Lots of Thoughts on Integer Operations
The Problem: How do you teach integer operations from scratch? And I mean really teach – to give a foundation. I don’t mean ‘memorize the rules and apply them’ – I mean to give a model or an intuition about numbers. How do you justify ‘subtraction means add the opposite’? How do you explain why a negative times a negative is a positive? Describe a real-world scenario that would require you to evaluate -3 – (-2). That last sentence, by the way, is one of the most useful exercises I took away from my pedagogy courses: try to figure out a scenario that would require a student to evaluate the type of expression you want them to evaluate. This is especially tricky for fractions and the subtracting a negative problem.
A Tale of Two Models: I’ve been quietly looking around the internet for resources on integer operations. I even asked the blogotwittersphere on twitter. From looking around, there seem to be two primary models for representing integer operations: a physical model and a number-line model.
The physical model is when you let numbers represent some sort of physical object. The idea of positive and negative numbers are that they are two objects which ‘cancel each other out’, which is where the algebra phrase ‘these two numbers cancel’ comes from. Addition represents the action of ‘bringing objects together’ while subtraction represents the action of ‘taking objects away’ (disclaimer: I stole this vocabulary from Kate Nowak’s video which is posted below). A physical model is amazing at explaining basically everything involving addition and subtraction of positive and negative numbers – it stumbles a little bit with multiplication and division of negatives. This model is especially useful at explaining why subtraction is like adding the opposite and is completely necessary when interpreting real-world situations.
On the other hand, a number line model is when you let numbers represent a continuum of some kind – they are positions on some sort of sliding scale. This is a more abstract representation because the numbers no longer represent things, which is a big deal to some students. In this representation, a negative number is simply a state that is the same distance but in the opposite direction from 0. In this way, numbers are almost akin to vectors – they have both a direction and a magnitude away from zero. Addition represents the action of adding two vectors – 2 + -3 would mean you start at 2 and move in the direction of -3. If you still consider subtraction an operation, then most people in this model use the rule that ‘subtraction means add the opposite’ and turn every problem into an addition problem, which is much easier to do on the number line. We say that 3 and -3 are ‘inverses’ not because the ‘cancel’, but because starting at 3 and moving in the -3 direction lands you at 0, which is the ‘identity’. The connection between signs and direction (‘positive direction’, ‘negative direction’) makes it easier to explain why a negative times a negative is a positive.
Best Explanation of a Physical Model I’ve seen: Kate Nowak’s youtube video where she teaches integer operations. After I watched this, I realized how powerful it is to read ‘+3’ as ‘three positives’ instead of ‘positive three’. Another thing I love about the video is she freely admits that the model doesn’t really explain why a negative times a negative is a positive – which is part of the point of this post. Somewhat ironically, most of the examples Kate uses at the beginning of the video are number-line models (temperature, credit balance, etc).
Best Explanation of a Number-Line Model I’ve Seen: Some guy on Reddit posted the best explanation for multiplication that I’ve ever read. His whole idea is that the ‘-‘ symbol is neither an operation nor a sign – it just represents ‘opposite’. This means if I were to interpret 3 – (-2), I would read this as: start at 3. The first ‘-‘ symbol tells you to face the opposite direction, so you turn and face the negative numbers. BUT, the second ‘-‘ symbol tells you to turn and face the opposite direction again, which means you’re back to facing forwards. Then you finally take 2 steps in that direction, which puts you at +5.
After the jump: a collection of people around the internet who use the different models.
People who use Physical Models:
James Tanton uses a ‘holes and piles’ analogy to describe how integer operations should work (if you click the link, skip towards the end of the post). It works exceptionally well because it has a real way of creating the need for the expression ‘-3 – (-2)’: you have 3 holes and you want to remove 2 holes – what will you have left? Well, removing a hole is like adding a pile, so you end up with only 1 hole left. In other words: ‘-3 – (-2) = -1’. 3 holes take away 2 holes leaves only 1 hole. Sidenote: Pretty much everything James Tanton does is gold. Just sayin.
