S S +, SD - IGNORE the signs!

What is -2 x 3?

When I was taught about multiplying and dividing 2 positive or negative numbers, I was told to remember...

1) A "+" times a "+" is a positive
2) A "+" times a "-" is a negative
3) A "-" times a "+" is a negative
4) A "-" times a "-" is a positive.

The same applies when dividing numbers.

Lots to remember. And statements 2 and 3 are really the same thing. So...

When multiplying or dividing 2 numbers, the first thing to do is...

IGNORE the signs.

They do not exist. For now.

For example... -2 x 3.

Treat this as 2 x 3 = 6. Then remember SS+, SD-

If the signs are the SAME (SS), your answer is positive (+). If the signs are are DIFFERENT (SD), your answer is negative (-). 

In our example, the signs are different so your answer has to be negative. Turn your answer of 6 into -6 and you are done.

Backwards Backwards

         5
-    - 3

What is this?  

1) 5 take away minus 3?

2) 5 minus minus 3?
3) 5 plus 3?

In effect, this is 5 plus 3. Why? Firstly, despite what your mother told you, two wrongs do make a right. Well, in math they do. In this case, the 2 negatives (wrongs) become a right (positive).

Think of the motion of the woman in the photograph. She is moving to the right and we generally think of a right movement in math as positive. A left motion in mathematics is considered negative. Left, in history, was considered evil or sinister.

If we ask the woman to turn around and walk backwards, or asked her to walk "backwards backwards", she'd still be moving to the right. (Just facing the other way but we don't worry about this - just the direction of travel.) 

Your two "minus" signs are your backwards backwards. They become a positive. Try walking backwards backwards for yourself. 

Pythagorean Triples - Work Smart, Not Hard!

A lot of triangle questions can be solved without using the Pythagorean Theorem. 

No need to worry about pesky squares and square roots! 

Pythagorean Triples to the rescue!

Some common triples are...

3:	4	:5
5:	12	:13
8:	15	:17
7:	24	:25
9:	40	:41

The first 2 triplets (3:4:5 and 5:12:13) are the most common ones that I have seen on math tests. 

If you have a triangle question that involves two sides being 5 and 12 units in length, then you know the third side is 13 units long. 

Some questions aren't quite so obvious. You might have a triangle with two sides being 6 and 8 units. 

There is no Pythagorean triple with a 6 and 8 in it. 

However, 6 and 8 are just double the 3 and 4 in the 3:4:5 ratio. The third side is double 5... 10! If your triangle question involves 26 and 24, the third side is 10.

Call a spade... something else

a)  5x + 3
+  2x + 6   =      ??????

b) 3x²  +  4x

+ 4x²  +  5x =      ??????

Some of my past students seemed to want to add the "x's" together, getting answers such as...

a)  5x + 3
+  2x + 6   =          7x² + 9        or....

b) 3x² + 4x

+ 4x² + 5x =              7x⁴ + 9 x²

My advice, always, is not to call a spade "a spade", but to use the good old cat and dog to help us. (Or any other animal you might choose.) For question a)...

Restate the question as 5 cats and 3 dogs ADDED to 2 more cats and 6 more dogs. You'll get 7 cats and 9 dogs. None of those cats gets squared or changed to a super cat, nuclear powered cat, robo-cat or anything else.  So your answer is...

7x + 9 

For question b) ... you'll simply have 3 elephants and 4 cats ADDED to 4 elephants and 5 cats. Your answer is...

7x² + 9 x

It is easy to get blinded by all the exponents when you get questions such as...

 7x² + 2y + 9x⁴  +  8x² + 3y + 3x⁴ 

It is much easier to imagine 15 elephants, 5 giraffes and 12 monkeys.

What the !!!!!!!!!!!!!!!!!!!!!! ????

Have you ever wondered what the ! symbol is doing on your calculator? 


You can use it to help solve these sorts of questions...





If you have a horizontal row of four blocks, how many different ways can you arrange the order of them in the row? 

Method 1:

Set up a grid and put the letters in. Something like RISK, RSIK, RKSI, etc. 

This is fairly time consuming and RISK(y). You will be prone to mistakes by forgetting some combinations. It is not practical for questions with larger numbers of blocks.

Method 2: Find the ! on your scientific calculator. It may be indicated as x! You may have to hit a shift or 2nd function as well, in order to activate this handy tool. So...

Enter 4 (your number of blocks) and activate the !    Answer is 24

Method 3: If you don't have a calculator... 4x3x2x1 = 24. 4 is the number of blocks and you multiply it by the 3 whole numbers that are smaller than 4 and greater than 0. If you had 5 blocks, you would work out 5x4x3x2x1 = 120