*By William McCallum*

**You will never have to subtract again**.

Students sometimes learn about addition and subtraction of integers using integer chips. These are circular chips, with a yellow chip representing +1 and a red chip representing -1. You start with the all-important rule that

$1+(\text{\u2013}1)=0$

, so you can add or remove a red-yellow pair without changing the number. To calculate the right hand side of the equation in the title,

, you put 3 yellow chips together with 5 red chips, then remove 3 red-yellow pairs, leaving 2 red chips. So

.

How do you calculate the left hand side, $3 – 5$? You want to take 5 yellow chips away from 3 yellow chips, which you obviously can’t do, so you add 2 more red-yellow pairs, then take away the 5 yellows, and you are again left with 2 reds, so $3 – 5 = \text-2$. So $3 – 5 = 3 + (\text-5)$.

I can see the value of integer chips in making calculations concrete, but I see some significant problems with them as well. One is that the step of removing or adding red-yellow pairs, although easy to do, is conceptually sophisticated and a little mysterious. Another problem is that integer chips don’t generalize well to working with rational numbers. Yet another is that although integer chips can be used to show *that* the equation in the title is true, they don’t leave a record of the calculations that enable you to reason about *why* it is true. Maybe it was a coincidence that it worked out with the numbers 3 and 5. There is a danger that “subtraction is addition of the opposite” could become a meaningless mantra. Finally, integer chips are problematic when you start thinking about contexts for negative numbers because in many contexts negative numbers represent the absence or removal of things, whereas red chips are clearly present things. See, for example, the discussion of contexts in Nik Doran’s blog post.

In order for students to see why the rules for operating with negative numbers are true, the IM 6–8 curriculum introduces negative numbers using number line diagrams. This makes sense because grade 6 students have been working with the number line for a few years and have moved on from discrete representations of addition and subtraction (which integer chips push them back to). In a number line diagram, negative numbers are just points to the left of zero. We don’t really care anymore if they are integers. To think about how to add and subtract them, let’s go back to adding and subtracting positive numbers. To represent $3 + 5 = 8$ I put line segments of lengths 3 and 5 on a number line diagram and end up at 8. (Notice that I have snuck some arrowheads onto the segments for reasons which become clear in a minute.)

Note that this diagram can also be used to represent $8 – 3 = 5$ because $8 – 3$ is the unknown addend in $3 + ? = 8$, that is, $8 – 3$ is the number you add to 3 to get 8. The diagram shows me that that unknown addend is 5. In fact every subtraction problem can be represented by an addition diagram. I’ll reveal the awesome power of this observation in a moment, but first, how do we represent addition of negative numbers? This is where the arrows come into play. Positive numbers are represented by arrows pointing to the right, negative numbers by arrows pointing to the left. So $3 + (\text-5)$ is represented the same way as $3 + 5$, except the arrow for -5 goes to the left, ending up at -2.

Addition diagrams always work the same way. You put an arrow from 0 for the first addend, then you put an arrow for the second addend starting at the tip of the first arrow, then figure out where you have landed on the number line.

What about subtraction? Remember (drum roll) that every subtraction problem can be represented with an addition diagram. Let’s try this out for $3 – 5$. The addition equation corresponding to $3 – 5 = ?$ is $5 + ? = 3$ (“3 – 5 is the number you add to 5 to get 3”). The addition diagram for this would start with an arrow to the right for 5, and would land on 3. What is the arrow that gets from the tip of the arrow for 5 to the point at 3? It is the arrow for -2.

Okay, that one was easy. What about the bête noire of all students, subtracting a negative number? Let’s try $3 – (\text-5)$. The addition equation corresponding to $3 – (\text-5) = ?$ is $(\text-5) + ? = 3$. In this case the missing arrow goes all the way from -5 to 3, so it is the arrow for 8.

I understand why a teacher might prefer integer chips to addition diagrams. Once you have learned the rules they are pretty simple. Addition diagrams may be more difficult initially because you have to think about how to express the subtraction problem as a missing addend problem when using them. But that thinking pays off later in a more durable understanding of operations with rational numbers (“$a-b$ is the arrow I add to $b$ to get $a$”). Integer chips are a way of doing; number line diagrams are a way of thinking.

**Next Step**

Question: How can you use addition diagrams to show why $3 + (\text-5) = 3 – 5$?

##### William McCallum

Bill McCallum, founder of Illustrative Mathematics, is a University Distinguished Professor of Mathematics at the University of Arizona. He has worked in both mathematics research, in the area of number theory and arithmetical algebraic geometry, and mathematics education, writing textbooks and advising researchers and policy makers. He is a founding member of the Harvard Calculus Consortium and lead author of its college algebra and multivariable calculus texts. In 2009–2010 he was one of the lead writers for the Common Core State Standards in Mathematics. He holds a Ph. D. in Mathematics from Harvard University and a B.Sc. from the University of New South Wales.

I like the idea of using a related addition to figure out the answer to a subtraction. I’m not convinced that it explains why

a-b=a+(-b) better or more convincingly than direct modeling with chips.

I use zero pairs in a slightly different way than what you describe.

To model a-b

1) Count out |a| chips of the color corresponding to the sign of a.

2) Add |b| sets of zero-pairs. This is a + |b|*(0) which still represents a.

3) Now take-away |b| chips of the color corresponding to the sign of b. Take these from the zero-pairs in step 2).

