Month: November 2014

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Mathematical Journeys: Subdivided Pie

Since it’s Thanksgiving week, let’s think about pie for a second. No, not mathematical pi, just actual real edible pies. For Thanksgiving I’m in charge of making dessert, so I’ll be bringing two pies, one pumpkin and one apple. Let’s say that I sliced the apple pie into 12 pieces, and the pumpkin pie, since it held together better, into 18.

Fast forward to the end of the evening. My pies were a big hit, and I have almost none left. In fact, all I have is three pieces of apple and four pieces of pumpkin. I want to combine the remaining slices into a single pie pan, so that they take up less space in the fridge. How do I figure out if my remaining pie will fit in one pan?

Well, let’s start by writing down the remaining amounts of pie in the form of fractions. Remember, one of the definitions of a fraction is parts of a whole, so let’s apply that definition to figure out our starting fractions.

The apple pie was cut into 12 pieces, and we have three out of twelve left. So our apple pie fraction is

 3
12

The pumpkin was cut into 18 pieces, and we have four left. So the pumpkin fraction is

 4
18

To figure out how much pie we have left and whether it will fit into one pan, we’ll have to add these two fractions together.

 3  +  4
12    18

Now, the first problem we run into is that our pies were cut into different amounts of pieces. We can’t accurately compare the two pies until we subdivide them into the same size pieces. So to do this, we’ll use a process that often confuses students when they first learn it: finding the Lowest Common Denominator.

I think this concept confuses students mostly because the terms used are long and complicated. So let’s talk about Lowest Common Denominator for a moment. We only specify Lowest because there are many different denominators that would work for this problem, but most of them are bigger than they need to be. We don’t want to work with numbers that are any larger than they need to be, so this process ensures that we don’t have to do any extra reducing after the fact.

We need to find a number that can divide evenly by both of our current denominators (though not necessarily at the same time). So all of the factors involved in either of our current denominators need to be present in our LCD. Let’s take the foolproof, long-way-round method and break each denominator down into its prime factors. You can do this with a factor tree:

12                18
3 x 4            3 x 6
3 x 2 x 2      3 x 3 x 2

So we see a lot of overlap here between the two denominators, right? Both contain a 3 and a 2. This is where the “lowest” part of the Lowest Common Denominator comes in. Worst case scenario, we could simply multiply each denominator by the other one and we’d come up with a number that both would divide by, right? But look at that overlap. That denominator might be common, but it certainly wouldn’t be the lowest – you’d be able to divide both top and bottom of each fraction by a 3 and a 2 (or by a 6, if you’re feeling efficient). We need all of those factors to be represented in the LCD, but we only need the duplicates to be represented once. What I like to do is rewrite the last lines of the factor trees one on top of the other, to make it easier to see the overlap:

12 =        3 x 2 x 2
18 = 3 x 3 x 2

So that 3 x 2 of overlap we only need once. So let’s rewrite out the prime factors that we need in our LCD:

LCD = 3 x 3 x 2 x 2

There. We have two 3’s, so the 18 will be happy, and two 2’s, so the 12 will be happy. What does that come out to?

LCD = 3 x 3 x 2 x 2 = 36

So our target number is 36. Now, let’s go back to our original problem:

 3 + 4
12  18

Now, remember that any number divided by itself is one? Good. And remember that one of the definitions of a fraction is a division problem you don’t want to do yet? Great. So we can multiply either of these fractions by a new fraction composed of a number over itself, and not change the actual value of the number. We’re simply rewriting it in a form that’s more useful to us. It’s basically the backwards application of the process for reducing a fraction. Only this time we’re not reducing; we’re making it more complex.

We’re going to do this in two separate cases. Just look at the first fraction for a moment. What number do we need to multiply the 12 by to get our target of 36?

If you’re stumped, just go back to the prime factor makeup of our target number and figure out what pieces are missing from the 12:

12 =           3 x 2 x 2
LCD = 3 x 3 x 2 x 2

So we’re missing a 3. So multiply that first fraction by 3/3, which, remember, is just a fancy way of writing “1”, so we can do that without messing up the value of the fraction:

3    x    3 =  9
12        3     36

Okay, good sign. Our denominator is 36, which was our target. Let’s take the second case. Look at the second fraction:

4
18

18 =     3 x 3 x 2
LCD = 3 x 3 x 2 x 2

So we’re missing a 2 here. Multiply by 2/2, which is just a fancy way of writing “1”:

4   x   2 = 8
18      2    36

Excellent! Hang on, we’re almost done. Now, what we’ve essentially found here are equivalent fractions for each of our originals that just happen to have the same denominator. Well, they don’t really “just happen” to have them; we did that on purpose to make our lives easier. So now we just plug these new forms of each fraction into the original problem:

9  +  8
36    36

And we have our pies subdivided into the same size pieces! Now we can add them normally to figure out if we have less than a whole pie’s worth of leftovers:

9   +   8 =    17
36    36       36

17 is less than 36, so we have less than a whole pie. We can safely put them both in the same pan without overflow.

