If you were told by a classmate that taking a cup of brackish lukewarm water three times a day for a month would heal your broken leg, you would probably laugh out loud. And even though your laughter might be a bit more subdued were the source of that statement the health segment of the evening news, I bet that you would nonetheless be quite skeptical. On many levels this would seem to you to be highly implausible, and so you might either absolutely dismiss it or yield to that nagging curiosity and check out some other sources.
On the other hand there are a host of other more reasonable sounding claims that you might be prepared to let slide. Examples might include the claim that an increase in oil prices pushes a decline in the Dow Industrial averages, or that there is a strong correlation between wealth and SAT scores. It doesn’t seem unreasonable to hold a healthy skepticism for these as well.
In the context of mathematics, to check for plausibility is to routinely check the reasonableness of any statement in a problem or its proposed solution, regardless of whether it seems true or false on initial impression; to be particularly skeptical of results that seem contradictory or implausible, whether the source be peer, teacher, evening news, book, newspaper, internet or some other; and to look at special and limiting cases to see if a formula or an argument makes sense in some easily examined specific situations.
Your classmate proposed traveling 240 miles to the beach at an average speed of 80 mph, and the next day traveling home at 40 mph. He pointed out that since you have an average speed of 60 mph, the total 480 miles there and back takes 8 hours. Yet when you went to the beach, it didn’t take 8 hours round trip, even though you two drove at precisely the speeds he had proposed. Why?
It certainly seems at first glance that your classmate is correct, but just because a result seems superficially correct doesn’t make it so. When there is an apparent contradiction in a problem, it often pays to make a quick check or two to see if what is being said is plausible. For example, how long does each half of the beach trip take?
Martin says, looking at the figure below, that the entire rectangle’s area divided by the area of the square inside is equal to $\dfrac {a} {b} + 1$. Melissa thinks that such a peculiarly simple answer is unlikely to be correct. Martin insists he’s right, and that in fact his formula would work for any values of $a$ and $b$ one might choose. How could Melissa check to see if Martin’s formula is at least plausibly correct?
One particular way that we check the reasonableness of a solution to a problem is to examine special and limiting cases. In the problem above, what would those cases be? Well, one special case would surely be when $a=b$, because you could quickly determine the right answer and check if Martin’s formula is correct there. What about if we examined when “$a$” was small and “$b$” was large—say $a=1$ and $b=100$, or even $a=1$ and $b=10000$? Does Martin’s formula make sense in those cases as well? Can you come up with another limiting case, and see if his formula is plausible for that case as well?
Once you have checked a solution for all the limiting cases you can think of, if it still seems like a reasonable solution, you might think about how to prove it always works yourself from first principles. Can you come up with Martin’s formula?
If one looks at $y=x^2$ on the calculator in “ZOOM SQR” mode (so that the scale in the $x$ and $y$ directions are the same and the graph looks most accurate), it appears that the graph is getting so steep, so quickly, that it will eventually become a vertical line. Does it? If so, estimate for what x value it becomes vertical; if not, explain why that can never happen.
While teaching in Brazil, Tony was approached by a fellow math teacher who said the following: “Hey Tony, do you know how to prove that all right triangles are similar? I was trying to show my students in class today and I couldn’t quite do it.” Could you have helped him out?
On a 3-D blueprint for an Olympic swimming pool, 1 ft. represents 16 actual feet. In order to determine how much water would be needed to fill the pool, Tim computes the volume from the blueprint, which is $4 \text{ ft}^3$, and then he multiplies by the scale factor of 16 to get $64 \text{ ft}^3$ of water to fill the pool. Tim is a little unsure if that is the correct amount, but it seems right. What do you think?
If Jorge told you that 3.16227765 was an exact solution to $x^2=10$, how could you determine without a calculator if he is correct, or slightly off?
Hero of Alexandria came up with a formula to determine the area of any triangle based solely on the lengths of its 3 sides. Below are 4 formulas, all of which purport to be Hero’s formula. In all 4 formulas, $a$, $b$, and $c$ are the lengths of the sides, and $s$ is the semi-perimeter, which is equal to half the perimeter.
$Area = \left(s-a \right)\left(s-b\right)\left(s-c\right)$
$Area = \sqrt {\dfrac {(s)(a)(b)(c)} {10}}$
$Area = \sqrt {(s)(s-a)(s-b)(s-c)}$
$Area = \sqrt {3(a+b+c)}$
Which of these formulas do you think is the right one? Why?
Try seeing if the formulas give the kind of answers you would expect for various “common” triangles you have experience with.
Test to see if “extreme” triangles (ones with very large or small values of some of the sides) also give reasonable results.
What units do you typically measure area in? Does that also help you in deciding which formulas are most plausible?
Problem continued on the next page
Now pick the formula you are most confident is the right one. Does the fact that it has passed all your “tests” prove that it is correct? If so, explain why. If not, explain how you could become convinced that Hero was in fact correct.
