Grandma's Secret Ingredient Pie!

We all remember our grandmother's special pie; she only made it once a year *around the holidays of course* and she never told anybody the secret ingredient.  What you didn't know is that all this time you have been adding and subtracting fractions!! Don't believe me? Ok. I'll show you!!

Let's pretend that this does not resemble a basketball, but is instead grandma's cherry pie.  She always cut it into 8 equal slices, as shown.  Now, when she gave you a slice, you were handed 1/8 of the pie.  Easy right?  Well, say you asked for another slice...How much would you have had then?  And what if you couldn't finish the first slice *It was a huge meal!* and so you shared it, how much would you have had??  This is where adding and subtracting fractions comes in.

Adding Fractions
When adding fractions, one things is very important: a common denominator.  What is that??  Well, in a fraction, you have two numbers, the top number and the bottom number.  The numerator is the top number; the denominator is the bottom number.  So basically we want to have the same number on the bottom.  After finding a common denominator, then we can add the numbers straight through:

1  + 3 = 4

8     8     8

In this case, we did not have to change the denominator because it was already common.  Let's try a different one:  

1 + 2 = ??

                                                           4    3
Here we want to find a common denominator.  To do this, we must find a common multiple of the numbers.  Let's list the first few multiples:

3: 3, 6, 9, 12, 15, 18                      4: 4, 8, 12, 16, 20, 24 

Right away I see one common multiple: 12! We want to change the denominator to 12, however without changing the value of the fraction.  This means that we will have to multiply the numerator by the same number that we multiply the denominator.

(1 x 3) + (2 x 4) -->  3 + 8  = 11
(4 x 3)    (3 x 4)      12  12    12

After we add the fractions we want to make sure to simplify the fraction.  In this case, the answer was already simplified. 

Subtracting Fractions
When it comes to subtracting fractions, the first few steps are the same as adding fractions: we want a common denominator.  The only difference is that we are subtracting the numbers instead of adding.  Lets try one:

5 - 2 = (5 x 2) - 2 = 10 - 2 = 8
8  16   (8 x 2)  16   16  16   16

We can further simplify the answer by dividing both the numerator and the denominator by 8.  So the simplified answer is 1
                                            2.

For an interactive fraction addition game, click here. For an interactive fraction subtraction game, click here.

Greatest Common Factor!!

Okay, so first let's define what a greatest common factor (GCF) is: the GCF of two or more numbers is the largest number that divides into both numbers. We are going to look at three simple ways to find the GCF: Venn diagrams, prime factorization, and the Euclidean Algorithm.

So we all remember Venn diagrams right? A Venn Diagram is a tool used to sort and compare information. When it comes to GCF, we are simply sorting the factors for two or more numbers.  For this first example, we are going to factor 36 and 42.  In the left circle we will list the factors of 36, and in the right circle we will list the factors of 42.  In the intersection, we will list the factors that are Common to both 36 and 42.

The factors of 36 are: 1, 2, 3, 4, 6, 9, 12, 18, and 36.  The factors of 42 are: 1, 2, 3, 6, 7, 14, 21, and 42.  The factors Common to both 36 and 42 are: 1, 2, 3, 6.  In this visual representation, it is easy to find the greatest, or largest, common factor.  We can see that the GCF is 6.





A second method to find the GCF is prime factorization.  With prime factorization, we use 'factor trees' to find which prime numbers we need to multiply to get a number.  For example, the prime factorization of 36 is: 2x2x3x3.  Let's use the numbers 24 and 72 for this example.

With these factor trees, we want to find two factors that multiply together to get the original number.  2x12=24.  However, we want to keep doing this until all factors are primes, so we have to factor 12, and then 4 and so on. In the end we find that: 24=2x2x2x3 and 72=2x2x2x3x3.  Now that we have the prime factorization, we want to find the "common multiples".  In this case, the common multiples are 2x2x2x3; after we multiply these, we find that the GCF is 24!  

A third method to finding the GCF is by using the Euclidean Algorithm.  For this algorithm, I found a great video that goes through each step in detail! Let's watch!

Well that is it for today.  For more examples and more practice on finding the GCF, click here!

Three Cheers for Ancient History!

I've always been the person that was fascinated by ancient history...It's just so ancient!  So when the class went over 'ancient numeration systems', I was quite surprised at my frustration.  When we learn to count, sometimes we take for granted how difficult it really is to learn.  So, I thought I'd share two of these today: the Egyptian Numeration system and the Babylonian Numeration system.

The Egyptian system is actually not so tough to pick up because the ancient Egyptians use a base of 10. Here is a little video that explains further:




Ok. So that isn't so hard right? Well, the Babylonian Numeration System is a bit more difficult.  First, they only use two symbols.  Second, instead of base 10, they use bas 60!!!



So does this make a little more sense?? It does take some practice to really understand these systems, believe me I know!  So, for more background information and more practice, click here!

Elementary, My Dear Watson!

