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Unit Title: Measurement, Quantities, and Relationships: The Foundation of MathematicsGrade Level: 6

Timeframe: Marking Period 1

Essential Questions

(1) Can we become confident in our knowledge of quantities, units, and values of measuring that are used in relationships?(2) Can we become effective and efficient at representing relationships in all four traditional representations—tables generalized to symbols, graphs, and narratives?(3) Can we become effective and efficient at asking and answering typical mathematical questions? (4) Can we become effective and efficient at talking about relationships given in any representation?(5) Can we become effective and efficient at reading and understanding others’ relationships and questions given in any representation?

New Jersey Student Learning Standards

Standards/Cumulative Progress Indicators (Taught and Assessed):

6.RP.3: Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.

6.RP.3a: Make tables of equivalent ratios relating quantities with whole number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.

6.EE.6: Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.

6.NS.5. Apply and extend previous understandings of numbers to the system of rational numbers. Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation.

6.RP.3 b: Solve unit rate problems including those 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.EE. 5: Reason about and solve one-variable equations and inequalities. Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.

6.NS.1: A. Apply and extend previous understandings of multiplication and division to divide fractions by fractions. 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?

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6.EE. 1: Apply and extend previous understandings of arithmetic to algebraic expressions. Write and evaluate numerical expressions involving whole-number exponents.

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).

21st Century Skills Standard and Progress Indicators:

CRP4. Communicate clearly and effectively and with reason.CRP8. Utilize critical thinking to make sense of problems and persevere in solving them.CRP11. Use technology to enhance productivity. CRP12. Work productively in teams while using cultural global competence.

Instructional Plan ReflectionDiagnostic AssessmentGrade 6 Math – Cumulative Assessment

SLO - SWBAT Student Strategies Formative Assessment Activities and Resources Reflection

CAR © 2009

6.RP.3: Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.

Do Now – Count around the room. Make combinations. [5 minutes]

Direct Instruction/Modeling

Small Group Instruction

Constructed responses from the Type 2-3 Bank and Carnegie Learning

PARCC Released Items

Performance Tasks

Standards Based Problems Aligned to Touchpoint and End of Unit Assessment

Exit Ticket - Standards Based Problems Aligned to Touchpoint and End of Unit Assessment

6.RP.3 – Touchpoint

Standards Based Constructed Response ExampleThe prices of packages of miniature cars at different stores are shown in the table. The miniature cars must be purchased in whole packages.

Part A In the table above, type the unit price for each miniature car at each store.

Part B Mr. Ming plans to buy 60 miniature cars. How much money would he save buying them at the cheapest unit price compared to the next cheapest unit price?

Part C The most expensive unit price was decreased by $0.10 per car. Using this new price, how many miniature cars could be purchased for $20?

Constructed Response ResourcesEdConnect 6.RP.3 - Type 2-3 Bank

Questions 1-31

Carnegie Learning Course 1

PARCC Released Items#17, 18, 24

https://prc.parcconline.org/system/files/Grade%2006%20Math

%20-%20Item%20Set.pdf

Performance Task ResourcesIllustrative Mathematics

Performance Tasks 6.RP.3https://

www.illustrativemathematics.org/content-standards/6/RP/A/3

PMI Math Labshttps://njctl.org/courses/math/6th-

grade-math/ratios-proportions-percents/

Direct Instruction ResourcesEngageNY Lessons 6.RP.3

https://www.engageny.org/search-site?search=6.RP.3

Learnzillion Lesson Plans Ratios Days 5-7

https://learnzillion.com/resources/64212-ratios

PMI SMART Presentations https://njctl.org/courses/math/6th-


CAR © 2009

https://njctl.org/courses/math/6th-grade-math/ratios-proportions-percents/










https://www.illustrativemathematics.org/content-standards/6/RP/A/3



https://prc.parcconline.org/system/files/Grade%2006%20Math%20-%20Item%20Set.pdf



6.RP.3 a: Make tables of equivalent ratios relating quantities with whole number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.






Performance Tasks



6.RP.3a – Touchpoint

Standards Based Constructed Response ExampleNancy and her family are driving 2,800 miles across the country from New York City to Los Angeles, California. Nancy’s dad drives 350 miles a day.

Part A. Find the two numbers that are missing from Nancy’s chart.

Part B. Plot the data from the table above on the graph below.

