Manga Math Mysteries: a Mystery with Fractions by Melinda Thielbar

4 th Grade 2 nd Nine Weeks Math Domain: Operations and Algebraic Thinking Cluster: Gain familiarity with factors and multiples. Common Core Standards: 4.OA.4 Find all factor pairs for a whole number in the range 1-100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1-100 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1-100 is prime or composite.

Key Vocabulary Prime Composite Factor Multiples Divisor Array Product Factor tree Division Multiplication (multiply) Factor pair quotient Mental computation

Habits of Mind  Persistence  Metacognition  Thinking Flexibly  Striving for Accuracy  Gathering Data Through all Senses  Questioning and Posing Problems  Applying Past Knowledge to New Situations

1 Domain: Operations and Algebraic Thinking Cluster: Gain familiarity with factors and multiples. Common Core Standard: 4.OA.4 Find all factor pairs for a whole number in the range 1-100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1-100 is a multiple of a given one-digit number. Determine whether a given whole number in the range 1-100 is prime or composite.

What does this mean? This standard requires students to demonstrate understanding of factors and multiples of whole numbers. This standard also refers to prime and composite numbers. A common misconception is that the number 1 is prime, when in fact; it is neither prime nor composite. Another common misconception is that all prime numbers are odd numbers. This is not true, since the number 2 has only 2 factors, 1 and 2, and is also an even number.

Math Practices: 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.

Essential Question:  How do you find factors and multiples of a number?  How do you determine if a number is prime, composite, or neither?  What strategies can I use to explain my thinking? Learning Targets (KUD) Criteria for Success for Mastery K: vocabulary associated with factoring; basic multiplication Students should be able to: facts; skip-counting; 2 U: the process used to determine if a number is prime, • List factor pairs for a variety of whole numbers from 1-100. composite, or neither • List factors to show if a number is prime or composite. D: determine factor pairs for a variety of whole numbers • Use skip counting to list multiples with any factor to create from 1-100; identify whole numbers as multiples of its that whole number. (Ex. Skip counting in multiples of 4: 4, 8, 12, factor; define prime as only having 2 factors; define 16, 20) composite as having 3 or more factors; I can:  define factors and multiples.  list all of the factor pairs for any whole numbers in the range 1-100.  determine multiples of a given whole numbers (1-100).  define prime and composite.  determine if a number is prime or composite. Examples Prime vs. Composite: A prime number is a number greater than 1 that has only 2 factors, 1 and itself. Composite numbers have more than 2 factors.

Students should understand the process of finding factor pairs so they can do this for any number 1 -100, Example: Factor pairs for 96: 1 and 96, 2 and 48, 3 and 32, 4 and 24, 6 and 16, 8 and 12.

Multiples can be thought of as the result of skip counting by each of the factors. When skip counting, students should be able to identify the number of factors counted e.g., 5, 10, 15, 20 (there are 4 fives in 20).

Example: Factors of 24: 1, 2, 3, 4, 6, 8,12, 24 Multiples: 1, 2, 3, 4, 5…24 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24 3, 6, 9, 12, 15, 18, 21, 24 4, 8, 12, 16, 20, 24 8, 16, 24 12, 24 3 24 To determine if a number between1-100 is a multiple of a given one-digit number, some helpful hints include the following: • all even numbers are multiples of 2 • all even numbers that can be halved twice (with a whole number result) are multiples of 4 • all numbers ending in 0 or 5 are multiples of 5 Textbook Resources Houghton Mifflin Harcourt Unit 1 Chapter 2: Basic Multiplication and Division 44A–44B, 44–46, 50A–50B, 50–51 Houghton Mifflin Harcourt Unit 3 Chapter 5: Multiples Math Expressions: 197, 198-204

Supplemental Resources STAMS Math Madness Buckle Down Houghton Mifflin Math Chapter Challenges North Carolina Mathematics Coach Math Intensive Intervention (Houghton Mifflin) Houghton Mifflin Harcourt Strategic Intervention Houghton Mifflin Harcourt Assessment Guide Houghton Mifflin Harcourt Problem Solving Practice Book Houghton Mifflin Harcourt Resource Book

Media Resources If You Were an Odd Number by Marcie Aboff If You Were an Even Number by Marcie Aboff Missing Mittens by Stuart Murphy

Web Resources 1. Study Jams http://studyjams.scholastic.com/studyjams/jams/math/multiplication-division/multiples.htm 2. Common Core State Standards-4th Grade-Explore Learning http://www.explorelearning.com/index.cfm?

