations. A.13 The student will express the square root of a whole number in simplest radical form and approximate square roots to the nearest tenth. A.14 The student will solve quadratic equations in one variable both algebraically and graphically. Graphing calculators will be

used both as a primary tool in solving problems and to verify algebraic solutions. A.15 The student will, given a rule, find the values of a function for elements in its domain and locate the zeros of the function both algebraically and with a graphing calculator. The value of f(x) will be related to the ordinate on the graph. A.16 The student will, given a set of data points, write an equation for a line of best fit and use the equation to make predictions. A.17 The student will compare and contrast multiple one-variable data sets, using statistical techniques that include measures of central tendency, range, and box-andwhisker graphs. A.18 The student will analyze a relation to determine whether a direct variation exists and represent it algebraically and graphically, if possible.

Curriculum Guide/Math/Algebra IA York County School Division EXPRESSIONS AND OPERATIONS

RELATIONS AND FUNCTIONS

EQUATIONS AND INEQUALITIES

STATISTICS

Virginia Standard/s of Learning: A.5 The student will create and use tabular, symbolic, graphical, verbal, and physical representations to analyze a given set of data for the existence of a pattern; determine the domain and range of relations; and identify the relations that are functions.

BIO.1, BIO.4, BIO.5, BIO. 8, Related Standard/s: S BIO.9

C/T 12.1, 12.2,

Technology Standard/s: 12.3, 12.4

Unit I: Patterns/Functions Content Component/s:

Suggested Time Frame:

• Analyzing Data for Patterns • Determining Functional Relationships

5 blocks/10 single periods

Assessment Sample/s: • Graph and comparison of a set of equations (inequalities) chosen by the teacher • Paragraphs explaining different methods for representing paired data, including one advantage and disadvantage of using each of the different methods • Unit quizzes/tests

Essential Knowledge & Skills/Mathematics Curriculum Framework/Algebra I/p. 7:

• use physical representations, such as algebra manipulatives, to represent quantitative data.

The student will: • analyze a table of ordered pairs for the existence of a pattern that defines the change relating input and output values. • write a linear equation to represent a pattern in which there is a constant rate of change between variables. • determine from a set of ordered pairs, a table, or a graph whether a relation is a function. • identify the domain and range for a relation, given a set of ordered pairs, a table, or a graph. M9A–1

Instructional Blueprint/s: (Strategies) The teacher will: • have students, in small groups, work with an equation of a line, describing all the different information that can be determined about it. The students will discuss the most efficient techniques of graphing the equation. • use a motion detector and CBLs to show students what a graph actually represents. • use the Wave Lesson and graphing calculator to introduce independent and dependent variables. • use the lesson Discovering Rates of Change and the graphing calculator to show the importance of slope. • provide opportunities for students to investigate patterns which arise from various geometric shapes to determine such things as, what effects the changing of a dimension will have on area or perimeter? Students develop formulas to illustrate the relations. • ask students to consider the statement, “Your shoe size is a function of the size of your foot.” Develop the idea that “is a function of” actually means “depends on.” Have students give examples of additional situations in which the value of one variable results in only one true value for a second variable. Extend this to have the students use mapping to prove the relationship of data in various examples. • use cooperative groups. Have a student name a number. Have a second student perform an operation on that number and give the second item in the sequence. Each succeeding student must decide what operation was used and give the next sequence item. Rotate and repeat. ¤ explain and model for students how to determine the domain of a function given its graph. Create a worksheet of the graphs of various relations and functions. Guide students in determining the domain and the range of each graph shown.

M9A–2

Essential Understandings/Mathematics Curriculum Framework/Algebra I/p. 7: • A set of data may be characterized by patterns, and those patterns can be represented in multiple ways. • Graphs can be used as visual representations to investigate relationships between quantitative data. • Algebra is a tool for describing patterns, making generalizations, and representing a relationship in which output is related to input. • A function is a relation for which there is a unique output for each input. • A relation can be represented by a set of ordered pairs. • The domain consists of the first coordinates of the ordered pairs. • The range consists of the second coordinates of the ordered pairs. • A relation is a function if each element in the domain is paired with a unique element of the range. Resources: Text: Algebra 1, Integrations, Applications, Connections, Glencoe, pp. 56-62, 252-314 Technology: • TI-83 Graphing Calculator and CBL • Algeblaster (S) • Graphlink (S) • Algebra I (ProOne) (S) • Equation Editor (Microsoft Office) (S) • Hyperstudio (S) • PowerPoint (S) • Best Grapher Notes: • ¤ indicates an instructional blueprint for a topic that has been included in previous math SOL tests and that is not covered in the math textbook.

Curriculum Guide/Math/Algebra IA York County School Division EXPRESSIONS AND OPERATIONS

RELATIONS AND FUNCTIONS

EQUATIONS AND INEQUALITIES

STATISTICS

Virginia Standard/s of Learning: A.17 The student will compare and contrast multiple one-variable data sets, using statistical techniques that include: measures of central tendency, range, and box-and-whisker graphs.

Related Standard/s: S BIO.1, BIO.8, BIO.9

C/T 12.1, 12.2,

Technology Standard/s: 12.3, 12.4

Unit II: Statistical Methods Content Component/s: • Problem Solving • Box-and-Whisker Plots • Measures of Central Tendency • Range

Suggested Time Frame: 5 blocks/10 single periods

Assessment Sample/s: • Box-and-Whisker Plot with mean, median, and mode displays • Chart describing/providing examples of various measures of central tendency • Unit quizzes/tests

Essential Knowledge & Skills/Mathematics Curriculum Framework/Algebra I/p. 19: The student will: • calculate the measures of central tendency and range of a set of data with no more than 20 data points. • compare measures of central tendency using numerical data from a table with no more than 20 data points. • compare and contrast two sets of data, each set having no more than 20 data points, using measures of central tendency and the range. • compare and analyze two sets of data, each set having no more than 20 data points, using box-and-whisker plots. M9A–3

Instructional Blueprint/s: (Strategies) The teacher will: • give the students a list of data in tabular form and have cooperative groups of students decide which plot (stem-and-leaf or box-andwhisker) of the data they prefer and provide a rationale for their selection. • give the students the football scores of the high school’s team for each of the past four years. From this set of data, ask students to compare and analyze the team’s performance using statistics and graphs (e.g., median, mode, box-and-whisker plots). Have students use statistics and graphs to check their analysis and make new conclusions. Discuss how summarizing data is helpful to analyze data. • give students a list of their previous class test scores. Have students analyze the data after calculating the measures of central tendency (e.g., mean, median, mode). Use the information obtained to make box-and-whisker plots. Discuss how summarizing data is helpful to analyze data. Students use their individual scores over a period of time. This activity could be adapted to use with spreadsheets. • provide students with bags of M&M’s. Students estimate the number in the bag before opening. Have students open the bag and record the amount on a small post-it-note. Use the post-it-notes to create a stem-andleaf plot. Students then use the information from this collection of data to create a boxand-whisker plot. Students use measures of central tendency to describe the data.

