Solving Quadratics Using Symmetry (Lesson 7.8)
Unit 1: Generalizing Patterns
Day 1: Intro to Unit 1
Day 2: Equations that Describe Patterns
Day 3: Describing Arithmetic Patterns
Day 4: Making Use of Structure
Day 5: Review 1.1-1.3
Day 6: Quiz 1.1 to 1.3
Day 7: Writing Explicit Rules for Patterns
Day 8: Patterns and Equivalent Expressions
Day 9: Describing Geometric Patterns
Day 10: Connecting Patterns across Multiple Representations
Day 11: Review 1.4-1.7
Day 12: Quiz 1.4 to 1.7
Day 13: Unit 1 Review
Day 14: Unit 1 Test
Unit 2: Linear Relationships
Day 1: Proportional Reasoning
Day 2: Proportional Relationships in the Coordinate Plane
Day 3: Slope of a Line
Day 4: Linear Equations
Day 5: Review 2.1-2.4
Day 6: Quiz 2.1 to 2.4
Day 7: Graphing Lines
Day 8: Linear Reasoning
Day 9: Horizontal and Vertical Lines
Day 10: Standard Form of a Line
Day 11: Review 2.5-2.8
Day 12: Quiz 2.5 to 2.8
Day 13: Unit 2 Review
Day 14: Unit 2 Test
Unit 3: Solving Linear Equations and Inequalities
Day 1: Intro to Unit 3
Day 2: Exploring Equivalence
Day 3: Representing and Solving Linear Problems
Day 4: Solving Linear Equations by Balancing
Day 5: Reasoning with Linear Equations
Day 6: Solving Equations using Inverse Operations
Day 7: Review 3.1-3.5
Day 8: Quiz 3.1 to 3.5
Day 9: Representing Scenarios with Inequalities
Day 10: Solutions to 1-Variable Inequalities
Day 11: Reasoning with Inequalities
Day 12: Writing and Solving Inequalities
Day 13: Review 3.6-3.9
Day 14: Quiz 3.6 to 3.9
Day 15: Unit 3 Review
Day 16: Unit 3 Test
Unit 4: Systems of Linear Equations and Inequalities
Day 1: Intro to Unit 4
Day 2: Interpreting Linear Systems in Context
Day 3: Interpreting Solutions to a Linear System Graphically
Day 4: Substitution
Day 5: Review 4.1- 4.3
Day 6: Quiz 4.1 to 4.3
Day 7: Solving Linear Systems using Elimination
Day 8: Determining Number of Solutions Algebraically
Day 9: Graphing Linear Inequalities in Two Variables
Day 10: Writing and Solving Systems of Linear Inequalities
Day 11: Review 4.4- 4.7
Day 12: Quiz 4.4 to 4.7
Day 13: Unit 4 Review
Day 14: Unit 4 Test
Unit 5: Functions
Day 1: Using and Interpreting Function Notation
Day 2: Concept of a Function
Day 3: Functions in Multiple Representations
Day 4: Interpreting Graphs of Functions
Day 5: Review 5.1-5.4
Day 6: Quiz 5.1 to 5.4
Day 7: From Sequences to Functions
Day 8: Linear Functions
Day 9: Piecewise Functions
Day 10: Average Rate of Change
Day 11: Review 5.5-5.8
Day 12: Quiz 5.5 to 5.8
Day 13: Unit 5 Review
Day 14: Unit 5 Test
Unit 6: Working with Nonlinear Functions
Day 1: Nonlinear Growth
Day 2: Step Functions
Day 3: Absolute Value Functions
Day 4: Solving an Absolute Value Function
Day 5: Review 6.1-6.4
Day 6: Quiz 6.1 to 6.4
Day 7: Exponent Rules
Day 8: Power Functions
Day 9: Square Root and Root Functions
Day 10: Radicals and Rational Exponents
Day 11: Solving Equations
Day 12: Review 6.5-6.9
Day 13: Quiz 6.5 to 6.9
Day 14: Unit 6 Review
Day 15: Unit 6 Test
Unit 7: Quadratic Functions
Day 1: Quadratic Growth
Day 2: The Parent Function
Day 3: Transforming Quadratic Functions
Day 4: Features of Quadratic Functions
Day 5: Forms of Quadratic Functions
Day 6: Review 7.1-7.5
Day 7: Quiz 7.1 to 7.5
Day 8: Writing Quadratics in Factored Form
Day 9: Solving Quadratics using the Zero Product Property
Day 10: Solving Quadratics Using Symmetry
Day 11: Review 7.6-7.8
Day 12: Quiz 7.6 to 7.8
Day 13: Quadratic Models
Day 14: Unit 7 Review
Day 15: Unit 7 Test
Unit 8: Exponential Functions
Day 1: Geometric Sequences: From Recursive to Explicit
Day 2: Exponential Functions
Day 3: Graphs of the Parent Exponential Functions
Day 4: Transformations of Exponential Functions
Day 5: Review 8.1-8.4
Day 6: Quiz 8.1 to 8.4
Day 7: Working with Exponential Functions
Day 8: Interpreting Models for Exponential Growth and Decay
Day 9: Constructing Exponential Models
Day 10: Rational Exponents in Context
Day 11: Review 8.5-8.8
Day 12: Quiz 8.5 to 8.8
Day 13: Unit 8 Review
Day 14: Unit 8 Test
Learning Targets
Given a quadratic in standard form, find its zeros by identifying the axis of symmetry and calculating the horizontal distance between the vertex and an x-intercept.
Tasks/Activity | Time |
---|---|
Activity Questions 1-2 | 8 minutes |
Debrief Questions 1-2 with Margin Notes | 8 minutes |
Activity Questions 3-4 | 20 minutes |
Debrief Questions 3-4 with Margin Notes | 5 minutes |
QuickNotes | 5 minutes |
Check Your Understanding | 8 minutes |
Activity: How Do We Find Zeros?
Lesson Handouts
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Answer Key
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Homework
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Experience First
It’s a big day in Algebra 1! Today students will learn a new way of solving for the x-intercepts/zeros of a quadratic that may be new to you as well! While the quadratic formula is probably one of the most well known formulas in high school mathematics, we’re going to introduce a new approach that we find to be more intuitive and strongly connects to the symmetry and graph of a quadratic function.
In the first part of the lesson, we review the idea that the x-intercepts/zeros are equidistant from the vertex/axis of symmetry. We give students scenarios where they are given the vertex and the horizontal distance to the intercepts in question 1, and the vertex and one zero in question 2, and students must reason about the location of the other intercept. This is a critical first step in understanding our variation on the quadratic formula. Be sure to debrief the front page with your students before having them move on to the back page. Make sure all students can explain the meaning of the distance referenced in question 2b. Have students do a quick turn and talk where they explain to their partner why this distance is important and how it’s related to finding the intercepts.
On page 2, students are given the equation for Janyce’s parabola. Note that students have already found the vertex and both x-intercepts of this parabola using symmetry. The goal in this part of the lesson is to identify this key information from an equation in standard form. Note that h2−ac represents the distance between the axis of symmetry and the x-intercept, and adding and subtracting this distance from the x-coordinate of the vertex will provide both intercepts. Here’s a visual of this approach. Proving this formula is outside the scope of an Algebra 1 class, but it is a worthwhile exercise for you as the educator. We provide a proof for you below. Monitoring Questions
Formalize Later
You may be wondering: why not just teach the quadratic formula? Here are a few reasons: Being the curious math teacher that you are, you’re probably wondering why this approach works. Check out the proof here.