Rational Numbers and Representations

1. Rational Numbers and Representations Fractions and Decimals Grades 3 -5 Workshop Longwood University Dr. Virginia Lewis Cathlene Hincker Miriah Eisenman

2. Before we get started • Instructor Introductions

3. First…Pre-Workshop Content Assessment • Remember, you are not be graded during this workshop! • Please answer the questions to the best of your ability. • At the end of our third day together, you will take a post-workshop assessment to see how this workshop has impacted your knowledge of Grades 3-5 fractions, decimals and representations.

4. Community of Learners • Complete 3 X 5 note card: • Name • Email • Where do you teach? • Number of years teaching & grade levels • Favorite mathematics topic • Why are you here? • What weaknesses/concerns do you have about your own understanding of fractions and decimals? • Introductions – Introduce another person in our class to everyone!

5. What do students need to know about fractions and decimals? • What is the essential knowledge?

6. Goals of the Workshop • To know more about rational numbers than you expect your students to know and learn. • An awareness of different models and representations to enhance thinking about rational numbers. • To become familiar with the connections between fractions, decimals, and place value.

7. Goals of the Workshop • To know what mathematics to emphasize and why in planning & implementing lessons. • To anticipate, recognize, and dispel students’ misconceptions about fractions and decimals. • Build on prior grades’ fraction ideas and know later-grade connections  vertical alignment.

8. The Standards National Council of Teachers of Mathematics Standards and the Virginia Standards of Learning

9. What do NCTM’s Principles and Standards for School Mathematics say? Understand numbers, ways of representing numbers, relationships among numbers, and number systems • Grades 3 -5 • develop understanding of fractions as parts of unit wholes, as parts of a collection, as locations on number lines, and as divisions of whole numbers; • use models, benchmarks, and equivalent forms to judge the size of fractions; • recognize and generate equivalent forms of commonly used fractions, decimals, and percents; National Council of Teachers of Mathematics. 2000. Principles and Standards for School Mathematics. Reston, VA: NCTM.

10. What do NCTM’s Principles and Standards for School Mathematics say? Compute fluently and make reasonable estimates • Grades 3 -5 • develop and use strategies to estimate computations involving fractions and decimals in situations relevant to students' experience; • use visual models, benchmarks, and equivalent forms to add and subtract commonly used fractions and decimals; National Council of Teachers of Mathematics. 2000. Principles and Standards for School Mathematics. Reston, VA: NCTM.

11. Fractions/Decimals in Virginia SOLs K-2 Kindergarten • K.5 Identify parts of a set and/or region that represent fractions for halves and fourths 1st • 1.3 Identify parts of a set and/or region that represent fractions for halves, thirds, and fourths and write the fractions 2nd • 2.3 Identify parts of a set and/or region that represent fractions for halves, thirds, fourths, sixths, eighths, and tenths (connects to decimals later) Virginia Department of Education. 2009. Mathematics Standards of Learning for Virginia Public Schools. Richmond, VA: Commonwealth of Virginia Board of Education.

12. What do students currently learn about fractions/decimals in your grade level? Organize into grade level groups Without looking at the SOLs brainstorm and record what you teach in your grade level about fractions and decimals. Look over the SOLs and adjust anything that needs adjusting on your chart paper. Record the SOL number next to each of your big ideas.

13. Identify on the grade level charts concepts that build on knowledge from previous grades.

14. Building on grades 3-5 In grade 6 • Describe and compare data, using ratios using the appropriate notations a/b, a to b and a:b. • Investigate and describe fractions, decimals, and percents as ratios • Identify a fraction, decimal, or percent from a representation • Demonstrate equivalent relationships among fractions, decimals, and percents • Compare and order fractions, decimals, and percents Virginia Department of Education. 2009. Mathematics Standards of Learning for Virginia Public Schools. Richmond, VA: Commonwealth of Virginia Board of Education.

15. Building on grades 3-5 More grade 6 • Demonstrate multiple representations of multiplication and division of fractions • Multiply and divide fractions and mixed numbers • Estimate solutions and then solve single-step and multistep practical problems involving addition, subtraction, multiplication and division of fractions • Solve single-step and multistep practical problems involving addition, subtraction, multiplication, and division of decimals Virginia Department of Education. 2009. Mathematics Standards of Learning for Virginia Public Schools. Richmond, VA: Commonwealth of Virginia Board of Education.

16. Building on grades 3-5 In grade 7 • Compare and order fractions, decimals, percents, and numbers written in scientific notation; • Identify and describe absolute value for rational numbers • Solve single-step and multistep practical problems, using proportional reasoning Virginia Department of Education. 2009. Mathematics Standards of Learning for Virginia Public Schools. Richmond, VA: Commonwealth of Virginia Board of Education.