4 method=cResource.dspStandardCorrelation&id=1502 3. 4th Grade Number Activities http://www.k-5mathteachingresources.com/ 4. Illuminations http://illuminations.nctm.org/lessonslist.aspx? grade=2&standard=1&standard=2&standard=3&standard=4&standard=5 5. Houghton Mifflin Harcourt School Math http://www.harcourtschool.com/search/search.html 6. Houghton Mifflin Harcourt School Math http://www.harcourtschool.com/activity/elab2004/gr5/8.html

5 Domain: Number and Operation in Base Ten Cluster: Use place value understanding and properties of operations to perform multi-digit arithmetic. Common Core Standards: 4.NBT.5. Multiply a whole number up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models. 4.NBT.6. Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

Key Vocabulary Algorithm Area model Dividend Reason Factor Partitioning Divisor Compare Multiple Compensation Quotient Distributive Property product

Habits of Mind  Persistence  Metacognition  Thinking Flexibly  Striving for Accuracy  Gathering Data Through all Senses  Questioning and Posing Problems  Applying Past Knowledge to New Situations

6 Domain: Number and Operation in Base Ten Cluster: Use place value understanding and properties of operations to perform multi-digit arithmetic. Common Core Standard: 4.NBT.5. Multiply a whole number up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

What does this mean? Students who develop flexibility in breaking numbers apart have a better understanding of the importance of place value and the distributive property in multi-digit multiplication. Students use base ten blocks, area models, partitioning, compensation strategies, etc. when multiplying whole numbers and use words and diagrams to explain their thinking. They use the terms factor and product when communicating their reasoning. Multiple strategies enable students to develop fluency with multiplication and transfer that understanding to division. Use of the standard algorithm for multiplication is an expectation in the 5th grade.

Essential Question:  What is the importance of place value in the distributive property in multi-digit multiplication?  How can place value properties aid computation?

7  What strategies can I use to explain my thinking?  What are efficient methods for finding products?

Learning Targets (KUD) Criteria for Success for Mastery K: vocabulary associated with multiplication; basic Students should be able to: multiplication facts (0-12), U: place value system; more than one way to solve a problem •• Multiply a whole number of up to four digits by a one-digit (multiplication) whole number, and multiply two two-digit numbers using, D: multiply a multi-digit number by a one-digit multiplier; modeling, and drawing visual representations of base ten blocks. demonstrate multiplication of two two-digit numbers using • Multiply a whole number of up to four digits by a one-digit rectangular arrays, place value, and the area model (lattice whole number, and multiply two two-digit numbers using and method); solve multiplication of two two-digit numbers using drawing visual representations of area models, and providing properties of operations and equations; explain my chosen justification of the strategy. strategy; • Multiply a whole number of up to four digits by a one-digit  whole number, and multiply two two-digit numbers using and I can: drawing visual representations of partitioning, and providing  multiply a multi-digit number by a one-digit whole justification of strategy. numbers. • Multiply a whole number of up to four digits by a one-digit  demonstrate multiplication of two two-digit numbers whole number, and multiply two two-digit numbers using and using rectangular arrays, place value, and the area model. drawing visual representations of compensation, and providing  solve multiplication of two two-digit numbers using justification of strategy. properties of operations and equations. • Construct other strategies of multiplication using place value  explain my chosen strategy. and mathematical properties to justify his/her strategy. • Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

8 Examples 1. 36 x 94 = (30 + 6) x (90 + 4) = (30 + 6) x 90 + (30 + 6) x 4 = 30 x 90 + 6 x 90 + 30 x 4 + 6 x 4

90 + 4

30 30 x 90 = 30 x 4 = 3 tens x 9 tens = 3 tens x 4 = 27 hundreds = 12 tens = 120 2700 + 6 6 x 90 = 6 x 4 6 x 9 tens = 24 54 tens 540

