Last week, in Fundamental Understanding of Mathematics LIV, we took a look at a word problem that didn't have a neat known number as its answer, and came up with a known number answer by adding more information to the problem statement. In the real world of problem solving, this would be the equivalent of digging around for more information; doing another, slightly different experiment; conducting another survey.
Sometimes, that's not possible. Sometimes, there are many answers to a problem. How do we deal with that?
Last week we asked about John's age, and found out that it was twice Matthew's age plus three years.
When we added an additional fact, it turned out that John was 7 years old. But let's suppose this was John and Matthew Doe. The original problem didn't tell us which John and Matthew we were talking about. Maybe it was John Smith and Matthew Smith. If Matt Smith is 22, then John Smith is 47.
In fact, given our original problem statement, Matthew could be any age, and there would be a corresponding age for John, that we could figure out by multiplying Matthew's age by two then adding three years.
Suppose Matthew was three years old, then John would be twice three, or six, plus three more years, or nine years old.
Suppose Matthew was five, then John would be 2(5)+3 = thirteen years old.
And so it goes. If we set Matthew's age to some specific known number, then John's age is some other specific known number.
So, for the purposes of this problem, those two numbers go together. Two and seven. Three and nine. Five and thirteen. What these pairs of numbers have in common is their relationship, which is set up by the problem statement. The other significant thing about these numbers is that they are related to the problem in a specific way: Two is Matthew's age, seven is John's age, and not the other way around.
In order to save some writing, we generally write the numbers in order. For a particular problem, like this one, the order is arbitrary, but we must state it anyway, so people can understand our solutions. We create a set template (Matthew's age, John's age) so when we write the numbers (2, 7) we know the first number corresponds to Matthew's age and the second number corresponds to John's age. This lets us condense
If Matthew's age is 2 then John's age is 7
If Matthew's age is 3 then John's age is 9
If Matthew's age is 5 then John's age is 13
to
(Matthew's age, John's age)
(2, 7)
(3, 9)
(5, 13)
Less writing, but we need to learn how to read the abbreviated notation. In typical math textbooks, the authors take an additional shortcut. When they use letters like a and b, or x and y for the unknowns, the template is assumed to be in alphabetical order -- (a, b) or (x, y). Sometimes they mention this, sometimes they assume you already know it or you'll figure it out on your own.
So, one way we deal with many answers to a problem is to invent this abbreviated notation for writing many answers. We call these ordered pairs, because they are pairs of numbers, and they are in some kind of order. Another way to write these answers is to put the ordered pairs into what's called a
T-chart, because the lines we draw to make the chart look something like the letter 't'.
But what about putting the numbers on a number line? This is a tricky problem, because the numbers are related to each other. It's not just "2" it's "2 with an associated 7." If we use a number line, all we are showing is a 2 and a 7. The number line doesn't relate those two numbers. One solution is to use geometry that does relate two numbers. Something like, say, a rectangle, which has both a length and a height. To keep the two numbers in order, we would have to insist that the length always be one of the numbers, and the height be the other. So, let's build rectangles, using John's age for the length, and Matthew's age for the height.
Now we have something we can put on a number line.
We put the rectangles on the number line so the length is the distance along the number line (John's age) and the height is how high the rectangle rises above the number line (Matthew's age.)
We don't put the rectangles side by side, rather we put them all with their lower left hand corner on zero. Why? Because the bottom side of the rectangle represents John's age, and John's age always starts at zero. Remember, we calculated that John is 7, or 9, or 13, or some other number. There's no reason that those different ages should be added to any of our other ages for John, and if we put the rectangles side by side, we would be adding one John's age to another John's age. That doesn't make sense. So we don't do it. The rectangles all start with their lower left corner on zero.
But how high are they? What is Matthew's age? I knew the number when I built the rectangles, but that information got lost when I put the rectangles on the number line. Let's put that information back in.
This is typically how heights are shown in bar graphs, and it works here. But we might anticipate a more crowded set of numbers, which would make this hard to read. So lets slide those height numbers over to the side.
Hmmm. Looks like the beginnings of another number line. Only this new number line goes up and down instead of left and right.
One of these numbers represents Matthew's age, the other one represents John's age, and it might be a good idea to put that information on our drawing, too.
Now we have all the information about John's age and Matthew's age, but we have some extra stuff, too, that we could get rid of to make the drawing simpler, maybe easier to read. Can we get rid of the rectangles? All we really need is the upper right corner, that tells us how far the rectangle extends to the right, and also how high the rectangle is: that corner tells us John's age and Matthew's age for that rectangle. Let's see what it looks like with a dot at the upper right corner, and no rectangle.
Well, the corners of the rectangles seem to be in the right place, but without the rectangle sides, it's hard to tell exactly where those corners are. Perhaps we should extend the marks on the number lines across the drawing, so we can read where those dots are at.
That's better. There's a lot of set up, but the actual notation for each ordered pair of numbers is pretty simple: a dot at a particular place on the drawing. One dot represents John age 7, Matthew age 2. Another dot represents John age 9, Matthew age 3. The third dot represents John age 13, Matthew age 5.
Now, it may be an optical illusion, but it seems as if those dots line up. Let's see if they do.
They do line up. Is that useful? It turns out it is useful, because if we calculate other possible ages for John and Matthew, they are also on that line.
For example, if Matthew is one, John is twice one plus three, or five. Sure enough, that dotted line crosses the intersection of Matthew age one and John age five. If Matthew is four, John is eleven. Yes, another match with the dotted line.
In fact, all possible ordered pairs of numbers that come from our problem solution (John is twice Matthew's age plus three years) are on that line, if the line is extended far enough.
So we can represent all possible answers to a problem that has, as a solution, a relationship between two unknown numbers, simply by drawing a line.
Have fun in the comments.