M9A–4

Essential Understandings/Mathematics Curriculum Framework/Algebra I/p.19: • Measures of central tendency can be used to characterize a set of data and to make predictions. • Statistical techniques can be used to organize, display, and compare sets of data. • Box-and-whisker plots can be used to analyze data. Resources: Text: Algebra 1, Integrations, Applications, Connections, Glencoe, pp. 25-32, 78-83, 339345, 427-434 Technology: • TI-83 Graphing Calculator • Algeblaster (S) • Graphlink (S) • Equation Editor (Microsoft Office) (S) • Hyperstudio (S) • PowerPoint (S) Notes:

Curriculum Guide/Math/Algebra IA York County School Division EXPRESSIONS AND OPERATIONS

RELATIONS AND FUNCTIONS

EQUATIONS AND INEQUALITIES

STATISTICS

Virginia Standard/s of Learning: A.2 The student will represent verbal quantitative situations algebraically and evaluate these expressions for given replacement values of the variable. Students will choose an appropriate computational technique, such as mental mathematics, calculator, or paper and pencil. A.3 The student will justify steps used in simplifying expressions and solving equations and inequalities. Justifications will include the use of concrete objects, pictorial representations, and the properties of real numbers, equality and inequality. Related Standard/s: E 9.8

C/T 12.1, 12.2,

Technology Standard/s: 12.3, 12.4

Unit III: Variables and Real Numbers Content Component/s: • Variables and Expressions • Exponents and Powers • Properties of Real Numbers • Operations with Real Numbers • Justification of Steps in Simplifying Expressions

Suggested Time Frame: 4 blocks/8 single periods

Assessment Sample/s: • Story using 4-8 key words which indicate the four operations. The students must demonstrate comprehension of the words and their functions in mathematics. Use a rubric to set scoring guidelines for students. • List of expressions which student has simplified using pictorial representations

and numbers and justification of each step using the properties of real numbers • Student-generated expressions to share with other groups • Paragraph explaining how you can show positive and negative numbers, variables and variable expressions with the algebra tiles • Unit quizzes/tests

Essential Knowledge & Skills/Mathematics Curriculum Framework/Algebra I/pp. 2 & 11:

• simplify expressions and solve equations and inequalities, using the order of operations. • solve equations, using the addition, multiplication, closure, identity, and inverse properties. • solve equations, using the reflexive, symmetric, transitive, and substitution properties of equality. • create and interpret pictorial representations for simplifying expressions and solving equations and inequalities.

The student will: • translate verbal expressions into algebraic expressions with three or fewer terms. • relate a polynomial expression with three or fewer terms to a verbal expression. • evaluate algebraic expressions for a given replacement set to include integers and rational numbers. • apply appropriate computational techniques to evaluate an algebraic expression. • simplify expressions and solve equations and inequalities, using the commutative, associative, and distributive properties.

M9A–5

Instructional Blueprint/s: (Strategies) The teacher will: • prepare the students for the game: “I Have. You Have.” Create small self-made cards with verbal expressions on one side and an algebraic expression on the other side. Tell the students to read the “I have…” side of the card. The student with the algebraic forms for that verbal expression answers “You have…”, then he/she reads his/her verbal expression. Play continues in that pattern. • provide 3 x 5 index cards. Write algebraic expressions on half of the cards and the equivalent verbal expressions on the other half. After shuffling the cards, distribute them to students. The student will then search for a match to his/her card. This is the “Concentration” format. • divide students into groups of four or five and ask them to make a list of key words that imply the four basic operations. After comparing the lists with other groups, each group will use the list to translate ten given verbal expressions to algebraic expressions. The groups will develop their own verbal expressions to translate. • place students in groups of three. Have one student write a mathematical expression. Have another student write the expression in words. Next, have a third student translate the words back to the expression. Compare the initial and final expressions. If they differ, verbalize each step to determine what was done incorrectly. • provide algebra tiles to verify steps in solving given equations. Students use the tiles to solve their own equations. • pair students. Provide students with a dictionary to define terms used in the properties (e.g., commutative). Students use numerical examples, objects, and pictures to demonstrate understanding of the properties. • prepare a set of cards. The names of the properties will be on one set of colored cards. Several examples of each property will be on cards of a different color. The cards should be shuffled. Students match the examples with the property name. M9A–6

¤ explain the identification of the sum of monomials represented by a model. Prepare a set of cards for a “Concentration” game with algebraic models of polynomial expressions on one card and the matching algebraic expressions on the other side. Students are assigned to find the matching pairs. ¤ provide a variety of examples of solving consumer problems using given formulas. Create a chart of formulas, some within word problems, for the students to algebraically evaluate for given values. Arrange for students to work with problems related to current consumer issues, sports, and music. ¤ explain the identification of a property that justifies a given algebraic manipulation. Distribute a solution key for a variety of algebraically-evaluated expressions. Place students in groups of 2-3. Tell the students to state the property that justifies each step. Essential Understandings/Math Curriculum Framework/Algebra 1/p. 2 & 11: • Algebra is a tool for reasoning about quantitative situations so that relationships become apparent. • Algebra is a tool for describing and representing patterns and relationships. • The numerical values of an expression are dependent upon the values of the replacement set for the variables. • There are a variety of ways to compute the value of a numerical expression and evaluate an algebraic expression. • The operations and the magnitude of the numbers in an expression impact the choice of an appropriate method of computation. • The representation and manipulation of expressions, equations, and inequalities can be modeled in a variety of ways, using concrete, pictorial, and symbolic representations. • Properties of real numbers and properties of equations and inequalities can be used to solve equations and inequalities and simplify expressions.

Resources:

Curriculum Guide/Math/Algebra IA Notes:

Text: Algebra 1, Integrations, Applications, Connections, Glencoe, pp. 6-69 • Algebra tiles

Real Numbers - some history - A history of real numbers. h t t p : / / w w w. r b j o n e s . c o m / r b j p u b / m a t h s / math008.htm

Technology: •Virginia TI-83 Graphing Calculator Standard/s of Learning: • Algeblaster (S) • Graphlink (S) • Algebra I (ProOne) (S) • Jasper Woodbury–”The General Is Missing (LD) • Equation Editor (Microsoft Office) (S) • Hyperstudio (S) •Related PowerPoint (S) Standard/s:

Content Component/s:

York County School Division

Algebra I, Real Numbers - A short test on real numbers. Student scores are reported back to students. http://library.advanced.org/11771/english/hi/ math/tests/alg/1.html • ¤ indicates an instructional blueprint for a topic that has been included in previous math SOL 12.2, tests and that is Standard/s: not covered inC/T the12.1, math textTechnology 12.3, 12.4 book.

Suggested Time Frame:

Assessment Sample/s:

M9A–7

M9A–8

Curriculum Guide/Math/Algebra IA York County School Division EXPRESSIONS AND OPERATIONS

RELATIONS AND FUNCTIONS

EQUATIONS AND INEQUALITIES

STATISTICS

Virginia Standard/s of Learning: A.1 The student will solve multistep linear equations and inequalities in one variable, solve literal equations (formulas) for a given variable, and apply these skills to solve practical problems. Graphing calculators will be used to confirm algebraic solutions.

Related Standard/s: E 9.8; M A.2

C/T 12.1, 12.2,

Technology Standard/s: 12.3, 12.4

Unit IV: Solving Linear Equations/Inequalities Content Component/s: • Solving Multistep Linear Equations with One or More Transformations • Literal Equations and Formulas • Problem Solving Assessment Sample/s: • Identification of property that justifies steps in a list of 5-6 solved equations and/or simplified expressions • Demonstration of how to find the distance, rate or time, given two of the unknowns, using the graphing calculator. Determination of each answer and Essential Knowledge & Skills/Mathematics Curriculum Framework/Algebra I/p. 10: The student will: • translate verbal sentences to algebraic equations and inequalities in one variable. • solve multistep linear equations and noncompound inequalities in one variable with the variable in both sides of the equation or inequality. • solve multistep linear equations and inequalities in one variable with grouping symbols in one or both sides of the equation or inequality. • solve multistep equations and inequalities in one variable with rational coefficients and constants.

Suggested Time Frame: 11 blocks/22 single periods

recording the literal equation (e.g., d = rt; d/r = t; d/t = r) • Poster designed to represent a real-life situation that can be translated into a numerical or algebraic word phrase to include order of operations. • Unit quizzes/tests • solve a literal equation (formula) for a specified variable. • apply skills for solving linear equations to practical situations. • confirm algebraic solutions to linear equations and inequalities, using a graphing calculator.

