Algebra 1 A

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Audubon Public Schools

Engaging Students ~ Fostering Achievement ~ Cultivating 21st Century Global Skills Written By: Patricia Martel

Approved May, 2011

Course Title: Algebra I A Unit Name: Operations on Numbers and Expressions Grade Level: 9 Content Statements

Successful students will be able to perform operations with real numbers and algebraic expressions, including expressions involving exponents, scientific notation, and square roots, using estimation and an appropriate level of precision. Reasoning skills will be emphasized, including justification of results.

Cumulative Progress Indicators (CPI) O:1 all indicators, O:2 all indicators

Overarching Essential Questions

What are some ways to represent, describe, and analyze patterns (that occur in our world)?

When is one representation of a function more useful than another?

How can we use algebraic representation to analyze patterns?

Why are number and algebraic patterns important as rules?

How are arithmetic operations related to functions? How can numeric operations be extended to algebraic objects?

Why is it useful to represent real-life situations algebraically?

What makes an algebraic algorithm both effective and efficient?

Overarching Enduring Understandings

Logical patterns exist and are a regular occurrence in mathematics and the world around us.

Algebraic representation can be used to generalize patterns and relationships.

The same pattern can be found in many different forms.

Relationships can be described and generalizations made for mathematical situations that have numbers or objects that repeat in predictable ways.

Functions are a special type of relationship or rule that uniquely associates members of one set with members of another set.

Algebraic and numeric procedures are interconnected and build on one another to produce a coherent whole. Rules of arithmetic and algebra can be used together with (the concept of) equivalence to transform equations and inequalities so solutions can be found to solve problems.

Variables are symbols that take the place of numbers or ranges of numbers; they have different meanings depending on how they are being used.

Proportionality involves a relationship in which the ratio of two quantities remains constant as the corresponding values of the quantities change.

Unit Essential Questions

How are verbal and algebraic models used to represent real life situations?

How can a problem be translated into an equation? How can you apply the rules of multiplication and division?

How can variables be used to solve problems dealing with consecutive integers?

How is the distributive property used in an algebraic

Unit Enduring Understandings

Use properties of number systems within the set of real numbers to verify or refute conjectures or justify reasoning.

Use rates, ratios and proportions to solve problems, including measurement problems.

Describe and distinguish among the various uses of variables, including:

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equation or expression?

What are number operations and algebraic expressions?

How can rates, ratios, percents, and proportions be applied to problem solving?

What are the steps to solving an equation that involves one or more transformations?

How can equations involving the addition and subtraction of polynomials be simplified and solved? How can equations be solved for different variables? How can numbers be represented using scientific notation?

Why does dividing by zero create an undefined result? How can fractions be multiplied and divided?

Why do you need a common denominator to work with some algebraic fractions and rational expressions? Why is it useful to know equivalent forms for rational expressions?

Why is simplifying an algebraic fraction like simplifying a numeric fraction?

How can the Pythagorean theorem be used to find the length of a side of right triangle?

How can we determine the difference between rational and irrational numbers?

How can general quadratic trinomials be factored? How can factoring help us to solve equations? How can problems be solved by factoring quadratic equations?

What is prime factorization? What does GCF mean?

Symbols for fixed unknown values (such as 3x – 2 = 7) Symbols for all numbers in properties (such as x +0 =x) Symbols for formulas (such as A = l * w)

Symbols for parameters (such as m and b for slope in y = mx + b)

Use matrices to represent and solve problems. Adding and subtracting matrices.

Multiplying a matrix by a scalar.

Apply the laws of exponents to numerical and algebraic expressions with integral exponents to rewrite them in different but equivalent forms or to solve problems.

Use the properties of radicals to convert numerical or algebraic expressions containing square roots in different but equivalent forms or to solve problems. Add, subtract and multiply polynomial expressions. Factor simple polynomial expressions.

Unit Rationale

Core content for Pre-Algebra includes a number of discrete skills and concepts, each related to broader mathematical principles. In teaching and learning Algebra I, it is important for teachers and students to comprehend the following big ideas and to connect the individual skills and concepts of Algebra I to these broad principles.

PATTERNS AND FUNCTIONS

Algebra provides language through which we describe and communicate mathematical patterns that arise in both mathematical and

non-mathematical situations, and in particular, when one quantity is a function of a second quantity or where the quantities change in predictable ways. Ways of representing patterns and functions include tables, graphs, symbolic and verbal expressions, sequences, and formulas.

