1. Unit Fractions In Action Presented by Dr. Stephen Sher & Dr. Melanie Maxwell

2. Today’s Objectives: Fractions Overview of Content StandardsMathematical Progressions Pedagogy Number Talks & Multiple Representations of Fractions Fraction Problems Professional Background Knowledge Operations Retain their Meaning Across Number Types Understanding Fractions: Specifically that Unit Fractions Rock!

3. The depressing thing about arithmetic badly taught is that it destroys a child’s intellect, and to some extent his integrity. Before they are taught arithmetic, children will not give their assent to utter nonsense; afterwards, they will. - Walter W. Sawyer, author of a Mathematician's Delight

4. Traditional Fraction Confusions ●Don’t see fractions as a single quantity or number ●Are not aware that the whole can change ●Don’t relate procedures to equivalent fractions and operations. ●May be used to seeing fractions in a limited number of ways- limited representations

5. Procedural fluency builds from an initial exploration and discussion of number concepts to using informal reasoning strategies and the properties of operations to develop general methods for solving problems. -National Council of Teachers of Mathematics, NCTM, 2014 ●Understanding can lead to student generalized shortcuts, especially when facilitated by good activities and discussions. ●Shortcuts are not useful without understanding because students do not know when they apply and when they don’t apply.

6. Recommendations from the Institute of Educational Sciences (IES) 1.Build on students’ informal understanding of sharing and proportionality 2.Fractions are numbers that expand the number system- Place on number line 3.Understand why procedures for computation make sense 4.Teachers understand computation & how to use varied pictorial and concrete representations

7. Essential Questions for Fractions 1)What is a fraction? 2)What are unit fractions and how do we use them? 3)What are equivalent fractions and how do we use them? 4)How can I use a number line to represent my work with fractions? 5)How do I represent fraction operations conceptually, to better understand ways to compute with fractions?

8. Approach to Fractions Seen as Key Shift in Common Standards

9. NF 3-5 * 1st and 2nd grade partition shapes into equal shares. * 1st and 2nd grade use fraction words halves, thirds, half of, a third of, etc. ½ ⅓ ⅕ ⅛ Unit fractions, one over any whole number, is foundational! Number Lines and Area Models

10. 3rd Grade From counting whole numbers to counting with unit fractions

11. Early Conceptual Benchmarks ●Define the whole and it can change ●Add unit fractions and place on a number line ●Larger unit fraction denominator means a smaller amount when the numerator is the same ○1/9 1/12 ●Use Graphics- diagrams, number lines & bar diagrams, area models, and objects ●Equivalence

12. The Whole or 1 Can Change

14. Varied Area representations

16. 3.NF. Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size

17. Build Understanding with Fraction Strips Students discover relationships and generalize findings Create number lines from fraction strips

18. Number Talk Order from least to greatest 585 698

19. Number Talk 2 ½ ÷ ½ ＝

21. 4th: Build from Unit Fractions to Sums “Armed with this insight, students decompose and compose fractions with the same denominator.” Learn Zillion Unit

22. 5th: Is the sum more or less than 1? How can I use unit fractions? ⅞ + ⅗ ⅔ + 7/10 ⅕ + ⅜

23. Operations: Multiplication & Division Operations retain their meaning!!! ●Multiply whole numbers by fractions - 4th ●Multiply fractions by fractions - 5th ●Divide whole numbers by fractions - 5th ●Divide fractions by whole numbers - 5th ●Divide fractions by fractions - 6th What is 5 x ⅓? Start work in each area with unit fractions.

24. Operations: Add and Subtract Let’s do some math! Use equivalent fractions to add and subtract. 4/8 + 1/4

25. Do Some Math ●Shading and showing fractional parts ●Comparing Fractions & other Math Reasoning Inventory problems ●Division of Fractions - Conceptual ●Word Problems - 4 operations ●Computation - 4 operations ●Multiply or Divide

26. Practice Fraction Work ●Visuals to represent and explain/justify your work. ● Apply and extend previous understandings of multiplication and division to multiply and divide fractions. Goal: Establish a mental picture of the operation.

27. Closure Share work from key problems Review Key Points: ●Unit fractions rock! (They are the rock.) ●Operations retain their meaning ●Understand Equivalence

28. Resources ●Mathematical Progressions ●IES Developing Effective Fractions Instruction, K-8