Authentic Lessons for 21st Century Learning

Bounce Wiggle Cross

Polynomial Graphs

K20 Center, JAMIE RENTZEL | Published: May 16th, 2022 by K20 Center

  • Grade Level Grade Level 11th, 12th
  • Subject Subject Mathematics
  • Course Course Algebra 2, Precalculus
  • Time Frame Time Frame 1-2 class period(s)
  • Duration More 60 minutes


Students will discover the three different types 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?



Teacher will show a short video of a roller coaster to engage students into the lesson. Students will be prompted to visualize the graph of the roller coaster's height over time and describe various characteristics of that graph and how they are connected to the roller coaster.


Students will be 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.


Students will come together as a whole group and teacher will "I Notice I Wonder" to gather students mathematical knowledge of polynomial graphs and then to incorporate the concepts the way the x-intercepts bounce, wiggle, or cross.


Students will use "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.


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.


  • Bounce Wiggle Cross Handout

  • "Card Sort" cards of graphs of polynomials

  • Graphing Calculator, one per group

  • Paper and Pencil


Show the roller coaster video on youtube and then ask students what mathematical way they could graph the roller coasters position over time. As the students are discussing, the teacher can model by drawing a polynomial on the whiteboard/screen of a sample polynomial graph. Then ask students to describe what is happening to the roller coaster as you move along the graph from left to right.

Sample Probing Questions: What does the x-axis represent in the scenario? What is the x-axis labeled? What is the y-axis labeled?


After the roller coaster discussion, briefly introduce that they will be digging deeper into the polynomial function to find out some more interesting tricks the equation can tell you about the graph. Be vague.

Students are then given a card sort with pictures of polynomial graphs zoomed in to see the x-intercepts. Teacher will ask students will be asked to sort the cards into three categories. The student choose the categories, there is no right or wrong, the student just needs to be able to justify.

After about 5 minutes, prompt student groups to label their categories and come with a brief explanation of why they chose those categories.


Gather the individual groups back to a whole group and then facilitate as each group shares out their categories and why.

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, scribe what the students "noticed" about the difference graphs. After the students begin to repeat or run low of observations, prompt them to share their "I wonder" statements about the graphs. What do you wonder about these graphs? The "I Wonder" statements will hopefully lead to the teacher incorporating the mathematical discussion of what causes the graph to bounce, wiggle or cross.

Teacher must remind students that "bounce wiggle and cross" are fun and easy way to remember the way the graph goes through the x-axis, but not the real terminology. The teacher they should use both the phrase ‘root behavior’ as well as the fun words to insure that students are connecting x-intercepts to roots to zeros, along with the bounce wiggle cross behaviors.


Students return to small groups and further explore polynomial functions and their roots given the equations. Each group is given the bounce, wiggle cross handout. One student per group is given a calculator in order to get a visual of the graph and confirm the groups discoveries. Teacher tells 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.

Teacher needs to be roaming 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).


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. Being 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 equation with technology.

Teacher questions/prompts for students: Discuss and summarize on the paper where the equation was correct/incorrect and why? What there enough information, or 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.