##
* *Reflection: Developing a Conceptual Understanding
Real World Applications of Quadratic Functions - Section 1: Warm Up

Students should be 100% at solving Quadratic Equations using the Quadratic Formula, but they are not. I really feel that students need to find a repeated method that works for them individually, where they are getting the solutions correct every time using this formula.

Some students may be more successful using technology, and others may feel more confident breaking it down into pieces by hand. Any method is fine if it is working. If it is not, students and teachers need to work together to find the mistakes of individual students.

**Focusing on the details of** **signs, identifying variables correctly, and simplifying correctly using Order of Operations are key to their success.**

I know that teachers including myself say, "Don't focus on the answer, focus on the method and the reasons and concepts of your ways." However, doesn't a wrong answer tell us that something went wrong, and we can work backwards to find the mistakes that are so important to correct. A correct answer also tells us that our method worked.

So, maybe we should tell students to focus on the answer as well to guide them in improving!

*A case where Practice does Prevail*

*Developing a Conceptual Understanding: A case where Practice does Prevail*

# Real World Applications of Quadratic Functions

Lesson 9 of 10

## Objective: SWBAT solve application problems of Quadratic Functions involving projectiles.

## Big Idea: To decide what formula to use based on the units of feet per second for velocity compared to meters per second.

*50 minutes*

#### Warm Up

*15 min*

I introduce Projectile Motion to students with this article about dropping a penny from the Empire State Building. It has been a myth over the years that if you drop a penny from the Empire State Building and hit someone in the head, that it would kill them. So, before reading the article, I pose this question to the class, **"Will a Penny that is dropped from the top of the Empire State Building kill someone if it hits them in the head?" **I allow students time to argue over the point for a few minutes and state their opinion.

After a few minutes of wait time and student responses, I move forward to read part of the article. I only read the first two paragraphs to the students. I obtained this article from the following website:

http://www.todayifoundout.com/index.php/2010/10/dropping-a-penny-from-the-top-of-the-empire-state-building-isnt-dangerous/

Then I ask the students how long does it take for the penny to hit the ground. I model this problem for the students and in the video below:

*expand content*

#### Power Point

*25 min*

After reviewing the Warm Up with students, I have them take notes from this Power Point. In the Warm Up, I introduce students to the formula for Projectile Motion when the height is in feet. So, at the first of the Power Point, I introduce students to a similar formula for height in meters. The only difference between the two formulas is the value of the leading coefficient due to gravity.

I already have students in homogeneous pairs based on their ability in the Quadratic Unit. For this lesson, I assign four students per group. To form a group of four, I place homogeneous pairs together that are about the same level.

After creating four students per group, I assign each group a problem to work from the Power Point. Again, these problems are from Purple Math that I reference in the Power Point. I hand each group a copy of the problem that they are to work. Depending on the number of groups, more than one group may have the same problem. I assign number five to a higher level group because it is a multi-step problem.

Since the answers to these problems are at the website that I reference in the Power Point, I do not allow students to access their computers or phones during this lesson. After students have presented their methods today, we will view the answers on the website in the Closure of this lesson. So each group should have one copy of their problem, their notes, and a graphing calculator.

I instruct each group to do the following:

- Draw a picture of the problem
- Select the Quadratic Equation that they will use for projectile motion
- Assign the values for the variables known
- Solve the Problem
- Show work

I allow about 20 to 25 minutes for students to complete a problem. If they finish early, I have them evaluate a different problem from another group that is also done. The two groups are to discuss the two problems, and provide feedback until time is called. Students may change their problem if they agree with the feedback, or leave it as they had it if they disagree. At the end of the 25 minutes, I have students share out their responses on the Power Point as I post the problem. Not all of the groups will present, but I ask for input from all groups.

