1. To start this unit, let students know that we will be studying polynomial and rational functions. Ask students to talk to a neighbor and see if they can define these two types of functions. If they cannot, encourage them to see if they can write down examples of both of them.
After a few minutes, gather some examples of polynomial functions. Then see if students came up with a definition. Press them to see if they truly understand polynomial functions. Ask them if a term in a polynomial function can have a non-integer exponent. Ask if y = sqrt(x) is a polynomial function. These are good questions to get them thinking about their knowledge of polynomial functions.
For rational functions, see if you can generate some examples. You can leave this definition open as we will clearly define the concept later on in the unit. Let students know that we will be focusing solely on polynomial functions for the next several units.
2. Tell your students that we will be working with quadratic functions today, a specific type of polynomial function. Today's activity uses a height equation for a Stomp Rocket, so you may want to show them this video to generate interest:
Students have been working on quadratic functions in Algebra 1 and Algebra 2. While I don't expect them to know everything about them at the drop of a hat, the important concepts of finding the vertex, solving using the quadratic formula, and converting from standard form to vertex form, should be familiar after the refresher during today's lesson.
Today we are going to use a jigsaw grouping strategy to give students time to reacquaint themselves with quadratic functions. This video (How to Organize a Jigsaw) contains information about a jigsaw grouping technique and the logistics of organizing it in your classroom.
Ideally you will want to have eight groups of four, but it's okay if things are not perfect. Make as many groups of four as you can. Each group of four is going to get one of the pages in the First Grouping Worksheet.docx. The group will have about 10 minutes to work on just the one page that they received. They should only see that one page and not the other three. If there are eight groups, then two groups should get page 1, two groups should get page 2, two groups should get page 3, and two groups should get page 4. If there are seven groups, for example, you can just give two pages to one group. Pages 2 and 3 are closely related, so I would suggest doubling those up.
Students in each group should be solving the problem on their page and making sure that every person in the group can explain their work to others. Students may need a refresher on how to solve some of these problems, so allow them to use their textbooks or the internet to find helpful resources. Again, the intent of this activity is to refamiliarize ourselves with these concepts.
After working on the first problem, have each group of four "count off" into the letters A, B, C, and D. Now we are going to jigsaw out and form new groups. If you have eight groups of 4, then there should be eight A's. So separate the A's into two new groups of four, the B's into two new groups of four, the C's into two new groups of four,and the D's into two new groups of four. If you think the lettering off will be too chaotic, you could go around to each group and write their letter on their first grouping worksheet. Then students know exactly what letter they should be.
Now there should be eight groups of four. Every group member should have completed a different page from the original worksheets. If you don't have exactly 32 students, be flexible and do whatever works. If a group is missing someone who did page 4, for example, they will be filled in once we do the whole class summary.
Once the new groups are assembled, take a deep breath and give students the Second Grouping Worksheet. Now the group member who did page 1 will explain their group's work to the new group and each member will fill in that section on their worksheet. Stress that group members should ask for clarification if needed. Then the group member who did page 2 will explain their work to the new group. This process is continued until all four problems have been explained.
If a group only has three people, they can attempt to try the fourth question by themselves. You may have two people who did the problem from page 1, and that is okay too.
Since this whole activity has been very student centered, we want students to be crystal clear on these concepts when they leave for the day. So for the group discussion, pull examples of each question that are correct and clearly explained so that you can pick up anyone that may have a misconception. Go through each of the four questions with the class and see if there are any questions. Hopefully at this point your students remember all of the important characteristics of quadratic functions.
After our quick summary of quadratic functions, here is an assignment (Assignment - Quadratic Functions.docx) for students to start in class and to finish up for homework. Again, encourage students to look online or ask a friend if they are a little rusty about some of these concepts.
Since we went through only quadratic functions for the hour, I want to connect it back to our work with polynomial functions. So as an exit ticket, these are some questions that will tie the lesson back to the main idea for the entire unit.
1. Why are all quadratic functions also polynomial functions?
2. What are some tools that you can use with a quadratic function but you cannot use with other polynomial functions?
3. What are some tools that you can use with a quadratic function and any polynomial function?