Let me start off by saying that Sherlock Holmes never actually said this, as many assume.  In Doyle's novels, these words can be found, but not in this exact order:

Sherlock: ...my dear Watson. (A little later) Watson: Elementary. 

However, this quote can be found in the 1929 film about Sherlock Holmes and has been made famous.  A person that has never read Sir Arthur Conan Doyle but has seen this movie and heard the quote many times before may conclude that this phrase is said by Holmes repeatedly.  This person has used inductive reasoning to come to this conclusion, though it is an incorrect conclusion.

Inductive reasoning is defined as"the process of forming conclusions on the basis of observations and experiments".  The conclusions made through the use of inductive reasoning are not always true, as found in the example above.  Here is a better example: "All sheep that I have seen are white, therefore all sheep are white." Again, this conclusion is wrong, but this does not mean that all conclusions made by inductive reasoning are wrong.

Another form of reasoning is deductive reasoning.  Deductive reasoning is defined as "the process of deriving conclusions from given statements".  With deductive reasoning, you are given a set of 'premises', or a general rule known to be true; with this set of premises, you can make a specific conclusion:

Premise: All men are mortal. Socrates is a man.

With these clues, I can deduce that Socrates is mortal.

A useful tool to see if conclusions are valid through your deductions is a Venn Diagram.  This slideshow will help me to explain:

Using Venn Diagrams with Deductive Reasoning

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You can also use deductive reasoning to solve logic problems.  Let's look at the following problem:

Use the given clues to match the jobs, majors, and method of transportations with the correct students
Students: Freshman, Sophomore, Junior, Senior
Work: Coffee Shop, Library, Theatre, Tutor
Major: English, Computer Science, Math, Psychology
Transportation: Bike, Bus, Car, Skateboard

A. The English major rides a bike
B. The tutor is a junior and won't ride a skateboard.
C. The sophomore takes the bus.
D. The freshman won't work in the coffee shop or drive a car.
E. The skateboarder is a psychology major.
F. The computer science major works in the library.
G. The math major works in the coffee shop and won't drive a car.
H. The senior drives a car.

Okay. So the first thing we want to do is to either create a chart, cut out slips of paper for each word, or any other sort of visual representation that works for you.  I chose to create a chart.  Then, you want to take each step one at a time, filling out any information given.

A. So, I pair English major and bike in the same column because of the first clue given.

Student
Work
Major	English
Transportation	Bike

B. I know that the junior is a tutor, and does not ride a skateboard, so I fill that out.

Student		Junior
Work		Tutor
Major	English
Transportation	Bike	(not a skateboard)

C. I also know that the sophomore takes the bus.

Student		Junior	Sophomore
Work		Tutor
Major	English
Transportation	Bike	(not a skateboard)	Bus

D. The freshman does not work in the coffee shop, and they do not ride a car.  I know that the freshman will either ride the bike or the skateboard.

E.  This is what my table would look like after reading the clues. 

Student		Junior	Sophomore				Senior
Work		Tutor			Library	Coffee Shop
Major	English			Psychology	Computer Science	Math
Transportation	Bike	(not skateboard)	Bus	Skateboard			Car

G. Now, I go back to clue B. I know that the junior does not ride the skateboard. Because the senior rides the car and the sophomore rides the bus, the only other option is the bike. So, I combine the column with the bike and the English major into the column with the junior's information.

Student	Junior	Sophomore				Senior
Work	Tutor			Library	Coffee Shop
Major	English		Psychology	Computer Science	Math
Transportation	Bike	Bus	Skateboard			Car

F. Now I look at clue D.  I knew that the freshman will either ride the bike or the skateboard. Now I know that the freshman rides the skateboard, so I will add freshman to that column.

Student	Junior	Sophomore	Freshman			Senior
Work	Tutor			Library	Coffee Shop
Major	English		Psychology	Computer Science	Math
Transportation	Bike	Bus	Skateboard			Car

G. Now, clue G states that the math major does not drive a car, so I know that the math major is not the senior.  It has to be the sophomore, because I already have a major for the junior and the freshman.

Student	Junior	Sophomore	Freshman		Senior
Work	Tutor	Coffee		Library
Major	English	Math	Psychology	Computer Science
Transportation	Bike	Bus	Skateboard		Car

H. This leaves the computer science major for the senior.

Student	Junior	Sophomore	Freshman	Senior
Work	Tutor	Coffee		Library
Major	English	Math	Psychology	Computer Science
Transportation	Bike	Bus	Skateboard	Car

I. Now all that is left is the job for the freshman.  For this, I will look at the list of jobs, and fill in the job that is left: theatre. I will also organize the information in order of freshman to senior.

Students	Freshman	Sophomore	Junior	Senior
Work	Theatre	Coffee Shop	Tutor	Library
Major	Psychology	Math	English	Computer Science
Transportation	Skateboard	Bus	Bike	Car

And this is the answer! So, remember that by taking one clue and one step at a time, we can use deductive reasoning to solve these logic problems. Sherlock Holmes would be proud!

For more practice on logic puzzles or fun puzzles to share with your students, click here!