Frank Noschese, in a completely coincidental fashion, uses a similar model to the holes and piles idea: “While I don’t teach math, I like thinking about +blocks and -holes. So “-3 + 2” is “3 holes plus 2 blocks” = 1 hole (-1). -3 – (-4)” is “3 holes, take away 4 holes” but how to take away holes? By adding blocks! 3 holes + 4 blocks = 1 block (+1)” (src)
@jacobaxdorph uses game-show tokens
@jsb16 also uses money with an explanation as to what the operations mean
People who use Number Line Models:
Dan Meyer appears to use a number line to introduce numbers, but I only know this because he posted his whole curriculum online and the integer operations section includes a number line. You can see the slides that cover this section here.
Khan Academy also uses a number line to introduce negative numbers. However, in a separate video, Khan discusses how to perform integer operations with negative numbers – that video is here. Right around 5:20, Khan encounters the bane of the number line model: subtracting a negative. He glosses over it by skipping straight to the rule – subtracting a negative is like adding a positive. The results of this can be seen in the highest-rated question in the comments just below the video: “I am still trying to grasp the 3-(-3)=6 I know that you say it means you cancel the negatives and make it a positive but I just can’t visualize why. It’s not enough for me to just except that that is what you do. I need to really understand it. I cant figure out how to recreate it on the number line. Can anyone show 3-(-3) looks on a number line?”. I haven’t read the answers, although some of them are quiet long and probably do a decent job of filling in the holes.
(Quick Personal Note: THIS IS THE PROBLEM I’VE HAD ALL YEAR! I decided to teach integer operations using a number line intuition (in fact, my slides are somewhere below), but I could NEVER find a quick answer to the ‘subtraction means add the opposite’ mantra. I don’t blame Khan for skipping over it because, frankly, it would complicate the model to explain why 3 – (-3) = 6 – it’s not what the model is meant for. But, I also don’t blame the student for asking – it’s a totally valid question that is much more easily answered with a physical model).
Kate MacInnis explains her number line model in the comments to this post: “In my number line models, adding has always meant move to the right and and subtracting has always meant move to the left. Then the negative means “opposite”. So 6-(-2) is starting at six, and moving two units opposite of left (aka right)”
David Coffey recommends a Cognitively Guided Instruction approach using temperature, golf scores, and altitude.
Jason Buell recommends a number line approach. He also admits money seems intuitive. It seems he has both a physical and number-line model in his head at all times – very clever.
Paul Salomon uses directional moves, baseball innings, and a hunger approach which I’m not totally sure how to categorize. In typing this, I also realize I’m not sure how baseball innings can be used to model integers,
@daveinstpaul uses a number line and temperature
A colleague here in Tucson uses a video game analogy: Positive and Negative numbers are stats on a sword. Addition is equipping a weapon and subtraction is removing the weapon. So 3 + 5 means you had 3 strength and you equipped a sword with +5 strength, so now you have 8 strength. 2 + -4 means you had 2 intellect and equipped a sword with -4 intellect, so now you have -2 intellect. The nice thing about this model is that it also has a valid representation of ‘-2 – (-3)’: You have -2 intellect and are removing a sword that has a -3 intellect stat. Since you’re removing the -3, your intellect goes up, leaving you with +1 intellect.
Lastly, earlier this year I spent a day teaching integer operations using the number line. All of my slides from this day are right here (my actual presentation was more dynamic, but I had to take out the bells and whistles so it could become a static pdf). I stole the idea of a number line representing Health Potions from another fellow math teacher and complete comic book nerd here in Tucson. One thing about my presentation is that I never tell them the shortcuts. Part of this is because when I taught this, I’m basically ignoring the idea that subtraction is an operation. In my mind, it’s an annoyance that ruins our intuition, not something that’s helpful for me to solve problems. The issue is: if I take this position, then I can’t teach the rule of ‘subtract the two numbers, take the sign of the bigger number’ – it’s a contradiction for me to say ‘there’s no such thing as subtraction – just add the opposite’ and come back later and say ‘hey! cool shortcut! subtract these things to get the answer!”
