Common Core Scope and SequenceSixth Grade

Quarter 1

Unit: 1 – Fractions & RatiosDomain: The Number System (NS) & Ratios and Proportional RelationshipsCluster: Understand ratio concepts and use ratio reasoning to solve problems Apply and extend previous understandings of multiplication and division to divide fractions by fractions

Standard Mathematical Practices1 Instructional Objectives Mathematical Task

6.RP.1 Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, “The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak.” “For every vote candidate A received, candidate C received nearly three votes.”

6.MP.2. Reason abstractly and quantitatively.

6.MP.6. Attend to precision.

Describe the relationship between two quantities using ratio vocabulary to demonstrate understand the concept of ratios

http://illustrativemathematics.org/standards/k8#4

(6.RP Games at Recess)(see website for commentary and solution)

The students in Mr. Hill’s class played games at recess.

6 boys played soccer 4 girls played soccer 2 boys jumped rope 8 girls jumped rope

Afterward, Mr. Hill asked the students to compare the boys and girls playing different games.

Mika said,“Four more girls jumped rope than played soccer.”Chaska said,“For every girl that played soccer, two girls jumped rope.”

Mr. Hill said, “Mika compared the girls by looking at the difference and Chaska compared the girls using a ratio.”

1.) Compare the number of boys who played soccer and jumped rope using the difference. Write your answer as a sentence as Mika did.

2.) Compare the number of boys who played soccer and jumped rope using a ratio. Write your answer as a sentence as Chaska did.

3.) Compare the number of girls who played soccer to the number of boys who played soccer using a ratio. Write your answer as a sentence as Chaska did.

6.RP.2 Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate language in the context of a ratio relationship. For example, “This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar.” “We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger.”

6.MP.2. Reason abstractly and quantitatively.

6.MP.6. Attend to precision.

Describe the relationship between two quantities using ratio vocabulary to demonstrate understand the concept of ratios

Understand the concept of unit rates as it relates to ratios

Use rate language in the context of a ratio relationship

http://illustrativemathematics.org/standards/k8#4

6.RP Mangos for Sale(see website for commentary and solution)

They were selling 8 mangos for $10 at the farmers market. Keisha said, “That means we can write the ratio 10 : 8, or $1.25 per mango.” Luis said, “I thought we had to write the ratio the other way, 8 : 10, or 0.8 mangos per dollar."

Can we write different ratios for this situation? Explain why or why not.

*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.MP.1. Make sense of problems and persevere in solving them.

6.MP.2. Reason abstractly and quantitatively.

6.MP.3. Construct viable arguments and critique the reasoning of others.

6.MP.4. Model with mathematics.

6.MP.7. Look for and make use of structure.

6.MP.8. Look for and express regularity in repeated reasoning.

Solve division of fraction by fraction real-world problems using models

Interpret quotients of fractions

http://illustrativemathematics.org/standards/k84

6.NS Baking Cookies(see website for commentary and solution)

Alice, Raul, and Maria are baking cookies together. They need ¾ cup of flour and 1/3 cup of butter to make a dozen cookies. They each brought the ingredients they had at home.Alice brought 2 cups of flour and ¼ cup of butter, Raul brought 1 cup of flour and 1/2 cup of butter, and Maria brought 1 ¼ cups of flour and ¾ cup of butter.

If the students have plenty of the other ingredients they need (sugar, salt, baking soda, etc.), how many whole batches of a dozen cookies each can they make?

6.NS.2 Fluently divide multi-digit numbers using the standard algorithm.