2. There are 25 dozen cookies in the bakery. What is the total number of cookies at the bakery?

Student 1 Student 2 Student 3 25 x12 25 x 12 25 x 12 I broke 12 up into 10 I broke 25 up into 5 I doubled 25 and cut and 2 groups of 5 12 in half to get 50 x 6 25 x 10 = 250 5 x 12 = 60 50 x 6 = 300 25 x 2 = 50 I have 5 groups of 5 in 25 250 +50 = 300 60 x 5 = 300 Textbook Resources Houghton Mifflin Harcourt Unit 1 Chapter 2: Basic Multiplication and Division 44A–44B, 44–46, 50A–50B, 50–51 9 Houghton Mifflin Harcourt Unit 3 Chapter 5: Multiply Lessons 1-4, pages 128-137, Lesson 6, pages 142-143, 149 Houghton Mifflin Harcourt Unit 3 Chapter 6: Divide Lessons 1-3, pages 154-160, Lesson 5, page 166-167, Lesson 7, page 170-171, Lesson 9-11, pages 174-182 Math Expressions: 499L-499N, 501, 502, 503, 504, 508, 509, 514, 515, 516, 518, 519, 520, 530, 531, 532, 536, 537, 538, 539, 540, 542, 543-545, 547, 548, 550-551, 552-554, 558,559, 560, 574-575, 576, 577, 578, 580-582, 583-586, 587, 588, 590, 591-592, 593, 595, 596, 598-600, 601, 602, 603, 604, 606-608, 609, 610, 612-613, 614, 615, 616 667J, 668-673, 674, 675, 677, 678, 680-681, 682, 685, 686, 687, 688, 689, 697, 690-692, 693, 694, 695, 696, 698-699, 700, 701, 702, 703, 704-707, 708, 709, 710, 711, 712-713, 714- 715, 716-717, 718, 719, 720, 726, 734 Houghton Mifflin (Turtle): Lesson 6 (p. 146-166), Lesson 7 (p. 172-181), Lesson 8.1-9.4 (p. 206-239)

Media Resources If You Were a Times Sign by Jerry Pallotta Double the Ducks by Stuart Murphy Doubling by John Burstein Numbers by Peter Patilla The Great Divide by Dayle Ann Dodds One Hundred Hungry Ants by Elinor Pinczes

Web Resources 1. Study Jams http://studyjams.scholastic.com/studyjams/jams/math/multiplication-division/distrib-property.htm 10 2. Common Core State Standards-4th Grade-Explore Learning http://www.explorelearning.com/index.cfm? method=cResource.dspStandardCorrelation&id=1502 3. 4th Grade Number Activities http://www.k-5mathteachingresources.com/ 4. Illuminations http://illuminations.nctm.org/lessonslist.aspx?grade=2&standard=1&standard=2&standard=3&standard=4&standard=5 5. Houghton Mifflin Harcourt School Math http://www.harcourtschool.com/search/search.html 6. Houghton Mifflin Harcourt School Math http://www.harcourtschool.com/activity/elab2004/gr5/8.html

Domain: Number and Operation in Base Ten Cluster: Use place value understanding and properties of operations to perform multi-digit arithmetic. Common Core Standard: 4.NBT.6. Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

What does this mean? In fourth grade, students build on their third grade work with division within 100. Students need opportunities to develop their understandings by using problems in and out of context. This standard calls for students to explore division through various strategies.

Essential Question:  What are different models of division?  What are efficient methods for finding quotients? 11  How does understanding multiplication help me develop strategies to divide  What questions can be answered using division?  What strategies can I use to explain my thinking?

