## Description

- Overview:
- This lesson unit is intended to help teachers assess how well students are able to understand what the different algebraic forms of a quadratic function reveal about the properties of its graphical representation. In particular, the lesson will help teachers identify and help students who have the following difficulties: understanding how the factored form of the function can identify a graphŐs roots; understanding how the completed square form of the function can identify a graphŐs maximum or minimum point; and understanding how the standard form of the function can identify a graphŐs intercept.

- Level:
- Lower Primary, Upper Primary, Middle School, High School
- Grades:
- Kindergarten, Grade 1, Grade 2, Grade 3, Grade 4, Grade 5, Grade 6, Grade 7, Grade 8, Grade 9, Grade 10, Grade 11, Grade 12
- Material Type:
- Assessment, Lesson Plan
- Provider:
- Shell Center for Mathematical Education, U.C. Berkeley
- Provider Set:
- Mathematics Assessment Project (MAP)
- Date Added:
- 04/26/2013

- License:
- Creative Commons Attribution-NonCommercial-NoDerivs 3.0
- Media Format:
- Downloadable docs, Text/HTML

## Standards

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# Common Core State Standards Math

Grades 9-12,Algebra: Seeing Structure in ExpressionsCluster: Write expressions in equivalent forms to solve problems

Standard: Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments.*

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Algebra: Seeing Structure in ExpressionsCluster: Write expressions in equivalent forms to solve problems

Standard: Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Algebra: Seeing Structure in ExpressionsCluster: Interpret the structure of expressions.

Standard: Use the structure of an expression to identify ways to rewrite it. For example, see x^4 – y^4 as (x^2)^2 – (y^2)^2, thus recognizing it as a difference of squares that can be factored as (x^2 – y^2)(x^2 + y^2).

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Algebra: Seeing Structure in ExpressionsCluster: Interpret the structure of expressions.

Standard: Interpret parts of an expression, such as terms, factors, and coefficients.*

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Functions: Interpreting FunctionsCluster: Analyze functions using different representations

Standard: Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Functions: Interpreting FunctionsCluster: Analyze functions using different representations

Standard: Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Functions: Interpreting FunctionsCluster: Analyze functions using different representations

Standard: Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.*

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Functions: Interpreting FunctionsCluster: Interpret functions that arise in applications in terms of the context

Standard: Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.*

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Functions: Interpreting FunctionsCluster: Analyze functions using different representations

Standard: Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Functions: Interpreting FunctionsCluster: Interpret functions that arise in applications in terms of the context

Standard: For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.*

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Functions: Interpreting FunctionsCluster: Interpret functions that arise in applications in terms of the context

Standard: Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.*

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Algebra: Seeing Structure in ExpressionsCluster: Write expressions in equivalent forms to solve problems

Standard: Factor a quadratic expression to reveal the zeros of the function it defines.

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Algebra: Seeing Structure in ExpressionsCluster: Write expressions in equivalent forms to solve problems

Standard: Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Algebra: Seeing Structure in ExpressionsCluster: Write expressions in equivalent forms to solve problems

Standard: Use the properties of an expression to identify ways to rewrite it. For example, see x4 - y4 as (x2)2 - (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 - y2)(x2 + y2).

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Functions: Interpreting FunctionsCluster: Analyze functions using different representations

Standard: Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = (1.02)^t, y = (0.97)^t, y = (1.01)^(12t), y = (1.2)^(t/10), and classify them as representing exponential growth and decay.

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Algebra: Seeing Structure in ExpressionsCluster: Interpret the structure of expressions.

Standard: Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1+r)^n as the product of P and a factor not depending on P.*

Degree of Alignment: Not Rated (0 users)

Cluster: Mathematical practices

Standard: Construct viable arguments and critique the reasoning of others. Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.

Degree of Alignment: Not Rated (0 users)

Cluster: Mathematical practices

Standard: Reason abstractly and quantitatively. Mathematically proficient students make sense of the quantities and their relationships in problem situations. Students bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.

Degree of Alignment: Not Rated (0 users)

Cluster: Mathematical practices

Standard: Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?” They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Functions: Interpreting FunctionsCluster: Understand the concept of a function and use function notation.

Standard: Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Algebra: Seeing Structure in ExpressionsCluster: Interpret the structure of expressions.

Standard: Interpret expressions that represent a quantity in terms of its context.*

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Functions: Interpreting FunctionsCluster: Understand the concept of a function and use function notation.

Standard: Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ≥ 1 (n is greater than or equal to 1).

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Functions: Interpreting FunctionsCluster: Understand the concept of a function and use function notation.

Standard: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Functions: Interpreting FunctionsCluster: Analyze functions using different representations

Standard: Graph linear and quadratic functions and show intercepts, maxima, and minima.*

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Functions: Interpreting FunctionsCluster: Analyze functions using different representations

Standard: Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.*

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Functions: Interpreting FunctionsCluster: Analyze functions using different representations

Standard: Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.*

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Functions: Interpreting FunctionsCluster: Analyze functions using different representations

Standard: Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.*

Degree of Alignment: Not Rated (0 users)

# Common Core State Standards Math

Grades 9-12,Functions: Interpreting FunctionsCluster: Analyze functions using different representations

Standard: (+) Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.*

Degree of Alignment: Not Rated (0 users)

Learning Domain: Algebra: Seeing Structure in Expressions

Standard: Write expressions in equivalent forms to solve problems

Indicator: Use the properties of an expression to identify ways to rewrite it. For example, see x4 - y4 as (x2)2 - (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 - y2)(x2 + y2).

