Come Undun

From  de.wikihow.com

On the Way to Grandma's House

Look familiar?

7 (x + 2) = 28

 

Solve for x

 

 

This is a Little Red Riding Hood question. Watch out for the...

Little Red Riding Hood decides to take 7 baskets... to Grandma's house. Each basket contains...

   an  X

and 2 apples (or whatever you'd like)

 

Therefore...



 

Little Red Riding Hood will carry a total of 7x and 14

 

 

Now we have a simple equation

 

7x + 14 = 28

 

 

We will employ the dump, shove and divide method to solve it.

 

 

1) DUMP 14 off each side of the equation so you have 7x = 14

 

 

2) SHOVE

the 7 under the 14

 

 

 

14

__

 

 7

 

 

 

 

3)

...14 by 7. Your answer is 2.

 

 

 

 

Done! Work smart, not hard!

 

 





  

 

 

 



Key Math Words - "English" words versus "Math" words

 

12 and 5...  What is the difference between these two numbers?

Well, you might say that 12 has two digits and 5 has only one. You might say that the digits in 12 are gold and the 5 is blue.

In the mathematical world, difference means subtraction. The difference, in this case, is 7.

Some key subtraction terms are...

reduce

minus

diminish

decrease

dropped      and...

fell

Some key addition terms are...

sum

total

raise

combine    and...

additional

Some key multiplication terms are...

product

per

Some key division terms are...

quotient

split   and...

shared

Knowing these terms can help with word questions, when you are trying to figure out which operation to perform.

Solving Equations - Think "Hockey Trades"

Look familiar?  

x + 31 = 64            

Solve for "x"




Think of these problems as hockey trades. Good old #31...

...likes playing for team "x". (They are in first place in the league.) We know that "x" is happy because there is a plus/positive sign to the left of his number. Think of the equation as...

x /+ 31/ = 64   

However, hockey being hockey, sometimes players get traded. Good old /+ 31/ is being sent down the lines (=) to the last place team. Do you think he will be happy when he gets there. No... (note the incredible graphics here at the RUGS Blog) He's now not happy and, therefore, is now negative /-31/. The equation will now look like this...

x = 64  /- 31/

64 - (minus, take away, etc.) 31 = 33

Therefore x = 33





You can also have some grumpy, negative players on the last place team who are suddenly traded to the top team. 



x /- 15/ = 64            

Solve for "x"


When grumpy (negative) #15 gets sent down the lines (=) to the new team, he becomes very happy (positive).

x = 64  /+ 15/

x = 79

"Than" is the Man

Word questions in Algebra. 4 words to...

...scare anyone. However, there is hope...

Remember, "than" is the man. "Than" is the key to solving these questions. For example...

3 numbers add up to 180. The smallest is 20 less than the largest. The third is 10 more than the smallest. Name the three numbers.

These numbers are referenced against the other numbers. The word "than" helps us here. I see the word "smallest" appear twice, so I am going to use that as my reference, or "x". The word that appears the most should become your reference.

So...

...       if...

...the smallest is 20 less than the largest, then the largest must be 20 more than the smallest, 
or x + 20. We now have...

1) The smallest = x
2) The largest = x + 20

We just need the third number.

The third is 10 more than the smallest. The smallest = x, so the third number must be x + 10.

We now have...

1) The smallest = x
2) The largest = x + 20
3) The third number = x + 10.

Still don't have an answer, though, do we?

We will have an answer, however, if we build an equation...

We know that the 3 numbers add up to 180, so...

x + x + 20 + x + 10 = 180

Group your " x's "...

3x + 20 + 10 = 180

Group the 20 and 10...

3x + 30 = 180

Drop 30 from each side..

3x = 150

x = 50

Still don't have an answer, though, do we?

Almost there - promise...

x = 50

Remember...

1) The smallest = x, so the smallest number is 50.

2) The largest = x + 20, so the largest is 70.

3) The third number = x + 10, so it is 60

50 + 60 + 70 = 180 

3 Consecutive Numbers...

Look familiar? The sum of three consecutive numbers is 36. What are the numbers?

Method 1) 

Guess. It is unlikely to be 1, 2 and 3...

...or 34, 35, and 36. 

Best strategy is to start with half of 36, which is 18. 

So try adding 16, 17 and 18. This gives us 51. Too big. 

Try adding 14, 15 and 16. This adds up to 45. 

Getting closer...

Eventually you will arrive at 11, 12 and 13. 

Method 2)

Algebra saves the day...

The first number is "x". The second number must be "x + 1" because it is 1 more than the first number. The third number is "x + 2". Put these together and you can make an algebraic formula...

x + x + 1 + x+2 = 36

3x + 3 = 36

3x -3 = 36 -3

3x = 33

x = 11

Your three numbers are 11, 12 and 13.

IER - Solve For...

                 E
     I =       __
                 R

Look familiar? Solve for E? Solve for R? The answer is simple, if you think of a triangle...
                                  
                                      E

IR

(X)

Now, just cover up the letter you are asked to "solve for..." with your thumb and...presto...

1)     E = I x R

2)     R = E/I   (E divided by I)

3)     I =  E/R  (E divided by R)

Note: What throws a lot of people is that there is no numerical answer to this question. The answer isn't 4 or 2.6 or whatever. These questions look a lot harder than they really are.

I Can't Do Algebra! That "x" is a box.

"Algebra" scares people. When alphabets start invading our math pages, we panic. However...

we all did algebra in Grade 2. It just wasn't called algebra.


Grade One math looked like this..



The teacher told us to add the 2 numbers together and put the answer in the box.We did as we were told, because in Grade One, we generally did as we were told.


Grade Two, Day One math looked like this...

Pretty much the same as Grade One's work. 

The teacher told us to add the 2 numbers together and put the answer in the box.We did as we were told, because in Grade Two, we generally did as we were told. (How long would this last?)

Grade Two, Day Two math looked like this...

Everyone will say they wrote 3 in the top box and 1 in the bottom box. However, you probably wrote 7 in both boxes. Why? Because...  

The teacher told us to add the 2 numbers together and put the answer in the box.We did as we were told, because in Grade Two, we generally did as we were told. (How long would this last?) Our brains might have been telling us that our answers were illogical and wrong but we didn't want to go against the teacher. Over time we learned to put 3 and 1 in the boxes. So, if we could do that in Grade Two, then why can't we do...

The "y" and "x" are just boxes. 

If  2 + y = 5, then y = 3. 

If 3 + x = 4, then x = 1.

There - not so scary after all.

Wipe, Cancel, Divide and Stick. When "powers" aren't all that powerful - dividing monomials

24 x³

 -----

 4 x²                        Confusing? Scary?

Not really, when you can WIPE, CANCEL, DIVIDE and STICK.

Step One - WIPE out the powers.  Rewrite them as...

24 x x x

  --------

 4 x  x 

Step Two - CANCEL

24 x x x

  --------

 4 x  x 

Step Three - DIVIDE  -     24/4 = 6

Step Four - STICK  the "x" on the end...

...   6x

Step Five - Done

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.