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cone volume problem

I was with a 2nd year high school class, preparing for our (US) state's standardized test. I asked the class how they would solve this, and they flipped through the sheets to find

$$V=\frac{1}{3}\pi r^2h$$

(To be clear, there is a 'formula' page that's given with the exam.) As they fumbled with their calculators, I asked them what the radius was. 3.5. and I wrote the numbers on the board.

$$V=\frac{1}{3}\pi (3.5)^2(9)$$

I attempted to show them how 3.5 squared is close enough to 12. Divide by 3 to get 4, and we are left with 36pi. Since 36x3 is 108, the only answer that can make sense is 115.

In my lecture, I offered 3 benefits of this process. (1) The speedy approach can save precious time on this type of question so more time remains for those requiring more thought and time. (2) If you still use the calculator, this is a good double check to be sure the answer makes sense, that you didn't hit the wrong key. (3) If you forget your calculator, the proctor won't always have an extra, and the whole exam can be done by hand.

Out of the 12 questions we did as a group, a full half lent themselves to this method, as number were easy to round, and the answers were different enough so cumulative rounding errors didn't inch results too far away from the true answer. After my second attempt, on the next appropriate question, to explain this approach, I realized it wasn't helpful to this class and I stopped.

I recall that as a student, estimating was a skill that we were taught as part of the regular math class. Is this no longer considered to be valuable?

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    $\begingroup$ You might be interested in Fermi estimates en.wikipedia.org/wiki/Fermi_problem also see lesswrong.com/lw/h5e/fermi_estimates . Used rather a lot in computational physics (to determine is a result is reasonable), probably other disciplines too. $\endgroup$ – Clumsy cat May 13 '17 at 18:36
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    $\begingroup$ @TheoreticalPerson Post an answer about that! That's probably the motherlode of insights into this topic. $\endgroup$ – Ben Kovitz May 13 '17 at 18:50
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    $\begingroup$ I'm a quant. I spend all day eyeballing, estimating, and mentally confirming the numbers on my screens. I would say the skill is absolutely invaluable. $\endgroup$ – Kaz May 13 '17 at 22:31
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    $\begingroup$ I agree 100% that estimation is useful, but I would be careful with this particular problem, because 132 is dangerously close to 115. If I was estimating, I would have complete confidence in eliminating A and D, not so much C. $\endgroup$ – Javier May 14 '17 at 22:54
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    $\begingroup$ Considered by whom? A particular secondary school system? If so, let us know where you live. Or are you asking us if WE consider it important? If so, yes, estimating helps me every day from sanity checking Excel to buying things at the grocery store to providing on the fly technical advice in business meetings to making good food in the kitchen. $\endgroup$ – Scott May 15 '17 at 0:14
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Yes, of course it is. You've identified many of the important reasons in your question. In practice in quantitative fields, we are all estimating as a first-pass on whether problems are soluble all the time. Estimation is incorporated throughout Common Core standards (e.g., Grade 3, Grade 4, Grade 7; there are more). And it was part of official state standards long before that (link).

That said, I am concerned that in some locations it may not in fact be taught. When I teach remedial arithmetic courses at my community college, the entire classroom of students will claim to have never heard the term before (either "estimate" or "approximate", etc.). It is, in fact, utterly bewildering to them. Therefore, in that course I make it the topmost priority and try to work estimation skills into the exercises every single day. Nevertheless, at the end of a semester, many students will say something like, "I never got this estimation thing".

Often when I share this with other people I get great skepticism ("but my child is learning that in the third grade right now"). We might theorize how it came to be: Maybe my students come from specially broken-down school systems. Or maybe many of the students have intellectual disabilities and were taught to rely on the calculator in the absence of basic numeracy. Or maybe they've just never built anything practical such that they saw the value in double-checking for errors. At any rate, it's that kind of gap for which Common Core is trying to give guidance, I think.

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I can't speak for a broad consensus, but I personally certainly think that estimation is still a valuable skill.

First, here's my rough, loose, quick way of solving the problem, using only easy mental arithmetic, no pencil, paper, or calculator. Seeing this will make my reasons below clearer.

