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3rd Grade Mathematics Unit #4: Exploring Fractional Value and Equivalence Pacing: 8 Weeks Unit Overview This unit is designed to build a true “fraction sense,” in much the same way we build number sense in the younger grades. All too often, students struggle with fractions because they try to treat them like whole numbers and have simply memorized algorithms or rules that are devoid of conceptual understanding. Students need multiple opportunities to explore and make sense of fractional values before they can begin working with them. Therefore a variety of physical and visual models must be incorporated throughout instruction, and the emphasis that fractions are, indeed, numbers must continually be reinforced through the use of a number line. This conceptual foundation begins with unit fractions (with a numerator of 1). In this unit, students will explore what happens to the size of each part when the denominator of the unit fraction increases. Then they will use unit fractions to compose and decompose non-unit fractions. Students will develop an understanding of the values of non-unit fractions by recognizing the relationship between the numerator and denominator, and they will come to understand that fractions with different numerators and denominators can actually have the same value. They will develop a language of equivalence and work with a variety of models (area models, linear models, etc) to represent and create equivalent relationships. Finally, students will continue to draw upon the part:whole relationship between the numerator and denominator to compare non-unit fractions by reasoning about the size or number of parts. At the end of the unit, students will apply their knowledge of fractions to represent and interpret fractional data on a line plot. Prerequisite Skills Fraction Unit fraction Fracitonal unit Quantity Value Part Whole Partition Precise/precision Halves Fourths Thirds Sixths • Decompose whole numbers into various sums Eighths • Create and interpret line plots with whole number Numerator data Denominator • Partition circles and rectangles into two, three, or four equal shares. • Understand how to use number lines that represent whole numbers. • Understand that a fraction is a whole divided into some number of equal parts. • Understand how to use numbers line to count and identify whole numbers. • Compare whole numbers, with or without number lines, using <, >, and =. • Identify fractions represented by models. 1 | P a g e Vocabulary Fraction tiles Fraction strips Plot Label Identify Represent Compose Decompose Equivalent Greater than Less than Number bond Equal/fair share Mathematical Practices MP.1: Make sense of problems and persevere in solving them MP.2: Reason abstractly and quantitatively MP.3: Construct viable arguments and critique the reasoning of others MP.4: Model with mathematics MP.5: Use appropriate tools strategically MP.6: Attend to precision MP.7: Look for and make use of structure MP.8: Look for and express regularity in repeated reasoning Common Core State Standards Progression of Skills nd 2 Grade Supporting Standards (20%) N/A 3.MD.4: Fractional Line Plots 3.G.2: Partition Shapes into Equal Areas Major Standards (70%) N/A 3.NF.1: Understand Unit Fractions 3.NF.2: Fractions on a Number Line N/A 3.NF.2a: Unit Fractions on a Number Line 3.NF.2b: Non-Unit Fractions on a Number Line 3rd Grade 4th Grade 3.NF.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. 3.NF.2: Understand a fraction as a number on the number line; represent fractions on a number line diagram. 3.NF.3: Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size. 4.NF.3: Understand a fraction a/b with a > 1 as a sum of fractions 1/b. 3.NF.3: Explain Equivalence of Fractions 3.NF.3a: Understand Equivalent Fractions 3.NF.3b: Explain, Recognize, and Generate Equivalent Fractions 3.NF.3c: Express Whole Numbers as Fractions 3.NF.3d: Compare Fractions 3.NF.3d: Compare two fractions with the same numerator or the same denominator by reasoning about their size. 3.MD.9: Generate measurement data by measuring lengths of several objects to the nearest whole unit, or by making repeated measurements of the same object. Show the measurements by making a line plot, where the horizontal scale is marked off in whole-number units. 2.G.3: Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, third of, etc., and describe the whole as two halves, three thirds, etc. 3.MD.4: Generate measurement data by measuring lengths using rulers marked with halves and fourths of an inch. Show the data by making a line plot, where the horizontal scale is marked off in appropriate units— whole numbers, halves, or quarters. 3.MD.4: Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Solve problems involving addition and subtraction of fractions by using information presented in line plots. 