1. Lucy West Education Consultant email: lucy@lucywestpd.com http://lucywestpd.comlucy@lucywestpd.comlucywestpd.com phone: 212-233-0419 cell: 917-494-1606

2. Lucy West: lucy@lucywestpd.comlucy@lucywestpd.com Power Point Will Be Posted on Web Site www.lucywestpd.com

3. Welcome Please sit together by school teams. Please make a name tent with your first name using MARKER and face it toward me Make yourself comfortable and get ready to have a productive day together.

4. Agenda Reflecting on Discourse at Our Schools Reflecting on and Adding to our Routines Repertoire Exploring Fractions Lunch Continuing our Exploration Reflections, commitments, next steps. The white paper—use this all day long please

5. Reflecting on our progress? Please consider the discourse at your school. How often do teachers meet to do math together, to plan a lesson in detail, to share student work, and to share what they have learned from implementing a shared lesson? What is enhancing the discourse and what is getting in the way of it? What might your next steps be? What supports might you need to put in place or receive from the network or outside vendors or…?

6. Revisiting Math Routines Rename the Number (equivalence, operations, informal assessment) Counting Around the Room (multiplication, patterns, estimation) Guess my rule/number/shape (logic, reasoning, questioning) Which of the above routines and additional routines have you regularly implemented in your class and how is that going? What questions, support, insights, concerns have arisen as a result?

7. Rename the Fraction Choose a common fraction such as 1/2, 3/4, 1/8, 1/3, 3/5, etc. Rename the fraction in as many different ways as you can. Example: 1/2.5.50 50% 2/4 4/8 20/40 500/1000 What can you say about these rational numbers? How are they the same or what is remaining constant and how are they different or what is changing? What are the big ideas related to this exploration?

8. Exploring Concepts in Understanding Fractions Compare these fractions. Are they equivalent or is one greater than the other and how do you know? See if you can figure this out without computing or finding the LCM or changing to a decimal. Just by reasoning. 5/6 7/8 Create another pair of fractions that would have the same characteristics as these two and explain how you know that one is greater than the other. What is your reasoning?

9. Pedagogical Content Knowledge Create a variety of visuals that conceptually compare 5/6 and 7/8. You must use at least one of the following models: Area model Number line model Other?

10. Eight CCSS Math Practices Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning.

11. Exploring Concepts in Understanding Fractions Compare these fractions. Are they equivalent or is one greater than the other and how do you know? See if you can figure this out without computing or finding the LCM or changing to a decimal. Just by reasoning. 4/7 6/13 Create another pair of fractions that would have the same characteristics as these two and explain how you know that one is greater than the other. What is your reasoning?

12. Landmark Fractions Zero, one and half are benchmark fractions. We can use them to help us estimate the value of other, less friendly fractions.

13. Which is greater? Consider the following pair of fractions. How can you determine which is greater without computing—by reasoning? What would you have to know to reason about these two fractions. 2/4 2/5 Create a second pair of fractions that have the same characteristics and explain how you know which is greater. Provide a visual to illustrate your thinking.

14. Comparing Fractions When two fractions have the same numerator, we can determine which fraction is greater by looking at the denominators. The fraction with the lowest denominator will be larger, because the size of the pieces are larger and because the numerators are equivalent, we know we have the same number of larger pieces as the fraction we are comparing it with. When two fraction have the same denominator, we can look to the numerator to see which fraction is greater. The fraction with the greatest numerator tells us we have more of the equal pieces.

15. Fractions: Ideas Fractions describe a part whole or part part relationship. The whole can define one thing, one group of things, or even part of one thing

16. Fractions: Ideas A given fraction can be different in size depending on the referent whole. One fourth of a pizza is larger than one fourth of a brownie.

17. Fractions: Ideas Fractions can be thought of as ratios. Fractions can be thought of as division.

18. Think of Ways to Represent 1/3 X 1/4 Think of contexts for 1/3 X 1/4 Consider how context impacts representation There was a quarter of a pan of brownies left. Lucy ate 1/3 of the portion that was left. What portion of the whole pan did she eat?

19. Fosnot and Dolk, Young Mathematicians at Work

21. Fractions as Divison Four children want to share three chocolate chip cookies. How can they distribute the cookies equally?

22. Fractions as Divison Thinking about the math:

23. Three Cookies

25. Common Core Standards: Rational Number Six children want to share five (5) chocolate bars fairly. How much chocolate does each child get?

26. Share 7 brownies among 4 kids

27. 8 brownies and four kids is no problem…

28. Four children plan to share seven brownies…

29. What are some ways your students would approach this problem?

30. SEVEN BROWNIES

31. Common Core Standards: Rational Number The Many Uses of Language in Fractions

32. Fractions in Everyday Language Half as … much, many, long, heavy, old In this room there are half as many women as there are men The bench is half the height of the table The street is 2 ½ times as wide as the sidewalk John earns half as much as Pete Copper is half as heavy as gold

33. Exploring Halves Create a design on your geoboard that shows 1/2. Record that design on geoboard paper. Make another design and record it. Repeat this process making as many different designs showing 1/2 as you can. The rules for this exploration: You must use the entire area of the geoboard Each half must be one contiguous piece--in other words you will end up with 2 pieces if you were to cut out the two halves

34. Exploring Fourths Create designs on your geoboard or geoboard paper that show four noncongruent fourths. Repeat and record several different solutions. The rules for this exploration: You must use the entire area of the geoboard Each fourth must be one contiguous piece--in other words you will end up with 4 pieces— each one different from the others if you were to cut out the fourths