Fun with Fractions: A Unit on Developing the Set Model: Unit Overview


 Eustace Gallagher
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1 Fun with Fractions: A Unit on Developing the Set Model: Unit Overview Number of Lessons: 7 Grades: 35 Number & Operations In this unit plan, students explore relationships among fractions through work with the set model. This early work with fraction relationships helps students make sense of basic fraction concepts and facilitates work with comparing and ordering fractions and working with equivalency. Math Content This unit exposes students to the set model and provides an opportunity for them to develop a thorough understanding of this model in multiple applications. In the set model, the unit is represented by the entire set, and subsets of the unit make up the fractional parts. For example, in a set of 16 marbles, 4 marbles comprise onefourth of the set of 16 marbles. The set of 16 in this example represents the whole or 1. This notion of referring to a collection of objects makes studying the set model difficult for young children. As students work with a variety of fraction models in contexts that promote reasoning and problem solving, they develop a more thorough understanding of fractions and their relationships. Each lesson is designed to last 45 minutes, but teachers may need to modify this time based on individual student needs. As subsequent lessons build on skills from earlier lessons, the plan should be followed in sequence.
2 Individual Lessons Lesson 1: Eggsactly with a Dozen Eggs During this lesson, students are given the opportunity to examine fractions as part of a set. This lesson helps students develop skill in problem solving and reasoning as they examine relationships among the fractions used to describe part of a set of 12. Lesson 2: Eggsactly with Eighteen Eggs During this lesson, students continue to examine fractions as part of a set. This lesson helps students develop skill in problem solving and reasoning as they examine relationships among the fractions used to describe part of a set of eighteen. Lesson 3: Eggsactly Equivalent During this lesson, students use twelve eggs to identify equivalent fractions. Construction paper cutouts are used as a physical model to represent various fractions of the set of eggs, e.g., 1/12, 1/6, 1/3, _, etc. Students investigate relationships among fractions that are equivalent. Lesson 4: Another Look at the Set Model using Attribute Pieces The previous lesson focused on the set model where all objects in the set are the same size and shape. Students also need work with sets in which the objects look different. In the real world,
3 we are often faced with fraction situations where the objects in the set are not identical. Consider a group of people where some are children, some are adults, some are male, some are female, etc. You might describe the group using fractions by showing the fraction of the group that is comprised of children or adults. For a button collection, we might describe the fraction of the button set that is blue, or the fraction of the button set that includes buttons with four holes. It is important that students see sets of objects as any collection, not just collections with identical members. For this lesson, students use fractions to describe a set of attribute pieces. Students develop skill in problem solving and reasoning as they think about their set and how to create new sets given specific fractional characteristics. Lesson 5: Finding Fractions in the Real World This lesson requires that students identify fractions in realworld contexts from a set of items that are not identical. Students develop skills in problem solving and reasoning as they think about their set and how to create new sets given specific fractional characteristics. Lesson 6: Class Attributes During this lesson, students create their own classroom survey or use the Classroom Survey to study the class and describe the set [class] in fractional parts. This lesson is integrated with other areas of the math curriculum, including data analysis and statistics.
4 Lesson 7: Looking Back and Moving Forward This lesson gives students another opportunity to explore fractions using the set model. This lesson is integrated with other areas of the math curriculum including data analysis and statistics. Standards Number & Operations for * Understand the placevalue structure of the baseten number system and be able to represent and compare whole numbers and decimals. * Recognize equivalent representations for the same number and generate them by decomposing and composing numbers. * Develop understanding of fractions as parts of unit wholes, as parts of a collection, as locations on number lines, and as divisions of whole numbers. * Use models, benchmarks, and equivalent forms to judge the size of fractions. * Recognize and generate equivalent forms of commonly used fractions, decimals, and percents.
