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Grade 6 UNIT 1: Ratios and Unit Rates Suggested Number of Days for Entire UNIT: 35

|Essential Question |Key Concepts |Cross Curricular Connections |

|How do we use ratios and unit rates to solve real-world mathematical|Representing and Reasoning About Ratios |Religion: Use the actual size of the parts of the Egyptian pyramids |

|problems? |Collections of Equivalent Ratios |to make a scale model using ratios and proportions. Find the |

| |Unit Rates |measurements of your own church and compare it to the pyramids. |

|Unit Vocabulary |Percent | |

|Ratio | |Science: Use ratios and proportions to analyze the amount of |

|Rate | |calories each student consumes in a day. Students will need to |

|Unit Rate |*Assessments |determine the amount of calories per serving. They would also set up|

|Value of a Ratio |Mid-Module Assessment: After Section B |a proportion based on how many servings they would eat to determine |

|Equivalent Ratios |(3 days -1 day for assessment, 1 day for assessment return, & 1 day |the calories for each item. Consider what is appropriate for |

|Percent |for remediation) |different body types. |

|Associated Ratios |End-of-Module Assessment: after Section D (3 days- 1, day for | |

|Double Number Line |assessment, 1 day for assessment return, & 1 day for remediation). |Other: Engage in researching the golden ratio. Construct a well |

|Ratio Table | |developed composition paper on the significance of the golden ratio |

| | |and where we see it in our everyday lives. |

| Unit Outcome (Focus) |

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|Students use reasoning about multiplication and division to solve ratio and rate problems about quantities. By viewing equivalent ratios and rates as deriving from, and extending, pairs of rows (or columns)|

|in the multiplication table, and by analyzing simple drawings that indicate the relative size of quantities, students connect their understanding of multiplication and division with ratios and rates. Thus |

|students expand the scope of problems for which they can use multiplication and division to solve problems, and they connect ratios and fractions. Students solve a wide variety of problems involving ratios |

|and rates. Students construct and analyze tables, such as tables of quantities that are in equivalent ratios. Use proportional reasoning to convert between measurements. |

UNIT 1 SECTION A: Representing and Reasoning About Ratios Suggested Number of Days for SECTION: 8

|Essential Question |Key Concepts |Standards for Mathematical Practice |

|How do we use ratios and unit rates to solve |Ratios |1. Make sense of problems and persevere in solving them |

|real-world mathematical problems? |Equivalent Ratios |3. Construct viable arguments and critique the reasoning of others |

| |Solving Problems by Finding Equivalent Ratios |6.Attend to precision |

| |Associated Ratios and the Value of a Ratio |7. Look for and make use of structure |

| |Equivalent Ratios Defined Through the Value of a Ratio |8. Look for and express regularity in repeated reasoning. |

|Comments |Standard No. |Standard |Priority |

| | |( Major Standard ( Supporting Standard ( Additional Standard | |

|Students develop fluidity in using multiple | | | |

|forms of ratio language and ratio notation. |6.RP.1 |Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. |( |

|They construct viable arguments and communicate |(DOK 2) | | |

|reasoning about ratio equivalence as they solve | | | |

|ratio problems in real world contexts (6.RP.3). | |Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent | |

|As the first topic comes to a close, students |6.RP.3a |ratios, tape diagrams, double number line diagrams, or equations | |

|develop a precise definition of the value of a | |a. Make tables of equivalent ratios relating quantities with whole number measurements, find missing values in the tables, and |( |

|ratio a:b, where b ≠ 0 as the value a/b, | |plot the pairs of values on the coordinate plane. Use tables to compare. | |

|applying previous understanding of fraction as | | | |

|division (5.NF.3). They can then formalize | | | |

|their understanding of equivalent ratios as | | | |

|ratios having the same value. | | | |

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UNIT 1 SECTION B: Collections of Equivalent Ratios Suggested Number of Days for SECTION: 7

|Essential Question |Key Concepts |Standards for Mathematical Practice |

|How do we use ratios and unit rates to solve |Tables of Equivalent Ratios |1. Make sense of problems and persevere in solving them |

|real-world mathematical problems? |The Structure of Ratio Tables: Additive and Multiplicative |3. Construct viable arguments and critique the reasoning of others |

| |Comparing Ratios Using Ratio Table | |

| |From Ratio Tables to Double Number Line Diagrams |4. Model with mathematics |

| |From Ratio Tables to Equations Using the Value of the Ratio |6.Attend to precision |

