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Standard 3 : Write, solve and graph quadratic equations, functions and inequalities in one and two variables.
Cluster Standards
This cluster includes the following benchmarks.
Visit the specific benchmark webpage to find related instructional resources.
- MA.912.AR.3.1 : Given a mathematical or real-world context, write and solve one-variable quadratic equations over the real number system.
- MA.912.AR.3.2 : Given a mathematical or real-world context, write and solve one-variable quadratic equations over the real and complex number systems.
- MA.912.AR.3.3 : Given a mathematical or real-world context, write and solve one-variable quadratic inequalities over the real number system. Represent solutions algebraically or graphically.
- MA.912.AR.3.4 : Write a quadratic function to represent the relationship between two quantities from a graph, a written description or a table of values within a mathematical or real-world context.
- MA.912.AR.3.5 : Given the x-intercepts and another point on the graph of a quadratic function, write the equation for the function.
- MA.912.AR.3.6 : Given an expression or equation representing a quadratic function, determine the vertex and zeros and interpret them in terms of a real-world context.
- MA.912.AR.3.7 : Given a table, equation or written description of a quadratic function, graph that function, and determine and interpret its key features.
- MA.912.AR.3.8 : Solve and graph mathematical and real-world problems that are modeled with quadratic functions. Interpret key features and determine constraints in terms of the context.
- MA.912.AR.3.9 : Given a mathematical or real-world context, write two-variable quadratic inequalities to represent relationships between quantities from a graph or a written description.
- MA.912.AR.3.10 : Given a mathematical or real-world context, graph the solution set to a two-variable quadratic inequality.
Cluster Information
Number:
MA.912.AR.3
Title:
Write, solve and graph quadratic equations, functions and inequalities in one and two variables.
Type:
Standard
Subject:
Mathematics (B.E.S.T.)
Grade:
912
Strand
Algebraic Reasoning
Cluster Access Points
This cluster includes the following Access Points.
- MA.912.AR.3.AP.1 : Given a one-variable quadratic equation from a mathematical or real-world context, select the solution to the equation over the real number system.
- MA.912.AR.3.AP.2 : Solve mathematical one-variable quadratic equations with integer coefficients over the real and complex number systems.
- MA.912.AR.3.AP.3 : Given a mathematical or real-world context, select a one-variable quadratic inequality over the real number system that represents the solution algebraically or graphically.
- MA.912.AR.3.AP.4 : Select a quadratic function to represent the relationship between two quantities from a graph.
- MA.912.AR.3.AP.5 :
Given the x-intercepts and another point on the graph of a quadratic function, select the equation for the function.
- MA.912.AR.3.AP.6 : Given an expression or equation representing a quadratic function in vertex form, determine the vertex and zeros.
- MA.912.AR.3.AP.7 : Given a table, equation or written description of a quadratic function, select the graph that represents the function.
- MA.912.AR.3.AP.8 : Given a mathematical and/or real-world problem that is modeled with quadratic functions, solve the mathematical problem, or select the graph using key features (in terms of context) that represents this model.
- MA.912.AR.3.AP.9 : Select two-variable quadratic inequalities to represent relationships between quantities from a graph or a written description.
- MA.912.AR.3.AP.10 : Select the graph of the solution set to a two-variable quadratic inequality.
Cluster Resources
Vetted resources educators can use to teach the concepts and skills in this topic.
Original Student Tutorials
- Identifying and Interpreting Parts of Quadratic Equations in Factored Form: Identify parts of quadratic equations in factored form and interpret them in terms of the context they represent in this interactive tutorial.
- Quadratic Function Part 2: Launches: Learn about different formats of quadratic equations and their graphs with experiments involving launching and shooting of balls in this interactive tutorial.
This is part 2 of a two-part series: Click HERE to open part 1.
- Quadratic Functions Part 1: Ball Games: Join us as we watch ball games and explore how the height of a ball bounce over time is represented by quadratic functions, which provides opportunities to interpret key features of the function in this interactive tutorial.
This is part 1 of a two-part series: Click HERE to open part 2.
- Highs and Lows Part 2: Completing the Square: Learn the process of completing the square of a quadratic function to find the maximum or minimum to discover how high a dolphin jumped in this interactive tutorial.
