Summary
Students will discover the three different types of x-intercepts that happen on polynomial functions.
Essential Question(s)
In what ways does identifying patterns help determine connections and expand mathematical understanding between functions and their graphs?
Snapshot
Engage
Students watch a short video of a roller coaster, prompting them to visualize the graph of the roller coaster's height over time. They will describe the various characteristics of that graph and how they are connected to the roller coaster.
Explore
Students will be divided into small groups and given a card sort with pictures of polynomial graphs zoomed in to see the x-intercepts. They will group the cards into three categories of their choosing.
Explain
Students will come together as a whole group and the teacher will use the "I Notice I Wonder" instructional strategy to gather students’ mathematical knowledge of polynomial graphs and then to incorporate the concepts of the way the x-intercepts bounce, wiggle, or cross.
Extend
Students will use a "bounce, wiggle, cross" handout to further explore and solidify the connection between the way the graph crosses the x-axis and the degree of each factor in the polynomial equation.
Evaluate
Students will create their own sketch of a polynomial and trade with their elbow partner to come up with the best equation to match it. Students will then summarize their sketch and equation, and discuss further the misconceptions, if any.
Materials
Card Sort cards of graphs of polynomials (attached; one per group)
Polynomial Behavior handout (attached; one per group)
Graphing calculators (one per group)
Paper and pencils
Engage
10 Minute(s)
Show the roller coaster video on YouTube and then ask students what mathematical way they could graph the roller coaster’s position over time. As the students are discussing, you can model by drawing a polynomial on a sample polynomial graph on the whiteboard/screen. Then ask students to describe what is happening to the roller coaster as you move along the graph from left to right.
Explore
20 Minute(s)
After the roller coaster discussion, briefly introduce that students will be digging deeper into the polynomial function to find out some more interesting tricks the equation can tell them about the graph.
Divide students into groups of 3 or 4. Distribute the Card Sort handout to the student groups and introduce the Card Sort instructional strategy. The handout has pictures of polynomial graphs zoomed in to see the x-intercepts. Ask students to sort the cards into three categories. The students choose the categories. There is no right or wrong, students just need to be able to justify their choices.
After about 5 minutes, prompt student groups to label their categories and come up with a brief explanation of why they chose those categories.
Explain
20 Minute(s)
Ask each group to share out their categories and why they chose them. Facilitate the discussion as they do so.
Once all groups have shared, use the I Notice I Wonder strategy. Based upon what the groups shared, have students think of at least one thing they noticed about the responses and one thing they are still wondering about.
On the white board/screen, write what the students "noticed" about the difference graphs. After the students begin to repeat or run low on observations, prompt them to share their "I wonder" statements about the graphs. These statements will hopefully lead you to incorporating the mathematical discussion of what causes the graph to bounce, wiggle or cross.
Remind students that "bounce, wiggle, and cross" are a fun and easy way to remember the way the graph goes through the x-axis, but not the real terminology. Use both the phrase “root behavior” as well as the fun words to ensure that students are connecting x-intercepts to roots to zeros, along with the bounce, wiggle, and cross behaviors.
Extend
Ask students return to their small groups and further explore polynomial functions and their roots, given the equations. Distribute the Polynomial Behavior handout to each group. Give a graphing calculator to one student in each group, in order to get a visual of the graph and confirm the group’s discoveries.
Tell the students to scan through the document with their groups and find out how to fill in the missing spots on the table. This is the time for students to solidify the connection between the degree between each factor of the polynomial and how it correlates to the root behavior; whether the graph bounces, wiggles or crosses at that given root.
Roam the room to observe the conversations and check for understanding about the connections between the equation, the graph, and the root behavior (bounce/wiggle/cross).
Evaluate
Once groups have completed the handout, get the attention of the whole class to give them their assignment.
Each student will take out a blank sheet of paper and sketch their own polynomial equation with various types of x-intercepts, as they choose. Ask them to be sure to label the roots.
When complete, the students will then trade with their Elbow Partner and then that student will create the equation that best fits the sketch. When finished, have the students trade back and check each others’ equations with technology.
Questions/prompts for students:
Discuss and summarize on the paper where the equation was correct/incorrect and why.
What there enough information, or is more needed?
What needs to be done to fix the equation if incorrect?
If correct, is there another equation that can be used to produce a similar graph?
Have the students then write down a summary about their graph, equation and any connections on the same sheet of paper. Each student turns in the sketch, equation, and summary at the bell as their exit ticket.
Resources
K20 Center. (n.d.). Card sort. Strategies. https://learn.k20center.ou.edu/strategy/147
K20 Center. (n.d.). Elbow partners. Strategies. https://learn.k20center.ou.edu/strategy/116
K20 Center. (n.d.). I notice, I wonder. Strategies. https://learn.k20center.ou.edu/strategy/180
Theme Park Review. (2011, May 19). Green Lantern POV Roller Coaster Front Seat Six Flags Great Adventure New Jersey SFGadv [Video]. YouTube. https://www.youtube.com/watch?v=60zrnOyBo3U