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.3 Impact Lesson 3.4 – Apply Properties , 190; Impact Lesson 4.1 – Add and Subtract Fractions, 215 ; Impact Lesson 4.3 – Multiply and Divide Decimals , 242-263 6.NS.4 Impact Lesson 3.4 – Apply Properties , 175-183; Impact Lesson 2.1 – Factors and Multiples (*In Course 2) , 74-91

https://www.georgiastandards.org/CommonCore/Common%20Core%20Frameworks/CCGPS_Math_6_6thGrade_Unit1SE.pdf

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

Chapter 4 Test Form A and B

Grade 6 Unit 1

Initial Performance Task: Back to School

1. Every school year (10 months) Michael spends about $540 on after-school snacks.

a. How much does he spend each month?

b. Assuming there are 20 school days in a month, how much does he spend per day on

snacks?

2. Michael has 3/4 of a yard of fabric to make book covers. Each book cover is made from 1/8 of

a yard of fabric. Will Michael have enough fabric to cover 7 books? If not, how many can he

cover? Show your mathematical thinking below:

3. Michael’s mom takes him to Staples for school supplies. Here is a portion of the receipt:

Item Price

Pencils (12 pack) $3.24

Scientific Calculator $15.89

3- Ring Binder $2.49

Student Planner $8.64

Gel Pens (6 pack) $5.58

a) Estimate the total cost of the items shown in the receipt. Show your work:

b) What is the price of one pencil? Show your work:

c) How much more did he spend on the calculator than the planner and pens combined?

d) After the first day of class, Michael realized he needed to buy 4 more binders. How much

more did he have to spend?

4. Every four school days, the cafeteria serves pizza. Every third school day, the cafeteria serves

salad. If they served both pizza and salad on the Wednesday, September 6th, what is the next

day and date that the cafeteria will serve both again? Show your mathematical thinking below:

5. Mrs. John, Michael’s teacher has 96 stickers and 72 mini-erasers that she gives out as prizes.

a) If each student in her class receives the same number of both stickers and erasers, and

there are none leftover, how many students are in Mrs. John’s class? Show how you know:

b) Using Greatest Common Factor and distributive property, find the sum of 96 + 72

6. Mrs. John has a special box that she fills with cubes whenever she notices a student being

helpful to another person. When the box is full, she throws her class a pizza party then starts

all over again. This box measures 8 inches x 7 inches x 4 inches.

a. What is the volume of the box? Show your work:

b. How many one inch cubes will it take to fill the box completely? How do you know?

c. If, on average, she adds 8 cubes too the box each day, how many school days will it

take for her class to earn a pizza party? Show your work:

Grade 6 Unit 1

Initial Performance Task: Back to School

Scoring Guide

Back to School Scoring Guide Points Section

Points

1.

a. Student gives correct answer of 54 and shows appropriate application of

division algorithm

b. Student gives correct answer of $2.70 and shows appropriate application of

division algorithm

1

1

2

2. Student gives correct answer of 6 book covers and shows both a model and the

appropriate equation. For example: 3/4 ÷ 1/8 = 6 and models the division using a

number line

2

2

3.

a. Student shows a correct estimate of $34 - $41 and shows how they estimated

each item (do not give credit for adding precisely then rounding the answer)

b. Student gives correct answer of .$27 per pencil and shows division algorithm

c. Student gives correct answer of $1.67 and shows correct addition and

subtraction algorithm

d. Student gives correct answer of $9.96 and shows multiplication algorithm (If

student adds, give ½ credit)

1

1

1

1

4

4. Student gives correct answer of Thursday, September 20 and shows appropriate

method. For example, making a diagram of a calendar and marking off the days

when both pizza and salad are served or finding the least common multiple (12)

and counting on 12 more school days to find the answer.

