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Lesson plan · Grade 11 · 50 minutes

Rollercoaster Polynomials

I can connect the zeros, multiplicities and leading coefficient of a polynomial to its graph, and use the remainder theorem to find factors.

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Standard
A-APR.B CCSS-Math
A-APR.C CCSS-Math
Shape
Explore first
Grouping
Pairs on devices, with whole-class pauses.
You need
The tool on the projector and on student devices. Graph paper for the exit check.
Tool
curioclass.org/tools/rollercoaster-polynomials/
  1. Hook5 min
  2. Explore17 min
  3. Make sense10 min
  4. Apply13 min
  5. Exit check5 min

Before class

  • Open Build on the projector. The default track is P(x) = −0.2(x + 4)(x − 4), one hill 32 m tall.
  • One grid square is 10 m each way, so the track is true to scale.
  • Decide whether pairs will try Commissions today or save it for homework.

Hook 5 min

  1. Show the default track. "The park says a coaster is just a polynomial. Where are the zeros? Where are the stations?"
  2. "Today you design tracks by choosing where they touch the ground, and you find out what the equation has to look like."

Explore 17 min

  1. Pairs drag the right footing from 4 to 2 and watch both forms of the equation change.
  2. Pairs set the zero at x = 2 to multiplicity 2, then 3. For each, they record how the track meets the ground: crosses, touches and turns back, or crosses flat.
  3. Pairs press Flip track and change a with the slider. "What changes, and what stays the same?" They read the end-behavior sentence each time.
  4. Pairs build a degree 4 track with two hills and one tunnel, then press Ride and listen for the tone at each ground contact.
  5. Pairs who want to see the algebra press Show the multiplication.

Make sense 10 min

  1. "Why does a zero of multiplicity 2 never let the track cross?" Push for the sign argument: a squared factor is never negative, so the sign doesn't change there.
  2. "Two tracks have the same zeros but different a. What stays the same?" (The zeros and the crossing behavior. The height and the direction change.)
  3. "How can you tell the end behavior from the factored form alone?" (The degree and the sign of a.)

Apply 13 min

  1. Switch to Survey. The plan comes in standard form only, and the track hides in fog.
  2. Pairs move the stake and press Survey here. Synthetic division gives the remainder, and a post that tall is planted at c. A remainder of 0 locks a footing.
  3. Ask: "Why does the remainder equal the height of the track at c?" Try c = 0: "What do you notice about the constant term?"
  4. Pairs use Check for a double footing when a zero might repeat, and keep going until the fog lifts.
  5. Pairs who finish try Plan 9: "It has only one footing. Where did the other two zeros go?"

Exit check 5 min

On graph paper, sketch y = (x + 3)(x − 1)²(x − 4) with the zeros labeled and the right end behavior. Then: "P(2) = 0 for some polynomial P. What do you know about its factors?" Sketches that cross at x = 1 show who needs Build again.

Support and stretch

  • Support: start with degree 2 and 3 tracks, with Show the multiplication open.
  • Support: give pairs a table with columns for zero, multiplicity and "crosses or touches."
  • Stretch: Commissions C5 (Big Three): "Why must the tunnel zeros sit close together?"
  • Stretch: match a ghost track in the blueprints (B1 to B4) by reading its zeros and a.

Short on time

  • Build with multiplicity 1, 2 and 3 at one zero (10 min).
  • One Survey plan as a class (10 min), then the exit sketch (5 min).

How each button works is in the teacher notes. From Curio Class, free under CC BY 4.0: change it, print it, share it.