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Parts of a Whole Thematic Unit Outline 3rd Grade 35 Days Christina Ferrara Overview In this 35-day Grade 3 module, students extend and deepen Grade 2 practice with equal shares to understanding fractions as equal partitions of a whole (2.G.3). Their knowledge becomes more formal as they work with area models and the number line. The module or unit is broken up into six main topics: Topic A: Partition a Whole into Equal Groups Topic B: Unit Fractions and their Relation to the Whole Topic C: Comparing Unit Fractions and Specifying the Whole Topic D: Fractions on the Number Line Topic E: Equivalent Fractions Topic F: Comparison, Size and Order Topic A opens this unit with students actively partitioning different models of wholes into equal parts (e.g., concrete models, fraction strips, and drawn pictorial area models on paper). They identify and count equal parts as 1 half, 1 fourth, 1 third, 1 sixth, and 1 eighth in unit form before an introduction to the unit fraction 1/b (3.NF.1). In Topic B, students compare unit fractions and learn to build non-unit fractions with unit fractions as basic building blocks (3.NF.3d). This parallels the understanding that the number 1 is the basic building block of whole numbers. In Topic C, students practice comparing unit fractions with fraction strips, specifying the whole and labeling fractions in relation to the number of equal parts in that whole (3.NF.3d). Students transfer their work to the number line in Topic D. They begin by using the interval from 0 to 1 as the whole. Continuing beyond the first interval, they partition, place, count, and compare fractions on the number line (3.NF.2a, 3.NF.2b, 3.NF.3d). In Topic E, they notice that some fractions with different units are placed at the exact same point on the number line, and therefore are equal (3.NF.3a). For example, 1/2, 2/4, 3/6, and 4/8 are equivalent fractions (3.NF.3b). Students recognize that whole numbers can be written as fractions, as exemplified on the number lines below (3.NF.3c). Topic F concludes the module with comparing fractions that have the same numerator. As they compare fractions by reasoning about their size, students understand that fractions with the same numerator and a larger denominator are actually smaller pieces of the whole (3.NF.3d). ESSENTIAL QUESTIONS • Why is it important to compare fractions as representations of equal parts of a whole or of a set? • How can I represent fractions in multiple ways? • Why is it important to understand and be able to use equivalent fractions in mathematics or real life? • How will my understanding of whole number factors help me understand and communicate equivalent fractions? • How are different fractions compared? ENDURING UNDERSTANDINGS Students will understand that: • Fractions are numbers • A fraction describes the division of a whole (region, set, segment) into equal parts • Fractions can be used to represent numbers equal to, less than, or greater than 1 • Fractional parts are relative to the size of the whole or the size of the set • The more fractional parts used to make a whole, the smaller the parts SKILLS Students will be able to: • Describe the difference between an equal part and a whole • Identify a numerator and a denominator • Solve story problems with fractions using a strategy/tool (number line, drawing, etc.) • Determine equivalent fractions • Illustrate whether one fraction is <, = or > than another • Place or locate fractions on a number line Common Core Standards CCSS.Math.Content.3.NF.A.1 Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b. CCSS.Math.Content.3.NF.A.2 Understand a fraction as a number on the number line; represent fractions on a number line diagram. CCSS.Math.Content.3.NF.A.2.a Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that each part has size 1/b and that the endpoint of the part based at 0 locates the number 1/b on the number line. CCSS.Math.Content.3.NF.A.2.b Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line. Common Core Standards (cont.) CCSS.Math.Content.3.NF.A.3 Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size. CCSS.Math.Content.3.NF.A.3.a Understand two fractions as equivalent (equal) if they are the same size, or the same point on a number line. CCSS.Math.Content.3.NF.A.3.b Recognize and generate simple equivalent fractions, e.g., 1/2 = 2/4, 4/6 = 2/3. Explain why the fractions are equivalent, e.g., by using a visual fraction model. CCSS.Math.Content.3.NF.A.3.c Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Examples: Express 3 in the form 3 = 3/1; recognize that 6/1 = 6; locate 4/4 and 1 at the same point of a number line diagram. CCSS.Math.Content.3.NF.A.3.d Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model. Focus Standards for Mathematical Practice Reason abstractly and quantitatively. Students represent fractions