6th Grade Mathematics Unit #1: Becoming a Sixth Grade Mathematician Acquiring Computational Fluency with Whole Numbers, Fractions and Decimals 33 Days Unit Overview Students build on their work in earlier grades in working with factors and multiples (4.OA.B.4, 5.NF.B.4b) as they formalize the concepts and uses of greatest common factor (including the distributive property) and least common multiple. Although students have learned all four operations with multi-digit whole numbers and decimals in earlier grades, in this unit they formalize the related standard algorithms as they build proficiency and fluency. This introductory unit also builds on students’ Grade 5 work with dividing fractions by whole numbers and whole numbers by fractions as students investigate dividing fractions by fractions. This unit sets the stage for students’ future work in finding unit rates involving ratios of fractions (7.RP.A.1). As students compute fluently with multi-digit numbers and find common factors and multiples, they need to look for and make use of structure (MP.7). They need to reason abstractly and computationally (MP.2) as they make sense of quantities and relationships in problem situations. Extending previous understandings of multiplication and division to divide fractions by fractions requires that students look for and express regularity in repeated reasoning (MP.8).
Prerequisite Skills Vocabulary Mathematical Practices
1) Understand that fractions and decimals are numbers that Factor Compute MP.1: Make sense of problems and persevere in solving represent quantities less than a whole Multiple Numerator them 2) Know the meaning of the terms “factors” and “multiples” GCF Denominator 3) Familiarity with number lines and bar models LCM Prime Factorization MP.2: Reason abstractly and quantitatively 4) Decompose fractions additively (3/4= 1/4+1/4+1/4) Divisor Mixed Number MP.3: Construct viable arguments and critique the reasoning 5) Multiply multi-digit numbers using the standard algorithm Dividend Decimal of others 6) Multiply and divide facts (up to 12 x 12) with fluency Quotient Perseverance 7) Multiply two fractions Factor Precision MP.4: Model with mathematics 8) Divide a whole number by a fraction and vice versa Product Critique MP.5: Use appropriate tools strategically Improper Algorithm Simplify Procedure MP.6: Attend to precision Reciprocal Efficiently MP.7: Look for and make use of structure Invert MP.8: Look for and express regularity in repeated reasoning Common Core State Standards Progression of Skills 5th Grade 6th Grade 7th Grade
5.NF.7: Apply and 6.NS.1: Interpret and 7.NS.3: Solve real- extend previousAccording to thecompute PARCC quotients Model of Contentworld Framework, and understandingsStandard of 3.NF.2fractions should, and serve solve as an opportunitymathematical problemsfor in- division todepth divide focus: unit word problems involving the four 6.NS.4: GCF fractions by whole involving division of operations with rational Additional numbers and whole fractions by fractions. numbers. and LCM Standards (10%) numbers by unit fractions 6.NS.3: Compute 5.NBT.6: Find whole- 6.NS.2: Fluently divide 7.NS.2: Apply and with Decimals number quotients of multi-digit numbers extend previous
whole numbers with up using the standard understandings of
6.NS.2: Divide Multi-Digit Whole Numbers to four-digit dividends algorithm. multiplication and and two-digit divisors, division and of fractions Major Standards using strategies based on to multiply and divide (70%) place value, the rational numbers. 6.NS.1: Interpret and Compute Quotients of properties of operations, Fractions and/or the relationship between multiplication and division. 5.NBT.7: Add, subtract, 6.NS.3: Fluently add, N/A multiply, and divide subtract, multiply, and According to the PARCC Model Content Framework, decimals to hundredths, divide multi-digit Standards 6.NS.1, 6.NS.2 and 6.NS.3 are culminating standards and the three using concrete models or decimals using the drawings and strategies standard algorithm for fluency expectations: based on place value each operation. N/A (4th Grade Skill) 6.NS.4: Find the N/A th “By the end of the year, 6 graders must be able to divide fractions, perform all greatest common factor four operations with decimals, and divide multi-digit whole numbers with fluency. of two whole numbers These are all culminating standards, which will not appear explicitly in future less than or equal to 100 and the least common grade levels and therefore necessitates an in-depth conceptual understanding and multiple of two whole th procedural fluency in order for students to be successful in 7 grade and beyond.” numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1-100 with a common factor as a multiple of a sum of two whole numbers with no common factor.
