Grade 6 Unit 1: Multiplying and Dividing (4 Weeks) Stage 1 – Desired Results Established Goals Unit Description Using the meanings of fractions, multiplication and division, and the relationship between multiplication and division, students will understand and explain why the procedures for dividing fractions make sense. They will interpret and compute quotients of fractions and solve word problems involving division of fractions by fractions. These skills can be applied in solving volume problems where the edge lengths have fractional values. By the end of 6 th grade, students are expected to fluently (with speed and accuracy) divide multi-digit numbers and compute with multi-digit decimals, using the standard algorithms. These skills should be solidified in this unit. The Mathematical Practices should be evident throughout instruction and connected to the content addressed in this unit. Students should engage in mathematical tasks that provide an opportunity to connect content and practices. Common Core Learning Standards 6.NS.1 Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. For example, create a story context for (2/3) ÷ (3/4) and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that (2/3) ÷ (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general, (a/b) ÷ (c/d) = ad/bc.) How much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 3/4-cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with length 3/4 mi and area 1/2 square mi? 6.G.2 Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems. 6.NS.2 Fluently divide multi-digit numbers using the standard algorithm. 6.NS.3 Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation. 6.NS.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. For example, express 36 + 8 as 4 (9 + 2). Bridge Guidance(Concepts taught in earlier grades): 5.NF.1 Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators. For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12. (In general, a/b + c/d = (ad + bc)/bd.). 5.NF.4: Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction. a. Interpret the product (a/b) × q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a × q ÷ b. For example, use a visual fraction model to show (2/3) × 4 = 8/3, and create a story context for this equation. Do the same with (2/3) × (4/5) = 8/15. (In general, (a/b) × (c/d) = ac/bd.) b. Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas. 5.NF.5: Interpret multiplication as scaling (resizing), by: a. Comparing the size of a product to the size of one factor on the basis of the size of the other fact, without performing the indicated multiplication.
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Grade 6 Unit 1: Multiplying and Dividing (4 Weeks)
Stage 1 – Desired Results
Established Goals Unit Description Using the meanings of fractions, multiplication and division, and the relationship between multiplication and division, students will understand and explain why the procedures for dividing fractions make sense. They will interpret and compute quotients of fractions and solve word problems involving division of fractions by fractions. These skills can be applied in
solving volume problems where the edge lengths have fractional values. By the end of 6th
grade, students are expected to fluently (with speed and accuracy) divide multi-digit numbers and compute with multi-digit decimals, using the standard algorithms. These skills should be solidified in this unit. The Mathematical Practices should be evident throughout instruction and connected to the content addressed in this unit. Students should engage in mathematical tasks that provide an opportunity to connect content and practices. Common Core Learning Standards
6.NS.1 Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. For example, create a story context for (2/3) ÷ (3/4) and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that (2/3) ÷ (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general, (a/b) ÷ (c/d) = ad/bc.) How much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 3/4-cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with length 3/4 mi and area 1/2 square mi? 6.G.2 Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems. 6.NS.2 Fluently divide multi-digit numbers using the standard algorithm. 6.NS.3 Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation. 6.NS.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. For example, express 36 + 8 as 4 (9 + 2).
Bridge Guidance(Concepts taught in earlier grades): 5.NF.1 Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators. For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12. (In general, a/b + c/d = (ad + bc)/bd.). 5.NF.4: Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.
a. Interpret the product (a/b) × q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a × q ÷ b. For example, use a visual fraction model to show (2/3) × 4 = 8/3, and create a story context for this equation. Do the same with (2/3) × (4/5) = 8/15. (In general, (a/b) × (c/d) = ac/bd.)
b. Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas.
5.NF.5: Interpret multiplication as scaling (resizing), by:
a. Comparing the size of a product to the size of one factor on the basis of the size of the other fact, without performing the indicated multiplication.
b. Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (n×a)/(n×b) to the effect of multiplying a/b by 1.
5.NF.6: Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem.
5.NF.7 5.NF.7: Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.
a. Interpret division of a unit fraction by a non-zero whole number, and compute such quotients. For example, create a story context for (1/3) ÷ 4, and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that (1/3) ÷ 4 = 1/12 because (1/12) × 4 = 1/3.
b. Interpret division of a whole number by a unit fraction, and compute such quotients. For example, create a story context for 4 ÷ (1/5), and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that 4 ÷ (1/5) = 20 because 20 × (1/5) = 4.
c. Solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem. For example, how much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 1/3- cup servings are in 2 cups of raisins?
Common Core Standards of Mathematical Practice
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.
