3rd Grade MathematicsUnit 5: Fractions

Teacher Resource Guide

2012 - 2013

In Grade 3, instructional time should focus on four critical areas:

1. Developing understanding of multiplication and division and strategies for multiplication and division within 100;Students develop an understanding of the meanings of multiplication and division of whole numbers through activities and problems involving equal-sized groups, arrays, and area models; multiplication is finding an unknown product, and division is finding an unknown factor in these situations.

2. Developing understanding of fractions, especially unit fractions (fractions with a numerator of 1); Students are able to use fractions to represent numbers equal to, less than, and greater than one. They solve problems that involve comparing fractions by using visual fraction models and strategies based on noticing equal numerators or denominators.

3. Developing understanding of the structure of rectangular arrays and of area;Students understand that rectangular arrays can be decomposed into identical rows or into identical columns. By decomposing rectangles into rectangular arrays of squares, students connect area to multiplication, and justify using multiplication to determine the area of a rectangle.

4. Describing and analyzing two-dimensional shapes;Students compare and classify shapes by their sides and angles, and connect these with definitions of shapes. They also relate their fraction work to geometry by expressing the area of part of a shape as a unit fraction of the whole.

3rd Grade 2012-2013 Page 1

Unit Time Frame Test ByTR

IMES

TER

1 1: Addition and Subtraction (Within 1,000)

7 weeks 8/27 – 10/12 October 12

2: Multiplication and Division: Models within 100 5 weeks 10/15 – 11/16 November 16

TRIM

ESTE

R 2 3: Geometry/Measurement 4 weeks 11/19-12/21 December 21

4: Multiplication and Division: Properties within 100 5 weeks 1/2 – 2/8 February 8

5: Fractions 8 weeks 2/11 – 4/12 April 12

TRIM

ESTE

R 3

6: Multiplication and Division: Application & Fluency within 100 7 weeks 4/15 – 5/30 May 30

3rd Grade Mathematics 2012 - 2013

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Unit 5: Fractions February 11-April 12 (8 weeks)

Big Ideas Essential QuestionsAll fractions are built from a unit fraction. For example : ¾ = ¼ + ¼ + ¼ How are all fractions related to a unit fraction?The denominator tells us the number of equal-sized parts in one whole. It tells what is being counted. The numerator tells us how many of the equal-sized parts we have. It is the counting number.

What do the numerator and the denominator in a fraction describe?

Identifier Standards Mathematical Practices

STAN

DARD

S

3.NF.1

3.G.2

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.

Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole.

1) Make sense of problems and persevere in solving them.

2) Reason abstractly and quantitatively.

3) Construct viable arguments and critique the reasoning of others.

4) Model with mathematics.

5) Use appropriate tools strategically.

6) Attend to precision.

7) Look for and make use of structure.

8) Look for and express regularity in repeated reasoning.

3.NF.2b

3.NF.2a

3.MD.4

Understand a fraction as a number on the number line; represent fractions on a number line diagram.

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.

Understand a fraction as a number on the number line; represent fractions on a number line diagram.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.

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.NF.3d

3.NF.3

Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.

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.

Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.a. Understand two fractions as equivalent (equal) if they are the same size, or the same point on a

number line.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.c. Express whole numbers as fractions, and recognize fractions that are equivalent to whole

numbers.

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Unit 5: Fractions February 11-April 12 (8 weeks)

Identifier Standards Bloom’s Skills ConceptsST

ANDA

RDS

3.NF.1

3.G.2

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.

Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole.

Understand (2) Understand(fraction 1/b and a/b as formed by parts the size of 1/b)

3.NF.2b

3.NF.2a

3.MD.4

Understand a fraction as a number on the number line; represent fractions on a number line diagram.

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.

Understand a fraction as a number on the number line; represent fractions on a number line diagram.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.

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.

Understand (2) Represent (fraction on a number line)

3.NF.3d

3.NF.3

Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.

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.

Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.a. Understand two fractions as equivalent (equal) if they are the same size, or the same point on a

number line.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.c. Express whole numbers as fractions, and recognize fractions that are equivalent to whole

numbers.

Analyze (4)

Remember (1)

Remember (1)

Evaluate (5)

Compare (two fractions w/ same numerator or same denominator)

Recognize(comparisons only valid w/ same sized whole)

Record (comparisons w/ symbols)

Justify (comparisons)

numeratordenominatorgreater than >less than <equal =

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Unit 5: Fractions February 11-April 12 (8 weeks)

Instructional Strategies for ALL STUDENTSReal-world context – For students to reach the level of rigor intended for the new Iowa Core, they must develop understanding of fraction concepts within real-world contexts. Therefore, it is essential to present students with problems within real-world contexts as much as possible.

Sharing and the Concept of Fractional Parts - The first goal in the development of fractions should be to help children construct the idea of fractional parts of the whole – the parts that result when the whole or unit has been partitioned into equal-sized portions or fair shares. Children seem to understand the idea of separating a quantity into two or more parts to be shared fairly among friends. They eventually make connections between the idea of fair shares and fractional parts. Sharing tasks are, therefore, good places to begin the development of fractions. For example, 4 children are sharing 3 cookies. How much will each child get? Problems of this type can be solved as early as kindergarten! In the section, Sharing and the Concept of Fractional Parts from Teaching Student-Centered Mathematics 3-5, Van de Walle & Lovin, Pearson, 2006, p. 132-133 show several of these sharing problems for beginning fraction concepts. These activities are listed below in the Lesson Bank. The pages are also included on the Wiki.

