p422-439-Gr3ON-Number-Unit6-fractions

Unit 6: Fractions Lesson Content Page 1 Fractions Introduction 442 2 Introduction to Fractions 424 3 Unit Fractions and Non-unit Fractions 428 4 Read Aloud: Jump, Kangaroo, Jump!: First Reading 432 5 Jump, Kangaroo, Jump!: Second Reading 436 6 and 7 Comparing Fractions 440 6 Equal Sharing with Area Models 447 7 Equal Sharing of One Whole 449 8 Equal Sharing of More Than One Whole 452 9 Naming Fractions Relative to Different Wholes 456 10 Introduction to Set Models 461 11 Investigating Set Models 465 12 Identifying Fractions in Set and Area Models 469 13 Equal Sharing with Set Models 472 14 Equal-Sharing Problems 476 Fraction Art 483

Fractions Introduction About the Research indicates that fractions are among the most difficult math concepts for students to understand in elementary grades, and that poor comprehension of fractions can interfere with later math progress (Petit, Laird, & Marsden, 2010, p. xi). It has also been shown that “improving students’ understanding of fractions occurs through more precise instruction rather than through significant increases in the time spent learning about fractions” (Ontario Ministry of Education, 2014, p. 21). Teachers are encouraged to plan intentional units and lessons, while using a variety of hands-on concrete objects, drawings, and tools to maximize learning experiences. Like whole numbers, fractions represent quantity. Although there are many meanings for fractions, students in grade three generally work with fractions as expressions of part-whole relationships, by composing and decomposing wholes and comparing fractional parts using various models. A frequent misconception about fractions is that they always represent quantities less than one, which overlooks the existence of both mixed fractions, such as one and one half, and improper fractions, such as nine fourths. Fractions and Sharing In grade three, students engage in fair-share (or equal-share) problems that can result in whole numbers, mixed numbers, and fractional amounts. As students compare the results of different sharing situations, they are investigating the relationships between the number of sharers and the amount to be shared. For example, when sharing the same whole, the greater the number of people sharing, the less each person receives. Representing Fractions Students can investigate a variety of representations as they divide whole objects and sets of objects and identify the parts in relation to the whole. Students expand on their knowledge of fractions using area models, finding out that fractional parts of an area must be equal in size but not necessarily equal in shape. They also work with set models, reaching the understanding that it is the number of objects in the set that establishes the whole rather than their size. According to the curriculum, fractions can be expressed as a count of unit fractions (e.g., 3 one fourths), as words (e.g., three fourths), as a combination of words and numbers (e.g., 3 fourths), and symbolically, ustsuindgensttsanudnadredrsftraancdtiothnealmneoatnatiinogno(fe.ega.c, h43 ). It is important that representation, as well as the connections between them. When using standard fractional 422 Number and Financial Literacy

notation, students can also investigate the repeated addition of unit fractions that have the same denominator. Equivalent Fractions Grade three students also learn about equivalent fractions: different ways of naming or representing the same part of a whole or group. For example, through investigation, they learn that one half is also equal to three sixth and five tenths. As they explore equivalent fractions represented in standard fractional notation, they can discover relationships between the numerator and denominator. In the case of one half, for example, the numerator is always half of the denominator. Lesson Topic Page 1 Introduction to Fractions 424 428 2 Unit Fractions and Non-unit Fractions 432 436 3 Read Aloud: Jump, Kangaroo, Jump!: First Reading 440 447 4 Jump, Kangaroo, Jump!: Second Reading 449 452 5 Comparing Fractions 456 461 6 and 7 Equal Sharing with Area Models 465 469 6 Equal Sharing of One Whole 472 476 7 Equal Sharing of More Than One Whole 483 8 Naming Fractions Relative to Different Wholes 9 Introduction to Set Models 10 Investigating Set Models 11 Identifying Fractions in Set and Area Models 12 Equal Sharing with Set Models 13 Equal-Sharing Problems 14 Fraction Art Fractions 423

1Lesson Introduction to Fractions Math Number Curriculum Expectations • B 1.6 use drawings to represent, solve, and compare the results of fair-share Previous Experience problems that involve sharing up to 20 items among 2, 3, 4, 5, 6, 8, and 10 with Concepts: sharers, including problems that result in whole numbers, mixed numbers, In grade two, students and fractional amounts divided various two- dimensional shapes and • B 1.7 represent and solve fair-share problems that focus on determining and three-dimensional objects into halves and fourths. using equivalent fractions, including problems that involve halves, fourths, Students should be and eighths; thirds and sixths; and fifths and tenths familiar with the terms ‘half,’ ‘fourth,’ ‘part,’ and Possible Learning Goals ‘whole.’ • Communicates an understanding of what they think fractions are and offers Teacher Look-Fors some examples • Equally partitions various two-dimensional shapes into halves, thirds, and fourths using various strategies • Distinguishes between shapes that are correctly divided into halves, thirds, or fourths, and those that are not correctly divided • Selects one or two appropriate strategies for equi-partitioning shapes into fractional parts • Identifies the whole, and explains/shows how the parts relate to the whole • Explains how shapes can be equally divided into fractional parts that are equal in size • Names some fractional parts using appropriate language PMraotcheesmseast:ical About the rrReeepflareescsoteinnnignti,gncgao,nndnepcrotivnign,g, communicating It is worthwhile to uncover students’ previous knowledge of, and any misconceptions they may have about, fractions at the beginning of the unit. Marilyn Burns cautions that children’s “understanding of fractions typically is incomplete and confused. For example, they often think of half as any part of a whole, rather than one of two equal parts, and they often refer to one-half being larger than another” (Burns, 2000, p. 223). Empson and Levi state that “early experiences with physically partitioning objects or sets of objects may be as important to a child’s development of fraction concepts as counting is to their development of whole number concepts” (Empson & Levi, 2011, p. 60). This is 424 Number and Financial Literacy

Math Vocabulary: why they recommend that tasks be focused on equally partitioning wholes pfqfrfoaotoauhnruucatrer,trertitwoehtfeohnsrhtuis,,hror,dthilthreo,hadr,ntlsesvwee,,eeqotcsfhufoo,otiauhrnfudlorig,rrtsutdrh,hurs,tse,ht,nwst,o (i.e., equi-partitioning) rather than on naming fractions after looking at pre-divided wholes with some parts filled in. The latter naming tasks become an exercise in counting pieces, rather than having students connect the fraction to the whole (Empson & Levi, 2011, p. 7). Several principles of fractions can be reinforced as students equi-partition area models. One such principle is that “the equal parts into which the whole is divided are equal but do not have to be identical” (Small, 2009, p. 199). It is a common misconception that equal fractional pieces need to be congruent (the same shape and size) when, in fact, equal fractional pieces can have different shapes and still have the same area. Another misconception that can arise is that fractional parts within a whole must be adjacent. For example, students might not realize that, in a rectangle divided into four equal parts, when two parts not touching one another are shaded, the shaded area is two fourths, or one half, of the whole. Addressing such ideas as they arise is critical to preventing the development of misconceptions. About the Lesson This introductory lesson is designed to evoke rich discussion so you can discover what your students know and wonder about fractions, and any misconceptions they may have. As students equi-partition area models, fundamental concepts are reviewed such as identifying the whole and parts, and identifying benchmark fractions such as halves and fourths. Materials: Minds On (20 minutes) paper squares, paper • Ask students what they think fractions are. Have them give some examples. rectangles (not squares), Digital Slide 53: Dividing Ask whether they use fractions in their daily lives and how. into Halves • Ask what ‘one half ’ means. Show a square piece of paper and ask students Time: 55 minutes how they could divide it in half. Ask if there is more than one way. Show the different ways they suggest to create halves by folding several congruent paper squares. Have students prove that the shape is divided in half. • Repeat using paper rectangles. • Ask students for an example of how either shape could be divided into two pieces but not be divided into halves. Have students explain their thinking. • Show Digital Slide 53: Dividing into Halves. Working with one image at a time, discuss how the items could be divided into halves. Then discuss ways that the items could be divided into two parts that are not halves. • Ask what fourths are (e.g., four equal parts). Show one way a paper square can be divided into fourths. Fractions 425

Working On It (15 minutes) • Students work in pairs. Provide each pair with four identical (congruent) paper squares and one paper rectangle. • Challenge students to find different ways to fold their shapes into fourths, using a fresh paper shape for each way they find. • Indicate to students that they need to justify how they know they divided the shapes into equal parts. Differentiation • For students who may not be ready for fourths, have them work on creating only halves. • For students who need more of a challenge, give them other shapes, such as regular hexagons, or shapes that cannot readily be divided into fourths, such as triangles and trapezoids. Assessment Opportunities Observations: Pay attention to any misconceptions that arise. For example, students may think that shapes are divided into fourths whenever they create four parts, even when the parts are not equal in area. Conversations: To further probe for understanding, ask some of the following prompts: – What is the whole? – How many parts is your shape divided into? How do you know they are the same size? – What do we call each part? – How are fourths different from halves? Consolidation (20 minutes) • Have student pairs meet with another pair to share their folded shapes. Each group proves to the other group that their partitions are equal fourths. • Meet with the class. Strategically select some pairs to share their solutions and their justifications of why they represent fourths. On one example, label each fourth using words (i.e., write ‘one fourth’). Ask how many parts make up the whole and what the whole can be called (four fourths). • Draw attention to the fact that the examples of fourths from the same-shaped piece of paper are not necessarily the same shape (e.g., A—a square folded once vertically and once horizontally versus B—a square folded vertically two times). Prove that the two examples of fourths are in fact the same size by cutting out one fourth from each example (e.g., a fourth from A and a fourth from B), and then cutting up one of those shapes and recomposing it so it fits on the other shape. Conclude that the fourths need to be the same size, but not necessarily the same shape. Introduce the term ‘congruent’ and 426 Number and Financial Literacy

explain it means ‘the same size and shape.’ Explain that fractional parts of a whole are the same size but are not congruent. • Co-create an anchor chart to highlight what students know about fractions: – Fractions are numbers that represent quantities. – Fractions can represent the equal-sized parts that a whole has been divided into. – Wholes can be divided in different ways to create equal-sized parts. – Fractional words help to describe the fraction (e.g., one half) and the whole (e.g., one whole or four fourths). • Add vocabulary used in the discussion to the Math Word Wall (e.g., part, whole, equal, fraction, half, halves, fourths, one fourth, congruent). Over the upcoming days, draw attention to the terms and ask what they mean. Ask questions such as, “If I eat one fourth of a cracker, what does that mean? What do you know about the pieces of the cracker? What do you know about the remaining pieces that I didn’t eat?” Encourage students to use fractional terms. • B uilding Social-Emotional Learning Skills: Critical and Creative Thinking: Ask students what they wonder about fractions or what they would like to learn. If they don’t have any ideas, tell them that they may get more ideas about fractions as they do additional activities in the coming days; they will be just like mathematicians, who get more curious and excited as they start to investigate an idea. Reinforce that although they don’t know everything about fractions YET, they will soon start to understand more with time, hard work, and interest. Encourage students to be “fraction detectives” for the next few days, by listening for fractional words people use or looking for fractions in their environment. This will help students realize that fractions play an integral role in their lives and are not just something they learn about at school. Fractions 427

