6th+Grade+Unit+4+Bits+and+Pieces+II+and+III

6th Grade Math Unit 4 Bits and Pieces II and III

Enduring understanding (Big Idea):

Students use the meaning of fractions, the meanings of multiplication and division, and the relationship between multiplication and division to understand and explain why the procedures for dividing fractions make sense. Students use these operations to solve problems involving fractions and decimals in real world context.

Essential Questions:

What kinds of models can be used to show computation with fractions? Will the strategies and algorithms we have developed apply to all fractional quantities? Do results from algorithms support those found with models? How do decimals compare to fractions that I know? What does “quotient” represent? When is it appropriate to use division to solve problems?

BY THE END OF THIS UNIT:

Students will know…

 how to solve real world problems using operations on fractions and decimals

Vocabulary: quotient, reciprocal, inverse operation dividend, division notation ÷, /, divisor, quotient, remainder, addend, sum, difference, factor, product, divisor, dividend, quotient, remainder

Learning Task :

 Mathematical Reflections

Performance Task:



Check-Up(s) / Partner Quiz

Project:

Unit Review:



Looking Back Looking Ahead

Students will be able to…



Use a model to show division of fractions.



Use my understanding of multiplication of fractions to explain division of fractions.



Divide fractions to find the quotient.



Interpret the meaning of the quotient.



Solve word problems using division of fractions.



Write an equation to solve a problem using division of fractions.



Write a story problem that will use division of fractions.



Divide multi-digit numbers using the standard algorithm.







Add multi-digit decimals using the standard algorithm.



Subtract multi-digit decimals using the standard algorithm.

Multiply multi-digit decimals using the standard algorithm.

Divide multi-digit decimals using the standard algorithm.

Mathematical Practices in Focus:

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

Standards on successive pages were unpacked by Utah State Office of Education, CMS-district specific modifications and resources for this unit were created by CMS teacher leaders.

Standards are listed in alphabetical /numerical order not suggested teaching order. PLC’s must order the standards to form a reasonable unit for instructional purposes.

Bits & Pieces II

Unit Plans

Investigation 1 does not cover

6 th grade common core, but is a good resources to support students’ conceptual understanding of fractions.

Similarly, Investigation 2 focuses on developing conceptual understanding of adding and subtracting fractions

5.NF

.


Investigation

Estimating Fractions

1.1

Getting Close

1.2

Folding Fraction Strips

Adding and Subtracting Fractions

2.1 Land Sections

2.2 Visiting the Spice Shop

2.3 Just the Facts

2.4 Designing Algorithms for Addition and

Subtraction

Suggested ACE Questions

1.1 ACE 1-15,19-22, 31-33,35

1.2 ACE 28-30, 37-40

2.1 ACE 2,29

2.2 ACE 4, 6-12, 30-35

2.3 ACE 14-18, 41-44

2.4 ACE 19-25, 46, 47

Unit Plans

Standard 6.NS.1

Investigation 3

Multiplying with Fractions

Investigation 4

Dividing with Fractions

Investigation

3.1

How much of the Pan Have we Sold?

3.2

Finding a Part of a Part

3.3

Modeling More Multiplication Situations

3.4

Changing Forms

3.5

Writing a Multiplication Algorithm

4.1 Equivalent Fractions and Equal Shares

4.2 Finding Equivalent Fractions

4.3 Comparing Fractions to Benchmarks

4.4 Writing a Division Algorithm


3.1 ACE 1-3, 36, 37

3.2 ACE 6-8, 10, 38, 39

3.3 ACE 11-14, 40-45

3.4 ACE 18-20, 46, 47

3.5

ACE 21-33

4.1 ACE 1,2,24-29

4.2 ACE 5-8, 30-34

4.3 ACE 11-14

4.4 ACE 15-23




Bits & Pieces III

Unit Plans

Standards 6.NS.2; 6.NS.3

Investigation 1

Decimals – More or Less!

Investigation 2

Decimal Times

Investigation 3

The Decimal Divide

Standard 6.RP.3c

Investigation 4

Using Percents

Investigation 5

More About Percents

Investigation

1.1

About How Much?