Learning Targets (KUD) Criteria for Success for Mastery K: vocabulary associated with division; basic multiplication Students should be able to: facts (0-12), U: place value system; more than one way to solve a problem  Divide a whole number of up to four digits by a one-digit (division ), whole number modeling and drawing visual representations of D: demonstrate division of a multi-digit number by one-digit base ten blocks, and providing justification of strategy. using place value, rectangular arrays, and area model;  Divide a whole number of up to four digits by a one-digit construct other strategies of division using mathematical whole number modeling and drawing visual representations of properties to justify my strategy; explain my chosen place value principles/distributive property, and providing strategies using visual representations. justification of strategy. Example: 260 ÷ 4 = (200 ÷ 4) = ( 60 ÷ 4) I can:  Divide a whole number of up to four digits by a one-digit  demonstrate division of a multi-digit number by one-digit whole number modeling and drawing visual representations of using place value, rectangular arrays, and area model. multiplication, and providing justification of strategy.  construct other strategies of division using mathematical Example: properties to justify my strategy. 4 x 50 = 200,  explain my chosen strategies using visual representations. 4 x 10 = 40, 4 x 5 = 20; 50 + 10 + 5 = 65 So 260 ÷ 4 = 65 = 65  Divide a whole number of up to four digits by a one-digit whole number modeling and drawing visual representations of open arrays or area models, and providing justification of strategy.  Construct other strategies of division using mathematical properties to justify his/her strategy. 12  Find whole-number quotients and remainders with up to four- digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

Examples 1. A 4th grade teacher bought 4 new pencil boxes. She has 260 pencils. She wants to put the pencils in the boxes so that each box has the same number of pencils. How many pencils will there be in each box?  Using Base 10 Blocks: Students build 260 with base 10 blocks and distribute them into 4 equal groups. Some students may need to trade the 2 hundreds for tens but others may easily recognize that 200 divided by 4 is 50.  Using Place Value: 260 ÷ 4 = (200 ÷ 4) + (60 ÷ 4)  Using Multiplication: 4 x 50 = 200, 4 x 10 = 40, 4 x 5 = 20; 50 + 10 + 5 = 65; so 260 ÷ 4 = 65 This standard calls for students to explore division through various strategies.

2. There are 592 students participating in Field Day. They are put into teams of 8 for the competition. How many teams get created? Student 1 Student 2 Student 3 592 divided by 8 592 divided by 8 592 I want to get to 592 There are 70 8’s in I know that 10 8’s = 80 -400 50 8 x 25 = 200 560 If I take out 50 8’s that is 400. 8 x 25 = 200 192 592 - 560 = 32 592 - 400 = 192 8 x 25 = 200 -160 20 There are 4 8’s in 32 I can take out 20 more 8’s which is 160 200 + 200 + 200 = 600 70 + 4 = 74 192 - 160 = 32 32 600 - 8 = 592 8 goes into 32 4 times -32 4 I had 75 groups of 8 and I have none left took one away, so there I took out 50, then 20 more, then 4 more are That’s 74 0 74 teams

13 Textbook Resources Houghton Mifflin Harcourt Unit 1 Chapter 2: Basic Multiplication and Division 44A–44B, 44–46, 50A–50B, 50–51 Houghton Mifflin Harcourt Unit 3 Chapter 5: Multiply Lessons 1-4, pages 128-137, Lesson 6, pages 142-143, 149 Houghton Mifflin Harcourt Unit 3 Chapter 6: Divide Lessons 1-3, pages 154-160, Lesson 5, page 166-167, Lesson 7, page 170-171, Lesson 9-11, pages 174-182 Math Expressions: 499L-499N, 501, 502, 503, 504, 508, 509, 514, 515, 516, 518, 519, 520, 530, 531, 532, 536, 537, 538, 539, 540, 542, 543-545, 547, 548, 550-551, 552-554, 558,559, 560, 574-575, 576, 577, 578, 580-582, 583-586, 587, 588, 590, 591-592, 593, 595, 596, 598-600, 601, 602, 603, 604, 606-608, 609, 610, 612-613, 614, 615, 616 667J, 668-673, 674, 675, 677, 678, 680-681, 682, 685, 686, 687, 688, 689, 697, 690-692, 693, 694, 695, 696, 698-699, 700, 701, 702, 703, 704-707, 708, 709, 710, 711, 712-713, 714- 715, 716-717, 718, 719, 720, 726, 734 Houghton Mifflin (Turtle): Lesson 6 (p. 146-166), Lesson 7 (p. 172-181), Lesson 8.1-9.4 (p. 206-239)