Divide multi-digit numbers fluently.

http://illustrativemathematics.org/standards/k8#4

6.NS Interpreting a Division Computation(see website for commentary and solution)

1. 3024÷162. 1280÷163. 144÷16

Vocabulary: interpret, reciprocal, multiplicative inverses, visual fraction model, multi-digit, greatest common factor (GCF), least common multiple (LCM), prime numbers, composite numbers, factors, multiples, distributive property, prime factorization, relatively prime, quotient, standard algorithmVocabulary: ratio, equivalent ratios, tape diagram, unit rate, part – to – part, part – to – whole, percent, double line diagram, constant speed

Explanations and Examples: (3)6.RP.1 A ratio is the comparison of two quantities or measures. The comparison can be part-to-whole (ratio of guppies to all fish in an aquarium) or part-to-part (ratio of guppies to goldfish).

Example 1:A comparison of 6 guppies and 9 goldfish could be expressed in any of the following forms: 6/9, 6 to 9 or 6:9. If the number of guppies is represented by black circles and the number of goldfish is represented by white circles, this ratio could be modeled as

These values can be regrouped into 2 black circles (goldfish) to 3 white circles (guppies), which would reduce the ratio to, 2/3 , 2 to 3 or 2:3.

Students should be able to identify and describe any ratio using “For every _____ ,there are _____” In the example above, the ratio could be expressed saying, “For every 2 goldfish, there are 3 guppies”.

6.RP.2A unit rate expresses a ratio as part-to-one, comparing a quantity in terms of one unit of another quantity.Common unit rates are cost per item or distance per time.

Students are able to name the amount of either quantity in terms of the other quantity. Students will begin to notice that related unit rates (i.e. miles / hour and hours / mile) are reciprocals as in the second example below. At this level, students should use reasoning to find these unit rates instead of an algorithm or rule.

In 6th grade, students are not expected to work with unit rates expressed as complex fractions. Both the numerator and denominator of the original ratio will be whole numbers.Example 1:There are 2 cookies for 3 students. What is the amount of cookie each student would receive? (i.e. the unit rate)Solution: This can be modeled as shown below to show that there is 2/3 of a cookie for 1 student, so the unit rate is 2/3: 1.

Example 2:On a bicycle Jack can travel 20 miles in 4 hours. What are the unit rates in this situation, (the distance Jack can travel in 1 hour and the

amount of time required to travel 1 mile)?Solution: Jack can travel 5 miles in 1 hour written as 5 mi. and it takes 1/5 of an hour to travel each mile written as 1/5 hr. 1 hr. 1 mi.Students can represent the relationship between 20 miles and 4 hours.

Explanations and Examples:3

6.NS.1 In 5th grade students divided whole numbers by unit fractions and divided unit fractions by whole numbers.Students continue to develop this concept by using visual models and equations to divide whole numbers by fractions and fractions by fractions to solve word problems. Students develop an understanding of the relationship between multiplication and division.

Example 1:Students understand that a division problem such as 3 ÷ 2/5 is asking, “how many 2/5 are in 3?” One possible visual model would begin with three whole and divide each into fifths. There are 7 groups of two-fifths in the three wholes. However, one-fifth remains. Since one-fifth is half of a two-fifths group, there is a remainder of 1/2. Therefore, 3 ÷ 2/5 = 7 1/2, meaning there are 7 1/2 groups of two-fifths. Students interpret the solution, explaining how division by fifths can result in an answer with halves.

Students also write contextual problems for fraction division problems. For example, the problem, 2/3 ÷ 1/6 can be illustrated with the following word problem:

Example 2:Susan has 2/3 of an hour left to make cards. It takes her about 1/6 of an hour to make each card. About how many can she

make?This problem can be modeled using a number line. a. Start with a number line divided into thirds.

b. The problem wants to know how many sixths are in two-thirds. Divide each third in half to create sixths.

c. Each circled part represents 1/6. There are four sixths in two-thirds; therefore, Susan can make 4 cards.

6.NS.2 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, 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.

Performance Tasks:There are several tasks in the following assessment. Task 4 speaks more directly to this unit. Feel free to use as much of the performance assessment available to meet the needs of your students.

http://www.sgsd.k12.wi.us/Curriculum/Math/Math_Middle_Performance_Assessments.htm

Information for the Teacher

Task Description

This assessment asks students to apply math skills by planning, designing, and drawing an ideal retail store. Students also research characteristics of their favorite retail store and learn what facilities and resources a store requires.

