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Algebra I Module 4, Topic A: Quadratic Expressions, Equations, Functions, and Their Connection to Rectangles

Classroom Resource Information

Title:

Algebra I Module 4, Topic A: Quadratic Expressions, Equations, Functions, and Their Connection to Rectangles

URL:

https://www.engageny.org/resource/algebra-i-module-4-topic-overview

Content Source:

EngageNY

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Lesson/Unit Plan

Overview:

Module 4, Topic A introduces polynomial expressions. In Module 1, students learned the definition of a polynomial and how to add, subtract, and multiply polynomials. Here, their work with multiplication is extended and connected to factoring polynomial expressions and solving basic polynomial equations (A-APR.A.1, A-REI.D.11). They analyze, interpret, and use the structure of polynomial expressions to multiply and factor polynomial expressions (A-SSE.A.2). They understand factoring as the reverse process of multiplication. In this topic, students develop the factoring skills needed to solve quadratic equations and simple polynomial equations by using the zero-product property (A-SSE.B.3a). Students transform quadratic expressions from standard form, ax² + bx + c, to factored form, f(x) = a(x - n)(x - m), and then solve equations involving those expressions. They identify the solutions of the equation as the zeros of the related function. Students apply symmetry to create and interpret graphs of quadratic functions (F-IF.B.4, F-IF.C.7a). They use the average rate of change on an interval to determine where the function is increasing or decreasing (F-IF.B.6). Using area models, students explore strategies for factoring more complicated quadratic expressions, including the product-sum method and rectangular arrays. They create one- and two-variable equations from tables, graphs, and contexts and use them to solve contextual problems represented by the quadratic function (A-CED.A.1, A-CED.A.2). Students then relate the domain and range for the function to its graph and the context (F-IF.B.5).

Content Standard(s):

