Pre-Session Table Task Write as many equivalent fractions as you can for the following fraction: 1 Based upon research by Dr. Cathy Bruce, Trent University and Curriculum and Assessment Policy Branch Comparing Fractions Math For Teaching 2 Based upon research by Dr. Cathy Bruce, Trent University and Curriculum and Assessment Policy Branch

Learning Focus Participants will: Explore the impact of using intentionally selected representations. Use number lines to develop fraction number sense. Compare fractions using a variety of strategies and tools. Connect research to practice. 3 Based upon research by Dr. Cathy Bruce, Trent University and Curriculum and Assessment Policy Branch

Session Norms Be engaged in the tasks and discussions as this enriches everyones experience. Be a learner it is fun! Actively seek connections between the research and your classroom experience. By better understanding student thinking we better understand the impact we can have on their learning. 4 Based upon research by Dr. Cathy Bruce, Trent University and Curriculum and Assessment Policy Branch Agenda

Intended Audience Activity: Pre session table task Overview of the Research Activity: Partitioning Rectangles Using number lines Understanding equivalency Understanding density Curriculum connections task Activity: Exploring mathies.ca 5 Student

Educator Student Educator/Student Educator/Student Educator/Student Educator Educator Based upon research by Dr. Cathy Bruce, Trent University and Curriculum and Assessment Policy Branch The FLP Across the Elementary Grades 6

Based upon research by Dr. Cathy Bruce, Trent University and Curriculum and Assessment Policy Branch Fractions Across the Grade 9 Courses Multiply/Divide Add/Subtract Compare Unit 7 Based upon research by Dr. Cathy Bruce, Trent University and Curriculum and Assessment Policy Branch

Fractions Across the Grade 10 Courses Multiply/Divide Add/Subtract Compare Unit 8 Based upon research by Dr. Cathy Bruce, Trent University and Curriculum and Assessment Policy Branch Fractions Across Grades 11 and 12 Multiply/Divide

Add/Subtract Compare Unit 9 Based upon research by Dr. Cathy Bruce, Trent University and Curriculum and Assessment Policy Branch The Fractions Research Team 20112014 Curriculum and Assessment Policy DSBN Branch HWDSB TRENT MATHEMATICS EDUCATION

RESEARCH COLLABORATIVE Cathy Bruce Shelley Yearley Tara Flynn Rich McPherson 10 KPRDSB OCDSB SCDSB

SMCDSB TLDSB YCDSB Sarah Bennett Based upon research by Dr. Cathy Bruce, Trent University and Curriculum and Assessment Policy Branch Overarching Research Question What instructional strategies support junior and intermediate grade students in the learning of fractions, an area of mathematics that is typically viewed as difficult to learn?

11 Based upon research by Dr. Cathy Bruce, Trent University and Curriculum and Assessment Policy Branch fractionsteaching.ca 12 12 Based upon research by Dr. Cathy Bruce, Trent University and Curriculum and Assessment Policy Branch Change in P Data and

Post-Assessment Change in Pre- and Post-Assessment Means, by Grade (n=360) Means, By Grade 100 Grade 6 90 Grade 7 80

Grade 8 70 Grade 9 Enriched 60 Grade 9 Locally Developed 50 Grade 9 Applied Grade 10 Academic

40 30 20 10 0 Pre-Assessment Mean 13 13 Post-Test Mean Based upon research by Dr. Cathy Bruce, Trent University and Curriculum and Assessment Policy Branch fractionslearningpathways.ca

14 Based upon research by Dr. Cathy Bruce, Trent University and Curriculum and Assessment Policy Branch Partitioning Rectangles Construct a large rectangle on graph paper provided. Partition your rectangle into fourths. Shade in one one-fourth of your rectangle. Now partition each of your fourths into thirds. What fraction of the area is shaded using the new fractional unit? Write a symbolic equation with these two

fractions. 15 Based upon research by Dr. Cathy Bruce, Trent University and Curriculum and Assessment Policy Branch Using Number Lines 16 Based upon research by Dr. Cathy Bruce, Trent University and Curriculum and Assessment Policy Branch What the research tells us There is a multitude of representations in North American textbooks (15-25) compared to

Japanese texts (4) The four are: number lines rectangular models volume models flats & rods 17 Based upon research by Dr. Cathy Bruce, Trent University and Curriculum and Assessment Policy Branch Math Teaching for Learning: Purposeful Representations 18

Based upon research by Dr. Cathy Bruce, Trent University and Curriculum and Assessment Policy Branch Purposeful Representations These representations are used consistently in Japanese resources with the purpose of developing students understanding of fraction as quantity and to emphasize the underlying concepts of i. expressing all fractions as a multiple of a unit fraction, ii. making comparisons based on like-units, and iii. identification of the whole 19