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Big Ideas Students Will… • Division is breaking down numbers Know/Understand Be Skilled At… into equal or fair parts. • How to use the standard algorithm for division to • Using the standard algorithm to compute multi-digit quickly solve division problems. division problems with procedural fluency • An algorithm is an efficient procedure that provides a structure to get a • How to use the standard algorithm for addition, • Flexibly, accurately, efficiently and appropriately solution to different types of problems. subtraction, multiplication, and division to quickly adding, subtracting, multiplying, and dividing multi- and accurately solve problems involving multi-digit digit decimals using the standard algorithm decimals. Dividing fractions results in a quotient • • Explaining the reasonableness of their results from that is larger than both the divisor and • That the distributive property may be used to adding, subtracting, multiplying and dividing multi- the dividend (connect this to the big express the sum of two whole numbers 1-100 with a digit decimals using their understanding of the th idea of multiplying fractions in 5 common factor as a multiple. standard algorithm. grade à the product is less than both factors) • That greatest common factor and least common • Using the distributive property to add numbers 1- multiple are ways to discuss number relationships in 100 with a common factor (e.g., 20 + 24 = 4 (5 +6)). • How does the procedure for dividing multiplication and division. fractions connect to visual • Finding greatest common factor of two whole representations (number lines, area/bar • That the answer of a fraction-by-fraction division numbers less than or equal to 100. models, etc.) and how does it relate to problem will result in a quotient smaller than either of the two fractions. the inverse operation of multiplication? • Finding the least common multiple of two whole numbers less than or equal to 12.
• That there are multiple strategies to find the quotient • The Greatest Common Factor (GCF) of of fractions including but not limited to visual • Applying and extending their understanding of two prime numbers is always 1 fraction models and equations. whole number operations to fractional numbers.
• All numbers have an infinite amount of • That a quotient represents a specific result • Developing visual models for multiplying and multiples depending on the context of the problem. dividing fractions (area models, arrays, bar and number line models) • That multiplying (a/b) x q is the same as a x q ÷ b. • Making use of benchmarks and other strategies to determine reasonableness of the solutions when operating with fractions
• Using knowledge of fractions, equivalences and the inverse relationship between multiplication and division to develop strategies
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Unit Sequence Student Friendly Objective Key Points/ Instructional Lesson Exit Ticket SWBAT… Teaching Tips Resources I. Classroom Rituals, Rules and Procedures My Math a. Class Structure – DO NOW, Fluency Drills, Material Chapter 3 Distribution, Organizing binders notes, etc “Am I Ready?” 1 b. Practice classroom procedures and articulate their purpose c. Explain the rules and expectations of our classroom My Math Chapter 4 Lessons 1 – 4 II. Pre-Assessments & Goal Setting (5.NF.4) 2 a. Fluency pre-assessment (assign levels) b. Set individual and class goals
III. Getting to Know your Fellow Mathematicians a. Survey the class (Personality Types, Learning Modality, Interests, 3 Students will learn the etc.) expectations and practices of b. Collect, display and analyze data a 6th grade mathematician. IV. Writing and Speaking like a Mathematician a. Accountable talk protocols b. Writing mathematical arguments
4 V. Investigating and Applying the 8 Mathematical Practices
VI. Explaining PARCC Task Types while reviewing 5th grade content (5.NBT.5, 5.NBT.6, 5.NF.4)
*Note: Goals IV and V should be accomplished through and with content th (i.e. critical 5 grade skills/prerequisites for this unit: 5 5.NBT.3: Read, Write and Compare Decimals 5.NF.4: Multiply Fractions
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6 Identify the Greatest Common • Distinguish between factors and 1) Create a venn diagram to My Math Factor (GCF) of up to three multiples (assess students’ ability to determine the greatest Chapter 1, Lesson 1 numbers within 100 by listing list factors and multiples of given common factor of 48 and 64 *Only require students to factors of each number in a numbers in the 4-square Do Now) list factors– prime venn diagram. • Encourage students to use reasoning 2) Is the GCF of a pair of factorization will be strategies. For instance: numbers ever equal to one introduced later o When listing factors for a of the numbers? Explain *only practice the GCF large number: cut the number with an example. problems – LCM comes in half and work down from tomorrow there, since a number can not 3) How can you find the GCF have a factor that is greater of 72 and 84 without listing https://learnzillion.com/le than half its value. all of their factors in a venn ssons/2499-find-gcf-by- o When finding the GCF of two diagram? Explain your listing-factor-pairs numbers that are close to one strategy. another (i.e. 72 and 75 – I a. Provide another pair of don’t need to go through the numbers that you would whole process because I know apply this strategy to in the GCF can not be greater order to efficiently than 3) identify the GCF. Why does this strategy work for this pair of numbers? 7 Use prime factorization to • Encourage students to explain how 1) Use prime factorization to determine the greatest common this method ensures greater accuracy determine the greatest common factor between a pair of and can be more efficient than listing factor of 96 and 54 numbers. all factor pairs as we did yesterday • Challenge students to continue applying reasoning strategies: they should still look for opportunities for reasoning to trump procedure – i.e. 98 and 90 can not have a GCF greater than 8 – so instead of prime factorization they may try dividing each by numbers 8 and less until they find the largest divisor that divides evenly into each)
