1. Fraction Sense Fran Gibson

2. 2 Sacramento County Office of Education | 08.22.11 Welcome and Introductions Please share the following information about yourself: – Name – Position and School District – Experience with California’s Common Core State Standards – One thing that you would like to learn from today’s presentation

3. 3 Sacramento County Office of Education | 08.22.11 Norms We all agree to: Learn about mathematics Work together Implement what we learn We will all work to: Learn Be Cooperative Be on Time Be Flexible Be Forgiving Be Responsible Be Respectful of Others Please turn all cell phones to off or silent and refrain from text messaging.

4. 4 Sacramento County Office of Education | 08.22.11 Learner Outcomes Today we will… Explore CCCSS that focus on naming, equivalence, comparing and ordering fractions on the number line. Develop an understanding of how fraction understanding progresses across grade-levels. Discuss implications of this structure on teaching with the current mathematics textbook. Identify starting points for beginning implementation of CCCSS.

5. 5 Sacramento County Office of Education | 08.22.11 Number Line Activity With a partner: Solve the 12 problems on your set of cards. Discuss the strategies used and identify the concepts, skills, and procedures needed to solve each. Order number lines in increasing order of difficulty. Be ready to justify your choices.

6. 6 Sacramento County Office of Education | 08.22.11 Instructional Implications As you watch the following two videos, keep the following questions in mind: – What strategies did the students use? – How did it compare to the strategies you used as adults? – What did the students understand? Be prepared to discuss and share out some of your thoughts. Teacher as Practioner

7. Video Example 1

8. Video Example 2

9. 9 Sacramento County Office of Education | 08.22.11 Looking for the Big Mathematical Idea With a partner: Summarize, in as short a statement as you can, the “BIG” mathematical idea that these problems represent. Investigate the Number and Operations – Fractions cluster and standards and determine how the problem supported the content of the cluster. Be ready to share your ideas with the larger group.

10. 10 Sacramento County Office of Education | 08.22.11 Cluster Progression On your own… 1.Using the “Number and Operations–Fractions” handout, pick one cluster within the domain for each grade level and read the standards. With your tablemates… 2.Discuss the following questions: – What do you notice? – What big ideas are repeated or similar? – What is different? 3.Be ready to share out with the entire group.

11. 11 Sacramento County Office of Education | 08.22.11 Fractions on a Number Line On your own… 1.Draw a number line, locate and identify 2/5. With your tablemates… 2.Discuss the following questions: – What process did you use to do this? – Could you use this process for any fraction? Why? 3.Be ready to share out with the entire group.

12. 12 Sacramento County Office of Education | 08.22.11 Equivalent Fractions- Part 1 With a partner: – Using the same number line, identify and mark 4/10. – Discuss the following questions. How many pieces are in the whole now? Did you use the same process? How was it the same? How was it different? Are the two marked fractions equal? How can you be sure? – Be ready to share out.

13. 13 Sacramento County Office of Education | 08.22.11 Equivalent Fractions- Part 2 3/7 = ?/28 How can you determine the number of 28ths that would be equivalent to 3/7 using the process from the previous activity? Share out solutions and strategies

14. 14 Sacramento County Office of Education | 08.22.11 Summarizing the Big Idea 3/7 = 12/28 Thought Process: – Three pieces that are broken into four equal pieces making twelve pieces. – Seven pieces that are broken into four equal pieces making twenty-eight pieces. What arithmetic problem could be used to get the number of pieces for the numerator? For the denominator?

15. 15 Sacramento County Office of Education | 08.22.11 Fraction Families On your own… 1.Make a list of fractions that could be built from thirds, based on our understanding of fraction equivalency on the number line. 2.Think about the following questions: Would sixtieths be on this list? How many equal pieces would each third be broken into to make sixtieths? How would you describe the family of fractions that can be generated from thirds? With your tablemates… 3.Discuss your responses to the previous questions. 4.Build fraction families for halves, fourths, and sixths. 5.Discuss whether or not the description you found with thirds still applies.

16. 16 Sacramento County Office of Education | 08.22.11 Comparing and Ordering Fractions With your tablemates… 1.Compare these two fractions: 5/8 and 5/11, without computing. 2.Discuss your reasoning to determine: Which is bigger? Which is smaller? Which is closest to ½? Which is closest to 1? 2.Be ready to share out with the entire group.

17. 17 Sacramento County Office of Education | 08.22.11 Practice Comparing Fractions On your own… 1.Using relationships, not common denominators, compare the fraction pairs in your notebook. 2.Think about the following: – What generalizations can you make about comparing fraction size? – What strategies did you use? – How do you know you are right? With your tablemates… 3.Discuss how you might get students to the same kind of exercise. 4.Be ready to share out with the entire group.

18. 18 Sacramento County Office of Education | 08.22.11 Conceptual Thought Patterns for Comparison Read through the conceptual thought patterns for comparison in your notebook. Highlight key words or phrases. Partner up with others and discuss the following: – What kinds of questions do you ask your students to help you understand their thinking about fractional relationships? – What are some instructional implications inherent in these thought patterns? Be ready to share out your thoughts.

19. 19 Sacramento County Office of Education | 08.22.11 Debrief of Conceptual Thought Patterns for Comparison What are the key ideas in each of the thought patterns? More of the same-size parts Same number of parts but parts of different sizes More and less than one-half or one whole Distance from one-half or one whole

20. 20 Sacramento County Office of Education | 08.22.11 Which Fraction is Greater? On your own… 1.Using the problem set in your notebook, determine which fraction in each pair is the greatest using one or more of the conceptual thought patterns. 2.Next to each pair of fractions, indicate the thought pattern or patterns you used. 3.Be ready to share out with the entire group.

21. 21 Sacramento County Office of Education | 08.22.11 Debrief of Which Fraction is Greater? Debrief of Which Fraction is Greater? Explain the thought process you used to determine which fraction in each pair was the greatest. 4/5 or 4/9 4/7 or 5/7 3/8 or 4/10 5/3 or 5/8 3/4 or 9/10 3/8 or 4/7 7/12 or 5/12 3/5 or 3/7 5/8 or 6/10 9/8 or 4/3 4/6 or 7/12 8/9 or 7/8

22. 22 Sacramento County Office of Education | 08.22.11 Connect to the Curriculum Investigate how your text uses the number line for fraction instruction. How might you use what you have learned in this session to enhance your lessons?

23. 23 Sacramento County Office of Education | 08.22.11 Resources  Websites ◦ Common Core Standards: www.corestandards.orgwww.corestandards.org ◦ California Common Core Standards: Visit the California Department of Education’s Common Core State Standards Web page at: http://www.cde.ca.gov/ci/cc The standards Frequently asked questions Additional resources

24. 24 Sacramento County Office of Education | 08.22.11 Reflection Complete the prompts on the Notebook page.

25. 25 Sacramento County Office of Education | 08.22.11 Feedback Form Complete and submit the Feedback Form included in the back of the Notebook.