Unit 1: Fractions and Decimals (no calculators) Time Frame: Quarter 1 about 25 days Connections to Previous Learning: By the end of grade 4, students are expected to fluently add, subtract, multiply and divide multi-digit whole numbers. In grade 5, procedural fluency is defined by the Common Core as skill in carrying out procedures flexibly, accurately, efficiently and appropriately. In the elementary grades, students were introduced to division through concrete models and various strategies to develop an understanding of this mathematical operation (limited to 4-digit numbers divided by 2-digit numbers). In 6th grade, students become fluent in the use of the standard division algorithm. This understanding is foundational for work with fractions and decimals in 7th grade. Students are also expected to fluently add, subtract, multiply and divide decimals to hundredths. 6th graders will extend these skills to include any decimals using standard algorithms for each operation. Focus within the Grade Level: Sixth graders will extend these skills to dividing a fraction by a fraction. There is not one single standard algorithm; however, a standard algorithm is a strategy that works every time. Some well-known algorithms include partial sums, differences, products and quotients, tradition algorithms for each operation (including regrouping). However, some students develop algorithms that are efficient, allow for fluent calculation and demonstrate understanding of both the value and the operations. If students can justify their reasoning for an algorithm, it may be categorized as a standard algorithm. Connections to Subsequent Learning: Students can add, subtract, multiply and divide using decimals and fractions to solve complex application problems. In seventh grade students will use percents and scale factors to determine percent of increase or percent of decrease, discounts and markups. Mathematical Practices - Practices to be explicitly emphasized are indicated in BOLD 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. Unit 1 Clover Park School District 6/5/15 Page 1
Students will be able to independently use their learning to Stage 1 Desired Results Transfer Goals Apply a sense of number size to their lives. UNDERSTANDINGS Meaning Goals ESSENTIAL QUESTIONS Students will understand that The two types of division quotative (partitive) and measurement are applied to fractions and decimals as well as to whole numbers. Multiplication and division are inverse operations. The relationship of the location of the digits and the value of the digits is part of understanding multi-digit operations. Division can be represented using multiple formats (manipulatives, diagrams, real-life situations, equations). Operations on decimals and whole numbers are based upon place value relationships. How is division related to realistic situations and to other operations? What role does place value play in multi-digit operations? How can division be represented and interpreted? In what ways can the area of a net be determined? Students will know Standard algorithms for addition, subtraction, multiplication and division of multi-digit decimals Acquisition Goals Students will be skilled at Compute quotients of fractions divided by fractions. (6.NS.1) Explain the meaning of a quotient determined by division of fractions, using visual fraction models, equations, real-life situations, and language. (6.NS.1) Divide multi-digit numbers fluently using the standard algorithm. (6.NS.2) Fluently add, subtract, multiply and divide decimals to solve problems. (6.NS.3) Identify the factors of any whole number less than or equal to 100. (6.NS.4) Determine the Greatest Common Factor of two or more whole numbers less than or equal to 100. (6.NS.4) Identify the multiples of two whole numbers less than or equal to 12 and determine the Least Common Multiple. (6.NS.4) 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 Unit 1 Clover Park School District 6/5/15 Page 2
Calculators 6.NS.1 6.NS.2 6.NS.3 6.NS.4 no no no no Materials Needed for Unit: Holt Course 1 Holt Course 1 Common Core Curriculum Companion Additional Materials Engage NY 6 th Grade Module 2 CMP 2 Bits and Pieces II Prerequisite Skills Add, subtract and multiply fractions. Divide fractions by whole numbers and whole numbers by fractions. Use area models for fraction or decimal computation situations. Fluently add, subtract, multiply and divide whole numbers. Stage 1 Established Goals: Common Core State Standards for Mathematics 6.NS.A Apply and extend previous understandings of multiplication and division to divide fractions by fractions. 6.NS.A.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. 6.NS.B Compute fluently with multi-digit numbers and find common factors and multiples. 6.NS.B.2 Fluently divide multi-digit numbers using the standard algorithm. 6.NS.B.3 Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation. 