concretely, pictorially, and abstractly and move back and forth between representations. Students also represent word problems involving fractions pictorially and then express the answer in the context of the problem. Construct viable arguments and critique the reasoning of others. Students reason about the area of a shaded region to decide what fraction of the whole it represents. Attend to precision. Students specify the whole amount when referring to a unit fraction and explain what is meant by equal parts in their own words. Look for and make use of structure. Students understand and use the unit fraction as the basic building block or structure of all fractions on the number line. Terminology New or Recently Introduced Terms • Unit fraction (fractions with numerator 1) • Non-unit fraction (fractions with numerators other than 1) • Fractional unit (half, third, fourth, etc.) • Equal parts (parts with equal measurements) • Unit interval (the interval from 0 to 1, measured by length) • Equivalent fraction (2 fractions that name the same size) • Copies (refers to the number of unit fractions in 1 whole) Familiar Terms and Symbols • Number line • Arrays • Halves, thirds, fourths, sixths, eighths (1/2, 1/3, 1/4, 1/6, 1/8) • Half of, one third of, one fourth of, etc. (1/2, 1/3, 1/4, 1/6, 1/8) • =, <, > (equal, less than, greater than) • Equal shares (pieces of a whole that are the same size) • Whole (e.g., 2 halves, 3 thirds, etc.) • Fraction (e.g., 1/3, 2/3, 3/3, 4/3) • Partition (divide a whole into equal parts) Unit Assessments Formative Assessments: Exit Slips or Post-It Notes Fraction Man Fraction Bingo Differentiated Graphic Organizers Classification Fraction Poster Observing Students in Groups Create and answer each other’s fraction word problems Write your own recipe Cooking Class Quests about each complete Topic Summative Assessments: Choice Final Assessment for unit: 1. Create poster, storyboard or comic strip about where we see and how we use fractions in real world 2. Write a creative, imaginary recipe using fractions 3. Create a mini storybook like Gator Pie by Louise Mathews, incorporating your knowledge of fractions Objectives: Lesson Plan: Candy Fractions and Classification Students will: • create fractions, understanding that the part becomes the numerator and the whole becomes the denominator • distinguish differences within one set and sort them into different groups • identify various fractions based on their own classifications Scientific Skills: Students will: • classify objects into categories • observe the teacher and their partner creating fractions with objects • communicate their findings to each other through written work and illustrations Materials • • • • • • • • • • • • Differentiated graphic organizers Hershey bars M&Ms (peanut and regular) Ziplock bags Rolls of assorted Life Savers Decks of cards Poster Paper Coloring Supplies Tape Pencils Post-its Index Cards Procedure Beginning Review: Determine students’ knowledge of the terms numerator and denominator and the part/whole relationship. Review the term equal and why the fractions are divided into equal groups. Give each student an index card with one fraction and have them label numerator and denominator. Students will keep index card on their desk as a reference during activity. Anticipatory Set: • Show the students the chocolate candy bars, each equal in size and weight. • Discuss sharing the candy with a friend. Break one candy bar into random sized pieces and the other into equal sections. • Ask if the pieces of the randomly broken candy bar are equal. Would it be fair to give unequal pieces to your friend? • Discuss how fractions are an easy way to break a whole into equal pieces so that everyone gets a fair share. • Break the Hershey Bar into each of its 12 little equal parts. Explain that since we have 12 equal parts, our denominator will be 12. • What will be our fraction to express one whole if the denominator is 12? Mini Lesson: • Put students in small groups of three. Each group will now get one roll of Life Savers and one Hershey Bar. • Students will unwrap the items while the teacher hands out differentiated graphic organizers. • Students will count the total number of parts in each “whole” item (Hershey bar and Life Savers). Teacher will guide students through graphic organizer. • Students will record their findings. Example questions on graphic organizer: 1. 2. 3. 4. 5. 6. 7. Break the Hershey bar into equal parts. How many pieces are there? ______ This number will be our denominator. What is the fraction for one whole? ______ Take two pieces of chocolate away from the group. Write a fraction for the remaining parts:_____ Separate the chocolate pieces into two equal groups. Both groups should have the same amount of chocolate. Write the fraction for each group: ______ How many lifesavers are there? _____ This number will be our denominator. Take away all the red ones. How many are left? _____ What is the fraction for the remaining lifesavers?