5 Eggsactly with a Dozen Eggs: Lesson 1 of 7 Length: 45 minutes Grades: 35 Number & Operations During this lesson, students are given the opportunity to examine fractions as part of a set. This lesson helps students develop skill in problem solving and reasoning as they examine relationships among the fractions used to describe part of a set of 12. Learning Objectives Students will be able to: * Demonstrate understanding that a fraction can be represented as part of a set, given a set of identical items (eggs) * Identify fractions when the whole (set) and part of the set is given * Identify fraction relationships associated with the set, such as one fourth is one half of one half Materials * An egg carton filled with 12 plastic eggs or markers for each pair of students * Resource Sheet Eggsactly Eggs Recording Sheet * Class Notes recording sheet Instructional Plan Introduce the set representation by having each pair of students examine an egg carton filled with plastic eggs [or some other marker if plastic eggs are unavailable or cost prohibitive]. Ask students how many eggs are in the set. (12) Suppose six are used to bake a cake. Have students remove six eggs. Students
6 should record their egg configuration on the Eggsactly Eggs Recording Sheet. Have students participate in a gallery walk examining other students egg cartons to see all the different ways students might have removed six. Ask students what all the egg cartons have in common. (There are six remaining.) What fraction of the entire set is 6? (6/12; Accept _ or other equivalent fractions.) [If students do not make the connection between equivalent fractions, e.g. 6/12 = _, they have an opportunity to develop these relationships in later lessons.] What fraction was removed? (6/12 or _ ) Have students label their recording sheet as 6/12. [Some students may choose to label their sheet with an equivalent fraction such as _. If so, this provides an excellent opportunity to introduce equivalent fractions.] Continue removing varying numbers of eggs. For example, suppose this time that we need eight eggs to bake our cake. Have students remove eight eggs. Students should record their egg configuration on the Eggsactly Eggs Recording Sheet. Have students go on another gallery walk to see all the different ways students might have removed eight. Ask students what all the egg cartons have in common. (There are four remaining.) What fraction of the entire set is 4? (4/12; Accept 1/3 or 2/6) What fraction was removed? (8/12 or 2/3 or 4/6) Have students label their recording sheet as 4/12 (Accept 1/3 or 2/6). Have students work in pairs to continue the investigation as different numbers of eggs are used. Students should be given time to investigate the variety of ways in which the eggs can be arranged. These arrangements should be recorded on the Eggsactly Eggs Recording Sheet and the sheet should be labeled according to the fraction. For example, students might use one full sheet to record all the ways to show _ of a dozen. Have students investigate the different ways they can arrange their eggs when given the fraction. For example, ask students to show 1/4 of a dozen? (Use the Eggsactly Eggs Recording Sheet to have students represent several different configurations all equivalent to 1/4 of a dozen.)
7 Have students identify fraction relationships associated with the set (e.g., _ of the set of 12 eggs is the same as 6/12 of the set OR When the numerator stays the same and the denominator increases, the fractions become smaller 1/3 is smaller in area than _.). Convene the whole class to discuss the activities in this lesson. The guiding questions may be used to focus the class discussion as they were used to focus individual student s attention on the mathematics learning objectives of this lesson. Questions for Students 1. What do you notice about the relationship between _ of a dozen and 1/4 of a dozen? (Students should be able to tell from their recordings that 1/4 is _ of _.) 2. What do you notice about the relationship between 1/3 of a dozen and 1/6 of a dozen? (Students should be able to tell from their recordings that 1/6 is _ of 1/3.) 3. What can you tell about the size of the fraction when the numerator is the same for both fractions, e.g., 1/4 and 1/6? (The smaller the denominator, the larger the fraction) 4. What can you tell about the size of the fraction when the denominator is the same for both fractions, e.g. 2/5 and 3/5? (The smaller the numerator, the smaller the fraction) Assessment Options At this stage of the unit, it is important to know whether students can: * Demonstrate understanding that a fraction can be represented as part of a set * Identify fractions when the whole (set) and part of the set are given * Identify fraction relationships associated with a set of twelve
8 Student recordings can be used to make instructional decisions about students understanding of fraction relationships. Areas needing additional work can be developed during subsequent lessons. More challenging experiences can be provided for those students who need them. You may choose to use the Class Notes recording sheet to make anecdotal notes about students understandings. Teacher Reflection 1. Which students understand that a fraction can be represented as part of a set? What activities are appropriate for students who have not yet developed this understanding? 2. Which students can identify fractions when the whole (set) and part of the set is given? What activities are appropriate for students who have not yet developed this understanding? 3. Which students/groups can articulate relationships between fractions? 4. Which pairs worked well together? Which groupings need to be changed for future lessons? 5. If groups were not successful working together, what was the source of the problem? (i.e., Were the problems behaviorrelated or were academic levels not matched appropriately in the groups?) 6. What parts of the lesson went smoothly? What parts should be modified for the future? Standards and Expectations Number & Operations 35 * Develop understanding of fractions as parts of unit wholes, as parts of a collection, as locations on number lines, and as divisions of whole numbers. * Use models, benchmarks, and equivalent forms to judge the size of fractions. * Recognize and generate equivalent forms of commonly used fractions, decimals, and percents.
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11 Eggsactly with Eighteen Eggs: Lesson 2 of 7 Length: 45 minutes Grades: 35 Number & Operations During this lesson, students continue to examine fractions as part of a set. This lesson helps students develop skill in problem solving and reasoning as they examine relationships among the fractions used to describe part of a set of eighteen. Learning Objectives Students will be able to: * Demonstrate understanding that a fraction can be represented as part of a set, given a set of identical items (eggs) * Identify fractions when the whole (set) and part of the set is given * Identify fraction relationships associated with the set (e.g., _ is _ of _) Materials * An egg carton designed to hold 18 plastic eggs or markers for each pair of students * Eggsactly Eggs recording sheet transparency * 18 Eggs in a Carton recording sheet * Class Notes recording sheet Instructional Plan Use the Eggsactly Eggs Recording Sheet as an overhead transparency to review fractions as part of a set of 12. For example, ask students how to show _ of a dozen. Accept equivalent fractions (6/12, 3/6, etc.) and all the arrangements of six eggs in a carton that holds twelve eggs.