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8 Determine the least common • All numbers have an infinite number 1) Create a venn diagram to My Math multiple (LCM) of up to three of multiples find the LCM of 7 and 12: Chapter 1, Lesson 1 numbers. Identify and explain • Students should deduce that *Only require students to patterns in multiples of given multiplying the two numbers together 2) Lyle listed the multiples of 4 list multiples numbers. will always result in a common and 9 all the way up to 100. He *Only practice the LCM multiple, but not necessarily the least noticed that the list of common problems common multiple (i.e. 9 x 3 = 27, but multiples has a pattern. the LCM of 9 and 3 is 18) Describe a pattern in the list of https://learnzillion.com/le • Emphasis on looking for patterns and numbers that Lyle might have ssons/2500-find-lcm-of- making use of structure seen. two-numbers-by-listing- • Suggested problem for exploratory the-multiples-of-each hook (to revisit again in closing) 3)Nikki says that the easiest way to find the LCM between Nina was finding multiples of 6. She two numbers is simply to said: “18 and 42 are both multiples of multiply the two numbers 6, and when I add them, I also get a together. For instance, 12 x 5 = multiple of 6: 18 + 42 = 60 “ 60 and the LCM between 12 and 5 is 60. Is Nikki right? Explain to Nina why adding two multiples Does her strategy work all the of 6 will always result in another multiple time? Explain and provide an of 6. example
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9 Model real world scenarios 2) Today’s hook should be an inquiry- 1) There are �� girls and �� https://learnzillion.com/le that require finding the GCF or based approach to a real world boys who want to ssons/2393-answer-word- LCM of 2 or more numbers. situation involving factors (see video) participate in a Trivia problems-by-listing- so students can connect this otherwise Challenge. If each team factor-pairs abstract concept to how we use it in must have the same number our every day lives http://s3.amazonaws.com of girls and boys, what is 3) Purpose of this lesson is for students /illustrativemathematics/i the greatest number of to “make sense of problems” and llustration_pdfs/000/000/ develop appropriate/efficient teams that can enter? How 258/original/illustrative_ strategies for modeling and many boys and girls will be mathematics_258.pdf?13 representing them. Based on their on each team? 72633257 models, students should determine whether the problem requires a GCF 2) The florist can order roses in “Geared Up 3 Act Task” or an LCM. They may then apply the bunches of one dozen and lilies (Appendix C) strategies learned yesterday to solve in bunches of 8. Last month she 4) Challenge students to connect their ordered the same number of Engage NY Module 2 models to the concept of the LCM so roses as lilies. If she ordered no Lesson 18 they can connect the more than 100 roses: (Appendix C) procedure/concept with the real world application a. What is the smallest number of bunches of each that she could have ordered? Explain b. What is another combination of roses and lilies she could have ordered? Explain:
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10 Express the sum of two whole • Recommendation: assess ability to 1) Rewrite the following “Let’s Distribute!” numbers using the distributive solve problems using the distributive addition problem using the (Appendix C) property and the greatest property in the 4 square do now for distributive property: common factor of the two “I’m ready to tackle today’s objective” https://learnzillion.com/le numbers. • Given a pair of addends, students 2) Henry uses a total of 36 blue ssons/2581-rewrite- should be able to identify if the two beads and 28 red beads to make addition-problems-as- numbers have a common factor. If 4 bracelets. Each bracelet has multiplication-problems- they do, they identify the common an equal number of blue beads using-the-distributive- factor and use the distributive property and an equal number of red property to rewrite the expression. They prove beads. Write a numerical that they are correct by simplifying expression that represents the both expressions. correct distribution of beads for If students are struggling the bracelets? Explain how you with the distributive Example: got your answer. property concept teach Use the greatest common factor and the this first: distributive property to find the sum of 36 https://learnzillion.com/le and 8. ssons/2556-write- 36 + 8 = 4 (9) + 4(2) multiples-of-a-sum- 44 = 4 (9 + 2) using-a-visual-model 44 = 4 (11) 44 = 44ü
• If students are struggling with this concept, use manipulatives to demonstrate how the distributive property can be used to turn addition problems into multiplication problems.