3.G.2: Partition shapes into parts with equal areas. Express the area of each part as a unit fraction N/A Standard 3.NF.2 should serve as opportunity for in-depth focus: future work with the number system. It is critical that students at this grade are able to place fractions on a number line diagram and understand them as a related component of their ever-expanding number system.” The key advance in fraction concepts between third and fourth grade is: “Fraction equivalence is an important theme within the standards that begins in grade 3. In grade 4, students extend their understanding of fraction equivalence to the general case, a/b = (n x a)/(n x b) (3.NF.3à4.NF.1). They apply this understanding to compare fractions in the general case (3.NF.3d à 4.NF.2).” 2 | P a g e 4.NF.1: Explain why a fraction a/b is equivalent to a fraction (n x a) / (n x b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. 4.NF.2: Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction N/A According to the PARCC Model Content Framework, “Developing an understanding of fractions as numbers is essential for N/A Big Ideas • Fractions are numbers with special names that tell how many parts of that size are needed to make the whole, written in the form a/b (when b is not zero). For example, thirds require three parts to make one whole; one part is one-third. • Unit fractions can be composed to represent a larger share of a whole. • A fraction’s value means nothing without specifying the whole. A fraction only tells us about the relationship between the parts and the whole. • The more equal sized parts that form a whole, the smaller the size of each part. The fewer equal sized parts that form a whole, the larger the size of each part • Two fractions are equivalent when they have the same value. Every fraction is equivalent to an infinite number of other fractions 3 | P a g e Students Will… Know/Understand… Be Able To… • Partitioning a shape means creating equal parts/groups • In a fraction, the numerator tells the number of equal parts being represented and the denominator tells the total number of equal parts. • On number lines, fractions lie in between consecutive whole numbers. Fractions may also represent whole numbers. • A fraction may be greater than, equal to, or less than 1. • Fraction 1/b is one portion of b equal parts, and is called a unit fraction. • The combined value of 1/b fractions equal the value of a/b. For example, 1/b + 1/b = a/b. • A fraction is a number that can be represented on a number line. • Two fractions can have the same value, but with different numerators and denominators. • Fractions that name the same part of a whole are equivalent. • Equivalent fractions are at the same point on a number line. • A larger denominator means a higher number of total parts, which means each unit part is smaller than they are when the denominator or total number of parts is lower. • Use models to represent unit fraction where each part is 1/a the area of the whole. • Partition a shape into equal and fractional parts. • Partition a number line into equal and fractional parts. • Recognize a fraction represented pictorially, symbolically, and verbally. • Use models and number lines to represent nonunit fractions. • Use the symbols >, =, and < to compare fractions. • Decompose whole numbers, mixed numbers and non-unit fractions into their sum of unit fractions. • Combine 1/b correctly to create a/b. • Model fractions greater than 1 using illustrations, concrete objects and written expression. • Compare fractions using illustrations, concrete objects, symbols and number lines. • Recognize simple equivalent fractions, looking at fractions with denominators of 2, 3, 4, 6, or 8. • Create simple equivalent fractions, looking at fractions with denominators of 2, 3, 4, 6, or 8. • Express whole numbers as fractions and recognize fractions that are equivalent to whole numbers. • Compare two fractions with the same numerator or denominator by reasoning about their size/value. • Model equivalent fractions using visual models and number lines. Unit Sequence Student Friendly Objective SWBAT… 1 Use concrete models to partition wholes into equal parts and to identify fractional units. Key Points/ Teaching Tips This should be a hands-on lesson in which students are able to apply their measurement, multiplication, and division skills to partitioning wholes. From the beginning, students must attend to precision when partitioning/dividing a whole into equal parts. They should check their work each time by counting the equal parts they created (not the lines they drew). In order to develop language skills around fractions, the first three lessons of this unit require students to name fractions using a numeral to show the number of equal parts represented and a word to show the fractional unit (i.e. 1 third). Students may benefit from a word wall for spelling. Students should become familiar with the pattern that it takes n-1 lines to draw n equal parts (i.e. it takes 2 lines to draw 3 equal parts; it takes 4 lines to draw 5 equal parts). They may also notice that as the whole is partitioned into more equal parts, the size of the unit fraction gets smaller. (This will be revisited later in the unit.) Exit Ticket Adapted from EngageNY Module 5, Lesson 1 Exit Ticket: 1. Name the fraction that is shaded: 2. A plumber has 12 inches of pipe. He cuts it into pieces that are each 3 inches in length. What fraction of the pipe would one piece represent? (Use your yellow strip from the lesson to help you.) 3. Estimate to partition the rectangle into: a. Thirds: b. Fourths: c. What pattern do you notice about the number of lines required to partition a rectangle into a given number of equal parts? 