6.RP.3: Use Ratio and Rate Reasoning to Solve Real-World and Mathematical Problems, E.G

Unit Title: Measurement, Quantities, and Relationships: The Foundation of Mathematics
Grade 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?

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 / Reflection
Diagnostic Assessment
Grade 6 Math – Cumulative Assessment
SLO - SWBAT / Student Strategies / Formative Assessment / Activities and Resources / Reflection

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 Example
The 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 Resources
EdConnect 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 Resources
Illustrative Mathematics Performance Tasks 6.RP.3
https://www.illustrativemathematics.org/content-standards/6/RP/A/3
PMI Math Labs
https://njctl.org/courses/math/6th-grade-math/ratios-proportions-percents/
Direct Instruction Resources
EngageNY 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-grade-math/ratios-proportions-percents/

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.

/ 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.3a – Touchpoint
Standards Based Constructed Response Example
Nancy 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 on1 gallon of gas. Make a table to show how many miles he drives on 12 gallons of gas.
Use, words, numbers, and/or pictures to show your work. / Constructed Response Resources
EdConnect 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 Resources
Illustrative Mathematics Performance Tasks 6.RP.3
https://www.illustrativemathematics.org/content-standards/6/RP/A/3
PMI Math Labs
https://njctl.org/courses/math/6th-grade-math/ratios-proportions-percents/
Direct Instruction Resources
EngageNY Lessons 6.RP.3
https://www.engageny.org/search-site?search=6.RP.3
Learnzillion Lesson Plans Ratios Days 5-10
https://learnzillion.com/resources/64212-ratios
PMI SMART Presentations https://njctl.org/courses/math/6th-grade-math/ratios-proportions-percents/

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.

/ 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.EE.6 – Touchpoint
Standards Based Constructed Response Example
Paul 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 apounds 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 Resources
EdConnect 6.EE.6 - Type 2-3 Bank
Questions 11-13
Carnegie Learning Course 1
PARCC Released Items
#3
https://prc.parcconline.org/system/files/Grade%2006%20Math%20-%20Item%20Set.pdf
Performance Task Resources
Illustrative Mathematics Performance Tasks 6.EE.6
https://www.illustrativemathematics.org/content-standards/6/EE/B/6
PMI Math Labs
https://njctl.org/courses/math/6th-grade-math/expressions/
Direct Instruction Resources
EngageNY 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
PMI SMART Presentations
https://njctl.org/courses/math/6th-grade-math/expressions/

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.

/ 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.NS.5 – Touchpoint
Standards Based Constructed Response Example
Corrine 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 Resources
EdConnect 6.NS.5 - Type 2-3 Bank
Questions 6-10
Carnegie Learning Course 1
PARCC Released Items
#12
https://prc.parcconline.org/system/files/Grade%2006%20Math%20-%20Item%20Set.pdf
Performance Task Resources
Illustrative Mathematics Performance Tasks 6.NS.5
https://www.illustrativemathematics.org/content-standards/6/NS/C/5
PMI Math Labs
https://njctl.org/courses/math/6th-grade-math/numbers-and-operations-6th-grade/
Direct Instruction Resources
EngageNY 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/

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?

/ 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.3b – Touchpoint
Standards Based Constructed Response Example
Sophia and Juwan went to the farmers’ market to buy some fruit. The prices of fruit are shown below.
Sophia bought 2.5 pounds of peachesand 1.4 pounds of apples.
Part A. How much did Sophia spend on the peaches? How much didshe 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 ofblueberries. / Constructed Response Resources
EdConnect 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 Resources
Illustrative Mathematics Performance Tasks 6.RP.3
https://www.illustrativemathematics.org/content-standards/6/RP/A/3
PMI Math Labs
https://njctl.org/courses/math/6th-grade-math/ratios-proportions-percents/
Direct Instruction Resources
EngageNY Lessons 6.RP.3
https://www.engageny.org/search-site?search=6.RP.3
Learnzillion Lesson Plans Ratios Days 5-10
https://learnzillion.com/resources/64212-ratios
PMI SMART Presentations https://njctl.org/courses/math/6th-grade-math/ratios-proportions-percents/

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.