2. Adding and Subtracting Fractions

3. Do not train children to learning by force and harshness, but direct them to it by what amuses their minds, so that you may be better able to discover with accuracy the peculiar bent of the genius of each. Plato

4.  Although children learn addition of whole numbers with ease, addition of fractions — though conceptually the same as addition of whole numbers — is much harder.  It requires knowledge of fraction equivalencies.  To add two fractions, you have to know that they must be thought of in terms of like units.  We take this for granted when we add whole numbers: 3 + 5 is really 3 ones + 5 ones  — but not when we add fractions: 3 halves + 5 fourths is, for purposes of addition, 6 fourths + 5 fourths. Why is adding fractions a difficult concept for students to grasp?

5. Equivalent fractions In the following picture we have ½ of a cake because the whole cake is divided into two congruent parts and we have only one of those parts. But if we cut the cake into smaller congruent pieces, we can see that = Or we can cut the original cake into 6 congruent pieces, a fraction can have many different appearances, these are called equivalent fractions

6. Now we have 3 pieces out of 6 equal pieces, but the total amount we have is still the same. Therefore, == If you don’t like this, we can cut the original cake into 8 congruent pieces, Equivalent fractions a fraction can have many different appearances, these are called equivalent fractions

7. then we have 4 pieces out of 8 equal pieces, but the total amount we have is still the same. === We can generalize this to = whenever n is not 0 Therefore, Equivalent fractions a fraction can have many different appearances, these are called equivalent fractions

8. Let’s Eat Pizza The pizza is currently 8 pieces What if I wanted to eat one eighth of the pizza? One fourth of the pizza? One sixteenth of the pizza? One twelfth of the pizza?

9. Addition of Fractions  The objects must be of the same type  We combine bundles with bundles and sticks with sticks.  Addition means combining objects in two or more sets  In fractions, we can only combine pieces of the same size  In other words, the denominators must be the same

10. Click to see animation + = ? Example: Addition of Fractions

11. Example: Addition of Fractions + =

12. + = Example: The answer is which can be simplified to Addition of Fractions

13. Addition of Fractions with equal denominators More examples

14. With different denominators In this case, we need to first convert them into equivalent fraction with the same denominator. Example: An easy choice for a common denominator is 3×5 = 15 Therefore, Addition of Fractions

15. When the denominators are bigger, we need to find the least common denominator by factoring. If you do not know prime factorization yet, you have to multiply the two denominators together. With different denominators Addition of Fractions

16. More Exercises: = = = = = = = = =

17. Subtraction of Fractions  Subtraction means taking objects away  Objects must be of the same type  we can only take away apples from a group of apples  In fractions, we can only take away pieces of the same size. In other words, the denominators must be the same.

18. Subtraction of Fractions Example: This means to take away (Click to see animation) take away equal denominators

19. Subtraction of Fractions Example: equal denominators

20. Subtraction of Fractions Example: equal denominators

21. Subtraction of Fractions Example: equal denominators

22. Subtraction of Fractions Example: equal denominators

23. Subtraction of Fractions Example: equal denominators

24. Subtraction of Fractions Example: equal denominators

25. Subtraction of Fractions Example: equal denominators

26. Subtraction of Fractions Example: equal denominators

27. Subtraction of Fractions Example: equal denominators

28. Subtraction of Fractions Example: equal denominators

29. Subtraction of Fractions Example: equal denominators

30. Subtraction of Fractions Example: equal denominators

31. Subtraction of Fractions Example: equal denominators

32. Subtraction of Fractions Example: equal denominators

33. Subtraction of Fractions Example: equal denominators

34. Subtraction of Fractions Example: equal denominators

35. Subtraction of Fractions Example: equal denominators

36. Subtraction of Fractions Example: equal denominators

37. Subtraction of Fractions Example: equal denominators

38. Subtraction of Fractions Example: equal denominators

39. Now you can see that there are only 8 pieces left, therefore Subtraction of Fractions Example: equal denominators

40. More examples: Did you get all the answers right? Subtraction of Fractions

41. Adding/Subtracting 3 88 13 84 1 ==-- 1 8 8 8 2 = Fraction Addition/Subtraction Fraction Simplification

42. 28 112 7 + + + = = = Common Denominator = ?????? 28 11 7 2 4 4 28 11 28 8 +118 = 28 19 Fraction Addition/Subtraction Adding/Subtracting

43. Fractions: Steps for Success 1. Know the fraction rules and how to apply them 2. Show your work and write out each step 3. Check your work for errors or careless mistakes