2

2

5.

a. Student gives correct answer of 24 students and shows the GCF by breaking

each number down into its prime factors and multiplying the common factors

(23 x 3 = 24)

b. Student shows that 96 + 72 = 168 by factoring out the GCF and expressing the

other factor as a sum:

96 + 72 = 168

12(4 + 3) = 168

12(7) = 168

168 = 168

1

1

2

6.

a. Student gives correct answer of 224 in3 and shows application of either V = lwh

or V = bh

b. Student gives correct answer of 224 cubes and makes a connection to the

above answer

c. Student gives correct answer of 28 school days and shows use of division

algorithm

1

1

1

3

Total Points 15 15

Novice Apprentice Practitioner Expert

0 - 4 points 5 - 8 points 9 - 12 points 13 - 15 points

Grade 6 Unit 1

Final Performance Task: Sara’s Birthday Party

1. Sara goes shopping at Party City for decorations. Here is a portion of her receipt:

Item

Balloons

Price

$38.80

Plates $12.99

Napkins $9.89

Cups $11.49

a) How much more do the balloons cost than plates, napkins and cups combined?

b) If Sara is planning on inviting 25 people to her party, about how much per person did she

spend on balloons? On the lines below, explain how you made your estimate.

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

________________________________________________________________________

2. Sara estimates that she has spent about 32,120 hours sleeping in her lifetime. If she sleeps 8

hours a night, how old is Sarah? Show how you solved:

3. Sara is making invitation cards for her upcoming birthday party. It takes her about 1/6 of an

hour to make each card but she only has 2/3 of an hour left to finish. About how many cards

can she make in this time?

Use any model to show your mathematical thinking in the space below:

4. Sara’s mom is making the goody bags for her party. If she places a lollipop in every 3rd bag

and a fuzzy pencil in every other bag, which bags will contain both a lollipop and a fuzzy pencil

if she makes 25 bags? Show how you solved:

5. Sara’ dad is buying treats for her party.

a. He bought 54 cookies and 36 munchkins. If each child must receive the same amount of

treats and all treats must be used, did Sara’s dad buy enough treats for 25 people? Why or

why not? Show your mathematical thinking in the space below:

b. Rewrite the expression 54 + 36 using GCF and Distributive Property. How does your new

expression relate to the maximum number of people that can have treats?

6. A diagram of Sara’s cake is pictured below:

6 in.

a) How many 1- ½ inch cubes can Sara cut from her cake? Show your mathematical thinking.

b) What is the volume of Sara’s cake? Show your work.

3 in.

7.5 in.

Grade 6 Unit 1

Final Performance Task: Sara’s Birthday Party

Scoring Guide

Sara’ Birthday Party Scoring Guide Points Section

Points

1.

a) Student gives correct answer of $4.43 and shows correct application of

addition and subtraction algorithms with decimal numbers. For example: adds

the cost of plates, napkins and cups correctly ($34.37) then subtracts this

amount from the cost of balloons.

b) Students should estimate that Sara spent about $1.50 per person for the

balloons. Example: If Sara spent $1.00 per person, she would have spent $25.

But, she spent about $14 more which is a little more than $.50 per person. So

all together she spent about $1.50 per person

1

1

2

2. Student gives correct answer of 11 years and shows the method through division.

Example: 32,210/365 = 88 hours and 88/8 = 11 years

2

2

3. Student gives correct answer of 4 cards and shows a model to demonstrate

understanding. For example, uses a number line that shows it will take 4 1/6 units

of an hour to “use up” her 2/3 of an hour. Or, student may model using an

equation such as 2/3 ÷ 1/6 = 4

2

2

4. Student gives correct answer of a lollipop and a fuzzy pencil in the 6th, 12th, 18th

and 24th bag. Students can model their understanding of common multiples

through pictures, diagrams and/or listing multiples of both 2 and 3, then indicating

which are common between the two.

2

2

5.

a) Student gives correct answer of “no” and offers explanation or shows how the

answer was determined. Example: I factored 54 and 36 into its prime factors

and found the greatest common factor to be 18 (or show the factoring).

Therefore, the maximum number of people that can get both treats is 18 (3

cookies and 2 munchkins each).

b) Student gives correct answer of 90 = 18(3 + 2) and explains that the GCF (18)

represents the max number of people

2

1

3

6.

a) Student gives correct answer of 40 cubes and shows their process through

diagrams or actual “cutting of the cake” into 1.5 inch cubes or finds the total

volume of the cake and divides it by the volume of each piece of cake (135 in3

÷ 3.375 in3= 40 pieces)

b) Student gives correct answer of 135 in3 and shows correct application of either

V = l w h or V = b h

1

1

2

Total Points

13

13

Novice Apprentice Practitioner Expert

0 - 4 points 5 - 7 points 8 - 10 points 11 - 13 points