Quadratic Modeling (DAY 2)

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SWBAT apply the concept of a quadratic function to 4 classic application problems: projectile motion, revenue maximization, area, and number puzzles.

Big Idea

Quadratic functions can help us solve problems involving projectiles, area, numbers and financial situations.


15 minutes

When solving application problems, I encourage my students to use the graphing calculator to help with the calculation part.  I want them to save their brain power for the modeling part of the activity [MP5].

The warm up Using the Graphing Calculator to Find Special Features of Parabolas is all about using the graphing calculator to find the minumum/maximum of a parabola and the input values that make the output equal to some value.  My students often have trouble with the following content standard:

F-IF.B.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.*

 Specifically, they need support in using input values to specify where something interesting(increasing, decreasing, maximum, etc) is happening with the output.  This warm-up is designed to reinforce this skill [MP2].

Direct Instruction

20 minutes

When my students leave Algebra 2, I want them to have the skills to apply the concept of a quadratic function to physics, economics, geometry, and other settings.  Giving my students big interesting problems like "Will it Hit the Hoop?" is one way I help them understand the modeling cycle.  For many students, rolling up their sleeves and figuring out how to solve a big problem is the best way to learn about modeling.

For a substantial number of students, though, applied problems are really hard to understand until they see some problems solved by an expert.  While I have experimented with lessons in which sets of students become "experts" in each problem type and then teach the class, I have found that I leave too many students behind with this approach.  I have decided that for most classes of Algebra 2 students, a day of direct instruction in quadratic word problems is really beneficial.

I distribute the problem set Quadratic Application Problems , which is a collection of  Classic Quadratic Modeling Applications. There are four categories of problems that I want my students to be familiar with and each category is represented in this problem set [MP4].

  1. Projectile Motion.  Students will often encounter problems which involve modeling vertical position as a function of time.  I spend some time explaining the physics involved in this problem and draw attention to the fact that we are modeling only the height of the object and that its vertical position is not part of the calculation.  Factoring in horizontal motion requires more information and students will not encounter this type of word problem until they take physics.
  2. Supply and Demand.  In these problems, a business has some knowledge about how its customers will respond to price increases and wants to set its price in order to maximize revenue.  These problems can be challenging because the input variable is defined in an unusual way (x= the number of times the business changes its price by a given amount).
  3. Geometry. This is a varied set of problems, but a classic is maximizing the area of that is fenced in by a given amount of fencing.
  4. Number.  In this type of problem, information is provided about the relationship between two or more numbers and the goal is to determine the numbers.  

Quadratic Application Problems is organized by problem type, with at least three questions from each of the four categories above grouped together.  I give students 10 minutes to silently look over the first problem in each set and jot down some ideas about how to solve it.  Then they take 5 minutes to talk to the people near them about the problems.  Finally we have a problem solving session that goes like this

  1. I lead us in a discussion about how to solve the first problem in the Projectile Motion set
  2. Students work independently to solve the second problem in the set.
  3. We discuss student solutions to the second problem and the merits of the different strategies they used.

We then repeat the process with the Supply and Demand set, the Geometry set and the Number set.

My goal in this lesson is to provide students with a small set of problems that they feel very confident solving.  I have seen that building my students' confidence in this way helps them be more persistent when confronted with novel problems [MP1].


Practice and Reinforcement

30 minutes

Next, students will work with their table partners to finish Quadratic Application Problems [MP4].  There are 18 application problems in all and I expect students to complete the set over the next few days [MP1].   I remind them that they can use the Desmos app on their tablet or computer if they are working on these problems at home and do not have access to a graphing calculator.  The solutions to these problems will be available on Edmodo.

Introduction to Cutting Corners Activity

25 minutes

Tomorrow, we will work on a challenging MAP task called "Cutting Corners."  Today I will introduce the task and give students time to attempt the activity independently.  Tomorrow they will work in assigned groups to build on their individual solutions. 

The task is about using geometry and quadratic equations to determine how a bus makes turns without veering into the bike lane. To help students visualize the scenario, we watch a short clip about busses and bike lanes.  


I distribute the 2 page Cutting Corners Task which has a large diagram and two questions for students to answer individually.  In the following day's lesson, students will have an opportunity to improve their individual solutions by working with a group.  The goal of the Cutting Corners activity is to represent the logistics of a bus making a turn using knowledge of geometry and quadratic equations. [MP4, MP2].

Students work for 15 minutes without asking questions and then turn their answers in to me.