Last week, in Fundamental Understanding of Mathematics LIII, we solved some word problems using our balance model. I thought I'd continue solving problems this week, but with a bit of a different spin. Let's put our balance model through it's paces.
One thing you may have noticed about last week's problems is they all had "answers." That is to say, the problem asked us to figure out some specific number and we found that number. One frustration students have with algebra is that not all algebra problems have such answers.
In three years, John will be twice as old as Matthew. How old is John?
Let's use our banana to represent John's age, and the apple to represent Matthew's age.
In three years, John will be
years old
and Matthew will be
years old.
The problem tells us that John's future age is twice Matthew's future age
We will use the distributive property of multiplication over division [ a(b+c) = ab + ac ]to get rid of the parentheses on the right hand side.
then we use our "subtract the same thing from both sides" rule to subtract 3 in order to get the banana (John's present age) by itself.
Now what? Well, the truth of the matter is: we're done. We have the banana, alone, on one side of the scale, and that banana represents John's age. We have what it is equal to (twice Matthew's age plus three) on the other side of the scale. That's the answer: John is twice Matthew's age plus three years.
Beginning algebra students get very frustrated with this kind of answer. "But I don't know how OOOOLD he is!" they wail. Or they think it's a trick question: "There is not enough information to answer the question," they pronounce smugly, having figured out the trick, anticipating bubbling in "none of the above" on a standard test form.
But we do have enough information to answer the question: John is twice Matthew's age, plus three years.
It is perfectly acceptable to answer a math question about the value of an unknown number in terms of another unknown number. It only seems unreasonable because if you went up to someone who knew John and asked, "How old is John?" John's friend would say something like, "Oh, John? He's fourteen." or "He's twenty-seven," or "That old fart? Going on sixty three, I reckon."
In common, everyday conversation, that question would be answered with a known number.
In a mathematical conversation, though, that question is answered by getting the unknown, by itself, on one side of an equal sign.
Some might object that we solved a similar problem last week, but got a known number for an answer. If you recall that problem, though (or if you go back and take a look at it now,) there was more information given in the problem statement. We had two unknowns, but we also had enough information to write two separate equations.
Let's see how that works:
In three years, John will be twice as old as Matthew. How old is John?
This problem statement gave us only one equation.
Let's add something to our original problem:
John is five years older than Matthew. In three years, John will be twice as old as Matthew. How old is John?
Now we have another bit of information relating our two unknowns: Matthew's age plus five is John's age.
Now that we have two equations, we can replace the banana in this equation with the equivalent apples plus number from the other equation:
Then we can solve for the apple and find Matthew's age. Subtract an apple and three from both sides to get:
Matthew is two years old. John is five years older, so he is seven. In three years, Matthew will be five and John will be ten, twice Matthew's age. All the problem conditions are satisfied: John is seven years old.
We start to see a pattern here. When we had a problem with one unknown number, we needed one equation to find the unknown number's actual value. When we had a problem with two unknown numbers, we needed enough information to write two equations in order to fix the values of both unknowns. With only one equation, the best we could do was give one unknown number in terms of the other unknown number.
One unknown -- one equation. Two unknowns -- two equations. It's suggestive, but too soon to tell what kind of pattern this might be. The obvious pattern is + 1 -- +1, that is, add an unknown to the mix, must have another equation to find fixed values. But there are other patterns that also fit: +1 -- x2 and x2 -- +1.
Maybe when we add one unknown, we need to double the equations. Or, maybe when we double the unknowns all we need is one additional equation. All three patterns fit the data we have so far. Which is correct, if any? Is the next step three unknowns, three equations; three unknowns, four equations; four unknowns, three equations; or something else entirely? Time will tell, or someone will write about it in the comments.
As long as we are talking about comments: have fun down there.