6.NS.B.4 Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole 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. Major Clusters Supporting Clusters Additional Clusters Explanations, Examples, and Comments Overview of Suggested Assessments & Tasks/Activities for Unit Vocabulary will be listed next to each standard - multi-digit, quotient, dividend, divisor, remainder, reciprocal, inverse, greatest common factor, least common multiple, prime numbers, composite, factor, multiple, distributive, prime factorization, visual fraction model Mid-Unit Assessment End-of-Unit Assessment Task P. 186 Engage NY Module 2 Teacher Materials Formative assessments for this unit Available Performance Tasks Georgia Department Of Education: Estimation is the Root of Fluency 6.NS.B.3 Finding Common Factors 6.NS.B.4 Back to School 6.NS.B.4 Unit 1 Clover Park School District 6/5/15 Page 3
Secret Number 6.NS.B.4 Let s Distribute 6.NS.B.2/6.NS.4 Where Does the Decimal Go? P. 143 6.NS.3 Illuminations The Factor Game 6.NS.4 http://illuminations.nctm.org/lesson.aspx?id=1010 Inside Mathematics Rabbit Costumes 6.NS.1 6.NS.B.2 Vocab - multi-digit, quotient, divisor, remainder, Students are expected to fluently and accurately divide multi-digit whole numbers. Divisors can be any number of digits at this grade level. As students divide they should continue to use their understanding of place value to describe what they are doing. When using the standard algorithm, students language should reference place value. For example, when dividing 32 into 8456, as they write a 2 in the quotient they should say, there are 200 thirty-twos in 8456 and could write 6400 beneath the 8456 rather than only writing 64. TI Activities Multiplication & Division of Decimals Dividing by Fractions Distributive Property Order of Operations Stage 3 MATERIALS BY STANDARD(S): Teacher should use assessment data to determine which of the materials below best meet student instructional needs. All materials listed may not be needed. This is the first time students are introduced to the algorithm of long division. In 5 th grade, they use strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. 6.NS.B.2 Holt Course 1 Lesson 1-3 Exponents Holt Course 1 Lesson 1-4 Order of Operations Holt Course 1 Lesson 4-1 Divisibility Holt Course 1 Lesson 1-1A Common Core Companion Divide Multi-Digit Whole Numbers EXTRAS Unit 1 Clover Park School District 6/5/15 Page 4
Example: If additional Materials are needed EngageNY, Module 2, Lessons 13-15 (3) http://www.engageny.org/sites/default/files/resource/attachments/g6- m2-teacher_materials.pdf Unit 1 Clover Park School District 6/5/15 Page 5
6.NS.B.3 The use of estimation strategies supports student understanding of operating with decimals. Example: First, students estimate the sum and then find the exact sum of 14.4 and 8.75. An estimate of the sum might be 14 + 9 or 23. Students may also state if their estimate is low or high. They would expect their answer to be greater than 23. They can use their estimates to selfcorrect. Answers of 10.19 or 101.9 indicate that students are not considering the concept of place value when adding (adding tenths to tenths or hundredths to hundredths) whereas answers like 22.125 or 22.79 indicate that students are having difficulty understanding how the four-tenths and seventy-five hundredths fit together to make one whole and 25 hundredths. Students use the understanding they developed in 5th grade related to the patterns involved when multiplying and dividing by powers of ten to develop fluency with operations with multi-digit decimals. 6.NS.B.3 Holt Course 1 Lesson 3-2 Estimating Decimals Review (1) Holt Course 1 Lesson 3-5 Multiplying Decimals Review (1) Holt Course 1 Lesson 3-5 Hands on Lab Explore Decimal Multiplication and Division (1) Holt Course 1 Lesson 3-6 Dividing Decimals by Whole Numbers (1) Holt Course 1 Lesson 3-7 Dividing by Decimals (1) Holt Course 1 Lesson 3-8 Interpret the Quotient (1) Holt Course 1 Lesson 4-4 Fraction and Decimal Conversion (1) Recommended Performance Tasks From Georgia Department Of Education: Estimation is the Root of Fluency Where Does the Decimal Go? P. 143 6.NS.3 Additional Resources if needed Holt Course 1 Lesson 3-1 Representing, Comparing, and Ordering Decimals - Review (1) Holt Course 1 Lesson 3-3 Hands on Lab: Explore Decimal Addition & Subtraction Review (1) Holt Course 1 Lesson 3-3 Adding and Subtracting Decimals Review (2) Unit 1 Clover Park School District 6/5/15 Page 6