2Lesson Unit Fractions and Non-unit Fractions Math Number Curriculum Expectations • B1.6 use drawings to represent, solve, and compare the results of fair-share problems that involve sharing up to 20 items among 2, 3, 4, 5, 6, 8, and 10 sharers, including problems that result in whole numbers, mixed numbers, and fractional amounts • B1.7 represent and solve fair-share problems that focus on determining and using equivalent fractions, including problems that involve halves, fourths, and eighths; thirds and sixths; and fifths and tenths • B 2.8 represent the connection between the numerator of a fraction and the repeated addition of the unit fraction with the same denominator using various tools and drawings, and standard fractional notation Teacher Possible Learning Goal Look-Fors • Investigates unit fractions and non-unit fractions using area models and Previous Experience with Concepts: develops the appropriate fractional language to describe them In grades one and two, students had some • Identifies unit fractions in whole area models and properly names them experience identifying • Identifies non-unit fractions in whole area models and properly names them and naming unit and • Learns how to count unit fractions in various ways, developing a sense of non-unit fractions of area models that are divided magnitude into halves and fourths. • Counts forward using unit fractions up to one whole and beyond (e.g., 5 one- PMraotcheesmseast:ical fifths) arRenefdlaesscottrinanitgne,ggsiaeenlsed,cptirnogvitnogo,ls ccoomnnmecutninicga,trinegpresenting, • Mentally visualizes or physically creates fractional parts and wholes based on counting unit fractions About the Generally, as students work with area models, they first learn about unit fractions (fractions with a numerator of 1, representing one of the equal units into which the whole is divided). This involves “reasoning about the relative contribution of the numerator and the denominator when only one part of the whole is considered” (Petit, Laird, & Marsden, 2010, p. 86). Students can also compare unit fractions by focusing on the size or number of the units, based on the number of equal units in the whole (e.g., for fractions of a same- sized whole, the fewer the units, the larger the units). Students can then extend this knowledge to non-unit fractions (numerators greater than 1), realizing that they are now considering more than one of the equal units. 428 Number and Financial Literacy

To help students perceive fractions as quantities, they can count the fractional parts of concrete models by unit fractions—1 one-fourth, 2 one-fourths, 3 one-fourths, 4 one-fourths—much like they count apples or other objects—1 apple, 2 apples, 3 apples, 4 apples. This helps students visualize two fourths and three fourths as being multiples of one fourth. Students can use the counting of unit fractions to transition to the repeated addition of unit fractions with the same denominator w(ei.lgl.,be14 i+nt41ro+du14c+ed14in=th44 e). This concept and standard fractional notation upcoming read aloud, Jump, Kangaroo, Jump. It is beneficial to count unit fractions up to and beyond 1 whole, such as 4 one-fourths, 5 one-fourths, 6 one-fourths, and so on, to reinforce the idea that fractions can be greater than 1 whole. This is important, since many students and adults think fractions are always less than 1. Such experiences also reveal that fractions can have more than one name (e.g., six fourths is the same as one whole and two fourths or one whole and one half). Mutshneaiivrttdehf,nraVftohcou,tcireoatihngb,,hufwtlihafht,rhoyn,l:eisn,itxhhta,hl,f, Once students have a strong understanding of unit fractions, they can tenth compose non-unit fractions with multiple unit fractions. They can also decompose wholes into unit fractions by equi-partitioning the whole. In Materials: an area model, this involves dividing the whole into equal-sized parts. Digital Slide 54: As students identify the unit fraction within the whole, they can also Fraction Strips, identify the complement fraction that makes up the rest of the whole BLM 60: Fraction Strips (e.g., one third is the unit fraction and two thirds is the rest that makes printed on 11 × 17 up the whole). This reinforces the part-part-whole relationship of paper (2–3 copies), fractions. chart paper, glue Time: 60 minutes About the Lesson In the lesson, pairs of students explore unit fractions and non-unit fractions using area models. Students are introduced to fractional language and written words up to ten tenths. Minds On (15 minutes) • Show Digital Slide 54: Fraction Strips. Explain that each row represents one whole and that some of the wholes have been divided into fractional parts. Ask students what pattern they see. (e.g., The wholes are divided into more and more fractional parts.) • Ask what fractional parts they see in the second row. Label each part ‘one half.’ Ask how many halves make up the whole. Explain that one part is known as a unit fraction because it identifies one of the equal units that make up the whole. Explicitly show how to count the number of unit fractions in the row (e.g., 1 one-half, 2 one-halves). • Draw attention to the row showing thirds and repeat the same line of questioning as for halves. Point to one third and ask how many thirds are Fractions 429

remaining (e.g., two thirds). Explain that two thirds is called a ‘non-unit fraction,’ and that it is made up of unit fractions. • Continue this line of questioning for the rest of the rows, labelling the first unit fraction in each row using words (e.g., one fourth). Have students identify the whole and how many unit fractions in each row make up the whole. Ask what pattern they see. (e.g., If you are looking at fifths, there are five unit fractions that make up the whole.) Working On It (20 minutes) • Using BLM 60, prepare fraction strips for fourths to tenths. (You will need one strip per student pair, and to have each fraction strip studied by at least two pairs of students.) • Students work in pairs. Provide each pair with one fraction strip, and have them glue it in the middle of a half piece of chart paper. • Inform students that they will brainstorm everything they observe about their fraction strip. Have partners share a few observations they can make about their fraction strips (e.g., the whole, the fractional part, the number of unit fractions) and discuss some of the ways they can show their thinking on their chart paper (e.g., words, colours, labels). They can also colour in parts of the strip and give the parts a fractional name. Differentiation • For students requiring further support, consider providing them with fraction strips already explored in the Minds On (e.g., halves or thirds) or strips having fewer fractional parts (e.g., fourths or fifths). • For some students, you may decide to extend the activity by having them work with and compare two fraction strips (e.g., fifths are smaller than fourths; there are more fifths in a whole than fourths in a whole). Assessment Opportunities Observations: Pay attention to how students are naming the parts. Can they connect the counting of the unit fractions to the fractional name? (e.g., 1 one- third, 2 one-thirds, so we call it two thirds.) Do they understand that fractions can represent the whole? (e.g., 3 one-thirds is three thirds or one whole.) Conversations: If students are having difficulty naming fractions, pose some of the following prompts: – What units is this whole divided into? (e.g., fifths) Count the first three unit fractions in this whole (e.g., 1 one-fifth, 2 one-fifths, 3 one fifths). What name can we give this fraction? (e.g., three fifths) Where are the one fifths in the three fifths? – What fraction of the whole is left? (e.g., two fifths) How many fifths are there altogether? (e.g., 5) How do you count them? How do we name the whole using fraction words? (e.g., five fifths) Is there another name we could give this fraction strip? (one whole) 430 Number and Financial Literacy

Consolidation (25 minutes) • Have pairs who investigated the same fraction strip meet together in a group. Have the group members discuss and compare what they discovered. • Meet as a class. For each fraction strip, have student groups share one idea that they recorded. As students present and explain their thinking, note their observations on the anchor chart started in the previous lesson. • Count some of the unit fractions together as a class while pointing to the fractional pieces, so all students can practise. • Ask students to examine what happens to the size of the pieces, including the unit fraction, when the same whole is divided into more pieces. For example, discuss how one third is larger than one fifth and why this makes sense. • Add any new ideas about fractions to the anchor chart. Add the following terms to the Math Word Wall: fifth, sixth, seventh, eighth, ninth, tenth, unit fraction, non-unit fraction. Also add some non-unit fractional names. Materials: Math Talk: BLM 61: Five Frames Math Focus: Counting unit fractions beyond one whole Teaching Tip Let’s Talk Integrate the math Select the prompts that best meet the needs of your students. talk moves (see page 8) throughout • Show BLM 61 and draw attention to the first five frame. What is the whole in Math Talks to maximize student the first row? How many equal parts does it have? What is the name of the participation and unit fraction? active listening. • Let’s count the unit fractions on the five frame together (e.g., 1 one-fifth, 2 one-fifths, 3 one-fifths, 4 one-fifths, 5 one-fifths). Is there another way to say 5 one-fifths? (e.g., one whole) • Draw attention to the upper two five frames together. If one five frame is the whole, how many wholes do we have now? (2) How can we count all the unit fractions? (1 one-fifth, 2 one-fifths, 3 one-fifths, ... 10 one-fifths) What other names can we give the two wholes? (10 one-fifths, ten fifths) I am looking at all of the fifths in the first whole and only one of the fifths in the second whole. How do I name the fraction I’m looking at? (six fifths) How can counting help you name the fraction? What is another name for the fraction? (e.g., one whole and one fifth) • Draw attention to the third five frame. How would we count on from 10 one- fifths to add this whole? (11 one-fifths, 12 one-fifths, …) • Who can show eleven fifths? What other name could we give to this fraction? (two wholes and one fifth) We can call this fraction eleven fifths or two wholes and one fifth. We are learning that fractions can have more than one name. • If someone told you that fractions are always less than one whole, what would you tell them? How could you use this model to prove that fractions can be more than one whole? Fractions 431

3Lesson Jump, Kangaroo, Jump!: First Reading Language Introduction to the Read Aloud Curriculum Expectations The read aloud text presents math concepts in a meaningful context that allows students to make connections to their everyday lives. During the first reading of Jump, Kangaroo, Jump!, students apply literacy strategies such as predicting, making connections, inferring, and synthesizing information to connect with the adventures of the animals in the story. The first reading can also prompt discussions about students’ personal experiences with similar activities. During the second reading, students act as mathematicians and apply the mathematical processes to discover and explore fractional concepts embedded in the story. The context can help students think about fractions as a regular part of their lives. The discussions that emerge from this reading further the learning about fractions of sets. There can also be discussions about students’ feelings about fractions and their self-confidence with solving related problems. Both readings are also valuable for assessing where students are in their understanding of fractions and how they interpret fractions within a context. The readings can reveal some of students’ misconceptions, which concepts need greater emphasis, and what differentiation may be necessary. Oral Communication • 1.3 identify a variety of listening comprehension strategies and use them appropriately before, during, and after listening in order to understand and clarify the meaning of oral texts • 1.6 extend understanding of oral texts by connecting the ideas in them to their own knowledge and experience; to other familiar texts, including print and visual texts; and to the world around them • 1.4 demonstrate an understanding of the information and ideas in a variety of oral texts by identifying the important information or ideas and some supporting details Reading • 1.5 make inferences about texts using stated and implied ideas from the texts as evidence 432 Number and Financial Literacy