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Model real world problems • Students should apply a variety of 1) Omar donated a total of 90 11 requiring the greatest common strategies to solve real world hot dogs and 72 bags of chips “Back to School” factor and/or least common problems, but the emphasis should for the class picnic. Each (Appendix C) multiple, expressing this remain on modeling and connecting student will receive the same relationship using the the procedure/concept to its real world amount of refreshments. All distributive property. application. refreshments must be used. • Continue challenging students to find a. What is the greatest number opportunities to apply reasoning over of students that can attend procedures and/or to continue looking the picnic? for patterns between numbers, as well b. How many bags of chips as common structures found in LCM will each student receive? problems vs. structures found in GCF c. How many hotdogs will problems so they can begin to each student receive? determine which approach to take by simply reasoning about the scenario 2) The elementary school lunch menu repeats every 20 days; the middle school lunch menu repeats every 15 days. Both schools are serving pizza today. In how may days will both schools serve pizza again?
Flex Day (Instruction Based on Data) 12 Recommended Resources: “Factors and Multiples Puzzle” (Appendix C) “Secret Number” (Appendix C)
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13 Apply the standard algorithm • Students should begin to rely on a 1. Compute the sum: My Math for adding and subtracting problem’s structure to determine Chapter 3, Lesson 1 decimals. whether or not it will be necessary for 486.07 + 9.8 them to borrow or regroup when Engage NY Module 2 adding/subtracting decimals 2) Compute the difference: Lesson 9 • Emphasize MP6 and the importance of (Appendix C) lining up the decimals and place 4.002 - .7 values when you add and subtract
3) Riley bought two types of chocolate that weighed a total of 7.55 ounces. There were 0.75 ounces of milk chocolate and the rest was dark chocolate. How much dark chocolate, in ounces, did Riley buy? Explain your answer.
14 Estimate sums and differences • Students are more likely to make a 1. Estimate the sum of 74.835 My Math before computing in order to mathematical mistake when decimals and 2.67: Chapter 3, Lesson 2 judge the reasonableness of are involved – encourage them to your work. round to whole numbers in order to 2. Estimate the sum of 14.4 identify a “ballpark” sum or difference and 8.75. so they can judge the reasonableness of their computations. (a) Do you expect the actual • The purpose of this lesson is for them result to be greater or less to realize the power of estimating as a than your estimate? Why? mathematician (not to see estimating as “another procedure”) (b) Calculate the actual sum Compare the result to your estimate to explain whether or not your answer is reasonable
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15 Apply the standard algorithm • Encourage students to estimate 1. 48.3 × 7.39 = My Math for multiplying decimals. products first in order to identify a Chapter 3, Lessons 3-4 “ballpark answer” against which they can judge the reasonableness of their 2) Place a decimal on the right Engage NY Module 2 product side of the equal sign to make Lesson 11 the equation true. Explain your (Appendix C) reasoning.
3.58×1.25=044750 16 Apply estimation and place 1) Estimate the quotient and Engage NY value strategies in order to explain which strategy you Module 2 divide multi-digit whole used: Lessons 12 - 13 numbers (Appendix C) 82,901 ÷ 54 =
Now find the quotient using place value strategies: 17 Apply the standard algorithm • According to the “Unpacked 1) Find the quotient: My Math to solve multi‐digit division Standards Guide:” 1,534 ÷ 26 = Chapter 3, Lesson 5 problems quickly and accurately. In 6th grade, students become fluent in the 2. Mrs. Phelps bought 4 boxes Engage NY use of the standard division algorithm, of crayons at the store to share Module 2 continuing to use their understanding of with her students. She had a Lesson 14 place value to describe what they are total of 256 crayons. (Appendix C) doing. Place value has been a major emphasis in the elementary standards. (a) Mrs. Phelps wants to give This standard is the end of this each of her students an equal progression to address students’ number of the crayons she understanding of place value. bought. There are 32 students in Mrs. Phelps's class. How many Example 1: crayons should each student When dividing 32 into 8456, students get? should say, “there are 200 thirty-twos in 8456” as they write a 2 in the quotient. (b) There are 64 crayons in each They could write 6400 beneath the 8456 box. How many more boxes of
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rather than only writing 64. crayons does Mrs. Phelps need if she wants each of her students to get 12 crayons? Explain your answer using diagrams, pictures, mathematical expressions, and/or words.
3) Yari was doing the long division problem shown below. When she finishes, her teacher tells her she made a mistake.Find Yari's mistake. Explain it to her using at least 2 complete sentences. Then, re-do the long division problem correctly.
18 Divide decimals by whole 1) Sandra saved up $873.25 to My Math numbers spend on Christmas presents Chapter 3, Lesson 7 for the toy drive at the community center. She bought 27 toy robots to donate. How much did each robot cost?
