# Tag Info

43

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'...

40

In a comment, the OP has suggested that he actually wants a practical example convincing students that the product of two negative numbers is positive. This is related to, but psychologically distinct from, the question asked. In particular, students often have different mental models for multiplication (a binary operator) and negation (an unary operator). ...

32

Draw a number line and label all the integers. Tell him that adding $x>0$ is moving $x$ units to the right and subtracting $x>0$ is moving $x$ units to the left. Tell him that adding $0$ is not moving at all. Tell him that adding $x<0$ is moving $-x$ units to the left and subtracting $x<0$ is moving $-x$ units to the right.

28

Again and again he finds $-4$ greater than $-3$. Ask him who is richer, he who has a smaller debt $($like $3$ rupees$)$, or he who has a bigger debt $($like $4$ rupees$)$, assuming both persons have no money, just debts. He has spent several years seeing $4$ greater than $3$. A debt of $4$ rupees is indeed bigger than one of $3$ rupees. But the one that ...

26

I’m more curious about incorrect things in them. Yet, this is the first thing I found. There's absolutely nothing "incorrect" about this. As Dave L Renfro noted in a comment: and whole number is rarely used as a precise designation outside of school mathematics there is no agreed-upon rigorous definition of the term, and in fact it's largely ...

24

Apologies, this should be a comment on the answer provided by @Jasper Loy but I don't have enough rep on this site. I just wanted to add that in my experience, struggling students have an easier time grasping negative numbers when the number line is oriented vertically rather than horizontally. I think we as humans naturally make the 'up=greater, down=less'...

20

I don't think that "textbooks" decided this, usage did. The term "integer" covers positive and negative, so it would be redundant for whole numbers to refer to that category. And there is an argument to be made for the term linguistically: a negative number is sort of the opposite of having a whole thing. But ultimately, there's not much ...

17

I would use the (real) number line. First introduce the concepts of positive numbers (distance to the right of zero) and negative numbers (distance to the left of zero). Then introduce the concept of negation. Instead of teaching it as flipping with respect to the vertical line at zero, I recommend teaching it as the rotation of $180^\circ$ (counter-...

14

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.

12

You seem to describe "whole numbers" in this American usage as describing $\mathbb {N}$, the set of natural numbers, whereas you expected it to describe $\mathbb{Z}$, the set of integers. As others have pointed out there is nothing "incorrect" about them, it's a language difference. Although it is worth knowing those language differences ...

11

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 ...

10

I want to post this answer just to give show any future prospectors what did work for me in this particular case! and also because it's tooo long for a comment (pardon my over emphasis) I mixed the concepts of number line first horizontal, but then as he found it difficult I showed him a vertical number line as suggested by @JVL, along with pictures of ...

9

This is redundant with several nice answers, but let me try it anyway, as (for me) the intuition is strong. A gambler (or spendthrift) loses \$10 per day. If they have \$30 today, how much did they have 5 days ago? So I'm using loses per day as one negative, and past time as another. It is intuitively clear that the further in the past, and the greater ...

9

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 ...

8

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 ...

8

I would say you're doing your student a disservice if you were to seriously disallow a negative denominator. A fraction is simply a ratio of two integers (where the denominator is not allowed to be zero). I disagree with @yoniLavi that we never need such fractions. Since division by negative numbers makes sense, such a fraction with a negative denominator ...

7

Talking about your proposed language alteration with your students would make more of a difference than just using it. You could help them see that math is partly about defining relationships carefully, and that language, with all its richness, isn't so good at that. In calculus, when I ask where the derivative is greater, and we're looking at negative ...

7

You might explain that BEDMAS is not the whole story when it comes to the order of operations. There is an operation called negation. It reverses the sign on numerical quantifies. It gives the additive inverse of any number, i.e. for all $x \in R$, we have $x + (-x) = 0$. Unfortunately, most textbooks use the same symbol for both subtraction and negation. ...

7

I'm going to answer with something of a polemical frame challenge: FOIL is evil, and probably shouldn't even be taught. Okay... that's a bit extreme. How's this: FOIL is a mnemonic that is, in my opinion, not all that useful, and should not be taught. In my own experience teaching college algebra and precalculus courses, students come to rely on FOIL ...

6

Perhaps teaching about other negative numbers would be easier, like $-1 > -2$ or $-100 > -200$. Some possible approaches: Suppose you have $5$ apples. If I took from you $3$ or $4$, in which case would you have more? Now suppose you have no apples and again I take $3$ or $4$ from you (that means, should you get a hold of some apple, you have to give ...

6

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 ...

6

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 ...

5

I'm not really understanding the aversion to using abstract analysis. You don't need to ruin it for your students and say "by the way, this is a result of Ring-theoretic axioms, which is waaaaaaay beyond the scope of anything you'll ever learn!" Just show them that (-1)*(1 - 1) = 0 (which they won't dispute), and then distribute the terms, giving you -1 + (...

5

You could have also proven this algebraically, by doing this: To show that $-(-x)=x$ is the same as showing $-(-x)-x=0$. We then simply say $$-(-x)-x=-1(-x+x)=0 \tag{QED}$$ Or without using the distributive law, it could go something like this: If the students already believe that $-1\times-1=1$, then $$--a=-1\times-1\times a=1\times a=a\tag{QED}$$ ...

5

I work with students with disabilities and often have to teach them about negative numbers. I tend to use an elevator analogy that I borrowed from a math methods class I took years ago; I create a fictional building that has 5 levels above ground, a ground floor (0), and then an underground parking structure with a P1, P2, P3, P4. Then I tell them we're ...

5

My opinion is that it should be exercised and thereafter done mentally (not written down) as part of reading an expression. This then allows students and practitioners to directly follow the order of operations, simplifying at each step, and then in the last step recall and apply only the sign rules for addition. Note that the issue of reading/writing ...

5

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). ...

5

It can definitely be confusing, and a clear distinction should be made when teaching. It's not always made, and students can in fact remain lost for a long time. I've seen students who thought $1 \cdot (-1) = 0$, unsure of the role of $"\cdot"$, $"-"$, and parentheses. As far as symbol, it's not too confusing if when teaching you use parentheses, e.g. "\$(-5)...

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