Materials: Assessment Opportunities Written by Stuart J. Murphy Observations: Note each student’s ability to: Illustrated by Kevin – Infer and make predictions from the text and illustrations in the story O’Malley – Make connections between the text/images in the story and what they Text Type: Fiction: know about the physical characteristics of the animal campers and Narrative–Fantasy their abilities in the event Time: 20–25 minutes – M ake connections with the feelings of the animals in the story – Reflect upon the story and synthesize the message of the story Read Aloud: Jump, Kangaroo, Jump! Summary: It’s Field Day for the animal campers, who compete in different sports that require various-sized teams. Students can identify fractions in the various sets discussed in the story. Students can also connect with how it feels to compete, and what it takes to be a good winner and a good loser. NOTE: Throughout this first reading, read aloud and discuss only the narration, not the pages or text featuring fractional notation. Select the prompts that best suit your students. Before Reading Predicting/inferring Activating and Building On Prior Knowledge Making connections • Show students the front cover of the book. Read the title and the names of Analysing the author and illustrator. Ask students what they think the story might be about. Ask what clues on the cover help them make a prediction. Ask if they think this book is fiction or non-fiction, and why. • Ask students if they have ever participated in jumping events and what feelings they had before, during, and after the events. • S etting a Purpose: Say, “We will now read the story to find out why the kangaroo is being encouraged to jump.” During Reading Inferring/predicting • Read pages 4 and 5. Discuss how the story says that today is Field Day. Ask Making connections what a Field Day is. Ask what else a Field Day might be called. Using prior knowledge • Ask students what the story might be about. Ask what new information they now know about the story that they didn’t know from just looking at the front cover. Provide an opportunity for students to change or add to their predictions about the text. • Ask students if they have ever participated in any events like this. • Show pages 6 and 7. Before reading any of the text, ask students which animals they recognize and which ones are unfamiliar to them. Fractions 433

Inferring/predicting • Read pages 6 and 7. Ask, “Now that you have heard the names of the Monitoring comprehension animals, how might you match up their names with their pictures? In what country do you think this story takes place? Why?” Using prior knowledge/ making connections • Ask how they think the animals are feeling right now and why they think so. • Use the following prompts about the elements of fictional texts to check for Using prior knowledge/ making connections comprehension: Monitoring comprehension – What is the setting of the story? Analysing/inferring Inferring/making – Who are the characters in the story? connections – What is the tone at the beginning of the story? Inferring/activating prior knowledge/making connections – What is the conflict or problem in the story? Inferring/making • Reread this sentence on page 7: “They warmed up as they waited for Ruby, connections the kangaroo counselor, to start the events.” Ask students what a warm-up is, Using prior knowledge/ and if they know another name for a warm-up. Ask what actions the animals inferring are doing in the picture that help describe what a warm-up is. Ask why the Inferring animals might be doing a warm-up. • Ask what they think a counsellor is and what a counsellor does. • Read page 8 only. (Page 9 will be read in the second reading.) Ask what the whistle is for. Ask how tug-of-war is played, and why it is important to have equal teams. Ask what factors might determine which team wins. • After reading pages 10 and 11, ask what “The campers tugged and grunted” means. Ask for synonyms for ‘tugged’ and ‘grunted.’ • Ask students what the picture tells them about the tug-of-war game. • R ead pages 12 and 13. Ask students how they think the animals on Kangaroo’s team feel right now, and how the other team feels about winning. Ask students how they know. Ask how they feel when they win and when they lose. • Read page 14. (Page 15 will be read in the second reading.) Ask students what a relay race is. Ask why working as a team is so important in a relay race. Ask what it means to ‘count off ’ to make equal teams. • Read pages 16 and 17. Ask students how they know that the competitors are trying hard. Ask what they think the other competitors on land are doing and why. Ask why it is important to cheer on your team during a race. • Read pages 18 and 19. Ask students how they think Kangaroo’s team is feeling at this time. Ask how the team that came in third is feeling. Ask, “What do you think Kangaroo is thinking right now? Why?” • Read page 20 only. (Page 19 will be read in the second reading.) Ask students how they think a canoe race works. Ask what they think it takes to win a canoe race. Ask for examples of what could happen that might cause someone to lose. • Read pages 22 and 23. Ask why the animals in the two front canoes are looking at each other. Ask what they might be saying to each other. Then ask students how they think the paddlers in the last two canoes are feeling, and whether they think these paddlers are giving up. Ask why. 434 Number and Financial Literacy

Inferring/predicting • Read pages 24 and 25. Ask, “Do you think Kangaroo has given up on Using prior knowledge/ winning a race? Why? What do you think the last race will be?” predicting • Read pages 26 and 27. Ask students to tell you what long jump is. Ask how Analysing/inferring you win at long jump. Ask for the meaning of the line in the story “This Monitoring comprehension time, you’re on your own.” Ask students to predict who they think will win Analysing and to explain why they think so. Analysing/inferring • Read pages 28 and 29. Ask students why they think Kangaroo is smiling and if they think that he believes he will be the winner. Ask why he is so confident this time. • After reading pages 30 and 31, ask, “What does ‘a camp record’ mean?” • Ask students why all of the animals are happy even though they lost. Ask what this tells them about the animals’ characters. • Ask students why the author might have chosen to make Kangaroo the winner of the last event. Ask what this tells the reader. After Reading Synthesizing • Ask students what they think the big idea of the story is, and what the Analysing/inferring author’s message might be. Ask why the author might have written this book. Using prior knowledge/ (e.g., He wanted to tell us about animals; he wanted to tell us about field day competitions; he wanted to tell us about cooperation.) making connections Inferring/predicting • Ask students for reasons why the author did not tell the reader who was the overall winner of the Field Day. Ask if this was important to the story, and why or why not. • Ask what physical characteristics of the animals helped them in the events. Ask which other animals might be good at the sporting events in the story. • Ask, “When we read through the text a second time, what might you notice that you didn’t the first time?” Further Practice • R eflecting in Math Journals: Remind students that each of the animal campers in the story participated in the events. Then pose one or two of the following prompts for students to reflect on and write about: – Which different animal might you want to be that could be a new camper at Field Day? Why did you choose this animal? – Which of the events might you be good at? Which ones might you struggle with? Why? Fractions 435

4Lesson Jump, Kangaroo, Jump!: Second Reading Math Number Curriculum Expectations • B1.6 use drawings to represent, solve, and compare the results of fair-share Teacher problems that involve sharing up to 20 items among 2, 3, 4, 5, 6, 8, and 10 Look-Fors sharers, including problems that result in whole numbers, mixed numbers, and fractional amounts • B 1.7 represent and solve fair-share problems that focus on determining and using equivalent fractions, including problems that involve halves, fourths, and eighths; thirds and sixths; and fifths and tenths • B2.8 represent the connection between the numerator of a fraction and the repeated addition of the unit fractions with the same denominator using various tools and drawings, and standard fractional notation Possible Learning Goals • Connects fractions to their lives by exploring how they are used in everyday contexts • Investigates fractions using a set model • Represents fractions as a count, with words, with words and numbers, and in standard fractional notation • Identifies fractions represented through various scenarios within the text • Composes and decomposes fractional parts with concrete materials • Names fractional parts using various fractional names, including standard fractional notation • Visualizes fractions using fractional names (e.g., three fourths) • Begins to recognize the difference between area and set models PMraotcheesmseast:ical About the ccaPoonrmnodnbmpelercuomtnviniicnsgaog,tl,rvinerinegpgfrle,esrceetiannstgion,ngi,ng Exploring math in a context helps to make the concepts meaningful. Students begin to realize that math is all around them, and that knowledge helps them recognize its significance in their lives. When students encounter problems within a context rather than in isolation, they can also use their intuition to solve them. The more opportunities we give students to ‘see’ the math in their environment and in their daily routines, the more they will value studying mathematics in school. 436 Number and Financial Literacy

Math Vocabulary: About the Lesson ttspwnhdiaxuieertrmdlnhtf,sotseh,fm,rrsfawao,itncuhhoatroaritto,llohefn,rs,a,helqaqpuluvaaearrtslt,,,ers, During the second reading of the text, students focus on the math and are challenged with recognizing and naming fractions of sets presented in the story. Many new concepts about fractions are introduced in this lesson, including the use of standard fractional notation to represent fractions. This is a good opportunity to assess what students know and what concepts are going to need more reinforcement. Materials: Assessment Opportunities Jump, Kangaroo, Observations: Note each student’s ability to: Jump!, concrete – Identify fractional parts and the whole using various fractional names, materials (e.g., two-sided including standard fractional notation counters, square tiles) – U se various strategies to determine the parts and whole of a fractional Time: 20–25 minutes amount – Be flexible in thinking about fractions and how they can be represented – M ake connections between the text and illustrations in the story and what they know about fractions NOTE: Choose the prompts that are most appropriate for your students. Before Reading Reflecting/connecting Activating and Building On Prior Knowledge Problem solving • Ask students what they already know about fractions. • Show the front cover of the book. Ask students what fractional amounts are represented by the characters on the front cover. (e.g., one fourth of the animals on the cover are jumping) • Ask what kind of a model they are using when they are looking at the animals. (e.g., a set model) • Setting a Purpose: Say, “We have read about Kangaroo and his friends competing in the events on Field Day. Now we are going to be math detectives and uncover the mathematics as we read the book again.” During Reading Representing • As you are reading, students can use concrete materials (e.g., two-sided Problem solving counters, square tiles) to represent the fractions they hear about in the story. Representing • Read pages 6 and 7. Ask how many animals are in the whole group. Ask what fraction of the animals are kangaroos, and then what fraction of the animals are not kangaroos. • Ask students what fraction of the whole group of animals are koala bears. Have them represent the whole and the fractional amount using concrete materials. Fractions 437

• Repeat naming and representing fractions for emus, dingoes, kookaburras, and platypuses. Reasoning and proving • Read page 8. Discuss how the 12 campers are to be split up and ask how Communicating/ representing many will be on each team. Ask students how they could split the group into two teams. (e.g., put one animal in one group and then one in the other, and Problem solving continue until there are no more animals) Reasoning and proving/ • Draw attention to the standard fractional notation for one half on page 9. problem solving Explain that the top number is known as the ‘numerator’ and the bottom Reasoning and proving number is known as the ‘denominator.’ Together, the numbers are read as ‘one half.’ Explain that the denominator represents the whole made up of two Communicating/ equal groups and the numerator represents the part, or one of the two equal representing groups. Ask how 6 can be one half of 12. Draw attention t2ootnhee-thwaolve21ssoorntwthoe page. Ask what they total if you add them together. (e.g., Problem solving halves or 22 ) Explain that 1 one-half plus 1 one-half equals 2 one-halves. Problem solving/ reasoning and proving • Ask how Ruby knows the groups are equal. Ask what would happen if there were 13 animals. • Ask students what fraction of the total group is on each team, then ask for tahneotnhuemr weraaytoorfawnrditdinegno‘smixintwateolfrthmse.’ aWn.riAtesk162haonwd have students explain what they know that one half and six twelfths are equal amounts. Explain trhepatre21seanntdth162e are known as ‘equivalent fractions’ because they both same amount. • Read page 14. Say, “On page14, we learn that the 12 campers are being split into three teams. Ruby divides the teams equally in thirds. What does this mean?” Have students work with their concrete materials to divide 12 into thirds. • Ask students what fraction of all the competitors are on each team for the 142 swimming relay race. Ask for another way of writing ‘four twelfths.’ Write in standard fractional notation and ask students what the numerator and denominator mean. Ask how they know that one third and four twelfths represent equal amounts or are equivalent fractions. Have students use their concrete materials to prove that these fractions are equivalent. • Draw attention to the standard fractional notation for one third on page 15. Ask what the numerator and denominator mean. Count the three thirds: 1 one-third, 2 one-thirds, 3 one-thirds. Ask what 13  +  13  +  13 equals and how 33 represents the whole. • Ask what fraction of the first team is made up of platypuses. (one fourth, 14 ) what the whole is this time. (4 animals on the first team) Ask what Ask of the second sfhoouwrthhso,w24 ). Ask fraction how they can team is made up of dingoes (two two students rename two fourths. Have them fourths and one half are equal using their concrete materials. • Read page 20. Have students use their concrete materials to divide the animals into fourths. Ask how they could name the fraction that one of the teams represents using words and fractional notation. (e.g., onnuemfoeruarttohr, a14n, dor dtherneoemtwinealfttohrsi, n132e)aHchavceassetu. dAesnktshoexwptlahienttwhoe meaning of the fractions can represent the same amount. 438 Number and Financial Literacy