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19 Find the quotient of two multi- • Inquiry Based Hook: “Sophia's dad 1) Find the quotient: My Math digit decimals paid $43.25 for 12.5 gallons of gas. Chapter 3, Lesson 8 What is the cost of one gallon of gas?” 3.0821 ÷ 17 = • Encourage estimation prior to computation to judge the Engage NY reasonableness of their work against a 2) Look at the problem below. Module 2 Lesson 15 “logical, ballpark quotient” Will the answer be greater (Appendix C) than or less than one whole? Explain how you can tell without even using the algorithm to divide:
33.8 ÷ 32.5
3) Place a decimal on the right side of the equal sign to make the equation true. Explain your reasoning
26.97÷6.2=04350
20 Flex Day (Instruction Based on Data) Recommended Resources: “Reasoning about Multiplication, Division and Place Value” (Appendix C) “Estimation is the Root of Fluency” (Appendix C) “Where Does the Decimal Go?” (Appendix C) My Math Chapter 3: Problem Solving Investigation (Pages 211 – 213) My Math Chapter 3 Mid-Chapter Check (Page 214) My Math Chapter 3 Review and Reflect (Pages 249 – 252)
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21 Create a model to represent the • From the unpacked standards 1. Create a model to illustrate: Engage NY division of a whole number by guide: Module 2 Lessons 1 – 2 a fraction and, conversely, the How many fifths are in 15? (Appendix C) division of a fraction by a Students understand that a division whole number 2 How many halves are in 3? My Math problem such as 3 ÷ is asking, 5 Chapter 4 Inquiry Lab Make observations using these 2 How many sixths are in 4? (Pages 301 – 304) “how many are in 3?” One models to infer what happens 5 to the size of the quotient when possible visual model would begin How many two-thirds are in 2? € you divide a whole number by with three whole and divide each into https://learnzillion.com/le a fraction or a fraction by a fifths. There are 7 groups of two- How many three-fourths are in 2? ssons/3584-interpret- € whole number (i.e. compare fifths in the three wholes. However, quotient-of-a-fractional- 22 the size of the quotient to the one-fifth remains. Since one-fifth is 2. The model represents the quotient division-problem size of the dividend) half of a two-fifths group, there is a of two fractions. Write an expression 1 2 that represents this model. Explain http://www.ixl.com/math remainder of . Therefore, 3 ÷ = 2 5 your reasoning. /grade-6/divide-by- 1 1 fractions-with-models 7 , meaning there are 7 groups of 2 2 two-fifths. Students interpret the https://learnzillion.com/le € € solution, explaining how division by ssons/3541-divide-a- fifths can result in an answer with whole-number-by-a- € € halves. fraction 3) What happens to the size of the • Pacing: 2 days quotient when you divide a whole https://learnzillion.com/le number by a fraction? Explain and ssons/3592-divide-a- use an example/visual to support fraction-by-a-whole- your thinking: number
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23 Use the standard algorithm to • Encourage students to connect the 1) Apply the standard algorithm to My Math divide whole numbers by inverse relationship between find the quotient: Chapter 4, Lesson 6 fractions. Judge the multiplication and division with 2 reasonableness of your the process of dividing a whole 18 ÷ /3 https://learnzillion.com/le quotient by applying what you number by a fraction (given the ssons/1290-understand- observed yesterday regarding procedure of multiplying by the 2) How much chocolate will each reciprocals-by-thinking- the size of the quotient relative inverse) – i.e. they should start to person get if 3 people share 1/2 about-inverse- to the original whole number connect the procedure to the lb of chocolate equally? Create a relationship-between- observations they made regarding model to represent and solve: multiplication-and- the size of the quotient. If you division are dividing by a proper fraction, 3) Explain: what is the effect on the *Note: the above video is when you flip to its reciprocal quotient when you divide a helpful for getting you will now be multiplying by a whole number by a proper students to understand number greater than 1 fraction vs an improper fraction? what a reciprocal is, • Students should begin to explain How does this change the result? which will help them to why and how the standard Based on what you know about develop their algorithm works (see video) the standard algorithm, explain understanding of why why this happens. these algorithms work
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24 Use area and bar models to • Pacing: 2 days 1) Create a diagram to find the Engage NY represent and solve problems Example from unpacked standards quotient: Module 2 Lessons 3 - 4 requiring the division of two guide: (Appendix C) fractions. Michael has 1 of a yard of fabric to 4/5 ÷ 3/8 2 My Math Make observations using these make book covers. Each book cover Chapter 4 Inquiry Lab models to infer what happens is made from 1 of a yard of fabric. 2) How many 3/4-cup servings are (Pages 313 - 316) to the size of the quotient 8 in 2/3 of a cup of yogurt? How How many book covers can Michael (compared to the two fractions wide is a rectangular strip of https://learnzillion.com/le make? Solution: Michael can make 4 you started with) when you land with length 3/4 mi and area ssons/1251-divide-a- book covers. divide two fractions. 1/2 square mi? fraction-by-a-fraction
25 3) Keisha is looking for patterns when dividing fractions. She takes the following two fractions:
3/4 and 2/5 She tries dividing them two different ways (using 3/4 as the divisor the first time, then 2/5 as the divisor the
second time). • Encourage students to connect their observations of the quotient a. Which of these fractions is today when dealing with two largest? fractions to what they observed b. What happens when you divide a when dividing a whole number by larger fraction by a smaller a fraction. fraction? • Encourage students to think about c. What happens when you divide a relative sizes of fractions before smaller fraction by a larger dividing (i.e. is the divisor or the fraction? dividend larger? What happens when you divide a larger fraction Use a model to illustrate and then by a smaller fraction? What explain happens when you divide a smaller fraction by a larger fraction?)
