Polygon or Polyhedron – A polyhedron is defined using polygons, so in the beginning students will understand the difference. After some time has passes though, students tend to get these similar sounding words confused. Remind them that polygons are two-dimensional and polyhedrons are three-dimensional. The extra letters in polyhedron represents it spreading out into three-dimensions.
The Limitations of Two Dimensions – It is difficult for students to see the two-dimensional representations of three-dimensional figures provided in books and on computer screens. A set of geometric solids is easily obtained through teacher supply companies, and are extremely helpful for students as they familiarize themselves with three-dimensional figures. When first counting faces, edges, and vertices most students need to hold the solid in their hands, turn it around, and see how it is put together. After they have some experience with these objects, student will be better able to read the figures drawn in the text to represent three-dimensional objects.
Assemble Solids – A valuable exercise for students as they learn about polyhedrons is to make their own. Students can cut out polygons from light cardboard and assemble them into polyhedrons. Patterns are readily available. This hands-on experience with how three-dimensional shapes are put together will help them develop the visualization skills required to count faces, edges, and vertices of polyhedrons described to them.
Computer Representations – When shopping on-line it is possible to “grab” and turn merchandise so that they can be seen from different perspectives. The same can be done with polyhedrons. With a little poking around students can find sights that will let them virtually manipulate a three-dimensional shape. This is another possible option to develop the students’ ability to visualize the solids they will be working with for the remainder of this chapter.
Using the Contrapositive – If students have already learned about conditional statements, point out to them that Example Six in this section makes use of the contrapositive. Euler’s formula states that if a solid is a polyhedron, then . The contrapositive is that if , then the solid is not a polyhedron. Students need periodic review of important concepts in order to transfer them to their long-term memory. For more review of conditional statements, see the second section of the second chapter of this text.
Each Representation Has Its Use – Each of the methods for making two-dimensional representations of three-dimensional figures was developed for a specific reason and different representation are most appropriate depending on what aspect of the geometric solid is of interest.
Perspective – used in art, and when one wants to make the representation look realistic
Isometric View – used when finding volume
Orthographic View – used when finding surface area
Cross Section – used when finding volume and the study of conic sections (circles, ellipses, parabolas, and hyperbolas) is based on the cross sections of a cone
Nets – used when finding surface area or assembling solids
Ask the students to think of other uses for these representations.
When students know and fully understand the options, they will be able to choose the best tool for each task they undertake.
Isometric Dot Paper – If students are having trouble making isometric drawings, they might benefit from the use of isometric dot paper. The spacing of the dots allows students to make consistent lengths and angles on their polyhedrons. After some practice with the dot paper, they should be able to make decent drawings on any paper. A good drawing will be helpful when calculating volumes and surface areas.
Practice – Most students will need to make quite a few drawings before the result is good enough to be helpful when making calculations. The process of making these representations provides the student with an opportunity to contemplate three-dimensional polyhedrons. The better their concept of these solids, the easier it will be for them to calculate surface areas and volumes in the sections to come.
1. In the next week look around you for polyhedrons. Some example may be a cereal box or a door stop. Make a two-dimensional representation of the object. Choose four objects and use a different method of representation for each.
These can make nice decorations for classroom walls and the assignment makes students look for way to apply what they will learn in this chapter about surface area and volume.
The Proper Units – Students will frequently use volume units when reporting a surface area. Because the number describes a three-dimensional figure, the use of cubic units seems appropriate. This shows a lack of understanding of what exactly it is that they are calculating. Provide students with some familiar applications of surface area like wrapping a present or painting a room, to improve their understanding of the concept. Insist on the use of correct units so the student will have to consider what exactly is being calculated in each exercise.
Review Area Formulas – Calculating the surface area and volume of polyhedrons requires the students to find the areas of different polygons. Before starting in on the new material, take some time to review the area formulas for the polygons that will be used in the lesson. When students are comfortable with the basic area calculations, they can focus their attention on the new skill of working with three-dimensional solids.
