1. Those Dynamic Fractions

2. Dynamic: Characterized by continuous change, activity, or progress
Fractions can change in many ways
They can use larger numbers or smaller numbers
They can be changed to decimals or percentages
But even when a fraction’s appearance changes, its value must remain the same.

3. Fractions can use bigger numbers or they can use smaller numbers…
1/2 can be called 2/4 or 3/6 or 50/100 but the top number is always half of the bottom number

4. 12/20 can be called
24/40,
3/5 or
6/10
but the top number is always a multiple of 3 and the bottom number is always a multiple of 5.

5. You try…
On your answer sheet, write some other names for ¼ in the first space.
Try to think of at least 3 other names.
Remember the denominator (bottom number) for these answers will always be 4 times bigger than the top number.

6. 1/2, 1/4 and 3/5 are called Lowest Terms because they cannot go any lower.
Most people prefer to talk about fractions in lowest terms. If a fraction is not in lowest terms, you have to reduce it.

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

8. To reduce 6/8, you divide the top and bottom numbers by the same thing.
Ask yourself, which times tables have both 6 and 8 for an answer.
The answer would be the twos times tables. 3X2 = 6 and 4X2 =8.

9. I usually write it like this:
6÷ 2 = 3
8÷ 2 = 4
6/8 reduces to 3/4
Now you try: reduce 10/14 and write the answer in the second space.
Remember to be fair to the numerator and denominator.

10. To reduce a fraction to lowest terms, you keep reducing it until you can’t reduce anymore.
How do you tell when you can’t reduce anymore? How do you tell when your fraction is in lowest terms?

11. There are 5 ways to tell:
When the top number is a 1
When the numerator and denominator are neighbors (like 2/3 or 5/6)
When both the numerator and denominator are prime numbers
When the numerator is a prime number and does not divide evenly into the denominator
When the numerator and denominator have no common factors.

12. Now you try…
How many of these fractions are in lowest terms? Write your answer on your answer sheet.
7/ / /12
1/ / /12
10/25 2/32 4/25

13. You don’t ALWAYS want fractions in lowest terms.
When you compare fractions or add and subtract them, you have to find common denominators.
Common denominators aren’t usually in lowest terms

14. When you reduce fractions, you divide.
When you find common denominators, you multiply.
To find common denominators, for 1/2 and 3/5, you look at the 2 and the 5 (because those are the denominators).

15. You look at the smaller denominator (the 2) and you start saying your twos times tables (2, 4, 6, 8, 10, 12, 14, 16, 18, etc.)
You stop when you find a number that the 5 will go into evenly – that would be 10 in this case.
Our common denominator for 2 and 5 is 10 because 10 is in both the 2 and the 5 times tables.

16. Copy these onto your answer sheet.
This is how I write it:
1 X  = _
2 X  = 10
AND
3 X  =_
5 X  = 10
Copy these onto your answer sheet.
I call it “equal sign and a line”.

17. Those little boxes look like little elevators, don’t they? 1 X  = _
Now your fraction
looks like this:
1 X 5 = 5
2 X 5 = 10
5/10 is an equivalent fraction name for 1/2. We know this is true because 5 is half of 10.
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.

18. Now let’s work with the 3/5… 3 X  = _ 5 X  = 10 3 X 2 = 6 5 X 2 = 10
Write 6/10 beside the 3/5 on your 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.

19. This is what you should now have written on your answer sheet
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.
1 X 5 = 5
2 X 5 = 10
3 X 2 = 6
5 X 2 = 10

20. 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!
2 X  = ?
3 X  = 12
1 X  = ?
12 X  = 12

21. 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!
2 X  = ?
3 X  = 12 4 4
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!
1 X  = ?
12 X  = 12 1 1

22. 2 X 4 = 8
3 X 4 = 12
Now we can:
Compare
Add
Subtract
1 X 1 = 1
12 X 1 = 12

23. Now you try. Write the answer on your answer sheet.
What would be common denominators for these fractions: 3 4 5 6
Did you do it?

24. Try these. Write these answers on your answer sheet.
What are some other names for these fractions?
1/2 1/4 2/10 6/21

25. And that’s how you make dynamic fractions!