Based upon research by Dr. Cathy Bruce, Trent University and Curriculum and Assessment Policy Branch Fractions on Stacked Number Lines 20 Based upon research by Dr. Cathy Bruce, Trent University and Curriculum and Assessment Policy Branch Further Evidence Two promising representations, as noted in the research to date, are bar or rectangular area models and number lines as a linear representation because both are more readily and accurately partitioned

evenly for odd and large numbers of partitions. (Watanabe, 2012) Number lines (linear measures) can effectively show proper and mixed fractions simultaneously and promote attention to the relationship of the numerator to the denominator. (Bruce, Chang, Flynn, Yearley, 2013) 21 Based upon research by Dr. Cathy Bruce, Trent University and Curriculum and Assessment Policy Branch Effective Questioning Johanning (2011) cautions that using visual models such as fraction strips and

number lines support students ability to visualize fractions and develop a sense of relative size. However, visual models are not enough. During instruction, students should routinely be asked to use their understanding of relative size to make sense of situations in which fractions are used operationally. (99) (Bruce, Bennett and Flynn, 2014) 22 Based upon research by Dr. Cathy Bruce, Trent University and Curriculum and Assessment Policy Branch Types of Number Lines

Empty number line Open number line 0 Closed number line 0 23 50 Based upon research by Dr. Cathy Bruce, Trent University and Curriculum and Assessment Policy Branch The research also tells us Students hold on to initial representations so what we

introduce is important! Circular area models are overrepresented (and typically used to introduce fractions) and pose difficulties (they are difficult to equi-partition into large or odd numbers AND students are not formally introduced to the concept of area of a circle until intermediate grades) In North America, numeric - symbolic manipulation is privileged prematurely 24 Based upon research by Dr. Cathy Bruce, Trent University and Curriculum and Assessment Policy Branch Why Number Lines?

Lewis (p.43) states that placing fractions on a number line is crucial to student understanding. It allows them to: Further develop their understanding of fraction size (PROPORTIONAL REASONING) See that the interval between two fractions can be further partitioned (DENSITY) See that the same point on the number line represents an infinite number of equivalent fractions (EQUIVALENCY) 25 25 Based upon research by Dr. Cathy Bruce, Trent University and Curriculum and Assessment Policy Branch

Fractions on Stacked Number Lines - revisited 26 Based upon research by Dr. Cathy Bruce, Trent University and Curriculum and Assessment Policy Branch Fractions on the Number Line 27 Based upon research by Dr. Cathy Bruce, Trent University and Curriculum and Assessment Policy Branch

Mapping to the FLP 28 Based upon research by Dr. Cathy Bruce, Trent University and Curriculum and Assessment Policy Branch Fractions on the Number Line Using a number line, complete the following tasks: 1. Place a fraction between 1 and 2. 2. Place two fractions between 1/12 and 9/12 3. Place a fraction between 2/5 and 2/3 4. Place a fraction between 1/3 and 2/3

5. Place four fractions between 1/3 and 2/3 29 Based upon research by Dr. Cathy Bruce, Trent University and Curriculum and Assessment Policy Branch Examining Student Thinking Select one sample and consider it in terms of the following: What are some assets (what knowledge and skills are demonstrated)? What are you wondering (what is unclear or unwritten?) What are some challenges? (what knowledge and skills appear to be fragile or absent?)

What other observations would you add? 30 Based upon research by Dr. Cathy Bruce, Trent University and Curriculum and Assessment Policy Branch Student Thinking Sample #1 31 Based upon research by Dr. Cathy Bruce, Trent University and Curriculum and Assessment Policy Branch Student Thinking Sample #2 32

Based upon research by Dr. Cathy Bruce, Trent University and Curriculum and Assessment Policy Branch Student Thinking Sample #3 33 Based upon research by Dr. Cathy Bruce, Trent University and Curriculum and Assessment Policy Branch Student Thinking Sample #4 34

Based upon research by Dr. Cathy Bruce, Trent University and Curriculum and Assessment Policy Branch Student Thinking Sample #5 35 Based upon research by Dr. Cathy Bruce, Trent University and Curriculum and Assessment Policy Branch Student Thinking Sample #6 36

Based upon research by Dr. Cathy Bruce, Trent University and Curriculum and Assessment Policy Branch Student Thinking Sample #7 37 Based upon research by Dr. Cathy Bruce, Trent University and Curriculum and Assessment Policy Branch Using the Number Line Linear representations such as the number line support the study of equivalent fractions, as any point on the line can represent and infinite number of equivalent fractions. Consider the

location of two on this number line. Linear representations such as the number line support the study of equivalent fractions, as any point on the line can represent an infinite number of equivalent fractions. Consider the location of two on this number line. Using whole number units, we would name it 2. However, we can use other units to name it, such as halves, in which case it would be 4/2 0 1 However, we can use other units to name it, such as halves, in which case it would be 2