This is part 2 of a 2 part series. Click HERE to open part 1.
- Highs and Lows Part 1: Completing the Square: Learn the process of completing the square of a quadratic function to find the maximum or minimum to discover how high a dolphin jumped in this interactive tutorial.
This is part 1 of a 2 part series. Click HERE to open Part 2.
- Solving Rational Equations: Cross Multiplying: Learn how to solve rational linear and quadratic equations using cross multiplication in this interactive tutorial.
- Finding the Zeros of Quadratic Functions: Learn to find the zeros of a quadratic function and interpret their meaning in real-world contexts with this interactive tutorial.
- Finding the Maximum or Minimum of a Quadratic Function: Learn to complete the square of a quadratic expression and identify the maximum or minimum value of the quadratic function it defines. In this interactive tutorial, you'll also interpret the meaning of the maximum and minimum of a quadratic function in a real world context.
- Graphing Quadratic Functions: Follow as we discover key features of a quadratic equation written in vertex form in this interactive tutorial.
Formative Assessments
- Comparing Quadratics: Students are asked to compare two quadratic functions, one given by a table and the other by a function.
- Quilts: Students are asked to write and solve an equation that models a given problem.
- Graphing a Quadratic Function: Students are asked to graph a quadratic function and answer questions about the intercepts, maximum, and minimum.
- Zeros of a Quadratic: Students are asked to identify the zeros of polynomials, without the use of technology, and then describe what the zeros of a polynomial indicate about its graph.
- Model Rocket: Students are asked to graph a function in two variables given in context.
- Hotel Swimming Pool: Students are asked to write an equation in two variables given a verbal description of the relationship among the variables.
- Rocket Town: Students are asked to rewrite a quadratic expression in vertex form to find maximum and minimum values.
- Jumping Dolphin: Students are asked to find the zeros of a quadratic function in the context of a modeling problem.
- Complete the Square - 1: Students are asked to solve a quadratic equation by completing the square.
- A Home for Fido: Students are asked to rewrite a quadratic function in an equivalent form by completing the square and to use this form to identify the vertex of the graph and explain its meaning in context.
- Launch From a Hill: Students are asked to factor and find the zeros of a polynomial function given in context.
- Complete the Square - 2: Students are asked to solve a quadratic equation by completing the square.
- Complete the Square - 3: Students are asked to solve a quadratic equation by completing the square.
- Complex Solutions?: Students are asked to explain how to recognize when the quadratic formula results in complex solutions.
- Which Strategy?: Students are shown four quadratic equations and asked to choose the best method for solving each equation.
Lesson Plans
- Solving Quadratic Equations by Completing the square: Students will model the process of completing the square (leading coefficient of 1) with algebra tiles, and then practice solving equations using the completing the square method. This lesson provides a discovery opportunity to conceptually see why the process of squaring half of the b value is considered completing the square.
- Solving Quadratics - Exploring Different Methods: Students will explore how different methods find the solutions (roots) to quadratic equations including, factoring, graphing, and the quadratic formula.
- Discovering Properties of Parabolas by Comparing and Contrasting Parabolic Equations:
- Teachers can use this resource to teach students how to derive the equation of a parabola in vertex form y = a(x – h)2 + k, when given the (x, y) coordinates of the focus and the linear equation of the directrix.
- An additional interactive graphing spreadsheet can be used as a resource to aid teachers in providing examples.
- Acting Out A Parabola: the importance of a vertex and directrix: Students will learn the significance of a parabola's vertex and directrix. They will learn the meaning of what exactly a parabola is by physically representing a parabola, vertex, and directrix. Students will be able to write an equation of a parabola given only a vertex and directrix.
- How High Can I Go?: Students will graph quadratic equations, and identify the axis of symmetry, the maximum/minimum point, the vertex, and the roots. Students will work in pairs and will move around the room matching equations with given graphs.
- Ranking Sports Players (Quadratic Equations Practice): In this Model Eliciting Activity, MEA, students will rank sports players by designing methods, using different indicators, and working with quadratic equations.
