# Module 1

Version 1.3.1

### 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... Read More

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  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.

Focus Standards for Mathematical Practice

MP.1          Make sense of problems and persevere in solving them.  Students make sense of and solve real-world and mathematical ratio, rate, and percent problems using representations, such as tape diagrams, ratio tables, the coordinate plane, and double number line diagrams.  They identify and explain the correspondences between the verbal descriptions and their representations and articulate how the representation depicts the relationship of the quantities in the problem.  Problems include ratio problems involving the comparison of three quantities, multi-step changing ratio problems, using a given ratio to find associated ratios, and constant rate problems including two or more people or machines working together.

MP.2          Reason abstractly and quantitatively.  Students solve problems by analyzing and comparing ratios and unit rates given in tables, equations, and graphs.  Students decontextualize a given constant speed situation, representing symbolically the quantities involved with the formula, $distance=rate×time$ .

MP.5          Use appropriate tools strategically.  Students become proficient using a variety of representations that are useful in reasoning with rate and ratio problems, such as tape diagrams, double line diagrams, ratio tables, a coordinate plane, and equations.  They then use judgment in selecting appropriate tools as they solve ratio and rate problems.

MP.6          Attend to precision.  Students define and distinguish between ratio, the value of a ratio, a unit rate, a rate unit, and a rate.  Students use precise language and symbols to describe ratios and rates.  Students learn and apply the precise definition of percent.

MP.7          Look for and make use of structure.  Students recognize the structure of equivalent ratios in solving word problems using tape diagrams.  Students identify the structure of a ratio table and use it to find missing values in the table.  Students make use of the structure of division and ratios to model $5$ miles/ $2$ hours as a quantity $2.5$ mph.