12 Have students investigate how their fractions would change if the egg carton holds 18 eggs instead of 12. Have students remove varying numbers of eggs to represent a fraction of the carton that remains. Students should also be given the fraction and asked to model using 18 eggs. Use of the 18 Eggs in a Carton recording sheet should be used as students discover the different representations for each of the fractions. Ask students how many eggs are in the set. (18) Suppose nine are used to bake a cake. Have students remove nine eggs. Students should record their egg configuration on the 18 Eggs in a Carton Recording Sheet. Have students participate in a gallery walk examining other students egg cartons to see all the different ways students might have removed nine. Ask students what all the egg cartons have in common. (There are nine remaining.) What fraction of the entire set is nine? (9/18; Accept _ or other equivalent fractions.) [If students do not make the connection between equivalent fractions, e.g., 9/18 = _, they will have an opportunity to develop these relationships in the next lesson.] What fraction was removed? (9/18 or _ ) Have students label their recording sheet as 9/18. [Some students may choose to label their sheet with an equivalent fraction such as _. This provides an excellent opportunity to work with equivalent fractions.] Continue removing varying numbers of eggs. For example, suppose this time that we need twelve eggs to bake our cake. Have students remove twelve eggs. Students should record their egg configuration on the 18 Eggs in a Carton Recording Sheet. Have students go on another gallery walk to see all the different ways students might have removed twelve. Ask students what all the egg cartons have in common. (There are six remaining.) What fraction of the entire set is six? (6/18; Accept 1/3 or 2/6) What fraction was removed? (12/18 or 2/3 or 4/6) Have students label their recording sheet as 6/18 (Accept 1/3 or 2/6).
13 Have students work in pairs to continue the investigation as different numbers of eggs are used. Students should be given time to investigate the variety of ways in which the eggs can be arranged. These arrangements should be recorded on the 18 Eggs in a Carton Recording Sheet and the sheet should be labeled according to the fraction. For example, students might use one full sheet to record all the ways to show _ of eighteen eggs. Have students investigate the different ways they can arrange their eggs when given the fraction. For example, ask students to show 1/4 of eighteen eggs. (Use the 18 Eggs in a Carton Recording Sheet to have students represent several different configurations all equivalent to 1/4 of eighteen eggs.) Identify fraction relationships associated with the set (e.g., _ of the set of 18 eggs is the same as 9/18 of the set OR When the numerator stays the same and the denominator increases, the fractions become smaller 1/3 is smaller in area than _.). Questions for Students 1. Have students think about the fractions that were constructed using a 12egg carton and the fractions that were constructed using an 18egg carton. Were any of the same fractions used? For example, we were able to show _ of 12 as well as _ of 18. How did the parts represented by these fractions differ? 2. We have worked with egg cartons that serve as models for fractions with denominators of 2, 3, 4, 6, 9, and 12. How many slots would an egg carton have to have in order to work with fractions such as 5/8? What about 3/5? Have students generate other fractions they might like to model. 3. What fraction relationships were you able to identify in this lesson?
14 Assessment Options At this stage of the unit, it is important to know whether students can: * Demonstrate understanding that a fraction can be represented as part of a set * Identify fractions when the whole (set) and part of the set are given * Identify fraction relationships associated with a set of eighteen Student recordings can be used to make instructional decisions about students understanding of fraction relationships. Areas needing additional work can be developed during subsequent lessons. You may choose to use the Class Notes recording sheet to make anecdotal notes about students understandings. Extensions Continue the activity by changing the size of the egg carton. [You can create larger cartons by cutting the desired number of eggcups and gluing them to other cartons.] For example, have students investigate how their fractions would change if the egg carton holds 6 eggs or 24 eggs. Teacher Reflection 1. Which students understand that a fraction can be represented as part of a set? What activities are appropriate for students who have not yet developed this understanding? 2. Which students can identify fractions when the whole (set) and part of the set is given? What activities are appropriate for students who have not yet developed this understanding? 3. Which students/groups can articulate relationships between fractions?
15 4. Which pairs worked well together? Which groupings need to be changed for future lessons? 5. If groups were not successful working together, what was the source of the problem? (i.e., Were the problems behaviorrelated or were academic levels not matched appropriately in the groups?) 6. What parts of the lesson went smoothly? What parts should be modified for the future? Standards and Expectations Number & Operations 35 * Use models, benchmarks, and equivalent forms to judge the size of fractions. * Recognize and generate equivalent forms of commonly used fractions, decimals, and percents. * Develop understanding of fractions as parts of unit wholes, as parts of a collection, as locations on number lines, and as divisions of whole numbers.
