##
* *Reflection: Developing a Conceptual Understanding
How long will it take? (Day 1 of 2) - Section 2: New Information

It was not clear to some of the learners why the quadratic 2x^{2 }= 32 was "separated" into two equations y = 2x^{2 }and y = 32, when graphing to find the solutions. Some did not understand why a line?...why a parabola? and why the intercepting points are the solutions? These were some of the questions that I can remember.

First, it may be a good idea to remind students that the points on a line or curve are the solutions to the equation of that line or curve. Take the two linear equations y = x + 5 and y = 2x and graph both on the same plane so that the learners see that they both intersect at the point with coordinates (5, 10). This is the only point that is a solution to both equations....the intersecting point. Ask them to substitute 5 into both equations to verify this. When x is 5, y is 10 in both equations.

I would then ask them if the following is true.... x + 5 = 2x

Students will see that this is true by substitution. (You may want to refer to the transitive property as well) Ask students to solve for x and they will get 5 as the answer.

In reference to our quadratic 2x^{2} = 32, the line y = 32 intersects the parabola in two places, where the abscissa is -4 and where it is 4. The ordinate is 32 at both of these and solving for x in the equation gives us +4 and - 4.

*Student Thinking about Graphical Solutions using a Systems Approach*

*Developing a Conceptual Understanding: Student Thinking about Graphical Solutions using a Systems Approach*

# How long will it take? (Day 1 of 2)

Lesson 5 of 7

## Objective: SWBAT solve equations equivalent to those of the form ax^2 = b

#### Launch

*10 min*

**Students need to recognize an important distinction between**

**the statements x**^{2 }= 36 and x = √36 .**There are two values of x**

**that satisfy x**^{2}= 36, which are 6 and −6.

**However, the symbol √36 refers to the principal square root of**

**36,****which is nonnegative, so the only value that satisfies x in**

**x = √36****is 6.**

#### Resources

*expand content*

#### New Information

*15 min*

Once I'm done with the Launch entrance slips I ask a volunteer to hand out a NEW INFO HANDOUT to each student.

This lesson assumes student knowledge of the meaning of square roots presented in my Powers and Exponents unit, which means learners can solve equations of the form x^2 = b. Students will learn here that with just one additional step, they can solve an equation of the form ax^2 = b.

*In Question 1, students should realize by analyzing the graph and the table that the equation has two solutions (-4, 32) and (4, 32). Have students prove this by checking each solution, showing the math.*

*In Question 2, students use Galileo's equation. Again, they should realize that there are two solutions, but only one that is meaningful. State that this is not always true. Sometimes both solutions in the problem are meaningful. Allow calculator use for this question.*

After making sure students have finished this work, I project the NEW INFO SOLUTION PAGE on the board so students can see, discuss, and ask any question before going on to the application section.

*expand content*

#### Application

*20 min*

I like to project the Application Problems (HowLongWillitTake) on the Smartboard. They can also be printed and given to the students. I allow learners to pair up. I make sure at least one of them has a calculator. The discussions among the learners can be quite interesting. I like to walk around and listen in and check if they are considering whether the solutions obtained are both feasible or whether one of them doesn't make sense. I always ask why they are discarding an answer and what the appropriate answer means with respect to the work problem.

Students always love to go to the board, so I ask volunteers to write their work up for all their classmates to see and ask any questions if they have them.

*expand content*

#### Closure

*5 min*

Today, I will close the lesson with a** GQ Talk**. This closing strategy is quite simple and can be effectively done in little time.

**G**: I call on students to verbally state what they think the General goals or ideas of the lesson were, in their own words. As they respond, I write their ideas on the board without discarding any of their responses.

**Q**: Once I have sufficient responses on the board, I then ask students to ask a Question about any idea in the lesson they feel uncertain about, and motivate other students in the class to answer the questions asked by their classmates. If I suspect someone may be struggling, I call on the learner and motivate him or her, to come forth with a question or indicate the area of confusion. This always sheds light on whether students have grasped what was intended to be taught in the lesson.

*expand content*

##### Similar Lessons

###### Racing Molecules

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- UNIT 1: Number Sense
- UNIT 2: Solving Linear Equations
- UNIT 3: Relationships between Quantities/Reasoning with Equations
- UNIT 4: Powers and Exponents
- UNIT 5: Congruence and Similarity
- UNIT 6: Systems of Linear Equations
- UNIT 7: Functions
- UNIT 8: Advanced Equations and Functions
- UNIT 9: The Pythagorean Theorem
- UNIT 10: Volumes of Cylinders, Cones, and Spheres
- UNIT 11: Bivariate Data

- LESSON 1: Linear vs Quadratic (Day 1 of 2)
- LESSON 2: Linear vs Quadratic (Day 2 of 2)
- LESSON 3: The Biggest Possible Area
- LESSON 4: Playing with Parabolas - Hands on
- LESSON 5: How long will it take? (Day 1 of 2)
- LESSON 6: How Long Will it Take? (Day 2 of 2)
- LESSON 7: Are Absolute Value Functions Linear?