4 . 2 4 2 0 4 We know then that 2 = . 2 38

1 2 A whole can be partitioned into an infinite number of equi -partitions. Therefore, there are essentially an infinite number of equivalent fractions for any point on the number line. The exploration of equivalence allows students to develop an understanding of equivalent fractions as simply being a different way of naming the sameupon quantity; it also supports them inBruce, viewingTrent the fraction

as a numeric value. Based research by Dr. Cathy University and Curriculum and Assessment Policy Branch Precise Instructional Decisions Introduce a new concept with a familiar representation, and introduce a new representation with a familiar concept. For example, use a number line with primary

students when first introducing unit fractions as they have experience with whole numbers on the line. Later, represent these same unit fractions when introducing a new model, such as a rectangle. Paying Attention to Fractions, page 22 39 Based upon research by Dr. Cathy Bruce, Trent University and Curriculum and Assessment Policy Branch Understanding Equivalency 40

Based upon research by Dr. Cathy Bruce, Trent University and Curriculum and Assessment Policy Branch Equivalence at a Young Age 41 Based upon research by Dr. Cathy Bruce, Trent University and Curriculum and Assessment Policy Branch Equivalence as Balance https://globalmathblog.wordpress.com/2015/ 42

Based upon research by Dr. Cathy Bruce, Trent University and Curriculum and Assessment Policy Branch Avoiding Misconceptions about Equivalency If students are prematurely or solely exposed to symbolic procedures for determining equivalent fractions they are prevented from connecting this procedure to the visual models. Being able to make such connections will allow them to connect equivalent fractions to their proportional reasoning skills and to whole numbers and decimals and support them in making sense of algebraic fractions in later grades. Paying Attention to Fractions, page 17 43

Based upon research by Dr. Cathy Bruce, Trent University and Curriculum and Assessment Policy Branch Using Models for Equivalence Construct a large rectangle on graph paper provided. Partition your rectangle into fourths. Shade in one one-fourth of your rectangle. Now partition each of your fourths into thirds. What fraction of the area is shaded using the new fractional unit? Write a symbolic equation with these two fractions. 44

Based upon research by Dr. Cathy Bruce, Trent University and Curriculum and Assessment Policy Branch Merging for Equivalence Write as many equivalent fractions as you can for the following fraction: 45 Based upon research by Dr. Cathy Bruce, Trent University and Curriculum and Assessment Policy Branch Research Connection Think Individually read pages 14-17 of PATF

Pair Discuss Share At the table, create a summary on chart paper of the key ideas presented Post around the room 46 46 Based upon research by Dr. Cathy Bruce, Trent University and Curriculum and Assessment Policy Branch Comparison Task 1 2

Which is closer to 1 28 or 19 ? 26 16 Explain your thinking.

47 Based upon research by Dr. Cathy Bruce, Trent University and Curriculum and Assessment Policy Branch Which is greater? Which is greater : 7 or 8 5 8

How do you know? 3 3 Which is greater : 12 or 10 How do you know? 48 48 Based upon research by Dr. Cathy Bruce, Trent University and Curriculum and Assessment Policy Branch Which is greater? (Continued)

Which is 8 greater 12 : 1 or2 ? Explain your answer in two different ways. For further information about the benefits of multiple solutions, see:

Journal for Research in Mathematics Education (NCTM) July 2014 Do Multiple Solutions Matter? Prompting Multiple Solutions, Interest, Competence and Autonomy Stanislaw Schukajlow and Andre Krug 49 49 Based upon research by Dr. Cathy Bruce, Trent University and Curriculum and Assessment Policy Branch Using Benchmarks Write each of these fractions on a sticky note. 1 2

3 Place each of the fractions on the number line, and prepare to justify your placements. 50 Based upon research by Dr. Cathy Bruce, Trent University and Curriculum and Assessment Policy Branch Comparing Fractions 51 Based upon research by Dr. Cathy Bruce, Trent University and Curriculum and Assessment Policy Branch

Understanding Density 52 Based upon research by Dr. Cathy Bruce, Trent University and Curriculum and Assessment Policy Branch Connections of Density of Fractions The concept of density of [fractions] is applied when making measuring tools for different purposes. For example, a carpenter building a barn would be content with accuracy to the nearest eighth of an inch. However, a cabinet

maker would not. The cabinet maker would need accuracy to thirty-seconds of an inch. However, thirty-seconds of an inch would be wholly inadequate for making electronic components as electronic components need to be accurate to microns (forty millionth of an inch). Imagine the repartitioning necessary to go from a thirty-second of an inch to a forty millionth of an inch. (Petit, Laird, & Marsden, 2010, p. 124) 53 Based upon research by Dr. Cathy Bruce, Trent University and Curriculum and Assessment Policy Branch

