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

Angle Properties Song

Angle Properties - Interior, exterior, corresponding...

Which is which? See...here for more info. (And another great song.)

Angles Song

Angles - Obtuse? Acute, Right? Reflex? 

Which is which? See...here for more info. (And a great song.)

Double/Double or Double/Double/Double

Are you multiplying by 4 or 8? 


Dividing by 4 or 8?


If you are multiplying by 4, think double/double.

So... 6 X 4? 

(No expense spared here at the RUGS Blog for our state of the art graphics.)

Double 6 = 12 and double 12 = 24. 

So 6 x 4 = 24.

4 x 8?

If you are multiplying by 8, think double/double/double.

Double 4 = 8 and double 8 = 16 and double 16 = 32. 

So 4 x 8 = 32.

If you are dividing by 4, think half/half.

So... 24 divided by 4? 

half of 24 is 12 and half of 12 is 6 

24 divided by 4 is 6

If you are dividing by 8, think half/half/half.

So... 24 divided by 8? 

half of 24 is 12 and half of 12 is 6 and half of 6 is 3

24 divided by 8 is 3

Adding/Subtracting Positive and Negative Numbers

There are a couple of visual/concrete models used to help student to add and subtract positive and negative numbers.

There's the thermometer...

...and the elevator...

...and the diver...

A diver rises and falls below the surface of the water. A diver can also climb out of the water on a set of stairs...

The water level is "0". So, imagine...

(-3) + (+4)

(-3) is the STARTING position of the diver. 3 steps below the water level.

+ means "and" or "then"

+ is the direction of travel.          + means up

4 is the distance of travel, in this case 4 steps up

So, think of the diver starting at 3 steps below the water surface. The diver then travels up 4 steps. The diver will emerge from the water and be one step above the water surface.

(-3) + 4 = 1 (or +1)

3 out of 4 - Again. What if there is no 3rd number?

Here's a quick review...



"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 MUADO - 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 MUADO, and get yourself the right answer.

So...

What if the question has no 3rd number? 

For example...

You need to answer 40 questions correctly on an exam in order to get the 80% passing mark. How many questions are on the exam?

Where's the 3rd number?

The third number is 100 - the % means 100.

Your two "measurements" are questions answered correctly (Q) and percent (%).

Q           %
40           80
?            100


Another example...

Pat can cut a lawn in 45 minutes? How long will it take Pat to cut 2.5 lawns?

Where's the 3rd number?

The third number is 1 - "a lawn" = 1.

Your two "measurements" are lawns (L) and time (T).

L           T
1           45
2.5         ?

What's the Chance, Part Deux

A quick review...

The GED often features one or two probability questions.

You will PROBABLY get one. Or two. 

One way to look at these questions is to think...

The chances of one thing happening...

                           ...divided by all possible chances.

For example, what is the chance of spinning a "yellow" when you use the spinner below?

The chance of the "one" thing is 1. (There is one yellow section.)

The "all possible chances" is 4. (There are 4 sections.)

The chances of one thing happening...

                           ...divided by all possible chances is..

1/4. 

Or... 1 out of 4 or...

.25 or...

25%

(These are all equivalents.)

So... what if you have 2 spinners?

What are the chances of both spinners landing on a section with the word "yellow"?

The chance for the first spinner is ¼

The chance for the 2nd spinner is ⅕

You now need to multiply these fractions.

¼  x ⅕ = 1/20. 

(Remember, 1 x 1 = 1 and 4  x  5 = 20)

What are the chances of both spinners landing on a section with the word "yellow"? 

 1/20. 

What's the Chance? Part 1

The GED often features one or two probability questions.

You will PROBABLY get one. Or two. 

One way to look at these questions is to think...

The chances of one thing happening...

                           ...divided by all possible chances.

For example, what is the chance of spinning a "yellow" when you use the spinner below?

The chance of the "one" thing is 1. (There is one yellow section.)

The "all possible chances" is 4. (There are 4 sections.)

The chances of one thing happening...

                           ...divided by all possible chances is..

1/4. 

Or... 1 out of 4 or...

.25 or...

25%

(These are all equivalents.)

Another example might be...

What is the chance of landing on a colour that has more than 3 letters, when you use the spinner above?

3/4. 

Or... 3 out of 4 or...

.75 or...

75%

So... what if you have 2 spinners? 

See the next post.

"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