EQUIVALENCE:

There are many different – but equivalent – forms of

Unit Overview

Number Sense and Operations Reasoning with real numbers Using ratios, rates, and proportions Using variables in different ways Using matrices

Using numerical exponential expressions Using algebraic exponential expressions Using numerical radical expressions Using algebraic radical expressions Algebraic Expressions

Operating with polynomial expressions Factoring polynomial expressions

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a number, expression, function, or equation, and these forms differ in their efficacy and efficiency in interpreting or solving a problem, depending on the context. Algebra extends the properties of numbers to rules involving symbols; when applied properly, these rules allow us to transform an expression, function, or equation into an equivalent form and substitute equivalent forms for each other. Solving problems algebraically typically involves

transforming one equation to another equivalent equation until the solution becomes clear. Authentic Learning Experiences

Connections A new teen recreation center is being proposed. In order to meet one of the requirements by the county council, determine the population density of several surrounding towns and propose a location for the center. Explain your reasoning to the council.

Connections: In civil engineering, surveyors are often asked to determine the approximate distance across a body of water. For example, find the distance across a lake that is bounded by four streets, each adjacent to the northern, western, southern, and eastern shore, respectively, and meeting at perpendicular intersections. Each street is one mile long. Find the approximate distance from the southwestern corner to the northeastern corner of the lake.

Connections: Represent the surface area of a rectangular shipping carton (with dimensions l, w, and h) as a polynomial expression. Use this expression to create a table of possible dimensions which would be associated with a particular surface area. For example, with only 10 square feet of cardboard, find the dimensions of some boxes (rectangular prisms) that could be constructed from the cardboard.

Representations: Compare different representations for an irrational number you would expect to encounter in everyday life, including a physical representation, common decimal approximations, common fraction approximations, and the value produced by a calculator. Discuss the relative accuracies of the approximations and suggest appropriate circumstances for the use of each. For example, students may identify and compare every day encounters with the ratio of the circumference of a circle to its diameter (π).

Representations: Use algebra tiles to model expressions and develop an understanding of factoring.

Representations: Abu Kamil used geometric models many centuries ago to solve algebraic problems. Use a sheet of paper to model (a + b)2_{and remove the b}2_{section from the corner of the paper. After cutting, rearrange} the pieces and explain how this represents the factorization of a2_{+ 2ab + b}2_.

Problem Solving: Many students will have encountered the parental solution to splitting the last brownie between two children: one cuts, the other chooses. Have students consider a way to extend this algorithm to three or more children.

Technology Integration: Using spreadsheet software, examine variables as a set of objects and find the image of a set of objects using a function to gain an output (NCTM Navigating through Algebra in Grades 9-12). For example, the corresponding values of f(n) = 3n are examined by using a single number substituted for n in the function, next by using the set of natural numbers less than or equal to 50 under this function, and finally to considering the variable n as the set of real numbers. Students identify a real world situation where a continuous function using the set of real numbers versus a single number may occur (e.g. the gravitational force on an object of a particular mass as it moves to higher altitudes which might include either mountain climbing or a space shuttle trip; the height of a candle as it burns over time).

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Technology Integration: Use the internet to research and present the data for the Political Party research project, as well as a graphing calculator or a spreadsheet to accomplish the task.

21st Century Skills and Themes Problem Solving: Embedded and ongoing

Technology: utilizing the internet to access and analyze information (Smartboard) See Authentic Learning Activities and Interdisciplinary Activities

Unit Learning Targets/Scaffolding to CPIs  activating prior knowledge

 offering a motivational context to pique student interest or curiosity in the subject at hand  breaking a complex task into easier, more "doable" steps to facilitate student achievement  showing students an example of the desired outcome before they complete the task  modeling the thought process for students through "think aloud" talk

 offering hints or partial solutions to problems  using verbal cues to prompt student answers

 teaching students chants or mnemonic devices to ease memorization of key facts or procedures  facilitating student engagement and participation

 displaying a historical timeline to offer a context for learning

 using graphic organizers to offer a visual framework for assimilating new information  teaching key vocabulary terms before reading

 guiding the students in making predictions for what they expect will occur in a story, experiment, or other course of action

 asking questions while reading to encourage deeper investigation of concepts  suggesting possible strategies for the students to use during independent practice

 modeling an activity for the students before they are asked to complete the same or similar activity  asking students to contribute their own experiences that relate to the subject at hand