| |From Ratio Tables, Equations, and Double Number Line Diagrams to Plots on the Coordinate|7. Look for and make use of structure |

| |Plane |8. Look for and express regularity in repeated reasoning. |

| |A Synthesis of Representations of Equivalent Ratio Collections | |

|Comments |Standard No. |Standard |Priority |

| | |( Major Standard ( Supporting Standard ( Additional Standard | |

|Students explore collections of equivalent | | | |

|ratios in real world contexts. They build ratio |6.RP.3a |Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent |( |

|tables and study their additive and | |ratios, tape diagrams, double number line diagrams, or equations | |

|multiplicative structure (6.RP.3a). Students | |a. Make tables of equivalent ratios relating quantities with whole number measurements, find missing values in the tables, and | |

|continue to apply reasoning to solve ratio | |plot the pairs of values on the coordinate plane. Use tables to compare. | |

|problems while they explore representations of | | | |

|collections of equivalent ratios and relate | | | |

|those representations to the ratio table | | | |

|(6.RP.3). Building on their experience with | | | |

|number lines, students represent collections of | | | |

|equivalent ratios with a double number line | | | |

|model. They relate ratio tables to equations | | | |

|using the value of a ratio. | | | |

UNIT 1 SECTION C: Unit Rates Suggested Number of Days for SECTION: 8

|Essential Question |Key Concepts |Standards for Mathematical Practice |

|How do we use ratios and unit rates to solve |From Ratios to Rates |Make sense of problems and persevere in solving them. |

|real-world mathematical problems? |From Rates to Ratios |Reason abstractly and quantitatively. |

| |Finding a Rate by Dividing Two Quantities |4. Model with mathematics |

| |Comparison Shopping – Unit Price and Related Measurement Conversions |Attend to precision |

| |Getting the Job Done – Speed, Work and Measurement Units |8. Look for and express regularity in repeated reasoning |

| |Problem-Solving Using Rates, Unit Rates, and Conversions | |

|Comments |Standard No. |Standard |Priority |

|Rate reasoning is central to the study of | |( Major Standard ( Supporting Standard ( Additional Standard | |

|algebra and is a critical foundation for | |( Standard ends at this grade ( Fluency Standard | |

|understanding slope in grades 7 & 8. | | | |

| | | | |

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| |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 |( |

| |(DOK 2) |ratio relationship. | |

| | | | |

| | | | |

| |6.RP.3b |Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent | |

| |(DOK 2) |ratios, tape diagrams, double number line diagrams, or equations |( |

| | | | |

| | |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? | |

| | | | |

| | | | |

| | |d. Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing | |

| | |quantities. | |

| |6.RP.3d | | |

| |(DOK 2) | | |

|Students build further on their understanding of| | | |

|ratios and the value of a ratio as they come to | | | |

|understand that a ratio of 5 miles to 2 hours | | | |

|corresponds to a rate of 2.5 miles per hour, | | | |

|where the unit rate is the numerical part of the| | | |

|rate, 2.5, and miles per hour is the newly | | | |

|formed unit of measurement of the rate (6.RP.2).| | | |

|Students solve unit rate problems involving unit| | | |

|pricing, constant speed, and constant rates of | | | |

|work (6.RP.3b). They apply their understanding | | | |

|of rates to situations in the real world. | | | |

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UNIT 1 SECTION D: Percent Suggested Number of Days for SECTION: 6

|Essential Question |Key Concept |Standards for Mathematical Practice |

|How do we use ratios and unit rates to solve |Percent and Rates per 100 |1. Make sense of problems and persevere in solving them |

|real-world mathematical problems? |A Fraction as a Percent |4. Model with mathematics |

| |Percent of a Quantity |7. Look for and make use of structure |

| |Solving Percent Problems |8. Look for and express regularity in repeated reasoning |

|Comments |Standard No. |Standard |Priority |

| | |( Major Standard ( Supporting Standard ( Additional Standard | |

| | |( Standard ends at this grade ( Fluency Standard | |

|Students are introduced to percent and find | | | |

|percent of a quantity as a rate per 100. | | | |

|Students understand that N percent of a quantity|6.RP.3c |Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent |( |

|has the same value as N/100 of that quantity. |(DOK 1) |ratios, tape diagrams, double number line diagrams, or equations. | |

|Students express a fraction as a percent, and | | | |

|find a percent of a quantity in real-world | |d. 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 | |

