Maximums+and+Minimums

Goal: Identify the max or min of a quadratic function and explain it in the context of the problem. (II.2.F.IF.4) 1. Students will be able to identify the maximum and minimum of a quadratic function. 2. Students will be able to explain what the maximum or minimum of the quadratic function represents within the context of the problem.


 * Develop Understanding**

Surface ideas: 1. maximum 2. minimum 3. parabola 4. quadratic

//Task #1: My Dog Skip//

Materials needed: String, rulers, graphing paper, geoboard (optional)

Task 1a: My Dog Skip Launch: Introduce the activity and discuss the directions with your students. Review how to find the area of a rectangle. Demonstrate how to find the area of a rectangle that has the same perimeter as your string length that does not have the maximum area.

Explore: As they complete the task watch for students filling out a table of values and graphing the parabolic curve.

Discuss: Select students to share what they found. 1. Make sure to talk about whether or not you could have rectangles with heights that are not integer values, such as a height of 3 1/2 half feet. As part of this discussion, a smooth parabolic graph should be involved. 2. Discuss the relationship between the maximum value of the graph as compared to the maximum area of the dog run.

Task 1b: My Dog Skip (part 2) Launch : Pass out and explain 2nd part of the worksheet

Explore: As students complete the task, watch for students who have the equation written factored form, vertex form, and/or standard form.

Discuss: Select students to share what they found. 1. Discuss what makes these equations quadratic equations. 2. How can you use the equation to find the vertex? 3. Compare and contrast the different methods used to find the maximum areas. 4. Discuss how things would change if you were had a different amount of fencing material or were enclosing the dog run against your house (only requiring 3 sides of fencing)


 * Solidify Understanding**

Surface ideas: 1. if a> 0 the quadratic function will have a min 2. if a< 0 the quadratic function will have a max 3. model will not perfectly match data 4. estimate future data from past data 5. correct graphing techniques

//Task #2: Hot day in Florida//

Materials needed: task sheet, calculator, graphing paper, pens, pencils, document camera

__Task 2a:__ Hot day in Florida Launch: Talk to student about what they know about temperature, how it is represented and how it changes throughout the day.

Explore: (Task 2a) Distribute the task sheet with the Florida Temperature problem on it. Allow students to come up with their own graphs and figure out as groups the best way to graph the data.

Discuss: 1. Choose a representation that is incorrectly graphed with respect to x=0 being midnight. Ask the class why this isn’t correct. 2. Discuss with the class how all of the x-values are negative time, because of the problems stipulation. 3. Show a correct representation of plotted temperature vs. time data 4. Discuss the estimation of temperature from 6p.m. to midnight

__Task 2b:__ Hot day in Florida Launch: Introduce the activity and discuss the directions with your students.

Explore: As a group develop a quadratic polynomial model for the data (and graph). Allow students to use calculators and textbooks and notes

Discuss: 1. Pick two different models to present to the class. Compare contrast reasoning behind the creation of each equation. 2. Bring up with class the variable squared and how that affects or is related to the graph being a parabola 3. Bring up whole picture versus part of the picture and what data is presented and what information can be concluded about the points of data that had to be estimated.(Domain) 4. Decide as a class which model would be most appropriate for the given data from the models presented. 5. Graph model and compare with 24 hour period.

__Task 2c:__ Hot day in Florida Launch: Introduce the activity and discuss the directions with your students.

Explore: Use the Model to predict temperatures for the next day, at 4am, noon, and 8pm.

Discuss: 1. Quickly have students give the three answers. See if there is conflicting opinions for solutions. 2. Bring up difference between model solution for noon and given data value for temperature at noon. 3. Recap with students: a. meaning of min and max from graph b. Quadratic model is best fit model for data and doesn’t match exactly c. We can generate models from data d. Compare model graph and –b/2a or model equation e. From model equation we can find model min or max


 * Practice Understanding**

Task 3: Music and Money Launch: Introduce and explain activity.

Explore: As students complete the task, watch for good examples and your “favorite no’s.”

Discuss: Select students to share what they found. 1. Make sure to discuss the values in context of the problem. 2. Discuss which gig is the most profitable? 3. You can also begin discussing transformations in terms of the operating costs.