Eureka Math » Grade 6 » Module 1

Module 1

Version 1.3.0 (2015-16)

Ratios and Unit Rates

In this module, students are introduced to the concepts of ratio and rate.  Their previous experience solving problems involving multiplicative comparisons, such as Max has three times as many toy cars as Jack, (4.OA.A.2) serves as the conceptual foundation for understanding ratios as a multiplicative comparison of two or more numbers used in quantities or measurements (6.RP.A.1).  Students develop fluidity in using multiple forms of ratio language and ratio notation.  They construct viable arguments and communicate reasoning about ratio equivalence as they solve ratio problems in real-world contexts (6.RP.A.3).  As the first topic comes to a close, students develop a precise definition of the value of a ratio a:b , where b0   as the value ab , applying previous understanding of fraction as division (5.NF.B.3).  They can then formalize their understanding of equivalent ratios as ratios having the same value. 

With the concept of ratio equivalence formally defined, students explore collections of equivalent ratios in real-world contexts in Topic B.  They build ratio tables and study their additive and multiplicative structure (6.RP.A.3a).  Students continue to apply reasoning to solve ratio problems while they explore representations of collections of equivalent ratios and relate those representations to the ratio table (6.RP.A.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 defined in Topic A.  Finally, students expand their experience with the coordinate plane (5.G.A.1, 5.G.A.2) as they represent collections of equivalent ratios by plotting the pairs of values on the coordinate plane.  The Mid-Module Assessment follows Topic B.

In Topic C, 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.A.2).  Students solve unit rate problems involving unit pricing, constant speed, and constant rates of work (6.RP.A.3b).  They apply their understanding of rates to situations in the real world.  Students determine unit prices, use measurement conversions to comparison shop, and decontextualize constant speed and work situations to determine outcomes.  Students combine their new understanding of rate to connect and revisit concepts of converting among different-sized standard measurement units (5.MD.A.1).  They then expand upon this background as they learn to manipulate and transform units when multiplying and dividing quantities (6.RP.A.3d).  Topic C culminates as students interpret and model real-world scenarios through the use of unit rates and conversions.

In the final topic of the module, students are introduced to percent and find percent of a quantity as a rate per 100.  Students understand that N  percent of a quantity has the same value as N100  of that quantity.  Students express a fraction as a percent and find a percent of a quantity in real-world contexts.  Students learn to express a ratio 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.A.3c).  The End-of-Module Assessment follows Topic D.

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Mid-Module Assessment

End-of-Module Assessment

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