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26 Represent and solve real world • Example from the unpacked Alisa had some juice in a bottle. problems involving fraction standards guide: Then she drank 3/8 liters of juice. If division using a number line 2 1 this was ¾ of the juice that was ÷ 3 6 originally in the bottle, how much juice was there to start? Use a a. Start with a number line divided number line to represent and solve into thirds. this problem: € €
0 1
b. The problem wants to know how many sixths are in two-thirds. Divide each third in half to create sixths.
0 1 27 Connect the procedure for • students should begin to explain It requires ¼ cup of a credit to play a My Math dividing fractions with visual why the algorithm works: use the video game for one minute. Chapter 4 Lesson 7 models (i.e. bar models and relationship between number lines). Apply this multiplication and division to (a) Emma has 7/8 credit. Can she “Understanding procedure to fluently divide explain that (2/3) ÷ (3/4) = 8/9 play for more or less than one Algorithms” fractions. because 3/4 of 8/9 is 2/3. (In minute? Explain how you know. (Appendix C) general, (a/b) ÷ (c/d) = ad/bc.) – see Learnzillion video (b) How long can Emma play the https://learnzillion.com/le video game with her 7/8 credit? ssons/1315-understand- why-dividing-by-a- number-gives-the-same- result-as-multiplying-by- its-reciprocal
https://learnzillion.com/le ssons/1323-solve- division-of-fractions- problems-using- reciprocals 17 | Page
28 Create real world scenarios to 2 Engage NY • For example, the problem, ÷ match problems requiring the 3 Module 2 Lessons 5 – 6 division of fractions in order to 1 (Appendix C) can be illustrated with the represent the various 6 interpretations of quotients. following word problem: € 2 € Susan has of an hour left to 3 1 make cards. It takes her about 6 of an hour to make each card. About€ how many can she make?
€ 29 Connect models of division • skip if behind on pacing Engage NY with fractions to the inverse Module 2 Lesson 7 operation of multiplication (Appendix C)
30 Apply prior knowledge to • This should be an inquiry-based One mug of hot chocolate uses 2/3 My Math deduce the most efficient lesson where students are able to cup of cocoa powder. How many Chapter 4 Lesson 8 algorithm for dividing mixed reason through a problem mugs can Sheila make with 3 ½ cups numbers. requiring division with mixed of cocoa powder? Engage NY numbers – some may choose to Module 2 Lesson 8 use models and visuals to figure it A) Solve the problem by (Appendix C) out, while others will realize they drawing a picture. can use what they already know B) Solve using the standard “Dividing Mixed (i.e. converting to improper algorithm Numbers in Context” fractions to turn this into a basic (Appendix C) division problem for easier solving)
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31 Represent and solve real world 1) For each of the following, create “Do It Yourself” problems requiring division of a real world scenario and then (Appendix C) fractions and/or mixed represent it using a visual model numbers. to solve: http://s3.amazonaws.com/ill ustrativemathematics/illustr ation_pdfs/000/000/407/ori Given a fraction division 15 ÷ 2/3 ginal/illustrative_mathemati problem, write a real world cs_407.pdf?1372633305 contextual problem that could ½ ÷ 7 be solved using the equation. Model and solve. 4/5 ÷ 1/8
32 Flex Day (Instruction Based on Data) Recommended Resources: “Pick a Number Any Number” (Appendix C) “Engage NY Module 2 Unit Assessment” (Appendix C) “Share my Candy Performance Task” (Appendix C) My Math Chapter 4 Review and Reflect (Pages 335 – 338) 33 MCLASS Beacon End of Unit Assessment Appendix B
Note: This assessment will be administered online*
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Appendix A: Unpacked Standards Guide Source: Public Schools of North Carolina NCDPI Collaborative Workspace Unpacking Standard What do these standards mean a child will know and be able to do? 6.NS.1 Interpret and compute 6.NS.1 In 5th grade students divided whole numbers by unit fractions and divided unit fractions by whole quotients of fractions, and solve numbers. Students continue to develop this concept by using visual models and equations to divide whole word problems involving division of numbers by fractions and fractions by fractions to solve word problems. Students develop an understanding of fractions by