Bottles and Divers
(Rates of Change)
High School Math Project
Download PDF Lesson |
Get Acrobat
A Printable version of this lesson is available in PDF format
This requires a free plug-in, Adobe Acrobat Reader.
You can find out if you need it by clicking on the PDF link.
Objective
Students calculate the rate of change between two points on a curve by determining
the slope of the line joining those two points. They then extend this method to
estimate the instantaneous rate of change at a given point by taking two points very
close to and on either side of the given point.
Overview of the Lesson
The concept of slope as the rate of change is an important one in the study of
algebra. This lesson has an introductory activity that uses bottles of various shapes
to help students understand the concept of rate of change. The lesson then develops
the concept of the average rate of change between two points on a curve. Finally, it
examines instantaneous rate of change: rate of change function is used to estimate
the rate of change at a particular point. This function is used along with the
calculator to make computing the rate of change fast and easy for students. The
development of this function helps students understand what is meant by
instantaneous rate of change and lays a firm foundation for the study of the
derivative in calculus.
Materials
- graphing calculator with overhead unit
- overhead projector
- class data chart on newsprint or blackboard
- a collection of various bottles, beakers, and flasks
For each group of four:
Procedure
- Graphically Speaking: The activity allows students to get hands-on experience
dealing with the abstract concept of rate of change. Each group of students is
given one container. They use a graduated cylinder to measure amounts of
water, then add the water to the container and measure the height of the
water. After this data is collected, students make a scatterplot of their data.
If possible, each group should conceal their container from the other groups
as they are collecting data and creating the scatter plot. To help them conceal
the containers as the students do their measuring, you may wish to make
screens of poster board folded in half and taped to a desk. A large box with two
sides cut away also works well.
The containers should represent a variety of shapes. Your science department
may have a selection of beakers, flasks, long stem funnels, and graduated
cylinders that could be used, or students could bring in a variety of bottles or
containers. In assigning the containers, keep in mind that containers' shapes
affect the difficulty level of the activity: beakers, graduated cylinders, and any
other type of prism are easier; an Erlenmeyer flask is moderately difficult; and
the Florence flask and the long stem funnel are generally more difficult, or at
least more time consuming.
As the groups complete their measurements and scatterplots, have them tape
these to the blackboard or a wall. When all groups are finished, lead the entire
class through a discussion of the graphs that are on display. For each graph
you might ask questions similar to the following. Have each group discuss
them before you ask for responses directed to the entire class.
- What do you think the shape of the container is like?
- Is the graph similar to any other graph that is displayed?
- How do you think the shape differs from the shape of the container
whose graph is similar to it?
- Could you write an algebraic rule for the graph?
After the groups have discussed these questions, have a representative from
one of the groups go to the board or the overhead and sketch the shape his or
her group believes the graph represents. Let other groups either confirm the
sketch or challenge it by giving their alternative and justifications for a
different sketch. As the class goes through this discussion, remind the group
whose shape is being discussed that they are not to give any hints. After the
sketch has been made and discussed, the group who did the plot should show
their container and either affirm the sketch or explain why the sketch is not
correct.
- The Bungee Jumper: Class discussion of the "Think About This Situation"
section of Introduction: The Bungee Jumper serves as the introduction to the
lesson on rates of change for functions with algebraic rules. This activity is
designed to engage the students and to have them talk about some ways in
which they could use an algebraic rule to help determine a rate of change.
Students could mention determining an average rate of change or using the
algebraic rule to create a table of values or a graph.
- The Diver Problem: Have students work in groups to complete the problem.
They might need to be reminded to consider the symbolic representation, the
graph, and the table of values as they explore this problem. After all of the
groups have had time to work and discuss the problem, have them present
their findings using the blackboard or newsprint. Discuss each part of the
problem, moving from small group discussions to whole class discussions.
Problem number 4 should be thoroughly discussed because it allows students
to generalize the procedure that the previous problems had been leading
them to discover. Students might need more guidance with this
generalization.
As students work with the rate-of-change estimate function with their
calculator, it will be helpful for them to understand that they can enter that
equation in Y1 as Y1 =  , where Y2 is the function for
which they are exploring rates of change. As they explore different functions,
they simply have to enter the new function in Y2.
|
