1. Building Foundations for Mathematics
Conceptual Understanding of Fractions

2. Estimate how much pie was eaten?
1 5
Estimate how much pie was eaten?

3. Using Fraction Models
Student proficiency with fractions is essential to success in algebra
Models help students clarify ideas that are often confused in a purely symbolic mode
Students need many concrete experiences with fractions to develop a deep understanding
The process should be introduced by using visual fraction models (area models, number lines, and so on) to build understanding before moving into the standard algorithm
Important to do the same activity using two different types of models

4. Using Fraction Models
Introduce multiple models for the same fraction. It is important that students don’t get stuck using only one model for a given fraction. Having multiple models at their disposal will help the flexibility of their thinking and allow them to switch models if needed.
Encourage students to think flexibly about “the whole”. When working with the set model, students see that ¼ looks different in a set of 4 items than it does in a set of 12 items.
Allow students to explore fractions as more than just “a part of a whole”.
Give students practice with decomposing fractions.
Estimate! When introducing new fractions that are unfamiliar to students, help them rely on benchmarks (0, ½, and 1).

6. Conceptual Models
Region: a whole which has been partitioned into equal parts and some of those parts are shaded. Linear: length of the whole is divided into equal lengths (easily connected to fractions on a number line) Set: a set of objects as a whole (unit). Work very well for developing algorithms later on. Area: rectangles to show a visual representation of the product of two fractions

7. Region Models
Region: a whole which has been partitioned into equal parts and some of those parts are shaded.
The most concrete model and most often used.
The region is the whole (the unit) and the parts are congruent (same shape and size)
When presenting the region model a variety of shapes should be used so students don’t always think of fractions as piece of a pie.
The rectangle is the easiest model for children to draw and partition.
The circle is easiest to see as a whole.

9. Show why 2 3 is the same as 4 6 and 8 12
Try It …
Show why is the same as and

10. Table Talk
How does using the region model support student conceptual understanding?

11. Linear Model
Linear: length of the whole is divided into equal lengths (easily connected to fractions on a number line)
Any unit of length can be partitioned into equal parts.
Rulers are examples of linear models.
Transfers easily to fractions on a number line
Fractions on a number line highlights the fact that some fractions occupy the same position on a number line and therefore represent the same quantity

14. Try It …
Jack and Jill ate two identical candy bars. Jack ate of his candy bar. Jill ate of her candy bar. Who has more candy bar left?

15. Table Talk
How does using the linear model support student conceptual understanding?

16. Set: a set of objects as a whole (unit).
Set Model
Set: a set of objects as a whole (unit).
Uses a set of objects as a whole (unit).
Without mentioning fractions students should have many opportunities to partition sets – background for both fractions and division
Leads to understanding of questions What is two-fifths of 15? Etc.
Can be the most confusing, but work very well for developing algorithms later on.
Establish important connections to real-world uses of fractions and with ratio concepts

17. Using the manipulatives …
Show 8 counters as a whole set
How many are in of the set?
Show 4 counters as one-half of a set
How big is the set?
Show 10 counters as a whole set
What fraction of the set is 6 counters?
Show 45 counters
If this is 5 6 of the set, how big is the whole set?

18. How does using the set model support student conceptual understanding?
Table Talk
How does using the set model support student conceptual understanding?

19. Area Model
Area: rectangles to show a visual representation of the product of two fractions
The rectangle is used to connect the area model used with whole numbers to the use with fractions
Important for students to see the two fractions being multiplied as the factors
Just as with whole numbers, the first factor tells how much of the second factor are needed ( 3x 5 = 3 sets of 5, x = of )
Important for students to understand the model so they can name the product (The unit, the way the parts are measured, must remain the whole.)

21. Find 1 3 of 2 6 Explain your reasoning
Try It …
Find 1 3 of 2 6 Explain your reasoning

22. Table Talk
How does using the area model support student conceptual understanding?

23. Subitizing with Fractions

24. Balanced Understanding
It is important to develop a balanced understanding connecting the model to symbols and words.

25. Representational
Concrete
Abstract

26. Final Thought
Convince your partner of the value of using a variety of fraction models for student conceptual understanding.