Some preparatory work will be necessary. Students will need access to information about stores such as the square footage, cost per square foot, number of employees, etc (see Tasks 3 and 4). The teacher can research this information in advance, or give students ballpark figures/estimations with which to work.

For the students who complete Tasks 1 through 5 ahead of the rest of the class, there are enrichment tasks provided at the end of the assessment. Other enrichment possibilities could include the following: students consider the implications of various design choices (one big store vs. two smaller ones); students trade results with students working in retail stores to assemble a larger set of data.

You can adjust the scale of work for this assessment. For example, you can ask students in Task 3 to research additional or fewer aspects than these specific questions entail.

Required Materials

Access to building professionals (for enrichment tasks)Drawing supplies and toolsGraph paper

Scoring key for the teacher

Answers will vary.Task 4: How much will your store cost?Prepare a table comparing your ideal store to your current store information used in Task 3.

The comparison should include the following items:

Square footage Rental or lease cost for space needed for your store Number of employees Cost of inventory merchandise Cost of equipment necessary to operate store (phones, cash registers, etc.) Cost per square foot Total costScoring Guide - Task 4

4 Exemplary

Criteria for the Proficient category have been successfully completed. The response includes more advanced work. For example, the student breaks down the cost of the new store and prepares sub-

estimates for each part of it, or compares costs of aspects of it to the real store. Other examples include:

3 Proficient

The student creates a table containing estimations and data from the real store. The student compares the estimations with the real data. The student determines an estimated cost for the ideal store. The number is reasonable. The items in the table are consistent with work in previous tasks or work presented in this task.

2 Progressing

Three of the criteria for a score of Proficient are met. More work is needed.

1 Not meeting the standard(s)

Less than three of the criteria for a score of Proficient are met.

The task should be repeated.

You will be given information on the average cost of a typical store. Use this information and the information in your table to estimate the total cost of your ideal store.

Unit: 2 –Solving problems with Fractions, Ratios, and RatesDomain: The Number System (NS) and Ratios and Proportional Relationships (RP)Cluster: Compute fluently with multi-digit numbers and find common factors and multiples. Understand ration concepts and use ratio reasoning to solve problems.Standard Mathematical

Practices1Instructional Objectives

Mathematical Task

*6.NS.3 Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.

6.MP.2. Reason abstractly and quantitatively.6.MP.7. Look for and make use of structure.6.MP.8. Look for and express regularity in repeated reasoning.

Add, subtract, multiply and divide multi-digit decimals fluently

http://illustrativemathematics.org/standards/k8 4

6.NS Buying Gas(see website for commentary and solution)

Sophia’s dad paid $43.25 for 12.5 gallons of gas. What is the cost of one gallon of gas?

6.NS Jayden’s Snacks(see website for commentary and solution)

Jayden has $20.56. He buys an apple for 79 cents and a granola bar for $1.76.

1. How much money did Jayden spend?2. How much money does Jayden have

now?

6.RP.3b Solve unit rate problems including those

Solve real-world problems involving

Solve:

involving unit pricing and constant speed. For example, if it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed?

6.MP.1. Make sense of problems and persevere in solving them.

6.MP.2. Reason abstractly and quantitatively.

6.MP.4. Model with mathematics

6.MP.5. Use appropriate tools strategically.

6.MP.7. Look for and make use of structure

unit pricing and constant speed

If Carla made $168 dollars after working a total of 14 hours, at that rate how much would she make after working a total of 45 hours.

At what rate is she being paid per hour? ($12)At that rate, how many hours would Carla have to work in a week to make $630 dollars? (52.5 hours)How many combinations of hours/days could Carla work in a week to make $630 dollars? Ex. 7 days 7.5 hours per day at $12 dollars per hour.

6.RP.3c Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent.