fractions, e.g., by using the relationship between multiplication and division. visual fraction models and equations to represent the problem. For Example 1: example, create a story context for 2 2 Students understand that a division problem such as 3 ÷ is asking, “how many are in 3?” One possible (2/3) ÷ (3/4) and use a visual 5 5 fraction model to show the quotient; visual model would begin with three whole and divide each into fifths. There are 7 groups of two-fifths in the use the relationship between three wholes. However, one-fifth remains. Since one-fifth is half of a two-fifths group, there is a remainder multiplication and division to 1 2 1 1 of . Therefore, 3 ÷ = 7 , meaning there are€ 7 groups of two-fifths.€ Students interpret the solution, explain that (2/3) ÷ (3/4) = 8/9 2 5 2 2 because 3/4 of 8/9 is 2/3. (In explaining how division by fifths can result in an answer with halves. general, (a/b) ÷ (c/d) = ad/bc.) How much chocolate will each person get € € € € if 3 people share 1/2 lb of chocolate equally? How many 3/4-cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular This section represents one-half of two-fifths strip of land with length 3/4 mi and area 1/2 square mi? 2 1 Students also write contextual problems for fraction division problems. For example, the problem, ÷ 3 6 can be illustrated with the following word problem:
Example 2: € € Susan has 2 of an hour left to make cards. It takes her about 1 of an hour to make each card. About how 3 6
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€ € many can she make?
This problem can be modeled using a number line. a. Start with a number line divided into thirds.
0 1
b. The problem wants to know how many sixths are in two-thirds. Divide each third in half to create sixths.
0 1
c. Each circled part represents 1 . There are four sixths in two-thirds; therefore, Susan can make 4 cards. 6
€ Example 3: Michael has 1 of a yard of fabric to make book covers. Each book cover is made from 1 of a yard of fabric. 2 8 How many book covers can Michael make? Solution: Michael can make 4 book covers.
yd
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Example 4: 1 2 Represent ÷ in a problem context and draw a model to show your solution. 2 3 Context: A recipe requires 2 of a cup of yogurt. Rachel has 1 of a cup of yogurt from a snack pack. How 3 2 much of the recipe can Rachel make?
Explanation of Model: The first model shows 1 cup. The shaded squares in all three models show the 1 cup. 2 2 The second model shows 1 cup and also shows 1 cups horizontally. 2 3 The third model shows 1 cup moved to fit in only the area shown by 2 of the model. 2 3 2 is the new referent unit (whole) . 3 3 out of the 4 squares in the 2 portion are shaded. A 1 cup is only 3 of a 2 cup portion, so only ¾ of the 3 2 4 3 recipe can be made.
1 1 2 2
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6.NS.2 Fluently divide multi-digit 6.NS.2 In the elementary grades, students were introduced to division through concrete models and various numbers using the standard strategies to develop an understanding of this mathematical operation (limited to 4-digit numbers divided by algorithm. 2-digit numbers). In 6th grade, students become fluent in the use of the standard division algorithm, continuing to use their understanding of place value to describe what they are doing. Place value has been a major emphasis in the elementary standards. This standard is the end of this progression to address students’ understanding of place value.
Example 1: When dividing 32 into 8456, students should say, “there are 200 thirty-twos in 8456” as they write a 2 in the quotient. They could write 6400 beneath the 8456 rather than only writing 64.
6.NS.3 Fluently add, subtract, 6.NS.3 Procedural fluency is defined by the Common Core as “skill in carrying out procedures flexibly, multiply, and divide multi-digit accurately, efficiently and appropriately”. In 4th and 5th grades, students added and subtracted decimals. decimals using the standard Multiplication and division of decimals were introduced in 5th grade (decimals to the hundredth place). At algorithm for each operation. the elementary level, these operations were based on concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction. In 6th grade, students become fluent in the use of the standard algorithms of each of these operations. The use of estimation strategies supports student understanding of decimal operations.