Key Terms

Binomial - polynomial with exactly two terms constant - number

consecutive integers - whole numbers that are all in a row

equation - contains numbers and/or variables and Must contain an Equal sign inequality - contains numbers and/or variables and uses four inequality symbols integer - positive or negative whole number

Monomial - variable or number or both separated only by multiplication numerical expression - numbers separated by mathematical operations

Order of operations - order that must be followed when there is more than 1 mathematical operation present; PEMDAS - Parenthesis, Exponents, Multiplication/Division, Addition/Subtraction

percent - something out of 100

percent increase - comparison of how much a value has increased compared to the original percent decrease - comparison of how much a value has decreased compared to the original Polynomial - one or more monomials separated by addition or subtraction

proportion - two fractions set equal to each other ratio - comparison between two values

rational number - can be written as a fraction

scientific notation - easier way to write really big or really small numbers

simple interest - percentage of every dollar the bank pays as a reward for allowing them to hold your money tax - percentage added on to every dollar for the government

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Term - a piece of a polynomial

Trinomial - polynomial with exactly three terms

tip - percentage added on to every dollar for a service performed unit rate - ratio that contains the number one

variable - letter or symbol that stands for a number

variable expression -variable or number or both separated by mathematical operations Instructional Strategies

Notes Discussion Small group work Experiments

Lecture and Demonstration Guided Practice

Real life demonstrations

Watch for Common Misconceptions: Students will frequently confuse 25_{(meaning 2 2 2 2 2) with 2 5}_{⋅ ⋅ ⋅ ⋅} _⋅ (meaning 5 + 5).

Customizing Learning/ Differentiation Special Needs

Students are engaged in small group work, where students of differing abilities and learning styles should be grouped together. Students act as peer coaches to support students with special needs.

ELL

Allow English Language Learners to play a very active role in selecting their hotspots to study. Many students’ families may have immigrated from countries or regions that feature hotspots. The process of sharing their own perspective or cultural ties to their native region’s biodiversity is invaluable to the group’s work.

Gifted Learners

Offer scientific journal articles as sources for research to gifted students. The vocabulary and writing style is very advanced, but gifted students might be able to garner the needed information and data from these primary sources.

Mainstream Learners

Throughout the unit during class time, plan and hold small learning sessions/work groups where students can selectively attend to learn more about a specific topic. Hold these sessions often, changing the topic every week. Topics can include, but not be limited to using maps, planning an interview, interpreting scientific data, reading graphs and charts, etc. Allow students to select the sessions they would like to attend, based on their perceived need, and they should plan the sessions into their research schedule ahead of time.

Formative Assessments Quizzes

Unit Test

Homework checks Questions during practice Progress Tests

Performance Assessment Task: Suppose that a drug company has established that a patient must have 40 mg of a certain prescription drug in the body for the drug to be effective. Moreover, the company’s studies indicate that anything in excess of 600 mg is toxic, and its research has shown that the body eliminates 10 percent of the drug every four hours. Imagine you are a doctor prescribing this drug for a patient. How often would you want your patient to take the drug, and in what quantity, to ensure effectiveness while avoiding toxicity? (NCTM

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Performance Assessment Task: Every Saturday you play basketball in the local community youth club. At the end of a season after a club tournament, the players in the club meet at a fast-food restaurant for a party. If hamburgers cost 59 cents each, find a way to determine the total cost of hamburgers when various numbers of players in the club each have a hamburger (NCTM Illuminations).

Performance Assessment Task: Assume that you are helping your school guidance counselor with the

scheduling of courses for the coming year. Use two matrices to represent the enrollment numbers of students, one for males and another for females, in three elective courses (e.g. music, technology, and art) for the past five years in your school. Create a matrix to show the total enrollment in these electives in your school during each of the five years. If you were scheduling for next year, how many students would you project to be in each elective? Explain your reasoning.

Interdisciplinary Connections

Find a linear relationship between the ratio of the length of a person’s thigh bone and his height, and use this to estimate the height of a person whose thigh bone has been found in an archeological dig.

Have students consider the advantages and disadvantages of different voting methods, including weighted voting (ranking 1st, 2nd, and 3rd choices), runoff elections, plurality voting (whoever gets the most votes wins), and majority voting (winner must receive more than half of the votes). Link this discussion to social studies and the study of government.

