Unit 4 Organizer:



|Grade 3 Mathematics Frameworks |

|Unit 4 |

|Fractions and Decimals |

Unit 4

FRACTIONS AND DECIMALS

(6 weeks)

TABLE OF CONTENTS

Overview 3

Key Standards & Related Standards 4

Enduring Understandings 6

Essential Questions 6

Concepts & Skills to Maintain 7

Selected Terms and Symbols 8

Classroom Routines 9

Strategies for Teaching and Learning 9

Evidence of Learning 10

Tasks 11

• Paper-Folding Fractions 12

• Pattern Block Fractions 18

• Eating Fractions 23

• A Bowl of Beans 28

• Pizza Party 35

• All in a Line 41

• Trash Can Basketball 46

• Sweet Decimal Fraction Bars 51

Culminating Task

• The Fraction Story Game 57

OVERVIEW

In this unit students will:

• Recognize that the numerator is the top number (term) of a fraction and that it represents the number of equal-sized parts of a set or whole; recognize that the denominator is the bottom number (term) of a fraction and that it represents the total number of equal-sized parts or the total number of objects of the set

• Explain the concept that the larger the denominator, the smaller the size of the piece

• Compare common fractions with like denominators and tell why one fraction is greater than, less than, or equal to the other

• Represent halves, thirds, fourths, sixths, eighths, tenths , and twelfths using various fraction models

• Recognize the first decimal place to the right of the decimal as the tenths place

• Add and subtract decimal fractions and common fractions with like denominators using physical models and/or pictures

Although the units in this instructional framework emphasize key standards and big ideas at specific times of the year, routine topics such as counting and ordering numbers, working with calendars and clocks, counting collections of coins, and patterning should be addressed on an ongoing basis through the use of a daily math meeting board, centers, and games. The first unit should establish these routines, allowing students to gradually understand the concept of number and to develop computational proficiency.

To assure that this unit is taught with the appropriate emphasis, depth, and rigor, it is important that the tasks listed under “Evidence of Learning” be reviewed early in the planning process. A variety of resources should be utilized to supplement, but not completely replace, the textbook. Textbooks not only provide much needed content information, but excellent learning activities as well. The tasks in these units illustrate the types of learning activities that should be utilized from a variety of sources.

STANDARDS ADDRESSED IN THIS UNIT

Mathematical standards are interwoven and should be addressed throughout the year in as many different units and activities as possible in order to emphasize the natural connections that exist among mathematical topics.

KEY STANDARDS

M3N1. Students will further develop their understanding of whole numbers and decimals and ways of representing them.

a. Identify place values from tenths through ten thousands.

b. Understand the relative sizes of digits in place value notation (10 times, 100 times, 1/10 of a single digit whole number) and ways to represent them including word name, standard form, and expanded form.

M3N5. Students will understand the meaning of decimal fractions and common fractions in simple cases and apply them in problem-solving situations.

a. Identify fractions that are decimal fractions and/or common fractions.

b. Understand that a decimal fraction (i.e. 3/10) can be written as a decimal (i.e. 0.3).

c. Understand the fraction a/b represents a equal sized parts of a whole that is divided into b equal sized parts.

d. Know and use decimal fractions and common fractions to represent the size of parts created by equal divisions of a whole.

e. Understand the concept of addition and subtraction of decimal fractions and common fractions with like denominators.

f. Model addition and subtraction of decimal fractions and common fractions with like denominators.

g. Use mental math and estimation strategies to add and subtract decimal fractions and common fractions with like denominators.

h. Solve problems involving decimal fractions and common fractions with like denominators.

RELATED STANDARDS

M3P1. Students will solve problems (using appropriate technology).

a. Build new mathematical knowledge through problem solving.

b. Solve problems that arise in mathematics and in other contexts.

c. Apply and adapt a variety of appropriate strategies to solve problems.

d. Monitor and reflect on the process of mathematical problem solving.

M3P2. Students will reason and evaluate mathematical arguments.

a. Recognize reasoning and proof as fundamental aspects of mathematics.

b. Make and investigate mathematical conjectures.

c. Develop and evaluate mathematical arguments and proofs.

d. Select and use various types of reasoning and methods of proof.