Media Resources If You Were a Times Sign by Jerry Pallotta Double the Ducks by Stuart Murphy Doubling by John Burstein Numbers by Peter Patilla The Great Divide by Dayle Ann Dodds One Hundred Hungry Ants by Elinor Pinczes

14 Web Resources 1. Study Jams http://studyjams.scholastic.com/studyjams/jams/math/multiplication-division/distrib-property.htm 2. Common Core State Standards-4th Grade-Explore Learning http://www.explorelearning.com/index.cfm? method=cResource.dspStandardCorrelation&id=1502 3. 4th Grade Number Activities http://www.k-5mathteachingresources.com/ 4. Illuminations http://illuminations.nctm.org/lessonslist.aspx? grade=2&standard=1&standard=2&standard=3&standard=4&standard=5 5. Houghton Mifflin Harcourt School Math http://www.harcourtschool.com/search/search.html 6. Houghton Mifflin Harcourt School Math http://www.harcourtschool.com/activity/elab2004/gr5/8.html

15 Domain: Numbers and Operations: Fractions Cluster: Extend understanding of fraction equivalence and ordering. Common Core Standards: 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. Use this principle to recognize and generate equivalent fractions. 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 such as ½. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.

Key Vocabulary Simplest Form Fraction Denominator Fraction Chain Equivalent Fraction Numerator Unit fraction Fraction Partners Benchmark

16 Habits of Mind  Persistence  Metacognition  Thinking Flexibly  Striving for Accuracy  Gathering Data Through all Senses  Questioning and Posing Problems  Applying Past Knowledge to New Situations

Domain: Numbers and Operations: Fractions Cluster: Extend understanding of fraction equivalence and ordering. Common Core Standard: 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. Use this principle to recognize and generate equivalent fractions.

What does this mean? This standard refers to visual fraction models. This includes area models, number lines or it could be a collection/set model. This standard extends the work in third grade by using additional denominators. (5, 10, 12 and 100). This standard addresses equivalent fractions by examining the idea that equivalent fractions can be created by multiplying both the numerator and denominator by the same number or by dividing a shaded region into various parts.

17 Essential Question:  How can you identify common equivalent fractions?  How can fractions be modeled?  What are common denominators?  How do you find a common denominator of a fraction?

Learning Targets (KUD) Criteria for Success for Mastery K: vocabulary associated with fractions Students should be able to: U: fractions are part of a whole D: generate equivalent fractions within the same whole • Demonstrate equivalent fractions using manipulatives, problem (things of the same size) solving, and computation. • Identify fraction models with denominators of 5, 10, 12, and I can: 100. (Include tenths and hundredths grids)  use visual representations to explain equivalent fractions. • Draw fraction models with denominators of 5, 10, 12, and 100.  explain why fractions are equivalent using models. (Include tenths and hundredths grids)  explain to find common denominators.  generate equivalent fractions by multiplying or dividing the numerator and denominator by the same number.  use visual models to justify why multiplying or dividing the numerator and denominator by the same number generates equivalent fractions. Examples

½ = 2/4 = 9/18

18 There is NO mathematical reason why fractions must be written in simplified form, although it may be convenient to do so in some cases. Textbook Resources Houghton Mifflin Harcourt Unit 5 Chapter 9: Fractions and Mixed Numbers 260A-260B, 260-262, 264A-264B, 264-267, 268A- 268B, 268-269 Math Expressions: 837K, 837N, 857-862, 910-912, 914-916, 917-922, 928-929, 934-938, 940, 944, 1145-1146 Houghton Mifflin (Turtle): 19.1-19.4

19 Media Resources Apple Fractions by Jerry Pallotta Decimals and Fractions by Rebecca Wingard-Nelson The Hershey’s Milk Chocolate Bar Fractions Book by Jerry Pallotta Fractions by Michele Koomen Fractions by Penny Dowdy Full House by Dayle Ann Dodds Go, Fractions! by Judith Stamper Eating Fractions by Bruce McMillan If You Were a Fraction by Trisha Shaskan Inchworm and a Half by Elinor Pinczes Manga Math Mysteries: A mystery with Fractions by Melinda Thielbar Web Resources 1. http://www.mail.clevelandcountyschools.org~ccselem/07BB59E1-00870B98?plugin=loft Fractions Unit from DPI 2. http://www.brainpop.com 3. http://www.gamequarium.com/fractions.htm 4. http://www.illuminations.nctm.org/Activities.aspx?grade=2 5. http://www.smartexchange.com 6. http://www.jc-schools.net/tutorials/interact-math.htm 7. http://www.glencoe.com/sites/common_assets/mathematices/ebook_assets/vmf/VMFinterface.html 8. wwwk-6.thinkcentral.com

20 Domain: Number and Operation in Base Ten Cluster: Extend understanding of fraction equivalence and ordering. Common Core Standard: 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 such as ½. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.