A Prism Does Will Not Always Be Sitting On Its Base – When identifying prisms, calculating volumes, or using the perimeter method for calculating surface area, it is necessary to locate the bases. Students sometimes have trouble with this when the polyhedron in question is not sitting on its base. Remind students that the mathematical definition of the bases of a prism is two parallel congruent polygons, not the common language definition of a base, which is something an object sits on. Once students think they have identified the bases, they can check that any cross-section taken parallel to the bases is congruent to the bases. Thinking about the cross-sections will also help them understand why the volume formula works.
1. A prism has a base with area and a height of . What is the volume of the prism?
2. A triangular prism has a height of . Its base is an equilateral triangle with side length . What is the volume of the prism?
3. The volume of a cube is . What is the cube’s surface area?
Understand the Formula – Many times students think it is enough to remember and know how to apply a formula. They do no see why it is necessary to understand how and why it works. The benefit of fully understanding what the formula is doing is versatility. Substituting and simplifying works wonderfully for standard cylinders, but what if the surface area of a composite solid needs to be found.
1. Find the surface area of the piece of pipe illustrated in the composite solid part of this section.
Make and Take Apart a Cylinder – Students have a difficult time understanding that the length of the rectangle that composes the lateral area of a cylinder has length equal to the circumference of the circular base. First, review the definition of circumference with the students. A good way to describe the circumference is to talk about an ant walking around the circle. Next, let them play with some paper cylinders. Have them cut out circular bases, and then fit a rectangle to the circles to make the lateral surface. After some time spent trying to tape the rectangle to the circle, they will understand that the length of the rectangle matches up with the outside of the circle, and therefore, must be the same as the circumference of the circle.
The Volume Base – In the past, when students used formulas, they just needed to identify the correct number to substitute in for each variable. Calculating a volume requires more steps. To find the correct value to substitute into the in the formula , usually requires doing a calculation with an area formula. Students will often forget this step, and use the length of the base of the polygon that is the base of the prism for the . Emphasize the difference between , the linear measurement of the length of a side of a polygon, and , the area of the two-dimensional polygon that is the base of the prism. Students can use dimensional analysis to check their work. Volume is measured in cubic units, so three linear measurements, or a linear unit and a squared unit must be fed into the formula.
1. The volume of a in tall coffee cup is approximately . What is the radius of the base of the cup?
Don’t Forget the – The most common mistake students make when calculating the volume of a pyramid is to forget to divide by three. They also might mistakenly divide by three when trying to find the volume of a prim. The first step students should take when beginning a volume calculation, is to make the decision if the solid is a prism or a pyramid. Once they have chosen, they should immediately write down the correct volume formula.
Prism or Pyramid – Some students have trouble deciding if a solid is a prism or a pyramid. Most try to make the determination by looking for the bases. This is especially tricky if the figure is not sitting on its base. Another method for differentiating between these solids is to look at the lateral faces. If there are a large number of parallelograms, the figure is probably a prism. If there are more triangles, the figure is most likely a pyramid. Once the student has located the lateral faces, then they can make a more detailed inspection of the base or bases.
Height, Slant Height, or Edge – A pyramid contains a number of segments with endpoints at the vertex of the pyramid. There is the altitude which is located inside right pyramids, the slant height of the pyramid is the height of the triangular lateral faces, and there are lateral edges, where two lateral faces intersect. Students frequently get these segments confused. To improve their understanding, give them the opportunity to explore with three-dimensional pyramids. Have the students build pyramids out of paper or light cardboard. The slant height of the pyramid should be highlighted along each lateral face in one color, and the edges where the lateral faces come together in another color. A string can be hung form the vertex to represent the altitude of the pyramid. The lengths of all of these segments should be carefully measured and compared. They should make detailed observations before and after the pyramid is assembled. Once the students have gained some familiarity with pyramids and these different segments, it will make intuitive sense to them to use the height when calculating volume, and the slant height for surface area.
