Standards Academy

1. Standards Academy Grades 3 and 4 Day 3

2. Welcome Back Reflection on yesterday: • 4 Point Evaluation • Parking Lot

3. Objectives • Build a Solid Understanding of the Fractions Progression Document • Examine Essential Fraction Strategies • Modeling Fractions on a number line • Equivalent Fractions • Addition and Subtraction of Fractions • Multiplication of Fractions • Study Core Standards Related to Fractions

4. Context and Models Region or Area Models • Helps students visualize parts of the whole. Length Models • Helps students visualize iteration, counting and measurement. Set Models • Helps build understanding of division and ratio concepts.

5. Various Models

6. Fraction Brainstorm Fold a paper into three columns. Title each column after a different fraction model. With a partner, generate some question types or activities that work well with that model. See how many you can list that have nothing to do with pizza or food.

7. Early fractions instruction generally focuses on the idea that fractions represent parts of a whole. Although the part-whole interpretation of fractions is important, too often instruction does not convey another simple but critical idea: fractions are numbers with magnitudes (values) that can be either ordered or considered equivalent. Many common misconceptions—such as that two fractions should be added by adding the numerators and then adding the denominators—stem from not understanding that fractions are numbers with magnitudes. Not understanding this can even lead to confusion regarding whether fractions are numbers. For example, many students believe that four-thirds is not a number, advancing explanations such as, “You cannot have four parts of an object that is divided into three parts.” Further, many students do not understand that fractions provide a unit of measure that allows more precise measurement than whole numbers; these students fail to realize that an infinite range of numbers exists between successive whole numbers or between any two fractions. Reliance on part-whole instruction alone also leaves unclear how fractions are related to whole numbers.

8. So Why the Number line? After reading this excerpt from the IES practice guide take a moment to: • Think: Why emphasize instruction on the number line? • Pair: Turn to a partner and discuss why. • Share: With the class discuss what you and your partner think.

9. Block Party • Read the Progression Document. • Take 3 min to read your card and think about its meaning. • When signaled, find the person who has the same card as you, share your thinking and clarify each other's understanding. • When signaled, mingle around the room finding a new partner. Share, explain and discuss your quote cards. • Repeat as many mingles as time allows.

10. Ah-ha! “Unit fractions are the basic building blocks of fractions in the same sense the number 1 is the basic building block of whole numbers; just as every whole number is obtained by combining a sufficient number of 1’s, every fraction is obtained by combining a sufficient number of unit fractions.” http://www.illustrativemathematics.org/fractions_progression

11. Fraction Strategies D B C A

13. Fraction Equivalence Snow Ball Fight! • Skim the progression document for information about fraction equivalence. • Find one statement or model to write down, andone questionboth pertaining to fraction equivalence. • Crumble your notes and toss them across the room. • Pick up one note and it take back to your partner or triads to discuss.

14. Ah-ha! “Fraction equivalence is founded in reasoning and experimentation with number lines and tape diagrams. Students will need numerous exposures and opportunities to discover and explore equivalence in order to form a solid conceptual understanding.” http://www.illustrativemathematics.org/fractions_progression

15. Equivalency Strategies A B

16. Reasonaboutthis..

17. Addition and Subtraction • Solve and review your task individually. Think about the following questions: • Can this be modeled with a number line? Is there a better model? • How could a student use their understanding of unit fractions to solve this? • Can this be seen as composed of unit fractions or decomposed to unit fractions? • Does the student need to convert to common units? • As a table discuss your various tasks. • Have each member describe their task. • Address the questions from step one. • Extend your discussion • What’s the math? How could it be improved? What is the relation to addition and subtraction of fractions? What is the core connection? What support/vocabulary/background would your students need?

18. Addition and Subtraction A B C E http://www.illustrativemathematics.org/fractions_progression

20. If only…….Let’s check out this oldie but goody

21. Multiplication • Read article individually. • Form groups to discuss the following questions: • What assumptions does the author of the text hold? • What do you agreewith? • What do you want to argue? • What do you want to aspireto? • Take turns making sure each person at your table has an opportunity to respond to each of the 4 A’s.

22. Multiplication A B C D http://www.illustrativemathematics.org/fractions_progression

23. Wrap Up & Reflect Questions or Concerns…….. Complete a 4 point evaluation: • What went well today? • What could be improved? • What do you need more support in? • What did you master?