The quiz contains four questions. Each deals with a topic in elementary school mathematics, however we analyze each question from a college perspective, showing the depth of understanding that is expected of college students. At the end,
we discuss the relationship between these problems and college readiness.
1. Many students know that you are “not allowed to divide by 0.” Why is 12 ÷ 0 undefined?
Your answer is correct! In college, it is expected that students provide reasoned answers based on definitions, previously known results and logic.
Answer a) is not a sufficient reason to explain why you cannot divide by 0, even if your high school teachers told you this many times. It is your job, as a college student, to understand and reason with concepts, not simply memorize rules. In fact,
answer b) is a memorized rule, not an explanation. Answer c) is also insufficient. There are many things involving 0 that are defined, for example, 30=1; why does that make sense? No, all of
the answers presented are insufficient for the college level. Here is a correct explanation. We must think about what division means. For example, why does 12 ÷ 4 = 3? Because
4 × 3 = 12. In other words, 12 ÷ 4 is the name for the number that you can multiply by 4 to obtain 12.
Using this definition of division, 12 ÷ 0 is the name for the number that you can multiply by 0 to obtain 12. However, there is no number you can multiply by 0 to obtain 12. Therefore
12 ÷ 0 is the name for a number that does not exist. Therefore 12 ÷ 0 is undefined, in other words, we give no meaning to
it.
Your answer is not correct. In college, it is expected that students provide reasoned answers based on definitions, previously known results and logic.
Answer a) is not a sufficient reason to explain why you cannot divide by 0, even if your high school teachers told you this many times. It is your job, as a college student, to understand and reason with concepts, not simply memorize rules. In fact,
answer b) is a memorized rule, not an explanation. Answer c) is also insufficient. There are many things involving 0 that are defined, for example, 30=1; why does that make sense? No, all of
the answers presented are insufficient for the college level. Here is a correct explanation. We must think about what division means. For example, why does 12 ÷ 4 = 3? Because
4 × 3 = 12. In other words, 12 ÷ 4 is the name for the number that you can multiply by 4 to obtain 12.
Using this definition of division, 12 ÷ 0 is the name for the number that you can multiply by 0 to obtain 12. However, there is no number you can multiply by 0 to obtain 12. Therefore
12 ÷ 0 is the name for a number that does not exist. Therefore 12 ÷ 0 is undefined, in other words, we give no meaning to
it.
2. We know that 12 ÷ 4 = 3. A common story problem used to explain why this is true is the following:
Suppose you have 12 candy bars and you have 4 friends to whom you want to give the candy bars. How many candy bars does each friend receive? Can you modify this story problem to explain why
12 ÷ 1/2 = 24?
Your answer is correct, The story problem used to explain 12 ÷ 4 = 3 does not work when
you are considering 12 ÷ 1/2 = 24, since the idea of a 1/2 a friend does not make sense (in this scenario, anyway). Concerning answer b), if we cannot determine a scenario for
12 ÷ 1/2 how would we know that inverting and multiplying is an algorithm that gives a correct answer? We can modify the story problem so that it makes sense both
for 12 ÷ 4 = 3 and 12 ÷ 1/2 = 24. Suppose you have 12 candy bars and you want to give each of your friends 4 candy bars. How many friends
can you give candy to? The correct answer is 3 friends. But this problem works for our new situation. Suppose you have 12 candy bars. You want to give 1/2 a candy bar to each friend. How many friends can you give candy to? The answer is 24 friends.
Your answer is not correct; the story problem used to explain 12 ÷ 4 = 3 does not work when
you are considering 12 ÷ 1/2 = 24, since the idea of a 1/2 a friend does not make sense (in this scenario, anyway). Concerning answer b), if we cannot determine a scenario for
12 ÷ 1/2 how would we know that inverting and multiplying is an algorithm that gives a correct answer? We can modify the story problem so that it makes sense both
for 12 ÷ 4 = 3 and 12 ÷ 1/2 = 24. Suppose you have 12 candy bars and you want to give each of your friends 4 candy bars. How many friends
can you give candy to? The correct answer is 3 friends. But this problem works for our new situation. Suppose you have 12 candy bars. You want to give 1/2 a candy bar to each friend. How many friends can you give candy to? The answer is 24 friends.