Part C. Nancy’s dad drives 30 miles on 1 gallon of gas. Make a table to show how many miles he drives on 12 gallons of gas.


Questions 1-31








PMI Math Labshttps://njctl.org/courses/math/6th-grade-math/ratios-proportions-percents/







CAR © 2009

















Use, words, numbers, and/or pictures to show your work.

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6.EE. 6: Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.






Performance Tasks



6.EE.6 – Touchpoint

Standards Based Constructed Response ExamplePaul sells apples at a farmer's market. He sells each pound of apples for $2, but he must pay $25 of the money he makes to the market. Part A. Write an expression that shows how many dollars, d, Paul will make from selling a pounds of apples. Make sure to include the amount he has to pay to the market in your expression.

Part B. Paul sold a total of 68 pounds of apples. Use the expression you wrote to find out how much money Paul made from selling apples.

Use words, numbers, and/or pictures to show your work.

Constructed Response ResourcesEdConnect 6.EE.6 - Type 2-3 Bank

Questions 11-13


PARCC Released Items#3




Performance Tasks 6.EE.6https://www.illustrativemathematics.org/content-standards/6/EE/B/6


grade-math/expressions/

Direct Instruction ResourcesEngageNY Lessons 6.EE.6

https://www.engageny.org/search-site?search=6.ee.6

Learnzillion Lesson Plans Representing Relationships 1-4

https://learnzillion.com/resources/64217



CAR © 2009

https://njctl.org/courses/math/6th-grade-math/expressions/








https://www.illustrativemathematics.org/content-standards/6/EE/B/6






6.NS.5. Apply and extend previous understandings of numbers to the system of rational numbers. Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation.






Performance Tasks



6.NS.5 – Touchpoint

Standards Based Constructed Response ExampleCorrine had $65.00 in her savings account. She deposited $10.00 that she earned mowing the lawn and $15.00 that she earned babysitting.

The following week, Corrine withdrew $20.00 from her savings account to go to the movies. She also withdrew $70.00 to buy a bicycle.

Corrine wants to buy a set of reflectors for her bicycle that costs $15.00.

Part A

Use positive and negative integers to show the beginning balance in Corrine's savings account, each deposit, each withdrawal, and the total amount remaining in Corrine's savings account after she buys the bicycle.

Part B

Explain why Corrine will or will not have enough money left to buy the reflectors.

Constructed Response ResourcesEdConnect 6.NS.5 - Type 2-3 Bank

Questions 6-10






Performance Tasks 6.NS.5https://www.illustrativemathematics.org/content-standards/6/NS/C/5


grade-math/numbers-and-operations-6th-grade/

Direct Instruction ResourcesEngageNY Lessons 6.NS.5

https://www.engageny.org/search-site?search=6.ns.5

Learnzillion Lesson Plans Grade 6 Unit 3 Lesson 5

https://learnzillion.com/resources/64214-multi-digit-

computation-and-finding-common-factors-and-multiples

PMI SMART Presentations

https://njctl.org/courses/math/6th-grade-math/numbers-and-

operations-6th-grade/ CAR © 2009

https://njctl.org/courses/math/6th-grade-math/numbers-and-operations-6th-grade/



https://learnzillion.com/resources/64214-multi-digit-computation-and-finding-common-factors-and-multiples








https://www.illustrativemathematics.org/content-standards/6/NS/C/5






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






Performance Tasks



6.RP.3b – Touchpoint

Standards Based Constructed Response ExampleSophia and Juwan went to the farmers’ market to buy some fruit. The prices of fruit are shown below.

Sophia bought 2.5 pounds of peaches and 1.4 pounds of apples.

Part A. How much did Sophia spend on the peaches? How much did she spend on the apples?

Part B. Sophia wants to spend $2.00 to buy 1.5 pounds of blueberries. Will this be enough money? Why or why not?

Part C. Juwan wants to spend as much of $5 as he can on fruit. Determine the number of pounds of apples Juwan can buy if he has already purchased 1.1 pounds of peaches and 1.2 pounds of blueberries.


Questions 1-31








PMI Math Labshttps://njctl.org/courses/math/6th-grade-math/ratios-proportions-percents/







CAR © 2009

















6.EE.5: Reason about and solve one-variable equations and inequalities. Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.






