1. Visual Fractions
Generally speaking, the progression used when teaching fractions starts with the use of manipulatives and other visual representations that make connections to real life things: pizza, money, lollies, etc.. As children begin to be able to articulate these understandings both verbally and with mathematical expressions, more abstract models such as area rectangles, fraction bars and grids are introduced, permitting students to expand their understandings in more abstract ways.

2. Fraction Origami
By simply taking a square piece of paper and folding it in half; write the fraction ½ ; fold in half again; write ¼ . Continue this process to create smaller fractions. Unfold the paper and trace lines where there are folds in the paper.
Notice that in the case of the example, the manipulative (the folding paper) becomes a model (a more abstract visual representation) as students draw over the folds and fill in the values.
Visual Fractions

3. Area Diagrams
Area rectangle modelling the fraction ¾
Visual Fractions

4. Using Area Rectangles to Create Equivalent Fractions
The more standard algorithms that most of us use to add, subtract, and multiply fractions, are to a child's perspective, a collection of rules, which at best "work", but at worst become all mixed up in students' minds. This is usually because these 'rules' have no underpinnings in the students' web of mathematical knowledge. In other words, if students don't understand why the "rules" for using these methods exist in the first place, it becomes very difficult to apply them at the right time, not to mention execute them consistently. For this reason, this "traditional" approach (just teaching them how to do things, but not why they are doing them the way they are) to teaching operations with fractions requires a "drill and kill" methodology.
Thankfully, the contemporary approach to teaching these methods has students explore these concepts in many different ways at a fundamental level.
Visual Fractions

5. Using a 100 grid
This diagram models
20/100
2/10
1/5
Visual Fractions

6. 3/5 x 1/2 x
As can be seen on the 100 grid to the left,
3/5 x ½ = 30/100 or 3/10
Visual Fractions

7. ¼ + 2/5 ¼ + 2/5 on the 100 grid is equivalent to 25/100 + 40/100.
65/100 = 13/20.
Visual Fractions

8. ¼ + 2/5 ¼ + 2/5 on the 100 grid is equivalent to 25/100 + 40/100.
65/100 = 13/20.
Alternatively,
5/20 + 8/20 = 13/20
Visual Fractions

9. 22/3  2/3
Visual Fractions

10. 22/3  2/3 = 8/3 x 3/2 = 4
Visual Fractions

11. What is 80/100 in lowest terms?2 5 3 8 6 4 7 10
Visual Fractions

12. What is 80/100 in lowest terms?4 5
Visual Fractions

13. What is ½ + 3/10 (in lowest terms)?2 5 3 8 6 4 7 10
Visual Fractions

14. What is ½ + 3/10 (in lowest terms)?4 5
Visual Fractions

15. What is ¾ x 2/5? 1 2 5 3 8 6 4 7 10
Visual Fractions

16. What is ¾ x 2/5? 3 10
Visual Fractions

17. What is 3¾  ¾ ?
Visual Fractions

18. What is 3¾  ¾ ?
15/4  ¾
= 15/4 x 4/3
= 5
Visual Fractions

19. What is 1¼ - 3/8 ? 1 2 5 3 8 6 4 7 10 x
Visual Fractions

20. What is 1¼ - 3/8 ? 7 8 x
Visual Fractions

21. References:
Visual Fractions