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18 Eggsactly Equivalent: Lesson 3 of 7 Length: 45 minutes Grades: 35 Number & Operations During this lesson, students use twelve eggs to identify equivalent fractions. Construction paper cutouts are used as a physical model to represent various fractions of the set of eggs, e.g., 1/12, 1/6, 1/3, _, etc. Students investigate relationships among fractions that are equivalent. Learning Objectives Students will be able to: * Demonstrate understanding that a fraction can be represented as part of a set, given a set of identical items (eggs) * identify fractions when the whole (set) and part of the set is given * Identify equivalent fractions (e.g., _ of the set of 18 eggs is the same as 9/18 of the set * Identify relationships inherent in equivalent fractions (e.g., _ can be multiplied by 2 to get the equivalent fraction 2/4 or 2/4 can be divided by 2 to get the equivalent fraction _.) Materials * An egg carton designed to hold 12 plastic eggs or markers for each pair of students * Construction paper cut to fit various fractions of an egg carton, e.g., 1/12, 1/6, 1/4, 1/3, _, etc. * Eggsactly Eggs recording sheet transparency * Class Notes recording sheet
19 Instructional Plan Lesson Plans Use the Eggsactly Eggs recording sheet as an overhead transparency to review fractions as part of a set of 12. For example, ask students how to show _ of a dozen eggs. Accept equivalent fractions (6/12, 3/6, etc.) and all the arrangements of six eggs in a carton that holds a dozen eggs. Give students paper cutouts that cover various parts of the egg carton, e.g., 1/12, 1/6, 1/4, 1/3, _ (see illustration below). Students need enough cutouts of each fraction to represent the whole. For example, students will need two 1/2s, 6 1/6s, 4 1/4s, and so forth. Have students investigate each cutout and identify the fraction that is represented by each. Guide students to label each cutout with the appropriate reduced fraction. For example Prompt students to begin looking for fractions that cover the same area, i.e., equivalent fractions. For example, ask students how many 1/12 pieces are needed to cover 1/6 (2). Have students record 1/6 = 2/12 on notebook paper. Ask students how many 1/6 pieces are needed to cover 1/3 (2). Have students record 1/3 = 2/6 on notebook paper. Have students work in pairs to continue identifying as many equivalent fractions as possible. Groups should record all equivalent fractions on notebook paper. When finished, have groups take turns reporting the equivalent fractions to the whole class. If any groups did not find the equivalent fraction being shared, they should add the new set to their list. Ensure that all of the following are identified: 1/6 = 2/12 1/4 = 3/12 1/3 = 2/6 1/3 = 4/12
20 = 2/4 = 3/6 = 6/12 Have students explore relationships between the equivalent fractions. For example, students might notice that crossmultiplying yields equivalent products, for example, _ and 6/8. By multiplying the numerator of the first fraction (3) and the denominator of the second fraction (8), students yield a product of 24. By multiplying the remaining two numbers, the numerator of the second number (6) and the denominator of the first number (4), students yield a product of 24. Students should investigate whether this rule holds true for all equivalent fractions. For example: If students do not find this relationship on their own, guide them to discover the relationship between cross products. Guiding questions are provided to facilitate this process. [This discovery introduces students to concepts that will be important to the study of formal algebra. For example, students may use cross multiplication to solve for a missing term in a proportion such as: x Students would first determine that 8 x 2 = 16 and then use algebra to solve 4 (x) = 16. It might be important to note why cross multiplication works. For example: 4 = OR a = c b d
21 When you crossmultiply, you're really multiplying each side by n/n, with n being the denominator of the other side: 4 x 4 = 2 x OR d x a = c x b d b d b If simplified, we are left with 16 = OR ad = bc bd bd Since both sides are being divided by the same quantity, you only consider the numerators, i.e. ad = bc or 16 = 16, which is what you would get if you crossmultiplied. Helping students develop this kind of conceptual understanding through reasoning can lay important groundwork for later mathematics.] Questions for Students 1. What do you notice about the relationship between _ of a dozen and 6/12 of a dozen? (Students should be able to tell from their recordings that _ and 6/12 are equivalent fractions.) 2. What do you notice about the relationship between 1/3 of a dozen and 4/12 of a dozen? (Students should be able to tell from their recordings that 1/3 and 4/12 are equivalent fractions.) 3. What do you notice about the relationship between 1/4 of a dozen and 3/12 of a dozen? (Students should be able to tell from their recordings that 1/4 and 3/12 are equivalent fractions.)