What does this mean? This standard calls students to compare fractions by creating visual fraction models or finding common denominators or numerators. Students’ experiences should focus on visual fraction models rather than algorithms. When tested, models may or may not be included. Students should learn to draw fraction models to help them compare. Students must also recognize that they must consider the size of the whole when comparing fractions (ie, ½ and 1/8 of two medium pizzas is very different from . of one medium and 1/8 of one large).

Math Practices: 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.

21 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.

Essential Question:  How can common equivalent fractions be identified?  How can fractions be modeled and compared with a whole?  What are common denominators?  How do you find common denominators?

Learning Targets (KUD) Criteria for Success for Mastery K: vocabulary associated with fractions, Students should be able to: U: only compare fractions referring to the same whole;  Label benchmark fractions (such as ½) D: compare two given fractions by generating equivalent  Construct fraction models (pictures or number lines) for the fractions with common denominators; compare two given purpose of comparing two. fractions by reasoning about their size or their location on a  Construct fraction models (pictures or number lines) for number line, or comparing them to a benchmark fraction; purpose of comparing the same fractional part of two record the comparison using symbols (<, =, and >) and justify different wholes (the “wholes” should not be congruent). each comparison.  Construct fraction models (pictures or number lines) for the purpose of comparing the two fractions of two different I can: wholes (the “wholes” should not be congruent).  explain that comparing two fractions is valid when they  List the multiples of 2 different denominators to find the refer to the same whole. least common multiple.  compare two given fractions by generating equivalent  Convert two different fractions to common denominators fractions with common denominators. using the least common multiple.  compare two given fractions by reasoning about their size  Compare two fractions with different numerators and or their location on a number line, or comparing them to a different denominators, arrays, and/or area models. benchmark fraction. 22  record the comparison using symbols (<, =, and >) and justify each comparison. Examples There are two cakes on the counter that are the same size. The first cake has ½ of it left. The second cake has 5/12 left. Which cake has more left? Student 1 Area model: The first cake has more left over. The second cake has 5/12 left which is smaller than 1/2.

Student 2 Number Line model: First Cake

0 1 ½

Second Cake

0 1 3/12 6/12 9/12 Student 3 verbal explanation: I know that 6/12 equals 1/2. Therefore, the second cake which has 5/12 left is less than ½.

Textbook Resources Houghton Mifflin Harcourt Unit 5 Chapter 9: Fractions and Mixed Numbers 260A-260B, 260-262, 264A-264B, 264-267, 268A-

23 268B, 268-269 Math Expressions: 837K, 837N, 857-862, 910-912, 914-916, 917-922, 928-929, 934-938, 940, 944, 1145-1146 Houghton Mifflin (Turtle): 19.1-19.4

Media Resources Apple Fractions by Jerry Pallotta Decimals and Fractions by Rebecca Wingard-Nelson The Hershey’s Milk Chocolate Bar Fractions Book by Jerry Pallotta Fractions by Michele Koomen Fractions by Penny Dowdy Full House by Dayle Ann Dodds Go, Fractions! by Judith Stamper Eating Fractions by Bruce McMillan If You Were a Fraction by Trisha Shaskan Inchworm and a Half by Elinor Pinczes Manga Math Mysteries: A mystery with Fractions by Melinda Thielbar