Determine a percent of a whole number as a rate out of 100

Solve problems to find the whole when given a part and a percent

http://illustrativemathematics.org/standards/k8 4

6.RP Finding a 10% Increase(see website for commentary and solution)

5,000 people visited a book fair in the first week. The number of visitors increased by 10% in the second week. How many people visited the book fair in the second week?

6.RP.3d Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.

Convert units of measurements using ratio reasoning (by multiplying and dividing quantities)

The ratio of feet to yards is 3:1.Solve for the following:

___ ft: 4 yards18 ft: __ yards1 ft: ___ yard(s)

Vocabulary: ratio, equivalent ratios, tape diagram, unit rate, part – to – part, part – to – whole, percent, double line diagram, constant speed, multi-digit, standard algorithm6.RP.3b Students recognize the use of ratios, unit rate and multiplication in solving problems, which could allow for the use of fractions and decimals.Example 1:In trail mix, the ratio of cups of peanuts to cups of chocolate candies is 3 to 2. How many cups of chocolate candies would be needed for 9 cups of peanuts.

Solution:One possible solution is for students to find the number of cups of chocolate candies for 1 cup of peanuts by dividing both sides of the table by 3, giving 2/3 cup of chocolate for each cup of peanuts. To find the amount of chocolate needed for 9 cups of peanuts, students multiply the unit rate by nine (9 • 2/3), giving 6 cups of chocolate.

Example 2:If steak costs $2.25 per pound, how much does 0.8 pounds of steak cost? Explain how you determined your answer.Solution:The unit rate is $2.25 per pound so multiply $2.25 x 0.8 to get $1.80 per 0.8 lb of steak.6.RP.3cThis is the students’ first introduction to percents. Percentages are a rate per 100. Models, such as percent bars or 10 x 10 grids should be used to model percents.

Students use ratios to identify percents.Example 1:What percent is 12 out of 25?Solution: One possible solution method is to set up a ratio table:Multiply 25 by 4 to get 100. Multiplying 12 by 4 will give 48, meaningthat 12 out of 25 is equivalent to 48 out of 100 or 48%.

Students use percentages to find the part when given the percent, by recognizing that the whole is being divided into 100 parts and then taking a part of them (the percent).

Example 2:What is 40% of 30?Solution: There are several methods to solve this problem. One possible solution using rates is to use a 10 x 10 grid to represent the whole amount (or 30). If the 30 is divided into 100 parts, the rate for one block is 0.3. Forty percent would be 40 of the blocks, or 40 x 0.3, which equals 12. See the weblink below for more information.http://illuminations.nctm.org/LessonDetail.aspx?id=L249

Students also determine the whole amount, given a part and the percent.

Example 3:If 30% of the students in Mrs. Rutherford’s class like chocolate ice cream, then how many students are in Mrs.Rutherford’s class if 6 like chocolate ice cream?

(Solution: 20)

Example 4:A credit card company charges 17% interest fee on any charges not paid at the end of the month. Make a ratio table to show how much the interest would be for several amounts. If the bill totals $450 for this month, how much interest would you have to be paid on the balance?Solution:

One possible solution is to multiply 1 by 450 to get 450 and then multiply 0.17 by 450 to get $76.50.

6.RP.3d A ratio can be used to compare measures of two different types, such as inches per foot, milliliters per liter and centimeters per inch. Students recognize that a conversion factor is a fraction equal to 1 since the numerator and denominator describe the same quantity. For example, 12 inches is a conversion factor since the numerator and denominator equal the same amount. 1 footSince the ratio is equivalent to 1, the identity property of multiplication allows an amount to be multiplied by the ratio. Also, the value of the ratio can also be expressed as 1 foot allowing for the conversion ratios to be expressed in a format so that units will “cancel”. 12 inchesStudents use ratios as conversion factors and the identity property for multiplication to convert ratio units.