22 4. What do you notice about the relationship between 1/6 of a dozen and 2/12 of a dozen? (Students should be able to tell from their recordings that 1/6 and 2/12 are equivalent fractions.) 5. What other equivalent fractions did you identify using 12 eggs? 6. What equivalent fractions did you identify using 18 eggs? Were any of the same fractions used? 7. What relationships do you see in the equivalent fractions identified in this lesson? Do you notice any patterns when you multiply one fraction s numerator with the other fraction s denominator? Assessment Options At this stage of the unit, it is important to know whether students can: * Demonstrate understanding that a fraction can be represented as part of a set * Identify fractions when the whole (set) and part of the set are given * Identify fraction relationships associated with the set including equivalent fractions (_ of the set of 12 eggs is the same as 6/12 of the set OR When the numerator stays the same and the denominator increases, the fractions become smaller 1/3 is smaller in area than _. OR When you multiply the numerator with an equivalent fraction s denominator, the products are equivalent) Collect recording sheets student complete in the activities of this lesson. Review these and use the information to make instructional decisions about students understanding of fraction relationships. Provide feedback to students about their misunderstandings and next steps for those who demonstrate a command of the content and processes. Areas needing additional work can be developed during subsequent lessons. You may
23 choose to use the Class Notes recording sheet to make anecdotal notes about students understandings. Extensions Continue the activity by using the egg carton that holds 18 eggs. Have students model and record all equivalent fractions. Have students investigate how their fractions would change if the egg carton holds 6 eggs or 24 eggs. Teacher Reflection 1. Which students understand that a fraction can be represented as part of a set? What activities are appropriate for students who have not yet developed this understanding? 2. Which students can identify fractions when the whole (set) and a part of the set is given? What activities are appropriate for students who have not yet developed this understanding? 3. Which students/groups can articulate relationships between the fractions? What activities are appropriate for students who have not yet developed this understanding? 4. Which students/groups can identify equivalent fractions? What activities are appropriate for students who have not yet developed this understanding? 5. Which pairs worked well together? Which groupings need to be changed for future lessons? 6. If groups were not successful working together, what was the source of the problem? (i.e. Were the problems behaviorrelated or were academic levels not matched appropriately in the groups?) 7. What parts of the lesson went smoothly? What parts should be modified for the future?
24 Standards and Expectations Number & Operations 35 * Recognize and generate equivalent forms of commonly used fractions, decimals, and percents. * Develop understanding of fractions as parts of unit wholes, as parts of a collection, as locations on number lines, and as divisions of whole numbers. * Use models, benchmarks, and equivalent forms to judge the size of fractions.
25 Another Look at the Set Model using Attribute Pieces: Lesson 4 of 7 Length: 2 45 minute periods Grades: 35 Number & Operations The previous lesson focused on the set model where all objects in the set are the same size and shape. Students also need work with sets in which the objects look different. In the real world, we are often faced with fraction situations where the objects in the set are not identical. Consider a group of people where some are children, some are adults, some are male, some are female, etc. You might describe the group using fractions by showing the fraction of the group that is comprised of children or adults. For a button collection, we might describe the fraction of the button set that is blue, or the fraction of the button set that includes buttons with four holes. It is important that students see sets of objects as any collection, not just collections with identical members. For this lesson, students use fractions to describe a set of attribute pieces. Students develop skill in problem solving and reasoning as they think about their set and how to create new sets given specific fractional characteristics. Learning Objectives Students will be able to: * Demonstrate understanding that a fraction can be represented as part of the set, given a set of items that are not identical (attribute pieces) * Describe a set of objects based on its fractional components * Identify fraction relationships associated with the set
26 Materials * Class Notes recording sheet * 3 x 5 index cards (5 for each pair of students) * Attribute Pieces(either commercial or homemade from the Attribute Pieces resource sheets Attribute Sheet 1 Attribute Sheet 2) Note: Attribute pieces should include the following attributes or characteristics Color: red, blue, and yellow Shape: square, rectangle, circle, hexagon, and triangle Thickness: thick and thin Size: large and small Directions for making paper attribute pieces: Homemade attribute pieces can be made by printing the Attribute Pieces resource sheet using a color printer. Printing and cutting both pages (1 and 2) of attribute pieces will produce exactly one full set of attribute pieces. These materials can be printed onto thick paper and/or laminated to increase their durability. Instructional Plan Part I Introduction to the Materials: If students have not used attribute pieces before, time should be given for them to freely explore. Point out that the pieces have four variable attributes: (1) color, (2) shape, (3) thickness, and (4) size. Tell students that they are going to become familiar with their attribute block set by grouping them in a variety of ways. Have students first group by color, then by shape, and finally by thickness and size. Choose two attribute pieces and ask students to look for similarities and differences. How are the pieces alike different? Can you find two pieces that are different in only one way? Select two other pieces that are different in one and only one way.
