Is it Prime (Or is it Memorex)?

A prime number is a natural number greater than 1, that has no positive divisors (factors) other than 1 and itself.

If you think of this definition in pictures of rectangles...

2 - ᥆᥆
3- ᥆᥆᥆
4 - ᥆᥆᥆᥆ or  ᥆᥆
                       ᥆᥆
5 - ᥆᥆᥆᥆᥆
6 - ᥆᥆᥆᥆᥆᥆  or ᥆᥆᥆
                             ᥆᥆᥆

Any number that can ONLY be represented by a straight line is prime.

Straight line is prime.

"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!

Math Doctor Video - New Version

See Math Doctor Link for the new Math Doctor Video - write out a 10 by 10 multiplication in under 1.5 minutes.

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.

 

 

 

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.

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.

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 

In the dog house

Look familiar? 

1) Pat scored 54 out of 200 on a test. What percent is this?

or...

2) 36 out of 300 people ride bikes to work. What percent is this?

The key to solving these types of questions comes in the words " out of ".

The " out of " number needs to be thrown " out of " the house. 200 has forgotten the wedding anniversary (yet again!) and has to go and sleep in the dog house. 200 is not allowed in the house.

54 is inside...

and 200 is out of the house, in the dog house...

So now we have...

 200 divided into 54. 

This is a job for Captain Zero and will be covered in the next blog post. 

Note: Do not get into the habit of assuming you are dividing the "bigger number" into the smaller one. You might get a question such as...

Pat scored 154 out of 150 on a test. Bonus marks were scored! What percent is this? 

If Pat scored more than 150, then the answer has to be more than 100%. In this case, 150 is your " out of " number.

          

Ah, yes, the good old Circular Lawn or Swimming Pool Question

You may get a question that not many people, outside of a math test, would worry about. Or think about.

The old circular lawn or swimming pool question.

As the incredibly professional-looking graphic above looks more like a pool than a lawn, let's go with this type of question...

A circular pool has a sunbathing deck in the middle. The distance from the centre to the inner edge of the water is 4m. The distance from the centre to the outer edge is 6m. What area does the water cover?

(Hint: It isn't 2 metres.) 

What is missing?

Imagine the pool deck is a like the centre of a pie that has been eaten.The water is the pie that hasn't been eaten. So, work out the area of the whole pie and subtract the area of the piece that was eaten. What is left is your answer.

The area of a circle is...

pi X radius squared

So...

pi x 6 squared         subtract        pi x 4 squared

62.83 square metres.

Note: You won't get the right answer by working out pi x 2 squared (2 being the difference between 6 and 4.)

Perhaps the easiest way to remember this is  

pi x big radius squared  

subtract 

pi x small radius squared 

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