M9A–9

Instructional Blueprint/s: (Strategies) The teacher will: • make a set of “Property Rummy” cards, which will include the properties of equality and a number of expressions which illustrate the properties. Have students work in small groups of three or four. The object of this card game is to form sets of three cards each. Each set must contain one property card and two expression cards that illustrate that property. To begin, each player is dealt eight cards. The remaining cards are placed face down and the top card is turned face up. Next, turns are taken by either drawing the top face up or face down card and then discarding one of the cards. When two sets can be formed from the cards in hand, they are placed in front of the player. The first player to do this wins the game. • arrange students in groups of four. Model non-numerical processes using a balance scale or a scale made using a meter stick and a triangular prism. Students find different ways to keep the scales balanced. Students are asked to add or subtract to find the “balance” for an unknown object. • provide students with red and white squares, which represent positive and negative integers, respectively, to model a one-step equation. A white square “cancels” a red one. Students solve equations by manipulating the squares until one side is empty. • divide the class into four or five small groups. Have each group create five word problems involving either an equation or inequality. The word problems should be practical (e.g., involving area, interest, deriving a minimum test score needed for a particular test average, the menu of a fast food restaurant). Have groups switch and solve each other’s problems. Answers are verified by the group that originally created the problems. • discuss the use of balance scales. Students are asked to find five different ways to keep the scales balanced. They are also asked to add or subtract to find the balance weight of an unknown object. • provide Quiz on a Card. Each group of 5 M9A–10

students will be given a set of 5 index cards with a numerical or algebraic word phrase printed on each card. Each group member will choose a card and write the phrase as an expression and evaluate the expression. ¤ provide examples of equations and how they are used to solve word problems. Split the class into groups of 2-3. Write several equations on the board, such as 3x + 5 = 17 or 5(x-1) + 10 = 40. Ask the groups to create a word problem based on student interest to match the equations, then solve the problem and answer the question. Have the groups exchange problems to verify that the solution and the created problem match the equation. Essential Understandings/Mathematics Curriculum Framework/Algebra I/p.10: • A solution to an equation or inequality is the value or set of values that can be substituted to make the equation or inequality true. • Equations and inequalities can be solved in a variety of ways. • The solution of an equation in one variable can be found by graphing each side of the equation separately and finding the x-coordinate of the point of intersection. • Practical problems can be interpreted, represented, and solved using linear equations and inequalities in one variable

Resources: Text: Algebra 1, Integrations, Applications, Connections, Glencoe, pp. 140-191 • Algebra tiles Technology: • TI-83 Graphing Calculator • Algeblaster (S) Standard/s of Learning: •Virginia Graphlink (S) • Algebra I (ProOne) (S) • Jasper Woodbury–”The General Is Missing (LD) • Equation Editor (Microsoft Office) (S) • Hyperstudio (S) • PowerPoint (S) Notes: Related Standard/s: A.1 Strategies • Lesson 5: Linear Equations - The index to linear algebra and how to graph linear equaContent Component/s: tions using the “Internet Academy.” Excellent source of problems and how to solve them in little increments. http://iasec.fwsd.wednet.edu/iamath/ page2d.htm#Lesson5 • Linear Inequalities - This is a complete lecture on slides. This address takes you to the text versionSample/s: from there you can go to the Assessment graphic version, but it takes a long time to download. http://mesa7.mesa.colorado.edu/~bornmann/ classes/math091/lecture/chap_02/ tsld014.htm • Assignment 10: More solving formulas This site offers a good explanation on how to work through solving formulas and what you can and cannot combine. http://198.85.210.30/~stonere/oph101/ lesn10c.htm A.1 Resources • Algebra Through Modeling - Tutorial on how to use the TI-82 to enter and organize data. http://www-cm.math.uniuc.edu/MathLink/algebra-module/ALG.index.html • Algebra Online - Free service designed to allow students, parents, and educators to communicate including free tutoring, chat room and message board. http://www.algebra-online.com • Algebra I - Linear Equations - Graphing capabilities for linear equations and quizzes. http://www.bremenbraves.com/algebra/ index.html

Curriculum Guide/Math/Algebra IA A.2 Strategies • The next generation of problem York County Schoolsolving Division This Pacific Tech site provides various problems and lesson plans about solving word problems and how to solve them step-bystep. Not just for algebra, either! http://www2.hawaii.edu/suremath/ intro_algebra.html • Algebra word problems - A question for Dr. Math asked by a student about where to put the number and the letters when converting word problems into algebraic equations. http://forum.swarthmore.edu/dr.math/problems/mike6.6.96.html • Problem Solving Methods - This site, (author - Alan M. Selby, of Montreal, Canada) provides different strategies for C/T attacking 12.1, 12.2, a Technology word problem. Standard/s: 12.3, 12.4 http://www.cam.org/~aselby/prob.html A.2 Resources • Math Archive - LotsSuggested of word problems within Time Frame: various categories. Solutions are included, unless they haven’t found one yet. http://bsuvc.bsu.edu/~d004ucslabs/ • Word problems galore from the Mathematics Problem Solving Task Centre of The Mathematical Association of Victoria. http://www.srl.rmit.edu.au/mav/PSTC/general/index.html • The Problem Solving Corner - poses weekly problems for Virginia schools. Students get personal responses and their names up on the Web if their solution is correct. http://www.wm.edu/education/Faculty/Mason/pscmain.html • Math Forum Student Center - Weekly problems and the famous Ask Dr. Math corner. http://forum.swarthmore.edu/students/ • Words Before Symbols - Background information on the rationale of algebra and algebraic thinking. http://www.cam.org/~aselby/volumee/html/ bk1bch03.html • Best Grapher • ¤ indicates an instructional blueprint for a topic that has been included in previous math SOL tests and that is not covered in the math textbook.

M9A–11

M9A–12

Curriculum Guide/Math/Algebra IA York County School Division EXPRESSIONS AND OPERATIONS

RELATIONS AND FUNCTIONS

EQUATIONS AND INEQUALITIES

STATISTICS

Virginia Standard/s of Learning: A.4 The student will use matrices to organize and manipulate data, including matrix addition, subtraction, and scalar multiplication. Data will arise from business, industrial, and consumer situations.

Related Standard/s: S BIO.1, BIO.6

C/T 12.1, 12.2,

Technology Standard/s: 12.3, 12.4

Unit V: Matrices Content Component/s: • Addition and Subtraction • Scalar Multiplication

Assessment Sample/s: • Matrix to organize the following information: On the first day of ticket sales, the yearbook staff sold 60 tickets to 6th graders, 112 tickets to 7th graders, and 120 tickets to 8th graders. On the second day of ticket sales, 85 tickets were sold to 6th graders, 86 tickets to 7th graders, and 126 tickets to 8th graders. Which matrix best organizes the ticket sales for the

Suggested Time Frame: 4 blocks/8 single periods

two days. Give 4 choices in the following form: 6th 7th 8th

Day 1

Day 2

• Unit quizzes/tests

Essential Knowledge & Skills/Mathematics Curriculum Framework/Algebra I/p. 17: The student will: • represent data from practical problems in matrix form. • calculate the sum or difference of two given matrices that are no larger than 4 x 4. • calculate the product of a scalar and a matrix that is no larger than 4 x 4. • solve practical problems involving matrix addition, subtraction, and scalar multiplication, using matrices that are no larger than 4 x 4. • read and interpret the data in a matrix representing the solution to a practical problem. M9A–13

Instructional Blueprint/s: (Strategies)

Resources:

The teacher will: • have students obtain data from the class (e.g., males, females vs. favorite television show broken into types) and display this in matrix form. Students perform operations on the matrix. Follow-up by using the graphing calculator to enter the date into matrix form and to perform the arithmetic with the matrices. • give students a stadium or auditorium seating chart and a make-believe ticket for a seat. They will determine their place in the seating matrix. • give the students a set of data that describes the number of boys and girls for each grade (9, 10, 11, 12) at “East H.S.” and “West H.S.” Students represent the data in matrix form using sex (male or female) as the rows and grade (9, 10, 11, and 12) as the columns. The School Board decides to combine the two schools into one. Using a matrix addition describe the total enrollment by sex and grade for the new school.

