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* *Reflection: Developing a Conceptual Understanding
Solving Quadratics by Factoring-Day 2 - Section 2: Practice

This lesson happened to fall on the day after a week long break. It worked out very well in order to give students a reintroduction to solving quadratic equations and give them time to practice and make connections between the algebraic and graphical representations.

I made it a point in the narrative to discuss question #17 and #22. As anticipated almost EVERY student tried to solve these equations by setting them equal to zero. From there, about half were successful at factoring them to find the correct solutions. I made it a point to pull the class back together and ask them what they notice about #17 and #22 that made them different from the other equations. Once students looked closer they realized that they were linear. This just shows that sometimes students go into "autopilot" and the thinking that you would like to see goes right out the window!

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*Developing a Conceptual Understanding: Welcome Back*

# Solving Quadratics by Factoring-Day 2

Lesson 5 of 21

## Objective: SWBAT solve a quadratic equation and using technology to check their answer graphically.

#### Launch

*10 min*

This launch is a **concept_attainment** and the process is explained via video at this link. This launch connects to the closure of the previous lesson where students simplified an expression and solved an equation.

**Technology Note**: Please view the launch presentation (Solve_Quadratics_Day2) in slide show mode. On Slide 2 there are many expressions lumped together at the top of the slide. In slide show mode they appear one at a time and then populate the table below.

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#### Practice

*25 min*

My intention in this practice section is to give my students an opportunity for lots of practice solving quadratic equations. As they work on Solve_Quadratics Day 2_Practice Students will work with their partner on this practice assignment. However, while they are practicing, I want them to be developing the conceptual understanding. In order to promote this I will question students about the meaning of what they are actually finding when they obtain a solution (MP2).

In order to get students to think about the solutions in multiple ways, I ask them to graph the associated function using technology as a check. As they visually inspect the roots of the function (see roots_of_quadratic.png -- the picture shown here is from desmos.com), they can see a pattern with respect to the location of their solutions.

**Technology Note:** The desmos.com website provides a particularly friendly interface for students. They can evaluate a root by clicking or tapping on it. A graphing calculator could also be used for this purpose.

Once students graph the function, I ask them to make a sketch next to their solution to the equation. I also request that they clearly label the roots.

Ultimately, there are too many questions here for students to get through in 25 minutes. That said, I provide a lot and I choose an appropriate number for my class on a particular day. It is always better to have extra problems. For my current class, I asked students to pick two questions from each row. At the end of class, I asked students to complete the remaining questions at home.

**Teaching Notes:**

- For questions like #14, it may be necessary to explain to students that in order to graph the associated function all of the terms should be on one side of the equal sign. If a students tries to graph the equation as written using desmos, they will be graphing the points of intersection of two functions: the quadratic equation and the line. For some students, this could be an interesting extension!
- Problem 17 and Problem 22 are both linear equations. I want to make sure that students don't get so caught up in a procedure, that they incorrectly apply that procedure without inspecting the equation first. Of course, both of these equations could be solved by setting the equation equal to zero and factoring. Depending on how many students solve the equation in this way, I may decide to address the fact that this is probably not the most efficient method of solving. I can imagine doing this by direct instruction or by having students share their solutions.

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#### Closure

*5 min*

Today's Ticket Out (Solve_Quadratics_Day2_Close) gives the students the opportunity to express their ideas in writing. Now that students have worked through several examples that look similar to this, they need to try to express their reasoning in a clear, coherent way (**MP3**).

I plan to post the vocabulary words: **roots (zeros), solutions, quadratic equation, quadratic function, and solve** on the board so that students can use these words in their explanation. I find that providing a list of vocabulary words can also help to spark student thinking.

#### 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