Communicating/ • Draw attention to the 341 s on page 21. Have students count the fourths: 1 one- representing one-fourths, 4 one-fourths. Ask what the total is if fourth, 2 one-fourths, Communicating they add all of the one-fourths together. Ask how this fraction (four fourths) Problem solving can be written in standard fractional notation. • Read pages 26 and 27. Ask how many groups there are if each animal competes alone (12). Ask what fraction of the group Kangaroo represents (one twelfth). Ask how this can be represented using standard fractional notation and what the numerator and denominator mean. • Read pages 30 and 31. Ask students what fraction of the events Kangaroo won. Ask what fraction of the events Kangaroo did not win. After Reading Problem solving/ • Ask students how they could divide the animals into sixths. They can use reasoning and proving their concrete materials. Ask how many animals are in each group (2) and fhHhrooaigwwchtitl13ohigneahsyn.tdcAhaos26nkwawhr13eroiwteoeqfmutahilavlinsathylaeesnanaatniffmrrimaaacclattsiliosoannrwesa(.is1n22a).l62sAo(4sak).ghTroouwurnp162boafacn4kdatno16imparaaeglese.q1Du5ii.vscaulesnst • B uilding Social-Emotional Learning Skills: Identify and Manage Emotions; Stress Management and Coping: Ask students how they feel after learning so much about fractions in the book. It is important that they can identify their emotions so they know how to deal with them. Reassure them that this is just the beginning of the unit and they have lots of time to understand all the ideas that were discussed while reading the book. Make a list of the concepts they found most confusing. Periodically revisit it throughout the lesson to highlight how their learning is progressing. You can draw attention to concepts that they understand more fully or you can add new ideas that they find confusing to the chart. This will help students manage their feelings throughout the unit. Discuss what strategies students can use when they are feeling stressed (e.g., take a break, do some stretches, ask a friend for help). Explain that learning is a process and you are there to help guide them. Further Practice • Reflecting in Math Journals: Orally pose one or more of the prompts below. Students can respond using diagrams, numbers, and/or words. – Show at least three things that you learned about fractions from reading Jump, Kangaroo, Jump! – What additional fractions could you make about Field Day? – Create your own drawing of something you may have seen and represent it as a fractional model. • Sorting Activity: Throughout the year, provide multiple opportunities for students to review fractions of sets. Sort the sets by a mystery rule and have students figure out the attribute that was used to sort them. They can also describe the sort using fractional language. (e.g., One half of the students’ shoes have laces.) Fractions 439

5Lesson Comparing Fractions Math Number Curriculum Expectations • B 1.6 use drawings to represent, solve, and compare the results of fair-share problems that involve sharing up to 20 items among 2, 3, 4, 5, 6, 8, and 10 sharers, including problems that result in whole numbers, mixed numbers, and fractional amounts • B1.7 represent and solve fair-share problems that focus on determining and using equivalent fractions, including problems that involve halves, fourths, and eighths; thirds and sixths; and fifths and tenths Teacher Possible Learning Goal Look-Fors • Compares unit fractions and non-units fractions using concrete materials Previous Experience with Concepts: and/or visual representations Students have reviewed equally partitioning • Equally partitions wholes into halves, fourths, eighths, thirds, and sixths with fractions represented by reasonable accuracy area models. They also have an understanding of • Identifies the unit fractions in a whole unit fractions and non- • Represents and identifies the appropriate number of unit fractions in a given unit fractions, using area models. non-unit fraction (e.g., three eighths is composed of 3 one-eighths) • Demonstrates a strategy for comparing fractions, and explains why one fractional part is bigger than another or why two fractional parts are the same • Has some understanding that the more equal pieces a whole is divided into, the smaller the pieces will be PMraotcheesmseast:ical About the rarReenepfdlareesscsotterinannitgntei,gngsgiaee,nlsed,ccptoirnnognvietnocgot,ilnsg, communicating John Van de Walle and LouAnn Lovin explain that to develop fractional number sense, students need to have “some intuitive feel for fractions. They should know ‘about’ how big a particular fraction is and be able to tell easily which of two fractions is larger” (Van de Walle & Lovin, 2006a, p. 262). As students compare fractions, they are investigating the relationship between the amount being shared and the number of people sharing. It is important for students to compare fractions using concrete models and drawings so they can not only visually determine which is bigger, but also by how much. Too often, students in later grades compare fractions using ‘tricks’ and procedures to manipulate the numerical form, but they do not really understand what they are doing. 440 Number and Financial Literacy

Math Vocabulary: Students can effectively compare fractions with visual representations. They uhsnaixilttfh,f,rthasicertdvioe, nnfo,thuw,rhtehoi,glefhi,tfhth, , can also apply their understanding of unit fractions to make comparisons. ninth, tenth For example, when asked to compare three eighths and three fourths, if they have discovered that one fourth is larger than one eighth, then three fourths, or three units of fourths, would naturally be larger than the same number of units of eighths. This follows the reasoning that, in each case, there is the same number of parts but the parts are of different sizes. About the Lesson In this lesson, students compare unit fractions and non-unit fractions in the context of “Who ate more?” This is fitting, since students are often first exposed to equal-sharing problems in relation to food. The fractions in the lesson are presented in both words and in standard notation in order to help students make connections between the two representations. As students work through this real-life problem, you have a good opportunity to reinforce the mathematical modelling process and its four components: • Understand the Problem • Analyse the Situation • Create a Model • Analyse and Assess the Model Use an anchor chart to highlight how students move back and forth among the components as they design and refine their model. For example, students may analyse the context (Analyse the Situation) before they are presented with the problem (Understand the Problem). As they test their model, they may find it necessary to reanalyse the situation in order to gather more information so they can select more-appropriate strategies and tools for their model. There are some suggestions in the lesson about how and when to reinforce the model, although these need to be adjusted so they reflect the way in which your students are working through the process. Materials: Minds On (20 minutes) congruent, long, thin, • Review the fraction strips on Digital Slide 54: Fraction Strips. Count one of paper rectangles; chart the rows (e.g., sixths) by unit fractions (1 one-sixth, 2 one-sixths, etc.) and paper; scissors; glue; use the count to name some fractions in various ways (e.g., 2 one-sixths, Digital Slide 54: Fraction 2 sixths, two )s.ix(Athnsa, l62y,soenthe ethSiirtdu)a.tRioenvi)ew the fractional name for the whole Strips, concrete (six sixths, 66 materials (optional) • Show students two congruent paper rectangles, one of which is divided into Time: 65 minutes fourths and the other into fifths. Fractions 441

• Have students imagine that the rectangles are chocolate bars. Ask what they notice about the sizes of the chocolate bars (e.g., they are the same). Explain that Tamara ate two pieces of the first chocolate bar and Owen ate two pieces of the second chocolate bar. Ask who ate more. Have students turn and talk with their partner. (Understand the Problem) • As a class, discuss students’ responses and their reasoning. Discuss why two fourths is a larger fraction than two fifths, even though both quantities are made up of the same number of pieces (e.g., because fourths are larger than fifths when the same whole is being considered). • Tell students that their challenge today will be to compare various fractional amounts to discover who ate more. (Understand the Problem) Working On It (20 minutes) • Have students work in pairs to figure out who ate more of chocolate bars that are the same size in the following problems. You can make the problems more relevant and engaging by replacing the names given with names of your students. Review the problems, and emphasize that the chocolate bars are all the same size. – Problem 1: Justin ate three fourths ( 34 ) of a bar and Mia ate three eighths ( 38 ) of a bar. – Problem 2: Justin ate four eighths ( 48 ) of a bar and Mia ate two fourths ( 24 ) of a bar. – Problem 3: Justin ate five eighths ( 85 ) of a bar and Mia ate one half ( 21 ) of a bar. – Problem 4: Justin ate two thirds ( 32 ) of a bar and Mia ate five sixths ( 56 ) of a bar. – Problem 5: Justin ate three fifths ( 53 ) of a bar and Mia ate six tenths ( 160 ) of a bar. – Problem 6: Have students create and solve their own problem using fractions of their choice. • Students can use any tools or strategies they want to represent the math. For example, they may use paper rectangles (or fraction strips from BLM 60) to represent their chocolate bars, or they may use concrete materials or drawings to make comparisons in different ways. (Create a Model) Differentiation • Adapt the problems if/as needed by changing the fractional amounts. • Students can compare the fractional amounts to a benchmark fraction, such as one half, rather than comparing the fractions to each other. • For students who may have difficulty equi-partitioning, provide pre-divided paper rectangles. • For students who need more of a challenge, give problems involving fractions that are close in value, such aans dfivoerdseixrtfhrsac(t65io) nans dfosrixtheriegehcthhsoc(o86l)a.tYeobuarcsa.n also have students compare 442 Number and Financial Literacy

Assessment Opportunities Observations: Pay attention to how students are partitioning their wholes. Are they folding the rectangles into parts, measuring out parts, or drawing partition lines freehand? Do they check their partitions by counting the number of pieces or seeing whether the pieces are the same size? Conversations: Pose some of the following prompts to help students uncover inaccuracies: – How does your model/drawing show that this fraction is larger than the other? – Are the pieces in your model the same size? How could you make sure they are? – Without looking at your model, which fraction do you think should be larger? Why? Could the way you split up your bar into fractions make it look different than what you were thinking? Consolidation (25 minutes) • Meet as a class. Discuss each of Problems 1 to 5, highlighting some of the following points: – Problem 1: The number of pieces eaten is the same for each chocolate bar, but the sizes of the pieces differ. Three larger pieces (fourths) will be more than three smaller pieces (eighths). – Problem 2: Both fractions are the same amount. Reinforce the term ‘equivalent fractions.’ Have students prove how they can be the same. Ask how else they could name this fractional amount (one half). – Problem 3: In Problem 2, students found that four eighths is equal to one half. They can use this information to figure out that since five eighths is larger than four eighths, then five eighths must be larger than one half. In this case, they are using one half as a benchmark fraction. – Problem 4: Students can make the connection that sixths are half the size of thirds. They can think about breaking the two thirds Justin ate into sixths, giving double the number of pieces; that is, four sixths. They can then compare the 4 one-sixths to the 5 one-sixths Mia ate. In this case, they create equal-sized pieces by dividing the pieces in one chocolate bar so they are the same size as the pieces in the other bar; they can then compare the number of equal-sized pieces. Highlight how two thirds and four sixths are equivalent fractions. – Problem 5: The two fractions are equivalent. Have students prove it using concrete materials and/or drawings. • Add some of these strategies to an anchor chart, with visual examples to further clarify what they mean. • Discuss which strategies and tools students found most effective for comparing fractions. Ask how they might solve these problems differently if they were to do them again. (Analyse and Assess the Model) Fractions 443