4) What do you have? You have the original |a| chips in the color of the sign of a, and you have |b| chips in the OPPOSITE color of the sign of b. In other words, we have a+ (-b).

For a number line approach I explain it this way.

a+b

1) Start at a.

2) Face right.

3) If b is positive, walk forward |b| steps.

If b is negative, walk backward |b| steps.

For a-b

change step 2) to face left.

What students find convincing may depend on their prior knowledge. Do they understand subtraction as “take away” only or also as “compare” (2.OA.1)? Do they understand subtraction as an independent operation or as related to addition (1.OA.4)?

John Baldwin and I wrote about this years ago: http://ime.math.arizona.edu/2007-08/Baldwin_Kessel_testing.pdf

Helium Balloons and Sandbags

When first introducing addition and subtraction with integers, my colleagues and I started our unit with “balloons and sandbags”. I am wondering if anyone else ever used this technique. Let me know. 🙂

Let’s start with a hot air balloon…

Adding positive numbers means – “putting on helium balloons to the hot air balloon”

Adding negative numbers means – “putting on sandbags to the hot air balloon”

The number line is placed vertically on the Smart Board with 0 being at the edge of a cliff. A hot air balloon hovers at the edge of the cliff at a specific integer. Adding a positive number would require a student to “put on” helium balloons to the hot air balloon which would cause the hot air balloon to rise. Adding a negative number would require a student to “put on” sandbags to the hot air balloon which would cause the hot air balloon to fall.

For example, the story for 3 + 5 is as follows: The hot air balloon is hovering at +3 above the cliff. Adding 5 means “putting on” 5 helium balloons to the hot air balloon which causes it to rise to +8. Therefore, +3 + +5 = +8.

The story for 3 + (-5) is as follows: The hot air balloon is hovering at +3 above the cliff. Adding (-5) means “putting on” 5 sandbags to the hot air balloon which causes it to fall to -2.

Therefore, 3 + (-5) = -2.

Now for subtraction.

Subtracting positive numbers means – “taking off helium balloons”

Subtracting negative numbers means – “taking off sandbags”

The story for 3 – 5 is as follows: The hot air balloon is hovering at +3 above the cliff. Subtracting 5 means to “take off” 5 helium balloons. This causes the balloon to fall to -2.

“Putting on” 5 sandbags [3 + (-5)] and “taking off” 5 helium balloons [3 – (+5)] causes the balloon to end up at -2.

Therefore, 3 + (-5) = 3 – 5.

This is just one of a few introductory lessons we used to help students visualize the result of adding and subtracting integers. Eventually students will begin to see patterns and develop algorithms for addition and subtraction of integers.

Great post! If I may add… Integer chips (or the ‘ones’ square from algebra tiles) engage students in sophisticated thinking (which I love about them!!) but also necessary for its transition and progression to using algebra tiles to solve equations. That is my purpose for the investigation.

Next thought… Subtraction is distance, for example 92-48 can be illustrated on the number line as starting at 48 and finding the distance from 48 to 92 (which is 44 to the right so positive 44). For 3 – 5 start at 5 and go to 3 to find the distance between 5 and 3 hence the distance is left 2 so the answer is -2. Now for 3 – (-5) start at -5 and go to 3 which is finding the distance from -5 to +3. We move to the RIGHT 5 places then 3 more for a total of 8 to the right which means +8.

Distance for subtraction has been a game changer for my students! And a nice progression from whole number subtraction on the number line of starting at the second number and counting up.

One final thought… range is subtraction (or adding up) and range is distance so subtraction is distance!!

Jonily

@mindsonmath

The nice thing about the chips is that subtraction is treated in a very intuitive way (literally “taking away”). The number line is much more flexible. But, the method you propose for subtraction on the number line seems much less intuitive than chips.

Why not define subtraction as the same thing as addition, except we flip the arrow representing the number being subtracted before we connect it with the other arrow? This definition of subtraction on the number line presents “8 – 3” in a way that is easily viewed as “taking away”. Not quite as concrete as chips, but close.

This definition also works very well for any other subtraction problem (and clearly illustrates that subtracting a negative is equivalent to adding a positive).

Chips seem to provide a more concrete and intuitive idea of subtraction being “taking away”. The method proposed for subtracting with the number line (converting to an addition problem) seems less intuitive.

Why not treat subtraction on the number line in the same way as addition – except you flip the arrow arrow around for the number being subtracted. Subtraction on the number line then would go along with the more concrete notion of subtraction being “take away”.

Last year, for the first time, I had a 7th grader say, “The number line model and the chip model are the same. This overlap of the arrows… that’s the zero pairs.”

It was a huge aha-moment for a number of students in the class, that finally gave meaning to the number line. For some, the number line model can be “doing” if they are just counting spaces without making meaning. Whatever way a person sees, or feels, the values in their head is their way of thinking. I agree that the number line provides for better transfer to rational numbers, giving a more precise way of showing the final value. But, in my head, I still feel rational numbers as zero pairs with partial chips.

The great powers is in seeing the connections between different models.

I like a number line, however, I am very cautious of using positive is “to the right” and negative is “to the left”. This operates on the premise that all number lines are orientated in the same direction; this doesn’t translate well to elevations or thermometers. When using the number line my students and I have found that it is important to emphasize the operation (more or less) with the integers. 8 + (-2) is 8 becoming ‘more negative’ by 2. 4 – (-5) is 4 becoming ‘less negative’ by 5. While direction can be implied our conversations tend to focus more on how the magnitude from zero is changing.