Text: Algebra 1, Integrations, Applications, Connections, Glencoe, pp. 70-138 Technology: • TI-83 Graphing Calculator • Algeblaster (S) • Graphlink (S) • Algebra I (ProOne) (S) • Equation Editor (Microsoft Office) (S) • Hyperstudio (S) • PowerPoint (S)

Essential Understandings/Math Curriculum Framework/Algebra 1/p. 17: • Matrices are a tool for organizing and displaying data. • A relationship exists between arithmetic operations and operations with matrices. • Matrices can be used to solve practical problems. • Only matrices of the same dimensions can be added or subtracted.

Notes: A.4 Resources Java Script Linear Algebra I - Computer program that will help with matrices as well as linear equations. Has explanations on how the program works, too. http://www.csus.edu/org/mathsoc/line_alg.html Peanut Software for Windows - Freeware from plotting, exploring geometry, statistics, fractals, discrete math, to matrices. http://www.exeter.edu/~rparris/ Math 45 - Linear Algebra - Perhaps a little advanced for Algebra I, but it may have general ideas on how to use every day material in linear and matrix algebra. http://www.redwoods.cc.ca.us/sciweb/instruct/ darnold/LinAlg/activity.htm Linear Algebra - Lots of modules about matrix operations. Has worksheets for “helper applications such as Mathcad, Maple, and Mathematica (which are quite necessary). http://www.math.duke.edu/modules/materials/ linalg/ Mathematics Archives: Industrial Mathematics - Links to sites involving industrial math. http://archives.math.utk.edu/topics/industrial/ Math.html

M9A–14

Curriculum Guide/Math/Algebra IA York County School Division EXPRESSIONS AND OPERATIONS

RELATIONS AND FUNCTIONS

EQUATIONS AND INEQUALITIES

STATISTICS

Virginia Standard/s of Learning: A.6 The student will select, justify, and apply an appropriate technique to graph a linear function and linear inequalities in two variables. Techniques will include slope-intercept, x- and y-intercepts, graphing by transformation, and the use of the graphing calculator. A.7 The student will determine the slope of a line when given an equation of the line, the graph of the line, or two points on the line. Slope will be described as rate of change and will be positive, negative, zero, or undefined. The graphing calculator will be used to investigate the effect of changes in the slope on the graph of the line. Related Standard/s: S BIO.1, BIO.2, BIO.8, BIO.9

C/T 12.1, 12.2,

Technology Standard/s: 12.3, 12.4

Unit VI: Linear Functions and Graphs Content Component/s: • Graphing Linear Functions • Slope-Intercept • X- and Y-Intercepts • Transformations

Suggested Time Frame: 6 blocks/12 single periods

Assessment Sample/s: • Examples of slopes in real situations • Examples of solutions to linear equations • Unit quizzes/tests

Essential Knowledge & Skills/Mathematics Curriculum Framework/Algebra I/pp. 12– 13: The student will: • graph linear equations and inequalities in two variables that arise from a variety of practical situations. • use the line y = x as a reference, and apply transformations defined by changes in the slope of y-intercept. • express linear functions or inequalities in slope-intercept form, and use the graphing calculator to display the relationship. • explain why a given technique is appropriate for graphing a linear function.

• recognize that m represents the slope in the equation of the form y = mx + b. • find the slope of the line, given the equation of a linear function. • calculate the slope of a line, given the coordinates of two points on the line. • find the slope of a line, given the graph of a line. • recognize and describe a line with a slope that is positive, negative, zero, or undefined. • describe slope as a constant rate of change between two variables. • compare the slopes of graphs or linear functions, using the graphing calculator.

M9A–15

Instructional Blueprint/s: (Strategies) The teacher will: • have students calculate the slope of several staircases in the school, as well as delivery ramps or handicap ramps. Students write the equation to draw the graph of the line that would represent ramps or stairs. Use the equation to draw the graph of the lines on the graphing calculator. Discuss how the changes in slope affect the “steepness” of a line. • use graphing calculators. Have students investigate the changes in the graph caused by changing the values of the constant and coefficient. This will allow the student to visually compare several equations at the same time. ¤ exhibit examples of tables and graphs of a line, showing students how to identify the table that matches the graph of a line. Create an activity related to a current topic (e.g., music, arts, sports) in which students graph various lines, by first making a table of values. Emphasize graphing of horizontal and vertical lines using this method. ¤ show students how to identify a graph with a translated equation of line. Pair students and tell them to translate graphs by a given amount. For example, “translate the graph of y = 3x + 4 up 2 units. Students will then provide the equation of the new graph. Essential Understandings/Math Curriculum Framework/Algebra I/P.12-13: • Linear functions and inequalities can be written in a variety of forms. • Linear functions and inequalities can be graphed, using a variety of techniques. • An appropriate technique for graphing linear functions and inequalities can be determined by the given information and/or the tools available. • Justification of an appropriate technique for graphing linear equations and inequalities is dependent upon the application of slope, xand y-intercepts, and graphing by transformations. • Linear equations and inequalities arise from a variety of practical situations. M9A–16

• The slope of a linear function represents a constant rate of change in the dependent variable when the independent variable changes by a fixed amount. • The slope of a line determines its relative steepness. • The slope of a line can be determined in a variety of ways. • Changes in slope affect the graph of a line. Resources: Text: Algebra 1, Integrations, Applications, Connections, Glencoe, pp. 252-316 Technology: • TI-83 Graphing Calculator • Algeblaster (S) • Graphlink (S) • Algebra I (ProOne) (S) • Equation Editor (Microsoft Office) (S) • Hyperstudio (S) • PowerPoint (S) • Best Grapher Notes: ¤ indicates an instructional blueprint for a topic that has been included in previous math SOL tests and that is not covered in the math textbook.

Curriculum Guide/Math/Algebra IA York County School Division EXPRESSIONS AND OPERATIONS

RELATIONS AND FUNCTIONS

EQUATIONS AND INEQUALITIES

STATISTICS

Virginia Standard/s of Learning: A.8 The student will write an equation of a line when given the graph of the line, two points on the line, or the slope and a point on the line. A.16 The student will, given a set of data points, write an equation for a line of best fit and use the equation to make predictions.

Related Standard/s: S BIO.1, BIO.2, BIO.4, BIO.5

C/T 12.1, 12.2,

Technology Standard/s: 12.3, 12.4

Unit VII: Equations of Lines Content Component/s: • Writing Equations of Lines • Standard Form • Problem Solving

Suggested Time Frame: 10 blocks/20 single periods

Assessment Sample/s: • • • •

Writing of equations given 2 points, or a point and slope Graph of line of best fit with specified data from teacher Transformation of a linear equation from standard to slope-intercept form Unit quizzes/tests

Essential Knowledge & Skills/Mathematics Curriculum Framework/Algebra I/pp. 14 & 18: The student will: • recognize that equations of the form y = mx + b and Ax + By = C are equations of lines. • write an equation of a line when given the graph of a line. • write an equation of a line when given two points on the line whose coordinates are integers. • write an equation of a line when given the slope and a point on the line whose coordinates are integers. • write an equation of a vertical line as x = c.

• write an equation of a horizontal line as y = c. • write an equation for the line of a best fit, given a set of six to ten data points in a table, on a graph, or from a practical situation. • make predictions about unknown outcomes, using the equation of a line of best fit.