Web Resources

24 1. http://www.mail.clevelandcountyschools.org~ccselem/07BB59E1-00870B98?plugin=loft Fractions Unit from DPI 2. http://www.brainpop.com 3. http://www.gamequarium.com/fractions.htm 4. http://www.illuminations.nctm.org/Activities.aspx?grade=2 5. http://www.smartexchange.com 6. http://www.jc-schools.net/tutorials/interact-math.htm 7. http://www.glencoe.com/sites/common_assets/mathematices/ebook_assets/vmf/VMFinterface.html 8. wwwk-6.thinkcentral.com

Domain: Numbers and Operations: Fractions Cluster: Build fractions from unit fractions by applying and extending previous understanding of operations on whole numbers. Common Core Standards: 4.NF.3 Understand a fraction a/b with a > 1 as a sum of fractions 1/b. a. Understand addition and subtraction of fractions as joining and separating parts referring to the same whole. b. Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions by using a visual fraction model. Examples: 3/8 = 1/8 + 1/8 + 1/8, 3/8 = 1/8 + 2/8, 2 1/8 = 8/8 + 1/8 + 1/8. c. Add and subtraction mixed numbers with like denominators by replacing each mixed number with and equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction. d. Solve word problems involving addition and subtraction of fractions referring to the same whole and having like

25 denominators by using visual fraction models and equations to represent the problem.

Key Vocabulary Unit Fraction Decompose Mixed Number Addition/Joining Operation Reason Improper Fraction Subtraction/Seperating Fraction Justify

Habits of Mind  Persistence  Metacognition  Thinking Flexibly  Striving for Accuracy  Gathering Data Through all Senses  Questioning and Posing Problems  Applying Past Knowledge to New Situations  Working Interdependently Domain: Number and Operation in Base Ten Cluster: Build fractions from unit fractions by applying and extending previous understanding of operations on whole numbers. Common Core Standard: 4.NF.3 Understand a fraction a/b with a > 1 as a sum of fractions 1/b. a. Understand addition and subtraction of fractions as joining and separating parts referring to the same whole. b. Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions by using a visual fraction model. Examples: 3/8 = 1/8 + 1/8 + 1/8, 3/8 = 1/8 + 2/8, 2 1/8 = 8/8 + 1/8 + 1/8. c. Add and subtraction mixed numbers with like denominators by replacing each mixed number with and equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction.

26 d. Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators by using visual fraction models and equations to represent the problem.

What does this mean? a. A fraction with a numerator of one is called a unit fraction. When students investigate fractions other than unit fractions, such as 2/3, they should be able to join (compose) or separate (decompose) the fractions of the same whole. b. Students should justify their breaking apart (decomposing) of fractions using visual fraction models. The concept of turning mixed numbers into improper fractions needs to be emphasized using visual fraction models. c. A separate algorithm for mixed numbers in addition and subtraction is not necessary. Students will tend to add or subtract the whole numbers first and then work with the fractions using the same strategies they have applied to problems that contained only fractions. d. Practice these addition and subtraction skills within the context of word problems.

Math Practices: 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. Essential Question:  How do I compose and decompose fractions of the same whole number?  What is a mixed number?  What procedure do you use to covert a mixed number to an improper fraction?  When is it appropriate to use an improper fraction or mixed number?

Learning Targets (KUD) Criteria for Success for Mastery K: vocabulary associated with fractions; procedure used to Students should be able to: convert mixed numbers and improper fractions; a. 27 multiplication facts  Recall a fraction 1/b (1/4) as the quantity formed by 1 part U: when to use mixed numbers and improper fractions when a whole is partitioned into b (4) equal parts; understand D: explain using visual representations why a/b = a x 1/b; a fraction a/b (3/4) as the quantity formed by a (3) parts of decompose fractions into unit fractions; solve word problems 1/b (1/4). that involve multiplying whole numbers and fractions using  Develop a logical argument why denominators stay the same visual representations; use inverse operations to check their when adding or subtracting fractions. work  Understand addition and subtraction of fractions as joining and separating parts referring to the same whole. I can: b.  explain why a/b = a x 1/b by using visual models to show  Relate mixed numbers and their equivalent improper how to decompose fractions into unit fractions and fractions. represent it as a multiple of unit fractions (e.g., ¾ = ¼ + ¼ c. + ¼ = 3 x ¼.)  Understand addition and subtraction of mixed numbers with  decompose a fraction (a/b) into a multiple of unit like denominators by replacing each mixed number with an fractions (a x 1/b) in order to show why multiplying a equivalent fraction, and/or by using properties of operations whole number by a fraction (n x (a/b)) results in (n x a)/b and the relationship between addition and subtraction. (e.g., 5 x 3/8 = 5 x (3 x 1/8) = 14 x 1/8 = 15/8). d.  solve word problems that involve multiplying a whole  Understand how to use visual fraction models and equations number and fraction with visual models and equations. to represent word problems involving addition and subtraction  use inverse operations to check my work. of fractions referring to the same whole and having like denominators.

