Algebra 1, Quarter 4, Unit 4.1

Multiple Representations for Quadratic Functions Overview Number of instructional days:

5 (1-day assessment)

(1 day = 45 minutes)

Content to be learned

Mathematical practices to be integrated

•

Reason abstractly and quantitatively.

Identify quadratic patterns using multiple representations. (1 day)

•

Solve problems involving quadratic functions using multiple representations. (3 days)

•

Work between and among different representations of quadratic functions, including written descriptions, tables, graphs, and equations. (embedded all 4 days)

§

Move back and forth between the true meaning of symbols and the abstract manipulation of symbols.

Construct viable arguments and critique the reasoning of others. §

Use prior material and properties of mathematics to solve more complex problems involving deductive and inductive reasoning.

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What are advantages and disadvantages of the different representations for quadratic functions in given problem situations?

Essential questions •

How can you determine that a pattern represents a quadratic function?

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How can a quadratic function be modeled, and how do the models relate to one another?

Cumberland, Lincoln, and Woonsocket Public Schools in collaboration with the Charles A. Dana Center at the University of Texas at Austin

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Algebra 1, Quarter 4, Unit 4.1 2010–2011

Multiple Representation for Quadratic Functions (5 days)

Written Curriculum Grade Span Expectations M(F&A)–10–1 Identifies, extends, and generalizes a variety of patterns (linear and nonlinear) represented by models, tables, sequences, or graphs to solve problems. (State) M(F&A)–10–2 Demonstrates conceptual understanding of linear and nonlinear functions and relations (including characteristics of classes of functions) through an analysis of constant, variable, or average rates of change, intercepts, domain, range, maximum and minimum values, increasing and decreasing intervals and rates of change (e.g., the height is increasing at a decreasing rate); describes how change in the value of one variable relates to change in the value of a second variable; or works between and among different representations of functions and relations (e.g., graphs, tables, equations, function notation). (State)

Clarifying the Standards Prior Learning Students have worked with linear relationships since grade 4. They identified, described, and compared situations that represent constant rates of change. In middle school, students were introduced to problemsolving situations involving slope and constant rate of change versus varying rates of change. Furthermore, students distinguished between linear and nonlinear relationships represented in tables, graphs, equations, or problem situations. Current Learning Students work between and among different representations of quadratic functions and relations. Students identify quadratic functions and problems using tables, graphs, and equations that will lead to solving quadratic equations this year. Future Learning Students will represent and analyze functions in several ways, analyze characteristics of functions (quadratic, exponential, logarithmic, and trigonometric), and apply knowledge of functions to interpret situations.

Additional Research Findings According to Principles and Standards for School Mathematics, high school students should have substantial experience in exploring the properties of different classes of functions. They should learn properties of functions that are quadratic (pp. 296–299).

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Cumberland, Lincoln, and Woonsocket Public Schools in collaboration with the Charles A. Dana Center at the University of Texas at Austin

Algebra 1, Quarter 4, Unit 4.2

Characteristics of Quadratic Functions Overview Number of instructional days:

9 (1-day assessment)

(1 day = 45 minutes)

Content to be learned

Mathematical practices to be integrated

•

Analyze constant, variable, or average rates of change in quadratic functions. (1 day)

Make sense of problems and persevere in solving them.

•

Identify the domain and range of a quadratic function. (1 day)

•

Work between different representations.

•

Algebraically determine the minimum and maximum values of a quadratic function. (1 day)

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Use technology to solve problems.

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Algebraically determine the x- and y-intercepts of a quadratic function. (1 day)

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Identify increasing and decreasing intervals of a quadratic function. (1 day)

•

Distinguish between independent and dependent variables of a quadratic function. (2 days)

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Describe how the change in the value of one variable relates to the change in the value of a second variable of a quadratic function. (1 day)

Attend to precision. •

Use labels of axes and units of measure correctly.

•

How does the change in the value of one variable of a quadratic function relate to the change in the value of a second variable of the function?

•

How do the numeric parameters of realworld applications affect the domain and range?

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How do you distinguish between the independent variable and dependent variable given a real-world scenario?

Essential questions •

How do the values of a, b, and c in the standard form of a quadratic function help approximate the shape and orientation of a parabola?

•

What are examples of real-world problem contexts that can be solved with quadratic functions?

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What is the significance of the x- and y-intercepts in the graph of a quadratic function?

•

Why is it important to know maximum and minimum values when interpreting scenarios depicted by quadratic functions?