M9A–17

Instructional Blueprint/s: (Strategies) The teacher will: • give students graphs of several lines and have students reproduce the graphs on the graphing calculator. Students describe the strategy used to find the equation. Sketch a flow chart that shows the steps. • divide students into pairs. One student is given a card with a graph or line which he/she describes as accurately and precisely as possible to his/her partner. The other student writes the equation of the line. Students then look at the graphs to check the answers. The partners then switch positions and repeat. • have students use a chalkboard grid to find slope and y-intercept of the line. Have students use the slope formula and one point to find the equation. Use the slope formula to discuss vertical and horizontal lines. Have students use graphing calculators to find slope and intercepts. • allow students to take on the role of “forensic mathematicians,” trying to determine how tall a deceased person was whose femur is 17 inches long. Students measure their own femurs and their heights, entering this data into a graphing calculator or computer and creating a scatterplot. They note that the data are approximately linear, so they use the built-in linear regression procedures to find the line of best fit. • provide activities from Real World Math with the CBL System and CBL units connected to TI-82 calculators to collect data. Students or groups of students link with others to have additional data. They use the stat calc functions to determine the best fit (e.g., model). • provide students with a board, which represents the first quadrant of a plane, and nails to represent the location of points to plot data. Using rubber bands wrapped around a rod, students connect the rubber bands to the nails. The “line” which is formed by the rod is a good representation of the “best fit line” for the data. ¤ model how to determine an equation for a line on a graph using technology, manipulatives, and the chalkboard. Provide M9A–18

students with a variety of graphs related to current topics. Tell students to review the graphs and find the equations of the lines shown. Students can use manipulatives to portray the graphs and equations. Essential Understandings/Math Curriculum Framework/Algebra 1/pp. 14 & 18: • The equation of a line defines the relationship between two variables. • The graph of a line represents the set of points that satisfies the equation of a line. • A line can be represented by its graph or by an equation. • The equation of a line can be determined by two points on the line or by the slope and a point on the line. • The graphing calculator can be used to determine the equation of a line of best fit for a set of data. • The line of best fit for the relationship among a set of data points can be used to make predictions where appropriate. Resources: Text: Algebra 1, Integrations, Applications, Connections, Glencoe, pp. 322-380 Technology: • TI-83 Graphing Calculator • Algeblaster (S) • Graphlink (S) • Algebra I (ProOne) (S) • Equation Editor (Microsoft Office) (S) • Hyperstudio (S) • PowerPoint (S) • Best Grapher Notes: ¤ indicates an instructional blueprint for a topic that has been included in previous math SOL tests and that is not covered in the math textbook.

Curriculum Guide

Algebra IB

7/04

Algebra IB Content Outline Topic I.

SYSTEMS OF EQUATIONS AND INEQUALITIES A. Solving Linear Systems Graphically and Algebraically B. Systems of Linear Inequalities C. Linear Programming D. Problem Solving

II.

III.

SOLs

Suggested Time Frame

A.9

9 blocks/ 18 single periods

POWERS AND EXPONENTS A. Properties of Exponents B. Scientific Notation C. Problem Solving D. Square Roots

A.10, A.13

6 blocks/ 12 single periods

POLYNOMIALS A. Polynomial Operations B. Factoring Quadratic Trinomials C. Problem Solving

A.11, A.12

8 blocks/ 16 single periods

IV.

QUADRATIC EQUATIONS A. Factoring Techniques B. Graphing Solutions

A.14

4 blocks/ 8 single periods

V.

DIRECT VARIATION A. Writing Equations B. Problem Solving

A.18

4 blocks/ 8 single periods

VI.

FUNCTIONAL RELATIONSHIPS A. Domain and Range B. Zeros C. Graphing

A.15

4 blocks/ 8 single periods

VII.

RATIONAL EXPRESSIONS, EQUATIONS AND THEIR APPLICATIONS (AFTER SOL TEST) A. Proportion and Percent B. Probability C. Simplifying Rational Expressions D. Operations on Rational Expressions E. Solving Rational Equations F. Problem Solving

10 blocks/ 20 single periods

Algebra I: Blueprint Summary Table No. of Items

Reporting Categories Expressions and Operations

12

Relations and Functions

12

Equations and Inequalities

18

Statistics

8

Total Number of Operational Items Field-Test Items** Total Number of Items

SOLs A.2; A-10; A.11; A.12; A.13 A.5; A.15; A.18 A.1; A.3; A.6; A.7; A.8; A.9; A.14 A.4; A.16; A.17

50 10 60

* These field-test items will not be used to compute students’ scores on the test.

Algebra IB Virginia Standards of Learning A.1

The student will solve multistep linear equations and inequalities in one variable, solve literal equations (formulas) for a given variable, and apply these skills to solve practical problems. Graphing calculators will be used to confirm algebraic solutions.

A.2

The student will represent verbal quantitative situations algebraically and evaluate these expressions for given replacement values of the variables. Students will choose an appropriate computational technique, such as mental mathematics, calculator, or paper and pencil.

A.3

A.4

A.5

A.6

A.7

The student will justify steps used in simplifying expressions and solving equations and inequalities. Justifications will include the use of concrete objects; pictorial representations; and the properties of real numbers, equality, and inequality. The student will use matrices to organize and manipulate data, including matrix addition, subtraction, and scalar multiplication. Data will arise from business, industrial, and consumer situations. The student will create and use tabular, symbolic, graphical, verbal, and physical representations to analyze a given set of data for the existence of a pattern, determine the domain and range of relations, and identify the relations that are functions. The student will select, justify, and apply an appropriate technique to graph linear functions and linear inequalities in two variables. Techniques will include slope-intercept, x- and y-intercepts, graphing by transformation, and the use of the graphing calculator. The student will determine the slope of a line when given an equation of the line, the

graph of the line, or two points on the line. Slope will be described as rate of change and will be positive, negative, zero, or undefined. The graphing calculator will be used to investigate the effect of changes in the slope on the graph of the line. A.8

The student will write an equation of a line when given the graph of the line, two points on the line, or the slope and a point on the line.

A.9

The student will solve systems of two linear equations in two variables both algebraically and graphically and apply these techniques to solve practical problems. Graphing calculators will be used both as a primary tool for solution and to confirm an algebraic solution.

A.10 The student will apply the laws of exponents to perform operations on expressions with integral exponents, using scientific notation when appropriate. A.11 The student will add, subtract, and multiply polynomials and divide polynomials with monomial divisors, using concrete objects, pictorial and area representations, and algebraic manipulations. A.12 The student will factor completely first- and second-degree binomials and trinomials in one or two variables. The graphing calculator will be used as a tool for factoring and for confirming algebraic factorizations. A.13 The student will express the square root of a whole number in simplest radical form and approximate square roots to the nearest tenth. A.14 The student will solve quadratic equations in one variable both algebraically and graphically. Graphing calculators will be

used both as a primary tool in solving problems and to verify algebraic solutions. A.15 The student will, given a rule, find the values of a function for elements in its domain and locate the zeros of the function both algebraically and with a graphing calculator. The value of f(x) will be related to the ordinate on the graph. A.16 The student will, given a set of data points, write an equation for a line of best fit and use the equation to make predictions. A.17 The student will compare and contrast multiple one-variable data sets, using statistical techniques that include measures of central tendency, range, and box-and-whisker graphs. A.18 The student will analyze a relation to determine whether a direct variation

Curriculum Guide/Math/Algebra IB York County School Division EXPRESSIONS AND OPERATIONS

RELATIONS AND FUNCTIONS

EQUATIONS AND INEQUALITIES

STATISTICS

Virginia Standard/s of Learning: A.9 The student will solve systems of two linear equations in two variables, both algebraically and graphically, and apply these techniques to solve practical problems. Graphing calculators will be used both as a primary tool of solution and to confirm an algebraic solution.

Related Standard/s: H/SS WHII.1

C/T 12.1, 12.2,

Technology Standard/s: 12.3, 12.4

Unit I: Systems of Equations and Inequalities Content Component/s: • Solving Linear Systems Graphically and Algebraically • Systems of Linear Inequalities • Linear Programming • Problem Solving

Suggested Time Frame: 9 blocks/18 single periods

Assessment Sample/s: • Real-life applications that can be solved by a system • Solution to a system of linear equalities by graphing • Model of real-life situation using linear programming • Unit quizzes/tests

Essential Knowledge & Skills/Mathematics Curriculum Framework/Algebra I/p. 15: The student will: • given a system of two linear equations in two variables that has a unique solution, solve the system by substitution or elimination to find the ordered pair which satisfies both equations. • given a system of two linear equations in two variables that has a unique solution, solve the system graphically to find the point of intersection. • determine whether a system of two linear equations has one solution, no solution, or infinite solutions.

