How to justify single digit answers to long problems?

Question

I have been teaching my student BODMAS and giving him problems like $$4\times4-6+8\div4+10\div2+5-4\times2$$

While solving the problems he asked me that, "So small answers for such big questions?"

I sadly had no answer other than "yes", but I sense that this might be interpreted as unfruitful in some students if not particularly mine, so is there any justification for this problem ?

Answer 1

A string of random multiplications, divisions, additions, and subtractions, even one involving single-digit numbers, isn't likely to have a nice, small answer. Ask your student to create some problems of his own and solve them. This will teach him two things:

1. that nice answers aren't typical,
2. that the teachers who created these problems actually took a great deal of care to ensure that the answers came out to nice numbers.

Then tell him that the reason they did that was so he could focus on learning order of operations without getting distracted by lots of hairy arithmetic.

Answer 2

Tell him that it is a common occurrence (esp. in science) that the complexity of the problem is a whole different matter than the complexity of its formulation. The issue here is with our perception, i.e. what seems easy or complex to us.

For example Fermat's last theorem can be worded very concisely, while its proof is very long and difficult. Similarly, a detailed answer to a short question like "why a plane can fly" involves a lot of physics, aerodynamics, flow equations and so on.

On the other hand some apparently long and tedious calculations might have simple answers, for example if some big number is even, or when the final result is to be multiplied by $0$, or calculating integrals of odd functions or limits with multiple insignificant terms, etc. It also happens outside math, e.g. in a restaurant when the waiter takes an order, the menu is actually a part of the question, yet the answer is probably relatively short.

The discrepancy between complexity of the problem and its formulation is rather common, so a more interesting question would be where our intuition comes from (i.e. why do we have it if it is wrong so often)?

I have no idea, but if I were to speculate:

• because of too little experience, with more experience we get better intuition (i.e. Fermat's last theorem wouldn't seem simple to an experienced mathematician);
• in ancient times $\text{bigger} = \text{stronger}$ was actually almost always true;
• human mind doesn't handle scale and complexities well, we mostly use thumb-rules that fail when any higher complexity is involved;
• in school students rarely face problems of high complexity, tasks are picked by the teacher and longer formulation often does mean more work.

I hope this helps $\ddot\smile$

Answer 3

The point of problems like these is not that a long expression can be reduced to a short one. Instead, the idea is that expressions need to be well-defined, i.e., non-ambiguous. To attain such a goal, one introduces the notion of an "order of operations"; for you, this means BODMAS, while those in the U.S. are probably more familiar with PEMDAS.

One way to get this point across would be to present a single expression and talk about different ways you could interpret it (i.e., depending on the order in which you carry out the various operations). You could then enter the expression into a calculator several times. Note that the calculator gives you the same answer every time, and ask the student if he thinks the calculator will ever give a different answer.

It won't. Why? Because the calculator is programmed to follow a particular order of operations; that way, two people with the same computation to carry out will not somehow end up with different results (assuming that they don't make any errors of their own). Just as there is an order of operations implemented in the calculator, there is one to which we subscribe, as well.

But the reasoning is the same: to eliminate ambiguity.

You might go further and ask him about different values an expression can take on depending on where you put the brackets (or, as those in the U.S. would say, parentheses). Even for a single operation: Does it matter where the brackets go? This can lead naturally to a discussion of binary operations for which associativity holds. For example, $a+(b+c) = (a+b)+c$ for any $a,b,c \in \mathbb{Z}$; that is to say, for the set of integers, addition is an associative binary operation. But what happens when the operation is subtraction rather than addition? Similarly, $a\times(b\times c) = (a \times b)\times c$ for any $a,b,c \in \mathbb{Z}$; that is to say, for the set of integers, multiplication is an associative binary operation. But what about division?

As to the student's astute observation: Expressions can be written in many ways. For example, the relatively "short" expression $3^{3^{3}}$ has only three numbers, but is equal to $7,625,597,484,987$. (If you raise it to $3$ once more, then you get an obscenely large number.) Meanwhile, you could write $1 - 1$ a bunch of times with a $+$ in between each of them; the sum would then be $0$. So here we could have a "long" expression that, when evaluated, gives a short answer.