Cumberland, Lincoln, and Woonsocket Public Schools in collaboration with the Charles A. Dana Center at the University of Texas at Austin

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Algebra 1, Quarter 4, Unit 4.2 2010–2011

Characteristics of Quadratic Functions (9 days)

Written Curriculum Grade Span Expectations M(F&A)–10–2 Demonstrates conceptual understanding of linear and nonlinear functions and relations (including characteristics of classes of functions) through an analysis of constant, variable, or average rates of change, intercepts, domain, range, maximum and minimum values, increasing and decreasing intervals and rates of change (e.g., the height is increasing at a decreasing rate); describes how change in the value of one variable relates to change in the value of a second variable; or works between and among different representations of functions and relations (e.g., graphs, tables, equations, function notation). (State)

Clarifying the Standards Prior Learning Students have worked with linear relationships since grade 4. They have identified, described, and compared situations that represent constant rates of change. In middle school, students were introduced to problem-solving situations involving slope and constant versus varying rates of change. Furthermore, students distinguished between linear and nonlinear relationships represented in tables, graphs, equations, or problem situations. Current Learning Students investigate specific characteristics of quadratic functions, such as domain, range, intercepts, and increasing and decreasing intervals. Students work between and among different representations of quadratic functions and relations. Future Learning Students will represent and analyze nonlinear functions in multiple ways, analyze characteristics of functions (quadratic, exponential, logarithmic, and trigonometric), and apply knowledge of functions to interpret situations.

Additional Research Findings Principles and Standards for School Mathematics states, “High school students should have substantial experience in exploring the properties of different classes of functions. They should learn properties of functions that are quadratic.” (pp. 296–299)

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Cumberland, Lincoln, and Woonsocket Public Schools in collaboration with the Charles A. Dana Center at the University of Texas at Austin

Algebra 1, Quarter 4, Unit 4.3

Combinations and Permutations Overview Number of instructional days:

10 (1-day assessment)

(1 day = 45 minutes)

Content to be learned

Mathematical practices to be integrated

•

Use counting techniques to solve problems involving o combinations and permutations. (3 days) o probability, odds, and compound probability. (3 days)

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Attend to precision.

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Calculate and compute accurately (including technology).

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Look and make use of structure.

Use organized lists, tables, tree diagrams, models, and the Fundamental Counting Principle to solve problems. (3 days)

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Recognize the importance of a symbol or mathematical tool and extend its use to other problems.

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How can you determine whether to use a combination or permutation in a given situation?

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Essential questions •

How are counting techniques useful in finding the number of possible outcomes of an event?

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How do lists, tables, tree diagrams, models, and the Fundamental Counting Principle relate to one another?

Cumberland, Lincoln, and Woonsocket Public Schools in collaboration with the Charles A. Dana Center at the University of Texas at Austin

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Algebra 1, Quarter 4, Unit 4.3 2010–2011

Combinations and Permutations (10 days)

Written Curriculum Grade Span Expectations M(DSP)–10–4 Uses counting techniques to solve contextualized problems involving combinations or permutations (e.g., organized lists, tables, tree diagrams, models, Fundamental Counting Principle, orsc others). (State)

Clarifying the Standards Prior Learning In grade 2, students used counting techniques, as well as diagrams, lists, and tree diagrams, to solve problems involving combinations. Students were introduced to permutations in grade 3 and used combinations and permutations in grades 4, 6, 7, and 8. Current Learning Students use counting techniques to solve contextualized problems involving combinations and permutations. Students make organized lists, tables, tree diagrams, and models, and use the Fundamental Counting Principle. Future Learning In grades 11 and 12, students will use counting techniques to solve problems involving permutations and combinations, using a variety of strategies (nCr, nPr, or n!), and will find unions, intersections, and complements of sets.

Additional Research Findings A Research Companion to Principles and Standards for School Mathematics states that students need to work with real data throughout elementary and middle grades in order to understand more complex issues (p. 213). According to Benchmarks for Science Literacy, “Students should know that there is no one right way to solve a math problem. Different methods have different advantages and disadvantages” (p. 28).

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Cumberland, Lincoln, and Woonsocket Public Schools in collaboration with the Charles A. Dana Center at the University of Texas at Austin

Algebra 1, Quarter 4, Unit 4.4

Inequalities Overview Number of instructional days:

5 (1-day assessment)

1 day = 45 minutes

Content to be learned

Mathematical practices to be integrated

•

Solve one-step inequalities. (1 day)

Model with mathematics.

•

Solve multistep inequalities. (2 days)

•

Use functions to model problem situations.

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Graph inequalities on number line. (1 day)

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Analyze the relationships of specific characteristics of functions.

Look for and make use of structure. •

Demonstrate understanding of functions by using the structure of functions.

•

Use the structure of functions to determine connections among graphs, tables, and equations.

Essential questions •

How does the solution from an inequality differ from equality?

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How do we graphically represent the solutions to inequalities?

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How do we solve one-step inequalities? How do we solve two-step inequalities?

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How do we apply inequalities to real-world situations?