Model-Eliciting-Activities, MEAs, allow students to critically analyze data sets, compare information, and require students to explain their thinking and reasoning. While there is no one correct answer in an MEA, students should work to explain their thinking clearly and rationally. Therefore, teachers should ask probing questions and provide feedback to help students develop a coherent, data-as-evidence-based approach within this learning experience.
- Radical Mathematical: In this lesson students will solve radical equations, showing how extraneous solutions may arise. Students will solve radical equations that model real-world relationships.
- Hip to be (completing the) Square: This lesson is an introduction to completing the square. Students will learn what it means to "complete the square" with a quadratic trinomial expression. They will practice both with manipulatives and mathematically, and will then use that information to find the maximum or minimum value of an expression using the vertex form of a quadratic. This lesson moves through all levels of Concrete, Representational, Abstract instruction.
- Where did the answers go? Oh, they're imaginary!: In this lesson, students will solve quadratic equations with imaginary (complex) solutions and be facilitated in the discovery of the Fundamental Theorem of Algebra.
- Parts and more Parts-- Parabola Fun: This is an entry lesson into quadratic functions and their shapes. Students see some real-life representations of parabolas. This lesson provides important vocabulary associated with quadratic functions and their graphs in an interactive manner. Students create a foldable and complete a worksheet using their foldable notes.
- Graphing Quadratics Made Easy: Vertex Form of the Equation: This lesson covers quadratic translations as they relate to vertex form of a quadratic equation. Students will predict what will happen to the graph of a quadratic function when more than one constant is in a quadratic equation. Then, the students will graph quadratic equations in vertex form using their knowledge of the translations of a quadratic function, as well as describe the translations that occur. Students will also identify the parent function of any quadratic function as
. - The Quadratic Quandary: Students will sort various quadratic equations by the method they would use for solving (ie. factoring, quadratic formula). Then as a class they justify their placements and eventually discover that there are many ways to solve and that some make sense in different situations, however there is no real "correct" method for each equation type.
Perspectives Video: Experts
- Jumping Robots and Quadratics: <p>Jump to it and learn more about how quadratic equations are used in robot navigation problem solving!</p>
- Using Mathematics to Optimize Wing Design: Nick Moore discusses his research behind optimizing wing design using inspiration from animals and how they swim and fly.
Download the CPALMS Perspectives video student note taking guide.
- Problem Solving with Project Constraints: <p>It's important to stay inside the lines of your project constraints to finish in time and under budget. This NASA systems engineer explains how constraints can actually promote creativity and help him solve problems!</p>
Perspectives Video: Professional/Enthusiasts
- Revolutionize Wing Design with Equations and Statistics: <p>Brandon Reese, a PhD candidate in the FAMU-FSU College of Engineering, discusses the significance of both Bernoulli's equation and statistical analysis for the design of a "smart wing."</p>
- Quadratic Equations and Robots: <p>Get in gear with robotics as this engineer explains how quadratic equations are used in programming robotic navigation.</p>
Perspectives Video: Teaching Ideas
- Solving Quadratic Equations using Babylonian Multiplication: Unlock an effective teaching strategy for teaching solving quadratic equations using Babylonian multiplication in this Teacher Perspectives video for educators.
- Solving Quadratic Equations by Completing the Square: Unlock an effective teaching strategy for teaching solving quadratic equations by completing the square in this Teacher Perspectives video for educators.
- Making Connections with Vieta's Formula: Unlock an effective teaching strategy for explaining the equation of the axis of symmetry using Vieta's formula in this Teacher Perspectives video for educators.
- Solving Quadratic Equation Using Loh's Method: Unlock an effective teaching strategy for solving quadratic equations in this Teacher Perspectives video for educators.
Problem-Solving Tasks
- The Lighthouse Problem: This problem asks students to model phenomena on the surface of the earth by examining the visibility of the lamp in a lighthouse from a boat.
- What functions do two graph points determine?: This problem solving task challenges students to find the linear, exponential and quadratic functions based on two points.
Tutorials
- Graphing Quadractic Functions in Vertex Form: This tutorial will help the students to identify the vertex of a parabola from the equation, and then graph the parabola.
- Graphing Quadratic Equations: This tutorial helps the learners to graph the equation of a quadratic function using the coordinates of the vertex of a parabola and its x- intercepts.