Look at these 6 numbers: 1, 3, 6, 9, 11 and 12. Their mean is 7. Subtracting the mean from each of the numbers and adding those together gives us $-6+-4+-1+2+4+5$, which equals 0. Juniper is unimpressed, and says that you would always get 0, regardless of the 6 numbers you chose. Sassafras disagrees, and says it is highly dependent on choosing the right 6 numbers; for example, in this case, exactly 3 were above the mean and exactly 3 were below, and also there were no decimals to complicate matters. Who is right?
Zargo says that instead of doing lots of intricate calculations, he can find the area of a rhombus by just multiplying the diagonal lengths together. See if you can determine in a minute if his method is plausible.
If you draw a line from the vertex of any triangle to the midpoint of the opposite side (i.e. the median), will it be perpendicular to that side, or would it bisect the vertex angle from which it was drawn?
A pollster interviewed 100 families, and reported that the mean number of children was 2.037 and the median was 1.8. I do not believe either of these figures. Do you? Why?
Brian thinks he remembers that the area of a parallelogram is equal to the product of consecutive sides, but he isn’t quite sure. You can’t remember whether he’s right either, but you know you can check his formula to see if it is plausible. Is it?
Show that the formula for the area of a triangle can be viewed as just a special case of the area of a trapezoid.
Which is bigger, $4^\frac {1} {4}$ or $10^\frac {1} {10}$? No calculators allowed! (Hint: try thinking about limiting cases.)
Hero’s formula gives us a formula for the area of a triangle based only on the lengths of its 3 sides (see problem 7 in this lesson). No one has yet come up with a formula for the area of a quadrilateral based only on the lengths of its 4 sides. Why do you think that is?
Veneeta graphs $y=x^2$, $y=x^2+4$ and $y=x^2-10$ on the same calculator screen using ZOOM STD. What does she notice about how their shapes compare to each other?
Grunchik changes the viewing window on his calculator so that it graphs $x$ values from -4 to 4, and $y$ values from -10 to 20. He then graphs the same three equations that Veneeta graphed. What does he notice about how their shapes compare to each other?
Veneeta and Grunchik compared their different answers, but aren’t sure what to make of them. What do you think?
Prashad says that in a quadrilateral ABCD he has examined, $\text{AB}+\text{BC}+\text{CD}+\text{DA}$ is equal to 1.8 times the diagonal AC. Evaluate whether what he says is possible or not.
When an object falls under gravity, its speed increases by a constant amount each second. Two stones are dropped at the same time from a cliff, but one of them is 10 feet higher up than the other at the time of dropping. As they fall, will the distance between them always be the same?
Later on, two stones are dropped at the same height from a cliff, but one stone is released one second before the other. As they fall, will the distance between them always be the same?
On the same screen as $y=x^2$, graph $y=x$ in ZOOM STANDARD. Then “ZOOM IN” once.
Note that the graph of $y=x$ is “above” the graph of $y=x^2$ for some values of $x$.
Hermione thinks that those values of $x$ will be the only ones where $y=x$ is above $y=x^2$. What do you think?
Bart is feeling a little sick. Having recently read about simpsonitis, a very rare and debilitating disease, and being somewhat hypochondriacal, he goes to see his physician, Dr. Kalvakian. The doctor checks him out and decides to administer a special blood test for detecting the disease.
This diagnostic test is 98% accurate (returns a positive result) for people who have simpsonitis and 95% accurate (returns a negative result) for people who do not have simpsonitis. Approximately 0.3% of people in the country actually have this disease.
Unfortunately, several days after taking the blood test, Bart receives a phone call from Dr. Kalvakian. The doctor tells him that he tested positive for the disease. Bart asks, “What’s the chance that I actually have simpsonitis? I mean, you said that the test was not 100% accurate.” Dr. Kalvakian replies, “Well, there’s a 98% chance that you have the disease, Bart.”
Bart initially has a cow, but then he decides to tell the brainy Lisa what Dr. Kalvakian said. What do you suppose Lisa told Bart in response?
Zollywog the crazy Geometry student has come up with a formula for the length of a median in a triangle! He claims that the length of the median m that bisects side a of a triangle (with other sides b and c, of course) is:
$m=\sqrt{\dfrac {2b^2+2c^2-a^2} {4}}$.
Could this possibly be true?
Felipe tells Bradley that he has just come up with a cool fact: a regular polygon of n sides, with distance $r$ from the middle of the polygon to any one of the vertices, will always have an area less than $4r^2$. Bradley is skeptical of Felipe, since as $n$ increases the area keeps getting bigger. Can you resolve their dispute?
Jillian says that, for any positive integer $x$, $\dfrac{{420\left( {x + 1} \right)!}}{x}$ will always be an integer. Explain why she is correct.