Performance Tasks




Standards Based Constructed Response ExampleA biologist is monitoring the number of bass fish, x, in a certain pond throughout the year. She calculates that the pond conditions are ideal when

the number of fish is

Part A. During each of her visits to the pond last year, the biologist estimated the number of bass fish to be 375, 600, 550, 450, and 575. Which estimates make the inequality for ideal pond conditions true?

Part B. While reviewing her work, the biologist changes the inequality for ideal pond conditions

to Which of the five estimates from part A satisfy the new inequality for ideal pond conditions?



Questions 1-6


Performance Task Resources Illustrative Mathematics

Performance Tasks 6.EE.5https://www.illustrativemathematics.org/content-standards/6/EE/B/5

PMI Math Labshttps://njctl.org/courses/math/6th-grade-math/equations-inequalities/



Learnzillion Lesson Plans Understanding, Writing, and

Solving Equations and Inequalities Lessons 1, 3, and 4

https://learnzillion.com/resources/64222-understanding-

writing-and-solving-equations-and-inequalities

PMI SMART Presentations https://njctl.org/courses/math/6th-grade-math/equations-inequalities/

6.NS.1: A. Apply and extend previous understandings of multiplication



Standards Based Constructed Response Example


Questions 1-7


CAR © 2009

https://njctl.org/courses/math/6th-grade-math/equations-inequalities/


https://learnzillion.com/resources/64222-understanding-writing-and-solving-equations-and-inequalities










and division to divide fractions by fractions. 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





Performance Tasks



Anton’s car uses gallon of gas to drive to and from his sister’s house.

Part A If Anton has gallon of gas, how many times can he drive to and from his sister’s house? Show or explain your work.

Part B If Anton has gallons of gas, how many times can he drive to and from his sister’s house? Show or explain your work.


#6 and 7https://prc.parcconline.org/

system/files/Grade%2006%20Math%20-%20Item%20Set.pdf


Performance Tasks 6.NS.1https://

www.illustrativemathematics.org/content-standards/6/NS/A/1

PMI SMART Presentations and Math Labs

https://njctl.org/courses/math/6th-grade-math/fraction-and-decimal-

computation/



Learnzillion Lesson Plans Dividing Fractions

https://learnzillion.com/resources/64216-dividing-fractions

PMI SMART Presentations https://njctl.org/courses/math/6th-grade-math/fraction-and-decimal-

computation/

CAR © 2009

https://njctl.org/courses/math/6th-grade-math/fraction-and-decimal-computation/










https://www.illustrativemathematics.org/content-standards/6/NS/A/1






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?

CAR © 2009

6.EE.1: Apply and extend previous understandings of arithmetic to algebraic expressions. Write and evaluate numerical expressions involving whole-number exponents.






Performance Tasks




Standards Based Constructed Response ExampleMs. Brown asked her class to evaluate –2(3)2 – 5 and –2(–3)2 – 5.

Mona’s work is shown below.

Part A Identify the error Mona made when she evaluated both expressions.

Part B Find the correct value for each of the two expressions. Show or explain your work for each of the two expressions.

Part C Explain why the expressions do or do not have the same value.


Questions 6-9






Performance Tasks 6.EE.1https://www.illustrativemathematics.org/content-standards/6/EE/A/1





Learnzillion Lesson Plans Algebraic Expressions Lesson 1

https://learnzillion.com/resources/64220-algebraic-

expressions



CAR © 2009



https://learnzillion.com/resources/64220-algebraic-expressions







https://www.illustrativemathematics.org/content-standards/6/EE/A/1












Performance Tasks




Standards Based Constructed Response ExampleThere are 72 tickets for a rehearsal for the class play and 96 cookies for the reception. Each student receives the same number of tickets as every other student. Each student also receives the same number of cookies as every other student.

Part A. What is the greatest possible number of students?

Part B. Write an expression that represents Part A showing how many tickets and cookies each student will receive.