27 In order to familiarize students with the various attribute pieces, have students make a different train. Tell the students that you will choose an attribute piece for the engine. Students should pick an attribute piece that is different in one (and only one) way. Have students tell how the block is different in one way. Continue adding to the difference train. Have students make other trains with two, three, or four differences between attribute pieces. Students should explain why they selected a particular attribute piece based on the number of varying attributes. Part II  Have students work in pairs to randomly select eight attribute pieces. For example, students might choose the following set. A capital T is used to denote a thick attribute piece while a lowercase t is used to denote a thin attribute piece. Have students generate a list of fractions to describe this set and record their description on a 3 x 5 index card. For example The set is * 2/8 or 1/4 red * 3/8 yellow * 3/8 blue * 5/8 thick * 3/8 thin * 2/8 or 1/4 triangular * 2/8 or 1/4 square * 2/8 or 1/4 rectangular * 2/8 or 1/4 hexagonal * 6/8 or 3/4 large * 2/8 or 1/4 small
28 On the reverse side of the card, students should cut out paper attribute pieces representative of their set and glue them to the card. Have students repeat this process until they have described between three and five attribute sets. Pair students and have them share their descriptions with a partner. The partner should attempt to construct the set based on the characteristics on the index card. When they think they have the correct answer, they should turn over the index card to check. [These selfchecking cards can be used for reinforcement at a center or for independent work.] Questions for Students 1. Choose two attribute pieces. How are they alike? How are they different? 2. Choose two other attribute pieces. How are they alike? How are they different? 3. Can you find two pieces that are alike in one and only one way? 4. Can you find two pieces that are alike in two and only two ways? 5. Can you find three pieces that are alike in three and only three ways? 6. Can you find two pieces that are different in one and only one way? 7. Can you find two pieces that are different in two and only two ways? 8. Can you find three pieces that are different in three and only three ways? 9. When using the cards you constructed in this lesson, is it possible to have more than one answer (different sets of attribute pieces) for a given set of characteristics? 10. What problem solving strategies did you use to determine what attribute pieces are in the set? (e.g., logical reasoning, guess and check. Accept all reasonable strategies.)
29 Assessment Options Lesson Plans At this stage of the unit, it is important to know whether students can: * Demonstrate understanding that a fraction can be represented as part of a set * Describe a set of objects based on its fractional components * Identify fraction relationships associated with the set Use the index cards with fractions that describe the set of attribute pieces to make instructional decisions about students understandings. You may use the Class Notes recording sheet to make anecdotal notes about student performances. Provide feedback to students about their progress toward meeting the learning objectives of this lesson. Teacher Reflection 1. Which students understand that a fraction can be represented as part of a set? What activities are appropriate for students who have not yet developed this understanding? 2. Which students can describe a set of objects based on its fractional components? What activities are appropriate for students who have not yet developed this understanding? 3. Which students/groups can articulate the relationships between fractions? 4. How are students recording fractions of the set in reduced form, or do all fractions use the number in the set as the denominator? [This information is helpful for documenting where students are in their understanding of reducing fractions.] 5. Which pairs worked well together? Which groupings need to be changed for future lessons? 6. If groups were not successful working together, what was the source of the problem? (i.e., Were the problems behaviorrelated or were academic levels not matched appropriately in the groups?)
30 7. What parts of the lesson went smoothly? What parts should be modified for the future? Standards and Expectations Number & Operations 35 * Use models, benchmarks, and equivalent forms to judge the size of fractions. * Develop understanding of fractions as parts of unit wholes, as parts of a collection, as locations on number lines, and as divisions of whole numbers. * Recognize and generate equivalent forms of commonly used fractions, decimals, and percents.
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33 Finding Fractions in the Real World: Lesson 5 of 7 Length: 2 45 minute periods Grades: 35 Number & Operations This lesson requires that students identify fractions in realworld contexts from a set of items that are not identical. Students develop skills in problem solving and reasoning as they think about their set and how to create new sets given specific fractional characteristics. Learning Objectives Students will be able to: * Demonstrate understanding that a fraction can be represented as part of the set, given a set of items that are not identical * Describe a set of objects based on its fractional components * Identify fraction relationships associated with the set * Reduce fractions to their lowest terms Materials * Chart paper * Markers * Class Notes recording sheet Instructional Plan In this activity, students identify the set model as it is used in the real world. They begin investigating fractions within their group, in the classroom environment, and in the outside world. This lesson may be extended to identify fractions in the home environment.
34 Organize students into groups of four or five. Have students draw a picture of the students in their group on chart paper. Students may draw stick people to represent those in their group. Encourage them to use a lot of detail in their drawings, e.g., include hair color, facial features, clothing, shoes, accessories, etc. Have them label their drawing with as many fractions as possible. For example, 2/5 of the group is wearing blue jeans 3/5 of the group has brown hair 1/5 of the group is wearing glasses. Each group should share their chart paper drawings with the whole class, explaining the fractions they discovered. As students share, have the class look for fractions that are in lowest terms. For example, students will probably recognize that 2/4 can also be represented as _. Have students look for fraction relationships between fractions that are not in lowest terms. Guide students to discover that a fraction in lowest terms has a numerator and denominator with no common factors other than one. This means that fractions can be reduced or can be divided by some number other than one. Have students find common factors. For example, suppose 2/4 of the students in the group wear glasses; 2 and 4 can be divided by 2. This leaves _, which is a fraction in reduced terms. Have students demonstrate this understanding by encircling the two students who wear glasses and then encircling the two students who do not wear glasses. Students can easily see that 1 out of 2 groups wear glasses. Have students continue to locate fractions that can be reduced. Students might begin to see a pattern. In some cases, the numerator is the first number circled and becomes the first group and other groups are made of the same size. For example, 2/8 have blue eyes. Students should circle the 2 as one group and make groups of 2 from the remaining 6. They can easily see that 2/8 can be reduced to.