M3P3. Students will communicate mathematically.

a. Organize and consolidate their mathematical thinking through communication.

b. Communicate their mathematical thinking coherently and clearly to peers, teachers, and others.

c. Analyze and evaluate the mathematical thinking and strategies of others.

d. Use the language of mathematics to express mathematical ideas precisely.

M3P4. Students will make connections among mathematical ideas and to other disciplines.

a. Recognize and use connections among mathematical ideas.

b. Understand how mathematical ideas interconnect and build on one another to produce a coherent whole.

c. Recognize and apply mathematics in contexts outside of mathematics.

M3P5. Students will represent mathematics in multiple ways.

a. Create and use representations to organize, record, and communicate mathematical ideas.

b. Select, apply, and translate among mathematical representations to solve problems.

c. Use representations to model and interpret physical, social, and mathematical phenomena.

ENDURING UNDERSTANDINGS

• Fractional parts are equal shares of a whole or a whole set.

• The more equal sized pieces that form a whole, the smaller the pieces of the whole become.

• When the numerator and denominator are the same number, the fraction equals one whole.

• When the wholes are the same size, the smaller the denominator, the larger the pieces.

• The fraction name (half, third, etc) indicates the number of equal parts in the whole.

• A decimal point can be used in a number to name a part smaller than one whole.

• If a whole is divided into ten equal parts, the parts can be named with tenths.

ESSENTIAL QUESTIONS

• How can we be sure fractional parts are equal in size?

• What does each term in a fraction represent?

• How does the number of equal pieces affect the name of a fraction?

• What do I know about a fraction that has the same numerator and denominator?

• What are the important features of a unit fraction?

• Why is the size of the whole important?

• How can I write a fraction to represent a part of a group?

• How can I represent a fraction of a discrete model (a set)?

• How are multiplication, division, and fractions related?

• How can I be sure fractional parts are equal in size?

• What do the numbers (terms) in a fraction represent?

• How does the number of equal pieces affect the fraction name?

• How can I write a fraction to represent a part of a group?

• How are multiplication, division, and fractions related?

• Why does the denominator remain the same when I add fractions with like denominators?

• How do we add fractions with like denominators?

• Why is the number 10 important in our number system?

• How are tenths related to the whole?

• How are decimals and fractions related?

• Why is the number 10 important in our number system?

• How can I write a fraction to represent a part of a group?

• When we compare two fractions, how do we know which has a greater value?

• How can you use decimal fractions to solve addition and subtraction problems?

• What happens to the denominator when I add fractions with like denominators?

• How are decimal fractions and common fractions used in problem-solving situations?

CONCEPTS/SKILLS TO MAINTAIN

It is expected that students will have prior knowledge/experience related to the concepts and skills identified below. It may be necessary to pre-assess in order to determine if time needs to be spent on conceptual activities that help students develop a deeper understanding of these ideas.

• Number sense and place value from tenths through ten thousands (comparing, ordering, naming/renaming)

• Counting collections of coins and bill, making fair trades for coins and bills

• Fluency with basic facts: addition, subtraction; developing fluency with multiplication

SELECTED TERMS AND SYMBOLS

The following terms and symbols are often misunderstood. These concepts are not an inclusive list and should not be taught in isolation. However, due to evidence of frequent difficulty and misunderstanding associated with these concepts, instructors should pay particular attention to them and how their students are able to explain and apply them.

The definitions below are for teacher reference only and are not to be memorized by the students. Teachers should present these concepts to students with models and real life examples. Students should understand the concepts involved and be able to recognize and/or demonstrate them with words, models, pictures, or numbers.

The websites below are interactive and include a math glossary suitable for elementary children. It has activities to help students more fully understand and retain new vocabulary. (i.e. The definition for dice actually generates rolls of the dice and gives students an opportunity to add them.) Note – At the elementary level, different sources use different definitions. Please preview any website for alignment to the definitions given in the frameworks.





← Common fraction: A number used to name a part of a group or a whole containing a fraction bar, a numerator, and a denominator.

← Decimal fraction: A fraction whose denominator is a power of ten (i.e. 1/10, 5/10, 10/10).

