Fractions and Building Fraction Sense in K-3

1. Welcome to …

2. TODAY’S AGENDA • Chickens and Eggs Task • Building Fraction Sense in K-3 • BREAK • Making Equal Sections • Davis – Organizing Shapes • LUNCH • Fraction- Comparisons • Fractions- Equivalence • Math Camp Task • BREAK • Extending Children’s Thinking • Fractions Walk-Across • SMP Reflections

3. Bring your ideas… • As a group of professionals we have made a commitment to helping children attain success in life and a voice in the world. • Many times the best part of these kinds of professional development is simply the chance to share ideas, raise questions, and work with other practitioners to improve our own understandings and practice. • Please bring your stories of children’s learningwith you.

4. Our Socio-mathematical Norms • Listen intently when someone else is talking avoiding distractions • Persevere in problem solving; mathematical and pedagogical • Solve the problem in more than one way • Make your connections explicit - Presentation Ready • Contribute by being active and offering ideas and making sense • Limit cell phone and technology use to the breaks and lunch unless its part of the task. • Be mindful not to steal someone else’s “ice cream” • Respect others ideas and perspectives while offering nurturing challenges to ideas that do not make sense to you or create dissonance. • Limit non-mathematical and non-pedagogical discussions

5. Presentation Norms • Presenters should find a way to show mathematical thinking, not just say it • Presenters should indicate the end of their explanation by stating something like “Are there any questions, discussion, or comments?” • Others should listen and make sense of presenters’ ideas. • Give feedback to presenters, extend their ideas, connect with other ideas, and ask questions to clarify understandings

6. Fractions – Problem Solving If 1½ chickens can make 1½ eggs in 1½ days then how long would it take 6 chickens to make a dozen eggs?

7. Fractions – Non Standard Measurement What kinds of mathematical questions might we engage students in at this point? Let’s make a list.

8. Fractions – Spatial Fraction Development How many ways can you divide a Geoboard into four equal sections? Draw them out on dot paper

9. Break Time

10. Fractions – Spatial Fraction Development Without measuring divide the paper strips into the following equal amounts: 2 equal amounts 3 equal amounts 4 equal amounts 5 equal amounts 6 equal amounts

11. Fractions – Spatial Fraction Development Use your findings from the paper strips to mark where fractions should be on the number line. After you have marked your fractions on the number line, what kinds of things might we notice? What kinds of questions might be explored or entertained by our students?

12. Fractions – Spatial Fraction Development Why include both part-whole models (paper strips and fraction tiles) and number lines in learning about fractions? “To locate a point on a number line requires active thinking and reasoning, not just counting parts in a whole that has already been subdivided into a particular number of parts. The number line provides a context for understanding fractions that is different from, and more versatile than, part-whole models with the fractional parts already subdivided.” - Developing Essential Understandings of Rational Numbers Grades 3 – 5 (NCTM, 2011)

13. Fractions – Spatial Fraction Development Davis Reading “Organizing Shapes”

14. Lunch

15. Fractions – Coca Cola Who likes Coca Cola?

16. Fractions – Valid Comparisons Three Half Objects Three Objects Revisited When comparing fractions as greater than, less than, or equal to, what condition must exist for these comparisons to be valid?

17. Fractions – Valid Comparisons In proper mathematical notation we would not write ½ > ½ or ½ > ¾ When comparing fractions we must make sure they are parts of the same whole.

18. Fractions – Equivalence Use fraction tiles to find different ways make a ½. For this exploration find which unit fractions can be used to make a half and which cannot and develop an explanation for why. Extension: Create a fraction tile drawing of a way to make ½ with a unit fraction that is not in the fraction tile kit.

19. Fractions – Equivalence Having students explore fraction equivalence through many experiences (paper strips, geoboards, sharing equal parts of a whole piece of food, folding paper (Davis), and fraction tiles) will allow them to make sense of and find relations among the many representations that are possible for fractions allowing them to reason out the Fundamental Law of Fractions.

21. Water Measurements • You have three cups of water: • 500 mL, 200 mL, and 300 mL • Mission: Using only the cups and water you have, you must get exactly 400 mL of water back in the 500 mL cup. • You are only allowed to use the exact measurements that are already on the cups to solve the problem. No estimating!

22. Break Time

23. Extending Children’s Mathematics • Use individual think time to work on both tasks. • “Jeremy is making cupcakes. He wants to put ½ cup of frosting on each cupcake. If he makes 4 cupcakes for his birthday party, how much frosting will he use to frost all of the cupcakes?” • “3 children want to share 5 candy bars so that each person gets the same amount. The candy bars are the same size. How much can each child have?”

24. Extending Children’s Mathematics • How do the problems compare? • What thinking and strategies did you use to solve each of them? • Were your strategies the same?

25. Extending Children’s Mathematics • Read pages 3 – 15 in ECM. • Use the discussion questions to share ideas with a partner. • Small group discussion • Whole group discussion

26. Susan Empson Video (continued)

27. Video Discussion • Reflect on the video. • What was significant for you? • How does it relate to your teaching practice? • Share your responses with a partner.

28. Walk-Across Groups

29. Develop a Walk-Across for Fractions K-5 Assignment: What is a “Walk Across for Fractions?” It’s a focused look at mathematical connections in the CCSSM: 1) You will demonstrate what connections you can see in the standards across the domains and grade levels. 2) You will explain how a standard connects with prior and/or subsequent standards regarding students’ development of fractional understanding?

30. Develop a Walk-Across for Fractions K-5 FIRST: To begin, you should give attention to each standard, regardless of domain, and consider whether or not it pertains to ones understanding of fractions. You should only include standards that DO pertain to either development of pre-fraction ideas or directly to fractions themselves. You may find many standards not in the fraction specific CCSSM domain that also act to build fraction sense. SECOND: Once you find connections among standards, articulate an explanation of how they are connected. Do some standards prepare students for future standards? How so? Show what students would be doing and thinking in one standard and explain how that doing and thinking prepares them to do and think about future mathematics. When explaining connections among standards use the names (for example 3.G.2). THIRD: There is a large creative element to this task. You may display and explain the connections in any creative media you choose. It can be as formal as a word document, excel sheet, or power point. Or as informal as a large scale painting, video, or drawings. The only delimiters that must be satisfied are 1) and 2) above. That will most likely require text or comments in some form or another.

31. Time of Reflection Take a few moments to reflect on SMP’s connected to the content tasks we did today. -- Name of the task and related SMP’s -- Evidence for the chosen SMP’s -- Jot down how you contributed to our shared community of professionals and what mathematical and/or pedagogical knowledge you are taking away from today.

32. Stay Safe • Please help us put the room in proper order. • Please leave your name tags for next time.