Materials: • If there is time, for Problem 6, have students meet with another pair, Digital Slides 55–56: exchange the problems they created, and solve them. They can then compare Fractional Parts of an their solutions. If time is short, have them exchange problems on another day. Area • B uilding Social-Emotional Learning Skills: Positive Motivation and Perseverance: Ask students what they have learned about fractions to this point. Remind them that they have been working hard as ‘fraction detectives,’ listening for fraction words, and that perhaps some of the words on the Math Word Wall reveal concepts they may still wonder about or that they still want to learn about. Take 5–10 minutes over the next few days to practise the vocabulary on the Word Wall. For example, show students a visual of a fraction and have them describe it using words and symbolic notation. Show them a fraction in standard fractional notation and have them explain what it means and provide an example. Offer regular practice so students experience how it can help them improve. Math Talk: Math Focus: Investigating how fractional parts in an area model need to be the same size but not the same shape About the Marian Small explains that, for area models, “the equal parts into which the whole is divided are equal but do not have to be identical” (Small, 2009, p. 199). In other words, they need to cover the same area and be the same size, but they do not have to be the same shape. Teaching Tip Let’s Talk Integrate the math Select the prompts that best meet the needs of your students. talk moves (see page 8) throughout • S how Digital Slide 55. Math Talks to maximize student • W hat can you tell me about this shape? How could we describe this in terms of participation and active listening. fractions? (e.g., It is divided into halves; it is divided into sixths; three sixths of the rectangle are red and three sixths are blue.) • H ow are three sixths and one half the same? • W hat does congruent mean? (e.g., the same size and same shape) Are the two halves congruent? What can you visualize to prove it? (e.g., Turn the red shape so it sits on top of the blue shape.) 444 Number and Financial Literacy

• S how Digital Slide 56. • What can you tell me about this shape? (e.g., It is a trapezoid; it is divided into two parts.) Do you think it is divided into halves? Why? Turn and talk to your partner. • How can you prove that the shape is divided into halves? How can you visualize that the two halves are equal? (e.g., Move one red half-square and rotate it so it is with the other red half-square.) Do the two halves need to be congruent? (e.g., No, they just need to be the same size.) • H ow are the fractions in this shape the same as and different from the fractions in the shape on the previous slide? • What have we learned about fractions today? (e.g., The equal parts need to be the same size, but they don’t have to be the same shape.) Materials: Math Talk: Digital Slide 57: More Math Focus: Investigating fractional area models with non-adjacent fractional Fractional Parts of an parts and parts that are not the same shape Area, paper, scissors About the Marian Small explains that fractional parts do not need to be adjacent within the whole (Small, 2009, p. 200). Students need to see wholes with non-adjacent parts so this concept can be explicitly discussed. They also need further examples of how equal parts need to be the same size, but not necessarily the same shape. Let’s Talk Select the prompts that best meet the needs of your students. • S how Digital Slide 57. • What do you think this might be? It is a flag. Where have you seen flags before? What shape is this flag? Turn and talk to your partner about how you might describe the fractions shown in the flag design. continued on next page Fractions 445

• W hat fraction of the flag is the yellow part in the top left? (one fourth) How do you know? What fraction is the rest of the flag? (three fourths) How much of the flag is yellow? Turn and talk to your partner. Does it matter if the two yellow parts aren’t touching? Why? (e.g., They still take up the same area of the whole whether they are touching or not.) • What fraction of the flag is the red section at the bottom left? (one eighth) How do you know? How can knowing that the yellow section above it is one fourth help you? If this red part is one eighth, what fraction is the rest of the flag? What fraction of the whole is the red section at the top of the flag? How do you know? How does the yellow section beside it help you know this? What fraction of the flag is red? Does it matter that the two red pieces are different shapes? How could we prove that the two shapes are the same size? Let’s trace the one shape, cut it out, and see if we can make it fit on the other shape. What did we find out? Does is matter if the two red parts are touching or not? • What fraction of the flag is blue? How did knowing what fraction of the flag is red help you? What fraction of the flag is not blue? • In what ways can you describe one half of the flag? Turn and talk to your partner and think of at least two ways (e.g., all the red and the blue sections; one yellow section, one red section, and one blue section; two yellow sections). How could you describe three fourths of the flag? • What have we learned about fractional parts today? (e.g., The equal parts of a whole do not have to have the same shape, but they do need to have the same size. We also know that the parts that make up a fraction do not have to be beside each other or touching.) 446 Number and Financial Literacy

and6 7Lessons Equal Sharing with Area Models Math Number Curriculum Expectations • B1.6 use drawings to represent, solve, and compare the results of fair-share Teacher problems that involve sharing up to 20 items among 2, 3, 4, 5, 6, 8, and 10 Look-Fors sharers, including problems that result in whole numbers, mixed numbers, and fractional amounts Previous Experience with Concepts: • B1.7 represent and solve fair-share problems that focus on determining and Students have composed area models from unit using equivalent fractions, including problems that involve halves, fourths, fractions to tenths, and eighths; thirds and sixths; and fifths and tenths compared fractional amounts of a whole, and Possible Learning Goals equally partitioned whole objects into halves and • Equally shares area models by equi-partitioning wholes into fractional parts fourths. • Develops an understanding that the more people share a whole, the less of PMraotcheesmseast:ical the whole each person gets rarReenepfdlareesscsotterinannitgntei,gngsgiaee,nlsed,ccptoirnnognvietnocgot,ilnsg, communicating • Equally shares whole area models among various numbers of people • Explains and/or shows how the strategy chosen creates equal shares • Explains and names the various fractional parts in various ways • Identifies the whole and explains how the fractional parts relate to the whole • Understands that the more people share a whole, the less of the whole each person gets About the As students equally share quantities, they need to know how to divide whole objects or sets of objects into equal parts. Since division is about sharing, Empson and Levi recommend introducing equal-sharing problems within familiar and meaningful contexts. These division problems involve connecting the number of items to be shared to the number of people who are sharing, and then distributing them in a fair manner (Empson & Levi, 2011, p. 6). To engage in equal-sharing problems, students need both conceptual and skill-based understanding. In previous lessons, students learned that in an area model, a whole can be divided into equal parts in many different ways. Students are also developing the foundational skill of partitioning to make equal-sized areas and have engaged in paper continued on next page Fractions 447

Math Vocabulary: folding as an effective strategy for partitioning. The ability to visualize dhfeoiavquliufdr,atehhl,sase,livqzseeuixs,at,plh,tashwr,itrheidtoiisgol,ehn,ths, before folding paper is important. Students must consider the number of folds needed to achieve regions of a particular area. This type of spatial reasoning, accompanied by physically folding the paper to confirm their predictions, helps students to better understand fractions, think proportionately, and decide how to equally partition area models. In Paying Attention to Fractions K–12, it is recommended that teachers in primary and junior grades avoid introducing circles to represent fractions (Ontario Ministry of Education, 2014, p. 23). This is because posing questions about the fractional parts of a circle requires students to understand the area of a circle, which is not covered until the intermediate grades. A circle also creates challenges when equi-partitioning for fractional amounts other than one half or one fourth. For these reasons, students will be equi-partitioning rectangles in Lessons 6 and 7. About the Lessons In Lesson 6, pairs of students investigate how to share a chocolate bar equally between various numbers of friends. Students can use paper folding or other strategies to partition the chocolate bar into equal portions. In Lesson 7, students divide more than one whole among different numbers of people. 448 Number and Financial Literacy

6Lesson Equal Sharing of One Whole Materials: Minds On (20 minutes) brown construction- • Show students a brown construction-paper rectangle, and explain that it paper rectangle, BLM 62: Chocolate Bars, represents a chocolate bar. Ask how the chocolate bar could be fairly shared chart paper, markers between two friends. Have students turn and talk to a partner. Ask how we know whether the chocolate bar has been fairly shared. Time: 60 minutes • Discuss the methods students used in a previous lesson to divide a paper rectangle. (e.g., paper folding, measuring the paper, estimating where they think the partition lines might be) • Ask whether the chocolate bar could be divided into more than two pieces and still be fairly shared. Ask how many pieces would work, how many pieces would not work, and why they think so. Discuss the difference between odd and even numbers and how, when representing even numbers, concrete materials can be evenly paired up, while for odd numbers there is always one left over. • Ask whether the bar could be cut into three pieces and fairly shared. (Yes, if the bar is cut so the three pieces are one half, one fourth, and one fourth. One person get the piece that is one half, while the other person gets two pieces of two fourths). Highlight the equivalent fractions of one half and two fourths. Working On It (20 minutes) • Students work in pairs or groups of three. Provide each group with BLM 62: Chocolate Bars. Have students cut out the five chocolate bars on the BLM. • Tell students they are going to explore how to equally share a chocolate bar between different-sized groups of friends. Select the numbers that are most appropriate for your students. Suggestions include: – 2 friends – 4 friends – 8 friends – 3 friends – 6 friends – 5 friends – 10 friends • Tell students they can equally partition the bars by either folding or measuring. They are to mark their partitions on the bars and glue the marked chocolate bars onto chart paper. They will then record the number of people who shared and write the names of the fractions on the chart paper. Fractions 449

Differentiation: • For students who need more support, have them share a chocolate bar among a number of friends that is easy for paper folding (e.g., 2, 4, or 8 people). • For students who need more of a challenge, encourage them to find as many different ways to share the bar as they can for a specific number of friends (e.g., find at least two ways to share with four friends). Provide them with extra copies of BLM 62. • Students can also create and solve their own equal-sharing problem. Assessment Opportunities Observations: Listen to how students are communicating their ideas to one another. Are they justifying their thinking? Are they listening and adding on to each other’s ideas? Conversations: Use some of the following prompts to further probe thinking and encourage students within a pair or group to verbally communicate their ideas. Ask each question to a different student in the group. • E xplain to the rest of the group how you shared the chocolate bar. • Tell the group in your own words what your group member just said. Do you have anything to add? How do you know your group member described a fair share? • Is there another way that you could have fairly shared the chocolate bar? Explain it to the rest of the group. How can you prove that both ways end up with a fair share? • E xplain to your group how you named the fractional pieces of your chocolate bar. • D o you agree with the explanation? Can you add anything to it? Consolidation (20 minutes) • Have student groups meet with another group to share their work and check whether they assigned the correct fractional names. • Meet as a class, and have some students explain or demonstrate the strategies they used for partitioning. Discuss which ways worked better and how they may have adjusted their strategy according to the problem. • Show examples of how the chocolate bars were divided into differently shaped fractions. (e.g., One chocolate bar is cut vertically while the other is cut horizontally, creating different shapes.) Ask whether it matters that the shapes of some of the fractions are different. Highlight that as long as the two sections of the whole take up the same area, they are the same fractional amount. • Highlight any equivalent fractions that arise during the discussions. Have students prove that they are equivalent. 450 Number and Financial Literacy

• Ask students to think about what happens to the size of the chocolate bar portions when the bar is shared with more people. • Add important ideas to the anchor chart begun and used in previous lessons. Include that the chocolate bar pieces get smaller as they are shared with more people, and that the same-sized whole can be divided in different ways. Further Practice • Independent Problem Solving in Math Journals: Have students repeat the activity by sharing with 5 or 10 friends. Fractions 451

