## Penny Falling.pdf - Section 2: Investigation

# A Fireworks Display

Lesson 1 of 18

## Objective: SWBAT practice making sense of a problem and explore input and output numbers connected by a formula.

*60 minutes*

#### Opening

*10 min*

The purpose of today’s lesson is to engage the students in the unit problem without telling them what it’s all about. For today’s warm up, I just want to get students thinking about projectile-motion problems. I show them the Punkin' Chuckin' Question on the board and ask students to vote on which graph they think best matches the pumpkin situation.

After they vote, I ask students to explain why they chose the graph they did. Students typically have a lot to say about this problem, so I expect discussion to be quite lively. I sometimes write down student questions that come up that they will not get an answer to right now. Some of those will likely be related to physics and this is a good opportunity for cross-curricular work. I let students know that I will come back to these questions and eventually let them know the correct answer.

#### Resources

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#### Investigation

*45 min*

Students will now work on groups on the main task for today. The task I like to use for this lesson is from the IMP Quadratics unit. It is called Victory Celebration and is on page 280 of the Year 2 IMP curriculum. Any projectile motion type problem where students are looking for the vertex and one x-intercept will work. A similar, if somewhat simplified problem, Penny Falling, is included in the resources for this section.

In their groups, students will sketch the problem on poster board, write the questions they are looking to answer clearly, and try their hand at solving them. If there is time, they can present their work to the class.

Issues that I will be watching for:

- If students are having difficulty getting started, I will ask them:

What do you know?

What do you need to find out?

- I make sure students understand the starting height of the projectile is 160 feet. This should be evident in their sketches.
- Students may feel like they need to “do the math” before they can draw the problem situation. I encourage them to just sketch out what they think the situation is, without crunching any numbers.
- If students are unfamiliar with function notation, I may need to review it before this activity.
- When students begin to work on finding some answers, they may struggle with what the formula actually means. I plan to keep asking them what the height is based on and encourage them to select any amount of seconds to pass to see what the height would be.

**DIFFERENTIATION:** Students may need help pulling out the key information. I will recommend to students that they use a highlighter to notate the key data and questions.

It is unlikely that students will finish answering the questions in the task. If they do,I will ask them to graph the height function, if they haven’t done so already.

#### Resources

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If I had a longer block of time for this lesson (90 minutes rather than 60 minutes), I would include a discussion with the whole class about their posters. I usually don't have time, so I do it at the start of the next class.

I begin the summary discussion by having a couple of the groups show their posters. I ask them to present where they made a sketch of the fireworks situation.

I have students share out the questions the city needs answered. They should have written questions like:

- How long will it take for the rocket to reach the top of its path?
- How high will the rocket be at its highest point?
- How long will it take the rocket to reach the ground?

Next, I have students share their thinking about how they started to answer some of these questions. Initially, we focus on how long it will take the rocket to reach its maximum height and what that height would be. I might write a table on the board with different values for t and have students come up and fill in what they found for h(t). I will watch for students who only want to work with whole numbers. When a student is restricting the domain in this way, I ask, "How can you be sure that the maximum height of the rocket would occur after exactly 3 seconds. How high would the rocket be if it had traveled 2.5 seconds? What about 3.2 seconds?" Then, I let him/her take the initiative on other values they would like to explore.

After several students present, I plan to have my students graph the function on a graphing calculator and discuss how the calculatorgraph confirms or changes their idea about the maximum height. Then, I will ask students, "What does the landing time for the rocket mean in terms of the graph?" I expect that my students will likely try to guess and check until h(t) comes out close to zero. I often help them to make the connection here by asking, "What equation could you set up whose solution would tell you when the rocket hit the ground?"

If students are graphing the solution, some may wonder if there is a “short cut” that will help them answer these questions without guessing and checking. You can let them know that in this unit they will be learning algebraic ideas that will help them to understand this problem and work more efficiently.

Because students have processed a lot of ideas at this point in the lesson, I may want to allow some reflection time here, rather than at the end of class. If I make this choice, I will have students do a free write about the Fireworks Problem (Victory Celebration, page 280 of the Year 2 IMP curriculum) where they answer the following prompts: How did you solve this problem? What strategy did you use?

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#### Closing + Homework

*5 min*

In a real sense, students have just started to explore this problem more deeply. I’ll want to allow them a chance to share out their thinking and the work they have done thus far in groups at the end of class. I will offer them the following questions to guide their peer discussions:

- How have you shown your thinking in your sketch?
- What questions arose as you worked?

Next, I will clarify my expectations for homework. I expect students haven’t had enough time to tinker around with the questions and work toward answering them in class today. I anticipate assigning Question 3 to be completed for homework in preparation for a class discussion tomorrow.

Before closing the lesson, I will let students know we will be discussing their posters and some answers they may have come up with to the questions tomorrow. I will reiterate to the class that in order to participate fully in the beginning of tomorrow's, they will need to complete their homework.

#### Resources

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This material is adapted from the IMP Teacher’s Guide, © 2010 Interactive Mathematics Program. Some rights reserved.

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- UNIT 1: Introduction to Algebra: Focus on Problem Solving
- UNIT 2: Multiple Representations: Situations, Tables, Graphs, and Equations
- UNIT 3: Systems of Equations and Inequalities
- UNIT 4: Quadratics!
- UNIT 5: Data and Statistics
- UNIT 6: Arithmetic & Geometric Sequences
- UNIT 7: Functions

- LESSON 1: A Fireworks Display
- LESSON 2: Rabbit Run -- Day 1 of 2
- LESSON 3: Rabbit Run -- Day 2 of 2
- LESSON 4: Properties of Parabolas Day 1 of 2
- LESSON 5: Properties of Parabolas Day 2 of 2
- LESSON 6: Using an Area Model to Multiply Binomials
- LESSON 7: Vertex Form to Standard Form
- LESSON 8: Standard Form to Vertex Form Day 1 of 2
- LESSON 9: Standard Form to Vertex Form Day 2 of 2
- LESSON 10: Where Will the Rocket Land? Putting it all Together
- LESSON 11: Completing the Square Methods & Practice
- LESSON 12: Factoring Day 1 of 2
- LESSON 13: Factoring Day 2 of 2
- LESSON 14: Choosing a Method to Find x-intercepts
- LESSON 15: Quadratics Portfolio Day 1 of 3
- LESSON 16: Quadratics Portfolio Day 2 of 3
- LESSON 17: Quadratics Portfolio Day 3 of 3
- LESSON 18: Is It a Home Run?