Example 1:How many centimeters are in 7 feet, given that 1 inch ≈ 2.54 cm.Solution:

Note: Conversion factors will be given. Conversions can occur both between and across the metric and English system. Estimates are not expected.

Performance Task:http://schools.nyc.gov/NR/rdonlyres/A9F735CB-47E4-40F8-884F-EA54D0AB5705/0/NYCDOEG6MathRatios_Final.pdf

Sample Task 11. Giovanni is visiting his grandmother who lives in an apartment building on the 25th floor. Giovanni enters the elevator in the lobby, which is the first floor in the building. The elevator stops on the 16th floor. What percentage of 25 floors does Giovanni have left to reach his grandmother's floor? Use pictures, tables or number sentences to solve the task. Explain your reasoning in words.

SolutionThe reasoning used to solve the parts of the problem may include:a. Understanding that the whole is 25 and either 16 or (25 – 16) floors is a part of the whole involved. Some students may consider a “missing 13 th

floor”. In that case, the whole is 24 and either 15 or (24 – 15) is a part of the whole involved.b. Recognition of the need to determine how many “out of 100”.i. Using a fraction and its conversion to an equivalent fraction, possibly by simplifying first.ii. Converting to decimal then to percent.iii. Creating and reasoning from a grid representation of the contextc. Recognition of the need either to work with 9/25 (or 9/24) or to subtract from 100% after changing 16/25 (or15/24) to percent.

Sample Task 2Use the recipe shown in the table to answer the questions below.

Grandma’s Recipe for Sugar Cookies1 ½ cups butter2 cups sugar4 eggs¾ teaspoon baking powder

1 ¼ cups flour¼ teaspoon salt

Use pictures, tables or number sentences to solve the task.a) How many cups of sugar are needed for each egg? How do you know?b) Your sister notices that she needs three times as much baking powder as salt in this recipe.What is the ratio of baking powder to salt? Explain how you know.

SolutionThe reasoning used to solve the parts of the problem may include:a. A ratio of cups of sugar to each egg is formed and simplified or scaled down to a denominator of 1.b. Scaling down the 2:4 ratio in tabular form.c. Recognizing the phrase “three times as much baking powder as salt” as a 3:1 ratio.d. Forming a ¾ : ¼ ratio; possibly scaling that ratio up to 2 ¼ : ¾ on the basis of the “three times as much” language.e. Drawing a picture and reasoning from the picture for either or both parts.

a. How many cups of sugar are needed for each egg? How do you know?b. Your sister notices that she needs three times as much baking powder as salt in this recipe. What is the ratio of baking powder to salt? Explain your reasoning in words.

3 teaspoons baking powder : 1 teaspoon salt

¼ ¼ ¼¼ ¼ ¼¼ ¼ ¼¼ ¼ ¼

Baking Powder Salt

Sample Task 3Suppose a rabbit starts from home and, after 3 seconds, it is 12 cm from home. Generate several “distance from home” and elapsed time values for other parts of the rabbit’s journey so that the rabbit travels the same speed throughout its journey. Describe several different patterns that you see in the table. Write a rule or description that tells us how far from home the rabbit is for any number of seconds he travels.

An automobile is traveling so that at the end of 5 hours it has gone 294 miles. Generate several “distance from home” and elapsed time values for other parts of the journey so that the automobile travels the same speed throughout its journey. Write a rule or description that tells us how far the automobile travels for any number of seconds it is on the road.

2 1 ½4 2 1

¼¼¼¼

An airplane will go 480 miles in 2 hours. Generate several “distance from home” and elapsed time values for other parts of the journey so that the airplane travels the same speed throughout its journey. Write a rule or description that tells us how far the airplane travels for any number of seconds it is in the air.

http://www.pdesas.org/module/content/resources/13333/view.ashx

Fontain Middle School’s art department has been put in charge of designing the mural below, to be displayed in the courtyard with their newly planted gardens. It currently has six sections outlined. You are on the planning committee that will make several decisions regarding the mural. You volunteered to calculate the budget.