Degree of Alignment: Not Rated (0 users)

Learning Domain: Algebra: Seeing Structure in Expressions

Standard: Write expressions in equivalent forms to solve problems

Indicator: Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Algebra: Seeing Structure in Expressions

Standard: Write expressions in equivalent forms to solve problems

Indicator: Factor a quadratic expression to reveal the zeros of the function it defines.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Algebra: Seeing Structure in Expressions

Standard: Interpret the structure of expressions.

Indicator: Interpret parts of an expression, such as terms, factors, and coefficients.*

Degree of Alignment: Not Rated (0 users)

Learning Domain: Algebra: Seeing Structure in Expressions

Standard: Interpret the structure of expressions.

Indicator: Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1+r)^n as the product of P and a factor not depending on P.*

Degree of Alignment: Not Rated (0 users)

Learning Domain: Functions: Interpreting Functions

Standard: Understand the concept of a function and use function notation.

Indicator: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Functions: Interpreting Functions

Standard: Understand the concept of a function and use function notation.

Indicator: Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n ‰ä´ 1 (n is greater than or equal to 1).

Degree of Alignment: Not Rated (0 users)

Learning Domain: Functions: Interpreting Functions

Standard: Understand the concept of a function and use function notation.

Indicator: Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).

Degree of Alignment: Not Rated (0 users)

Learning Domain: Algebra: Seeing Structure in Expressions

Standard: Interpret the structure of expressions.

Indicator: Use the structure of an expression to identify ways to rewrite it. For example, see x^4 - y^4 as (x^2)^2 - (y^2)^2, thus recognizing it as a difference of squares that can be factored as (x^2 - y^2)(x^2 + y^2).

Degree of Alignment: Not Rated (0 users)

Learning Domain: Algebra: Seeing Structure in Expressions

Standard: Interpret the structure of expressions.

Indicator: Interpret expressions that represent a quantity in terms of its context.*

Degree of Alignment: Not Rated (0 users)

Learning Domain: Functions: Interpreting Functions

Standard: Analyze functions using different representations

Indicator: (+) Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.*

Degree of Alignment: Not Rated (0 users)

Learning Domain: Functions: Interpreting Functions

Standard: Analyze functions using different representations

Indicator: Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.*

Degree of Alignment: Not Rated (0 users)

Learning Domain: Functions: Interpreting Functions

Standard: Analyze functions using different representations

Indicator: Graph linear and quadratic functions and show intercepts, maxima, and minima.*

Degree of Alignment: Not Rated (0 users)

Learning Domain: Functions: Interpreting Functions

Standard: Analyze functions using different representations

Indicator: Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.*

Degree of Alignment: Not Rated (0 users)

Learning Domain: Functions: Interpreting Functions

Standard: Analyze functions using different representations

Indicator: Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.*

Degree of Alignment: Not Rated (0 users)

Learning Domain: Algebra: Seeing Structure in Expressions

Standard: Write expressions in equivalent forms to solve problems

Indicator: Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments.*

Degree of Alignment: Not Rated (0 users)

Learning Domain: Algebra: Seeing Structure in Expressions

Standard: Write expressions in equivalent forms to solve problems

Indicator: Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Functions: Interpreting Functions

Standard: Analyze functions using different representations

Indicator: Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Functions: Interpreting Functions

Standard: Analyze functions using different representations

Indicator: Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Functions: Interpreting Functions

Standard: Analyze functions using different representations

Indicator: Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.*

Degree of Alignment: Not Rated (0 users)

Learning Domain: Functions: Interpreting Functions

Standard: Interpret functions that arise in applications in terms of the context

Indicator: Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.*

Degree of Alignment: Not Rated (0 users)

Learning Domain: Functions: Interpreting Functions

Standard: Interpret functions that arise in applications in terms of the context

Indicator: For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.*

Degree of Alignment: Not Rated (0 users)

Learning Domain: Functions: Interpreting Functions

Standard: Interpret functions that arise in applications in terms of the context

Indicator: Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.*

Degree of Alignment: Not Rated (0 users)

Learning Domain: Functions: Interpreting Functions

Standard: Analyze functions using different representations

Indicator: Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Functions: Interpreting Functions

Standard: Analyze functions using different representations

Indicator: Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = (1.02)^t, y = (0.97)^t, y = (1.01)^(12t), y = (1.2)^(t/10), and classify them as representing exponential growth and decay.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Mathematical Practices

Standard: Mathematical practices

Indicator: Reason abstractly and quantitatively. Mathematically proficient students make sense of the quantities and their relationships in problem situations. Students bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize"Óto abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents"Óand the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Mathematical Practices

Standard: Mathematical practices

Indicator: Construct viable arguments and critique the reasoning of others. Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and"Óif there is a flaw in an argument"Óexplain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.

Degree of Alignment: Not Rated (0 users)

Learning Domain: Mathematical Practices

Standard: Mathematical practices

Indicator: Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, "Does this make sense?"ť They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.

Degree of Alignment: Not Rated (0 users)

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# Tags (9)

- Mathematics
- Algebra and Calculus
- Algebra
- CCSS
- Common Core Math
- Common Core PD
- Graphs
- ODE Learning
- Quadratic Functions

## Comments