Planning a Lesson in 3 Moments for Mathematics

Instructional Unit: Rates of Change

Focus: Covariational Reasoning Using Bottles of Different Shapes


Have students first explore rates of change with dynamic software such as GeoGebra. Some pre-made files already exist (, Then have students do some reading on rates of change and also explore adding water to different shaped bottles. Eventually, provide students with different shaped bottles and ask them to imagine that you are adding water at a constant rate and to draw a graph of the volume of water with respect to height of water. This can lead to discussions on rates of change and ultimately derivative (see and

Text Options:
  • Covariational Reasoning
  • Students can match bottle shapes to given functions or draw bottle shapes based on given functions or draw functions based on given bottle shapes

Mathematics Standards:

PREPARE, INTERACT, & EXTEND (PIE) Instructional Strategies (across content)