17. Building on grades 3-5 In grade 8 • Simplify numerical expressions involving positive exponents, using rational numbers, order of operations, and properties of operations with real numbers; and • Compare and order decimals, fractions, percents, and numbers written in scientific notation • Describe orally and in writing the relationships between the subsets of the real number system • Solve practical problems involving rational numbers, percents, ratios, and proportions Virginia Department of Education. 2009. Mathematics Standards of Learning for Virginia Public Schools. Richmond, VA: Commonwealth of Virginia Board of Education.

18. The Common Core Standards • You will find a copy of the grades 3-5 standards for fraction instruction in your packet • While Virginia is not currently participating in the Common Core it is interesting to see these standards too. • http://www.corestandards.org/math

19. But there’s more to the standards • In your group • Read the Introduction to the Standards of Learning • Highlight anything of interest you would like to discuss • Read your assigned Process Standard. Be prepared to summarize for the class what this standard encompasses.

20. What representations do you currently use during fraction and decimal instruction?

21. Multiple Representations • How many different ways can you represent ¾?

22. Fraction Models • Area Models for ¾.

23. Fraction Models • Set Models for ¾.

24. Fraction Models • Measurement Models for ¾.

25. Problems and Models • In Mrs. Park’s class there are 24 students. One third of the students play soccer. • My dad made me a pan of brownies for my birthday. I ate 5/8 of my pan. • I tried to run from my school to my favorite ice cream shop. I ran 9/10 of the way before I stopped because I was tired. Adapted from page 98 of McNamara, J. & Shaughnessy, M. 2010. Beyond Pizzas and Pies. Sausalito, CA: Math Solutions.

26. Using representations to solve problems • It takes 18 minutes for John to walk home from school. He has walked 2/3 of the way at a constant speed. How many minutes has he been walking? • There are 18 marbles in Suzanne’s marble collection. 2/3 of the marbles are green. How many green marbles does Suzanne have? • Joanna filled 18 bowls with 2/3 cup of flour in each. How much flour did Joanna use? Adapted from page 130 of Van de Walle, J.A., Bay-Williams, J.M., Lovin, L. H., & Karp, K.S. 2014. Teaching Student-Centered Mathematics Developmentally Appropriate Instruction for Grades 6-8 (2nd ed). Boston, MA: Pearson Education, Inc.

27. What are the meanings of fractions? • Fractions as part of a whole or part of a set • Fractions as quotients- the result of division • Fractions as ratios or rates – comparing quantities with like (ratio) or unlike (rate) units • Fractions as operators – Stretches or shrinks the magnitude of another number • Fractions as measures – Rational number thought of as a unit fraction to be repeated Lamon, Susan J. 2007. Rational Numbers and Proportional Reasoning Toward a Theoretical Framework for Research. In Frank K. Lester, Jr (Ed.), Second handbook of research on mathematics teaching and learning (pp. 629 – 667). Charlotte, NC: Information Age Publishing, Inc.

28. Fractions as part of a set

29. Fractions in our Class • I need 10 volunteers to come to the front of the room. • What fraction are girls? • What fraction are wearing jeans? • What other fraction questions can we ask and answer? • Why is the denominator for every fraction 10? What does the denominator represent? • Why is the numerator different? What does the numerator represent?

30. Use Student Data to Write Fractions • Which of the following statements is true for you? • My last name is longer than my first name. • My last name and first name are the same length. • My last name is shorter than my last name. • What fraction of the class has last names that are longer? • What fraction of the class has last names that are shorter? • What fraction of the class has names that are the same length?

31. Use Student Data to Write Fractions • How many syllables are in your first name? • What fraction of the class has one syllable in their first name? • What fraction of the class has fewer than three syllables in their first names? • What fraction of the class has more than three syllables in their first names?

32. Fractions as a part of a whole • Use color tiles to build a rectangle that is ½ red, ¼ yellow, and 1/8 green and 1/8 blue. Make a drawing of your rectangle. • Can you find another rectangle that also satisfies the requirements? Make a drawing of your rectangle • Use color tiles to build a rectangle that is 1/6 red, 1/2 green, 1/3 blue. Make a drawing of your rectangle. • Can you find another rectangle that satisfies these requirements? Make a drawing of your rectangle. • Why do you think we used the ½, ¼, and 1/8 fractions in one rectangle and the ½, 1/3, and 1/6 in another rectangle? Adapted from pg 280 Burns, M. 2007. About Teaching Mathematics a K-8 Resource (3rd ed). Sausalito, CA: Math Solutions

33. Why is the whole important? If the whole is a group of 12 color tiles, what is ½? If the whole is a group of 10 color tiles, what is ½?