  1. This problem would be a whole lot easier if it were a rectangular prism. The base would be 7 ⨉ 7 = 49 in2, which is almost 50. Multiplying that by the height of 9 would yield a volume of 450 in^2. Now we just have to "shave off" everything that isn't part of the cone. Notice that this already shows that answer D is hopeless.

  2. OK, let's shave it down to a cylinder. That's pretty easy. The base is a circle, which has an area of a little more than 3/4 of the square with the same "diameter" (i.e. a circumscribed square). 3/4 of 49…ecch, let's make this easier and say 3/4 of 48…that's 36. So the cylinder has a volume of a little more than 9⨉ that…10⨉36 = 360, subtract 36, that's 324.

  3. The idea of the volume formula for the cone is just that it's 1/3 of the cylinder. So, to shave away the part of the cylinder that's not the cone, just divide by 3. 324/3 = 300/3 + 24/3 = 100 + 8. So the volume of the cone is a little more than 108 in3. Looks like answer B.

Now here are some reasons why being able to estimate is a valuable skill:

  1. As you said, the estimate is a double-check on the precise calculation. Really, you should never do a precise calculation without also doing an estimate. Precise calculations, especially if a calculator is involved, can easily be off by orders of magnitude if you make a tiny mistake, like a wrong keypress. You should always know the "ball park" before you do the calculation, or your common sense can't function. Notice how considering a rectangular prism set an upper bound on reasonable answers. Just from looking at the picture, you should be thinking that the right answer is a little less than a third of that. Estimates can be wrong, too, but the point is: when the calculation disagrees with your estimate, you know something is wrong. Without an estimate, you can't tell.

  2. When students are taught math as a bunch of formulas and procedures to memorize, they often can't think fluidly or flexibly about it. When they meet a problem for which they don't know a formula or procedure, they're instantly stumped and give up. Estimating gives the student easy practice in thinking mathematically. Most of mathematical reasoning consists of seeing the same thing as it is involved in different relationships, each of which sheds some light on it or constrains it in some way. Estimating gives you practice in feeling around for helpful relationships, and in distinguishing between important and unimportant. It's easier than usual because you don't need to be exact. It gives you practice in the ubiquitous mathematical technique of solving a much easier problem than you were given and then figuring out how to correct for the difference. And it shows you that mathematics makes sense.

  3. Estimating teaches through direct experience that there is not just one right way to solve a problem. Notice that your way of reaching the same estimate was more efficient than mine. That's to be expected. I don't mess with geometry much. I just fumbled around a bit, but I was still able to find the answer completely in my head. I don't need to be boned up on solid geometry to find a passable way to solve the problem. I can cook something up that's good enough, without being an expert. There are many, many paths, and they all lead to the same answer.

  4. Consequently, estimating builds confidence. The more you estimate, the more you see math as a system of relationships that tend to guide you to the thing you're looking for. If you miss one relationship, no big deal, math is abundant in relationships. And if you're off by a little bit, no big deal. You see that math is best approached loosely, lightly, playfully. This is the opposite of the feeling that students usually get: that math is a minefield—make one false step and it's over.

  5. In real life, you almost never need an exact answer. Estimating is the main practical skill of mathematics beyond arithmetic. In real life, outside of fields like accounting, usually you can't even get very precise measurements. The first couple digits and the order of magnitude are usually all you need to know—often, all you can know. Also in real life, you seldom encounter problems fully spelled out. Usually you have to figure out what available information would enable you to make a good estimate. You can't really do that effectively without experience estimating.

For inspiration, read about Enrico Fermi. He is said to have been able to solve pretty much any physics problem in his head to one significant digit and the order of magnitude. And yes, he taught this skill to his students.

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In general, yes, estimation is a great way to get a quick sanity check.

In this particular example it looks like they chose values such that using $22/7$ as an approximation for $\pi$ lets you quickly cancel stuff out,

$$\begin{eqnarray*} V = {\frac{1}{3}}\times{\frac{22}{7}}\times{\frac{7}{2}}\times{\frac{7}{2}}\times9 = {\frac{11\times7\times3}{2}} = 115.5 \end{eqnarray*}$$

