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Equivalent Ratios
Lesson 4 of 20
Objective: Students will be able to determine whether or not two ratios are equivalent
Big Idea: Are All Things Truly Created Equal? Understanding the concept of equivalent ratios versus equivalent quantities.
Curriculum Reinforcer
Today's Curriculum Reinforcer gives students practice with previously taught concepts. Today's spiraled review focuses on division with whole numbers and fractions; calculations with decimals; factors and multiples; and the distributive property.
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Engagement
To start off today's lesson, I will have my students watch a short clip from the movie, Honey I shrunk the Kids, where one of the characters makes the statement, "That's like bench pressing a bulldozer!" The reason that I want my students to watch this clip is to introduce them to the concept of equivalent ratios, which is a prerequisite to understanding the concept of proportionality (MP2).
After watching the clip, I will ask students to tell me, "How much would an ant be able to lift if it were the same weight as you?" After they contemplate and arrive to their answers, we will use this discussion as a segue into today's lesson on equivalency.
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Starting from the discussion resulting from our viewing of a clip from Honey I Shrunk the Kids, I will introduce my students to the concept of equivalence in the context of ratios and proportions. During this presentation, I want my students to realize that using equivalent ratios does NOT mean that you are dealing with equal quantities. For example, think of a recipe for Bread_Rolls. Maybe the recipe makes one dozen rolls, but you need to make 3 dozen rolls. What can you do? You will scale the recipe so that while you are baking enough rolls. But, in order for the rolls to be baked successfully, the ratio of the ingredients in the larger batch should be equivalent to the ratio in the recipe for one dozen rolls. In other words, there will be a proportional relationship between the two batches (12rolls vs. 36rolls).
In this lesson I teach my students how to reason about situations where there is an equivalent ratio. To model for my students, I will work through three examples that show different methods for determining whether or not two ratios or rates are equivalent. The methods that I will demonstrate are as follows (see Equivalent Ratios):

Multiplicative reasoning (scaling up or down)

Simplifying

Cross products
After showing my students these different methods, I will give my students and opportunity to try out each of these methods on their own.
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Try It Out
To determine whether or not my students understood the lesson taught to them today, I will have them complete a brief exercise that contains 5 problems dealing with equivalency.
While my students are completing this exercise, I will be circulating the classroom answering questions and observing my students to determine if there are any common misconceptions that need to be addressed before moving on to allowing them to work independently. One misconception that I am anticipating is for students to determine that a set of ratios or rates are not equivalent because one is not an integer factor or multiple of the other. For example. many students do not think that 2 to 6 and 5 to 15 are equivalent ratios because 5 is not a multiple of two and 15 is not a multiple of 15. But, both of them simplify to 1 to 3 so they ARE EQUIVALENT. I will use this misconception to drive home the importance of knowing more than one method of determining equivalency (MP2, MP6).
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Independent Exploration
Now that my students have experience with equivalent ratios, I want them to work on problems involving equivalency on their own. This exercise asks students to determine if two ratios (or rates) are equivalent. In order to get a sense of how my students are approaching each problem, I ask them to EXPLAIN THEIR REASONING on each problem (MP2, MP3). I want my students to not only be able to tell me "yes" or "no" as to whether or not a set of ratios or rates are equivalent but, I am also looking for my students to be able to articulate the reason why they came to their conclusions.
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Closing Summary
To close out this lesson, we will go over the Independent Exploration in two steps. In the first step, my students will get with a partner so that they can compare and contrast their answers. While they are with their partners, they should be discussing what method(s) they used to determine whether or not the ratios/rates in question were equivalent. They also should be discussing which method they like and why, as well as their understanding of the lesson.
In the second step of the closing, each partner will present one of the problems from the independent exploration. They will share parts of their discussion with the class. During this time, I will be asking probing questions to get students to think deeper about the lesson. Also, I will create a bulleted list on a piece of chart paper of important and/or interesting points brought up during the presentation of the problems.
Before departing for the day, my students will complete the following Ticket Out the Door:
Are the following ratios equivalent? Why or Why not?
2/5 = 4/12 and 8/12 = 2/3
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