1.2

Adding and Subtracting Decimals

1.3

Using Fractions to Add and Subtract Decimals

1.4

Decimal Sum and Difference Algorithms

2.1 Relating Fractions and Decimal Multiplication

2.2 Missing Factors

2.3 Finding Decimal Products

2.4 Factor- Product Relationships

3.1

Deciphering Decimal Situations

3.2 The Great Equalizer

3.3

Exploring Dividing Decimals

3.4 Representing Fractions as Decimals

4.1 Determining Tax

4.2 Computing Tips

4.3 Finding Bargains

5.1 Clipping Coupons

5.2 How Much Can We Spend?


1.1 ACE 1-6, 37-38

1.2 ACE 8-18

1.3

ACE19-21

1.4

ACE23-32, 45, 46

2.1

ACE 1-3, 34, 39

2.2

ACE 7-10, 12-16

2.3

ACE 17-19, 22-24, 40, 44

2.4 ACE 25, 27-31, 33, 45

3.1 ACE 1-4, 28, 29

3.2 ACE 5-14

3.3 ACE 15-19, 22-24

3.4 ACE 27

4.1 ACE 1-3, 13

4.2 ACE 4-6, 31,32

4.3 ACE 8-12, 20-22, 35-39

5.1 ACE 1-6, 16-18

5.2 ACE 7, 19; 8-14




CORE CONTENT

Cluster Title: The Number System: Apply and extend previous understandings of multiplication and division to divide fractions by fractions.

Standard 6.NS.1:

Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. For example, create a story context for (2/3) ÷ (3/4) and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that (2/3) ÷

(3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general, (a/b) ÷ (c/d) = ad/bc.) How much chocolate will each person get if 3 people share 1/2 lbs. of chocolate equally? How many 3/4-cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with length 3/4 mi. and area 1/2 square mi.?

Concepts and Skills to Master



Understand how to set up a problem based on the context of the problem.



Be able to interpret what the quotient represents.



Recognize that what is known or not known is based on the type of division needed (partitive—Total / # of groups = size of groups—or quotative or measurement—Total / size of group = # of groups) model.



Create a story context using division of fractions.



Understand that multiplication and division are inverse operations regardless of the class of numbers.



Compute the division of fractions.



Solve a story context using division of fractions.



Model division of fractions with manipulatives, diagrams (e.g., bar model, number line) and story contexts.



Write equations representing authentic problems involving fractions.

SUPPORTS FOR TEACHERS

Critical Background Knowledge

Know that multiplication and division are inverse operations.

Know that division is either fair sharing (partitive) or repeated subtraction (quotative).

Convert between improper fractions and mixed numbers.

Division by whole numbers.

Division of a whole number by a fraction.

Model division with manipulatives, diagrams and story contexts.

Academic Vocabulary quotient, reciprocal, inverse operation




Suggested Instructional Strategies

Use this problem: How many servings of popcorn are in 4½ cups if each person receives 3/4 cup of popcorn

The teacher provides 4½ cups of popcorn. Students use a 3/4 cup measuring cup to solve the problem. Record solutions as a group.

1. Think-Pair-Draw-Share: Put students in pairs. Have one solve the problem using a picture/diagram and the other solve using the algorithm. Then they get together and compare.

2. Think-Pair-Share: Students solve the problem on their own, then get together and discuss how their solutions are the same and how they are different.

3. Four Corners: Give students a problem and the quotient. Give each corner in your room a label and have students go to the corner they think would be the correct label for the quotient.

Resources



Connected Mathematics Bits and Pieces II, Investigation 4



Math Forum - Teacher Tutorial

– http://mathforum.org/dr.math/faq/faq.divide.fractions.html



Dividing Fractions - Teacher Tutorial - http://www.tpub.com/math1/5g.htm



Visual Fractions - “Divide Fractions” - Interactive Applets and Game

- http://www.visualfractions.com/divide.htm



UEN - “Modeling Multiplication and Division of Fractions” Lesson - http://www.uen.org/Lessonplan/preview.cgi?LPid=23394



LearnAlberta - “Improper Fractions and Mixed Numbers” Video

Lesson - http://www.learnalberta.ca/content/mesg/html/math6web/index.html?page=lesson s&lesson=m6lessonshell02.swf

 Illuminations “Feeding Frenzy” - Unit Rates; Multiply/Divide

Fractions http://illuminations.nctm.org/LessonDetail.aspx?id=L781



UEN - “Sticky Note Math” Lesson - http://www.uen.org/Lessonplan/preview?LPid=15443



UEN - “Dividing Fractions” Lesson - http://www.uen.org/Lessonplan/preview?LPid=5301

Sample Formative Assessment Tasks

Skill-based task

Use representations to show that 1/4 divided by 1/2 is 1/2, that

2/3 divided by 2/5 is 5/3, that 2/3 divided by 3/4 is 8/9, and that

1½ divided by 6/4 is 1.

Problem Task



You have 5/8 pound of Skittles. You want to give your friends 1/4 lb. each. How many friends can you give Skittles to? Explain your answer.



You have a 3/4-acre lot. You want to divide it into 3/8-acre lots. How many lots will you have? Draw a diagram to justify your solution.