6.NS.B.4 Vocab - greatest common factor, least common multiple, prime numbers, composite, factor, multiple, distributive, prime factorization In elementary school, students identified primes, composites and factor pairs (4.OA.4). In 6th grade students will find the greatest common factor of two whole numbers less than or equal to 100. Examples: What is the greatest common factor (GCF) of 24 and 36? How can you use factor lists or the prime factorizations to find the GCF? Solution: 2² 3 = 12. Students should be able to explain that both 24 and 36 have 2 factors of 2 and one factor of 3, thus 2 x 2 x 3 is the greatest common factor.) What is the least common multiple (LCM) of 12 and 8? How can you use multiple lists or the prime factorizations to find the LCM? Solution: 2³ 3 = 24. Students should be able to explain that the least common multiple is the smallest number that is a multiple of 12 and a multiple of 8. To be a multiple of 12, a number must have 2 factors of 2 and one factor of 3 (2 x 2 x 3). To be a multiple of 8, a number must have 3 factors of 2 (2 x 2 x 2). Thus the least common multiple of 12 and 8 must have 3 factors of 2 and one factor of 3 (2 x 2 x 2 x 3). Rewrite 84 + 28 by using the distributive property. Have you divided by the largest common factor? How do you know? 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. 6.NS.B.4 Holt Course 1 Lesson 4-2 Factors and Prime Factorization (1) Holt Course 1 Lesson 4-3 Greatest Common Factor (1) Holt Course 1 Lesson 5-1 Least Common Multiple (1) Suggested Performance Tasks: Georgia Department Of Education Finding Common Factors 6.NS.B.4 Back to School 6.NS.B.4 Secret Number 6.NS.B.4 Let s Distribute 6.NS.2/6.NS.4 Illuminations The Factor Game 6.NS.4 http://illuminations.nctm.org/lesson.aspx?id=1010 Additional Materials if needed Engage NY Unit Module 2 Lesson 16-18 *this link provides activity for applying greatest common factor as it relates to the distributive property. (see example given starting on p. 147) http://www.engageny.org/sites/default/files/resource/ attachments/g6-m2-teacher_materials.pdf o 27 + 36 = 9 (3 + 4) o 31 + 80 63 = 9 x 7 63 = 63 There are no common factors. I know that because 31 is a prime number, it only has 2 factors, 1 and 31. I know that 31 is not a factor of 80 because 2 x 31 is 62 and 3 x 31 is 93. Unit 1 Clover Park School District 6/5/15 Page 7
6.NS.A.1 Vocab dividend, reciprocal, inverse, visual fraction model Contexts and visual models can help students to understand quotients of fractions and begin to develop the relationship between multiplication and division. Model development can be facilitated by building from familiar scenarios with whole or friendly number dividends or divisors. Computing quotients of fractions build upon and extends student understandings developed in Grade 5. Students make drawings, model situations with manipulatives, or manipulate computer generated models. Example: Mary read 573 pages during her summer reading challenge. She was only required to read 399 pages. How many extra pages did Mary read beyond the challenge requirements? Students may use several approaches to solve the problem including the traditional algorithm. Examples of other methods students may use are listed below: 399 + 1 = 400, 400 + 100 = 500, 500 + 73 = 573, therefore 1+ 100 + 73 = 174 pages (Adding up strategy) 400 + 100 is 500; 500 + 73 is 573; 100 + 73 is 173 plus 1 (for 399, to 400) is 174 (Compensating strategy) Take away 73 from 573 to get to 500, take away 100 to get to 400, and take away 1 to get to 399. Then 73 +100 + 1 = 174 (Subtracting to count down strategy) 399 + 1 is 400, 500 (that s 100 more). 510, 520, 530, 540, 550, 560, 570, (that s 70 more), 571, 572, 573 (that s 3 more) so the total is 1 + 100 + 70 + 3 = 174 (Adding by tens or hundreds strategy) 6.NS.A.1 *Review if Needed *Holt Course 1 Lesson 4-6 Mixed Numbers and Improper Fractions *Holt Course 1 Lesson 5-7 Multiplying Fractions by Whole Numbers (1) *Holt Course 1 Lesson 5-7 Multiplying Fractions (1) *Holt Course 1 Lesson 5-8 Multiplying Mixed Numbers (1) Holt Course 1 Lesson 5-7 Lab Model Fraction Multiplication (.5) Holt Course 1 5-7 Multiplying Fractions (1) Holt Course 1 5-8 Multiplying Fractions and Mixed Numbers Holt Course 1 Lesson 5-9A Common Core Companion TE (p. 100) & SE (P. 12) Hands on Lab: Model Fraction Division in Context (1) Holt Course 1 Lesson 5-9 Dividing Fractions & Mixed Numbers (1) Additional Resources, if necessary Holt Course 1 Lesson 5-9 Lab Model Fraction Division (1) CMP 2 Bits and Pieces II Investigation 1 Estimating with Fractions CMP 2 Bits and Pieces II Investigation 2 Adding and Subtracting Fractions CMP 2 Bits and Pieces II Investigation 3 Multiplying with Fractions CMP 2 Bits and Pieces II Investigation 4 Dividing with Fractions Suggested Performance Tasks: Inside Mathematics Rabbit Costumes (6.NS.1) Computation algorithm. A set of predefined steps applicable to a class of problems that gives the correct result in every case when the steps are carried out correctly. Computation strategy. Purposeful manipulations that may be chosen for specific problems, may not have a fixed order, and may be aimed at converting one problem into another. (Progressions for the CCSSM; Number and Operation in Base Ten, CCSS Writing Team, April 2011, page 2) Example: 3 people share 1 pound of chocolate. How much of a pound of chocolate does each person 2 Unit 1 Clover Park School District 6/5/15 Page 8
