1. TOOL CHECK

2. Adding and Subtracting FRACTIONS Unit 4 Lesson 4

3. First of all, what makes up a Fraction? A fraction has two parts to it: A fraction has two parts to it: A Numerator (the top number) A Numerator (the top number) And a Denominator (the bottom number) And a Denominator (the bottom number)

4. How do you ADD FRACTIONS? First of all, you need a “common denominator”. This means the bottom numbers of each fraction must be the same. First of all, you need a “common denominator”. This means the bottom numbers of each fraction must be the same. ½ + ¾ Cannot be added together... Yet. 2/4 + ¾ Can be added because the denominators are “common” (the same)

5. How did you do? To start any problem, you first need to determine if you CAN add them together as they are. Or…if you need to change them somehow to add them.

6. Can These Be Added? A. ¾ + ¼ B. ½ + 5/8 C. 3/16 + 5/16 D. 1 ½ + 3 ½ E. 10 3/16 + 3 5/8 F. 15/16 + 3 3/8 G. 2 7/8 + 2 3/8

7. Making a Common Denominator

8. How to make a common denominator. Here’s what you do if the denominators are different: You first need to find a number that BOTH denominators can divide into evenly. Find the common denominator for: Find the common denominator for: 2 and 4 2 and 4 ANSWER: 4 ANSWER: 4 16 and 4 16 and 4 ANSWER: 16 ANSWER: 16 4 and 8 4 and 8 ANSWER: 8 ANSWER: 8

9. HINT Did you notice that the common denominator was ALWAYS the bigger of the two denominators? Did you notice that the common denominator was ALWAYS the bigger of the two denominators? Just remember that this rule ONLY applies in woodworking. Not in your math class. Just remember that this rule ONLY applies in woodworking. Not in your math class.

10. Converting the Fractions

11. Converting the Fraction Step #1 Let’s try an example together! Let’s try an example together! ½ + ¾ The ½ needs to be converted to match the bigger denominator. The ½ needs to be converted to match the bigger denominator. So…(what number) x 2 = 4? So…(what number) x 2 = 4? Answer: 2 Answer: 2 Simple huh? Simple huh?

12. Converting the Fraction Step #2 Take the answer (2) and multiply it by both the numerator and denominator. Take the answer (2) and multiply it by both the numerator and denominator. 2 x ½ (OR) 2 x 1 = 2 2 x 2 = 4 2 x 2 = 4 Do you agree that ½ = 2/4? So now…2/4 + 1/4 can be added together.

13. Adding the Fractions

14. Adding the Converted Fraction Now…what do we do with 2/4 + 1/4? Now…what do we do with 2/4 + 1/4? All that’s left is adding ONLY the numerators. The denominator IS NOT added. It stays the same. All that’s left is adding ONLY the numerators. The denominator IS NOT added. It stays the same. So… 2/4 + 1/4 = 3/4 THE ANSWER!!! So… 2/4 + 1/4 = 3/4 THE ANSWER!!!

15. Conclusions All addition problems take the same steps to solve. All addition problems take the same steps to solve. The common denominator will ALWAYS be the bigger denominator of the two. The common denominator will ALWAYS be the bigger denominator of the two. Don’t be afraid of the problem if it has big numbers. It’s easy! Don’t be afraid of the problem if it has big numbers. It’s easy!

16. Subtracting Fractions

17. Subtraction Subtracting fractions begins exactly the same way as adding fractions. Subtracting fractions begins exactly the same way as adding fractions. The first thing you have to do is figure out if you CAN subtract them as they are. The first thing you have to do is figure out if you CAN subtract them as they are. If not, you will need to convert a denominator so you can. If not, you will need to convert a denominator so you can.

18. How did you do? Remember that all you need to know is if they are able to be subtracted. Remember that all you need to know is if they are able to be subtracted. If not, we need to convert one of the fractions. If not, we need to convert one of the fractions.

19. Can these be subtracted? 1 ½ - ¾ 1 ½ - ¾ 15/16 – 3/16 15/16 – 3/16 3 5/8 – 1 ½ 3 5/8 – 1 ½ 5 2/4 – 3 ¼ 5 2/4 – 3 ¼ 10 5/8 – 7 15/16 10 5/8 – 7 15/16 3 ¼ - 1 ¼ 3 ¼ - 1 ¼ 7 7/8 – 3 13/16 7 7/8 – 3 13/16

20. Make a common denominator

21. Let’s do one together 1 ½ - ¼ 1 ½ - ¼ You can see that one of them needs to be converted so you can subtract them. You can see that one of them needs to be converted so you can subtract them. What will the common denominator be? What will the common denominator be? ANSWER: 4 ANSWER: 4

22. Step #1Step #2 Identify the common denominator. Identify the common denominator. 1 ½ - ¼ 1 ½ - ¼ ANSWER: 4 ANSWER: 4 Since ¼ already has a denominator of 4 you don’t need to change it. Since ¼ already has a denominator of 4 you don’t need to change it. But ½ needs to be converted to 4’ths. But ½ needs to be converted to 4’ths.