Mathematics MA2019 (2019) Grade: 9-12 Algebra I with Probability	4. Interpret linear, quadratic, and exponential expressions in terms of a context by viewing one or more of their parts as a single entity. Example: Interpret the accrued amount of investment P(1 + r)^t , where P is the principal and r is the interest rate, as the product of P and a factor depending on time t. Unpacked Content Evidence Of Student Attainment: Students: Make sense of algebraic expressions by identifying structures within the expression which allow them to rewrite it in useful ways to assist in the solution of given problems. Interpret the meaning of the parts of an expression. For example, see that 3 + (x-2)² is a sum of a constant and a square, that the square contains the expression x-2, and that the value of the expression is always greater than 3. Justify their selection of a form for an expression by explaining which features of the expression are revealed by the particular form and how these features aid in resolving a problem situation. Teacher Vocabulary: Linear expression Quadratic expression Exponential expression Equivalent expressions Knowledge: Students know: How to recognize the parts of linear, quadratic and exponential expressions and what each part represents. When one form of an algebraic expression is more useful than an equivalent form of that same expression to solve a given problem. That one or more parts of an expression can be viewed as a single entity. Skills: Students are able to: Use algebraic properties to produce equivalent forms of the same expression by recognizing underlying mathematical structures. Interpret expressions in terms of a context. View one or more parts of an expression as a single entity and determine the impact parts of the expression have in terms of the context. Understanding: Students understand that: Making connections among the parts of an expression reveals the roles of important mathematical features of a problem. Diverse Learning Needs: Essential Skills: Learning Objectives: ALGI.4.1: Define linear, quadratic and exponential functions. ALGI.4.2: Classify an expression as linear, quadratic or exponential from a table. ALGI.4.3: Classify an expression as linear, quadratic or exponential from an equation. ALGI.4.4: Classify an expression as linear, quadratic or exponential from a graph. ALGI.4.5: Define terms, factors, and coefficients. ALGI.4.6: Identify factors in linear, exponential and quadratic expressions. ALGI.4.7: Identify coefficients in linear, exponential and quadratic expressions. ALGI.4.8: Identify terms in linear, exponential and quadratic expressions. ALGI.4.9: Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient). ALGI.4.10: Recognize one or more parts of an exponential expression as a single entity. ALGI.4.11: Recognize one or more parts of a quadratic expression as a single entity. ALGI.4.12: Recognize one or more parts of a linear expression as a single entity. Prior Knowledge Skills: Recognize ordered pairs. Identify ordered pairs. Recognize linear equations. Recall how to solve problems using the distributive property. Define linear functions, nonlinear functions, slope, and y-intercept. Alabama Alternate Achievement Standards AAS Standard: M.A.AAS.11.4 Identify an algebraic expression involving addition or subtraction to represent a real-world problem.
Mathematics MA2019 (2019) Grade: 9-12 Algebra I with Probability	5. Use the structure of an expression to identify ways to rewrite it. 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). Unpacked Content Evidence Of Student Attainment: Students: Make sense of algebraic expressions by identifying structures within the expression which allow them to rewrite it in useful and more efficient ways. Teacher Vocabulary: Terms Linear expressions Equivalent expressions Difference of two squares Factor Difference of squares Knowledge: Students know: Algebraic properties. When one form of an algebraic expression is more useful than an equivalent form of that same expression. Skills: Students are able to: Use algebraic properties to produce equivalent forms of the same expression by recognizing underlying mathematical structures. Understanding: Students understand that: Generating equivalent algebraic expressions facilitates the investigation of more complex algebraic expressions. Diverse Learning Needs: Essential Skills: Learning Objectives: ALGI.5.1: Define equivalent expressions. ALGI.5.2: Rewrite an exponential expression in an alternative way. ALGI.5.3: Rewrite a quadratic expression in an alternative way. ALGI.5.4: Rewrite a linear expression in an alternative form. ALGI.5.5: Understand that rewriting an expression in different forms in a problem context can shed light on the problem. ALGI.5.6: Recall properties of exponents. Prior Knowledge Skills: li>Give examples of the properties of operations including distributive, commutative, and associative. Recall how to find the greatest common factor. Combine like terms of a given expression. Recognize the property demonstrated in a given expression. Simplify expressions with parentheses (Ex. 5(4 + x) = 20 + 5x). Simplify an expression by dividing by the greatest common factor (Ex. 18x + 6y= 6(3x + y). Define linear expression, rational, coefficient, and rational coefficient. 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). Alabama Alternate Achievement Standards AAS Standard: M.A.AAS.11.5 Solve simple algebraic equations using real-world scenarios with one variable using multiplication or division.
Mathematics MA2019 (2019) Grade: 9-12 Algebra I with Probability	6. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. a. Factor quadratic expressions with leading coefficients of one, and use the factored form to reveal the zeros of the function it defines. b. Use the vertex form of a quadratic expression to reveal the maximum or minimum value and the axis of symmetry of the function it defines; complete the square to find the vertex form of quadratics with a leading coefficient of one. c. Use the properties of exponents to transform expressions for exponential functions. 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 or decay. Unpacked Content Evidence Of Student Attainment: Students: Make sense of algebraic expressions by identifying structures within the expression which allow them to rewrite it in useful ways to assist in the solution of given problems. Produce the useful equivalent forms of expressions, Factor a quadratic expression with leading coefficient of one to reveal the zeros of the function it defines and complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. Use the vertex form of a quadratic expression to reveal the maximum or minimum value and the axis of symmetry of the function it defines. Justify their selection of a form for an expression by explaining which features of the expression are revealed by the particular form and how these features aid in resolving a problem situation. Teacher Vocabulary: Quadratic expression Zeros Complete the square Roots Zeros Solutions x-intercepts Maximum value Minimum value Factor Roots Exponents Equivalent form Vertex form of a quadratic expression Knowledge: Students know: Techniques for generating equivalent forms of an algebraic expression, including factoring and completing the square for quadratic expressions and using properties of exponents. When one form of an algebraic expression is more useful than an equivalent form of that same expression to solve a given problem. Skills: Students are able to: Use algebraic properties including properties of exponents to produce equivalent forms of the same expression by recognizing underlying mathematical structures. Factor quadratic expressions. Complete the square in quadratic expressions. Use the vertex form of a quadratic expression to identify the maximum or minimum and the axis of symmetry. Understanding: Students understand that: Making connections among equivalent expressions reveals the roles of important mathematical features of a problem. Diverse Learning Needs: Essential Skills: Learning Objectives: ALGI.6.1: Convert an expression to an alternative format. ALGI.6.2: Recognize the best format for a specific application. ALGI.6.3: Match equivalent expressions written in different forms. a. ALGI.6.4: Define factor, quadratic expression and zero product property. ALGI.6.5: Factor a quadratic expression. ALGI.6.6: Use the zero product property to reveal the zeros in the function. ALGI.6.7: Solve a one-step equation. ALGI.6.8: Solve a two-step equation. ALGI.6.9: Determine the Greatest Common Factor (GCF). b. ALGI.6.10: Define maximum and minimum value. ALGI.6.11: Explain the steps for completing the square. ALGI.6.12: Given a quadratic expression in which the square has already been completed, determine the maximum or minimum values. c. ALGI.6.13: Define roots. ALGI.6.14: Find the equation using the distributive property. ALGI.6.15: Locate and identify the roots on a graph using the x-intercepts. ALGI.6.16: Take given roots and convert into a one-step equation set equal to zero. Prior Knowledge Skills: Identify how many solutions the linear equation may or may not have. Recall how to solve problems using the distributive property Explain the distributive property. Recall solving one-step equations. Alabama Alternate Achievement Standards AAS Standard: M.A.AAS.11.5 Solve simple algebraic equations using real-world scenarios with one variable using multiplication or division.

Tags:

coefficients, equations, equivalent, expressions, factors, functions, graph, integers, intercepts, linear function, polynomials, quadratic function, rate of change, variables

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There are ten lessons in this topic.

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Hannah Bradley

ALEX Classroom Resource

Algebra I Module 4, Topic A: Quadratic Expressions, Equations, Functions, and Their Connection to Rectangles