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    $\begingroup$ Please don't write mixed fraction. Pretty please. $\endgroup$ – Jessica B May 14 '17 at 7:55
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    $\begingroup$ @JessicaB What is the trouble with the mixed fraction? $\endgroup$ – Ben Kovitz May 14 '17 at 9:09
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    $\begingroup$ "Mixed fractions" are terrible notation — they look like multiplication, but are actually addition, in a weird, unnecessary exception to the usual rules of mathematical notation. It only takes one extra symbol to write $115 + \frac{1}{2}$, and that avoids all the ambiguity and confusion of mixed fractions. $\endgroup$ – Daniel Hast May 14 '17 at 15:09
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    $\begingroup$ @DanielHast Weird: to me, a mixed fraction doesn't look like multiplication, it looks like a mixed fraction—that is, like a number (so you're done) rather than a calculation (so you're not done). But math is positively overflowing with ambiguous notations that require context to sort out—and that the uninitiated can't sort out. Do you object to function notation, like $f(x)$, for the same reason: it looks like multiplication? I could sympathize with that: that one had me very confused for a long time when I was 13. $\endgroup$ – Ben Kovitz May 14 '17 at 19:02
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    $\begingroup$ @BenKovitz I can tell $f(x)$ is not multiplication because $f$ and $x$ are different types of object that cannot be multiplied. If it worked out they did need to be multiplied, you could use $f\cdot x$ or $xf$. With mixed fractions, there is no way of telling that a different convention is being used. I read this answer as 'half of 115' until I scrolled to another answer and saw 115.5. $\endgroup$ – Jessica B May 15 '17 at 6:41
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I have found that estimating is a very valuable skill, particularly for finding orders of magnitude of solutions without having to use a calculator. It comes in very useful for helping students work out problems on the fly.

Unfortunately, I have found that most students now have no feel for this skill, opting instead to use a calculator for any calculation, even those that could easily be worked out exactly in one's head. When we get into calculations involving quantities with uncertainties, this lack of understanding of really how precisely one can express an answer becomes even more pronounced. Students seem to always think the correctness of the answer corresponds to the number of decimal places that are written down, and they seemingly have no feel for whether a possible answer is feasible or not without working it out with a calculator.

I think we should sometimes require students to write exams without a calculator. That's how many of us learned not to require one for every little calculation. Estimating the answer in their head beforehand would also help students to realize when an answer from their calculator might be questionable.

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First, an even easier estimate: $$V = \frac{1}{3} \pi r^2 h \approx \frac{1}{3} \frac{22}{7} \left( \frac{7}{2} \right)^2 9 = \frac{2 * 11 * 7^2 * 3^2}{3 * 7 * 2^2} = \frac{1}{2}*11*7*3=115.5.$$ Never underestimate the power of "engineer's $\pi$" ! Admittedly, some of the cancellations make this problem even easier.

I would whole-heartedly agree with the other answers that YES estimation is vitally important. I'll try to flush out a few more reasons:

Daily Life

While the old teacher reply that "you might not have a calculator with you" has been somewhat invalidated by most people carrying around a computer in their pocket, many daily problems just don't require precise answers or have vaguely defined inputs. For example,

  • I'm throwing a party. How many pizzas to order / cases of ...soda...to buy / pounds of ice? Especially since I have a few flaky friends that might not show up or might bring along dates.
  • I'm driving on the interstate and have a quarter tank left. Do I need to pull off at the next exit or can I make it to the next town that's 100 miles down the road? What if the gas at the next exit is overpriced?
  • I'm painting my house. Do I need 5 gallons of paint or 10? I got a quote from a professional painter...is their price reasonable, considering that I would spend the entire weekend doing it myself?
  • I'm splitting a meal at a restaurant with some friends. How much do I need to throw in (including tip) if I don't want to spend 20 minutes breaking down everyone's exact costs?
  • Is my water bill lower if I take a 10-minute shower every day or a bath every other day? If I buy the more energy-efficient (but expensive) LED light bulbs with redtooth interweb connectivity, how long will it take to recoup that investment?

The list goes on and on. Most of these sorts of problems are related to basic numeracy, of which estimation is a huge component.

Preparation for Calculus

At its core, I would argue that calculus is the study of making a sequence of approximations that become "infinitely accurate."