23. Step #2 (continued) How do you convert ½ into 4ths? How do you convert ½ into 4ths? (what number) x 2 = 4? (what number) x 2 = 4? ANSWER: 2 ANSWER: 2 Now, multiply both the numerator (top number) and the denominator (bottom number) by 2. Now, multiply both the numerator (top number) and the denominator (bottom number) by 2. 1 x 2 = 2 1 x 2 = 2 2 x 2 = 4 2 x 2 = 4

24. Step #3 So now ½ has been converted to 2/4. So now ½ has been converted to 2/4. Now we have: 1 2/4 – ¼ Now we have: 1 2/4 – ¼ Go ahead and subtract ONLY the numerators. What did you get? Go ahead and subtract ONLY the numerators. What did you get? ANSWER: 1 ¼ ANSWER: 1 ¼

25. BORROWING!!! Generally, borrowing is the most difficult thing to do in subtracting fractions. Generally, borrowing is the most difficult thing to do in subtracting fractions. There are 4 simple steps to follow and it works for ANY fraction in ANY problem. There are 4 simple steps to follow and it works for ANY fraction in ANY problem. Don’t worry, it’s easy once you learn the steps. Don’t worry, it’s easy once you learn the steps.

26. Here is the problem Let’s say that you got a problem like this: Let’s say that you got a problem like this: 3 ¼ - 15/16 3 ¼ - 15/16 First step: They can’t be subtracted as they are. First step: They can’t be subtracted as they are. Second step: What is the common denominator? ANSWER: 16 Second step: What is the common denominator? ANSWER: 16 Third step: Convert a fraction. Third step: Convert a fraction.

27. Let’s go through it With a common denominator of 4 we need to figure out: (what number) x 4=16? With a common denominator of 4 we need to figure out: (what number) x 4=16? ANSWER: 4 ANSWER: 4 SO: 4 x 1 = 4 SO: 4 x 1 = 4 4 x 4 = 16 4 x 4 = 16

28. Oops! What’s this? The problem now reads like this: The problem now reads like this: 3 4/16 – 15/16 3 4/16 – 15/16 Normally you would now subtract. The problem is that 4 – 15 would be a negative number. We can’t have that! Normally you would now subtract. The problem is that 4 – 15 would be a negative number. We can’t have that! THUS, BORROWING IS NEEDED! THUS, BORROWING IS NEEDED!

29. Borrowing In this problem: In this problem: 3 4/16 – 15/16 Borrowing is having to increase the value or amount of 4/16 so that it’s bigger than 15/16. Borrowing is having to increase the value or amount of 4/16 so that it’s bigger than 15/16. In other words, we need to make 4/16 bigger so that we CAN subtract. In other words, we need to make 4/16 bigger so that we CAN subtract.

30. Here’s how to do it 3 4/16 needs to be changed somehow. 3 4/16 needs to be changed somehow. We’re going to take 1 whole number from the 3 and add it to 4/16. We’re going to take 1 whole number from the 3 and add it to 4/16. Would you agree that: Would you agree that: 2 + 1 4/16 = 3 4/16? NOW COMES THE TRICKY PART. NOW COMES THE TRICKY PART.

31. The tricky part 2 + 1 4/16 needs to be changed a bit before we can subtract from it. 2 + 1 4/16 needs to be changed a bit before we can subtract from it. Lets take 1 4/16 and “fix” it. Lets take 1 4/16 and “fix” it. Because 16 is the common denominator we need to write 1 in 16ths. Because 16 is the common denominator we need to write 1 in 16ths. We can write 1 as: We can write 1 as: 2/2 = 1 3/3 = 1 4/4 = 1 And so forth up to: And so forth up to: 16\16 = 1 SO NOW: 16 + 4 = 20 16 16 16

32. Recap 3 ¼ -15/16 = 3 ¼ -15/16 = 3 4/16 – 15/16 = 3 4/16 – 15/16 = ( 2 + 1 + 4/16) – 15/16 = ( 2 + 1 + 4/16) – 15/16 = (2 + 16/16 + 4/16) – 15/16 = (2 + 16/16 + 4/16) – 15/16 = (2 + 20/16) – 15/16 = (2 + 20/16) – 15/16 = All of these expressions are equal to each other. All of these expressions are equal to each other.

33. Let’s pause and try a couple problems. Ready for an easy test?

34. What fraction would you turn 1 into to complete the problem? 1 + 3/16 1 + 3/16 1 + 1/8 1 + 1/8 1 + 9/16 1 + 9/16 1 + ½ 1 + ½ 1 + ¾ 1 + ¾ 1 + 5/8 1 + 5/8

35. Back to the problem Now, instead of: Now, instead of: 2 + 1 4/16 we have: 2 20/16 If we rewrite the problem now we have: If we rewrite the problem now we have: 2 20/16 – 15/16 Now it’s just a simple subtraction problem! Now it’s just a simple subtraction problem!

36. Don’t forget 2 20/16 – 15/16 Remember that you only subtract the numerator, not the denominator. Remember that you only subtract the numerator, not the denominator. The answer: 2 5/16 The answer: 2 5/16 WHEW! WHEW!

37. Questions??