• write a system of two linear equations that describes a practical situation. • interpret and determine the reasonableness of the algebraic or graphical solution of a system of two linear equations that describes a practical situation.

M9B–1

Instructional Blueprint/s: (Strategies) The teacher will: • give the students an equation such as 5x + 3y = 15 to find two or more equations that satisfy each of these requirements: (1) The graphs of the given equation and a second equation intersect at a single point; (2) The graphs of the given equation and a second equation intersect at more than one point. • have students use a chalkboard grid and a graphing calculator to solve systems and to check systems solved by elimination and substitution. Have students use systems to solve problems involving price increases and profit margins, banking problems, investments, different rates, and different principles. • divide students into groups. Have each group solve a system of equations by a pre-described method. Make sure that all methods are assigned. Have students display their solutions to the class and discuss the most appropriate method for solving the system. • play “SYSTO” with the class. Make bingo cards (5 x 5) in the squares of which students will write ordered pairs. The ordered pairs come from a teacher-generated list of solutions to systems of equations. The teacher shows the students the systems of equations, one-by-one. After solving, students mark their cards with the solution. The first student to fill a row, wins. • have the students find points of intersection on the TI-83 calculators and connect this concept to systems with “no solutions” or “many solutions.” Essential Understandings/Math Curriculum Framework/Algebra 1/p. 15: • A system of linear equations with exactly one solution is characterized by the graphs of two lines whose intersection is a single point, and the coordinates of this point satisfy both equations. • A point shared by two intersecting graphs and the ordered pair that satisfies the equations characterize a system of equations with only one solution. • A system of two linear equations with no soM9B–2

lution is characterized by the graphs of two lines that do not intersect but are parallel. • A system of two linear equations having infinite solutions is characterized by two graphs that coincide (the graphs will appear to be the graph of one line), and all the coordinates on this one line satisfy both equations. • Systems of two linear equations can be used to represent two conditions that must be satisfied simultaneously. Resources: Text: Algebra 1, Integrations, Applications, Connections, Glencoe, pp. 382-444, 450-492 Technology: • TI-83 Graphing Calculator • Algeblaster (S) • Graphlink (S) • Algebra I (ProOne) (S) • Jasper Woodbury–”The General Is Missing (LD) • Equation Editor (Microsoft Office) (S) • Hyperstudio (S) • PowerPoint (S) Notes: A.9 Resources Mathematics of Cartography - Using the theme of maps, an investigation into the mathematical concepts of points, lines, areas, coordinates, and linear algebra. http://math.rice.edu/~lanius/pres/map Technology: • Best Grapher

Curriculum Guide/Math/Algebra IB York County School Division EXPRESSIONS AND OPERATIONS

RELATIONS AND FUNCTIONS

EQUATIONS AND INEQUALITIES

STATISTICS

Virginia Standard/s of Learning: A.10 The student will apply the laws of exponents to perform operations on expressions with integral exponents, using scientific notation when appropriate. A.13 The student will express the square root of a whole number in simplest radical form and approximate square roots to the nearest tenth.

Related Standard/s: E 9.8

C/T 12.1, 12.2,

Technology Standard/s: 12.3, 12.4

Unit II: Powers and Exponents Content Component/s: • Properties of Exponents • Scientific Notation • Problem Solving • Square Roots

Suggested Time Frame: 6 blocks/12 single periods

Assessment Sample/s: • Model of a paper repeatedly folded into thirds, showing powers of three • Explanation of the use of negative and zero exponents in algebraic expressions • Detailed solution using scientific notation to a specific problem related to a content area other than math • Unit quizzes/tests

Essential Knowledge & Skills/Mathematics Curriculum Framework/Algebra I/p. 3: The student will: • identify the base, exponent, and coefficient in a monomial expression. • simplify monomial expressions and ratios of monomial expressions in which the exponents are integers, using the laws of exponents. • express numbers, using scientific notation, and perform operations, using the laws of exponents. • estimate the square root of a non-perfect square to the nearest tenth by - identifying the two perfect squares it lies between;

- finding the square root of those two perfect squares; and - using those values to estimate the square root of the non-perfect square. • find the square root of a number, and make a reasonable interpretation of the displayed value for a given situation, using a calculator. • express the square root of a whole number less than 1,000 in simplest radical form

M9B–3

Instructional Blueprint/s: (Strategies) The teacher will: • have students investigate the following situations involving the equation y = 2x. 1. Compare the total money earned if a person makes a penny the first day, and doubles the amount every day to the amount that a person makes who earns the same amount every day. 2. Fold a piece of paper in half repeatedly and investigate how many sections there are. 3. Compute population over a period of time (e.g., from two cats, two mosquitoes). 4. Investigate large and small numbers from the almanac, a science book, or other appropriate data and write in scientific notation. • have students work in learning groups of student ecologists. Each group chooses two species of animals to research. Each group prepares a chart to generate population polynomials. Each chart should include: 1. Average number of offspring per litter. 2. Estimated number of females per litter (use 50 total offsprings). (This number will serve as x.) 3. Columns for each generation of females: first (1), second (x), third (x2), fourth (x3). 4. The fifth column will be the total (1 + x + x2 + x3). Each group finds the number of animals that can be expected after four generations. • give students a list of numbers. Have them identify which two perfect squares each number is between. Students estimate the square root and check estimates with calculators. • have students work in pairs and write number on cards. They write the square or square root on an equal number of cards. Have students find matches in a game format like “Concentration” or “Jeopardy.” • present “Square Root Price is Right.” If a student is given a radical expression such as: , , , he/she must give 85 , the nearest integral square root, “without going over.” • form a number line using students to represent whole numbers. Give other students M9B–4

numbers that are not perfect squares. Have these students determine between which two students to stand to represent the square root of their number. Students can determine where the person should move to get a good estimate of the square root. Essential Understandings/Math Curriculum Framework/Algebra 1/pp. 3 & 6.: • Repeated multiplication can be represented with exponents. • The laws of exponents can be investigated using patterns. • The base and the exponent impact the magnitude of the expression. • A relationship exists between the laws of exponents and scientific notation. • The square root of a perfect square is an integer. • The square root of a non-perfect square lies between two consecutive integers. • The inverse of squaring a number is determining the square root. • A radical in simplest form is one in which the radicand has no perfect square factors other than one. • The square root of a product is the product of the square roots. Resources: Text: Algebra 1, Integrations, Applications, Connections, Glencoe, pp. 494-519, 556-606 Technology: • TI-83 Graphing Calculator • Algeblaster (S) • Graphlink (S) • Algebra I (ProOne) (S) • Equation Editor (Microsoft Office) (S) • Hyperstudio (S) • PowerPoint (S) Notes: Technology: • Best Grapher

Curriculum Guide/Math/Algebra IB York County School Division EXPRESSIONS AND OPERATIONS

RELATIONS AND FUNCTIONS

EQUATIONS AND INEQUALITIES

STATISTICS

Virginia Standard/s of Learning: A.11 The student will add, subtract, and multiply polynomials and divide polynomials with monomial divisors, using concrete objects, pictorial, and area representations, and algebraic manipulations. A.12 The student will factor completely first- and second-degree binomials and trinomials in one or two variables. The graphing calculator will be used as a tool for factoring and for confirming algebraic factorizations. Related Standard/s: E 9.8

C/T 12.1, 12.2,

Technology Standard/s: 12.3, 12.4

Unit III: Polynomials Content Component/s: • Polynomial Operations • Factoring Quadratic Trinomials • Problem Solving

Suggested Time Frame: 8 blocks/16 single periods

Assessment Sample/s: • Drawing of a rectangle with sides labeled with binomials and explanation of how the area of the rectangle can be derived • Chart explaining the factoring of polynomials • Games invented involving polynomials and their factors • Unit quizzes/tests

Essential Knowledge & Skills/Mathematics Curriculum Framework/Algebra I/pp. 4–5: The student will: • model sums, differences, products, and quotients of polynomials with concrete objects and their related pictorial representations. • relate concrete and pictorial representations for polynomial operations to their corresponding algebraic manipulations. • find sums and differences of polynomials. • multiply polynomials by monomials and binomials by binomials symbolically. • find the quotient of polynomials, using a monomial divisor. • use the distributive property to “factor out” all

common monomial factors. • factor second-degree polynomials and binomials with integral coefficients and a positive leading coefficient less than four. • identify polynomials that cannot be factored over the set of real numbers. • use the x-intercepts from the graphical representation of the polynomial to determine and confirm its factors.

