##
* *Reflection: Coherence
Irrational Zeros of Quadratic Functions - Section 2: Investigation

I really enjoyed having the students work on this investigation. It allowed me to ensure that they were constantly and consistently making connections back to what the zeros of a polynomial function represent graphically (the x-intercepts). When students were algebraically finding the roots they received instant gratification when they could graph the function on the calculator and see where the roots were. Then, they solidified their understanding by sketching the graph and labeling the roots of the function on that graph. Even when students attempted number 5, they could see right away why there would be three solutions by looking at the graph. By having students make this connection repeatedly it really helped them to make connections between the algebraic and graphical representations.

*Making Connections*

*Coherence: Making Connections*

# Irrational Zeros of Quadratic Functions

Lesson 10 of 21

## Objective: SWBAT use the technique of completing the square to solve quadratic equations.

#### Launch

*10 min*

I begin today's lesson by having students work on **Slide 1** of Day 2_Launch with a partner. I ask the students to complete the task individually and compare answers. Once students have had a few minutes to work, I will call on two students to post their work on the board so that other students can compare with what they obtained as the solutions.

**Teaching Note**: I will be projecting the slide, so I will have students post their work on either side of the projection. This arrangement will enable students to refer to the work when completing the compare and contrast on the next slide.

After discussing the students' work on Slide 1, I will give students time to come up with as many ideas as they can for both the similarities and differences for the two methods of solving (**MP3**), factoring and completing the square. In order to give students a good starting point, I will suggest that students start with the "process" or steps. This is a more concrete idea that students can extract from the work that is posted on the board.

On this comparison task, I expect some of my students will be able to generalize their thinking to other problems. Other students may struggle a little more with the uses of each strategy, possibly because the warm up question can be solved using either method. This will be a good point of conversation: a quadratic equation that cannot be solved by factoring over the integers requires a different method. Once students have a couple of minutes to work, I will ask them to Think-Pair-Share to compare their work and try to add new ideas to their list. Finally, I will have several students share their ideas. I will keep a record of all of the ideas on on the board so that we can reference the list throughout the class.

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#### Investigation

*25 min*

During my quadratics unit, I want to continually connect the algebraic, numeric, and graphical solutions to an equation. This investigation (complete_the_square_day2_investigation) allows students to make a connection between the algebraic and the graphical representations of a quadratic equation.

My students will use either a graphing calculator or the Desmos online graphing calculator during this investigation. Both of these tools allow students to calculate the roots for a quadratic equation, even if they are irrational. I find that irrational roots can be confusing to students because they are not as comfortable with decimal values as they are with integer values. That said, I intend that during this investigation my students will successfully make meaning out of the decimal solutions they are finding algebraically(MP2).

My students will work with their partner on this investigation. I encourage students to check each others work. They should also verbally explain to their partner what they are trying to do in each step in the process of solving (MP3).

In questions #5 and #6, I want my students to connect the idea of factoring to the zeros of a polynomial function. The graphical representation will help students identify their algebraic solutions, reflect, and make this connection.

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#### Closure

*5 min*

The complete_the_square_day2_close provides an opportunity for students to use technology to work through a common misconception. I chose the numbers in this problem deliberately, to try to bring the misconception to the surface.

Some students may attempt to factor this equation thinking that 6 and 2 could be used in the binomials. However, I will ask students to graph the function. When they do they will notice that x = -6 and x = -2 are not the solutions. I hope that this observation will prompt students to look back at their work and make corrections (MP6).

At the same time, I expect other students to notice that the equation cannot be solved by factoring. I hope that they will employ completing the square. For these students, the graph will serve as conformation that their solution is correct.

#### Resources

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- LESSON 1: Overview of Quadratics-Essential Vocabulary
- LESSON 2: Quadratic Functions and Roots
- LESSON 3: Equations involving Factored Expressions
- LESSON 4: Solving Quadratics by Factoring-Day 1
- LESSON 5: Solving Quadratics by Factoring-Day 2
- LESSON 6: Applications of Quadratics Day 1
- LESSON 7: Simplifying Radical Expressions
- LESSON 8: Solving Quadratic Equations with Perfect Squares
- LESSON 9: Completing the Square Day 1
- LESSON 10: Irrational Zeros of Quadratic Functions
- LESSON 11: The Quadratic Formula-Day 1
- LESSON 12: Comparing the Three Methods of Solving Quadratics
- LESSON 13: Three Methods of Solving Quadratics and Word Problems
- LESSON 14: Identifying Roots and Critical Points-Need to Edit
- LESSON 15: Graphing Quadratic Functions Day 1
- LESSON 16: Key Features of Quadratic Functions
- LESSON 17: Sketching Polynomial Functions
- LESSON 18: Vertex Form of a Quadratic Function
- LESSON 19: Transformations with Quadratic Functions
- LESSON 20: Modeling With Quadratic Functions
- LESSON 21: Projectile Problems & Review