# #WithMathICan Explore Quadratic Models

#### 1 Hook

Tell students they are to explore how to find maximum area. Have the following supplies available and out for students to use: scissors, string, markers, rulers, graphing calculators or access to Desmos. Put students into groups of three. Put the problems up on the SMART Board. Ask students to collaborate to determine a strategy to solve the problem in their groups for five minutes. Then hand out the problem set and ask students to solve the problems with justification.

As students are working, assist students by asking clarifying questions if they are stuck.

Sample questions:

If they don't know how to start: Ask "How can you model the sides?"

If they are struggling finding area: Ask "What shape do you have? How do we find the area of that shape?"

If they are struggling determining the maximum area: Ask "Can you make a table with various values? Did you try some values in between what you have?"

If they have completed a model and a table: Ask "Can you write an equation to model the situation?"

If they get an equation: Ask "How can you find the maximum point?"

In a group, discuss and determine a solution method and justify using multiple methods. You may use the tools provided to assist you in your exploration.

#### 2 Direct Instruction

Using the problems from the Exploration Activity, have students present multiple solution methods to finding the maximum area in each problem. Ask for a numerical solution, a graphical solution (highlight maximum on Desmos), and an algebraic solution. Allow students to share their thought process including misconceptions and how they changed their thinking. As needed, help students make connections between representations. Make sure students have access to numerical, graphical and algebraic solutions. Encourage students to ask questions on any points of confusion.

Students will present their solution methods and incorporate other solution methods into their notes.

#### 3 Guided Practice

Students will complete a modeling problem as a member of a group. There are five different modeling problems so groups will compare and contrast their work at the conclusion of the problem solving.

Supplies needed: Markers, graph paper, posters or large sheets of paper.

Put students into small groups (3-4) and distribute the Quadratic Modeling Problems, one per group. Allow students to work together. Remind students to use the simpler problems from the exploration to help them.

As students are working, monitor groups and ask students clarifying questions. Sample questions:

If students are having trouble writing an algebraic equation, ask "How did you calculate the area numerically? or How is this problem similar to the exploration problems?"

If students are having trouble identifying an appropriate domain, ask "For what values of x can you find the area?, For what values of x can you not find the area?, "How could a different representation help you find the domain?"

If students are having trouble graphing, ask "What type of parent function do you have?, What are some important point to have for this particular type of function?, What tool do you have to help you graph?"

If students are having trouble finding other ways to calculate the maximum, ask "What does the maximum point represent on a quadratic function?, Do you know a way to calculate the maximum/minimum point?, Can you use another representation to help you solve this problem?"

In a group, discuss and determine a solution method and justify using multiple methods. You may use the tools provided to assist you in your problem.

#### 4 Gallery Walk/Class Discussion

Have students post their posters. Give each student 4-8 sticky notes. Ask each student to write a question or an "ah-ha" moment they have for something on each group's poster as they do the gallery walk (they do not have to put their name on their question). After students have had an opportunity to post their questions and "ah-ha" moments, use the post-its to guide the class discussion specifically highlighting points of confusion that have been clarified, connections between multiple "ah-ha" moments.

Students will do a gallery walk comparing and contrasting the modeling problems from each group. Students will respond to the other group's problems by writing questions and "ah-ha" moments on sticky notes on at least one per poster. Participate in the class discussion.

#### 5 Independent Practice/Extension

Ask students to complete independent practice items. For the extension problems, students will need access to Desmos or other graphing utility.

Complete independent practice items. For the extension problems, students will need access to Desmos or other graphing utility.