MCC6.RP.1 (DOK 2)
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.”
MCC6.RP.2 (DOK 2)
Understand the concept of a unit rate a/b associated with a ratio a: b with b ≠ 0 (b not equal to zero), 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."
MCC6.RP.3 (DOK 2)
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?
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.
6th Grade– CCGPS Math
LFS Unit 2: Rate, Ratio and Proportional Reasoning Using Equivalent Fractions
Douglas County School System
6th Grade Math Unit 2 10/15/2012
Rate, Ratio and Proportional Reasoning Using Equivalent Fractions Page 5
K-U-D Unit 2: Rate, Ratio and Proportional Reasoning
Using Equivalent Fractions
understand…By the end of the unit, I want my students to understand…
ratio concepts and use ratio reasoning to solve real world problems.
Know / Do
· The concept of a ratio is a way of expressing relationships between quantities. (RP.1)
· A rate is a special ratio that compares two quantities with different units of measure. (RP.2)
· Unit rates are the ratio of two measurements in which the second term is one (e.g., x miles per one hour). (RP.2)
· When using rates a/b, “b” cannot be 0 (because division by 0 is undefined). (RP.2)
· Rate language (per, each, or the @ symbol). (RP.2)
· Tools such as tables of equivalent ratios, tape diagrams, double number line diagrams, and equations support the development of ratio and rate reasoning. (RP.3a and 3b)
· Pairs of values from a table can be plotted on the coordinate plane. (RP.3a)
· The connections between tables and plotted points on the coordinate plane allow for extended reasoning and synthesis of the concept of ratios and rates. (RP.3a)
· Rate problems compare two different units, such as miles to hours. (RP.3b)
· A unit occurs when at least one of the units is one. (RP.3b)
· The connections between tools allow for extended reasoning and synthesis of the concept of ratios and rates (e.g., How do tape diagrams and double number lines show rate reasoning given the same context?). (RP.3b and 3c)
· A percent is a rate per 100 and can be represented using tools such as tables of equivalent ratios, tape diagrams, double number line diagrams, and equations. (RP.3c)
· Percentage-based rate problems compare two different units where one of the units is 100. (RP.3c)
· Measurement units employ ratio reasoning (e.g., If 3 feet is equal to one yard, then 6 feet is equal to 2 yards). (RP.3d)
Vocabulary:
(RP.1): Rate; Ratio; Relationship; Rational Number
(RP.2): Unit Rate;
(RP.3)
a: Equivalent Ratios; Proportion; X Y Table; Ordered Pairs; Coordinate Plane;
c: Percent; Tape Diagram
d: Tape Diagram; Convert; Customary Units of Measure; Metric Units of Measure; Double Number Line Diagrams / · Distinguish when a ratio is describing part to part or part to whole comparison. (RP.1) DOK2
· Describe ratio relationships between two quantities. (RP.1) DOK1
· Translate relationships between two quantities using the notation of ratio language (1:3, 1 to 3, 1/3). (RP.1) DOK1
· Communicate relationships between two quantities using ratio notation and language. (RP.1) DOK2
· Solve problems involving ratios. (RP.2) DOK1
· Correctly use ratio notation and models to represent relationships between quantities. (RP.2) DOK2
· Make, complete, and read a table of equivalent ratios. (RP.3a) DOK2
· Use a table to compare ratios. (RP.3a) DOK2
· Determine missing values using ratio reasoning. (RP.3a) DOK2
· Identify relationships in ratio tables. (RP.3a) DOK2
· Plot pairs of values from a table to a coordinate plane. (RP.3a) DOK1
· Solve real-world problems using rate reasoning (RP.3b) DOK2
· Calculate the unit rate. (RP.3b) DOK1
· Write a percent as a rate over 100. (RP.3c) DOK1
· Find the percent of number using rate methods developed in 6.RP.3b. (RP.3c) DOK1
· Given the parts and a percent, determine the whole using tools: tape diagrams, double number lines). (RP.3c) DOK2
· Represent the relationship of part to whole to describe percent using models. (RP.3c) DOK2
· Convert customary units using ratio tools and methods. (RP.3d) DOK1
· Convert metric units by multiplying or dividing by powers of ten. (RP.3d)DOK1
· Represent relationships between measurement units using tables of equivalent ratios, tape diagrams, double number line diagrams, and equations. (RP.3d) DOK2
SLM Unit 2: Rate, Ratio and Proportional Reasoning Using Equivalent Fractions
Key LearningConcept / Concept / Concept
Lesson EQ’s / Lesson EQ’s / Lesson EQ’s
1. / 1.