Now if I could just figure out how to fit that turkey in the fridge…

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Literature Spotlight: Appearances Deceive

War of the Worlds, by H.G. Wells, is classic science fiction. Written in 1924, it depicts the catastrophic and totally unexpected near-extinction of humanity by aliens from Mars. One of the main themes running through War of the Worlds is the idea that mankind’s assumptions about their world, the universe and the nature of life are constantly being challenged. The main reason the martians’ landing is so catastrophic to humankind is because the humans, by and large, have been lulled into a false sense of security. They believe they are capable of overcoming anything, that they are the most powerful beings in the universe, and as such are completely unprepared for the martians’ attack.

Humans at the beginning of H.G. Wells’s novel are portrayed as very self-satisfied. Even when confronted with the landing of the first martian cylinder, humanity is quick to dismiss the event as a mere curiosity. The story on the eve of the first day was “dead men from Mars,” (P. 14) and villagers from the area headed to the commons to see the cylinder as if it were a sideshow attraction. Almost immediately they are in over their heads and their assumptions are being proven wrong. The martians look nothing like humans, as everyone was subconsciously expecting them to. After this initial shock wears off, though, humanity quickly regains its mis-guided sense of security. The gravitational pull of earth is much stronger than the aliens’ native Mars, and they are sluggish in the atmosphere. Seeing this humanity continues to maintain a smug attitude about it all, claiming with certainty that there is no way the martians can get out of their pit. It never occurs to anyone that, just as humans have built technology to compensate for their weaknesses, the aliens might have done so as well. The reveal of the “fighting-machines” and the Heat-Ray throw this assumption into chaos, as the martians begin to slaughter the humans with no more thought than we might have to stepping on an anthill.

Stepping on an anthill brings up another facet of this theme: War of the Worlds is full of imagery depicting the humans as insects under the martians’ feet. The artilleryman states that “It never was a war, any more than there’s war between man and ants.” (P. 238) This imagery emphasizes the power the martians seem to hold over humanity, and humanity’s inability to deal with them at all. After all, how would an ant colony retaliate against a bulldozer? This imagery also points out the idea that humans barely register the existence of insects, just as the martians must barely be registering the humans’ attempts at retaliation. The introduction of the black gas furthers this theme by bringing up fumigation imagery; the idea that the martians spread a noxious cloud of gas across the land, then use jets of steam to disperse the gas itself, is reminiscent of humans smoking out a wasp nest or any other unwanted infestation. This imagery reduces all the splendor of humanity to a mere nuisance, something that must be dealt with to make the planet livable for the martians.

Fueling this theme is the fact that the humans actually do begin to display animalistic behavior in the wake of the martians’ destruction. Houses are broken into and looted, morals are abandoned, and it becomes every man for himself. Even our narrator succumbs to these animalistic tendencies, in a powerfully-moving scene where he murders an insane companion to keep from being discovered by the martians. This scene shows just how precarious our position really is, and how little it takes to unseat even the most sturdy of morals. Seeing humanity’s reaction makes it practically unthinkable that they could have resisted even if they knew what they were up against.

This theme comes full circle when the final destruction of the martians comes not at the hands of humans, but of the lowest of life forms – bacteria. In essence, what humanity could not do, the common cold did. This further emphasizes the powerlessness of the humans – the martians would have perished regardless of what humanity did to stop them. They were taken out by an entirely different force, and so humanity doesn’t really have a role in this drama at all. You could remove all of the humans and the story would have played out in much the same way.

Despite this, though, War of the Worlds has a truly hopeful theme arise in its resolution. Throughout the novel there are descriptions of the sheer size of the martians’ fighting-machines that highlight the futility of fighting something so much larger. But the destruction of all-powerful aliens by microscopic organisms reminds us that size does not equal power, and that the smallest of beings can still have a crucial impact. It also reminds us not to judge based on appearances or assumptions, and to keep our hubris in check. For in the vastness of the universe there are bound to be thousands of species more powerful than us, but that does not mean we are powerless to fight them.