3. Sally and Mary have a race that lasts 5 minutes. Below is a graph of the race. When does Mary pass Sally?
Your answer is correct. Many students think that Mary catches up to Sally after 4.5 minutes. However, note that the labeling of the y axis is speed, not distance run. Therefore,
the first time that Mary is going the same speed as Sally is after 4.5 minutes. It is also important to notice that the units for the x axis are minutes, but the units for the y axis are feet per second. For most of the race, Mary is running 4 ft./sec. slower than Sally. Therefore, Mary is falling 4 feet further behind Sally after each second. This means that Mary is falling 240 feet further behind Sally after each minute. So after 4 minutes, she is 4*240 = 960 feet behind Sally.
The last half minute when Mary runs faster than Sally is not enough for her to catch up to Sally. In fact, if you actually did the calculations, you would find that after 5 minutes Mary is 960 feet behind Sally.
Your answer is incorrect. The correct answer is e. Many students think that Mary catches up to Sally after 4.5 minutes. However, note that the labeling of the y axis is speed, not distance run. Therefore,
the first time that Mary is going the same speed as Sally is after 4.5 minutes. It is also important to notice that the units for the x axis are minutes, but the units for the y axis are feet per second. For most of the race, Mary is running 4 ft./sec. slower than Sally. Therefore, Mary is falling 4 feet further behind Sally after each second. This means that Mary is falling 240 feet further behind Sally after each minute. So after 4 minutes, she is 4*240 = 960 feet behind Sally.
The last half minute when Mary runs faster than Sally is not enough for her to catch up to Sally. In fact, if you actually did the calculations, you would find that after 5 minutes Mary is 960 feet behind Sally.
4. Consider the number 5 and the infinite repeating decimal 4.999. Which number is bigger? Why?
Your answer is correct, but do you know why they are equal? It has to do with what we mean by an infinite decimal. The infinite decimal 4.999
means the following: a) Look at the numbers 4.9, 4.99, 4.999, 4.999 and so on.
b) Ask yourself, what number do these numbers get closer and closer to? They get closer and closer to 5. c) The number 4.999 is defined to be the number that
4.9, 4.99, 4.999, 4.999 gets closer and closer to. Therefore
4.999 is defined to be 5. The two numbers are equal because of the meaning we give an infinite decimal.
Your answer is incorrect, the two numbers are equal. It has to do with what we mean by an infinite decimal. The infinite decimal 4.999
means the following: a) Look at the numbers 4.9, 4.99, 4.999, 4.999 and so on.
b) Ask yourself, what number do these numbers get closer and closer to? They get closer and closer to 5. c) The number 4.999 is defined to be the number that
4.9, 4.99, 4.999, 4.999 gets closer and closer to. Therefore
4.999 is defined to be 5. The two numbers are equal because of the meaning we give an infinite decimal.
All of the above questions deal with topics taught in grammar school, however at the college level, students are expected to understand exactly what the mathematical objects they are using mean and be able
to draw inferences from them. The first problem above requires a student to understand the definition of division to ascertain whether 12 ÷ 0 makes sense or not. In a similar way, the last problem
requires the student to understand what exactly is meant by an infinite decimal in order to infer the value of 4.999.
The third problem asks the student to understand exactly what the graph says and what it does not say. For the labeling of the graph, seeing the graph for Mary go above the graph for for Sally does not mean that Mary is passing Sally.
Finally, the second problem gives a simple example of a model related to a particular mathematical expression. It is often the case in mathematics that models break down under new conditions. It is common for mathematicians to search for new models
which work under the previous and the new conditions.
A college ready student is comfortable with these concepts. Many students view mathematics as simply a mechanism for determining what the variable x equals in an equation and only care about obtaining the “correct answer,”
whether they understand what they are doing or not. In college, it is important to have algebraic skills, but that is only part of the package that is expected of students. Conceptual knowledge is equally important, in fact, strong conceptual knowledge will
allow you to employ the right technique at the appropriate time to obtain the “correct answer.”