4 | P a g e Instructional Resources Engage NY Module 5, Lesson 1 (Appendix C) 2 Use fraction strips to identify and count fractional units and to model real-world situations. Students will create fraction strips while attending to patterns and strategies for folding equal parts. Student should understand that halves and thirds are useful tools for drawing fourths, eighths, and sixths. Sample explanation: A whole partitioned into fourths has two times as many equal parts as a whole partitioned into halves. Students should be able to demonstrate this visually. When counting equal parts, students should be encouraged to say the name of the fractional unit each time (i.e. “1 fourth, 2 fourths, 3 fourths” instead of “1, 2, 3 fourths”). 5 | P a g e Adapted from EngageNY Module 5, Lesson 2 Exit Ticket: 1. Fill in the blanks: There are ____ equal parts in all. ____ are shaded. 2. Michael bakes a piece of garlic bread for dinner. He shares it equally with his three sisters. Show how Michael and his three sisters can each get an equal share of the garlic bread. 3. Use the relationship between halves, fourths, and eighths to describe a strategy for drawing equal eighths. You may use the three fraction strips drawn below to help support your explanation. Engage NY Module 5, Lesson 2 (Appendix C) 3 Use pictorial area models to partition wholes into equal parts with precision. Students should recognize that wholes must be partitioned into equal parts in order to be identified as fractional units. 1. Each shape is 1 whole. Estimate to equally partition the following images to show the given fractional unit: a. Halves Sample PARCC EOY assessment question: Which shapes have parts that are 1/8 area of their whole shape? Drag and drop the three correct shapes into the box. b. Fourths c. Thirds 2. Which shapes are partitioned into eighths? Circle the three correct shapes below: 3. Two of the shapes above are not partitioned into eighths. Explain why. 6 | P a g e Engage NY Module 5, Lesson 3 (Appendix C) 4 Use fraction notation to label unit fractions. Students should become familiar with the following vocabulary: “unit fraction,” “numerator,” and “denominator.” Requiring students to label each equal part with the unit fraction will support their conceptual understanding of the iteration of fractional units and the value of each part. Sample PARCC EOY assessment question (answer already completed): Use the More or Fewer buttons as many times as needed to divide the circle into 6 equal parts. Then shade 1/6 of the area of the circle. Divide the figure into the correct number of equal parts by using the More and Fewer buttons. Then shade by selecting the part or parts. Adapted from EngageNY Module 5, Lesson 5 Exit Ticket: 1. Use the shape below to fill in the chart: My Math Chapter 10, Lesson 1 Engage NY Module 5, Lesson 5 (Appendix C) Total Number of Equal Parts Total Number of Equal Parts Shaded Unit Form Fraction 2. Using the two rectangles below, partition one into 5 equal parts. Partition the other into 8 equal parts. Label the unit fractions and shade 1 equal part in each rectangle. 3. What is a unit fraction? How are all unit fractions alike? How are they different? 5 Use fraction strips and <, >, or = to compare unit fractions by reasoning about their size. By the end of this lesson, students should be able to articulate that as the number of equal parts in the whole increases, the size of the fractional unit decreases. Students should begin to understand that the greater the denominator (the more number of equal parts a whole is partitioned into), the smaller the size of the unit fraction. This will be revisited once more explicitly in this unit. 7 | P a g e 1. Camille and Peter were reading a book. Camille read 1/4 of the book, while Peter read 1/3 of the same book. Who read a greater part of the book? Explain. 2. Write <, >, or = in the blanks below to compare the fractions: 1 eighth _____ 1 tenth 1/7 ______ 1/6 3. Explain how the size of the denominator affects the size of the unit fraction. Engage NY Module 5, Lesson 10 (Appendix C) 6 Compare unit fractions with different sized models representing the whole. This objective provides the opportunity for teachers to present students with a real-world situation as an engaging, inquiry-based hook. 1. Is ¼ of the smaller waffle equal to ¼ of the larger waffle? Explain. Engage NY Module 5, Lesson 11 (Appendix C) Students must be able to articulate that the size of the fraction is relative to (depends on) the size of the whole. To compare fractions, the wholes should be the same size. 7 Build non-unit fractions from unit fractions. Students should continue to familiarize themselves with the vocabulary of “numerator” and “denominator.” They should be able to write a fraction in word or numeral form. Students must be able to distinguish between the unit fraction and the fraction that the question is asking about (i.e. the fraction shaded). 