Problem Solving Learning Lesson: Bicycle

thomas_brown, Mar 2, 2025

A Problem Based Learning Activity that focus on the problem "Which Way did the Bicyclist travel?"

Table of Contents

  1. Bicyclist riding in a perfect circle
    1. Area (By Peeling!) PBL
    2. Exploring lines Tangent to a circle PBL
    3. Area of an Annulus
  2. Direction of the bicyclist rider and the area an annulus?
    1. A bicyclist travels in a perfect circle
  3. Which way did the bicyclist Travel?
    1. A Pair of Bicycle Tracks and the Direction of the Bicyclist

1.

Bicyclist riding in a perfect circle

Finding the area between two concentric circles. Also, establishing how the length of a bicycle wheelbase of a bicycle is constant. In addition, the back wheel path follows the front wheel path, consisting of a line that's tangent to the back wheel path to the front wheel path.

  • 1. Area (By Peeling!) PBL
  • 2. Exploring lines Tangent to a circle PBL
  • 3. Area of an Annulus

1.1.

Area (By Peeling!) PBL

Area of the circle when the radius is equal to one unit.
1) What is the area of the circle if the radius is equal to one unit?
– thomas_brown, Tim Brzezinski

1.2.

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1.3.

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2.

Direction of the bicyclist rider and the area an annulus?

An activity to enable students to solve the problem of "which way did the bicyclist travel?"

  • 1. A bicyclist travels in a perfect circle

2.1.

A bicyclist travels in a perfect circle

Riding in a perfect circle ""The Best Circular Bike Ever Made" 2003 (normal speed)"
Point 'P" represents the back wheel and Point "C" and Point "D" represent the front wheel of a bicycle, Move Point "P"
A bicyclist riding in a perfect circle
1) Can it be determined which direction a bicyclist traveled?[br][br][br][br][br][br]
Riding in a perfect circle
What does length of PD and PC represent on a bicycle?
Finding the area of the green shaded region the length PD and PC are equal to 1 unit respectively.
Area of the green shaded region?
– thomas_brown

3.

Which way did the bicyclist Travel?

Activity to understand how to find the direction of the bicyclist and referencing modeling a graph with an appropriate function.

  • 1. A Pair of Bicycle Tracks and the Direction of the Bicyclist

3.1.

A Pair of Bicycle Tracks and the Direction of the Bicyclist

A pair of bicycle tracks used in this activity courtesy of G'Day Math https://gdaymath.com/lessons/gmp/1-9-which-way-did-the-bicycle-go-and-other-bicycle-ponderings/
Concerning the above diagram, which curve represents the path of the back wheel? Dark Blue or the Black Curve?
1) The Dark Blue Curve or The Black Curve
Concerning the above figure, which curve represents the path of the front wheel? Dark Blue or the Black Curve?
2) The Dark Blue Curve or the Black Curve
Concerning the black and blue curves from above diagram.
3) What type of function would be a good fit for the black curve? Pick all that apply
Concerning the above pair of bicycle tracks.
4) What do you notice when you move point A left to right, in particular the distance from points to the bicycles' path of the front wheel? AB and AC line segments
Line segment AB and AC are lines segments that are tangent at point A on the black curve.
5) What does line segment AB or AC represents concerning a bicycle?
Concerning the activity below: The black curve represents the original path of the back wheel and the blue curve represents the path of the front wheel.
Again, slowly move point back and forth. Pay attention to the distance from point A to point C.
Concerning the above activity: What do you notice and what do you wonder?
6) Is the bright green path an adequate description of the original front wheel's path (dark blue curve)? What do you notice and what do you notice?
Concerning the above activity:
Move point A slowly in both directions. The vector AB represents the bicycle moving from right to left.
What do you notice?
7) Does the path of the red curve match the actual front wheel's path? Explain your reasoning.
The bicycle tracks
Which Direction did the bicyclist ride?
8 )Which Direction did the bicyclist ride?[br][br][br][br]
Which direction did the bicyclist ride?
9)[br][br]Explain your reasoning for your decision in the direction of the bicyclist.
– thomas_brown

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