Questions 1-4


PARCC Released Items#10-11




Performance Tasks 6.NS.4https://www.illustrativemathematics.org/content-standards/6/NS/B/4

PMI Math Labshttps://njctl.org/courses/math/6th-grade-math/factors-and-multiples/



Learnzillion Lesson Plans Grade 6 Unit 3 Lessons 5-10



PMI SMART Presentations https://njctl.org/courses/math/6th-grade-math/factors-and-multiples/

CAR © 2009

https://njctl.org/courses/math/6th-grade-math/factors-and-multiples/









https://www.illustrativemathematics.org/content-standards/6/NS/B/4






Summative Assessment

Grade 6 Math – End of Unit 1 Assessment in EdConnect

CAR © 2009

Unit Title: Relationships Between Quantities: Ratios and RatesGrade Level: 6


Essential Questions




6.RP.A.1: Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities6.RP.A.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 .6.RP.3 Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.

a. Make tables of equivalent ratios relating quantities with whole number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.b. Solve unit rate problems including those 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?c. 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.d. Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.

6.EE.A.3: Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression 3 (2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6 (4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y6.EE.A.4: Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them). For example, the expressions y + y + y and 3y are equivalent because they name the same number regardless of which number y stands for6.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.



CAR © 2009


Instructional Plan ReflectionDiagnostic AssessmentGrade 6 Math – Cumulative AssessmentGrade 6 Math – End of Unit 1 AssessmentUnit 1 Touchpoints


CAR © 2009

6.RP.A.1: Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities






Performance Tasks




Standards Based Constructed Response ExampleMatthew has 30 candies. Twelve of them are blue. The rest of the candies are red.

Part A What is the ratio of blue candies to red candies? Show your work. Reduce the ratio to simplest form.

Part B Matthew eats 2 of the red candies. What is the new ratio of blue candies to red candies? Show your work. Reduce the ratio to simplest form


Questions 1-5



Performance Tasks 6.RP.1https://www.illustrativemathematics.org/content-standards/6/RP/A/1




https://www.engageny.org/search-site?search=6.RP.3.1





CAR © 2009














6.RP.A.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.






Performance Tasks




Standards Based Constructed Response ExampleFor every 6 cups of flour, 3 cups of sugar are needed to make cookies.

Part A What is the unit ratio of flour to sugar? Write your answer as a fraction.

Part B Using the same ratio, how many cups of sugar are needed for 3 cups of flour?

Part C Using the same ratio, how many cups of flour are needed for 6 cups of sugar?


Questions 2-10












Learnzillion Lesson Plans Rates Lessons 4-9

https://learnzillion.com/resources/64213-rates-including-

percent



6.RP.3: Use ratio Do Now – Count around 6.RP.3c – Touchpoint Constructed Response Resources

CAR © 2009




https://learnzillion.com/resources/64213-rates-including-percent














and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.

the room. Make combinations. [5 minutes]





Performance Tasks



6.RP.3d – Touchpoint

Standards Based Constructed Response ExampleSaving Water

In one U.S. town, the average daily indoor water usage rate is 69.3 gallons per person. Approximately 16% of all water used in the home goes toward washing dishes, and 26% goes toward flushing toilets. Use this information to answer the following questions, showing all work.

Part 1. On average, how many gallons of water are used per person every week? Is that more or less than 3,000 gallons per year? Explain how you know.

Part 2. How much water, to the nearest gallon, is used for washing dishes per person each year?

Part 3. Replacing old toilets with low-flush versions will use 35% less water for flushing. How much water, to the nearest gallon, will changing to low-flush toilets save the average person per day? Compare the number of gallons of water used per person per year using old toilets versus the low-flush versions. Is this a significant difference? Why or why not?

Part 4. Carson and his family have a 1,500-gallon reserve water tank. There are four people living in Carson’s house. How many days would the water in the tank last if they used it at the average daily indoor water usage rate? How much longer would the tank last if Carson's family uses a low-flush toilet?

EdConnect 6.RP.3 - Type 2-3 BankQuestions 1-31












Learnzillion Lesson Plans Rates Lessons 10-14

https://learnzillion.com/resources/64213-rates-including-

percent



CAR © 2009



















6.EE.A.3: Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression 3 (2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6 (4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y






Performance Tasks




Standards Based Constructed Response ExamplePart A Write an expression for the following calculation: subtract 4 from 7, then multiply by z.

Part B Write a different but equivalent expression for the expression you wrote in Part A. Explain why the two expressions are equivalent.


Questions 1-6








Learnzillion Lesson Plans Algebraic Expressions Lessons 5-10; 12; 14

https://learnzillion.com/resources/64220-algebraic-

expressions



CAR © 2009
















6.EE.A.4: Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them). For example, the expressions y + y + y and 3y are equivalent because they name the same number regardless of which number y stands for






Performance Tasks




Standards Based Constructed Response ExamplePart A Use the distributive property to write an expression that is equivalent to 7y + 35. Show or explain your work.

