Lesson and Assessments - Modeling with Quadratic Functions

Progression of Skills (outline)

Before the lesson: 

• Students may have learned about key features of quadratic functions and/or may have learned strategies for factoring and/or solving quadratics. Students may also complete this toolkit as their first learning about quadratic functions.
• Students should understand how to read graphs and how to interpret functions.

During the lesson:

• Students will develop an understanding of how quadratics model real-world situations and help us solve problems, why we use quadratic functions, and how different representations of quadratic functions relate to the real-world model.

After the lesson: 

• Students will be prepared to learn to formally manipulate and solve quadratics using various methods and will interpret their solutions in real world contexts.

Lesson Activity 1: Using Quadratic Equations to Model Area (Starbursts)

Warm up/Intro

• Intro Activity - Starburst Growth (15-20 minutes)
  • Have students work in small groups to discuss and complete the activity. Whiteboards are a great resource for this.
  • Intro Activity - Starburst Growth Answer Key It is not important that students get the “right” answers on their own, but you should prompt them with questions to think about anything they did not get correct. Try not to tell them how to do it (ie. “multiply the rows by the columns.”). Instead, ask them about the patterns they see and if they can connect those patterns to math they already know.
  • Students who have not previously used quadratic functions may struggle to write an equation at the end of the Starburst activity. Encourage them to think about what mathematical operations would result in the number relationships on the table. It is ok if they are not able to write an equation! We want them to notice that this is a different pattern of growth.
• Watch out for the row for Figure 10 on the table. This is where students will have to start thinking about multiplying the numbers.
• Using the slides and/or answer key, discuss with the students the type of change we see in the number of Starbursts. 
• Slide 4: Emphasize that as the figure number grows by 1, we are adding 1 to each n value in the equation.
• They do not need to fully understand how to write an equation to model this situation, but they should be able to make connections between the situation, how to find successive numbers of Starburst and how that relates to the quadratic equation.

To access Activities 2-6, use these slides:

• Lesson Slides Project the slides or let students look at the slides on your screen
• Modeling with Quadratics Student Worksheet print outs for student work
• The first four slides connect Activity 1 to the remaining activities

Lesson Activity 2: Using Quadratic Equations to Model Area (Backyard Fence)

Modeling with Quadratics Student Worksheet (Activity 2)

Directions for Intervention Activity (20-30 minutes): 

1. If you have multiple students, divide them into partners or groups of 3. They should show their work on a whiteboard or piece of paper. Review norms for working together, sharing and respecting ideas and making sure all group members contribute.
2. Slide 5: Ask students to read the situation and work with their group to answer the questions. Encourage them to include several different diagrams of the garden and to show their calculations. They can approach finding the largest area in multiple ways, but the most likely is that they will guess and check by trying multiple lengths and widths. If they do not reach the maximum area of 156.25 sq. ft., challenge them to keep trying.
   • Formula reminder if needed: Perimeter is p=2l=2w, area is A=lw. Only share if students do not remember or are using an incorrect formula, or if they are using lengths and widths that will not add up to a total perimeter of 50.
3. Slide 6: Have students plot points and answer the questions on the slide. Important to note: We are relating length to area. Width will not appear on the graph.
4. Slide 7: This slide shows a table, graph, and several equations which all model the situation of the garden area. Collect thoughts, prompt students to think about how the different representations are connected. Where do we see the same numbers? Do some forms tell us different information from others?

Progress Monitoring:

• Are students making connections between the real world situations and the quadratic models?

Lesson Activity 3: Using Quadratic Equations to Model Vertical Motion (Desmos)

Warm up/Intro

Desmos Will it Hit the Hoop? (15-20 minutes)

• Preview the Teacher Guide (at the top right of the screen) to prepare for the activity. You will need to create a (free) teacher account in order to assign the activity to your students. Alternatively, students can complete the activity in preview mode without creating an account.
• Assign to your class or use a single session code. The Desmos activity may be completed as a single student, but if a class of students all completes the activity, it will share some data with them about their results as a class.
• Use pacing to restrict students not to go beyond screen 18 until you are ready for them to see the answers.
• Students may initially think it is purely a guessing game. As they move to modeling the path of the ball with a parabola, encourage them to be more precise with their predictions and to think about what the graph of the parabola is telling them about the situation. It is not intended that they will understand the formal math at this point.
• Worthwhile extension if you have time: Use Teacher Snapshots to share and discuss answers on screens 27 - 30. Focus on key features of the quadratic and how they relate to the real-world situation.

Lesson Activity 4: Using Quadratic Equations to Model Vertical Motion (T-shirt Launcher)

Modeling with Quadratics Student Worksheet (Activity 4) 

Directions for Intervention Activity (25-30 minutes):

1. If you have multiple students, divide them into partners or groups of 3. They should show their work on a whiteboard or piece of paper. Review norms for working together, sharing and respecting ideas and making sure all group members contribute.
2. Slide 9: Read through the situation description and ask students to answer the prompt on whiteboards or paper. If needed, provide coaching to write the linear equation to model this situation.
3. Slide 10: Review the data in the first table (this should match what students created on the last slide). Ask students to help you complete the equation in the far right column (this can be used to find the height for any time, t). Show the students the second table, showing the actual height of a t-shirt launched in the real world. Elicit student thoughts on why the two tables are different. We want them to realize that gravity is making the rate of change non-linear in the real world.
4. Slide 11: Students should graph points for each table and may connect the points of each table with a straight (linear) or curved (quadratic) line. Briefly discuss the differences between the shapes of the two graphs. You do not need to create math to match the graphs at this point; we just want to realize that gravity is slowing the increase and making our graph curve.
5. Slide 12: We need to add the effects of gravity, which are modeled by -16t2 (where distance is in feet and time is in seconds). Students may or may not know this fact, so please tell them now. Then we add -16t2 to our linear equation to make a quadratic model for vertical motion.
6. Slide 13: Based on the quadratic model we just built (above on the slide), what do we know about the situation modeled by the second equation on the slide? Allow students to talk and reason through this. You can prompt by asking about what each number could indicate, and relating it to the numbers in the first equation and what they model. We want to know that the t-shirt is launched from a height of 3 feet, at a velocity of 75 ft/second. The gravity term is constant, so it does not change.

Lesson Activity 5: Making Connections across Representations of Quadratic Models (Card Sort)

Card Sort Matching Activity Print one copy for each small group and cut the cards apart before giving to students. The rows are correctly matched on the file, so you may use this as the answer key.

Directions for Intervention Activity (10-15 minutes):

1. Slide 14: This slide shows an example of matching situation, function and graph. Discuss how we know that the function and graph model this situation. Some things to make sure to draw out from students:
   1. The gummy bear is thrown from 3 feet, which is the y-intercept of the graph and the constant term (without a variable) in the function.
   2. We are told the bear hits the ground in less than a second. Our parabola crosses the x-axis at slightly less than 1 second.
   3. There is a positive starting vertical velocity of 12 ft/second, and we see that our graph goes up, before the bear starts to fall. We also see this velocity in the linear term, 12t, of the function.
2. Give each student or group of students a set of cards for the card sort.
3. Explain that the cards contain real-world situations, graphs and equations. 
4. Their task is to match the different representations of quadratics to the situations they model.
5. As students work, ask them to explain their reasoning for why they are grouping cards together.