35 Prompt students to look for common factors that might be used to subdivide the group. For example, 6/12 can be divided by 3 or 6. If students divide the twelve into groups of 3, reducing the fraction to 2/4, they should be encouraged to continue looking for factors until there are no common factors other than one. Next, take students on a classroom scavenger hunt. Students should find as many fraction examples as possible. Again, students should be affirmed if they find examples representative of models other than the set model. Groups should record their findings on notebook paper. Students should be given an opportunity to share their fractions with the whole class. Students should add any additional examples to their list if they were not on their original one. As students share, have the class look for fractions that are in lowest terms. Have students look for fraction relationships between fractions that are not in lowest terms. Finally, take students on a fraction scavenger hunt on the playground or some other outdoor area. [This provides an opportunity for students to apply their knowledge in a new context and reinforces their understanding of the application of fractions to multiple environments.] Students should find as many fraction examples as possible. Again, students should be affirmed if they find examples representative of models other than the set model. Groups should record their findings on notebook paper. Students should be given an opportunity to share their fractions with the whole class. Students should add to their list any additional examples that were not on their original one. As students share, have the class look for fractions that are in lowest terms. Have students look for fraction relationships between fractions that are not in lowest terms.
36 Questions for Students Lesson Plans 1. What equivalent fractions did you identify in your small group? 2. What equivalent fractions did you identify in the classroom? 3. What equivalent fractions did you identify on the playground? 4. What relationships do you see in the equivalent fractions identified in this lesson? 5. Do you notice any patterns when reducing fractions? 6. How do you know if a fraction is in lowest terms? Assessment Options At this stage of the unit, it is important to know whether students can: * Demonstrate understanding that a fraction can be represented as part of a set * Describe a set of objects based on its fractional components * Identify fraction relationships associated with the set * Reduce a simple fraction to lowest terms Use chart paper drawings from the beginning of this lesson to make instructional decisions about students understandings. Student notes about fractions found in the classroom and outside environment also provide insightful information about students progress toward learning goals. You may use the Class Notes recording sheet to make anecdotal notes about student performances. Provide feedback to students about their progress toward meeting the learning objectives of this lesson.
37 Extensions For homework, ask students to find as many fraction examples as possible at home. Teacher Reflection 1. Which students understand that a fraction can be represented as part of a set? What activities are appropriate for students who have not yet developed this understanding? 2. Which students can describe a set of objects based on its fractional components? What activities are appropriate for students who have not yet developed this understanding? 3. Which students/groups can articulate the relationships between fractions? 4. How are students recording fractions of the set in reduced form, or do all fractions use the number in the set as the denominator? [This information is helpful for documenting where students are in their understanding of fractional relationships and should be documented.] 5. Which pairs worked well together? Which groupings need to be changed for future lessons? 6. If groups were not successful working together, what was the source of the problem? (i.e. Were the problems behaviorrelated or were academic levels not matched appropriately in the groups?) 7. What parts of the lesson went smoothly? What parts should be modified for the future?
38 Standards and Expectations Number & Operations 35 * Understand the placevalue structure of the baseten number system and be able to represent and compare whole numbers and decimals. * Recognize and generate equivalent forms of commonly used fractions, decimals, and percents. * Develop understanding of fractions as parts of unit wholes, as parts of a collection, as locations on number lines, and as divisions of whole numbers.
39 Class Attributes: Lesson 6 of 7 Length: 2 45 minute periods Grades: 35 Number & Operations During this lesson, students create their own classroom survey or use the Classroom Survey to study the class and describe the set [class] in fractional parts. This lesson is integrated with other areas of the math curriculum, including data analysis and statistics. Learning Objectives Students will be able to: * Demonstrate understanding that a fraction can be represented as part of the set, given some number of items * Use fractional components to describe their class attributes, for example, gender, hair color, and so forth * Identify fraction relationships associated with the set * Reduce fractions to their lowest terms Materials * Five to six envelopes [one envelope for each small group] * Scrap paper the size of a standard adhesive note [enough for each envelope to contain one slip for each student in the class] * Classroom Survey * Class Notes recording sheet Instructional Plan This activity may require two sessions to complete each of the tasks. First, the students collect data on the class and use that data to generate fractions that describe the class. Because this unit emphasizes data analysis, the students use knowledge of graphing and statistics to complete this lesson.