M9B–5

Industrial Blueprint/s: (Strategies) The teacher will: • have students prepare a chart on factoring polynomials. Each chart should include: 1) Type of factoring, 2) Number of terms (2, 3, 4, or more), 3) Examples of each type, and 4) Personal notes. • have students use area principles and Algeblocks to show operations with polynomials. • have students make a model block out of cardboard. The block will be named x and distances will be measured using x and showing remainder in inches (e.g., the length of the board could be 7x – 7 inches). Find lengths and areas in the classroom. • have students invent a game that involves matching polynomials and their factors. Suggested format could include Concentration, Jeopardy, Tic Tac Toe, or Old Maid. • have students use Algeblocks to show that the factors they find will give the area desired. • have students, working in pairs, (each group receives 16 cards) write two numbers on each of the eight cards, and the GCF of the two numbers on the other eight cards. Groups then exchange cards and place them on a flat surface. Students take turns trying to match pairs of numbers with the GCFs. If a match is made, the player keeps that pair of cards. The player with the most sets of cards at the end is the winner. • provide instruction on polynomials. Have students work in cooperative groups and have each group member write a polynomial on an index card. Cards should be collected, shuffled, and placed in a pile. Students pick a card from the pile and identify the degree of each polynomial. ¤ use models, manipulatives, and the chalkboard to teach students how to determine the width of a rectangle given its area and length. Provide a collection of rectangles cut from colored paper. Give each student several different colored rectangles. Have students find area from dimensions or one dimension given area and the other dimension. Begin with numeric values and progress through polynomial expressions for the area and dimension M9B–6

Essential Understandings/Math Curriculum Framework/Algebra 1/p. 4-5: • A relationship exists between arithmetic operations and operations with polynomials. • Polynomials can be represented in a variety of forms. • Operations with polynomials can be represented concretely, pictorially, and algebraically. • Polynomial expressions can be used to model real-life situations. • The distributive property is the unifying concept for polynomial operations. • Factoring reverses polynomial multiplication. • There is a relationship between the factors of a polynomial and the x-intercepts of its related graph. • Some polynomials cannot be factored over the set of real numbers. • Polynomial expressions in a variable x and their factors can be used to define functions by setting y equal to the polynomial expression or y equal to a factor, and these functions can be represented graphically. Resources: Text: Algebra 1, Integrations, Applications, Connections, Glencoe, pp. 520-549 Technology: • TI-83 Graphing Calculator • Algeblaster (S) • Graphlink (S) • Algebra I (ProOne) (S) • Equation Editor (Microsoft Office) (S) • Hyperstudio (S) • PowerPoint (S) Notes: A.11 Resources Learning About Algebra Tiles - Algebra tiles provide a useful way to introduce operations on polynomials. There are patterns of make-yourown tiles included. http://www.ucs.mun.ca/~mathed/t/rc/alg/tiles/ tiles1.html Technology: • Best Grapher • ¤ indicates an instructional blueprint for a topic that has been included in previous math SOL tests and that is not covered in the math textbook.

Curriculum Guide/Math/Algebra IB York County School Division EXPRESSIONS AND OPERATIONS

RELATIONS AND FUNCTIONS

EQUATIONS AND INEQUALITIES

STATISTICS

Virginia Standard/s of Learning: A.14 The student will solve quadratic equations in one variable both algebraically and graphically. Graphing calculators will be used both as a primary tool in solving problems and to verify algebraic solutions.

Related Standard/s: S CHEM.1

C/T 12.1, 12.2,

Technology Standard/s: 12.3, 12.4

Unit IV: Quadratic Equations Content Component/s: • Factoring Techniques • Graphing Solutions

Suggested Time Frame: 4 blocks/8 single periods

Assessment Sample/s: • Demonstration of the use of one method to find the roots of a quadratic equation and the use of other methods (at least one) to check accuracy of roots • Unit quizzes/tests

Essential Knowledge & Skills/Mathematics Curriculum Framework/Algebra I/p. 16: The student will: • solve quadratic equations algebraically or by using the graphing calculator. When solutions are represented in radical form, the decimal approximation will also be given. • verify algebraic solutions, using the graphing calculator. • identify the x-intercepts of the quadratic function as the solutions(s) to the quadratic equation that is formed by setting the given quadratic expression equal to zero.

M9B–7

Instructional Blueprint/s: (Strategies)

Resources:

The teacher will: • have students graph y = x2, y = x2 –4, and y = (x + 2)2 + 5. Students should identify the minimum point of each graph. Have students predict the minimum point of y = x2 + 7. Students should use the graphing calculator to verify their results. • present the following problem: The length of each side of a baseball diamond is 90 feet. What distance must a catcher throw the ball to pick off a base runner stealing second? Give the students a diagram and supply them with the Pythagorean Theorem. Solve the equation for “c.” ¤ create an activity in which students learn to calculate a zero of a given function. Explain using manipulatives and the chalkboard xintercepts as “zeros” and “roots.” Give students one term and tell them to provide the synonyms for that term (e.g., Teacher says, “Zero;” student responds, “x-intercept.”). Provide practice for students with a variety of problems that require them to calculate zero of a given function.

Text: Algebra 1, Integrations, Applications, Connections, Glencoe, pp. 608-652 Technology: • TI-83 Graphing Calculator • Algeblaster (S) • Graphlink (S) • Equation Editor (Microsoft Office) (S) • Hyperstudio (S) • PowerPoint (S) Web Site: • www.pbs.org/wgbh/nova/proof/puzzle/ baseball.html

Essential Understandings/Math Curriculum Framework/Algebra 1/p. 16: • The zeros or the x-intercepts of the quadratic function are the real root(s) or solution(s) of the quadratic equation that is formed by setting the given quadratic expression equal to zero. • Quadratic equations can be solved in a variety of ways. • A quadratic equation can have two solutions, one solution, or no solution. • A solution to a quadratic equation is the value or set of values that can be substituted to make the equation true.

M9B–8

Notes: Technology: • Best Grapher • ¤ indicates an instructional blueprint for a topic that has been included in previous math SOL tests and that is not covered in the math textbook.

Curriculum Guide/Math/Algebra IB York County School Division EXPRESSIONS AND OPERATIONS

RELATIONS AND FUNCTIONS

EQUATIONS AND INEQUALITIES

STATISTICS

Virginia Standard/s of Learning: A.18 The student will analyze a relation to determine whether a direct variation exists and represent it algebraically and graphically, if possible.

Related Standard/s: E 9.8

C/T 12.1, 12.2,

Technology Standard/s: 12.3, 12.4

Unit V: Direct Variation Content Component/s:

Suggested Time Frame:

• Writing Equations • Problem Solving

4 blocks/8 single periods

Assessment Sample/s: • Project involving real life situations with direct variation. Cartoon will be drawn for each of the three situations and as a caption, the variation will be stated in words and algebraically • Unit quizzes/tests

Essential Knowledge & Skills/Mathematics Curriculum Framework/Algebra I/p. 9: The student will: • given a table of values, determine whether a direct variation exists. • write an equation for a direct variation, given a set of data. • graph a direct variation from a table of values or a practical situation.