|contexts. Students learn to express a ratio | |involving finding the whole, given a part and the percent. | |

|using the language of percent and to solve | | | |

|percent problems by selecting from familiar | | | |

|representations, such as tape diagrams and | | | |

|double number lines, or a combination of both | | | |

|(6.RP.3c). | | | |

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|Possible Activities |

|EQUIVALENT RATIO BINGO: Create a list of 30+ problems involving ratios (make sure they all have different answers). Write the problems and answers on index cards. Write the answers on the board. Have |

|students randomly write 24 of the answers on a blank bingo card (cards can be reused or exchanged). Place the index cards face down. Take turns choosing a card and reading the problem to the students. You |

|may want to also write the problem on the board so that students can reference it later if they need more time. Students should mark off the correct answer on their game boards using chips or paper scraps. |

|Traditional bingo rules apply. Make sure you have the winning students read off their answers. Bingo card can be downloaded at Click on Math Games and download the Equivalent |

|Fractions Game for game boards. |

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|EQUIVALENT RATIO BOARD GAME: (2 person game) One player throws two dice and makes a proper fraction with the numbers rolled (smaller number as numerator). The player then finds an equivalent ratio on the |

|board and covers it with a marker. If an opponent’s marker is already there, it may be removed. If a player rolls doubles, he or she loses a turn. The first player to get three markers in a row wins! Create|

|boards of equivalent ratios using the roll of a dice or download a premade board online (see below).• Ex: Roll a 6 and 2. Fraction is 2/6. Equivalent ratio can cover 4/12 or 1/3. Download pre-made boards |

|at |

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|GAMES AT RECESS: 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. For the entire problem, visit |

|Resources |

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|VISUALIZE PART-TO-TOTAL RATIOS USING PICTURES In this lesson you will learn to visualize a part-to-total ratio using pictures. |

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|THE GOLDEN RATION: In this lesson, one of a multi-part unit from Illuminations, students learn and understand the concept of ratio. They learn about the Golden Ratio, used by the painter Leonardo da Vinci |

|in his work, while exploring different ratios to see whether the Golden Ratio holds true. |

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|Engage NY Grade 6 Module 1 Link: |

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|Possible Activities |

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|CONVERTING UNIT RATES ACTIVITY: Have students collect data on unit rates by timing themselves doing a variety of activities (saying the alphabet, hops on one foot in one minute, etc.). Have them use these rates|

|to find out how many hops on one foot they can do in one hour; or how many times they can say the alphabet in 45 minutes, etc.? |

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|COMPARISON SHOPPING: Bring in the Sunday ads for different grocery stores items (student can also view ads online). Challenge the students to comparison shop to find the best deal using unit cost. A short movie|

|explaining comparison shopping can be viewed online (see right). |

|Ex: 5 oranges for $3.50 or 2 for $1.75. Which is the better deal? |

|Ex: $4.50 for 24 ounces of pasta and $5.00 for 30 ounces of pasta. Which is the better deal? |

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|ROAD TRIP ACTIVITY: (groups of 3-5) Each group of students will choose and research a city in the Unites States. They will plan to travel to the city via car. The students will research which car they will |

|purchase (or rent) for the drive and how many gallons of gas the car gets per mile. The student will then go on a virtual road trip! They can calculate how many miles they traveled and how much gas they bought.|

|They can chart their unit rate of miles per gallon as well as graph their course. They can find the ratio of miles to the gallon as well as distance travelled per day. Extend: Students can track expenses: How |

|much total money they spent, including hotels and food, etc. They can also receive speed bumps along the way, such as your team received a flat tire on the 64thmile of day 2. What do you do? What is the added |

|cost? |

|PRICE PER POUND AND POUNDS PER DOLLAR The grocery store sells beans in bulk. The grocer's sign above the beans says, “5 pounds for $4.” At this store, you can buy any number of pounds of beans at this same |

|rate, and all prices include tax. For entire problem, visit: |

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|Resources |

|COMPARISON SHOPPING: A short movie explaining comparison shopping. Go to Click on Math, select Free Movies, and choose Comparing Prices. |

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|UNDERSTAND RATES AS A TYPE OF RATIO In this lesson you will learn how rates are part of the ratio family by reviewing the qualities of a ratio |

