Those Dynamic FractionsThose Dynamic Fractions

Dynamic: Dynamic:

Characterized by continuous change, Characterized by continuous change, activity, or progress activity, or progress

Fractions can change in many waysFractions can change in many ways– They can use larger numbers or smaller They can use larger numbers or smaller

numbersnumbers– They can be changed to decimals or They can be changed to decimals or

percentagespercentages

But even when a fraction’s But even when a fraction’s appearanceappearance changes, its value changes, its value mustmust remain the same. remain the same.

Fractions can use bigger numbers or Fractions can use bigger numbers or they can use smaller numbers…they can use smaller numbers…

1/2 can be called 2/4 or 3/6 or 50/100 1/2 can be called 2/4 or 3/6 or 50/100 but the top number is always half of but the top number is always half of the bottom numberthe bottom number

12/20 can be called 12/20 can be called

24/40, 24/40,

3/5 or 3/5 or

6/10 6/10

but the top number is always a but the top number is always a multiple of 3 and the bottom number multiple of 3 and the bottom number is always a multiple of 5.is always a multiple of 5.

You try…You try…On your answer sheet, write some On your answer sheet, write some

other names for ¼ in the first space.other names for ¼ in the first space.

Try to think of at least 3 other names. Try to think of at least 3 other names. Remember the denominator (bottom Remember the denominator (bottom

number) for these answers will number) for these answers will always be 4 times bigger than the always be 4 times bigger than the top number.top number.

1/2, 1/4 and 3/5 are called 1/2, 1/4 and 3/5 are called Lowest Lowest TermsTerms because they cannot go any because they cannot go any lower. lower.

Most people prefer to talk about Most people prefer to talk about fractions in lowest terms. If a fraction fractions in lowest terms. If a fraction is not in lowest terms, you have to is not in lowest terms, you have to reducereduce it. it.

To reduce a fraction, you divide. To reduce a fraction, you divide.

And just as your mother and father try And just as your mother and father try to keep things fair between you and to keep things fair between you and your brothers and sisters, you have your brothers and sisters, you have to keep things fair between the to keep things fair between the numerator (top number) and numerator (top number) and denominator (bottom number).denominator (bottom number).

To reduce 6/8, you divide the top and To reduce 6/8, you divide the top and bottom numbers by the same thing.bottom numbers by the same thing.

Ask yourself, which times tables have Ask yourself, which times tables have both 6 and 8 for an answer.both 6 and 8 for an answer.

The answer would be the twos times The answer would be the twos times tables. 3X2 = 6 and 4X2 =8.tables. 3X2 = 6 and 4X2 =8.

I usually write it like this:I usually write it like this:66÷ 2 = ÷ 2 = 338÷ 2 = 48÷ 2 = 4

6/8 reduces to 3/46/8 reduces to 3/4

Now you try: reduce 10/14 and write the Now you try: reduce 10/14 and write the answer in the second space.answer in the second space.

Remember to be fair to the numerator and Remember to be fair to the numerator and denominator.denominator.

To reduce a fraction to To reduce a fraction to lowest termslowest terms, , you keep reducing it until you can’t you keep reducing it until you can’t reduce anymore.reduce anymore.

How do you tell when you can’t reduce How do you tell when you can’t reduce anymore? How do you tell when anymore? How do you tell when your fraction is in lowest terms?your fraction is in lowest terms?

There are 5 ways to tell:There are 5 ways to tell:

1.1. When the top number is a 1When the top number is a 1

2.2. When the numerator and denominator When the numerator and denominator are neighbors (like 2/3 or 5/6)are neighbors (like 2/3 or 5/6)

3.3. When both the numerator and When both the numerator and denominator are prime numbersdenominator are prime numbers

4.4. When the numerator is a prime number When the numerator is a prime number and does not divide evenly into the and does not divide evenly into the denominatordenominator

5.5. When the numerator and denominator When the numerator and denominator have no common factors.have no common factors.

Now you try…Now you try…How many of these fractions are in How many of these fractions are in

lowest terms? Write your answer on lowest terms? Write your answer on your answer sheet.your answer sheet.

7/21 3/5 6/127/21 3/5 6/12

1/18 3/8 11/121/18 3/8 11/12

10/2510/25 2/322/32 4/254/25

You don’t ALWAYS want fractions in You don’t ALWAYS want fractions in lowest terms.lowest terms.

When you compare fractions or add When you compare fractions or add and subtract them, you have to find and subtract them, you have to find common denominators.common denominators.

Common denominators aren’t usually Common denominators aren’t usually in lowest termsin lowest terms

When you reduce fractions, you divide.When you reduce fractions, you divide.