Examples a. 1 ¼ - ¾ = Δ 4/4 + ¼ = 5/4 5/4 – ¾ = 2/4 or ½.

Example of word problem: Mary and Lacey decide to share a pizza. Mary ate 3/6 and Lacey ate 2/6 of the pizza. How much of the pizza did the girls eat together? Possible solution: The amount of pizza Mary ate can be thought of a 3/6 or 1/6 and 1/6 and 1/6. The amount of pizza Lacey ate can be thought of a 1/6 and 1/6. The total amount of pizza they ate is 1/6 + 1/6 + 1/6 + 1/6 + 1/6 or 5/6 of the whole pizza. 28 b. 3/8 = 1/8 + 1/8 + 1/8

= 3/8 = 1/8 + 2/8

c. Trevor has 4 1/8 pizzas left over from his soccer party. After giving some pizza to his friend, he has 2 4/8 of a pizza left. How much pizza did Trevor give to his friend? Possible solution: Trevor had 4 1/8 pizzas to start. This is 33/8 of a pizza. The x’s show the pizza he has left which is 2 4/8 pizzas or 20/8 pizzas. The shaded rectangles without the x’s are the pizza he gave to his friend which is 13/8 or 1 5/8 pizzas.

X X X X X X X X X X X X X X X X X X X X e. A cake recipe calls for you to use ¾ cup of milk, ¼ cup of oil, and 2/4 cup of water. How much liquid was needed to make the cake?

Textbook ResourcesMilk Oil Water Houghton Mifflin Harcourt Unit 5 Chapter 9: Fractions and Mixed Numbers Lessons 8 – 12, pages 282-298 Math Expressions: 837I, 841-844, 845, 847, 854, 894-898, 873L, 837M, 841-842, 843, 8484, 849, 852, 854-855, 886-887, 888, 902, 910, 873L, 873M, 841, 842, 842, 843, 847, 848-849, 850, 851, 852, 854, 855, 903, 904-905, 906, 907, 908, 910, 912, 848-849, 852 Houghton Mifflin (Turtle): Chapter 20, Lessons 1-2

Supplemental Resources STAMS Math Madness 29 Buckle Down Houghton Mifflin Math Chapter Challenges North Carolina Mathematics Coach Math Intensive Intervention (Houghton Mifflin) Houghton Mifflin Harcourt Strategic Intervention Houghton Mifflin Harcourt Assessment Guide Houghton Mifflin Harcourt Problem Solving Practice Book Houghton Mifflin Harcourt Resource Book

Media Resources Apple Fractions by Jerry Pallotta Decimals and Fractions by Rebecca Wingard-Nelson The Hershey’s Milk Chocolate Bar Fractions Book by Jerry Pallotta Fractions by Michele Koomen Fractions by Penny Dowdy Full House by Dayle Ann Dodds Go, Fractions! by Judith Stamper Eating Fractions by Bruce McMillan If You Were a Fraction by Trisha Shaskan Inchworm and a Half by Elinor Pinczes Manga Math Mysteries: A mystery with Fractions by Melinda Thielbar Web Resources 1. http://www.jamit.com.au/fraction-games.htm 2. http://www.brainpop.com 3. http://www.gamequarium.com/fractions.htm 4. http://www.illuminations.nctm.org/Activities.aspx?grade=2 5. http://www.smartexchange.com 6. http://www.jc-schools.net/tutorials/interact-math.htm 7. http://www.glencoe.com/sites/common_assets/mathematices/ebook_assets/vmf/VMFinterface.html 8. www-k6.thinkcentral.com