Unit 1

Ratios and Unit Rates

Math

Unit Length and Description:

35 days

Students begin their sixth grade year examining the concepts of ratio and rate. They use ratio language and ratio notation, and formalize understanding of equivalent ratios. Students solve ratio problems in real world contexts using various tools, such as tape diagrams, double number line diagrams, tables, equations and graphs. Students bridge their understanding of ratios to the value of a ratio, and then to rate and unit rate, discovering that a percent of a quantity is a rate per 100. The unit concludes with students expressing a fraction as a percent and finding a percent of a quantity in real world concepts.

Standards:

 RP – Ratios and Proportional Relationships Understand ratio concepts and use ratio reasoning to solve problems. 6.RP.A.1 Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities.  For example, “The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak.” “For every vote candidate A received, candidate C received nearly three votes.” 6.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. For example, “This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar.” “We paid \$75 for 15 hamburgers, which is a rate of \$5 per hamburger.” 6.RP.A.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. NS- The Number System Apply and extend previous understandings of multiplication and division to divide fractions by fractions 6.NS.B.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). Standards for Mathematical Practices 1.   Make sense of problems and persevere in solving them. 2.   Reason abstractly and quantitatively. 3.   Construct viable arguments and critique the reasoning of others. 4.   Model with mathematics. 5.   Use appropriate tools strategically. 6.   Attend to precision. 7.   Look for and make use of structure. 8.   Look for and express regularity in repeated reasoning. Instructional Outcomes:   6.RP.A.1: ·         I can write ratio notation - __:__, __ to __, __/__. ·         I can explain how order matters when writing a ratio. ·         I can demonstrate how ratios can be simplified. ·         I can demonstrate how ratios compare two quantities; the quantities do not have to be the same unit of measure. ·         I can recognize that ratios appear in a variety of different contexts; part-to-whole, part-to-part, and rates. ·         I can generalize that all ratios relate two quantities or measures within a given situation in a multiplicative relationship. ·         I can analyze context to determine which type of ratio is represented. 6.RP.A.2: ·         I can identify and calculate a unit rate. ·         I can use appropriate math terminology as related to rate. ·         I can analyze the relationship between a ratio a:b and a unit rate a/b where b≠0. 6.RP.A.3 ·         I can make a table of equivalent ratios using whole numbers. ·         I can find the missing values in a table of equivalent ratios. ·         I can solve real-world and mathematical problems involving ratio and rate, for example, by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations. 6.RP.A.3a ·         I can make a table of equivalent ratios using whole numbers. ·         I can find the missing values in a table of equivalent ratios. ·         I can plot pairs of values that represent equivalent ratios on the coordinate plane. ·         I can use tables to compare proportional quantities. 6.RP.A.3b ·         I can apply the concept of unit rate to solve real-world problems involving unit pricing. ·         I can apply the concept of unit rate to solve real-world problems involving constant speed. 6.RP.A.3c ·         I can demonstrate how a percent is a ratio of a number to 100. ·         I can find a percent of a number as a rate per 100. ·         I can solve real-world problems involving finding the whole, given a part and a percent. 6.RP.A.3d ·         I can apply ratio reasoning to convert measurement units by multiplying or dividing in real-world and mathematical problems. 6.NS.B.4 ·         I can identify the factors of two whole numbers less than or equal to 100 and determine the Greatest Common Multiple. ·         I can identify the multiples of two whole numbers less than or equal to 12 and determine the Least Common Multiple. ·         I can apply the Distributive Property to rewrite addition problems by factoring out the Greatest Common Factor.

Enduring Understandings:

·   A ratio expresses the comparison between two quantities. Special types of ratios are rates, unit rates, measurement conversions, and percents.

·   A ratio or a rate expresses the relationship between two quantities. Ratio and rate language is used to describe a relationship between two quantities (including unit rates.)

·   A rate is a type of ratio that represents a measure, quantity, or frequency, typically one measured against a different type of measure, quantity, or frequency.

·   Ratio and rate reasoning can be applied to many different types of mathematical and real-life problems (rate and unit rate problems, scaling, unit pricing, statistical analysis, etc.).

Essential Questions:

·   How can you use mathematics to describe change and model real-world situations?

·   When is it useful to be able to relate one quantity to another?

·   How are ratios and rates similar and different?

·   What is the connection between a ratio and a fraction?

·   How do you use equivalent rates in the real world?