Part B Provide a numerical example that would prove 4y and y + y + y + y are equivalent. Show or explain your work.

Part C Using the terms , write two expressions that would each simplify to Show or explain your work.


Questions 1-2








Learnzillion Lesson Plans Algebraic Expressions Lessons 11-15https://learnzillion.com/

resources/64220-algebraic-expressions

PMI SMART Presentations


CAR © 2009













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






Performance Tasks




Standards Based Constructed Response ExampleMembers of the school orchestra are organizing a trip to a concert.

If tickets cost $14 each, how many tickets can be purchased with $500?

The concert hall offers a special price of $12 per ticket. How many more tickets can be purchased with the $500 if the cost is $12 per ticket? Show all of your work.



Questions 1-5


PARCC Released Items#6 -8









Learnzillion Lesson Plans Grade 6 Unit 3 Lesson 4





CAR © 2009


















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






Performance Tasks




Standards Based Constructed Response ExampleThe table shows the costs of food items at a county fair.

Part A How much does it cost to buy 3 bottles of water, 2 corn on the cob, and 2 hamburgers? Show or explain your work.

Part B How much more does it cost to buy a bottle of water and a hamburger than to buy a hot dog and a lemonade? Show or explain your work.


Questions 1-9







PMI Math Labshttps://njctl.org/courses/math/6th-grade-math/fraction-and-decimal-

computation/






PMI SMART Presentations https://njctl.org/courses/math/6th-grade-math/fraction-and-decimal-

computation/

CAR © 2009
























Performance Tasks




Standards Based Constructed Response ExampleThere are 72 tickets for a rehearsal for the class play and 96 cookies for the reception. Each student receives the same number of tickets as every other student. Each student also receives the same number of cookies as every other student.

Part A. What is the greatest possible number of students?

Part B. Write an expression that represents Part A showing how many tickets and cookies each student will receive.



Questions 1-4







PMI Math Labshttps://njctl.org/courses/math/6th-grade-math/factors-and-multiples/






PMI SMART Presentations https://njctl.org/courses/math/6th-grade-math/factors-and-multiples/

CAR © 2009
















Unit Title: Expanding Our Understanding Of NumbersGrade Level: 6


Essential Questions




6.EE.B.7. Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers.

6.EE.B.8. Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams.

6.EE.C.9. Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time.

6.NS.C.6. Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates. 6.NS.C.6a. Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a number is the number itself, e.g., – (–3) = 3, and that 0 is its own opposite. 6.NS.C.6b. Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes. 6.NS.C.6c. Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane.

6.NS.C.7: Understand ordering and absolute value of rational numbers. 6.NS.C.7a. Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram. For example, interpret –3 > –7 as a statement that –3 is located to the right of –7 on a number line oriented from left to right. 6.NS.C.7b. Write, interpret, and explain statements of order for rational numbers in real-world contexts. For example, write –3 o C > –7 o C to express the fact that –3 o C is warmer than –7 o C. 6.NS.C.7c. Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a realworld situation. For example, for an account balance of –30 dollars, write |–30| = 30 to describe the size of the debt in dollars. 6.NS.C.7d. Distinguish comparisons of absolute value from statements about order. For example, recognize that an account balance less than –30

CAR © 2009

dollars represents a debt greater than 30 dollars.

6.NS.8: Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.





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6.EE.B.7. Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers.






Performance Tasks




Standards Based Constructed Response ExampleEach ticket for a concert costs $12. The total amount of money earned from ticket sales for the concert was $4,548. Let n represent the number of tickets sold.

Part A Write an equation to represent this scenario.

Part B How many tickets were sold to the concert? Show or explain your work.

Part C For the next concert, the performers want to increase the total amount of money earned from ticket sales at the previous concert by $600 without raising the cost per ticket. Let n represent the number of tickets sold and write an equation to represent this new scenario.

Part D How many more tickets will they need to sell for the concert described in Part C? Show or explain your work.


Questions 21-29






Performance Tasks 6.EE.7https://

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6.EE.B.8. Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams.






Performance Tasks




Standards Based Constructed Response ExampleOceanwave Parasailing Tours has 3 boats it uses to take customers out on the ocean to parasail. Each boat can hold fewer than 9 customers.