Finding a Fraction Between Two Fractions 1. What fraction could come between those two fractions? Convince a partner that your choice is correct. 1 2 4 54 ? 5

2 7 Based upon research by Dr. Cathy Bruce, Trent University and Curriculum and Assessment Policy Branch And now Identify all the fractions between 1/5 and . 55 55 Based upon research by Dr. Cathy Bruce, Trent University and Curriculum and Assessment Policy Branch

Emphasizing the Density of Fractions Tirosh, Fischbein, Graeber, & Wilson (1998) found that only 24 percent knew that there was an infinite amount of numbers between and , 43 percent claimed that there are no numbers between and , and 30 percent claimed that is the successor to (pp.8-9) Petit, Laird & Marsden, p. 131 56 Based upon research by Dr. Cathy Bruce,

Trentresearch University and Curriculum and Assessment Policy Branch Based upon by Dr. Cathy Bruce, Trent University and Curriculum and Assessment Policy Branch

Connecting Unit Fractions and Comparing Fractions 57 Based upon research by Dr. Cathy Bruce, Trent University and Curriculum and Assessment Policy Branch Punctuated Learning DRAFT Punctuated Instruction Map for Grade 5/6 Created by Ontario Classroom Teachers

Based on research with Dr. Cathy Bruce and the Curriculum and Assessment Policy Branch 58 Based upon research by Dr. Cathy Bruce, Trent University and Curriculum and Assessment Policy Branch What Students Said About Punctuated Fractions Instruction One teacher asked her 19 grade 4 students: Did you like learning math this way? Student A: 2 students said no

1 student response was unclear 16 students said yes Student B: I like learning math this was because were not just learning one specific thing at one time, were learning a bunch of things together. This is a easier and more fun way of learning. I have definitely learned a lot more that way. 59 I do not like learning math when one week we do this and the other week we that. I like

doing math when we do 1 part at a time. Student C: I do not like learning this way, I love it because it makes me focus on more than one thing. It also helps me by if I do one unit at a time I might forget it or what we are learning. Based upon research by Dr. Cathy Bruce, Trent University and Curriculum and Assessment Policy Branch Comparing Fractions across the K12 Curriculum Identify an expectation from the grade you teach from the page provided.

Read the section above the expectation. Identify a question or task that you have used in your classroom which addresses that expectation. Discuss with your table mates. 60 60 Based upon research by Dr. Cathy Bruce, Trent University and Curriculum and Assessment Policy Branch Resources

61 Developed by Ontario teachers Informed by current educational research Field tested in Ontario classrooms Responsive to feedback Free Free from advertisements Based upon research by Dr. Cathy Bruce, Trent University and Curriculum and Assessment Policy Branch

Exploring Representations Mobile Device A.Install from Store 1) Relational Rods+ by mathies 2) Set by mathies 3) Colour Tiles by mathies Laptop Check to see that you can access mathies.ca Learning Tools

4) Rekenrek by mathies 5) Puffin Academy (optional - to access our older stuff) - see http://mathclips.wikispaces.com/Using +Flash+on+Mobile 6) 62 Adobe AIR (for Android). Based upon research by Dr. Cathy Bruce, Trent University and Curriculum and Assessment Policy Branch

Exploring mathies.ca Learning Tools 1. 2. 3. 4. 63 Log on Select Learning Tools Select Select a Topic, Fractions comes up first Choose a tool and open it Based upon research by Dr. Cathy Bruce, Trent University and Curriculum and Assessment Policy Branch

Supports Scroll to the bottom of the page for a link to our email list. http://mathclips.ca/WhatsNewEmailList.html 64 64 Based upon research by Dr. Cathy Bruce, Trent University and Curriculum and Assessment Policy Branch Time to Explore Play with the tools to see how they might be

used to solve some of the fraction tasks we have worked on earlier or others of your choosing Think about some classroom activities that could be supported with these tools Note that the Fraction Number Line tool is not yet available Have fun! 65 65 Based upon research by Dr. Cathy Bruce, Trent University and Curriculum and Assessment Policy Branch Feedback and Comments How do the mathies tools support conceptual

understanding? How can you include them in your instruction? 66 66 Based upon research by Dr. Cathy Bruce, Trent University and Curriculum and Assessment Policy Branch Visit fractionsteaching.ca 67 67 Based upon research by Dr. Cathy Bruce, Trent University and Curriculum and Assessment Policy Branch