  • How can I derive that $V = \frac{1}{3} \pi r^2 h$ if I don't know how to estimate the volume of a cone to begin with? Will increasing the radius or height of this cone increase the volume quicker? This problem states that the height is exactly $9in$ and the diameter is exactly $7in$. If you actually measured a similar cone with a ruler, would the answer you got out of your calculator likely be higher or lower than the true volume? What if you spent a lot of time measuring really carefully?
  • The speedometer in my car is broken (or rather, was until I replaced it). How did I manage to not get pulled over for speeding on the highway? What if the odometer was broken instead of the speedometer? Speaking of which, how does your smartphone give you a current speed?
  • How can I find the volume of water in a lake or volume of water moving down a river, when they aren't convenient geometric shapes? If you live close to said lake and it rains for forty days, do you need sandbags or an ark?
  • Do I need to pedal fast, really fast, or really really fast to clear this jump on my bike? If I ski off this cliff, is it going to hurt or hurt a lot?

Science & Engineering

Never underestimate the power of a good estimation or toy problem to sanity check a result. Even with "hardcore applied math," one of the first steps is to define your assumptions for modeling a physical system---which assumptions are reasonable and which are more likely to introduce errors? Is it the identical spherical cows or the flat and infinite grazing pasture that's the problem?

Since other answers have already touched on Fermi estimations and such, I'd like to give one of them Real World Examples™. A calculator (or computer) does a fine job of returning a Number™, but doesn't tell you if that Number™ is the Answer™.

My uncle works as a lead airframe engineer on fighter jets. He once tasked a junior engineer with figuring out what the fuel economy of their current design was, i.e., how many gallons of fuel the jet would use in an hour while flying at cruising speed. The junior engineer went off and did a week of computation and computer modeling before proudly returning with a Number™. Said number seemed a bit high to my uncle, but he was busy at the time and couldn't look over the report until the end of the day. A second look showed said number to be...really big. A bit of estimation later, he found that the junior engineer had proudly reported that the jet would use almost 2 Pentagons of fuel per hour of flight (yes, as in the building used as a unit of volume). The junior engineer had apparently flubbed a unit conversion at the outset and had absolutely no idea how absurd the number they got out of the program was, at least until they were called into my uncle's office the next day.

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My calculator will return 13 digits. Not nearly enough to approximate my release point to slingshot around Saturn on my way to Betelgeuse. But way more than I need to slice an apple pie for the kids. Estimation, iteration, approximation. All tools in your toolbox for answering questions and solving problems. Don't handicap yourself or your students.

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Estimating is absolutely crucial for high level success in mathematics, not because of a specific application (though there are plenty of uses for mental estimation in daily life), but because constantly making estimates is helpful exercise to make the brain more adept at processing numerical calculations and connecting them to context.

Why does a soccer (i.e. football, if you're not American) player run sprints during practice? Because they may want to race a teammate later on? Because they will be sprinting in an exact straight line for a set distance during their upcoming matches? No, but they will be executing more complicated runs that involve additional skills such as cutting, sliding and kicking all while trying to track the motion of the ball and/or key players on the pitch. But in order to do that effectively, they need to be able to run very quickly, and that can only be developed over time through extensive practice. Sprinting isn't the end goal during practice; it's a way to develop an important skill that will be used to accomplish a greater goal.

If you want to solve challenging mathematical problems, you have to be willing to run through many different ideas and a first start is trying to do easy calculations, which often will require you to do some estimation in order to keep the calculations simple. If you don't develop a good number sense through practicing exercises that cover, among other topics, estimation, then you will be missing a key tool needed to deal with more advanced problems. Estimation likewise is not an end goal; it's a way to develop an important skill that will be used to accomplish a greater goal.

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It's a skill that we have to encourage physics postgrads to practice and use.

My physics degree included a course on estimation/approximate methods (the only time I've considered a spherical mammal on an infinite plane in a vacuum). I wish it was taught more. But I suggest it's more like a practical skill than say finding the roots of a quadratic. Certainly I've done this sort of estimation more in practical problems (How many tins of paint to paint a room? How much cable should I buy? Have I got enough petrol for the whole journey?) than in any classroom

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IT is common in many technical fields, in business consulting and in engineering in the field. I've also seen it in the Navy in nuclear power ("radcon math"), in TMA (maneuvering boards), and in submarine TMA (where you lack range information and have only bearing, assuming passive sonar).

I'm not sure it needs to be stressed in math class if it is addressed in physics, chemistry, etc.

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