Part A. Write an inequality to describe the number of people that can go parasailing with Oceanwave at one time. Let prepresent the number of people.

Part B. If 14 people want to parasail, what is the smallest number of boats the company could use?



Questions 1-7



Performance Tasks 6.EE.8https://njctl.org/courses/math/6th-grade-math/equations-inequalities/




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6.EE.C.9. Use variables to represent two quantities in a real-world problem that change in relationship to




Standards Based Constructed Response ExampleMolly and four friends want to buy tickets to an amusement park. They start with a total of $20, but they all have to do chores at home to earn the rest of the money. The list of ordered pairs below represents the number of chores performed (c) and the total


Questions 1-9



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one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time.




Performance Tasks



money collected (m).

Part A. Identify which variable is the dependent variable and which is the independent variable.

Part B. Plot the ordered pairs on a coordinate graph like the one below. Label the graph.

Part C. Write an equation for the line shown in your graph.

Part D. How much money do Molly and her friends earn for each chore they perform? (Assume that each chore pays the same amount of money.) If Molly and her friends perform 55 chores, how much money will they earn?

Part E. If each ticket to the amusement park costs $32.80, how many total chores will Molly and her friends need to perform to earn the rest of the

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money they need to buy five tickets?


6.NS.C.6. Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates. 6.NS.C.6a. Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a number is the number itself, e.g., – (–3) = 3, and that 0 is its own opposite. 6.NS.C.6b. Understand signs of numbers in






Performance Tasks




Standards Based Constructed Response ExampleThe coordinate plane shows the locations of three buildings in relation to Jessie’s house. Jessie’s house is located at the origin. Each coordinate unit represents 1 block.

Part A What is the ordered pair that represents the school? Explain how to find the coordinates for your


Questions 1-14






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ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes. 6.NS.C.6c. Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane.

answer.

Part B What is the ordered pair that represents the hospital? Explain how to find the coordinates for your answer.

Part C What reflection transforms the point that represents the school into the point that represents the hospital? Explain how you know your reflection is correct.



6.NS.C.7: Understand ordering and absolute value of rational numbers. 6.NS.C.7a. Interpret




Standards Based Constructed Response ExampleMs. Justice owns a small company. She recorded the account balance for her company for the first half of the year in the table below.


Questions 16-31



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statements of inequality as statements about the relative position of two numbers on a number line diagram. For example, interpret –3 > –7 as a statement that –3 is located to the right of –7 on a number line oriented from left to right. 6.NS.C.7b. Write, interpret, and explain statements of order for rational numbers in real-world contexts. For example, write –3 o C > –7 o C to express the fact that –3 o C is warmer than –7 o C. 6.NS.C.7c. Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or




Performance Tasks



Part A. Write an inequality that compares the greatest account balance with the least account balance.

Part B. The balance in July was $200 greater than in June. Write a numerical inequality that compares the balances of July and April.

Part C. Ms. Justice stated that the monthly balance for her company was greater in June than it was in March. Was her statement correct? Explain.



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negative quantity in a realworld situation. For example, for an account balance of –30 dollars, write |–30| = 30 to describe the size of the debt in dollars. 6.NS.C.7d. Distinguish comparisons of absolute value from statements about order. For example, recognize that an account balance less than –30 dollars represents a debt greater than 30 dollars.

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6.NS.8: Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.






Performance Tasks




Standards Based Constructed Response ExampleThe map below shows the location of Terry’s house.

Part A. What appear to be the coordinates of Terry’s house?

Part B. Naomi lives 4 blocks south and 6 blocks west of Terry’s house. Plot a point on the grid to show where Naomi lives. Label the point with the letter N.

Part C. Naomi lives 5 blocks from the library. She can walk straight down the street to the library. List all the ordered pairs that could be the location of the library.


Questions 6-13







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Unit Title: Statistics and GeometryGrade Level: 6


Essential Questions




6.SP.A.1. Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. For example, “How old am I?” is not a statistical question, but “How old are the students in my school?” is a statistical question because one anticipates variability in students’ ages.

6.SP.A.2. Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape.

6.SP.A.3. Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number.

6.SP.B.4. Display numerical data in plots on a number line, including dot plots, histograms, and box plots.

