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# The Number-Line Puzzle

November 12, 2017

Have you ever worked on a thousand-piece-puzzle? When I was doing one, I felt getting started on it a little daunting: not knowing where to start; randomly trying to put some pieces together until an image started coming together. When I finished, I couldn’t stop staring at it. As if, I had created a masterpiece. I felt proud.

Reflecting on my experience solving the puzzle, I realized that what kept me motivated were the small moments of success I found in connecting parts that I had been stuck on. Often that involved physically changing my perspective; looking at the same puzzle from a different angle.

One might argue that math is like solving a puzzle, too, with similar frustrations of getting started and feeling stuck along the way. However, enjoying math doesn’t mean you have to solve a new problem or even work on a very complicated problem. Sometimes, re-imagining and rethinking how to approach an existing problem can give you completely different insights. The problem can be even as simple as adding and subtracting numbers

Let’s imagine subtracting numbers. Take the subtraction problem (3) - (5) for an example. In how many different ways can you show that the answer is (-2)? Have you ever used a number-line to show addition and/or subtraction? If we show (3) - (5) on a number-line, we would start at 3, but then how would we show subtraction? We may consider the idea of subtraction as “moving back”, “losing”, or “taking away”; in all those cases, we could say that subtraction means to move left of the original number.

But, does the analogy of moving-to-left always work when subtracting numbers? Let’s consider another problem (-1) - (-3). How would we model this problem on a number line while continuing with the analogy of moving-to-the-left? In the visual below, we see that moving 3 numbers to left gives us a misleading answer: (-4). However, if we move 3 numbers to the right, we arrive at the correct answer: (2).

You might already know with your prior knowledge that (-1) - (-3) = (-1) + 3 = 2 and not -4. In other words, we know that subtracting a negative number is the same as adding. But still, how do we explain that on a number line? How do we explain the change of the direction moving away from the original number?

Since the concept of direction doesn’t work very well in all examples, we could use a different explanation. For example, in our problem (-1) - (-3), we know that our starting position is (-1) and that after we are done with the subtraction, we should get another number on the number-line. However, to get the three negatives, we must take away the three negative intervals after (-1). Furthermore, if we go to the right of (-1), the interval between each consecutive number is negative 1; whereas, if we go to the left of (-1), the interval between each consecutive number is positive 1.

Now that we have seen the concept of interval differences, we can demonstrate and evaluate the problem (-1) - (-3) on a number-line as follows:

Using the visual of number-line for addition and subtraction gives us a new insight:

When adding or subtracting, we don’t add or subtract the numbers to the left or right of the starting number, but rather, we add or subtract the intervals between the numbers.

The great thing about concept of adding and subtracting intervals is that it is not limited to subtracting negative numbers only. We can apply this concept to the whole range of integers (positive and/or negative numbers). Let’s consider the example 5 - 3. Using the interval explanation, we could show the problem as:

Using the number-line really helps us investigate more deeply about what it means to add and subtract numbers; however, that is not the only way to think about addition or subtraction. Do you have any strategies that help you think about a math concept differently? We would love to hear from you.

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