INSTRUCTIONAL UNIT 1

By Craig Rimpley

THREE DIMENSIONAL SOLIDS, VOLUME AND SURFACE AREA:

This is an introduction to three dimensional objects and the mathematical connections that can be explored within this subject. First we need to understand the differences between surfaces and solids. Both surfaces and solids are considered to be three dimensional figures but there are some specific differences.

DEFINITIONS:

A SURFACE IS THE BOUNDARY OF A THREE DIMENSIONAL OBJECT.
Example: a beach ball is a surface.
A SOLID IS THE UNION OF THE BOUNDARY OF AN OBJECT WITH THE SPACE OR REGION CONTAINED BY THE BOUNDARY.
Example: a bowling ball is a solid.

Both the bowling ball and beach ball have surfaces. We can see the differences between these two types of objects by looking at the space within each surface. The beach ball is filled with air that can be removed and replaced with ease. The bowling ball surface is filled with bowling ball all the way through, hence the name solid.

The shapes we will work with over the next few days are cylinders. We have done investigations with rectangles and squares. The infamous "box problem" was the last project or problem presented. So our students have some general knowledge of volume that should have been acquired duing this exercise.

We can start to explain surface by going back to the "box problem". The students can grasp an understanding of surface area by moving backwards in the "box problem". If we open our box up into the flat piece of paper we are looking at the surface area of our box. Once we have made this connection between a three dimensional object and surface area we are ready to explore a similar problem.

"THE CYLINDER PROBLEM"

Given a standard piece of paper make cuts so that you can fold the paper into a cylinder. The piece of paper should look like the following:

Once the students have completed their cylinders they need to compare with each other and ask the questions: Which of the cylinders that were made in class today has the largest volume and surface area and which has the smallest. If the construction of a cylinder is to time consuming to be part of the class then the instructor can bring prepared examples of each step of the construction and quickly present this to the class. The construction of the box needs to be part of the preparation of this lesson, so the students have a physical understanding of what was done.

Now we have two problems that are very similar mathematically, but visually different. If we compare and investigate each problem our students should be able to realize the likeness and differences between the two.

Most geometry textbooks are kind enough to give us the formulas for surface area and volume of a right cylinder. The text I referenced broke surface area into :
LATERAL AREA
The lateral area of a right cylinder is the product of its height and the perimeter of its base.
BASE AREA
This is simply the area of the circle that is the base of our right cylinder or PI*RADIUS^2.
The formula for volume of a right cylinder is the product of its height and the area of its base. We can use these formulas to explain the differences between volume and area. Notice the differences between the two formulas: surface area is a two dimensional measurement since we have multiplied two lengths together, and volume is the product of an area and a length which creates a three dimensional measurement.

We need to present all this material to our students before we begin to investigate these problems using technology. The first day of this lesson should be this introduction. We need to make sure that the concepts presented above are understood. Students can easily multiply two numbers together with their calculators and at the same time have no idea what it is what they were computing. ASK QUESTIONS THAT EMPHASIZE THE MATHEMATICAL CONCEPTS OF AREA AND VOLUME RATHER THAN THE COMPUTATIONS!

HOMEWORK : If the cylinder cut out was given as a presentation and not an activity than have the students create the "net" or cut outs needed to create the cylinder and fold them as a class at the beginning of the next class. Assign problems from the text you are using that call for the student to find surface areas and volumes of right cylinders and rectangular solids. The following is a homework hand out that could replace or accompany the work from your text.

CYLINDERS: SURFACE AREAS AND VOLUMES

FORMULAS:

1)SURFACE AREA OF A RIGHT CYLINDER = (height of cylinder ) X (perimeter of base of cylinder) + (area of base or bases)

2) VOLUME OF A RIGHT CYLINDER = (height of cylinder ) X (area of base of cylinder)

PROBLEMS:
1) What measures the union of the boundary of an three dimensional object and the space within the boundary, surface are or volume?

2) What is the formula for lateral area of a right cylinder?

3) A right cylinder has a radius of R and a height of H. What are the formulas for surface area and volume of a right cylinder in terms of radius and height?

4) Find the surface areas and volumes of each of the cylinders with the given characteristics:
a) a cylinder with radius 9 feet and a height of 6 feet?
b) a cylinder with a diameter of 6 feet and a radius of 9 feet?
c) a cylinder with radius 9 yards and height of 15 yards?
d) a cylinder with diameter 15 yards and height of 20 feet?
e) which of the cylinders above will hold the most water if filled to capacity?

5) sketch the "net" of a right cylinder.
a) Can you write the formulas that can be used to find surface area and volume of our sketch using only one variable? hint: the radius and height are dependent on each other in this situation!!!!

6) If the city needs to build a water tower that is the shape of a right cylinder that will hold 100,000 cubic feet of water, what are possible dimensions of the new water tower? (I am looking for the height and radius of our water tower.)

7) How can we minimize the surface area of our water tower and still hold the 100,000 cubic feet of water?

The follow up to this exercise sheet should be done in the computer lab or at least the demonstrations of technology should be presented to the class!

We can investigate each of the problems using GSP. A sketch of each "net" should be done so that the radius of our base or length and height of our box can be adjusted. If the computers are out of reach , overheads can be made from these sketches and used in the classroom until the computer lab can be used. The following are examples of overheads taken from the cylinder sketch:

If you would kike to access the GSP sketch that created these examples CLICK HERE !!!

We can also use GSP to create three dimensional objects that can manipulated by "parent segments" that will calculate surface areas and volumes. These types of sketches are interesting to play with, but probably a bit difficult for a high school student to create on his or her own.

We can use this exercise to teach kids how to use GSP and how to create sketches that are of use. This will take time and the students will need play with the program in order to realize some comfort with and understanding of Geometer's Sketch Pad.

Another possible lab exploration that will maximize the technology our students see is to set up the lab with GSP on a few computers, Algebra expressor on a few computers, and a spread sheet program on other computers. We can let split the class into three group and let each group investigate each program. If the class has a set of graphing calculator that can be taken into the computer lab the students could graph the equations and check their work with Algebra expressor. The same type of exercise could be done with the TABLE function that is part of most graphing calculators (TI-82) and the spread sheet program.

If the computer lab is unavailable these problems are great exercises for the graphing calculator that can be supported by a computer or not, depending on the availability of computers.

HOMEWORK: Have your students create a problem that is similar to the box / cylinder problem that is to be solved in class the next day, or the next day in the computer lab. Have them also create and solve a word problem that is similar to my water tower problem.

SUMMARY:

This is a two to three day lesson that focuses on the surface area and volume of right cylinders. We have made the assumption that our students have a general understanding of algebra and have explored the "box problem". I envision this lesson being useful in a college bound geometry class or as a closing section that focuses on the connections between different disciplines within mathematics. The lesson grants the educator room for changes that are inevitable, since every class and teacher is different. The use of technology I think is a must. At the very least the graphing calculator needs to be involved if the students are going to realize all the connections between geometry, algebra, and real life.

I have included examples of the sketches, the Algebra Xpresser file, and the spreadsheet that I suggested.

If you would like to access the Algebra Xpresser to see a graph of the "cylinder problem"

If you would like to see the Excell spreadsheet I used to solve the "cylinder problem "