7Lesson Equal Sharing of More Than One Whole Materials: Minds On (20 minutes) coloured tiles, BLM 63: What Fraction?, • Review how students shared one chocolate bar among 4 people. Then, ask BLM 64: Mini Chocolate Bars, chart paper what happened when they shared the same size of chocolate bar among 8 Time: 60 minutes people. (e.g., Everyone got less; everyone got half as much; everyone got equal but smaller pieces.) MmfraaixtcehtdioVnforac(fcaotbriomunlaa, lriymt:eprmrospearre optional) • Provide pairs with 8 coloured tiles. Tell students that, today, they are going to share more than one chocolate bar. Explain that one whole is still one chocolate bar and that one coloured tile will represent one chocolate bar. • Pose the following problem: – If 8 chocolate bars are shared among 4 people, how much does each person get? • Students can turn and talk to a partner. They can use the coloured tiles to solve the problem. Discuss their strategies for distributing the ‘bars.’ (e.g., deal out the 8 tiles one at a time to each person; mentally solve the problem and give 2 tiles to each person) • Pose the next problem: – If 4 chocolate bars are shared among 8 people, what fraction of a chocolate bar does each person get? • Ask how this problem is different from the previous one. (e.g., There are not enough whole chocolate bars for everyone, so the chocolate bars will need to be cut up into pieces.) Have students turn and talk to a partner to solve the problem. Discuss students’ solutions and their strategies. Ask whether coloured tiles are the most appropriate manipulative for this problem. Ask what might work better. (e.g., paper rectangles that can be cut up) Working On It (20 minutes) • Students work in pairs to solve all or some of the problems on BLM 63: What Fraction? The problems are intentionally clustered into pairs to allow for good discussion in the Consolidation. Ensure that all students complete one common pair of problems (e.g., Problems 1 and 2) so you can have a whole group discussion in the Consolidation. • Give pairs BLM 64: Mini Chocolate Bars with 24 congruent rectangles (representing chocolate bars). Have them cut out the rectangles to model the problems, glue them on chart paper, and then record an explanation of how they solved the problems and the fractional names they include. 452 Number and Financial Literacy

Differentiation • Select the problems that best suit the needs of each pair. • For students who need more of a challenge, have them create their own equal-sharing problem and solve it on a separate piece of paper. They can then exchange the problem with another pair and solve each other’s problems. Assessment Opportunities Observations: • eHaocwh are students equi-partitioning their chocolate bars? Do they divide chocolate bar into the number of people in the group and deal out the pieces? Do they deal out as many whole chocolate bars as possible and then equi-partition the remaining ones? Are they making partitions that are reasonably accurate? • Hfraocwtiodnoalthlaenygdueasgceri(bee.g.t,heeacshhapreerssothnatgeetascohnpe earnsodnogneetst?hiDrdo they use chocolate bar, each person gets four thirds of a chocolate bar, or each person gets one chocolate bar and part of another chocolate bar)? Conversations: If students struggle to equally share more than one whole, pose some of these prompts: – What does it mean to equally share? Are there enough chocolate bars so everyone can get one? – Visualize cutting all of the chocolate bars in half. How many halves would there be? Could you equally share that number of halves? How could you visualize dealing them out to all the people? (e.g., Count the halves as I pretend to deal them out to the three people.) Did it work out evenly? – How else might you divide the chocolate bars if you are sharing with three people? (e.g., Divide each into three pieces.) What do we call the pieces? (thirds) Try cutting the chocolate bars into thirds and see if it works. Consolidation (20 minutes) • Meet as a class. Strategically choose two or three different solutions for the common pair of problems that all students solved. • Possible prompts for Problems 3 and 4 include: – How are the problems the same? (e.g., same numbers) – How are they different? (e.g., One problem has at least one whole chocolate bar for each person and the other one doesn’t.) – For Problem 3, how did you divide 6 chocolate bars among 4 people? Sample solutions: • G ive each person one whole chocolate bar and then half of another bar. Fractions 453

• G ive each person one whole chocolate bar, and then divide the remaining two chocolate bars into fourths and give each person two fourths. • D ivide each chocolate bar into fourths and give each person six fourths. – How do you describe the fractions in each case? (e.g., one and one half; one and two fourths; six fourths) One and one half is known as a ‘mixed fraction,’ because part of it represents the whole and the other part represents part of one whole. Which other fraction is a mixed fraction? Do they represent the same amount? How do you know? – Is six fourths a mixed fraction? Six fourths is known as an ‘improper fraction,’ because it has more than four fourths. How could you recompose six fourths into a mixed fraction? (e.g., make a whole by putting four fourths together) Does six fourths represent the same amount as the other two fractions? – For Problem 4, how did you divide 4 chocolate bars among 6 people? Sample solutions: • D ivide each chocolate bar into six equal pieces, and everyone gets one piece from each of the four chocolate bars or four sixths. • D ivide three chocolate bars in half, and give each person one half. Divide the fourth chocolate bar into sixths and everyone gets one sixth. Each person gets one half and one sixth. • D ivide each chocolate bar into thirds so there are twelve thirds, and then each person gets two thirds. – Are all of the fractional amounts the same? How can you prove that? (e.g., We can cover the pieces and they all take up the same area.) – Did we find any equivalent fractions? (e.g., two thirds and four sixths) How can you prove that they are equivalent? – If you were sharing the chocolate bars, which way would you like to get your share? Why? • Conclude the lesson by discussing how there are different ways to share the chocolate bars, but in each case, the share is the same because the fractional parts take up the same area. • Ask whether people get more when there are more people or fewer people sharing the same amount. 454 Number and Financial Literacy

Materials: Math Talk: coloured tiles Math Focus: Creating wholes from fractions greater than one Teaching Tip Let’s Talk Integrate the math talk moves (see Select the prompts that best meet the needs of your students. page 8) throughout Math Talks to • W hat shapes can pizzas be, besides circles? Today, we are going to work with maximize student participation and pizzas that are rectangles. active listening. • T hink about this problem: “There are some pizzas divided into fourths. The group of students eats twelve fourths.” Do you think there was more than one pizza eaten? Why? Turn and talk to your partner. Find out how many pizzas were eaten. You can use coloured tiles to represent each fourth. • How many pizzas were eaten? What is an equal statement that we can make? (e.g., twelve fourths equals 3 wholes) How can we count the pieces of pizza? (1 one-fourth, 2 one-fourths, … , 12 one-fourths, or one fourth, two fourths, … , twelve fourths). How can we count them so we are identifying the whole each time? (e.g., one fourth, two fourths, three fourths, four fourths or one whole, etc.) • Let’s change the problem. The pizzas are divided into eight slices. What fraction would that be? (eighths) If the people ate twelve eighths, how many pizzas were eaten? How would we count twelve eighths? Work with your partner. You can use the coloured tiles, but this time each piece can represent one eighth. • What did you find? (e.g., one whole pizza and four eighths, one and a half pizzas) How can it be one and four eighths and also one and one half? Show me with your coloured tiles. What equal statement can we make? (e.g., twelve eighths equals one and four eighths or one and one half) • What do you notice about the number of pizzas that were eaten in the first problem compared to the number eaten in the second problem? Why? (e.g., There were double the amount of pizzas eaten because fourths are double the size of eighths.) • You may want to repeat this line of questioning with a different fraction, such as twelve sixths or eleven thirds. Fractions 455

8Lesson Naming Fractions Relative to Different Wholes Math Number Curriculum Expectations • B1.6 use drawings to represent, solve, and compare the results of fair-share problems that involve sharing up to 20 items among 2, 3, 4, 5, 6, 8, and 10 sharers, including problems that result in whole numbers, mixed numbers, and fractional amounts • B1.7 represent and solve fair-share problems that focus on determining and using equivalent fractions, including problems that involve halves, fourths, and eighths; thirds and sixths; and fifths and tenths Spatial Sense • E1.2 compose and decompose various structures, and identify the two- dimensional shapes and three-dimensional objects that these structures contain Teacher Possible Learning Goals Look-Fors • Composes different wholes and names fractions according to their Previous Experience with Concepts: relationship to the whole Students have worked with unit and non-unit • Recognizes, describes, and names equivalent fractions and proves their fractions in area models and have named fractions equivalency using words. • Uses fractional parts (pattern blocks) to compose wholes (e.g., hexagons) • Accurately uses fractional terms and symbols to name parts of wholes and the whole • Names fractional parts according to the defined whole • Identifies the whole if the part is defined • Flexibly adjusts to a new definition of the whole • Identifies equivalent fractions and justifies their reasoning • Recognizes some proportional relationships between fractional parts and wholes (e.g., if the whole is the hexagon, the trapezoid is one half, but if the whole is two hexagons, one hexagon is one half, because the whole is twice as big and so the half must be twice as big too) 456 Number and Financial Literacy

PMraotcheesmseast:ical About the rarReenepfdlareesscsotterinannitgntei,gngsgiaee,nlsed,ccptoirnnognvietnocgot,ilnsg, communicating Students need opportunities to use a variety of concrete objects, drawings, and other visuals to explore and represent fractions. The overuse of any MhohranhaeleoxtvhamewgsVbh,oouonhcsl,ae/atlr,frbh,aeuotpqlhmaeuirrzabdyoli:s,id,tr,siaixnthgsle,, one representation (e.g., square area models) may lead to difficulties in understanding other types of models (e.g., number line representations or volume models) (Ontario Ministry of Education, 2016, p. 43). Pattern blocks are effective concrete materials since the whole can be represented by any of the blocks, thereby altering the fractional value of all of the other blocks. For example, if in one activity the hexagon is defined as the whole, the trapezoid represents one half and the rhombus represents one third. If, in another activity, two hexagons are defined as the whole, the trapezoid now represents one fourth and the rhombus represents one sixth. This change helps students develop flexibility in their thinking and further reinforces proportional reasoning. It also reinforces the big idea that a fraction cannot be named unless the whole is identified. Pattern blocks are also effective for exploring equivalent fractions in area models, since the pieces are proportional to each other and students can easily make comparisons by overlaying one fractional part on top of another. The concrete representations help students name the fractions and prove their equivalence. About the Lesson In this lesson, students play a game that involves covering outlined wholes with pattern blocks. They play the game two times, each with a different whole. This reinforces the relationship of all fractions to the defined whole and how naming fractions is dependent on what the whole is. Materials: Minds On (15 minutes) pattern blocks • Show students a yellow pattern block. Ask what shape it is and how they (hexagons, blue rhombi, trapezoids, triangles), know. Tell them that they are going to build some fractions using pattern BLM 65: Cover It!, blocks and the hexagon will be the whole. BLM 66: Cover It! Spinners, pencils, paper clips • Students work in pairs. Give each pair a variety of pattern blocks comprised Time: 55 minutes of hexagons, trapezoids, blue rhombi, and triangles. Challenge students to cover the hexagon with like blocks in as many ways as possible. Tell them to be prepared to name the fractional parts. • Share students’ findings as a class. Encourage them to use fractional language to name and describe the fractions. (e.g., It takes two trapezoids to cover the hexagon, so each trapezoid is one half.) • Show students two yellow hexagons. Tell them that this is now the whole. Ask them what block would represent one half (one hexagon). Ask what the trapezoid represents. Ask why in the first problem the trapezoid represented one half and in this problem it now represents one fourth. (e.g., The whole is different so the fractional pieces change too.) Emphasize that it is important to know what the whole is in order to name the fractional parts. Fractions 457

