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

Shares

Sharing can be so difficult, especially when it isn't 50:50. Does this question look familiar?


Chris and Pat paint a house. Chris works for 5 hours and Pat works for 3. They are paid $240.00. How much should each get paid?

Of course Pat would say $120.00 but that is not the answer. 

The key to solving is looking at the 2 numbers that AREN'T the total - in this case the 5 and 3. These are the "shares", and they add up to 8. 

If you divide the $240.00 by 8, you will get $30.00. Each share is worth $30.00. Chris worked 5 shares, so will be paid 
$30.00 x 5 - $150.00. 

Pat worked 3 shares, so will be paid $30.00 x 3 - $90.00. 

If you add $150.00 and $90.00, you will get $240.00

Note! Be careful with these types of questions on multiple choice tests. The question might ask, "How much more does Chris earn than Pat?" or something similar. 

Up in the Sky! It's a "Y"

Which is the "x" axis and which is the "y"?

1) Well, "y" is the sky, so it is the vertical axis.

2) A pirate ship is sailing ACROSS the ocean to look for treasure. The captain sees an island on the horizon."X" marks the spot on the treasure map. "X" is the horizontal axis.

Age Problems - Method 2 - Substitution

Chris is 3 times as old as Pat.               Together their ages add up to 32.                          How old is each of them?

Method 2 - "Game Show" Higher or Lower Method

a) Pat is your "reference" or starting point. Look for the word "as" or "than". (The first sentence could have been written "Chris is 3 times older than Pat.") The word associated with "as" or "than" becomes your reference. 

b) Pick 1 to represent Pat's age. 

c) If Pat is 1, then Chris has to be 3 because "Chris is 3 times as old as Pat."

d) "Together their ages add up to 32..." so...
1 + 3 will not work, as it doesn't = 32. The answer is too low. Pick a higher number to represent Pat's age. 

e) Pick 10 to represent Pat's age. 

f) If Pat is 10, then Chris has to be 30 because "Chris is 3 times as old as Pat."

g) "Together their ages add up to 32..." so...

10 + 30 will not work, as it doesn't = 32. The answer is too high. Pick a lower number to represent Pat's age. 


h) Eventually, by going higher and lower, you'll figure out that Pat is 8. Chris is 24 because Chris is 3 times older than Pat.

i) Check! Always check! "Together their ages add up to 32."       8 + 24 = 32.

Classic "Age" Problems - A couple of ways to solve these questions - Method 1

Chris is 3 times as old as Pat.               Together their ages add up to 32.                          How old is each of them?

Method 1 - 

a) Create an equation. Pat is your "reference" or starting point. Look for the word "as" or "than". (The first sentence could have been written "Chris is 3 times older than Pat.") The word associated with "as" or "than" becomes your reference. 

b) Pick a letter to represent Pat's age. We'll use "x".

c) If Pat is "x", then Chris has to be 3x.

d) "Together their ages add up to 32..." so...
x + 3x = 32

e) Combine your x's. (That's a lot of alimony). 
4x = 32

f) This means 4 times "something" = 32. The answer is 8. One "x" = 8

g) Pat is 8, because Pat is one "x". Chris is 24 because Chris is 3 times older than Pat.

h) Check! Always check! "Together their ages add up to 32."       8 + 24 = 32.

Complementary? Supplementary?

My students often can't remember which angles are complementary and which ones are supplementary. I always tell them to think of a really cheap wedding, where the complementary (free) drinks are limited to 90. Complementary angles add up to 90 degrees. "C" for "cheap" and "c" for complementary. A "superb" wedding (supplementary) would offer 180 free drinks. supplementary angles add up to 180 degrees. "S" for "superb" and "s" for supplementary. Failing that, I tell them that "c" comes before "s" in the alphabet and 90 comes before 180.

complementary

supplementaryfrom ClipArt, etc.