Vocabulary / Vocabulary / Vocabulary
Douglas County School System
6th Grade Math Unit 2 10/15/2012
Rate, Ratio and Proportional Reasoning Using Equivalent Fractions Page 5
Domain: /Cluster:
Ratios and Proportional Relationships /Understand ratio concepts and use ratio reasoning to solve problems
MCC6.RP.1 /What does this standard mean?
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.” / 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). Students need to understand each of these ratios when expressed in the following forms: , 6 to 15 or 6:15. These values can be reduced to , 2 to 5 or 2:5; however, students would need to understand how the reduced values relate to the original numbers.Examples and Explanations / Mathematical Practice Standards
A ratio is a comparison of two quantities which can be written as a to b, , or a:b.
A rate is a ratio where two measurements are related to each other. When discussing measurement of different units, the word rate is used rather than ratio. Understanding rate, however, is complicated and there is no universally accepted definition. When using the term rate, contextual understanding is critical. Students need many opportunities to use models to demonstrate the relationships between quantities before they are expected to work with rates numerically.
A comparison of 8 black circles to 4 white circles can be written as the ratio of 8:4 and can be regrouped into 4 black circles to 2 white circles (4:2) and 2 black circles to 1 white circle (2:1).
Students should be able to identify all these ratios and describe them using “For every…., there are …” / 6.MP.2. Reason abstractly and quantitatively.
6.MP.6. Attend to precision.
Suggested Instructional Strategy
Using a variety of situations, describe relationships using ratio, for example:
1. Part to part: Compare the number of girls to boys in the classroom using the different symbols for ratio (girls: boys, girls to boys, /, girls out of boys). Then compare the number of boys to girls in the same way.
2. Part to whole: Compare the number of girls to the whole class. Do the same thing for the boys in the class.
Skill Based Task / Problem Task
There are four dogs and three cats. What is the ratio of dogs to cats and cats to dogs? / The newspaper reported, “For every vote candidate A received, candidate B received three votes.” Describe possible election results using at least three different ratios. Explain your answer.
Instructional
Resources/Tools / Ratio coloring activity: http://www.softschools.com/math/ratios/ratio_coloring_game/
Internet Resources:
https://ccgps.org/6.RP.html
Douglas County School System
6th Grade Math Unit 2 10/15/2012
Rate, Ratio and Proportional Reasoning Using Equivalent Fractions Page 6
Domain: /Cluster:
Ratios and Proportional Relationships /Understand ratio concepts and use ratio reasoning to solve problems
MCC6.RP.2 /What does this standard mean?
Understand the concept of a unit rate a/b associated with a ratio a:b with b ¹ 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 ¾ cup of flour for each cup of sugar.” “We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger.”[1] / A unit rate expresses a ratio as part-to-one. For example, if there are 2 cookies for 3 students, each student receives of a cookie, so the unit rate is :1. If a car travels 240 miles in 4 hours, the car travels 60 miles per hour (60:1). Students understand the unit rate from various contextual situations.Examples and Explanations / Mathematical Practice Standards
A unit rate compares a quantity in terms of one unit of another quantity. Students will often use unit rates to solve missing value problems. Cost per item or distance per time unit are common unit rates, however, students should be able to flexibly use unit rates to name the amount of either quantity in terms of the other quantity. Students will begin to notice that related unit rates are reciprocals as in the first example. It is not intended that this be taught as an algorithm or rule because at this level, students should primarily use reasoning to find these unit rates.
In Grade 6, 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.
Examples:
· On a bicycle you can travel 20 miles in 4 hours. What are the unit rates in this situation, (the distance you can travel in 1 hour and the amount of time required to travel 1 mile)?
