Don't Call it Algebra (And Nobody Will Know!)

The word Algebra can be intimidating to GED students.

Many people have this conception of Algebra...

I always tell my students that they did algebra back in Grade 2 but they just didn't know it.

Grade 1 -    1 + 1 = ▭

We followed the directions and added the 2 numbers together and wrote "2" in the box.

1 + 1 = 2

Grade 2  Day 1

1 + 1 = ▭

We followed the directions and added the 2 numbers together and wrote "2" in the box.

Grade 2  Day 2

1 + ▭ = 4

We followed the directions and added the 2 numbers together and wrote "5" in the box.

1 + 5 = 4

It looked wrong, and we knew it was wrong, but we followed the directions that said we had to add the two numbers together and put the answer in the box.

After a leap in our thinking, we were able to say that...

1 + "something" = 4

That "something" is 3. 

1 + 3 = 4

1 + ▭ = 4        The ▭ = 3

So how about 1 + x = 4?

"x" is just a ▭                  x = 3

I tell my students that they did Algebra in Grade 2 so they can certainly do it now in GED study.

"3 out of 4" Questions - Rate and Proportion

This is a modification from an earlier post. I realised there might be a few Heavy Metal fans out there...fans of the band MUAADON.

"3 out of 4" Questions - Rate and Proportion

Look or sound familiar? 



"A 10 metre flag pole casts a shadow of 6 metres. A 12 metre pole would cast a shadow that is ______ metres in length."

"A motorcyclist rides 125 miles in 3 hours. At this rate, how much distance will be covered in 5 hours?"

A factory produces 3000 widgets every 45 minutes. How many widgets will be produced in 7 hours?


These questions, I like to call, are "3 out of 4" questions. You are given 3 numbers and you need to find the 4th. How to solve? Make yourself a 4-way relationship box. 


Let's look at question 1 again. 

"A 10 metre flag pole casts a shadow of 6 metres. A 12 metre pole would cast a shadow that is ______ metres in length."

Your two "measurements" are height (H) and shadow length (SL).

H           SL
10            6
12            ?

Remember MUAADON - Multiply Up And Across, Divide by the Other Number.

12 x 6 = 72
72/10 = 7.2

You could have set your box up another way.

H      10    12
SL     6      ?

6 x 12 = 72
72/10 = 7.2

Same answer...


Tricks to watch out for!

In question 3, the two "measurements" are time and widgets. However, the first "time" is in minutes and the second one is in hours. You need to either convert the 45 minutes to .75 hours, or convert the 7 hours to 420 minutes. Then set up your box, remember MUAADON, and get yourself the right answer. Rock on!

Come Undun

From  de.wikihow.com

Do you know "Y" you were intercepted Part Three

"Do you know "Y" you were intercepted?" Part Three

 Which line has the equation          y = x + 6

Ok, smarty pants. What if 2 lines cross at the same spot and both have positive slopes? What then? (In this case, lines 5 and 8)

 

 

 

Then, we have to work out the slope of lines 5 and 8.

 

 

Think of jogging. If you want to go for a run up a slope in the morning, you must rise first before you run.

So...    to calculate the slope of a line...
use the formula...             slope = rise
                                                        run

Rise divided by run.

 

Rise is the vertical distance between two points on a slope. Run is the horizontal distance between two points on a slope.

 

Pick any 2 spots on line #5. Try to pick spots at the corners of the underlying grid.

 

slope = rise   2   = 1    run    2

 

Pick any 2 spots on line #8. Try to pick spots at the corners of the underlying grid.

 

 

slope = rise   4   = 2    run    2

 

Which line has the equation          y = x + 6

 

(Remember that this question can be written as y = 1x + 6. By convention, in algebra, the "1" in front of a letter or variable can be dropped for convenience.)

 

The slope of the line is associated with the value in front of the "x" in the given equation. The free floating number (in this case the "6"), is the place on the Y axis where the line crosses or intercepts.

 

Has to be line 5 for your answer, as it has a slope of 1. You are done.

 

Work SMART not HARD.

 

 

 

Do you know "Y" you were intercepted Part Deux

OK - so maybe the last post was a bit easy.

Perhaps there might be a harder question on the GED test?

A quick review... Which line has the equation

 

 y = x + 6

 

 

First of all, image you have been driving along and are suddenly "intercepted" by the police.

"Do you know "y" you were intercepted?" is probably the question you will be asked.

 

The key to this question is the "y" intercept, or the location at which the line crosses (intercepts) the "y" or vertical axis. (For some reason, nobody really cares about the poor old "x" axis. As with all "ex's", they are best left forgotten.)

 

y = x + 6 

 

The free floating number here is the 6. The only line that crosses/intercepts the "y"axis at 6 is line 5. Job done. You are finished.

 

                                 

 

OK, but what about...

 

 

 

 

 

 

Looks like 2 lines intercepting the "y" axis at 6. (Lines 5 and 7) So how now, brown cow?

 

Well, the next clue is the slope of the line. The free floating number corresponds with the intercept while the value associated with the "X" determines the slope. In this case, the slope is positive 1.

 

How can that be? (In algebra, values of positive 1 are dropped, being "understood" to be positive 1.) The formula for this line could have been written as y = 1x + 6

 

So we now need to find which line has a slope of positive 1.

 

By convention, lines that "rise" from the lower left to the upper right are considered to have a positive slope, Lines that "fall" from the upper left to the lower right have negative slopes. That rules out line 7.

 

 

Work SMART, not HARD!

Do you know "Y" you were "intercepted"?

Look familiar? - Which line has the equation

 

 y = x + 6

These questions look harder than they are.

 

First of all, image you have been driving along and are suddenly "intercepted" by the police.

"Do you know "y" you were intercepted?" is probably the question you will be asked.

 

The key to this question is the "y" intercept, or the location at which the line crosses (intercepts) the "y" or vertical axis. (For some reason, nobody really cares about the poor old "x" axis. As with all "ex's", they are best left forgotten.)

 

y = x + 6 

 

The free floating number here is the 6. The only line that crosses/intercepts the "y"axis at 6 is line 5. Job done. You are finished.

 

Work SMART, not HARD!



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!

 

 





  

 

 

 



Trick Question!

The sum of three consecutive positive odd numbers is 69.

What is the difference between the biggest and smallest of these three numbers?

HINT: Don't work out the three numbers! (They are 21, 23 and 25, by the way). The difference between the largest and smallest of any three given consecutive positive odd numbers is 4. Try it...


1,3,5


3,5,7

etc.

How much precious exam time would you have used up trying to work out that question?

Work SMART, not HARD.

"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

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.

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. 

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.