Determine the political composition of the United States Congress during the 20th Century by decade, first researching the types of party affiliations (e.g. Republican) and the associated numbers (of party representatives) in the House of Representatives and the Senate. Use matrices to represent tabular information for each, with a third matrix that displays the numbers of the U.S. Congress members by party affiliation for each decade. Using an historical timeline, identify significant laws that were enacted during each decade in the activity above. Shirley is interested in estimating how much she will be earning in the future if she stays with the same company. Shirley’s salary is $35,000 per year. Let x represent her annual percent increase expressed as a decimal. Express Shirley’s next year’s salary as a polynomial. Extension: What polynomial expression would represent Shirley’s salary in two years?

Use algebra tiles to represent polynomial expressions in multiple ways, comparing them from an artistic viewpoint. For example, while (2 + x + 2)(3+ x + 3), (x + 4)(x + 6), and x2 + 10x + 24 are all algebraically equal, the algebra-tile representations of the first two algebraic expressions would vary greatly in artistic symmetry. Resources

New Jersey Algebra I Content Document Scientific Calculator

Common Core Content Standards-Mathematics TI Smart software

Graphing Calculators

Kuta worksheets - http://www.kutasoftware.com/free.html

Tutorials on www.brightstorm.com

Suggested Activities for Inclusion in Lesson Planning

Defining, giving examples of, distinguishing between, and using numbers, and their properties, from each of the following number sets: Whole numbers, Integers, Rationals, and Reals.

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Comparing and ordering real numbers, including determining between which two consecutive whole numbers the value of a square root lies.

Showing that a given interval on the real number line, no matter how small, contains both rational and irrational numbers.

Establishing simple facts about rational and irrational numbers using logical arguments and examples. Providing counterexamples to refute a false conjecture.

Using dimensional analysis for unit conversion.

Solving problems using derived measures (e.g. percent change and density).

Solving problems involving scale factor (e.g. similar figures, scale drawings, map scales). Solving applications related to proportional representation.

Devising and using strategies for making fair decisions.

Identifying constant and variable terms in algebraic expressions, equations, and inequalities. Discriminating between x2_{and 2x (using appropriate applications and algebraic manipulatives).} Identifying and explaining equality of matrices.

Representing, computing, and solving problems using numbers in scientific notation. Translating to expressions with only positive exponents.

Translating to expressions with variables appearing only in the numerator.

Adding, subtracting, multiplying, dividing, and manipulating numerical or algebraic expressions with square roots. Results may be required to be given in exact form.

Using the distance formula, based on the Pythagorean Theorem, to solve problems.

When taking square roots of variable expressions, absolute values must be included when appropriate.

For applications, this includes using and interpreting appropriate units of measurement, and the appropriate level of precision.

Extending the use of the distributive property to division of a polynomial by a monomial. Factoring expressions within and without a context.

State Assessment Assumption: All radical expressions represent real numbers.

State Assessment Limitation: Expressions under radicals will be limited to monomials. When rationalization of a denominator is required, the radical in the denominator will contain no variables.

Performing operations on polynomials within and without a context.

State Assessment Limitation: Multiplication is limited to a monomial multiplied by a polynomial or a binomial multiplied by a binomial.

State Assessment Limitation: Items involving radicals will be limited to square roots. Students will not be expected to produce formal proofs.

Scope of State Assessment Content: Factoring out common monomial factors Factoring perfect-square trinomials Factoring differences of squares

Factoring out quadratics of the form that factor over the set of integers The factoring process may require more than one step.

Unit Timeline 18-22 weeks

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Audubon Public Schools

Engaging Students ~ Fostering Achievement ~ Cultivating 21st Century Global Skills Written By: Patricia Martel

Course Title: PreAlgebra Unit Name: Linear Relationships Grade Level: 9 Content Statements

Successful students will be able to solve and graph the solution sets of linear equations, inequalities and systems of linear equations and to use words, tables, graphs, and symbols to represent, analyze, and model with linear functions. In contextual problems students graph and interpret their solutions in terms of the context. They apply such problem solving heuristics as: identifying missing or irrelevant information; testing ideas; considering analogous or special cases; making appropriate estimates; using inductive or deductive reasoning; analyzing situations using symbols, tables, graphs, or diagrams; evaluating progress regularly; checking for reasonableness of results; using technology appropriately; deriving independent methods to verify results; and using the symbols and terms of mathematics correctly and precisely. Function notation should be introduced and used regularly but not exclusively.