40 Before the students can represent class characteristics using fractions, classroom data must be collected. To begin the data collection process, have the students brainstorm a list of the many ways the class or group might be described, for example, by gender, hair color, height, those who own pets, and so forth. This list can be used to create a classroom survey for data collection or you may choose to survey the class with the Classroom Survey. Organize the students into five or six groups and have each group select a question from the survey. Give each group an envelope that contains as many scrap pieces of paper as there are students in the class. Have each group record its question andanswer choices, if appropriate, on the envelope. For example, if the students ask about gender, they should include two choices, male or female. If it is not possible to identify all the possible choices, the students should leave their question in an openended format. For example, a question about types of pets should be left openended, as one might not be able to anticipate the variety of pets represented in the class. Conduct the survey by passing the envelopes around the room and giving each student a chance to respond. Before starting the survey, have the students remove all the paper from their group s envelope and leave it at their table. They will use these slips to record and submit their answer to each survey question. Begin the survey by having group members respond to the question on their envelope first, writing their answer on a slip of paper and placing it in the envelope. When the groups are finished with that question, they should pass their envelope to the next group, and so forth, until all the students have had a chance to respond to all the questions. [If the students in your class would benefit from getting up and moving around the room, instruct the students to leave the envelopes at each table and move from table to table to answer the questions.]
41 Once data are collected, the groups should tally the responses in their envelope, record the number and represent the quantity as a fraction, for example, 12 out of 24 students (12/24 or _ ) have blue eyes. Have each group reduce their fractions to lowest terms by finding the greatest common factor. For example, suppose 18/24 [or 3/4] of the class owns a pet. The greatest common factor for 18 and 24 is 6. The students might find it helpful to list all the factors for the numerator and the denominator, 18 and 24 in this example, and locate the greatest common factor. This can be done strategically by checking in order each pair of factors that when multiplied yield a particular product. For example, to exhaust all the factors of 18, one would begin with 1 x 18, then 2 x 9, then 3 x 6. Since 4 is not a factor, the student would move on to 5 and then to 6. Six has already been generated with 3 x 6. When the student begins to duplicate factors, they know they have exhausted the list. For organizational purposes, it is helpful to write the sets of factors in the following manner: For 18, the students should begin listing on opposite sides (following the format below) with 1 x 18, then 2 x 9 on the inside of the other factors, then 3 x 6 in the middle. When factors begin to repeat, e.g., 6 x 3, the students know that the list of factors has been exhausted. This list of factors for 18 would be recorded as follows: Step One 1 18 Step Two Step Three For 24, the list of factors would evolve as follows:
42 Next, ask the students to list the factors for both numbers one on top of the other so they can easily recognize the common factors. The greatest common factor should be circled. For example: The students should divide the numerator and denominator by the greatest common factor to reduce the fraction. For example, for 18/24, the students should divide by 6 to reduce the fraction to 3/4. If it is necessary to divide the lesson into two segments, this might be a logical beginning point for the second part of the lesson. Have group members organize their data in a chart and share it with the class. The students should record all fractional representations and may choose to record appropriate statistics on their chart, for example, mode for categorical [qualitative] data and mean, median, range, and mode for numerical [quantitative] data. Groups may choose to create their graph using the Create a Graph Web site from the National Center for Education Statistics (nces.ed.gov/nceskids/graphing/bar.asp). The students should choose the type of graph they want to create by using the pulldown menu. Although this site is userfriendly, a classroom demonstration of how to use it should prove helpful. An example of a bar graph created on this site is shown below: Once the students have created their graph, they should label the data in fractional parts and reduce all fractions to lowest terms. For example, this chart should be labeled with dog being 15/26, cats being 8/26 or 4/13, birds being 2/26 or 1/13, and 1/26 iguana. Ask students to share their graphs with the class and discuss how they used fractions in collecting the data depicted on each graph. If necessary, remind students to
43 consider what the fractions represent, how the data was collected, how categories were established, and how finding the lowest common factor simplified the process of reducing the fraction. Questions for Students 1. How can you take classroom data and record it as a fraction? 2. What can you tell about your class based on the data from your survey? 3. What fractions need to be reduced? 4. Do you notice any patterns when reducing fractions? 5. How do you know whether a fraction is in lowest terms? 6. If a fraction that needs to be reduced has more than one common factor, is it better to divide by the smallest factor or the largest factor? Assessment Options At this stage of the unit, it is important to know whether the students can do the following: 1. Demonstrate understanding that a fraction can be represented as part of a set 2. Describe a set of objects on the basis of its fractional components 3. Identify fraction relationships associated with the set 4. Reduce a simple fraction to lowest terms Use the students' graphs with fractional representations to make instructional decisions about the students' understanding. You may use the Class Notes recording sheet to make anecdotal notes about the students' performance. Give the students feedback about their progress toward meeting the learning objectives of this lesson.
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