Teaching Tip Working On It (25 minutes) You may want • Student pairs will play a game that involves making fractions. students to play each • Give each pair a variety of pattern blocks (hexagons, blue rhombi, trapezoids, round of the game on different days. triangles), two copies of BLM 65: Cover It! (one for each partner), BLM 66: Cover It! Spinners, and a pencil and paper clip for the spinners. • Tell students they will play two rounds of a game called Cover It! with their partner, and that each player uses his/her own game boards. • The rules of play using Game Board 1 are as follows: – The whole is one hexagon, and the goal is to cover or fill all the wholes on the game board. The winner is the player who covers all or most of the wholes first. – Players take turns spinning both spinners. The pattern block spinner (top) determines which block to use and the number spinner (bottom) determines how many of that block to use. – When it is their turn, students place the blocks on any hexagon outline on Game Board 1, naming the fractions they have created (keeping in mind what the defined whole is). They can place their selected blocks on more than one hexagon, but once a hexagon has a block on it, the rest of the hexagon must be filled with the same type of block. Once blocks are placed, they cannot be moved. – If a player spins both spinners and is unable to cover any remaining spaces, he/she loses a turn. • The rules of play using Game Board 2 are the same, but the whole is now two hexagons. Differentiation • Reduce or increase the number of hexagon outlines students need to cover on BLM 65. Some students may benefit from working with only one hexagon on the BLM. • For a further challenge, students can cover the wholes with more than one type of pattern block and name the fractions accordingly. (e.g., I have two green triangles, which is covering two sixths or one third of the hexagon, and one blue rhombus, which is covering one third or two sixths of my hexagon.) Assessment Opportunities Observations: • Pay attention to any misconceptions that may arise (such as using a combination of pattern blocks to cover the hexagon when this is not allowed by the rules). • Check that students are saying the names of the fractional parts according to the defined whole. 458 Number and Financial Literacy

Conversations: If students are having difficulty naming the fractions, pose some of these prompts: • Whexhaagtofrnacsttiilolnnoeef dthsetohebxeacgoovneirsedco? vVeirseudarliizgehtwnhoawt ?yoWuhnaetefdratcoticoonveorf the the rest of the hexagon. How could you describe those pieces as fractions? Is there a different way to cover the rest of the hexagon? What name would we give to those fractional pieces? • Foforthroeuwnhdo2le: What is the whole this time? (two hexagons) Outline half with your finger. What shape fits there? (one hexagon) What fraction does the hexagon represent? Consolidation (15 minutes) • Meet as a class. Ask students to reflect on the game they just played. Ask what strategy they used so they could cover as many wholes as possible. • Discuss how the fractional names assigned to the pattern blocks changed when the whole changed. Probe to see if students saw any relationships (e.g., what was half of the whole on Game Board 1 only covered one fourth of the whole on Game Board 2, so it only covered half as much as in the previous game). • Ask students what would happen if they played the game again and the trapezoid was the whole. Ask how they would name the pieces differently. Ask whether they think the game would be easier or harder to play with smaller wholes. Materials: Math Talk: 8 red (trapezoid) pattern blocks per pair Math Focus: Composing whole fractions from a given fractional amount using an area model, and discovering how the whole changes when the fractional part Teaching Tip is given a new value Integrate the math Let’s Talk: talk moves (see page 8) throughout Select the prompts that best meet the needs of your students. Math Talks to maximize student • Construct the configuration (right) with pattern participation and active listening. blocks, and show it to students. What do you know about this shape? (e.g., it is an octagon because it has 8 straight sides) What shapes make up the octagon? • If this represents three sixths of a whole, what might the whole look like? Students can work with a partner to build possible wholes from the fractional part they were shown. How many red trapezoids would we need in total? How did you know how many trapezoids were needed to build the whole? Is there continued on next page Fractions 459

more than one way the whole could look? (e.g., the pieces can be arranged in various ways) Let’s compare some of the ways that you have made. Does it matter what the whole looks like? (e.g., No, as long as it has six trapezoids.) • Direct students’ attention back to the original configuration. Is there another fractional name we could give this rather than three sixths? (e.g., one half) • Let’s imagine that this isn’t three sixths anymore. Imagine that it is one fourth. What might the whole look like? Students can work with a partner to build possible wholes from the fractional part they were shown. How many red trapezoids would we need in total now? How did you know how many trapezoids were needed to build the whole? Is there more than one way the whole could look? • What is important to know before you can build one whole from a fractional part? (e.g., You have to know what fraction of the whole the part represents.) 460 Number and Financial Literacy

9Lesson Introduction to Set Models Math Number Curriculum Expectations • B1.6 use drawings to represent, solve, and compare the results of fair-share Teacher problems that involve sharing up to 20 items among 2, 3, 4, 5, 6, 8, and 10 Look-Fors sharers, including problems that result in whole numbers, mixed numbers, and fractional amounts Previous Experience with Concepts: • B 1.7 represent and solve fair-share problems that focus on determining and Students have identified, created, and named using equivalent fractions, including problems that involve halves, fourths, fractional parts in area and eighths; thirds and sixths; and fifths and tenths models. Possible Learning Goal • Identifies and names fractional parts of a set, using colour as the defining attribute • Recognizes the whole as being the number of items in a set • Recognizes fractional parts of a set as being the number of objects in a subset of the whole compared to the number of items in the entire set • Identifies different attributes that can be used to define the subset • Uses two-sided counters to represent various parts of a whole • Demonstrates and explains fractions as parts of a set, using a model • Determines the whole set when given information about fractional parts • Uses appropriate fractional language to describe the parts and the wholes PMraotcheesmseast:ical About the rarReenepfdlareesscsotterinnanigtntei,gngsgiaee,nlsed,ccptoirnnognvietnocgot,ilnsg, communicating As grade three students investigate fractions using set models, they need to have a clear understanding of how they differ from area models. In an area model, one shape represents the whole, which is divided into fractional regions that are equal in area. In a set model, a collection of items represents the whole and the fractional parts are made up of subsets of the collection. The subsets can be defined by any attribute of the objects, such as colour. While the fractional parts in an area model must be of equal size, set models must have an equal number of objects in each fractional part. For example, items in a set can be different sizes, as long as there are the same number of items in each subset. When working with sets of objects, it is important to be explicit about what constitutes the whole and what attribute defines the fractional parts (e.g., one third of the set is green), since any attribute can continued on next page Fractions 461

Mswmeahtot,odhlpeeaV,l,rostac,etatftrrbmaibucoultaidtoreeynl:a, lapreaart, be used. To develop understanding, it is beneficial to identify the unit fraction within a set and count the unit fractions, as was done previously Materials: with area models, in order to help quantify the non-unit fractions. two-sided counters, small cups or About the Lesson containers, BLM 67: Fractions and Wholes of In this lesson, students investigate the part-whole relationship of fractions Sets represented in a set by shaking and spilling two-sided counters and Time: 60 minutes naming the fractional parts according to the attribute of colour. Minds On (20 minutes) • Ask students what they think a ‘set’ is and have them identify examples from around the room (e.g., a set of books, a set of crayons). Explain that the items in a set usually share some common attribute, such as a set of books that have a common author, theme, or topic. • Show students 5 two-sided counters. Ask why the counters might be called a set and how many counters make up the whole. Ask what makes up one part of the set (one counter) and how we could describe one counter using fractional language. (one fifth) Together, count the counters as unit fractions (e.g., 1 one-fifth, 2 one-fifths, … , 5 one-fifths). Ask what the whole is (five fifths). • Ask students whether they think it matters whether the items in a set are touching or spread far apart. (e.g., No, because there is the same number of counters in the set either way.) • Ask students to look at the counters and think about what is different about them (colour). Show students 5 counters, 2 that are red and 3 that are yellow. Ask what fractions they see if they are looking at colour as the defining attribute. Ask what the whole of the set is. • Have students work in pairs. Give each pair 5 two-sided counters. Ask them to create a set that is one-fourth yellow. Tell them that they do not have to use all of their counters to represent the fraction. • Ask what the whole is (four counters). Ask what the parts are and how they are defined. Ask what represents one fourth. Ask what the other part of the whole is (e.g., three-fourths red). Repeat with some other fractions, such as two-thirds red and then four-fifths yellow. In each case, ask what the other part that makes up the whole is. 462 Working On It (20 minutes) • Students work in partners. Provide each pair with 10 two-sided counters and a small cup. • Partner A puts 10 counters in the cup, shakes the cup, and spills out the counters. Partner A then states the fractions created by the spilled counters, including the whole (e.g., four-tenths yellow, six-tenths red, ten tenths is the whole). Number and Financial Literacy

• Partner B checks Partner A’s thinking. • Partner A represents the counters by colouring in circles on BLM 67: Fractions and Wholes of Sets (they may need 2 copies). She/he also records the fractional names of the parts and the whole. • To continue, partners switch roles and repeat. Each student should get 2–4 turns at shaking and spilling the counters. Differentiation • For some students, you may decide to have a different number of counters based on friendlier or more challenging numbers (e.g., some groups get 5 and some get 10). • For a further challenge, have students vary the number of counters they put in the cup each time so the whole changes regularly. Assessment Opportunities Observations: • Can students name the fractions using appropriate language? • pAarret-awnhyosleturdeelanttisonresphripes?eFnotirnegxtahmepplea,rtw-phaernt relationship instead of the there are two red counters and eight yellow counters, do they name the fraction as two eighths? Conversations: If students have difficulty naming the fractions according to part-whole relationships, pose some of the following prompts: – Explain how this represents two eighths (e.g., two counters are red and eight counters are yellow). – Show me the whole set. How many counters are there? If all 10 counters were red, what would be the name for one counter? (one tenth) What does the ‘one’ in the fractional word mean? (e.g., one counter) What does the ‘tenth’ represent? (e.g., the whole, which is 10 counters) What is the whole in your combination of yellow and red counters? (10, so tenths) So, if you describe how many red counters there are in relation to the whole, or all of the counters, what would you say? (e.g., two tenths) Why wouldn’t two eighths work? (e.g., That is representing two parts.) Consolidation (20 minutes) • Meet as a class. Ask what the whole is in each case if they always used 10 counters. Ask which fraction came up the most. Ask if any combinations of fractions never came up (e.g., ten tenths are yellow and nine tenths are red). • Show 10 counters, with 5 red and 5 yellow. Ask how they can name the set in terms of colour (e.g., five tenths are red and five tenths are yellow). Ask how else they could name the fraction, other than five tenths. (one half) Discuss why both fractional names mean the same thing and how they are considered equivalent fractions. Fractions 463

• Put 8 counters in the cup and spill them out. Have students identify the fraction of one colour. Ask why they are now using eighths rather than tenths to describe the fraction. (e.g., The whole has changed.) • Co-create a highlights and summary anchor chart about sets. Include some of the following ideas: – A set is a collection of items that have something in common. – Fractions can be represented using a set model. – It is important to identify the whole, which is how many items are in the set. – Parts of the set can be identified by an attribute (e.g., colour). – The fractional part is defined by how many items with an attribute are in the set, compared to the total number of items in the set. – Items in a set can be close together or far apart. 464 Number and Financial Literacy

