must have the exact same variable combination. Here theπis 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.
Similar Solids
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.
Key Exercise:
- 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.
Additional Exercises:
- The ratio of the surface area of two cubes is 25 : 49. What is the ratio of their volumes?
Answer:
(
√
25 )^3 :(
√
49 )^3
( 5 )^3 :( 7 )^3
125 : 343
Chapter 2. Geometry TE - Common Errors