Solution: You can travel 5 miles in 1 hour written as and it takes of a hour to travel each mile written as . Students can represent the relationship between 20 miles and 4 hours.
· A simple modeling clay recipe calls for 1 cup corn starch, 2 cups salt, and 2 cups boiling water. How many cups of corn starch are needed to mix with each cup of salt? / 6.MP.2. Reason abstractly and quantitatively.
6.MP.6. Attend to precision.
Suggested Instructional Strategy
1. Show examples of rates: 300 miles on 10 gallons of gas, $15 for 5 ounces, $30 for 6 hours.
2. Connect rates from number 1 with their unit rates: 30 miles per gallons, $3 per 1 ounce, $5 per 1 hour.
3. Convert rates from fraction form to written form using per, each, or @. Example 300 10 = 30 miles per gallon of gas.
4. Quick write: Students brainstorm examples of unit rates in the real world (e.g., 4 candy bars per $1, 55 miles per hour, 6 points per touchdown).
Skill Based Task / Problem Task
Identify (given examples) the difference between a ratio and a rate. / Is the following example a ratio or rate? [60 heartbeats per minute] Explain your answer.
Instructional
Resources/Tools / UEN- Lesson “Ratio, Rate, and Proportion”
Activities 1 and 2 from http://mypages.iit.edu/~smart/dvorber/lesson3.htm
CCGPS Internet Resources:
https://ccgps.org/6.RP_K1IV.html
Douglas County School System
6th Grade Math Unit 2 10/15/2012
Rate, Ratio and Proportional Reasoning Using Equivalent Fractions Page 6
Domain: /Cluster:
Ratios and Proportional Relationships /Understand ratio concepts and use ratio reasoning to solve problems
MCC6.RP.3 /What does this standard mean?
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. / Ratios and rates can be used in ratio tables and graphs to solve problems. Previously, students have used additive reasoning in tables to solve problems. To begin the shift to proportional reasoning, students need to begin using multiplicative reasoning. To aid in the development of proportional reasoning the cross-product algorithm is not expected at this level. When working with ratio tables and graphs, whole number measurements are the expectation for this standard.
Examples and Explanations / Mathematical Practice Standards
For example, At Books Unlimited, 3 paperback books cost $18. What would 7 books cost? How many books could be purchased with $54. To find the price of 1 book, divide $18 by 3. One book is $6. To find the price of 7 books, multiply $6 (the cost of one book times 7 to get $42). To find the number of books that can be purchased with $54, multiply $6 times 9 to get $54 and then multiply 1 book times 9 to get 9 books. Students use ratios, unit rates and multiplicative reasoning to solve problems in various contexts, including measurement, prices, and geometry. Notice in the table below, a multiplicative relationship exists between the numbers both horizontally and vertically. (Red numbers indicate solutions.)
Students use tables to compare ratios. Another bookstore offers paperback books at the prices below. Which bookstore has the best buy? Explain how you determined your answer.
To help understand the multiplicative relationship between the number of books and cost, students write equations to express the cost of any number of books. Writing equations is foundational for work in 7th grade. For example, the equation for the first table would be C = 6n.
The numbers in the table can be expressed as ordered pairs (number of books, cost) and plotted on a coordinate plane. Students are able to plot ratios as ordered pairs. For example, a graph of Books Unlimited would be: / 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.
Suggested Instructional Strategy
1. Have students make a table given a ratio situation. They should plot those points on a coordinate plane and draw conclusions about what’s happening in the ratio situation.
2. Give students a table with missing values and have them identify the missing values.
3. Have students study ratio relationships in a table.
Skill Based Task / Problem Task
Analyze the table below to determine the missing values.
Fill in the missing values on the table below.
Swimmers / 20 / 30 / 40 / 60 / 90 / 100
Life Guards / 2 / 3 / 4 / 6
/ Graph the information from the table on the coordinate plane and explain the relationship of swimmers to life guards.
Instructional
Resources/Tools / CCGPS Internet Resources:
https://ccgps.org/6.RP.3.html
http://www.youtube.com/watch?v=d625kdtsUIw
UEN: Price-Earnings ratio http://www.uen.org/Lessonplan/preview.cgi?LPid=25290
Douglas County School System