When you find common denominators, When you find common denominators, you multiply.you multiply.

To find common denominators, for 1/2 To find common denominators, for 1/2 and 3/5, you look at the 2 and the 5 and 3/5, you look at the 2 and the 5 (because those are the (because those are the denominators).denominators).

You look at the smaller denominator (the 2) You look at the smaller denominator (the 2) and you start saying your twos times and you start saying your twos times tables (2, 4, 6, 8, 10, 12, 14, 16, 18, etc.)tables (2, 4, 6, 8, 10, 12, 14, 16, 18, etc.)

You stop when you find a number that the 5 You stop when you find a number that the 5 will go into evenly – that would be 10 in will go into evenly – that would be 10 in this case.this case.

Our common denominator for 2 and 5 is 10 Our common denominator for 2 and 5 is 10 because 10 is in both the 2 and the 5 because 10 is in both the 2 and the 5 times tables.times tables.

This is how I write it:This is how I write it:

11 X X = = __2 X 2 X = 10 = 10

ANDAND

33 X X = =__

5 X 5 X = 10 = 10

Copy these onto your answer sheet.Copy these onto your answer sheet.

I call it “equal sign and a line”.

Those little boxes look like little elevators, don’t Those little boxes look like little elevators, don’t they?they?

11 X X = = __2 X 2 X = 10 = 10

Now your fraction Now your fraction looks like this:looks like this:

11 X 5 = X 5 = 5 5 2 X 5 = 102 X 5 = 10

5/10 is an equivalent fraction name for 1/2. We 5/10 is an equivalent fraction name for 1/2. We know this is true because 5 is half of 10.know this is true because 5 is half of 10.

On your answer sheet, write 5/10 beside the 1/2.On your answer sheet, write 5/10 beside the 1/2.

Ask yourself, 2 times what number equals 10?

5, right? Right. So 5 goes in the elevator. The elevator goes up to the second floor and the 5 is still there. So, 1 times 5 is what?

5, right? Right. So 5 goes on top of the 10.

Now let’s work with the 3/5…Now let’s work with the 3/5…

33 X X = = __5 X 5 X = 10 = 10

33 X 2 = X 2 = 66

5 X 2 = 105 X 2 = 10

Write 6/10 beside the 3/5 on your Write 6/10 beside the 3/5 on your answer sheet.answer sheet.

This time, ask yourself, 5 times what number equals 10?2, right? Right. So 2 goes in the elevator. The elevator goes up to the second floor and the 2 is still there. So, 3 times 2 is what? 6, right? Right. So 6 goes on top of the 10.

11 X 5 = X 5 = 5 5 2 X 5 = 2 X 5 = 1010

33 X 2 = X 2 = 665 X 2 = 5 X 2 = 1010

This is what you should now have written on your answer sheet. 1/2 and 3/5 now have common denominators.

Now we can compare them (which one is bigger? Smaller?)

Or we can add them. If we put the bigger one on top, we could subtract them.

And when we get done, we might need to reduce the answer.

THAT’S how you make dynamic fractions.

22 X X = = ? ?

3 X 3 X = 12 = 12

11 X X = = ??12 X 12 X = = 1212

Let’s try another one. Find common denominators for 2/3 and 1/12 (What number is in both the 3 and the 12 times tables?)

3X1 = 3

3X 2 =6

3X 3 = 9

3 X 4 = 12

3 X 5…

Hey, wait a minute… 12 works for both of them!

The common denominator is 12!

11 X X = = ??

12 X 12 X = 12 = 12

22 X X = = ? ?

3 X 3 X = 12 = 12

Next, I figure out the numerators (top numbers).

Those boxes look like elevators.

3 times what number equals 12? That would be 4. Put four in the elevator and send it upstairs… 2 X 4 = 8. The top number is 8!

Next, I figure out the numerator here.

Those boxes look like elevators.

12 times what number equals 12? That would be 1. Put 1 in the elevator and send it upstairs… 1 X 1= 1. The top number is 1!

4

4

1

1

22 X X 44 = = 8 8

3 X 3 X 44 = 12 = 12

11 X 1 = X 1 = 11

12 X 1 = 1212 X 1 = 12

Now we can:

Compare

Add

Subtract

Now you try. Write the answer on your Now you try. Write the answer on your answer sheet.answer sheet.

What would be common denominators for What would be common denominators for these fractions:these fractions:

33 44

55 66

Did you do it?Did you do it?

Try these. Write these answers on Try these. Write these answers on your answer sheet.your answer sheet.

What are some other names for these What are some other names for these fractions?fractions?

1/21/2 1/41/4 2/102/10 6/216/21

And that’s how you And that’s how you make dynamic make dynamic

fractions!fractions!