← Denominator: The number (term) below the fraction bar that represents the number of equal fractional parts into which a whole has been divided.

← Equivalent sets: Sets containing an equal number of objects.

← Increment: the amount or degree by which something changes.

← Numerator: The number (term) above the fraction bar in a common fraction representing the number of equal parts of a whole or group under consideration.

← Term: The number in the numerator and denominator of a fraction.

← Unit Fraction: Any common fraction with a numerator of one.

← Whole number: Zero or any counting number.

CLASSROOM ROUTINES

The importance of continuing the established classroom routines cannot be overstated. Daily routines must include such obvious activities such as taking attendance and lunch count, creating and discussing daily graphs, and calendar activities at a math meeting board. They should also include less obvious routines, such as how to select materials, how to use materials in a productive manner, how to put materials away, and how to access classroom technology such as computers and calculators. An additional routine is to allow adequate time for children to explore new materials before attempting any directed activity with these materials. The regular use of routines is important to the development of students’ number sense, flexibility, and fluency, which will support students’ performances on the tasks in this unit and throughout the school year.

STRATEGIES FOR TEACHING AND LEARNING

• Students should be actively engaged and developing their own understanding.

• Mathematics should be represented in as many ways as possible by using graphs, tables, pictures, symbols, and words.

• Students should be asked to explain their thinking and defend their opinions through conversation, demonstration, and by showing their work.

• Mathematics should be represented in as many ways as possible by using graphs, tables, pictures, symbols, and words.

• Appropriate manipulatives and technology should be used to enhance student learning.

• Students should be given opportunities to revise their work based on teacher feedback, peer feedback, and metacognition which includes self-assessment and reflection.

• Students need to write in mathematics class to explain their thinking, to share how they perceive topics, and to justify their work to others.

EVIDENCE OF LEARNING

By the conclusion of this unit, students should be able to demonstrate the following competencies:

• Identify and give multiple representations for the fractional parts of a whole (area model) or of a set, using halves, thirds, fourths, sixths, eighths, tenths and twelfths.

• Recognize and represent that the denominator determines the number of equally sized pieces that make up a whole.

• Recognize and represent that the numerator determines how many pieces of the whole are being referred to in the fraction.

• Represent and compare fractions with denominators of 2, 3, 4, 6, 10, or 12 using concrete and pictorial models.

• Model, represent, and solve addition or subtraction problems using decimal fractions or common fractions with like denominators.

• Solve problems involving fractions.

TASKS

The following tasks represent the level of depth, rigor, and complexity expected of all third grade students. These tasks or a task of similar depth and rigor should be used to demonstrate evidence of learning. It is important that all elements of a task be addressed throughout the learning process so that students understand what is expected of them. While some tasks are identified as a performance task, they also may be used for teaching and learning (learning task).

|Task Name |Task Type |Content Addressed |

| |Grouping Strategy | |

|Paper-Folding Fractions |Learning Task |Naming fractional pieces, area model |

| |Individual/Partner Task | |

|Pattern Block Fractions |Learning Task |Unit fractions, Identifying the whole |

| |Partner/Small Group Task | |

|Eating Fractions |Learning Task |Naming fractions of a set |

| |Individual/Partner Task | |

|A Bowl of Beans |Performance Task |Naming fractional pieces, discrete model |

| |Individual/Partner Task | |

|Pizza Party |Performance Task |Adding, subtracting fractions with like denominators |

| |Individual/Partner Task | |

|All in a Line |Learning Task |Create number line, using tenths |

| |Partner/Small Group Task | |

|Trash Can Math |Learning Task |Tenths, representing tenths as a decimal fraction and as a |

| |Partner/Small Group Task |decimal |

|Sweet Decimal Fraction Bars |Performance Task |Sharing equal parts, tenths |

| |Individual/Partner Task | |

|Culminating Activity |Performance Task |Create fraction game using story problems |

|The Fraction Story Game |Partner/Small Group Task | |

LEARNING TASK: Paper-Folding Fractions

STANDARDS ADDRESSED

M3N5. Students will understand the meaning of decimal fractions and common fractions in simple cases and apply them in problem-solving situations.