____ Take away everything but the red ones. Write the fraction for the red lifesavers? Guided Practice: • Hand out ziplock bags that contain mixed M&Ms (peanut and regular). • Have students examine the M&Ms. • Ask them in what ways we can sort the M&Ms (color, peanut vs. regular, broken/disfigured). • Students will sort their M&Ms. They will then create fractions for the various sorted groups. Independent Practice: • Break up students into pairs and hand out deck of cards. • Students will continue the previous process of sorting by different attributes with the deck of cards. • Encourage students to try sorting by colors, suits or numbers. • Pairs will write their fractions on their graphic organizer. Assessment: • In pairs, students will make a poster, depicting ways they created fractions with all materials today. Students can draw, use symbols or write sentences to demonstrate what they did. Students must include fractions. Ending Review: • Collect graphic organizers. Tape student posters around the room. • Students will conduct a gallery walk and look at each other’s work. As they walk around the room, students must write down every fraction they see written on a poster on a large post-it that they will carry. • When they have looped around the room, students stick their post-It on the “What Stuck With Me Today!” board. • Group will come together to discuss how the class seemed to classify their deck of cards. • Encourage students to bring in items that can be broken into fractions to share with the class tomorrow. Differentiation • Group students in high/low performing groups • Teachers and TAs walk around the room, assisting students or groups that need help • Scaffolded instruction; sorting with Life Savers as a whole class, M&Ms with small groups and a deck of cards in pairs • Lesson is visual and kinesthetic • Graphic organizers are differentiated and keep students organized • Assessment is visual and kinesthetic. It is also open to choice and creativity in terms of how they can present their findings with words and illustrations. • Gallery Walk is visual and kinesthetic • Writing down the fractions on post-its enforces repetition and also helps solidify concept FIELD TRIP “Celebrations in the Kitchen” • 30 students in 3rd grade and 6 adults • Each group will make their own batch of cookies. The cookie recipe here is nut and dairy free and all products are kosher. • Students will work with fractions as they complete each step in the recipe. • Students will determine where fractions are listed on tools such as a measuring cup, in order to pour the correct amount of liquid that the recipe calls for. • Students will also be introduced to the concept of adding fractions. If we add ¼ cup of sugar plus ¼ cup of chocolate chips, we just added (2) ¼ cup of ingredients. When this happens you simply say 2/4. Before we start, ask the students to predict what will happen to the outcome of my product, or cookies, if I don’t follow the directions and measure each fraction correctly. For example, what if I put in 2 cups of almond milk instead of ½ cup? What will happen? The teachers will also make a batch where they do not follow the directions, so that the students can see what happens to the cookies. Integration With Other Topics Science - Students will use measuring tools such as rulers, beakers, measuring cups and spoons in order to measure fractions. Besides cooking class field trip, students can measure different objects in ½, ¼, 1/8, 1/16 inches. Children’s Literature – I will use the book Gator Pie by Louise Mathews for one of the lessons. The book is very simple, which is good when learning a new concept. The two main characters are alligators that find a giant pie and decide to split it evenly between the two of them. As more gators gather, the two main character need to figure out how many equal pieces they need to cut. The book mentions fractions ½, 1/3, ¼, 1/8, and 1/100. Social Studies – I will use the idea of fractions in the context of Social Studies by discussing wholes that have a set number of parts such as the 7 continents, the 13 colonies or the 50 states. Is each continent 1/7 of all the continents? Why or why not? Students need to determine that because they are not equal parts in size that these cannot be measured as equal fractions or equal parts of the whole. Resources • Gator Pie by Louise Mathews • Pizza Fractions website: http://www.bgfl.org/bgfl/custom/resources_ftp/client_ftp/ks2/maths/fractio ns/index.htm • “Celebrations in the Kitchen” - 63 E. Old Country Rd, Hicksville, N.Y. 11801 http://celebrationsinthekitchen.com/ • CommonCore App • Johnney’s Math Page website: http://jmathpage.com/ • Khan Academy: khanacademy.org • Engageny.com • Superteacherworksheets.com