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. 2 | Page 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 3 | Page 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 4 | Page 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.