43
votes
Explaining the order of negative integers
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 ...
- 4,878
40
votes
How to explain that a negative number multiplied by a negative number is a positive number, and that $-(-x)=x$?
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 ...
- 19.4k
32
votes
Accepted
How to teach someone that $-3>-4$?
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 ...
- 440
28
votes
How to teach someone that $-3>-4$?
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 ...
- 397
26
votes
Why don’t American school textbooks recognize negative numbers as whole numbers?
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 ...
24
votes
How to teach someone that $-3>-4$?
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 ...
- 341
20
votes
Why don’t American school textbooks recognize negative numbers as whole numbers?
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 ...
- 1,258
17
votes
How to explain that a negative number multiplied by a negative number is a positive number, and that $-(-x)=x$?
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 ...
- 9,932
14
votes
Explaining the order of negative integers
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 ...
- 141
12
votes
Why don’t American school textbooks recognize negative numbers as whole numbers?
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 ...
- 291
11
votes
Explaining the order of negative integers
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 ...
- 2,258
10
votes
How to teach someone that $-3>-4$?
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 ...
- 1,105
9
votes
How to explain that a negative number multiplied by a negative number is a positive number, and that $-(-x)=x$?
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 ...
- 27.9k
9
votes
Accepted
Explaining the order of negative integers
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 ...
- 206
8
votes
Explaining the order of negative integers
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 (...
- 89
8
votes
Negative Denominator in Fractions; Importance and Applications
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 ...
- 381
7
votes
Should we rename the greater than sign?
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 ...
- 16.9k
7
votes
Exponents with Negative Base; with or without Parentheses
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 ...
- 781
7
votes
Why in the FOIL Method the terms are taken with their signs?
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 ...
- 6,442
6
votes
How to teach someone that $-3>-4$?
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 ...
- 8,729
6
votes
Explaining the order of negative integers
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 ...
- 20.6k
6
votes
Explaining the order of negative integers
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 ...
- 16.1k
5
votes
How to explain that a negative number multiplied by a negative number is a positive number, and that $-(-x)=x$?
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 ...
5
votes
How to explain that a negative number multiplied by a negative number is a positive number, and that $-(-x)=x$?
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 ...
- 241
5
votes
How to teach someone that $-3>-4$?
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 ...
- 1,016
5
votes
Subtraction to Addition Conversion; how important is it?
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 ...
- 20.6k
5
votes
Explaining the order of negative integers
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. (...
- 159
5
votes
Is a clear distinction made between signs and operators?
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 \...
- 191
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