a. Identify fractions that are decimal fractions and/or common fractions

b. Understand that a decimal fraction (i.e. 3/10) can be written as a decimal (i.e. 0.3)

c. Understand the fraction a/b represents a fraction that is a equal sized parts of a whole that is divided into b equal sized parts.

d. Know and use decimal fractions and common fractions to represent the size of parts created by equal divisions of a whole.

e. Recognize and describe a dime as having the value of one tenth of a dollar

M3P1. Students will solve problems (using appropriate technology).

a. Build new mathematical knowledge through problem solving.

b. Solve problems that arise in mathematics and in other contexts.

c. Apply and adapt a variety of appropriate strategies to solve problems.

d. Monitor and reflect on the process of mathematical problem solving.

M3P3. Students will communicate mathematically.

a. Organize and consolidate their mathematical thinking through communication.

b. Communicate their mathematical thinking coherently and clearly to peers, teachers, and others.

c. Analyze and evaluate the mathematical thinking and strategies of others.

d. Use the language of mathematics to express mathematical ideas precisely.

M3P4. Students will make connections among mathematical ideas and to other disciplines.

a. Recognize and use connections among mathematical ideas.

b. Understand how mathematical ideas interconnect and build on one another to produce a coherent whole.

c. Recognize and apply mathematics in contexts outside of mathematics.

M3P5. Students will represent mathematics in multiple ways.

a. Create and use representations to organize, record, and communicate mathematical ideas.

b. Select, apply, and translate among mathematical representations to solve problems.

c. Use representations to model and interpret physical, social, and mathematical phenomena.

ESSENTIAL QUESTIONS

• How can we be sure fractional parts are equal in size?

• What does each term in a fraction represent?

• How does the number of equal pieces affect the name of a fraction?

• What do I know about a fraction that has the same numerator and denominator?

MATERIALS

• “Paper-Folding Fractions” student recording sheet, copied on light-colored paper

• “Paper-Folding Fractions, Paper Strips” student sheet, copied on white paper or a set of 11 strips (7[pic] inch by [pic]inch) of paper

• Crayons

• Dark, thin-point markers

• Glue stick or tape

GROUPING

Individual/Partner Task

TASK DESCRIPTION, DEVELOPMENT AND DISCUSSION

In this task students explore fractions by folding strips of paper. They then describe the fractions they created and write its name in fractional form.

Background Knowledge

Before the activity, be sure the children all understand the concept of equal parts. Use pieces of different shaped paper (piece of construction paper, coffee filter, 8 ½ inch square cut from a piece of copy paper, 1/2 sheet of copy paper cut vertically, etc.) to demonstrate folding into equal-sized pieces. For some of the students to understand “equal-sized” you may have to cut and match the pieces, demonstrating that they are the same size.

Task Directions

Students will follow the directions below from the “Paper-Folding Fractions” student recording sheet.

You will be making fraction pieces using strips of paper. You will need 11 strips of paper. Fold your strips as follows:

□ 2 strips – fold in halves

□ 2 strips – fold in thirds

□ 2 strips – fold in fourths

□ 2 strips – fold in sixths

□ 2 strips – fold in eighths

□ Leave one strip unfolded

Once you have folded your strips, follow the directions below:

1. Choose four strips that represent different fractions.

2. Mark over the fold lines with a dark marker on each strip.

3. Shade in part of the fractional pieces on the strip (For example, shade three out of four parts).

4. Glue each strip on the back of this paper.

5. For each strip of paper glued

a. Write a sentence that tells the number of shaded parts and the total number of parts (i.e. I have four pieces and three are shaded).

b. Write the fraction name for the shaded part and the un-shaded part of each strip (i.e. ¾ shaded, ¼ un-shaded).

6. Keep the remaining fraction strips for additional tasks.

Comments

This lesson is an introduction to fractional pieces, writing fractions, and representing fractions with folded strips of paper.

• Copying the student sheet onto a piece of light-colored paper will make it easier to see the fraction strips when they are glued on the back of the student recording sheet.

• Be sure students are folding their paper into equal-sized pieces to represent each fractional piece.