1. Write a ratio in simplest form to represent the portion of the mural each student is scheduled to paint. Put the information in a table.

2. Calculate the percent of the mural that each piece represents. Add this information to your table from part A.

3. The committee decided to reduce the number of sections on the mural to three sections. Outline new dividing lines to create a total of three larger sections, by combining sections that are next to each other. The sections do not need to be equal in size. Label the three new sections on the mural map as 1, 2, and 3. Make a new table showing the new percents of the total mural that each piece covers. Include extra space in your table for parts C and D.

4. The total cost of the paint for the mural is $360. The cost of paint for each section is determined by size. For instance, the paint for a section that takes up 20% of the mural will cost 20% of $360. How much will the paint for parts 1, 2, and 3 cost? (Remember that the total paint cost must equal $360.) Include this information in the table from part C.

5. Additionally, the committee believes that the amount of money spent on supplies for each painter should be based on the amount of paint each painter will be using. For each of the three painters, calculate 15% of his/her paint cost to determine how much money will be allotted for his/her supplies.

6. Write a short report to share at the next committee meeting describing how your calculations were determined. Attach copies of your tables, calculations, and the mural map with the new outlines to your report.

Performance Assessment Scoring Rubric:

PointsDescription

4 Accurately calculated all ratios and percents for the original six mural sections and presented

them clearly in a table.

Divided mural into a total of three sections by combining adjacent pieces. Explained

reasoning for divisions as creative or mathematical.

Accurately calculated the percents for the three redrawn sections. May have extended data

to show decimal and/or fractions as well.

Used multiple methods to accurately make all dollar amount calculations for paint and

supplies for each of the three sections.

Demonstrated advanced understanding of the mathematical ideas and processes related to

the meaning of percent and percent of a number.

Written report is thorough, detailed, and insightful.Worked beyond the problem requirements, possibly by checking steps and/or incorporating

PointsDescription

technology.

3

Calculated most of the ratios and percents for the original six mural sections with high degree

of accuracy (1–2 errors) and presented them clearly in a table.

Divided mural into a total of three sections by combining adjacent pieces.

Accurately calculated the percents for the three redrawn sections.

Accurately made most dollar amount calculations for paint and supplies for each of their three

sections. May have one or two minor errors.

Demonstrated solid understanding of the mathematical ideas and processes related to the

meaning of percent and percent of a number.

Written report is thorough.Met all of the problem requirements.

2 Calculated the ratios and percents for the original six mural sections with more than three

errors (or several missing) and presented them in a table.

Divided mural into a total of three sections without combining adjacent pieces.

Accurately calculated two of the three percents for the redrawn sections.

Made dollar amount calculations for paint and supplies for each of the three sections with one

or two major errors or three or four minor errors.

Demonstrated some understanding of the mathematical ideas and processes related to the

meaning of percent and percent of a number.

Written report is too brief, missing some requested components.

PointsDescription

Partially met the problem requirements.

1

Calculated ratios or percents for the original six mural sections with some errors or calculated

both ratios and percents with mostly incorrect answers. May have an incomplete or difficult-

to-read table.

Divided mural into a total of three sections but did not include the full mural or show the

divisions clearly or accurately on the mural map.

Major errors calculating the percents for the three redrawn sections.

Approximately half of the dollar amount calculations for paint and supplies are incorrect or

missing.

Demonstrates substantial lack of understanding of portions of the problem related to

percents.

Written report is significantly incomplete or has inaccuracies throughout.Did not meet several of the problem requirements.

0 Table of ratios and percents for original six sections is incomplete or missing, or

answers are all incorrect.

Did not divide the mural into a total of three sections, left the original six sections in

place.

Percents for the three redrawn sections missing.

Most dollar amount calculations for paint and supplies are incorrect or missing.

Demonstrated a complete lack of understanding of the problem.

PointsDescription

Written report is missing. Did not meet any of the problem requirements.