Some Final Thoughts: When I first started thinking about this problem – building intuition about integer operations – I think I came into it thinking there was just one right answer, that there was a ‘best’ way to understand our number system and the operations we perform in it. It was some combination of the number line and a purely abstract way of thinking about the number system and algebra in general. In particular, I didn’t want to deal with subtraction at all – I used to tell my students “In elementary school, you learn that ‘+’ and ‘-‘ are operations. In middle school, you learn that ‘+’ and ‘-‘ are signs of numbers. In high school, you learn that they’re really the same thing”. I don’t say this too much anymore because I don’t know if I believe it anymore – I still believe there isn’t a place for subtraction in the purely abstract world of algebra (our number system is just a collection of numbers with inverses and identities where the only operations are addition and multiplication), but the idea of subtraction representing a real-world action is something that separates it from representing a negative number. This is something I’m still fleshing out in my mind.
In any case, now I’m thinking that it’s okay to have both models floating around in your head as long as you’re aware that they each help solve different types of problems. Which means it’s okay for me to teach both models to my students as long as I say ‘Ahh – this is a good problem to use the number line on’ or ‘Great – this is a perfect problem to use algebra tiles for’. I also think students need the physical model before they can start to understand the number line model, which is a problem I’m still facing this year. Anyway, One of the things that really cemented this for me was hearing my friend Jeff talk about his undergraduate career at the University of Arizona College of Optics, so I think I’ll end this rather long and involved post with a paraphrased quote from him:
“You spend the first year thinking of light as a straight line ray which lets you use simple geometry to analyze how light works. Then you learn that’s all wrong, so you spend your second year learning that light is a wave, which is how you explain diffraction. Then you learn that’s all wrong, so you spend your third year learning that light is a particle with some quantum physics, which is where photons come from. Then, in your final year, you get this grab-bag of tricks that lets you use all the models at once, which is where you get fancy things like the particle-wave duality of light. So really, you have to look at light 3 ways at once”
Update 2/6/2016: In much the same way that I spent lots of time thinking about the models we use to teach integers, James Tanton has spent a lot of time thinking about the models we use to teach fractions. His examination of Fractions through the grade levels is another great read in thinking carefully about the models we use when teaching fundamentals: http://www.jamestanton.com/wp-content/uploads/2014/01/Pamphlet-Fractions-2016.pdf
Mathy McMatherson 
Absolutely! While reading the beginning of your post, I was already planning a response that would say that you need to teach the students multiple models and help students learn to decide which model would be appropriate for a given situation. I attended a seminar recently that cited some research along the lines that students who could move fluidly between three or more models for integer operations were much more likely to be successful, and not just in their integer operations– they’re apparently more likely to be successful in pretty much all their future math. (I didn’t jot down the reference unfortunately.)
In my number line models, adding has always meant move to the right and and subtracting has always meant move to the left. Then the negative means “opposite”. So 6-(-2) is starting at six, and moving two units opposite of left (aka right). Similar to your random guy on Reddit, but I got that from a master’s-level math-ed class.
One quick note: I’ve never entirely been a fan of the 2-sign version of solving integer problems, where you change 6 – 3 into 6 + (-3). Every student walks into algebra understanding 6-3, but suddenly we want them to change how they handle one of the most basic algorithms they’ve been seeing since they were 5? Why do we make it more complicated? I like using one sign, so students change 6+(-3) into 6-3 before they start thinking about it. That’s where the health potion idea came from: if we can make them understand why adding a negative is the same as subtraction and subtracting a negative is the same as adding, we change every integer problem into something that looks very similar to what they’ve been seeing for years. They just need to figure out that 3-6 is going to drop them back below 0, and that intuition came pretty quickly for me when I learned it.
I’m guessing you’ve already seen this post, but in case you haven’t you may enjoy my omnibus link collection of explanations why a negative times a negative is a positive.
The thing that bothers me about the backward/forward faceforward/turnaround explanation (which is what I use in my picture answer, so I’m not that much against) is that the negative is used in two different ways for if it is applied to the first number or if it is applied to the second number. That seems to be the beef with my students who remain unconvinced but I haven’t found a good way to patch it up.
Jason – I hadn’t seen your post!! Excellent!
Regarding the ‘negative means two things depending on where the number is’, I sorta came up with an answer to that. It came after I started to really connect numbers as vectors on a number line.