1. A square pyramid is placed on top of a cube. The cube has side length . The slant height of the triangular lateral faces of the pyramid is . What is the surface area of this composite solid?
Note: The top face of the cube is covered by the base of the pyramid so neither square is included in the surface area of the composite figure.
Mix’em Up – Students have just learned to calculate the surface area and volume of prisms, cylinders, and cones. Most students do quite well when focused on one type of solid. They remember the formulas and how to apply them. It is a bit more difficult when students have to choose between the formulas for all four solids. Take a review day here. Have the students work in small groups during class on a worksheet or group quiz that has a mixture of volume and surface area exercises for these four solids. The extra day will greatly help to solidify the material learned in the last few lessons.
1. A cone of height sits on top of a cylinder of height . Both cone and cylinder have radius with length . Find the volume and surface area of the composite solid.
2. A cone of radius is carved out of a square prism of the same height. The square base of the prism has area and height . What is the volume of this composite solid?
Expand on Circles – Students learned about circles earlier in the course. Review and expand on this knowledge as they learn about spheres. Ask the students what they know about circles. Being able to demonstrate their knowledge will build their confidence and activate their minds. Now modify the definitions that the students have provided to fit the three-dimensional sphere. Students will learn the new material quickly and will remember it because it is now neatly filed away with their knowledge of circles.
Explore Cross-Sections – One of the goals of this chapter is to develop the students’ ability to think about three-dimensional objects. Most students will need a significant amount of practice before becoming competent at this skill. Take some time and ask the students to think about what the cross-sections of a sphere and a plane will look like. Explore trends. What happens to the cross-section as the plane moves farther away from the center of the circle? A cross-section that passes through the center of the sphere makes the largest possible circle, or the great circle of the sphere.
Cylinder to Sphere – It would be a good exercise for students to take the formula for the surface area of a cylinder and derive the formula for the surface area of a sphere. It is just a matter of switching a few variables, but it would be a good exercise for them. During the lesson, ask them to do it in their notes, wait a few minutes and then do it on the board or ask one of them to put their work on the board. It should look something like this:
Note to students that in the last line the like terms where combine. This can only be done because both terms had the factor . The coefficients could have been different, but to combine terms using the distributive property they must have the exact same variable combination. Here the is being treated as a variable even though it represents a number. This is a more complex application of like terms than students are used to seeing.
Limits – The formula for the volume of a sphere is developed using the idea of a limit. Explain this to the students or the logic might seem fuzzy to them. The limit is a fundamental concept to all of calculus. It is worthwhile to give it some attention here.
Surface Area is Squared – Surface area is a two-dimensional measurement taken of a three-dimensional object. Students are often distracted by the solid and use cubed units when calculating surface area or mistakenly cube the ratio of linear measurements of similar solids when trying to find the ratio of the surface areas. Remind them, and give them many opportunities to practice with exercises where surface area and volume are both used.
Don’t Forget to Adjust the Ratio – There are three distinct ratios that describe the relationship between similar solids. When the different ratios and their uses are the subject of the lesson, students usually remember to use the correct ratio for the given situation. In a few weeks when it comes to the chapter test or on the final at the end of the year, students will frequently forget that the area ratio is different from the volume ratio and the linear ratio. They enjoy writing proportions and when they recognize that a proportion will be used, they get right to it without analyzing the ratios. One way to remind them is to have them use units when writing proportions. The units on both sides of the equal sign have to match before they can cross-multiply. Give them opportunities to consider the relationship between the different ratios with questions like the one below.
1. If a fully reduced ratio is raised to a power, will the resulting ratio be fully reduced? Explain your reasoning.
Answer: Yes, two numbers make a fully reduced ratio if they have no common factors. Raising a number to a power increases the exponent of each factor already present, but does not introduce new factors. Therefore, the resulting two numbers will still not have any common factors.
These concepts frequently appear on the SAT. It will serve the students well to practice them from time to time to keep the knowledge fresh.
1. The ratio of the surface area of two cubes is . What is the ratio of their volumes?