The salient point is that a well-defined expression for your purpose refers to exactly one number. There are many equivalent ways to write a single expression; some short, some long. In fact, our decimal notation is really a way of abbreviating, so that, e.g., $365 = (3\times 100) + (6 \times 10) + (5 \times 1)$. In fact, I could have written the previous expression without brackets, given that BODMAS makes it unambiguous.

As a final note: Students are often asked to "simplify" expressions. The idea of such exercises is to remove ambiguity, and, the hope is, it will be clear for any particular class and teacher what constitutes a correct simplification. For example, many teachers ask that students "rationalize denominators" and re-write something like $\displaystyle \frac{1}{\sqrt{2}}$ as $\displaystyle \frac{\sqrt{2}}{2}$, thereby ensuring the denominator is an integer. Generalizing far beyond your student's query, one might ask whether this notion of simplifying is itself unambiguous. You can find more on this question in a MathOverflow post here, though I mention this more for the curious reader than as a matter of responding to your student.

Answer 4

I'd offer him another example, namely take 2 * (any 20 digit number) and show the result. But. When you divide the result by the original number you get 2.

Similarly, a string of +, *, and / can land you anywhere, a single digit, to many digits.

When looking particularly at * and /, it's not too tough to explain these are inverse functions, and Y*4/4=Y, so if you average equal numbers of multiplications and divisions, and the numbers you are using, you'll land close to where you started.

Answer 5

I like Will Orrick's answer in this thread. I'd like to propose a possible investigation for students to help them answer their own question of why "So small answers for such big questions?"

Begin with $2 + 2 = 4$ and ask:

Is it possible to make this expression on the left any length and still have the result come to 4?

Have a student or group of students add pairs (and/or triplets and/or singletons) of operations that return the result to 4 every time. You'll end up with a special case (in which each pair brings it back to 4) but perhaps your student or students, as a result of this investigation, will have some of their own conjectures about whether there is any necessary relationship between the length of the expression on the left and the result on the right.

For more variety in your pairs of operations, you could specify a range of single digit results, or $0...5$ or whatever. Instead of $4$.

Answer 6

Ask your student if he would rather have a long answer. If he says "yes," ask him why, and that should end the whole business.

Alternatively, you can point out that the question is actually quite short. It asks the student to simplify a single expression. One task, one answer. If the answer were "longer," it wouldn't be a good answer. Outside of the classroom, people ask questions for the explicit purpose of taking a long idea and making it short. That's why people hire other people to do math in the first place, because they don't have time to sort through the long stuff.

A related alternative is to explain that much of the point of mathematics is to find understandable order in complexity. Ask your student whether the long form or the simplified form is easier to understand and work with.

Answer 7

My background is in computer programming, so in that sense the mathematical operations, say:

4×4−6

Get transformed by a compiler into stack operations: Put 4 on the stack. Put 4 on the stack. Multiply the top two things on the stack. Store the new thing on the stack. Place a 6 on the stack. Subtract the last two things on the stack.

So when all is said and done, you'd have one thing left on the stack, your answer. I find myself wondering what your student would make of an RPN calculator. Or at least diagramming out everything that's happening in order to solve this problem?

Note too, that since all the operations are binary - each time you do an operation it takes two numbers and only gives you back one. Perhaps this distinction between a binary and unary operation would be useful to consider. Like the negation operation, -5 - takes a 5 and negates it. Whereas 3+5 takes two things and gives you back one. So if you have nothing but binary operators on integers, in the end you're just going to be left with a single answer.

Answer 8

This question is clearly not mathematically motivated. The student might accept a mathematical answer on a cognitive level, but it will not touch or him on the aesthetic or emotional level where this question comes from.

So give some non-mathematical analogy, for example:

Sports Let's compare two games: basketball and association football.

In Basketball with equally good teams nearly every move, effort and engagement will result in points. The more effort, engagement or moves, the more points for either team. So at the end of a game you may have 93 to 98 points (or whatever). In short: many moves and large results.

In association football with equally good teams, most moves, effort and engagement will not result in points. They have no obvious impact at all. Only a very small number of moves will result in goals and consequently points. So at the end of a game, you may have 1 to 0 points. In short: many moves and small results.

Math is so flexible, it can be anything: many numbers/operations and large results or many numbers/operations and small results.