# Explaining the order of negative integers

Today I happen to have an interesting discussion with a primary school kid.

I asked him "Which is the smallest one - digit integer?" He instantly replied $-1$. I told him that he's wrong and the answer is $-9$. He then replied then "why"? He constantly kept saying that $9$ is greater than $1$ and therefore, the smallest integer should also be -$1$. To him number with greater value is greater. He doesn't care about $+$ or $-$ sign.

I then explained things to him by drawing number line and then explained him that all the numbers to the right hand side of a particular number on the number line are greater. He then again asked "why?"

At the end I was speechless and couldn't satisfy his curiosity because I had no proper trick to put this into his brain. Kindly suggest me some methods to explain him everything he wants to know. I just want him to feel satisfied.

• It helps to meet the student where he is. His meaning of smallest is different than yours. What an interesting discussion to have. – Sue VanHattum May 26 '16 at 13:54
• "Smallest negative" is not a well-defined term, both "most-negative" and "closest to 0" are valid definitions. For example, it's reasonable to call negative one million a "big" negative number, which would make -1 the "smallest". So expecting him to understand why he's wrong is unfair, because he's not. – BlueRaja - Danny Pflughoeft May 26 '16 at 18:49
• Perhaps you could visualize it with an elevator in a skyscraper with many floors but above and below ground - and the actual ground as zero? Should show that -9 is lower than -1, which in turn is lower than +5... – Baard Kopperud May 26 '16 at 19:46
• When students are taught to add integers, the sign of sum, they may have been told, is that of the "largest" number. So, if we add -9 + 6, the sign of the sum would be negative. I myself may have, ummm... been guilty of this crime of imprecision some years ago. I guess you really have to teach absolute value or magnitude before arithmetic operations on integers. – Dan Christensen May 26 '16 at 20:56
• "Smallest" seems to be an issue and I can see why. Having an object with a -10C charge is not "smaller" then having a -1C charge. – cspirou May 29 '16 at 9:25

He wants to know the why, not per se the logic behind it.

So give him a reason he can understand. Explain to him that many hundreds of years ago the concept of zero didn't exist. There was no number for having nothing of something.

Then, someone thought of the zero, a simple value, that meant you had nothing.

But, things can have less than nothing.

Draw a simple number line:

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10


There was a worker, who wanted to buy a new tool. He has 5 coins. The tool costs 10 coins.

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
-----------------------------------------^----------|


How much extra coins does the worker need to be able to buy the tool?

The person selling the tool can lend him the remaining 5 coins. This means he owes the person selling still 5 coins.

The worker hands the salesman his five coins. His money in his pocket is now zero

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
-------------------------------^---------------------


The worker gets his tool. But now owes the salesman 5 coins, which he will have to pay out of his future salary. Until this is paid to the sales man he will not be able to have money in his pocket. He has -5 coins.

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10
-----------------^-----------------------------------


If he gets a coin for his work, he will have to pay the sales man first, making his coins -4. He can only get coins in his pocket again when his count reaches zero.

So what is less, 0 coins or -5 coins?

• This one seems to be a good approach. – Heisenberg May 26 '16 at 7:28

To my mind, the problem is the word smallest. If you asked me which is smaller, $-1$ or $-9$, I'd ask you to clarify in what sense. Colloquial use of small refers to magnitude rather than ordering. It is not true that $-9$ is smaller than $-1$ in magnitude, although it makes sense to say that $-9$ is less than $-1$ although it is bigger in magnitude. The kid's perspective is captured by dropping the "in magnitude" and speaking with less precision. When comparing electric charges $1$ and $-2$, which would you say is smaller? One says that $sin(t)$ has smaller amplitude than $-2cos(t)$ although there is no order relation between the two.

Explain the difference between order and size - one refers to relative position, the other to the distance from the origin - that by "smaller" you mean "less than" in reference to ordering, and not magnitude. The kid is likely to reject this (to his mind) misuse of "small", so an alternative is to not use the problematic word "smaller" except to refer to magnitude. In mathematics words have meanings that are precise and limited, and often different from their meanings in ordinary speech. "Less than" also has a mathematical meaning that doesn't conform well with ordinary speech, where "less" also often refers to size, but its use in relation to ordering is palatable. Communicate that in mathematical speech, "less than" usually refers to order.

It seems to me that what you asked wasn't really right. -9 is the "lowest one-digit integer" but (at least it can reasonably argued) 0 is the smallest. Maybe making this difference explicit would clear confusion: "big" negative numbers are a long way less than zero and therefore the lowest.

All 8 answers so far seem to have missed the following issue:

Ask the student why he picked $-1$ rather than $1$ or $0$.

If he changes his mind and says that $0$ is the smallest, then he is using a valid notion of "smallest", namely smallest in magnitude. Then it is interesting to discuss when this is useful (distance from origin) and when it is not useful.

If however, he says that $-1$ is strictly smaller than $1$, ask him why? Invariably the only answer he can give is that $-1$ is negative. Then the next question is, why are negative numbers smaller than positive numbers? It is entirely possible that the student was rote-taught this, but his teacher never gave an explanation as to why, and therefore it was messily mixed with the intuitive notion of magnitude and produced a weird ordering:

$-1 < -2 < -3 < \cdots < 0 < 1 < 2 < 3 < \cdots$.