M9B–9

Curriculum Guide/Math/Algebra IB York County School Division EXPRESSIONS AND OPERATIONS

RELATIONS AND FUNCTIONS

EQUATIONS AND INEQUALITIES

STATISTICS

Virginia Standard/s of Learning: A.15 The student will, given a rule, find the values of a function for elements in its domain and locate the zeros of the function both algebraically and with a graphing calculator. The value of f(x) will be related to the ordinate on the graph.

A.5; S BIO.1, BIO.2, BIO.8, Related Standard/s: M BIO.9

C/T 12.1, 12.2,

Technology Standard/s: 12.3, 12.4

Unit VI: Functional Relationships Content Component/s: • Domain and Range • Zeros • Graphing

Suggested Time Frame: 4 blocks/8 single periods

Assessment Sample/s: •

Organized pair of related data such as lunch items in school and their prices shown by a mapping or graph of the picture relationship between the data. Test data to see if a function exists using the vertical line test. • Representation of the above data in table form. Give four choices of graphs. Ask which graph could be used to represent the situation. • Graph depicting which of the following tables is best represented by the line shown in the graph? Give four choices in this form. x y • Unit quizzes/tests

Essential Knowledge & Skills/Mathematics Curriculum Framework/Algebra I/p. 8: The student will: • for each x in the domain of f, find f(x). • identify the zeros of the function algebraically and confirm them, using the graphing calculator.

M9B–11

Instructional Blueprint/s: (Strategies) • The teacher will: • have students provide series of data to place in a mapping. The domain could be the students’ names, the range could be their homeroom class. Have students provide additional data that could be represented by a mapping or graph. • ask the students to identify the relationship between consecutive numbers of a sequence and numbers written on the chalkboard: 5, 3, 6, 2, 7, 1, 8, 0. • have students use the graphing calculator and enter data and equations to determine patterns. Ask the students to make up a functional formula, generate a data table and bring the data table on a separate piece of paper to class. Students are asked to find a rule and express it as a formula. Have the students graph some of these formulas on a coordinate plane and decide if they are functions (use the vertical line test). • ask the students to examine a table written on the chalkboard and discover a pattern in terms of how y changes when x changes. Explain in your own words how to find y in terms of x. x 0 1 2 3 4

y 5 8 11 14 17

Assume that the pattern continues indefinitely. Use the rule you have found to extend the data to include negative numbers for x. Did you find a unique value for y given a particular value for x? Use a formula to describe the pattern you have found. • tell students that on the first day they will have 1 problem for homework, 1 on the second day, 2 the third day, 3 the fourth day, 5 the fifth day, 8 the sixth day and 13 the seventh day. If this pattern is continued how many problem will be given on the eleventh day? Students will make a set of ordered pairs and then graph the set to determine if a function M9B–12

•

•

•

exists. present the following problem to the students: Cosetta built a square pen in the corner of her backyard. The side of her backyard is 24 feet longer than the side of the pen. Find the dimensions of the pen and Cosetta’s backyard if the difference of their areas is 1152 square feet. give students, in small groups, graphs of various curves or random points. Have them determine if a relationship exists between the first and second coordinate, and if so, express it algebraically. Check for a function using the vertical line test, mapping or on the graphing calculator. This can be extended to determine the relationship to data in word problems (e.g., include a utility bill, renting a car, buying quantities instead of single items, service job rates). draw a coordinate plane on a flat surface outside or on the floor. Students pick a number on the x-axis to be used as a domain element. Give them a rule for which they will find the range value for this number. Students will move to this point on the plane. “Connect” the students using yarn or string. Analyze the different types of graphs obtained. divide students into groups. Give each group a folder. Have them write a relationship on the outside as well as a list of five numbers to be used as domain elements. They should write the domain and range elements in set notation on a piece of paper and place it in the folder. Folders are passed to each group. When the folder is returned to the group with which it began, the results will be analyzed and verified.

Essential Understandings/Math Curriculum Framework/Algebra 1/p. 8: • An equation represents the relationship between the independent and dependent variables. • The object f(x) is the unique object in the range of the Standard/s function f that associated with the Virginia of is Learning: object x in the domain of f. • For each x in the domain of f, x is a member of the input of the function f, f(x) is a member of the output of f, and the ordered pair [x, f(x)] is a member of f. • An object x in the domain of f is an x-intercept or a zero of a function f if and only if f(x) = 0. Related Standard/s: Resources: Text: Algebra 1, Integrations, Applications, Connections, Glencoe, pp. 287-302 Content Component/s: Technology: • TI-83 Graphing Calculator • Algeblaster (S) • Graphlink (S) • Equation Editor (Microsoft Office) (S) • Hyperstudio (S) Assessment Sample/s: • PowerPoint (S)

Curriculum Guide/Math/Algebra IB Notes: York County School Division A.5 Resources Function Basics - Description and illustrations of functions and graphing. http://tqd.advanced.org/2647/algebra/ funcbasc.htm

C/T 12.1, 12.2,

Technology Standard/s: 12.3, 12.4

Suggested Time Frame:

M9B–13

M9B–14

Curriculum Guide/Math/Algebra IB York County School Division EXPRESSIONS AND OPERATIONS

RELATIONS AND FUNCTIONS

EQUATIONS AND INEQUALITIES

STATISTICS

Virginia Standard/s of Learning: Topics in this unit are designed to provide extension and enrichment in the study of Algebra I. They are to be taught after the administration of the Algebra I SOL Test in the spring. Depending upon the amount of time remaining in a given school year, topics, required research, and use of resources may vary.

C/T 12.1, 12.2,

Related Standard/s: M A.1, A.2

Technology Standard/s: 12.3, 12.4

Unit VII: Radicals, Expressions, Equations and Their Applications (after SOL Test) Content Component/s: • • • •

Proportion and Percent Probability Simplifying Rational Expressions Operations with Rational Expressions

Suggested Time Frame: • Solving Rational Equations • Problem Solving

10 blocks/ 20 single periods

Assessment Sample/s: • Matching activity using a set of 5 word sentences representing real-life situations and a set of 5 equations translated from the word sentences • Project depicting the use of percents to solve real-life situations • Unit quizzes/tests

The student will: • solve equations using proportions. • using percents, solve real-life situations. • find the probability of an event. • simplify rational expressions. • add, subtract, multiply and divide rational expressions. • solve rational expressions used in real-life settings.

M9B–15

Instructional Blueprint/s: (Strategies)

Notes:

The teacher will: • give sets of real-life, percent-based, situations. Students use proportions to problem solve. • instruct students to use proportions to create scale drawings of their houses. • provide a pre-unit test for diagnostic purposes. Have students complete the test, using a teacher-made key, and prepare a correction and error analysis for each error

Math091 - Intermediate Algebra - Chapter 6: Rational expressions: A whole lecture on rational expressions and how to solve them. http://mesa7.mesa.colorado.edu/~bornmann/ classes/math091/lecture/chap_06/tsld001.htm

Resources: Text: Algebra 1, Integrations, Applications, Connections, Glencoe, pp. 192-238, 658-750 Technology: • TI-83 Graphing Calculator • Algeblaster (S) • Graphlink (S) • Equation Editor (Microsoft Office) (S) • Hyperstudio (S) • PowerPoint (S)

M9B–16

Review Algebra I - Operations on real numbers and problems to boot. http://www.xnet.com/~fidler/triton/math/review/ mat055/mat055.htm Resources Lesson 3: Simple Equations - The index to simple equations lessons. Excellent source of problems and how to solve them in little increments. http://iasec.fwsd.wednet.edu/iamath/ page2d.htm#Lesson3 Lesson 4: Equations and Inequalities - The index to solving equations and inequalities lessons. Excellent source of problems and how to solve them in little increments. http://iasec.fwsd.wednet.edu/iamath/ page2d.htm#Lesson4