• Typically, students have some difficulty folding the strips into thirds. Have students help each other with this. One way to assist with folding thirds is to “roll” the paper so that there are thirds represented, and then crease the paper.

• This is an opportunity for students to make sense of the meaning of denominator. They should recognize that as the number of equal parts increases, the denominator increases. Also, when the whole is the same, as the number of equal parts increases, the size of each piece decreases.

• This is an opportunity for students to make sense of numerator. The numerator identifies the number of pieces being considered or counted.

• Students should also recognize that when the numerator and denominator are the same, one whole is represented.

In addition to the activity above, students can make a fraction page by gluing one of each of the six different strips in order. Students may place the largest fractional pieces at the top and the smaller fractional pieces below. (i.e., 1 whole strip, a strip folded into halves, a strip folded into thirds, a strip folded into fourths, etc). If they line them up one above the other on the page, they can create a visual representation of how the sizes of different fractional pieces are related.

Questions/Prompts for Formative Student Assessment

• Are you unfolding your strip of paper to check your work?

• Is your strip folded into equal parts? How do you know?

• How many parts did you create? How many are shaded? How do you write that as a fraction?

Questions for Teacher Reflection

• Do students recognize that the parts of the fraction (fractional pieces) need to be the same size?

• Do students recognize that the numerator for the shaded fraction added to the numerator for the unshaded fraction equals the total number of fractional pieces in one whole?

• Do students recognize that folding halves in half makes fourths? Fourths folded in half makes eighths? Thirds folded in half make sixths?

DIFFERENTIATION

Extension

• Students can write comparative comments about their fraction strips. (More is shaded on [pic] of a strip than on[pic]of a strip. There are more pieces on a strip cut into thirds than on a strip cut into halves.)

• Students shade all possible combinations for a fraction (i.e. fourths could have [pic]shaded). Once all combinations are found the strips can be glued and labeled as above.

• Students can draw and label additional fractions using sets of objects, e.g. colored circles.

• Have students create one whole using a combination of fractional pieces.

Intervention

• Provide students with paper strips with small marks or dotted lines to indicate fold lines.

• Allow students to copy from a sentence frame for the required sentence (e.g., I have _____ pieces and _____ are shaded. _____ of the strip is shaded; _____ of the strip is unshaded.) Alternatively, provide a photocopy of a student recording sheet that includes the sentence frames.

TECHNOLOGY CONNECTION

Allows students to practice naming fractional parts of a whole.

Name _______________________________________ Date_____________________________

Paper-Folding Fractions

You will be making fraction pieces using strips of paper. You will need 11 strips of paper. Fold your strips as follows:

□ 2 strips – fold in halves

□ 2 strips – fold in thirds

□ 2 strips – fold in fourths

□ 2 strips – fold in sixths

□ 2 strips – fold in eighths

□ Leave one strip unfolded

Once you have folded your strips, follow the directions below:

1. Choose four strips that represent different fractions.

2. Mark over the fold lines with a dark marker on each strip.

3. Shade in part of the fractional pieces on the strip (For example, shade three out of four parts).

4. Glue each strip on the back of this paper.

5. For each strip of paper glued,

a. Write a sentence that tells the number of shaded parts and the total number of parts (i.e. I have four pieces and three are shaded).

b. Write the fraction name for the shaded part and the un-shaded part of each strip (i.e. ¾ shaded, ¼ un-shaded).

6. Keep the remaining fraction strips for additional tasks.

Paper-Folding Fractions

Paper Strips

Cut apart the strips below. Use them for the Paper-Folding Fractions task.

.

| |

Player #1

________________ |Number of Tosses |Prediction for Number of “Baskets” |Number of “Baskets”

(Use tallies) |Score as a fraction |Score as a decimal | | |10 | | | | | |Player #2

________________ |Number of Tosses |Prediction for Number of “Baskets” |Number of “Baskets”

(Use tallies) |Score as a fraction |Score as a decimal | | |10 | |

| | | |

1. Create a poster to display your group’s results. Your poster should include the following.

a. Represent the number of good throws for each partner as a decimal fraction and express a comparison of decimal fraction scores using a >, , ................
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