Models for Fractions – There is substantial evidence to suggest that the use of models in fraction tasks is important. Students should represent fraction concepts with physical materials and drawings in many different ways. Not only should students use these models, but they should also explore fractional concepts with a wide variety of models so that fractions don’t simply become “pie pieces.” Some fraction models may challenge students to be more reflective about the fraction concepts than others. It is important to note that the new Iowa Core places a much heavier emphasis on number line models. This is consistent with the idea that students should represent fractions with several different models. The number line is not a model that has been typically used in the past, but it has strong connections to operations with fractions in later grades. Below are examples of the three different types of fraction models: region or area models, length or measurement models, and set models. They are shown to emphasize the variety of models within each type and that students need experience with the variety that each type of model has to offer as well as each of the different types of models.

Region/Area Models

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Unit 5: Fractions February 11-April 12 (8 weeks)

Length/Measurement Models Set Models

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Unit 5: Fractions February 11-April 12 (8 weeks)

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Unit 5: Fractions February 11-April 12 (8 weeks)

Routines/Meaningful Distributed PracticeDistributed Practice that is Meaningful and PurposefulPractice is essential to learn mathematics. However, to be effective in improving student achievement, practice must be meaningful, purposeful, and distributed.

Meaningful: Builds on and extends understanding Purposeful: Links to curriculum goals and targets an identified need based on multiple data sources Distributed: Consists of short periods of systematic practice distributed over a long period of time

Routines are an excellent way to achieve the mandate of Meaningful Distributed Practice outlined in the Iowa Core Curriculum. The skills presented during routines do not necessarily reinforce the lesson concept for that day. Routines may be used to address a need for small increments of exposure to a skill or review of skills already taught. Routine activities may be repeated several days in a row, allowing for a build-up of conceptual understanding, or can be visited and re-visited over a period of time. Routines can be inserted as the schedule allows; in short intervals throughout the day or as a lesson opener or closer. Selection of the routine should be made based on informal teacher observation and formative assessments.

Skill StandardAddition and Subtraction 3.NBT.2Multiplication and Division Facts 3.OA.7Graphing 3.MD.3One-step word problems involving masses or volumes 3.MD.2Other skills students need to develop based on teacher observation and formative assessments.

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Unit 5: Fractions February 11-April 12 (8 weeks)

Lesson Progression

Lessons Teacher DirectionsTeacher’s

EditionPages

StandardsAddressed

Basic Fraction Concepts Introduce the unit with Expressions Lesson 1. Do Sharing Tasks, Correct Shares, and Finding Fair Shares activities for several days after the Expressions lesson. Each activity should be done multiple times and can be done simultaneously. See teacher notes for the activity on Wiki.

782

3.NF.1, 3.G.2

Math Expressions: Unit 11, Lesson 1, Activities 1-3

Wiki: Sharing Tasks

----Wiki: Correct Shares

Wiki: Finding Fair SharesCounting Fractional Parts (Numerator, Denominator) Combine Lesson 2 with the Fractional Parts Counting, and More,

Less, or Equal to One Whole activities. See teacher notes for the activities on the Wiki. Each activity should be done multiple times over several days. Be sure to include the number line as one of the models in the Fractional Parts Counting activity.Having students skip count by unit fractions, similar to skip counting by whole numbers is another helpful activity. Students can practice for short blocks of time during a transition. When skip counting, use improper fraction names instead of mixed fractions names.

792

3.NF.1, 3.NF.2, 3.G.2

Math Expressions: Unit 11, Lesson 2, Activities 1-2

Wiki: Fractional Parts Counting ----

Wiki: More, Less, or Equal to One Whole ----

Math Expressions: Unit 11, Lesson 18, Activity 2 916

Comparing and Equivalent Fractions The Expressions lessons involve making fraction strips and playing a game. Limit the fraction strips and game to halves, fourths, eighths, thirds, and sixths.The activities Choose, Explain, Test and Different Fillers should be done after the Expressions Lesson 9. They should be done multiple times over several days.The website activity and Expressions Lesson 10 should be done after these activities.

842

3.NF.1, 3.NF.2, 3.NF.3, 3.G.2

Math Expressions: Unit 11, Lesson 9, Activities 1-2Wiki: Choose, Explain, Test ----Wiki: Different Fillers ----Website:http://illuminations.nctm.org/LessonDetail.aspx?ID=L543Fun With Fractions: Investigating Equivalent Fractions

----

Math Expressions: Unit 11, Lesson 10 ----

Wiki: Measurement ActivityAfter measuring, have a class discussion and collect students’ measurements of various items. Display the data on a line plot.See teacher notes for the activity on the wiki.

---- 3.NF.1, 3.NF.2, 3.NF.3, 3.G.2, 3.MD.4

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Unit 5: Fractions February 11-April 12 (8 weeks)

Math Expressions Activities(use as centers, re-teaching/extension support, etc.)

Activity Standards Activity StandardsActivity Card 11-1, Intervention, On-Level, Challenge 3.NF.1, 3.G.2 Partner Games:

Make a Poundp. 44 (grade 3 kit)Activity Card 11-2, Intervention, On-Level 3.NF.1, 3.NF.2, 3.G.2

Activity Card 11-9, Intervention, On-Level, Challenge 3.NF.1, 3.NF.2, 3.NF.3, 3.G.2

Activity Card 11-10, Intervention, On-Level 3.NF.1, 3.NF.2, 3.NF.3, 3.G.2

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