Completing the Square (Lesson 6.5)
Unit 1: Sequences and Linear Functions
Day 1: Recursive Sequences
Day 2: Applications of Arithmetic Sequences
Day 3: Sum of an Arithmetic Sequence
Day 4: Applications of Geometric Sequences
Day 5: Sequences Review
Day 6: Quiz 1.1 to 1.4
Day 7: Linear Relationships
Day 8: Point-Slope Form of a Line
Day 9: Standard Form of a Linear Equation
Day 10: Quiz 1.5 to 1.7
Day 11: Unit 1 Review
Day 12: Unit 1 Test
Unit 2: Linear Systems
Day 1: Linear Systems
Day 2: Number of Solutions
Day 3: Elimination
Day 4: Larger Systems of Equations
Day 5: Quiz 2.1 to 2.4
Day 6: Systems of Inequalities
Day 7: Optimization Using Systems of Inequalities
Day 8: Quiz 2.5 to 2.6
Day 9: Unit 2 Review
Day 10: Unit 2 Test
Unit 3: Function Families and Transformations
Day 1: Interpreting Graphs
Day 2: What is a function?
Day 3: Translating Functions
Day 4: Quiz 3.1 to 3.3
Day 5: Quadratic Functions and Translations
Day 6: Square Root Functions and Reflections
Day 7: Absolute Value Functions and Dilations
Day 8: Equations of Circles
Day 9: Quiz 3.4 to 3.7
Day 10: Unit 3 Review
Day 11: Unit 3 Test
Unit 4: Working with Functions
Day 1: Using Multiple Strategies to Solve Equations
Day 2: Solving Equations
Day 3: Solving Nonlinear Systems
Day 4: Quiz 4.1 to 4.3
Day 5: Combining Functions
Day 6: Composition of Functions
Day 7: Inverse Relationships
Day 8: Graphs of Inverses
Day 9: Quiz 4.4 to 4.7
Day 10: Unit 4 Review
Day 11: Unit 4 Test
Unit 5: Exponential Functions and Logarithms
Day 1: Writing Exponential Functions
Day 2: Graphs of Exponential Functions
Day 3: Applications of Exponential Functions
Day 4: Quiz 5.1 to 5.3
Day 5: Building Exponential Models
Day 6: Logarithms
Day 7: Graphs of Logarithmic Functions
Day 8: Quiz 5.4 to 5.6
Day 9: Unit 5 Review
Day 10: Unit 5 Test
Unit 6: Quadratics
Day 1: Forms of Quadratic Equations
Day 2: Writing Equations for Quadratic Functions
Day 3: Factoring Quadratics
Day 4: Factoring Quadratics. Part 2.
Day 5: Solving Using the Zero Product Property
Day 6: Quiz 6.1 to 6.4
Day 7: Completing the Square
Day 8: Completing the Square for Circles
Day 9: Quadratic Formula
Day 10: Complex Numbers
Day 11: The Discriminant and Types of Solutions
Day 12: Quiz 6.5 to 6.9
Day 13: Unit 6 Review
Day 14: Unit 6 Test
Unit 7: Higher Degree Functions
Day 1: What is a Polynomial?
Day 2: Forms of Polynomial Equations
Day 3: Polynomial Function Behavior
Day 4: Repeating Zeros
Day 5: Quiz 7.1 to 7.4
Day 6: Multiplying and Dividing Polynomials
Day 7: Factoring Polynomials
Day 8: Solving Polynomials
Day 9: Quiz 7.5 to 7.7
Day 10: Unit 7 Review
Day 11: Unit 7 Test
Unit 8: Rational Functions
Day 1: Intro to Rational Functions
Day 2: Graphs of Rational Functions
Day 3: Key Features of Graphs of Rational Functions
Day 4: Quiz 8.1 to 8.3
Day 5: Adding and Subtracting Rational Functions
Day 6: Multiplying and Dividing Rational Functions
Day 7: Solving Rational Functions
Day 8: Quiz 8.4 to 8.6
Day 9: Unit 8 Review
Day 10: Unit 8 Test
Unit 9: Trigonometry
Day 1: Right Triangle Trigonometry
Day 2: Solving for Missing Sides Using Trig Ratios
Day 3: Inverse Trig Functions for Missing Angles
Day 4: Quiz 9.1 to 9.3
Day 5: Special Right Triangles
Day 6: Angles on the Coordinate Plane
Day 7: The Unit Circle
Day 8: Quiz 9.4 to 9.6
Day 9: Radians
Day 10: Radians and the Unit Circle
Day 11: Arc Length and Area of a Sector
Day 12: Quiz 9.7 to 9.9
Day 13: Unit 9 Review
Day 14: Unit 9 Test
Learning Targets
Rewrite quadratic equations as perfect squares.
Solve quadratic equations by completing the square.
Tasks/Activity | Time |
---|---|
Activity | 20 minutes |
Debrief Activity with Margin Notes | 10 minutes |
QuickNotes | 10 minutes |
Check Your Understanding | 15 minutes |
Activity: Completing the Square
Lesson Handouts
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Answer Key
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Homework
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Experience First
The first half of the unit focuses on solving quadratics by graphing and factoring. The second half of the unit will focus on solving by completing the square and using the quadratic formula. In today's lesson we're working to get students to understand why we would want an equation to be written as a perfect square (so we can square root) and if it's not a perfect square, how do we turn it into one (add something so that it is). Now I'm going to be honest, completing the square with EFFL is sort of hard. Your students can do it, but it might take a little more teacher monitoring than most lessons. We're going to start with some problems where students solve for x by square rooting an equation. Then we will give them a problem where they can't square root to solve because there is an x2 and x term, BUT we can rewrite the problem so that it is a perfect square. Depending on how your students are doing with this, you may want to pause the class here to discuss questions #1-2. After going over question #2, give students some time to work on question #3. If they struggle, that's ok! There's nothing wrong with a little struggle. We want them to think about how to fill in the rectangle diagram IF the factors were the same. Once they've filled in the x2 and $x$ terms, hopefully they will see what must go in the last spot. As they are working on #3, these guiding questions may be helpful. Quiding Questions: You may want to pause the group after #3 to go over their work before giving them time to work on #4. But if they're catching on, you could have them finish question #4 also before debriefing. You'll notice that for both question #3 and 4, the constant was already on the other side of the equation. We'll do examples in the Check Your Understanding where this is not the case.
Formalize Later
As mentioned, you may want to break up your debrief instead of waiting until students have completed the entire activity. Your students may need a little more scaffolding (but maybe not!). Use your professional discretion. You know your kids best. If you do want to break it up, we'd suggest debriefing after question #2 and possibly again after #3. During your debrief, you should highlight why we would want an equation to be written as a perfect square (so we can square root) in the first place. Point out how they were able to solve all of the equations in question #1 because we were able to square root. So if we don't have a perfect square, let's turn it into one. Color coding the value that is being added to create the perfect square can be really helpful. Also, we like to add a space to the equation after the bx term to signify that something needs to be added there. You'll notice that we aren't identifying on this first page that the value being added is (1/2b)2. We're actually not going to talk about this until the end of the Check Your Understanding problems. We think that using a rectangle diagram to figure out what to add is more helpful than memorizing (1/2b)2. That being said, you could add it in to the margin notes or to the QuickNotes. We added it at the bottom of the CYU.