Lesson objective: Understand that multiplicative relationships exist both between and within equivalent ratios.

Students bring prior knowledge that ratios show multiplicative relationships between quantities (6.RP.A.3.a ). This prior knowledge is extended to equivalent ratios as students use relationships "between" and "within" these ratios to find missing values. Students are not required to learn or use the terms "between" and "within" in this lesson. In a later lesson in this unit students will apply this reasoning to converting measurements.

The "between" relationship students may notice and use is that if one quantity is scaled up or down in a certain way, the other quantity must be scaled up or down in that same way in order to maintain the original ratio relationship. The "within" relationship is shown in the table as the multiplicative relationship between the quantities. It is connected to the unit rate that students study in Unit 2. The "within" relationship is the same for all pairs of quantities that form a ratio that is equivalent to the original ratio.

This work helps students deepen their understanding of *equivalence* as they identify the "between" and "within" relationships in equivalent ratios.

A conceptual challenge students may encounter is thinking that there is no equivalent ratio relationship if the factor used to scale up or down ("between") or the multiplicative comparison of the quantities ("within") does not result in a whole number quotient. For instance, 10:12 and 15:18 are equivalent ratios, even though there is not a whole number that we can multiply by 10 to get 15. Whole numbers are used in this lesson to support the concept, but be watchful for students who overgeneralize the use of whole numbers.

The concept is developed through work with a table that will help students organize the information and facilitate noticing important relationships.

Students engage in Mathematical Practice 5 (Use appropriate tools strategically) as they use ratio tables to identify the relationships within and between equivalent ratios.

**Key vocabulary:**

- equivalent ratios: two or more ratios that have the same multiplicative relationship or quotient

The label (C) at the end of the lesson title means that this lesson is a Conceptual Understanding lesson. The goal of this lesson is to introduce students to a new concept through their work on a task. Students will be able to start working on the task using prior knowledge. The new learning in this lesson is formalized in the next lesson that focuses on Fluency or Procedural Skills, labeled (FP). Following that lesson, students have the opportunity to apply their new understanding and skills in an Application lesson, labeled (A). Throughout these lessons, there are opportunities to develop new understanding, formalize and practice new learning, and apply the understanding in a new problem situation. Each lesson may be used on its own, but LearnZillion recommends that students experience the set of lessons together in the order in which they are presented on our site.