ESL Language Standards Standard 1 1. Identify and use reading and listening strategies to make text comprehensible and meaningful. 7.Present information clearly in a variety of oral and written forms for different audiences and purposes related to all academic content areas. 12. Convey information and ideas through spoken and written language, using conventions and features of American English appropriate to audience and purpose. 15. Apply self-monitoring and self-correcting strategies for accurate language production and oral and written presentation, using established criteria for effective presentation of information. 16. Apply learning strategies to acquire information and make texts comprehensible and meaningful. 9. Apply learning strategies to examine and interpret a variety of materials. Standard 4 5. Explain actions, choices, and decisions in social and academic situations.
Big Ideas 1. The meanings of each operation on fractions and
decimals are the same with the meanings of the operations on whole numbers.
Essential Questions 1. How does knowing the meaning of operations with whole numbers help you solve problems with fractions and decimals? 1. How do partition and measurement concepts relate to operations with whole numbers and fractions? 1. What determines the size of the quotient when dividing fractions?
2. Basic facts and algorithms for operations with rational numbers use notions of equivalence to transform calculations into simpler ones.
1. How does multiplying and dividing fractions relate to multiplying and dividing whole numbers? 2. How do I choose the most efficient strategy to solve problems with fractions and decimals depending on the context?
Content (Students will know….) A. The standard algorithms for computing with multi-digit whole numbers and decimals fluently B. Sums of numbers that share a common factor can be rewritten using the distributive property C. Fractions can be divided using the partition and measurement methods D. Division of fractions can be represented with models, equations, and stories E. Dividing fractions is related to multiplying and dividing whole numbers F. Volume of right rectangular prisms
Skills (Students will be able to…) A1. Fluently divide multi-digit whole numbers using the standard algorithm A2. Fluently add, subtract, multiply, and divide with multi-digit decimal numbers B1.Identify the greatest common factor of two whole numbers less than or equal to 100. B2. Identify the least common multiple of two whole numbers less than or equal to 12. B3. Apply distributive property to express the sum of two whole numbers (1-100) with a common factor. C1. Choose the appropriate method to solve problems involving division of fractions:
a whole number divided by fraction
a fraction by a whole number
a fraction divided by fraction
a mixed number divided by fraction C2. Divide fractions using the partition method C3. Divide fractions using the measurement method C4. Choose and apply the most appropriate method when dividing fractions C5. Use estimation to determine relative size of the quotient. D1. Use visual fraction models to represent division of fractions
Number line
counters
area models
circle/pie D2. Write equations to represent division of fractions D3. Write a story problem to represent division of fractions E1. Apply previous understandings of multiplication and division facts of whole numbers to divide fractions. E2. Use fraction multiplication facts to solve related division equations with fractions. F1. Find the volume of the right rectangular prism using unit cubes. F2. Discover the volume formula for the right rectangular prism using unit cubes. F3. Apply formulas for volume of right rectangular prism using V=lwh and V=Bh with whole numbers and unit fractional edge lengths
Terms/ Vocabulary Sum, difference, quotient, dividend, divisor, reciprocal, product, factor, multiple, greatest common factor, least common multiple, distributive property, volume, algorithm, base (of a prism), right rectangular prism, unit cube
Stage 2 – Assessment Evidence
Performance Task Initial Assessment: Back To School Final Assessment: Sara’s Birthday Party
Other Evidence Teacher observations, conferencing, teacher designed formative assessment pieces, student work, exit slips, journals, reflections, etc.
Stage 3 – Learning Plan
Impact Mathematics CCLS Aligned Lessons: The following lessons will support some of the essential questions aligned in this unit map. 6.NS.1 Impact Lesson 4.2 – Multiply and Divide Fractions , 226-232, parts of 233-241 6.G 6.G.2 Impact Lesson 7.3 – Surface Area and Volume , 434-448 ; Impact Lesson 7.4 – Capacity 451-453 6.NS.2 Fluently divide multi-digit numbers using the standard algorithm, Impact Lesson 1.1 – Patterns in Geometry, 23 . Interpreting a Division Computation: www.illustrativemathematics.org
Incorrect Division: www.illustrativemathematics.org Long division and why it works: http://www.homeschoolmath.net/teaching/md/long_division_why.php
6.NS.4 Factors: pg 9 Back to school Pgs. 10-11 Secret Number pg. 12 Let's Distribute pg. 14 6.NS.1 Dividing Fractions pg. 15 Fractional Divisors pg. 16 6.NS.2 Understanding Algorithm pg. 17 Do it Yourself pg. 19 6.NS.3 Estimation and Fluency pg 20
Elementary and Middle School Mathematics by John A. Van De Walle-Chapter 16 pgs. 264-279 (Partition and Measurement concept) Resource Masters-Impact Mathematics:
Lesson 4.2 Problem-solving Practice-Multiply and Divide fractions pg 19
Lesson 4.2 Enrichment-Multiply and Divide fractions pg. 20
Lesson 4.2 Quick Quiz-Multiply and Divide Fractions pg 21