Word Problems

"Adam is driving 35 miles per hour to the east while Jane is walking 4 miles per hour to the west. If they meet after ..." "Why doesn't this make any sense to me" I pondered as I watched my high school freshman math teacher, Mr. Plesa, lecture about word problems.

Throughout high school, I had a very difficult time parsing those sentences into numbers and vice versa. But why? Was it the learning disability akin to dyslexia but for numbers, dyscalculia?

Why did every word problem look like this to me?

Well, I don't think I have or had dyscalculia (nor Julia Gulia for that matter). Chances are, I was just "rubbish" at math. Even though I was being taught by one of the best teachers I had ever known, the gaps in my mathematical skill set were just too numerous to comprehend these problems.

Take a look at this problem from Khan Academy:

If you don't know where to start to solve a problem like this, then don't fret. Like Michael Jackson said, "You are not alone"(That's your earworm for the day, enjoy it).

There's a few ways to approach this problem and all start with working with the basic equation of motion, $\mathrm{distance = rate\times time}$. Which means that $\mathrm{rate = \frac{distance}{time}}$ and $\mathrm{time = \frac{distance}{rate}}$

Take, for instance, the solution steps from the Khan exercises:

I still have trouble making the problem less abstract by thinking about it this way.

Contrast this to the Udacity approach (via their Visualizing Algebra course) of setting up a table; remembering that $\mathrm{time = \frac{distance}{rate}}$ and we are using $t$ as the variable for distance traveled by train:

We are given that the total time is $\mathrm{8 \; hours}$,  so we can set up the equation as:

$ \mathrm{(time\; by \; train)+(time\; by\; car)=(total \;time)}$

$$ \frac{t}{40} + \frac{304-t}{24} = 8$$

The prime factorization of 40 is $2^3 \cdot 5$ and of 24 is $2^3 \cdot 3$, so the least common multiple (LCM) is $2^3\cdot 3\cdot 5 = 8\cdot 15 = 120$.  So we'll multiply the entire equation by 120 to get rid of the denominators and leave us with much easier to work with integers:
$$  \frac{t}{40}\cdot 120 + \frac{304-t}{24}\cdot 120 = 8 \cdot 120 \Longrightarrow$$
$$  3t + 1520-5t = 960 $$
And now we can solve for distance traveled by train, $t$:
$$ 560-2t = 0 \Longrightarrow$$
$$ 560 = 2t \Longrightarrow$$
$$ t = 280 $$

So we find that the distance traveled by train is equal to 280 miles.  We got to the same answer but took a slightly different approach. (If you don't believe me that $t$ has the unit $\mathrm{miles}$, try working out the equation with the units as a challenge.)

Perhaps somewhat strangely, I find it much easier to start tackling problems like these with the table method. This, despite the fact that the Khan approach seems less complex to me with regards to the actual algebra (There are no fractions so no need to bother finding a LCM). Yet, by using the table method, I can instantly begin working on the problem without having much difficulty in translating the words to an equation. Please let me know which one you find easier to conceptualize.