*expand content*

#### Closure

*10 min*

The closure for this lesson comes from the students presenting their solutions, and the class discussion that it creates. There were several mistakes that students made today on the problems, but several were corrected by drawing a picture, correcting a sign, identifying what the question was asking, and being able to label the parts of the Projectile Quadratic Equations correctly.

Below are some samples of student work:

In problem three, this group makes the mistake of solving the Quadratic Formula for the time it takes the object to hit the ground (equal zero). Instead, the group needs to find the Vertex (Maximum height) and the time it reaches its maximum height.

I provide this group with the formula to find the Vertex, this group contains some of my ESL Learners. After providing the group with the formula, the group does find that the object reaches its maximum height at two seconds.

When substituting the two back into the formula, the group gets the correct solution of 144 and does not provide the units of feet. The group also continues simplifying the formula for y incorrectly in the Vertex Formula. So this group is still having some conceptual understanding problems which drawing a picture on the board did help during the discussion. Pointing out the maximum height and the time along the x-axis that it reached it.

In problem four , most of the students label this problem incorrectly. They do not understand that the problem should be set equal to 34.3 instead of zero. Once this group did understand how to set up the problem, they were able to label the initial velocity and the initial height correctly. After re-defining the problem, the students were able to identify **a, b, and c **correctly for the Quadratic Formula as well.

Students did have difficulty understanding that the object was at or above 34.3 feet from one second to seven seconds until we drew a picture on the board during the discussion. The picture made the question more clear, and students could easily see that the time it was at or above 34.3 feet was six seconds. The difference of seven seconds minus one second.

In problem five, I assign this problem to a high level group because it is a multi-step problem. However, this group still makes a small sign error that makes their problem wrong. This group understands the concept, and finds the time it takes for each boy's book to drop into the pool. They set up the equation correctly, and make the substitutions into the formula correctly. They even answer the question by finding the difference of their times.

However, when solving for Herman's book they get solutions for the times of five seconds and negative two seconds. The five is actually a negative solution, and the other solution is positive two seconds. Herman's book takes two seconds to hit the pool. This is a difference of 1.16 seconds instead of 1.84 seconds.

Even though the Quadratic Equation was stated in **problem two** for the students within the problem, students still struggled with sign errors. On number two, there was a **positive and negative solution**, so students easily understood that time could not be negative. Therefore students chose the correct answer if they completed the problem correctly.

In **problem three **it was more difficult for students to state the solution even if they solved the problem correctly because there were **two positive solutions**. Students had to substitute each solution back into the function to find that 3.06 seconds was the correct time that it took to hit the ground (equal zero).

Again, the objective of this lesson is for students to find a method that works for them, whether it be with the use of technology or not. By looking at these mistakes, drawing pictures for conceptual understanding, and focusing on precision when working with positive and negative integers, students will be more successful.

*expand content*

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- UNIT 1: Introduction to Functions
- UNIT 2: Expressions, Equations, and Inequalities
- UNIT 3: Linear Functions
- UNIT 4: Systems of Equations
- UNIT 5: Radical Expressions, Equations, and Rational Exponents
- UNIT 6: Exponential Functions
- UNIT 7: Polynomial Operations and Applications
- UNIT 8: Quadratic Functions
- UNIT 9: Statistics

- LESSON 1: Introduction to Quadratic Functions
- LESSON 2: Graphing Quadratic Functions in Standard Form f(x)=ax^2+bx+c.
- LESSON 3: Graphing Quadratic Functions in Vertex Form f(x)=a(x-h)^2 + k.
- LESSON 4: Graphing Quadratic Functions in Intercept Form f(x)= a(x-p)(x-q)
- LESSON 5: Comparing and Graphing Quadratic Functions in Different Forms
- LESSON 6: Completing the Square of a Quadratic Function
- LESSON 7: The Quadratic Formula in Bits and Pieces
- LESSON 8: Solving Quadratic Functions Using the Quadratic Formula
- LESSON 9: Real World Applications of Quadratic Functions
- LESSON 10: Analyzing Polynomial Functions