Example 1: First estimate the sum of 12.3 and 9.75. Solution: An estimate of the sum would be 12 + 10 or 22. Student could also state if their estimate is high or low.
Answers of 230.5 or 2.305 indicate that students are not considering place value when adding.
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6.NS.4 Find the greatest common In elementary school, students identified primes, composites and factor pairs (4.OA.4). In 6th grade factor of two whole numbers less students will find the greatest common factor of two whole numbers less than or equal to 100. than or equal to 100 and the least For example, the greatest common factor of 40 and 16 can be found by common multiple of two whole 1) listing the factors of 40 (1, 2, 4, 5, 8, 10, 20, 40) and 16 (1, 2, 4, 8, 16), then taking the greatest numbers less than or equal to 12. common factor (8). Eight (8) is also the largest number such that the other factors are relatively Use the distributive property to prime (two numbers with no common factors other than one). For example, 8 would be express a sum of two whole multiplied by 5 to get 40; 8 would be multiplied by 2 to get 16. Since the 5 and 2 are relatively numbers 1–100 with a common prime, then 8 is the greatest common factor. If students think 4 is the greatest, then show that 4 factor as a multiple of a sum of would be multiplied by 10 to get 40, while 16 would be 4 times 4. Since the 10 and 4 are not two whole numbers with no relatively prime (have 2 in common), the 4 cannot be the greatest common factor. common factor. For example, 2) listing the prime factors of 40 (2 • 2 • 2 • 5) and 16 (2 • 2 • 2 • 2) and then multiplying the common express 36 + 8 as 4 (9 + 2). factors (2 • 2 • 2 = 8).
2 Factors of 16 2 2 5 Factors of 40 2
The product of the intersecting numbers is the GCF
Students also understand that the greatest common factor of two prime numbers is 1.
Example 1: What is the greatest common factor (GCF) of 18 and 24?
2 3 Solution: 2 ∗ 3 = 18 and 2 ∗ 3 = 24. Students should be able to explain that both 18 and 24 will have at least one factor of 2 and at least one factor of 3 in common, making 2 ∗ 3 or 6 the GCF. Given various pairs of addends using whole numbers from 1-100, students should be able to identify if the two numbers have a common factor. If they do, they identify the common factor and use the distributive property to rewrite the expression. They prove that they are correct by simplifying both expressions.
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Example 2: Use the greatest common factor and the distributive property to find the sum of 36 and 8. 36 + 8 = 4 (9) + 4(2) 44 = 4 (9 + 2) 44 = 4 (11) 44 = 44ü
Example 3: Ms. Spain and Mr. France have donated a total of 90 hot dogs and 72 bags of chips for the class picnic. Each student will receive the same amount of refreshments. All refreshments must be used. d. What is the greatest number of students that can attend the picnic? e. How many bags of chips will each student receive? f. How many hotdogs will each student receive?
Solution: a. Eighteen (18) is the greatest number of students that can attend the picnic (GCF). b. Each student would receive 4 bags of chips. c. Each student would receive 5 hot dogs.
Students find the least common multiple of two whole numbers less than or equal to twelve. For example, the least common multiple of 6 and 8 can be found by 1) listing the multiplies of 6 (6, 12, 18, 24, 30, …) and 8 (8, 26, 24, 32, 40…), then taking the least in common from the list (24); or 2) using the prime factorization. Step 1: find the prime factors of 6 and 8. 6 = 2 • 3 8 = 2 • 2 • 2 Step 2: Find the common factors between 6 and 8. In this example, the common factor is 2 Step 3: Multiply the common factors and any extra factors: 2 • 2 • 2 • 3 or 24 (one of the twos is in common; the other twos and the three are the extra factors.
Example 4: The elementary school lunch menu repeats every 20 days; the middle school lunch menu repeats every 15 days. Both schools are serving pizza today. In how may days will both schools serve pizza again?
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Solution: The solution to this problem will be the least common multiple (LCM) of 15 and 20. Students should be able to explain that the least common multiple is the smallest number that is a multiple of 15 and a multiple of 20. One way to find the least common multiple is to find the prime factorization of each number: 2 2 ∗ 5 = 20 and 3 ∗ 5 = 15. To be a multiple of 20, a number must have 2 factors of 2 and one factor of 5 (2 ∗ 2 ∗ 5). To be a multiple of 15, a number must have factors of 3 and 5. The least common multiple of 20 and 15 must have 2 factors of 2, one factor of 3 and one factor of 5 ( 2 ∗ 2 ∗ 3 ∗ 5) or 60.
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