34. Blocking Out Fractions: An AIMS activity • This activity can be purchased in PDF form from Aims Education Foundation at the following web address. • http://www.aimsedu.org/item/DA6539/Blocking-Out-Fractions/1.html

35. Grade 3 Sol Practice Item Made Available by the Virginia Department of Education (VDOE) How does instruction which focuses on the role of the whole help students interpret this question?

36. Unitizing • Is the ability to think of a quantity in different sized chunks • Fold an isosceles triangle in half three different times. Then unfold the triangle… • Can you see eighths? • Can you see fourths? • Can you see halves? • Students need to be able to think flexibly about fractions!

37. What is Partitioning? Partitioning is the act of breaking the whole into parts Students need experiences where partitioning in different ways results in equivalent amounts For example the following partitions have different representations but all are equivalent to 1 whole.

38. Partitioning the Whole • How can you divide this square into halves? Is there more than one way to divide your square? • Write a number sentence to represent how the halves combine to make a whole. • How can you divide this square into fourths? Is there more than one way to divide your square? • Write a number sentence to represent how the fourths combine to make a whole. Adapted from pg 52 of Schuster, L., and Anderson, N.C. 2005. Good Questions for Math Teaching . Sausalito, CA: Math Solutions.

39. Partitioning the Whole • How can you divide this square into eighths? • Write a number sentence to represent how the eighths combine to make a whole. • How can you divide this square into halves, fourths, and eighths? • Label and color-code each fractional part. • Write a number sentence to represent how the parts combine to make the whole. Adapted from pg 52 of Schuster, L., and Anderson, N.C. 2005. Good Questions for Math Teaching . Sausalito, CA: Math Solutions.

40. Egg Carton Fractions • Can you make halves? Thirds? Fourths? Sixths? Twelfths? • Can you make 2/3? How many twelfths are the same as 2/3? How many sixths are the same as 2/3? • Can you make ¾? How many twelfths in ¾? • Put two cartons together and make: • 1 ½ • 1 5/6

41. Equivalent names using red and yellow counters • You need 18 counters, 6 red and 12 yellow. • The 24 counters make up the whole. • Can you group the counters to “see” • 6/18 • 12/18 • 4/6 • 1/3 Adapted from pp. 153-154 in Van de Walle, J.A. and Lovin, L.H. 2006. Teaching Student-Centered Mathematics Grades 3-5. Boston, MA: Pearson Education, Inc.

42. Find the fraction name for each piece.

43. Fractions, Decimals and the Open Number Line • Draw an open number line like this: 0 2 • Locate 0.50 on the number line with your finger. • Locate ¼ on the number line with your finger. • Locate 0.75 on the number line with your finger. • Locate 1 ½ on the number line with your finger • What could this task reveal to a teacher using it for formative assessment?

44. Fractions as Operators Find the input-output rules for these function machines. Adapted from pg 83 of Chapin, S. H. and Johnson, A. 2000. Math Matters Grades K-6 Understanding the Math You Teach. Sausalito, CA: Math Solutions Publications.

45. A focus on Visualization • Can you see 4/10 of something? What is the whole? • Can you see 1 ½ of something? What is the whole? • Can you see 2/3 of 3/5? What fraction would that be? Adapted from pg 329 of Van de Walle, J. A., Karp, K. S., and Bay-Williams, J. M. 2013. Elementary and Middle School Mathematics Teaching Developmentally (8th ed.) Boston, MA: Pearson Education, Inc.

46. Is this 3/8?

47. Wrapping Up Wholes and Parts • Students need opportunities to think about the whole • Use materials where the size of the whole changes • Students need to encounter unequally partitioned areas and number lines • Students need to design their own strategies for partitioning areas and number lines

48. Decimal Basics

49. Cube activity: What’s special about base 10? • Grab two handfuls of unifix cubes. Divide the cubes into the specified group size. • Record the number of groups and the number of leftovers in your chart. For each different sized group do three trials. • What do you notice when we make groups of size 10? What is so special about base 10? Adapted from pp. 47-50. Cohen, S. C., Lester, J. B., and Yaffee, L. 1999. Building a System of Tens Casebook. Parsippany, NJ: Dale Seymour Publications.

50. Ways to Build a number with Base Ten Blocks • Build 156 with your base ten blocks • With your partner build and record all the possible ways you were able to make 156. • How many different ways are there? • How did you know when you had found them all?