Building the Foundation to Algebra

1. Building the Foundation to Algebra Rational Numbers

2. Goals • Examine the different interpretations of fractions. • Explore the meaning of "half". • Represent fractions using regional parts of a whole model. • Develop conceptual understanding of fractions as division.

3. Collection Box: ¾ • Individually, create at least 3 representations of ¾. Use pictures, diagrams, symbols, etc. • Share representations with people at your table. • Create a poster of your table’s representations.

4. Collection Box: ¾ Gallery Walk • One person from each group “mans” the group’s poster to answer questions. • Rest of group members view other posters. • Most common representations? • Most unusual/surprising representations?

5. Fraction Interpretations • Part-Whole • Parts of a region • Parts of a set or group • Measurement • Quotient • Ratio • Ratio • Rate • Multiplicative Operator

6. Fraction Interpretations • Part-Whole • Parts of a region • Parts of a set or group • Measurement • Quotient • Ratio • Ratio • Rate • Multiplicative Operator

7. Collection Box: ¾ • Identify the fraction interpretation illustrated by each of your collection box entries. • Denote the interpretation with a colored pencil. • Red: Part-whole region • Blue: Part-whole set • Green: Measurement • Orange: Ratio • Purple: Rate • Brown: Operator • Which interpretations were most common? Least common? • Which do you typically address in your mathematics curriculum?

8. Key Fraction Concepts • Identifying the “Whole”, “One”, or “Unit” • Relationships • Whole to Part • Part to Whole • Regions • Sets • Equal size pieces • Congruent • Area • Equivalent Fractions • Comparing Fractions

9. “Fraction” Sense • Magnitude/Quantity • Making sense of symbols • Ordering and comparing • Benchmarking • Equivalence • Representation • Physical • Pictorial • Words • Symbols • Sense-Making • Estimation • Operation sense • Interpreting fractions in context • How would you rate your own “Fraction” Sense? • How would you rate your students’ “Fraction” Sense?

10. Fractional Parts of Regions “I’ll take a large pizza with half-onion, two-thirds olives, nine-fifteenths mushrooms, five-eighths pepperoni, one-eighth anchovies, and extra cheese on five-ninths of the onion half.” Close to Home by John McPherson, 1993

11. Making Halves

12. Making Halves • Each person: Find at least three different ways to show halves on your geoboard. • Record each of your halves on geoboard paper. • Share your work with others in your group explaining how you know your ways show halves. • As a group, pick one example to present to the entire group.

13. Share methods with class

14. Making Fourths

15. Are These Eighths?

16. Looking through Teacher Lenses • How would you characterize the level of this task: High or low cognitive demand? • What mathematical ideas are embedded in the task? • What makes this worthwhile mathematics?

17. Comic Strip Fractions

18. Comic Strip Fractions • You decide to create a comic strip for your school’s newspaper. • To do this, you cut strips of paper that are a little narrower than the width of a newspaper page. • The strip represents one whole comic. • For your first comic, you want to have one frame. • Label this strip “one whole.”

19. Comic Strip Fractions • Get a new strip. • Fold it in half. • How many equal parts do you have? • Label each of the parts with the appropriate fraction. • Fold a new strip of paper in half. • Without opening up the strip, fold the strip in half again. • Predict the number of frames and check your prediction. • Label each of the parts with the appropriate fraction.

20. Comic Strip Fractions • Get a new strip. • Fold it in half a total of three times. • Predict the number of frames and check your prediction. • Label each of the parts with the appropriate fraction. • Repeat the folding in half process with a new strip of paper. • Fold the strip in half a total of four times. • Predict the number of frames and check your prediction. • Label each of the parts with the appropriate fraction.

21. Comic Strip Fractions Diane was puzzled about the way the folding activity contradicted what she was thinking. When Diane folded her “whole” strip into halves then halves again she got fourths just as she expected. But when she folded her strip a third time into halves, she expected to get 6ths because 3 times 2 is 6. When she folded it 4 times she expected 8ths because 2 times 4 is 8. She was surprised to find out that she was wrong! How would you explain to Diane the mathematical relationship between the number of folds and the number of pieces?

22. Comic Strip Fractions • thirds • fifths • sixths • ninths • tenths • twelfths Make strips to show the fractions listed below. Describe the folds you used to make each strip.

23. Comic Strip Fractions Making Connections • Which strips helped you make other strips? • Explain the underlying mathematical relationships between these strips.

24. Comic Strip Fractions • Arrange your strips in rows so that all of the left edges are lined up and the strips are ordered from the strip with the largest parts to the strip with the smallest parts. • Write as many number sentences as you can that relate the sizes of your fraction pieces. • We will be using these fraction strips throughout this workshop, so be sure to keep them in their envelope (Your “Fraction Kit”).

25. Looking through Teacher Lenses • How would you characterize the level of this task: High or low cognitive demand? • What mathematical ideas are embedded in the task? • What makes this worthwhile mathematics?

26. Sharing Quesadillas

27. Sharing Quesadillas • As part of your school’s international foods festival, a classmate brings quesadillas that he made for the entire class. • However, he only brought 21 quesadillas for the 28 students in your class. • Because your class normally works in groups of four, your teacher suggests that you give the same number of quesadillas to each group of four students. • How many quesadillas should each group receive?

28. Sharing Quesadillas • Each group must decide how to share their quesadillas equally among the group members. • How would you share the quesadillas equally among the group members? • With a partner, find two differentways to solve the problem. • Use a picture or diagram in your solutions.

29. Sharing Quesadillas • Compare solutions with the other people at your table. • Take turns sharing your solutions. • Are all solutions the same? • If not, do all solutions give the same answer? • Do all solutions work? • Choose a solution to share with the class. Explain the solution that you chose.

30. Sharing Quesadillas • Paula wants to have at least one piece that is one half of a quesadilla, so she starts by dividing all of the quesadillas in half. • Dwayne says that because each group has three quesadillas, he will divide each quesadilla into thirds. • Clifton wants to divide each quesadilla into eighths because he says that each person will get more pieces. • Juanita decides that she will divide the quesadillas into fifths The pieces may be tiny, but they won’t make as big a mess.

31. Sharing Quesadillas • Analyze each student’s method and determine: • Does the method work? • Why or why not? • What would you have to do to make the method work? • Write a number sentence that describes the amount of quesadilla each person gets using his or her method. Is this the same amount as in your group’s solution?

32. Sharing Quesadillas • Darnell claims that it doesn’t matter what number of pieces the quesadillas are initially cut into—any number will work. • Investigate Darnell’s method. Is he correct? Why or why not? Use mathematics to explain whether or not he is correct.

33. Sharing Quesadillas • Bobbie Jo wonders if there is an easy way to figure out the amount each person gets. She wants a way that would work even if the number of quesadillas and/or number of group members changed. • Try several other combinations of quesadillas and group members. How can you easily figure out the amount each person gets?

34. Looking through Teacher Lenses • How would you characterize the level of this task: High or low cognitive demand? • What mathematical ideas are embedded in the task? • What makes this worthwhile mathematics?

35. Sharing Quesadillas Recap Fractions as Division is critical to a student’s understanding of fractions. Students find this meaning of fractions unusual. It is different from the meaning that has been carefully developed in the earlier grades—that of fractions as amounts or parts of wholes, not as operations. ¼ of 24, 24/4, and 24 ÷ 4 all mean exactly the same thing. They are all expressions for 6.Van De Walle, 2004