2. Tatiana ate ½ of a small carrot. Louis ate ¼ of a large carrot. Who ate more carrot? Use words and pictures to explain your answer. 3. How can fractions be compared? 4. Adapted from EngageNY Module 5, Lesson 6 Exit Ticket: 1. Estimate to equally partition the strip and shade the answer. Write the unit fraction inside each shaded unit: 2 fifths = 2. Complete the chart: 3. Carrie drew this design on a rectangular piece of paper: Marianne drew this shape on a rectangular piece of paper of the same size: Marianne says the unit fraction in her shape is bigger. Is she correct? Why or why not? 8 | P a g e Engage NY Module 5, Lesson 6 (Appendix C) 8 Use fraction notation to describe visual models. This lesson is designed to provide students with additional practice manipulating fraction notation and applying it in real-world contexts. Sample PARCC assessment question: 1. The picture below shows Mark’s flower garden. What number should replace the question mark in the fraction to show the part of Mark’s garden that is covered with flowers? 6/? Sample PARCC EOY assessment question: 2. A flower garden is divided into equal parts. The color of the flowers planted in each part of the garden is shown. Circle three statements that are true: a. There are red or yellow flowers in 1/6 of the garden. b. Purple flowers are planted in 7/8 of the garden. c. Pink flowers are planted in 1/8 of the garden. d. Each part of the garden is 1/8 of the whole garden. e. There are yellow flowers in 3/6 of the garden. f. Red flowers are planted in 3/8 of the garden. 3. Choose one statement from #2 that was not true and rewrite the fraction to make it true. 9 | P a g e My Math Chapter 10, Lesson 2 9 Use visual models and <, >, or = to compare non-unit fractions with the same numerator. In this lesson, students should extend their reasoning about the relative sizes of unit fractions to non-unit fractions with the same numerators, but different denominators. They should be able to apply this understanding to simple word problems. Adapted from EngageNY Module 5, Lessons 28 & 29 Exit Tickets: 1. Shade the models to compare the following fractions. Circle the smaller fraction. 2. Jess and Sam made matching pies for a party. Jess saved 3/4 of her pie and Sam saved 3/8 of her pie. Who saved the most pie? Draw a picture to support your answer. 3. For either #1 or #2, use words to explain your answer. 10 11 Engage NY Module 5, Lessons 28 & 29 (Appendix C) *Note: do not use the number lines for this lesson Flex Day (Instruction Based on Data) Recommended Resources: Identify and represent shaded and non-shaded parts of one whole as fractions. Identify the number of given fractional units in one whole. EngageNY Module 5, Lesson 4 (Appendix C) “Fraction Barrier Game & Grid” (Appendix C) “Exploring Fraction Kits” (Appendix C) “Equal Parts on the Geoboard” (Appendix C) “Fractions with Color Tiles” (Appendix C) “Strategies for Comparing Fractions” (Appendix C) “Who Ate More?” (Appendix C) Students should understand that even though parts Adapted from EngageNY Module 5, Lesson 7 Exit may not be shaded (or in real life, for example, may Ticket: not be full), they are often still considered part of 1. There are _____ sixths in one whole. the whole. The EngageNY Module 5, Lesson 7 2. The fraction strip is 1 whole. Write Application Problem provides a real-life example. fractions to label the shaded and unshaded parts. Students may benefit from using fraction tiles and visual models to express how many fractional units it takes to create one whole. 3. Justin mows part of his lawn. Then the lawnmower runs out of gas. He has not mowed 9/10 of his lawn. What part of his lawn is mowed? 10 | P a g e Engage NY Module 5, Lesson 7 (Appendix C) 12 Use number bonds to decompose one whole into unit and non-unit fractions. In addition to abstract models, students should practice drawing pictures and number bonds to represent real-world situations. 1. Draw a number bond that shows the shaded and the unshaded parts of the shape below. Then show each part decomposed into unit fractions. Engage NY Module 5, Lesson 8 (Appendix C) 2. Complete the number bond. Draw a shape that has shaded and unshaded parts that match the completed number bond. 13 Given a fractional part, students will construct the whole. Students may benefit from the use of tracing paper for this lesson. This lesson should help students develop the conceptual understanding how unit fractions are iterated to construct a whole. Students should practice a few problems from the EngageNY Homework before the Exit Ticket, so they have been exposed to those question types. 11 | P a g e 3. Show each part of your number bond in #2 decomposed into unit fractions. Use words to explain what your number bond shows about the whole shape you created. Adapted from the EngageNY Module 5, Lesson 12 Exit Ticket & Homework: 1. The shape below represents the unit fraction. Draw a possible picture representing 1 whole. 2. The shape below represents 1/4. Estimate to draw the corresponding whole, label the unit fractions, then write a number bond that matches the drawing. Engage NY Module 5, Lesson 12 (Appendix C)