Rules:
(1) Always start at 0 facing the positives
(2) A negative in any context means ‘turn around’
(3) A number in any context means ‘step foward that many steps’
(4) Reset and face the positive numbers at the end of any movement
So -3 – (-2) Becomes the following:
(a) Start at 0 facing the positives (Rule 1)
(b) see a negative sign, so turn around and face the negatives (Rule 2)
(c) walk 3 steps forward (so you are now at -3) (Rule 3)
(d) Reset and face the positives (Rule 4)
(e) Negative sign, so turn around (face the negatives) (Rule 2)
(f) Another negative sign, so turn around (face the postives) (Rule 2)
(g) Take two steps forward (now at -1) (Rule 3)
(h) Done.
So, in essence, I’m treating that first number as a vector from 0, which is consistent with how I treat the second number.
Reflection: This feels like the computer sciency way of explaining integer operations – a machine with rules that moves along a continuous strip. Like some weird Turing machine.
Here’s other examples using physical models.
http://blog.mathpl.us/?p=644
This is a really useful collection of teaching techniques for number operations. I’m going to share it with my colleagues.
For what it is worth, I use the James Tanton model (which I remember learning from Stand and Deliver…) for integer operations with addition and subtraction, but I find an area model a bit more useful for multiplication.
You might look at curriculum materials published by the Math Learning Center in Portland, OR. The integer lessons are from Math Alive (older name VIsual Math), and as a stand-alone series of activities in a series called Math and the Mind’s Eye. They use an area model for multiplication and division with positive and negative edges (colored black and red, respectively.) In this model the area is the product of the lengths of the edges of the rectangle. Rectangles are built with square tiles which are black on one side and red on the other.
A negative edge involves a second move in addition to standard + x + multiplication, It involves flipping the rectangle over (all rectangles start out positive (black)). Thus with two black edges nothing flips and it’s just regular mult. With one red edge (pos x neg), the black rectangle flips once to red. With two red edges (neg x neg) the rectangle flips from black to red and then back to black again.
The mathematical integrity in this model is that it emphasizes that negative 3 can be interpreted as “the opposite of two”.
With division, you know the area and one edge of the rectangle and the answer is the length of the other edge. The above multiplication discussion is used to determine the necessary value.
I’ve always taught BOTH the physical and number line models. Some kids seem to meed the models more than others. I ask them to understand both, but they can use either in a defense of a particular procedure or answer.
Coming to the party quite late! (Fawn just posted a link to this. . .)
3-(-3) on a number line asks “what is the difference between 3 and -3?”
Students often only think of subtraction as “take away” but if you want to find out “How much older is John, age 18, than Sam, age 12?” you do it with subtraction. Of course, age doesn’t work with integers, but you can use temperature.
For neg x pos and neg x neg:
I use context all the time. John is walking backwards at 2 ft/sec. Where will he be in 5 sec? (-2 x 5 = -10. . .10 feet behind where he is now.) Where was he 3 seconds ago? (-2 x -3 = 6. . .6 feet in front of where he is now.)
Hope that helps.
I was going to suggest something like what Cindy W just posted:
Say I have $20 and every day I buy a $3 coffee. 5 days from now, I will be down $15 (or -15) from where I am right now. But how much money did I have 4 days ago? I’m spending (negative) each day going backwards for 4 days… meaning I had $12 more 4 days ago than I have now.
I had a student years ago for whom I drew a little guy with a + on one side and a – on the other. She’d face him the right way, and sometimes get the right answer…..
On a number line, each signed integer has one direction. So 3 – (-3) would start with a vector 3 units long pointing to the right (positive direction). The next vector would be pointing 3 units to the left starting at the arrow tip of the first vector. But then, since you are subtracting (and subtracting means backing up), you back up the vector for -3 by 3 units to the right. Now both vectors are nose to nose (arrow to arrow) for a total length of 6 units.
Great post! I recommend John Van De Walle’s book Elementary and Middle School Mathematics Teaching Developmentally. He shows a colored counter model and a number line model. In the 6th edition on page 499 he asks which model to use? and his answer is both “Students should experience both models and, perhaps even more important, discuss how the two are alike. A parallel development using both models at the same time may be the most conceptual approach.”