Along the way, the student should be asked to order the above seven numbers, which should reveal clearly his perspective. Ultimately, exploring the students' perspectives is crucial to a successful mathematical education.

Finally, for the original question of what is the smallest 1-digit number, the official answer is wrong because the correct answer is $0$. $-1$ is not a 1-digit number; it does not only consist of one digit! If you want to argue with me, simply do a Google search and see that almost everyone says "$k$-digit number" to exclude negative integers (there may be some who include zero, and some who do not, but nobody includes negative integers).

• And please don't ever assume that our goal as a teacher is to teach students to accept what we think we know. You said "I had no proper trick to put this into his brain.", which frankly shows an incorrect attitude. Tricks never provide proper understanding. – user21820 May 28 '16 at 7:00
• Excellent observation, which we can bring home for general application: try to use what the student really says, and what it means about his concept image. – Benoît Kloeckner Sep 15 '16 at 12:15
• @BenoîtKloeckner: Right; when students are quiet you can hardly tell what they understand and what they don't, but when you make them tell you what they think and why they think so, all their misconceptions are made clear, often even misconceptions that one as a teacher might not even have expected to have a bearing on the topic at hand. – user21820 Sep 16 '16 at 1:33
• @Saksham agreed with how tricks never provide proper understanding. Go here. – Mr Pie Sep 9 '18 at 0:19

So $-9$ is "smaller" than $-1$?

By that terminology, a bank account overdrawn by $9$ dollars (balance $-9$) would be said to carry a "smaller debt" than a bank account overdrawn by $1$ dollar (balance $-1$).

The concept of "smaller" and "bigger" corresponds naturally to magnitude.

In the complex plane, it corresponds to modulus. A complex number $z_0$ is smaller than $z_1$ if $|z_0| < |z_1|$.

We should avoid calling $-9$ "smaller".

Between the integer $-9$ and $-1$, $-9$ is the more negative of the two, and has the greater magnitude.

We can also say that $-9$ is less than $-1$ (the pronunciation of $-9\lt -1$). If you carry a bigger debt, you have less money (are less rich) than if you carry a smaller debt. A good way to explain why we call this "less" when the magnitude is getting bigger is that in measurements of measured or counted quantities like money, voltage or temperature, zero is an arbitrary point. For instance, just because you have zero dollars in a bank account, that doesn't mean you are broke: you hold money in other places, such as other bank accounts, investments or cash. If all your other assets are held equal, then a $-9$ account makes you poorer than a $-1$ account, even if you have a billion dollars. The zero is just relative to that account. In temperature, $-9^o C$ is less warm than $-9^o C$ because zero is also arbitrarily set; both temperatures are greater than absolute zero. If these measurements are expressed as absolute temperature, such as $^oK$, then the lower one indeed has the lower magnitude.

If he's ignoring the + or - signs, then that needs to get remediated. My approach is to:

• Define negative numbers as running to the left on the number line. (In other words, the "-" means "in the reverse direction"). Exercise this thoroughly first, finding numbers like +5 or -3 on a marked number line.
• Define the relation lesser-than as meaning further left on the number line. Exercise this, comparing various integers drawn on the number line.
• If one wants to consider just the number part, then introduce the absolute-value operator for that (that is, magnitude: distance from the origin). Exercise writing various expressions using absolute-value notation.

Applications of integers may help: Temperature, elevations, football, money. Which is colder: -1 or -9 degrees (a thermometer may help)? Which is lower: 50 feet below sea level or 100 feet below sea level? Which football team has done better: One who lost 2 yards on the drive, or one who lost 5 yards? Whatever is familiar to your student. The reason we need signed numbers is to distinguish different directions in cases like these.

• Negative numbers as debt were understandable to kids in seventh grade, but I'm not sure when it becomes understandable. One kid has 3 €, another has 1 € of debt, third one has 9 € of debt. Who has the least money? – Tommi May 26 '16 at 5:17
• I tried all the methods mentioned by you but nothing worked. – Heisenberg May 26 '16 at 5:39
• -1. As I see it, since the issue is the kid's definition of "smaller"; imposing the "right" definition is not as good an approach as it would be to discuss what "smaller" means. – Martin Argerami May 26 '16 at 11:01

I will only add to the other excellent answers that even the words "less than" (the conventional name of the $<$ sign) can reinforce the (incorrect) notion that $<$ is used to compare the magnitude of two numbers. For non-negative numbers this is perfectly correct and reasonable, but as soon as negative numbers enter the conversation it becomes problematic.

For this reason, it may sometimes be helpful to avoid pronouncing "$a<b$" as "$a$ is less than $b$" and instead pronounce it as "$a$ comes before $b$". (Or, if you want to be fancier, as "$a$ precedes $b$." This focuses the attention on $<$ as a symbol denoting order, not size. So we would say that $-9$ comes before $-1$, which comes before $3$.