Cumberland, Lincoln, and Woonsocket Public Schools in collaboration with the Charles A. Dana Center at the University of Texas at Austin

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Algebra 1, Quarter 4, Unit 4.4 2010–2011

Inequalities (5 days)

Written Curriculum Grade-Level Expectations/Grade-Span Expectations (5 days) M(F&A)–10–4 Demonstrates conceptual understanding of equality by solving problems involving algebraic reasoning about equality; by translating problem situations into equations; by solving linear equations (symbolically and graphically) and expressing the solution set symbolically or graphically, or provides the meaning of the graphical interpretations of solution(s) in problem-solving situations; or by solving problems involving systems of linear equations in a context (using equations or graphs) or using models or representations. (State)

Clarifying the Standards Prior Learning Students worked with equivalence between two expressions in third grade. Thereafter, they worked with one-step linear equations and multistep linear equations. In seventh and eighth grade, students translated problem-solving situations into equations. Students also manipulated equations, solving for different variables (literal equations). Current Learning Students solve real-world problems symbolically and graphically and interpret the meaning of the results. Future Learning Students will solve equations and systems of equations or inequalities and interpret the solutions algebraically and graphically. They will factor, complete the square, use the quadratic formula, and graph quadratic functions to solve quadratic equations. Students will also solve and interpreting solutions of equations involving polynomial, rational, and radical expressions and analyze the effect of simplifying radical or rational expressions on the solution set of equations involving such expressions.

Additional Research Findings Principles and Standards for School Mathematics indicates that students should represent and analyze mathematical situation and structure using algebraic symbols and use mathematical models to represent and understand quantitative relationships. (pp. 394–395) A Research Companion to Principles and Standards for School Mathematics has information on this topic in Stasis and Change: Integrating Patterns, Functions, and Algebra through the K–12 curriculum. (pp. 136–149) A Research Companion to Principles and Standards for School Mathematics states that graphs, diagrams, charts, number sentences, formulas, and other representations play an increasingly important role in mathematical activities. (pp. 250–261)

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Cumberland, Lincoln, and Woonsocket Public Schools in collaboration with the Charles A. Dana Center at the University of Texas at Austin

Algebra 1, Quarter 4, Unit 4.5

Inequalities and the Coordinate Plane Overview Number of instructional days:

7 (1-day assessment)

1 day = 45 minutes

Content to be learned

Mathematical practices to be integrated

•

Graph linear inequalities on the coordinate plane. (2 days)

Make sense of problems and persevere in solving them.

•

Graph systems of linear inequalities on the coordinate plane. (2 days)

•

Use systems of equations to plan a solution pathway for a problem situation.

•

Apply linear inequalities to real-world mathematical situations. (2 days)

•

Monitor and evaluate progress in solving a problem and change course if necessary.

Use appropriate tools strategically. •

Determine appropriate tools to use for solving systems of linear inequalities.

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Detect possible errors by strategically using estimation and other mathematical knowledge.

•

How are systems of equations different from systems of inequalities?

Essential questions •

How does graphing inequalities differ from graphing functions?

•

What does the graph of a system of inequalities actually show?

Cumberland, Lincoln, and Woonsocket Public Schools in collaboration with the Charles A. Dana Center at the University of Texas at Austin

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Algebra1, Quarter 4, Unit 4.5 2010–2011

Inequalities and the Coordinate Plane (7 days)

Written Curriculum Grade-Level Expectations/Grade-Span Expectations (5 days) M(F&A)–10–4 Demonstrates conceptual understanding of equality by solving problems involving algebraic reasoning about equality; by translating problem situations into equations; by solving linear equations (symbolically and graphically) and expressing the solution set symbolically or graphically, or provides the meaning of the graphical interpretations of solution(s) in problem-solving situations; or by solving problems involving systems of linear equations in a context (using equations or graphs) or using models or representations. (State)

Clarifying the Standards Prior Learning Students demonstrated conceptual understanding of equality in the lower elementary grades by finding what makes an open sentence true through adding, subtracting, or multiplying. In upper elementary grades, students solved one-step linear equations involving whole numbers through adding, subtracting, multiplying, or dividing. In middle school, students showed equivalence between two expressions by using models or different representations of the expressions, solving formulas for a variable requiring one transformation (e.g., d = rt; d/r = t), solving multistep linear equations with integer coefficients, and translating a problem situation into an equation. Students also showed that two expressions were or were not equivalent by applying commutative, associative, or distributive properties, order of operations, or substitution and by informally solving problems involving systems of linear equations in a context. Current Learning This content should be taught at the reinforcement level. Students find the equation of a line when given specific information about the line. They apply the knowledge from this unit in future units this year. Future Learning This GSE will be revisited in grades 10–12 and advanced math, where the students will learn about solving a variety of more complex functions.

Additional Research Findings A Research Companion to Principles and Standards for School Mathematics indicates that graphs, diagrams, charts, number sentences, formulas, and other representations play an increasingly important role in mathematical activities. (NCTM, pp. 250–261) According to NCTM Research Companion, graphs, diagrams, charts, number sentences, formulas and other representations play an increasingly important role in mathematical activities. (pp. 250–261)

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Cumberland, Lincoln, and Woonsocket Public Schools in collaboration with the Charles A. Dana Center at the University of Texas at Austin