Last week, in Number Sense 025, we took up that classic word problem of two trains passing in the night. Upon reflection, jumping so far ahead was a mistake. So I'm going to pretend it was just a one-of-a-kind lump in the pudding, and continue on from Number Sense 24, where we completed a model of an addition problem, and discovered that we could model addition fact families with a sort of abstract version of a number line.
In teaching (actually reteaching) fact families as a springboard into modeling and then writing equations from word problems, I was struck by the difficulty some of my students have with interpreting the equal sign properly. Follow me over the Kroissant.
The initial project was, given one of the facts in the fact family, to write the other three. As a refresher, a "fact family" is two addition facts and two subtraction facts that are related in that they all use the same numbers.
For example:
1 + 2 = 3
2 + 1 = 3
3 - 2 = 1
3 - 1 = 2
is probably the simplest fact family around.
So, I give students an addition fact, say, 5 + 4 = 9, and expect them to come up with the remaining three members of the "family."
Once the initial confusion over why 4 + 5 = 9 is a different addition fact than 5 + 4 = 9 (because 4 + 5 is the same as 5 + 4, so it's the same fact!!!) students quickly caught on that simply switching the first two numbers gave them another fact family member. This simple pattern, though, breaks down when subtraction is involved.
They would find one or the other correct subtraction fact: 9 - 5 = 4 or 9 - 4 = 5. But then, in trying to find the second subtraction fact, the first mistake was in simply using the pattern that worked for addition: switch the first two numbers.
9 - 5 = 4
5 - 9 = 4
oops!
Here's where I think it gets interesting. First I point out that 5 - 9 is not equal to 4. What they wrote only fits part of the fact family bill. Yes, it uses the three family numbers, and yes, it is a second subtraction, but no, it isn't an actual fact.
I don't tell them it is wrong because you can't subtract a large number from a small number, besides giving away the pattern, that isn't true. You can subtract a large number from a small number. A few students will recall negative numbers, and say, well then, it's 5 - 9 = -4. Well, that is a fact, but now we are not using the original three family numbers. -4 and 4 are on different spots on the number line, so they are different numbers. Our fact family uses 4, not -4.
There are two common attempts at solutions to this new puzzle. Either
9 - 5 = 4
9 - 5 = 4
as the two subtractions, or
9 - 5 = 4
and 15 - 9 = 4
You'd think that this second attempt would be immediately disqualified since 15 is definitely not in the original three numbers, but there is an overriding idea that makes 15 - 9 = 4 the only possible alternative subtraction fact.
If 9 - 5 = 4 is a true member of the 9, 5, 4 fact family, then, for the subtraction problem, 4 is the right answer!
Therefore, for the second subtraction fact, 4 must also be the right answer, in order for that second fact to be true, as well.
This is because the equal sign has taken on a non-mathematical meaning in elementary school, probably as a result of years of addition and subtraction handouts and problem sets with the form
1 + 2 = __ 2 + 5 = __ 3 + 7 = __
The operational definition of
= __.
is
answer goes here =>___.
And, of course, once you have the right answer, you don't change it, because any other answer would be a wrong answer.
Let's take another look at our model from Number Sense 24.
Here are three numbers, drawn originally on a number line. The length of the bar represents the number. There is a large number, the green bar. A small number (bottom right) and an in-between number (bottom left). The two bars on the bottom just touch (no overlap), so their total length is a sum of the two numbers.
When we place our green number and the two number sum as we did above, it is easy to see that the overall lengths are the same. We use an equal sign to show this sameness.
It could also be stated this way:
because
it doesn't really matter which comes first, and which comes second. The lengths (which is what we are making a statement about) don't change because we changed their order.
Notice, though, that there is no “answer.” The equals sign simply lets us make a statement about the lengths of three bars: The brownish bars' lengths, added together, are the same as the length of the green bar.
There is no 'answer'. There are only statements about numbers that are true.
It is this shift in thought that allows us to move beyond arithmetic to do algebra. Elementary school arithmetic teaches about counting and quantity, about combining two numbers to find a third number, using addition, subtraction, multiplication or division, sometimes exponents. In elementary school arithmetic there is indeed an answer. Students are given two numbers, assigned an operation, and they apply an algorithm to find that answer.
In algebra, though, we must give up our search for an answer and focus on a search for truth. In algebra, we are given, not numbers to combine, but true statements about numbers. Our job as mathematicians is to manipulate or combine those true statements to produce other true statements.
We already produced two true statements from our original first true statement. Let's find some more.
Here is another one, produced by changing the order of the top two bars, the darker brown bar comes first, which gives us:
As we did earlier, we can start with the green bar on top,
and get
I think that exhausts the possibilities of making true statements from that model using addition. Let's try subtraction. Subtraction involves taking away an amount. If we begin with our model...
Break off a piece of the green bar that is exactly the same length as the bar below it
And then take those pieces away
We are left with a piece the same length as the other short bar.
In other words (without the words)
I'd like to show this in a slightly different way, beginning with our equation, rather than the bars, side by side.
This is one of our true addition equations. Now I am going to appeal to your sense of balance. If we take the same amount away from both sides of the equal sign, will it still be a true statement? (Notice that is what we did in the first example, when we broke off a length of the green bar, and then removed that green piece from the top, and removed the light brown piece with the same length from the bottom)
If I can do that (spoiler alert: I can) then we have this situation:
(Ok, I had to shrink those bars to make them fit on the page, so they don't match the earlier pictures)
Now, let's take a look at the right side of that equation for a second
It starts with a short, dark brown bar.
Then I add a longer light brown bar and then I take the same longer light brown bar away!
What am I really adding to that short dark brown bar? Nothing! Nada! Zero! Ziltch!
So, adding nothing to the short dark brown bar, I'm left with just the short dark brown bar.
So, let's reattach the right side of the equation, after realizing that all there is left is the short dark brown bar...
This is the same diagram as the one we got earlier, so that should also help convince us that what we did in manipulating the original true statement results in another true statement.
We can find a second true subtraction statement by taking the short bar away from both sides
To recap:
I set up a true statement, an equation, using a number line bar model
Through various manipulations, I produce other true statements, or equations, about the same numbers.
I could produce four more by arranging to have the green bar on the right side of the equal sign, but it wouldn't really add anything to the discussion.
So I began with one true statement about the lengths of three bars, and wound up with four true statements about those bars. Sounds remarkable trivial, tedious, simpleminded and not too useful, doesn't it?
But let's suppose that I didn't start out with three bars, carefully selected to fit together neatly in our model. Let's suppose I only know two numbers and the third number was...
a
Now our set of statements looks like this
How can we tell whether these are true or not. They're full of goats! A goat doesn't fit neatly on a number line. (A goat would probably start eating the number line...)
Well, one thing we do know about those statements is that, if any one of them is true, then the rest of them are also true.
So if we are told: taking the goat away from the green bar is the same as the brown bar (the last diagram) then we know the other three diagrams are also true statements.
If we are told: The green bar is the sum of the brown bar and the goat (the first diagram) then we know the other three diagrams are true.
Once we have that bit of assurance that at least one of these statements is true, then we know the rest are true.
This can be useful, because in one of the diagrams, the goat is alone on one side of the equal sign, and on the other side are two known numbers and an operation. Hey, two known numbers and an operation? That's arithmetic! We know how to do that!
So there is the algebra mindset in a nutshell. We must give up our search for the answer, and, by manipulating true statements in ways that produce other true statements, explore the universe of true statements. Until, at journey's end, we discover a true statement that sets the unknown equal to some bit of arithmetic.
And arithmetic will let us find answers again.
Have fun in the comments. (Now where did that goat get to?)