get? Solution: Each person gets 1 lb of chocolate. 6 Manny has 1 2 yard of fabric to make book covers. Each book is made from 1 8 yard of fabric. How many book covers can Manny make? Solution: Manny can make 4 book covers. Unit 1 Clover Park School District 6/5/15 Page 9
Unit 1 Clover Park School District 6/5/15 Page 10
Evaluative Criteria/Assessment Level Descriptors (ALDs): 6.NS.A (SBAC Target B) Level 4 students should be able to use visual models in settings where smaller fractions are divided by larger fractions. They should also understand and apply the fact that a fraction multiplied or divided by 1 in the form of a/a is equivalent to the original fraction. Level 3 students should be able to apply and extend previous understandings of multiplication and division to divide a fraction by a fraction and be able to connect to a visual model. Level 2 students should be able to apply and extend previous understandings of multiplication and division to divide a whole number by a fraction between 0 and 1, divide a mixed number by a whole number, and be able to connect to a visual model. Level 1 students should be able to apply and extend previous understandings of multiplication and division to multiply a fraction by a fraction, divide a fraction by a whole number, and be able to connect to a visual model. They should understand the effect that a fraction greater than or less than 1 has on a whole number when multiplied and use or create visual models when multiplying a whole number by a fraction between 0 and 1. Stage 2 - Evidence OTHER ASSESSMENT EVIDENCE: Formative assessments (available on P: drive) Spaced Learning Over Time (SLOT - entrance & exit slips) Checking for understanding (CFU s) Lesson Quizzes Classroom Assessments SBA Examples (none at this time) Claim 1 Item Specs 6.NS.B (SBAC Target C) Level 4 students should be able to make generalizations regarding multiples and factors of sets of numbers (e.g., state that a particular set of numbers is relatively prime). Level 3 students should be able to fluently divide multi-digit numbers and add, subtract, multiply, and divide multi-digit decimal numbers. They should be able to find the greatest common factor of two numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Level 2 students should be able to divide multi-digit whole numbers and add and subtract multi-digit decimal numbers. They should be able to find common factors of two numbers less than or equal to 100 and multiples of two numbers less than or equal to 12. Level 1 students should be able to add, subtract, and multiply multi-digit whole numbers and decimals to hundredths. They should be able to use the distributive property to express the sum of two whole numbers with a common factor. Unit 1 Clover Park School District 6/5/15 Page 11
LEARNING ACTIVITIES: Stage 3 Learning Plan Sample Summary of Key Learning Events and Instruction that serves as a guide to a detailed lesson planning NOTES: **Days may change depending on any tasks/activities or assessing you choose to do. 6.NS.B.2 Day 1: Exponents Holt, Course 1, Lesson 1-3 Day 2: Order of Operations Remember to use GEMA, not PEMDAS Holt, Course 1, Lesson 1-4 Day 3: Divisibility (1) Holt Course 1 Lesson 4-1 Day 4-6: Divide Multi-Digit Whole Numbers Holt Course 1 Lesson 1-1A Common Core Companion 6.NS.B.3 Optional: Estimating Decimals Review (1) Holt Course 1 Lesson 3-2 Day 7: Multiplying Decimals Review (1) Holt Course 1 Lesson 3-5 Day 8: Hands on Lab Explore Decimal Multiplication and Division (1) Holt Course 1 Lesson 3-5 Day 9: Dividing Decimals by Whole Numbers (1) Holt Course 1 Lesson 3-6 Day 10: Dividing by Decimals (1) Unit 1 Clover Park School District 6/5/15 Page 12
Holt Course 1 Lesson 3-7 Stage 3 Learning Plan Sample Day 11: Interpret the Quotient (1) Holt Course 1 Lesson 3-8 Day 12: Fraction and Decimal Conversions Holt, Course 1, Lesson 4-4 6.NS.B.4 Day 13: Factors and Prime Factorization (1) Holt Course 1 Lesson 4-2 Day 14: Greatest Common Factor (1) Holt Course 1 Lesson 4-3 Day 15: Least Common Multiple (1) Holt Course 1 Lesson 5-1 6.NS.A.1 Day 16: Model Fraction Multiplication (1) Holt Course 1 5-7 Lab (Review if needed) Day 17: Multiplying Fractions (1) Holt Course 1 5-7 Day 18: Multiplying Fractions and Mixed Numbers Holt Course 1 5-8 Day 19: Hands on Lab: Model Fraction Division in Context (1) Holt Course 1 Lesson 5-9A Common Core Companion Day 20: Dividing Fractions & Mixed Numbers (1) Holt Course 1 Lesson 5-9 Daily Lesson Plan Format Learning Target: Warm-up: Activities: Whole Group: Small Group/Guided/Collaborative/Independent: Unit 1 Clover Park School District 6/5/15 Page 13
Whole Group: Checking for Understanding (before, during and after): Assessments: Stage 3 Learning Plan Sample Unit 1 Clover Park School District 6/5/15 Page 14