6.SP.B.5. Summarize numerical data sets in relation to their context, such as by: 6.SP.B.5a. Reporting the number of observations. 6.SP.B.5b. Describing the CAR © 2009

nature of the attribute under investigation, including how it was measured and its units of measurement. 6.SP.B.5c. Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered. 6.SP.B.5d. Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered

6.G.A.1. Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.

6.G.A.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.G.A.3. Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems.

6.G.A.4. Represent threedimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems.





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6.SP.A.1. Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. For example, “How old am I?” is not a statistical question, but “How old are the students in my school?” is a statistical question because one anticipates variability in students’ ages.

6.SP.A.2. Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape.

6.SP.A.3. Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number.






Performance Tasks



6.SP.A – Touchpoint

Standards Based Constructed Response ExampleThere are four students on the chess team. • The shortest student is 55 inches tall.• The tallest student is 66 inches tall.• The mean height of the four students is 61 inches. • The mean absolute deviation of the heights of the four students is less than 4 inches. What could be the heights, in inches, of the two other students? Explain how you found your answers or show your work.

Constructed Response ResourcesEdConnect 6.SP.1 - Type 2-3 BankEdConnect 6.SP.2 - Type 2-3 BankEdConnect 6.SP.3 - Type 2-3 Bank






Performance Tasks 6.SP.1https://

www.illustrativemathematics.org/content-standards/6/SP/A/1

Illustrative Mathematics Performance Tasks 6.SP.2

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content-standards/6/SP/A/2



content-standards/6/SP/A/3

PMI Math Labshttps://njctl.org/courses/math/6th-grade-math/statistical-variability/

Direct Instruction ResourcesEngageNY Lessons 6.SP.1

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6.SP.B.4. Display numerical data in plots on a number line, including dot plots, histograms, and box plots.

6.SP.B.5. Summarize numerical data sets in relation to their context, such as by: 6.SP.B.5a. Reporting the number of observations. 6.SP.B.5b. Describing the nature of the attribute under investigation, including how it was measured and its units of measurement.






Performance Tasks

Standards Based Problems Aligned to

6.SP.B – Touchpoint

Standards Based Constructed Response Example The dot plot shows the numbers of songs downloaded last week by Ms. Aung’s students.

Part A How many students are in Ms. Aung’s class? Show or explain your work.

Part B What is the difference between the mean and median number of songs downloaded per student?

Constructed Response ResourcesEdConnect 6.SP.4 - Type 2-3 BankEdConnect 6.SP.5 - Type 2-3 Bank






Performance Tasks 6.SP.4https://

www.illustrativemathematics.org/content-standards/6/SP/B/4



content-standards/6/SP/B/5

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6.SP.B.5c. Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered. 6.SP.B.5d. Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered

Touchpoint and End of Unit Assessment


Show or explain your work for finding the mean, median, and difference.

Part C Describe what the overall shape of the data means for the number of songs Ms. Aung’s students downloaded last week.


grade-math/data-displays/

Direct Instruction ResourcesEngageNY Lessons 6.SP.4



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6.G.A.1. Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.

6.G.A.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






Performance Tasks



6.G.A – Touchpoint

Standards Based Constructed Response ExampleA neighborhood is represented on a standard coordinate plane. There is a house located at the ordered pair (–2, 3). There are two stores located in the same area: Store M at (3, 3) and Store N at (–2, –1)

Part A Which store is closer to the house? Show or explain your work.

Part B What would be the coordinates of the ordered pair for a second house that is directly 3 units to the right of Store N in a horizontal direction? Show or explain your work.

Part C In order for the stores to be the same distance from the house at (–2, 3),where would Store M need to be located? Show or explain your work.

Constructed Response ResourcesEdConnect 6.G.1 - Type 2-3 Bank

Questions 6-8EdConnect 6.G.3 - Type 2-3 Bank

Questions 6-13Carnegie Learning Course 1




Performance Task ResourcesIllustrative Mathematics Performance Tasks 6.G.1


content-standards/6/G/A/1

Illustrative Mathematics Performance Tasks 6.G.2










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https://njctl.org/courses/math/6th-grade-math/geometry/

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problems.

6.G.A.3. Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems.

6.G.A.4. Represent threedimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems.

grade-math/geometry/

Direct Instruction ResourcesEngageNY Lessons 6.G.1

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Grade 6 Math – End of Unit 4 Assessment in EdConnectGrade 6 Math – Cumulative AssessmentDiagnostic Assessment

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