Unit 5:
Quadratic Functions
In Algebra 1, students demonstrate fluency by creating, comparing, analyzing, solving, providing multiple representations, and identifying the structures of exponential & quadratic functions building on the understanding of linear functions from grade 8 to apply linear, quadratic and exponential models to data and unfamiliar problems.
The Story of Alg. 1 - Unit 5: Quadratic Functions (Up to 44 days)
There are a lot of overlapping big ideas in this unit, especially between “connecting standards” in gray and others. Students will need to understand polynomial operations as they will be necessary for working with and finding the zeros of a quadratic equation/function. Students will need to apply these relationships to various contexts including area, which is extremely helpful in understanding like terms, multiplication, and factoring through the use of algebra tiles. Once students begin to examine quadratic growth and rates of change, it should be compared and contrasted to the two types of functions that have come before, linear and exponential. This can be done quickly and simply through a Clothesline Math activity comparing y=2x, y=2^x and y=x^2 where each function is on its own clothesline and students place the x value (input) on the appropriate clothesline location of the y value (output). Comparing quadratic growth to linear growth is one of the chief goals of A-REI.7, linear & quadratic systems, so the theme of systems continues in this unit. Creating, using, and solving quadratic equations allows for the incorporation of other contexts besides area. Frayer models can continue to help students to see the connections between multiple representations and are preferred over separating representations. In fact, students will be asked to compare two quadratic functions to one another in different representations in addition to comparing two different types of functions to one another in different representations. Depth can be explored in this unit by looking more closely at expressions, equivalent expressions, and by justifying solution steps in addition to solving and explaining problems with key features of quadratic functions. Time permitting, additional depth can be found in solving quadratics with complex roots; although, this is a focus of Algebra 2 and time is probably better spent elsewhere.
Unit 5 Notes
In this unit, students examine a linear & quadratic system comparing the two types of growth (A-REI.7). Arithmetic operations on polynomials including area models with & without algebra tiles for multiplying & factoring are a major focus of the unit and are necessary components for developing fluency with quadratic functions. Tiles can be introduced through polynomial combination to build procedural proficiency from conceptual understanding.
Vocabulary, Tools & Developmental Notes from SBAC
A-CED.A Create equations that describe numbers or relationships. (Target G)
Tasks for this target will require students to create equations and inequalities in one variable to solve problems. Other tasks will require students to create and graph equations in two variables to represent relationships between quantities.
F-IF.C Analyze functions using different representations. (Target M)
Tasks for this target will ask students to graph functions (linear, quadratic... exponential... ) by hand or using technology and compare properties of two functions represented in different ways. Some tasks will focus on understanding equivalent forms that can be used to explain properties of functions and may be associated with Algebra Target H.
A-REI.D Represent and solve equations and inequalities graphically. (Target J)
Tasks for this target will require students to interpret a line or curve as a solution set of an equation in two variables, including tasks that tap student understanding of points beyond the displayed portion of a graph as part of the solution set. Some of these tasks should explicitly focus on non-integer solutions (e.g., give three points on the graph of y = 7x + 2 that have x-values between 1 and 2).
Other tasks for this target will require students to approximate solutions to systems of equations represented graphically, including linear... exponential... (often paired with Target L - Interpret functions that arise in applications in terms of the context).
F.BF.A Build a function that describes a relationship between two quantities. (Target N)
Tasks for this target will require students to write a function (recursive or explicit, as well as translate between the two forms) to describe a relationship between two quantities.
A-APR.A.1 Perform arithmetic operations on polynomials. (Target F)
Tasks for this target will require students to add, subtract, and multiply polynomials.
A-REI.B Solve equations and inequalities in one variable. (Target I)
Tasks for this target will require students to solve linear equations and inequalities in one variable and solve quadratic equations in one variable. Tasks asking students to choose the appropriate method will contribute evidence to Claim 2 and 4.
A-REI.7 Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. (Claims 2, 3 & 4)
Problem solving, which of course builds on a foundation of knowledge and procedural proficiency, sits at the core of doing mathematics. Proficiency at problem solving requires students to choose to use concepts and procedures from across the content domains and check their work using alternative methods. As problem solving skills develop, a student’s understanding of and access to mathematical concepts becomes more deeply established. (Claim 2)
This claim refers to a recurring theme in the CCSSM content and practice standards: the ability to construct and present a clear, logical, convincing argument. For older students this may take the form of a rigorous deductive proof based on clearly stated axioms. For younger students this will involve more informal justifications. Assessment tasks that address this claim will typically present a claim or a proposed solution to a problem and will ask students to provide, for example, a justification, and explanation, or counter-example. (Math Content Specifications, p.63). Communicating mathematical reasoning is not just a requirement of the Standards for Mathematical Practice—it is also a recurrent theme in the Standards for Mathematical Content. For example, many content standards call for students to explain, justify, or illustrate. (Claim 3)
Students can analyze complex, real-world scenarios and can construct and use mathematical models to interpret and solve problems. (Claim 4)
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