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|TRAVEL IN THE SOLAR SYSTEM: This lesson, from Illuminations, affords students the opportunity to think about two aspects of the time required to complete space travel within the solar system. First, students |

|consider the amount of time that space travelers must spend on the journey. Second, students think about what kinds of events might occur on Earth while the space travelers are on their journey. Thinking about |

|both situations improves students' concept of time and distance as well as their understanding of the solar system. |

|Possible Activities |

| PROPORTIONS: Challenge students to discover proportions (equivalent ratios) and to plan a lesson with examples and models to present to the class. Resources found online at . Click on|

|Lessons then scroll down to Middle and choose How to Teach Proportions. |

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|I HAVE, WHO HAS? An activity for the entire class to practice solving problems. The students are each given a card or two. Each card contains a number on the top and a problem to solve on the bottom. The |

|answer to the problem leads to the next card; if cards are created they need to be completed all at once. This is a great game to reinforce mental math with unit rates or percent to decimal to fraction |

|conversions. Game cards can also be downloaded online (see right). |

|Ex: “I have ___, who has a ratio equivalent to 3:7? The student with 9:21 on their card responds: I have 9:21, who has ratio equivalent to 2:9?” The student with 4:18 on their card responds… and the game |

|continues until all of the cards are called. |

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|ADVANCED PAPER POOL: This reproducible worksheet, from an Illuminations lesson, presents a table on which students record their predictions about the behavior of pool balls on pool tables of different |

|dimensions. Three related questions accompany the table. |

|Resources |

|PREPARING TO TEACH PROPORTIONAL REASONING: What do your students already know about ratio, proportion, and scale? Do they have the knowledge and skills necessary to understand the mathematics of |

|proportional reasoning? Teaching Module 1 focuses on eight skills students should master before exploring Scale City: |

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|Engage NY Grade 6 Module 1 Link: |

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|Possible Activities |

|GO TO THE STORE: Write on the whiteboard, cut out, or print pictures of items that are for sale. Explain to the students that they have to buy any combination of four items. The students are then challenged|

|to find the total of the items plus sales tax. Extend: Can they calculate their total if select items are on sale (i.e. 25% off)? |

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|GRID AND PERCENT IT: In this Illuminations lesson, students solve various types of percent problems using a 10 x 10 grid, a common model for visualizing percents. This model offers a means of representing |

|the given information as well as suggesting different approaches for finding a solution. |

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|PERCENT SLOPE TOOL: This reproducible activity, from an Illuminations lesson, provides a template by which students can create a tool for calculating the slope of real-world inclines. |

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|Resources |

|FIND THE PART WHEN THE PERCENT AND TOTAL ARE KNOWN: In this lesson you will learn to calculate the part when you know the percent and the total by using a double number line: |

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|BIG MATH AND FRIES: This Illuminations lesson is designed to enlighten students about how to calculate percent of calories from fat, carbohydrates, and protein. The calculations are made to determine if a |

|person can follow the Zone Diet with only McDonald's food items. |

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|Engage NY Grade 6 Module 1 Link: |

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|Possible Activities |

|I HAVE, WHO HAS? An activity for the entire class to practice solving problems. The students are each given a card or two. Each card contains a number on the top and a problem to solve on the bottom. The |

|answer to the problem leads to the next card; if cards are created they need to be completed all at once. This is a great game to reinforce mental math with unit rates or percent to decimal to fraction |

|conversions. Game cards can also be downloaded online (see right). |

|Ex: “I have ___, who has a ratio equivalent to 3:7? The student with 9:21 on their card responds: I have 9:21, who has ratio equivalent to 2:9?” The student with 4:18 on their card responds… and the game |

|continues until all of the cards are called. |

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|MEASURING UP: This unit explores the concepts of proportional reasoning, ratio, and indirect measurement. Students engage in a variety of activities that involve taking their own measurements, exploring |

|different ratios, and examining similar figures. Students convert measurements into customary and metric units. These activities immerse students in problem solving, reasoning, and making connections to |

|real-life situations. |

|Resources |

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|CONVERT MEASUREMENT UNITS USING RATIO TABLES: In this lesson you will learn to convert measurement units by using ratio tables. |

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|CELL SIZE AND SCALE: This simple interactive from the University of Utah's Genetic Science Learning Center gives you the opportunity to see how various small things compare to one another. |

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|DISCOVERING GALLON MAN: In this lesson, one of a multi-part unit from Illuminations, students experiment with units of liquid measure used in the customary system of measurement. They also practice making |

|conversions of length and weight. |

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|Engage NY Grade 6 Module 1 Link: |

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