Cumulative Progress Indicators (CPI) L1.a, b, d, L2.a, c, d, e

Overarching Essential Questions

How can change be best represented mathematically? How can we use mathematical language to describe change?

How can we use mathematical models to describe change or change over time?

How can patterns, relations, and functions be used as tools to best describe and help explain real-life situations?

How are patterns of change related to the behavior of functions?

How are functions and their graphs related?

How can technology be used to investigate properties of linear functions and their graphs?

How can systems of equations be used to solve real-life situations?

Overarching Enduring Understandings

Graphs and equations are alternative (and often equivalent) ways for depicting and analyzing patterns of change.

Functional relationships can be expressed in real contexts, graphs, algebraic equations, tables, and words; each representation of a given function is simply a different way of expressing the same idea. The value of a particular representation depends on its purpose.

A variety of families of functions can be used to model and solve real world situations.

Unit Essential Questions

What does it mean if two lines intersect?

What method would be most appropriate to solve the system of equations?

What is the slope of the line?

Is the solution to a system of equations reasonable? Does the system have one, no or infinitely many

Unit Enduring Understandings

Recognize, describe and represent linear relationships using words, tables, numerical patterns, graphs and equations.

Describe, analyze and use key characteristics of linear functions and their graphs. Key characteristics include constant slope and x- and y-intercepts.

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solutions?

What is a system of linear equations and how do you solve it?

What does it mean when the system does not have a single solution?

What is slope and how is it related to solutions of a system of linear equations?

Graph the absolute value of a linear function and determine and analyze its key characteristics. Key characteristics include e vertex, slope of each branch, intercepts, domain and range, maximum, minimum, transformations, and opening direction.

Recognize, express and solve problems that can be modeled using linear functions. Interpret solutions in terms of the context of the problem.

Solve single-variable linear equations and inequalities with rational coefficients. Linear equations may have no solution (empty set), an infinite number of solutions (identity) or a unique solution.

Graph and analyze the graph of the solution set of a two-variable linear inequality.

Solve systems of linear equations in two variables using algebraic and graphic procedures.

Recognize, express and solve problems that can be modeled using single-variable linear equations; one- or two-variable inequalities; or two-variable systems of linear equations.

Unit Rationale

Relationships between quantities can be represented in compact form using expressions, equations, and inequalities. Representing quantities by variables gives us the power to recognize and describe patterns, make generalizations, prove or explain conclusions, and solve problems by converting verbal conditions and constraints into equations that can be solved. Representing quantities with variables also enables us to model situations in all areas of human endeavor and to represent them abstractly. LINEARITY

In many situations, the relationship between two quantities is linear so the graphical representation of the relationship is a geometric line. Linear functions can be used to show a relationship between two variables that has a constant rate of change and to represent the relationship between two quantities which vary proportionately. Linear functions can also be used to model, describe, analyze, and compare sets of data.

Unit Overview Linear Functions

Representing linear functions in multiple ways Analyzing linear function

Using linear models

Linear Equations and Inequalities Solving linear equations and inequalities Graphing linear inequalities

Solving systems of linear equations

Modeling with single variable linear equations, one-or two-variable inequalities or systems of equations

Authentic Learning Experiences

Connections: A landscaping contractor uses a combination of two brands of fertilizers, each containing a different amount of phosphates and nitrates. In a package, brand A has 4 lb. of phosphates and 2 lb. of nitrates. Brand B contains 6 lb. of phosphates and 5 lb. of nitrates. On her current job, the lawn requires at least 24 lb. of phosphates and at least 16 lb. of nitrates. How much of each fertilizer does the contractor need? (Students represent the given conditions as inequalities and use the intersection of their regions as the set of feasible answers.)

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investigate how many of each can be transported on a single trip. Use variables to represent the unknowns (e.g. x for cars and y for buses) and develop the graph of the inequality, using either paper-and-pencil or a graphing calculator. Recognizing that the solutions have to be whole numbers, students should identify the points whose coefficients are non-negative integers and in the first quadrant on or below the line.

21st Century Skills and Themes Problem Solving: Embedded and ongoing

Technology: utilizing the internet to access and analyze information (Smartboard) See activities in Authentic Learning experiences and interdisciplinary activities Unit Learning Targets/Scaffolding to CPIs

 activating prior knowledge