You have a 3/4-acre lot. You want to divide it into 2 sections. How many acres in each section will you have? Draw a diagram to justify your solution.

 How wide is a rectangular strip of land with length 3/4 mi. and area 1/2 square mi.?




CORE CONTENT

Cluster Title: The Number System: Compute fluently with multi-digit numbers and find common factors and multiples.

Standard 6.NS.2 : Fluently divide multi-digit numbers using the standard algorithm.




Identify when it is appropriate to use the standard algorithm.



Use the standard algorithm to compute multi-digit division problems with procedural fluency.

Note: Procedural fluency is defined as skill in carrying out procedures flexibly, accurately, efficiently and appropriately ( Adding It Up , National Research Council).



Divide multi-digit numbers using the standard algorithm.

SUPPORTS FOR TEACHERS




Understand the meaning of division.



Understand place value of multi-digit numbers.



Know that division is the inverse of multiplication.



Illustrate and explain the relationship between calculations and models for multiplying and dividing multi-digit numbers.



Divide with single-digit numbers.



Use compatible numbers to make an estimation to determine reasonableness of answers.



Use the standard algorithm for division.



Read division notation.



Model division with manipulatives, diagrams, and story contexts

Academic Vocabulary dividend, division notation ÷, /, divisor, quotient, remainder





1. Think Aloud: Do the problem with a partner while explaining and telling what you are thinking and doing.

2. Have students identify in a problem set when they would use mental math and when they would use the standard algorithm.

3. Connect students’ existing strategies for division with the standard algorithm.

4. As a starter activity, use division problems that can reasonably be solved by using mental math (e.g., 105/25), estimation (e.g.,

150 ÷ 12, 227 ÷ 30), and reasoning (e.g., when I think of 105 divided by 25, I think of 4 sets of 25 with 5 left over, the 5 left over is 5/25 which is 1/5, so the answer is 4 1/5). Model for the students your thinking as you work through the problem. (Note:

This strategy would not apply to complex division problems for which the algorithm is most appropriate [e.g., 4567 ÷ 192]).


Skill-based task

248 divided by 18

Resources



Textbook Correlation: Bits & Pieces II and III

Problem Task

I spent $504 on 28 tickets for a rock concert. How much did I spend on each ticket?




CORE CONTENT

Cluster Title: The Number System: Compute fluently with multi-digit numbers and find common factors and multiples

Standard 6.NS.3: Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation




Understand the role of place value in the operations of addition, subtraction, multiplication and division



Identify when it is appropriate to use the standard algorithm



Add, subtract, multiply, and divide multi-digit decimals.



Model the operations of addition, subtraction, multiplication, and division with manipulatives, diagrams, and story contexts for multi-digit decimals.

SUPPORTS FOR TEACHERS




Understand decimal place values



Know basic facts for addition, subtraction, multiplication and division



Add, subtract, multiply and divide single-digit decimals



Model the operations of addition, subtraction, multiplication, and division with manipulatives, diagrams and story contexts for single digit decimals.

Academic Vocabulary addend, sum, difference, factor, product, divisor, dividend, quotient, remainder





Connect students’ knowledge of various strategies to the standard algorithm.

Have students look at student work that contains a common misconception and look at errors and discuss how to correct the error.

Resources



Connected Mathematics Bits and Pieces III, Investigations 1-3



NLVM - Base Block Decimals - http://nlvm.usu.edu/en/nav/frames_asid_264_g_3_t_1.html?from=category_g_3

_t_1.html



NLVM - Circle 3 - Adding Decimals - Puzzle - http://nlvm.usu.edu/en/nav/frames_asid_187_g_3_t_1.html?open=instructions&from=c ategory_g_3_t_1.html



LearnAlberta - “Solving Problems with Decimals” Video Lesson - http://www.learnalberta.ca/content/mesg/html/math6web/index.html?page=lessons&les son=m6lessonshell05.swf



LearnAlberta - “Addition and Subtraction of Decimals” Video Lesson - http://www.learnalberta.ca/content/me5l/html/math5.html?goLesson=7



Math Play - Jeopardy - Computation Game – http://www.math-play.com/Decimals-Jeopardy/decimals-jeopardy.html



LearnAlberta - Multiplication and Division of Decimals - Video Tutorial - http://www.learnalberta.ca/content/me5l/html/math5.html?goLesson=10


Skill-based task:

1. 242.134 + 308.02

2. 38.9 – 14.334

3. 11.82 x 2.81

4. 341.8 ÷ 1.2

Problem Task

The school had a bake sale and raised $75.55. If each cookie cost

$0.05, how many cookies were sold? Explain how you got your answer



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