10Lesson Investigating Set Models Math Number Curriculum Expectations • B1.6 use drawings to represent, solve, and compare the results of fair-share problems that involve sharing up to 20 items among 2, 3, 4, 5, 6, 8, and 10 sharers, including problems that result in whole numbers, mixed numbers, and fractional amounts • B 1.7 represent and solve fair-share problems that focus on determining and using equivalent fractions, including problems that involve halves, fourths, and eighths; thirds and sixths; and fifths and tenths Teacher Possible Learning Goal Look-Fors • Uses various attributes to define how a set will be subdivided into fractional Previous Experience with Concepts: parts and names the fractional parts Students have previously explored fractions • Recognizes fractional parts of a set using various attributes represented as area • Identifies various parts of a whole set, using appropriate fractional language models and have started • Develops an understanding of fractions of a set, where the fractional parts to learn about fractions represented as a set (using may change depending upon the attributes which are being considered similar items and one (e.g., shapes, sizes, shading) attribute, e.g., colour). • Explains or shows that the size of the objects in a set does not matter when PMraotcheesmseast:ical other attributes are being considered rssaPetenrrplaodertbeceplstegrienominegvtsiinst,noogcgol,o,vlrnisnengafel,necdrctetiniangsg,o,ning communicating About the Math Vocabulary: A set model involves a collection of individual objects representing the sppeaatrr,ttsssu,, bfarstatecrtit,biowuntheaolle, whole, and subsets of those objects that share a certain attribute representing the fractional parts (Ontario Ministry of Education, 2014, pp. 4–5). The subsets can be defined by any attribute, and the items in the set can be different sizes when another attribute is being considered. Having fractional parts of a set represented by objects of different sizes can be especially confusing for students since they have learned that fractional parts of an area model must be the same size. In set models, it is the number of objects that constitutes the fractional part and not their sizes. Consider a set of six triangles, three red and three orange, that are all different sizes. We can say that one half of the triangles are red, regardless of their size, because colour is the attribute used to categorize the set. It is important to identify the attribute that is being used to define the fraction. About the Lesson Students investigate creating fractions with sets to reinforce the idea that items of a set can be different sizes. Fractions 465

Materials: Minds On (20 minutes) two-sided counters, • Review the set model used previously with two-sided counters, and discuss chart paper how we can use fractions to name parts of the whole set. Review what Time: 60 minutes ‘attribute’ means. • Have eight students come to the front of the room. Ask why the students could be considered a set (e.g., they are all in grade three). Ask whether the students all need to be the same size to be part of this set. Explain that while the parts of an area model need to be the same size, the parts in a set do not. In this case, what makes the students a set is that they are all in grade three and their size doesn’t matter. • Ask what the whole is (eight eighths). Have four of the students in the set crouch down and four stand up. Ask what fraction of the students are crouching and what fraction are standing. Discuss how four eighths could also be described as one half of the group. • Select six objects from around the room that constitute a set. For example, you might select a set of student’s school supplies, such as a backpack, calculator, pencil case, pencil, ruler, and notebook. • Have students study the attributes of the objects and ask what makes them a set. Ask how they might describe the whole (e.g., six objects, six sixths). Ask how they could make a subset by looking at the attributes of the objects (e.g., two of the items are red). Ask how they would describe that subset as a fraction (e.g., two sixths). Discuss several of their suggestions and highlight that they can use different attributes to define the fractional parts. • Ask whether the size of the items matters, like it did with area models. Ask what students use to determine the fractional part of the whole (e.g., the number of items in the subset compared to the numbers of items in the entire set). Working On It (20 minutes) • Have students work in groups of 3 or 4. Each group collects 5 to 10 objects from around the room to create a set. Remind them that the items must have something in common to make them a set. Students then think of different attributes they could use to create subsets, or fractional parts, of the whole. • Have students identify the whole and a subset using pictures or words and then write the fractional name for the subset. They can record their ideas on chart paper. • After representing their fractions in two or three ways, students leave out their sets in preparation for a gallery walk during the Consolidation. Differentiation • For students who needs more or less of a challenge, choose objects for them or define the number of objects they work with. • For students who need more of a challenge, have them think of less-obvious attributes to define their subsets. For example, they may create a subset according to what the objects can do rather than by their appearance. 466 Number and Financial Literacy

Assessment Opportunities Observations: • Can students determine and name the whole and the parts of the whole in a set? • Can students select an attribute to use to distinguish fractional parts of a set? • Are students using the number of objects in the subset in relation to the whole to define the fraction? • Are students naming some of the fractions in more than one way? (e.g., two sixths could also be considered one third of the set) Conversations: Pose some of the following prompts if students have difficulty naming the fractional parts: – W hat attribute are you using to classify items in the set (or group)? – H ow can you show this subset as a fraction? What is the fractional name of the whole? What determines that? (e.g., the number of objects in the whole set) – W hat other attributes can you use to make different parts of the whole set? – C ould these two pencils be considered a fractional part of all the items, even though they are different sizes? What attribute are you considering? (e.g., both are pencils) Consolidation (20 minutes) • Have groups display their sets but not the subsets recorded on chart paper. Have a gallery walk. As a class, visit each of the sets of items. • At each stop, the home group names a fraction but does not identify what subset of the set the fraction describes. The rest of the students suggest the subset they may be describing. Then the home group reveals their work and students can discuss the attribute used, what the fractional names of the whole and parts are, and whether there are any other ways of naming the same fraction. Discuss whether size is important when it is not the defining attribute and what determines the size of the fractional part (e.g., the number of items in the subset in relation to the number of items in the whole set). • Add any new ideas that emerge about set models on the highlights and summary anchor chart created in the previous lesson. • Building Social-Emotional Learning Skills: Positive Motivation and Perseverance: Remind students that in this lesson they learned that fractional parts of a set do not have to be the same size, but earlier they learned that the fractional parts of an area model do have to be the same size. Ask whether they think this is confusing. Tell students that as they learn more and more about mathematics, there are often new ideas that make you question what you previously learned. Explain that mathematicians will often think they understand a concept and then they discover new information Fractions 467

Materials: that makes them question their original thinking. Mathematicians continue to investigate in order to clarify their ideas. This takes time, patience, and BLM 68: Colouring to perseverance. They will also find that things will get clearer the more they Show Fractions study fractions. Further Practice • Independent Practice: Have students complete BLM 68: Colouring to Show Fractions. Materials: Math Talk: Digital Slide 58: Shoes Math Focus: Composing whole fractions from a given fractional amount using a set model Teaching Tip Let’s Talk: Integrate the math talk moves (see Select the prompts that best meet the needs of your students. page 8) throughout Math Talks to • Show students Digital Slide 58: Shoes. What set do you see? What makes this maximize student participation and a set? What is the whole? (e.g., three pairs of shoes or six individual shoes) active listening. Why can the whole be described in both ways? • Let’s look at the whole as being three pairs of shoes. How can you describe the whole set using fractional language? (three thirds) What parts can you see and how can you describe them using fractions? (e.g., One third of the pairs has laces and two thirds do not; one third is brown, one third is grey, and one third is black.) • Let’s change the whole so it is six individual shoes. What fractions do you see this time? (e.g., two sixths of the shoes have laces) • H ow are two sixths and one third the same? • D o the shoes need to be the same size to be part of the set? Why? • H ow would the fractions change if we were only looking at the brown and grey shoes? 468 Number and Financial Literacy

11Lesson Identifying Fractions in Set and Area Models Math Number Curriculum Expectations • B1. demonstrate an understanding of numbers and make connections to the Teacher way numbers are used in everyday life Look-Fors Possible Learning Goal Previous Experience with Concepts: • Identifies and names several fractions and their wholes in area and set models Students have previously explored fractions • Identifies whole sets and their fractional parts, using various attributes to represented as area and define the fractional parts set models. • Understands that the fractional parts of a set may change, depending upon the attributes which are being considered (e.g., shapes, sizes, shading) • Identifies whole objects or areas and their fractional parts, recognizing that the fractional parts must be of equal size • Names various fractional parts in an area model and explains or shows that they are equally partitioned (halves to eighths) and how they relate to the whole • Uses appropriate language to describe fractional parts and their wholes • Explains the difference between area and set models and gives examples of each PMraotcheesmseast:ical About the rssaPetenrrplaodertbeceplstegrienominegvtsiinst,noogcgol,o,vlrnisnengafel,necdrctetiniangsg,o,ning communicating Fractions play an important role in our lives and are readily evident in the world around us. Once students start learning about fractions using various models, they begin to recognize fractional amounts in their environment. For example, they may see a quilt that is divided into sixteenths, describe how one pattern repeats four times, apply what they have learned about area models, and conclude that the pattern takes up four sixteenths or one fourth of the quilt. Similarly, they may recognize that three out of nine pieces of fruit are red, apply what they know about set models, and conclude that three ninths or one third of the fruit is red. It is beneficial to present students with different models of fractions at the same time, to determine whether they can differentiate between them and apply the principles they have learned. For example, presenting examples of set and area models in the same context can uncover whether students have internalized concepts and can flexibly apply them in new situations. Fractions 469

Mpthaairrttdh,swV, hofooculaerbt,hueslqa,urfyaif:lt,hhsa, lf, About the Lesson sixths In partners or groups, students examine an image depicting a real-life event. Together, they identify and name fractions in area and set models. Materials: Minds On (15 minutes) “Fair Fractions!” (pages • Show students “Fair Fractions!” in the big book. Discuss what event the image 18–19 in the Number and Financial Literacy represents, what activities students see, and any specific rides or activities big book and little they are unfamiliar with. Ask students if they have ever been to a similar books), BLM 69: Fair event and what they liked about it. Fractions, chart paper Time: 60 minutes • Pose some of the following prompts: – Ask students what they see that could be represented with fractions. – Point out the people lined up to get on the octopus ride. Ask how many people there are. Ask whether they could be considered a set. What do they have in common? (e.g., they all want to get on the ride) Ask students how they could describe the whole. (e.g., nine ninths) Ask how the fractional parts could be counted. (e.g., 1 one-ninth, 2 one-ninths, 3 one-ninths, ..., 9 one-ninths). Ask students to look for and describe parts of the whole. (e.g., three of the nine people are wearing hats) How can they describe those parts as fractions. (e.g., three ninths, or one third, are wearing hats and sixth ninths, or two thirds, are not) Have students identify the kind of model they just looked at. – Ask students whether they see any area models of fractions. (e.g., the striped awning on the Ring Toss booth). Ask what the whole is and what the parts are. Ask students what they know must be true about the parts in order for them to be represented by fractions. (e.g., They must take up the same area.) Working On It (20 minutes) • Have students work in pairs or small groups (up to four). Give each group a copy of the Number and financial Literacy little book and BLM 69: Fair Fractions. • Tell students they will look for fractions in the picture, including both area and set models. They can glue their BLM on a half piece of chart paper, and then circle the fractions they find and annotate them with explanations on the surrounding chart paper. Have students determine whether each of the fractions they identify is an area or a set model. Differentiation • ELLs may need help with the vocabulary about events and items at a fair so they can describe them on their chart paper. 470 Number and Financial Literacy