(Truth in advertising: I have never used this way of speaking with elementary school students, but I have used it with graduate students in a real analysis class, when discussing ordered fields other than $\mathbb{R}$. In that context the order may really be based on something other than magnitude, so the phrase "less than" / "greater than" is not just confusing, it is outright inaccurate.)

• Nice suggestion. Maybe one could also read it as "$a$ is below $b$", or "$a$ is lower than $b$". – Michael Bächtold Mar 24 '19 at 21:30

Help him first to understand that subtracting 1 make numbers lower, lower and lower. Each time you subtract 1, you have less than before. Just like adding 1 will make them higher, higher and higher. (In my opinion the terms low and high will be less confusing as kids already have the association bigger figure (absolute) => bigger number in their heads). More you add, the more you have. More you subtract, the less you have.

As you go subtracting 1, even though the figure increases (by figure I mean the absolute number), the number gets lower, lower and lower. It means, you have less, less and less.

The owing concept might help him to understand that you can take -1 from quantities even when you go below zero, and that means "you have even less" than before. So if they have 5 apples and need to give you 5, they will then have 0. But they can also have 5 apples and owe you 8. You will subtract 8 from them. So after they give you all (5), the next 3 apples they get must be given to you as well, because they are due. It means that having -3 apples is worst than having 0, because it means less apples (even though the absolute number is bigger).

After they understand that even having 5 apples, you can try to subtract 8 (so they own 3 apples to you), ask if they would prefer that you subtract 8 or 20 (and how much that would represent). They should pick 8. Because if they pick 20, they will have so, so much less apples because of owing you... And that may help to understand that having -8 somehow means to have more than -20.

Just an insight, a slap dash explanation for you to work on. But hope it helps.

There is a great board game that I use with my third graders which teaches negative numbers. It's called Midnight Party or Ghost Party. In the game the students can get captured and get sent to a place with negative points of different amounts. The later they get captured, the less negative points they get. So the first piece captured lands on the -10 space and the second piece would get -9 and so on. The kids have to add up their negative points at the end of the round. Everyone understands negative points are bad and would rather have fewer negative points. In addition they can land on one or two spaces that have positive points and they understand that more positive points are good.

You can try to get the game or just create your own board game with a path on oak tag. There should be a path with a few positive spaces and spaces for capture. There should be a place where they are sent and get negative points. If it is just the two of you, you should each have 5 pieces so that there is something to add up.

I am confident that after playing this game a few times, the nine year old start to understand that negative points are bad and therefore it's better to have -1 than -8, just as it's better to have 8 than 1. This should help him make the leap that you are looking for.

We want inequalities to be a property of the relative position of numbers — it shouldn't change if we translate the whole picture by the same amount. More precisely, we want inequality to have the property that, for any numbers $r, s, t$, if $r + t < s + t$, then $r < s$. This is clearly true for positive integers, and once we've decided to keep this property, there's only one way to extend the definition to zero and negative integers.

Similarly, we want inequalities to be preserved by scaling by a positive number, i.e., if $rt < st$ and $0 < t$, then $r < s$. This also leaves us with only one choice for how to extend the definition of inequality to rational numbers.

The general pattern here is: to extend a definition to a new setting, identify some key properties you want to preserve. (A related example is exponentiation: we define $r^s$ for positive real $r$ and arbitrary integers $s$ by saying we want $r^{s+t} = r^s r^t$, and similarly for rational $s$ by saying $(r^s)^t = r^{st}$.) It's important to note that there is a new definition being made here, and not just exploring consequences of an old definition.

If a student is using a different definition, then it should be pointed out that this is due to different definitions, not an error in reasoning — which is a good opening to discuss the advantages of one definition over the other. Usually, when there are two conflicting definitions, they're trying to capture different notions.

• I think that the last paragraph is spot-on. – Carsten S May 28 '16 at 16:26

The student is correct: -9 is larger in magnitude that -1 (in conventional notation |-1|<|-9|). It's possible that the student isn't yet making the distinction between larger in magnitude and larger in the usual ordering, or just that the student finds the absolute value comparison more natural. But the student is expressing a legitimate view of ordering on the integers, it just happens not to be the one the teacher had in mind.

I think we turn students off to math when we tell them their (valid but underdeveloped) intuitions are "wrong" when we really mean that they aren't using the official language for expressing them.

I'd prefer to emphasize that these are both meaningful orderings, and so they have different names because they make sense in different situations.

Good opportunity for a lesson.
For each of the following pictures that you draw on the board ask and discuss with student (or class as the case presents) after you draw each. "Which is bigger?"

1. Stick figure of short fat and tall skinny person.
2. normal size (i.e. same font) 6 and 7
3. Tall Big 6 tiny 7

From this will come the need to clarify terms. And introduce the word "Magnitude" in terms of non-math terms. i.e. storms, hurricanes and/or earthquakes. This will not solve the 'problem' until "magnitude"is defined for the math person..i.e. in terms of distance from zero on the number line.

Now Discuss the following 'drawings' i.e. number line with the following points labeled: -7, -6, -1, 0, 1, 6 7

Have students make sentences using 2 of the numbers on the line with either the of words <, >, magnitude. Do not repeat a number.