Understanding Ratios

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Objective

Students will be able to demonstrate and describe their understanding that ratios are a comparison of two quantities that depict a relationship between those two quantities.

Big Idea

What Do You Mean a Ratio is Not a Fraction? Understanding the meaning of ratio.

Curriculum Reinforcer

5 minutes

The Curriculum Reinforcer often serves as a warmup for the day's learning. Today, it will also help my students to retain skills and conceptual understanding from earlier lessons. During this unit on Ratios, students will practice skills from our Number Sense Unit. In other words, my strategy for this unit is to use Spiraled Review to help my students retain what they learned during our first unit of the year. As their knowledge of ratios progresses, however, I will also include practice problems on Ratios & Proportions to strengthen the learning in this unit as well. 

Engagement

5 minutes

As an opening challenge I will have students simplify fractions by writing them in an equivalent form. This is a prerequisite skill to finding the base ratio between two given quantities. 

The fractions that I ask my students will simplify are as follows:

                        15/27                      12/30                      32/40

When working with ratios I encourage my students to simplify quantities whenever possible.

I also like to present tasks like this with an anchor context. For example, I might tell my students, "There are 15 girls and 12 boys in a classroom. So, 15/27 of the class are female students." I want my students to be able to reason from this to the simplified fraction "5/9 of the class are female students." Moreover, I want them to also recognize that for every 5 girls in the classroom there are 4 boys. I sometimes ask a question like, "If you were running the school, which comparison would be more useful to you?" For me, this is an important benefit of using ratios  to describe relationships. Ratios make it relatively easy to make comparisons and solve (or recognize) different problems.

Instruction & Teacher Modeling

10 minutes

Next, I will use the ratios that my students have just simplified to dig a little deeper into the meaning of the concept of ratios. I will do this by giving each ratio an anchor scenario:

  • As in the prior section I will present 15/27 a comparison of girls:students in a classroom, but we will discuss it as the ratio of girls-to-boys (15:12)
  • 12/30 will be presented as a ratio of lily pads:frogs in a pond
  • 32/40 will be bikers-to-skateboarders in an X-Games competition

 

I will simplify each ratio, emphasizing how the ratio continues to describe the relation between the quantities. For example, the ratio 12/30 simplifies to 2:5. This tells us that for every 2 lily pads, there are 5 frogs. In making this point, I will ask my students to think about the pros and cons of simplifying the ratio from the given data. 

In order to help my students construct a mental image of what happens when we simplify ratios, I will create a model as we discuss each scenario. For example, I will draw lily pads and frogs and consider how to simplify the ratio by equally distributing the frogs (MP2, MP4). To make this more interactive, I like to use clip art and prepare a PowerPoint slide that allows the students to move the images if they want to share how they think about simplifying the problems.

After showing how to determine the ratio between 2 different quantities, I will then demonstrate the different ways a ratio can be presented. During this presentation, I need to ensure that the students understand the following (MP7).

  • Ratios can be part-to-part
  • Ratios can be part-to-whole
  • Ratios can be whole-to-part
  • Ratios can be improper (e.g., whole-to-part)
  • Fractions are ratios, but ratios do not have to be a fraction
  • The difference between a fraction and a ratio
  • Ratios compare any positive rational number, even decimal, mixed numbers, and fractions
  • Simplifying ratios is useful for communication and problem solving, so it is helpful to present them in simplest form

See my Understanding Ratios video for more insight into how I present this information to my students. 

Writing Ratios to Make Comparisons

10 minutes

To allow my students an opportunity to demonstrate their understanding of the concept of ratios, they will practice writing comparison statements using part-to-part and part-to-whole ratios. I present them with the following two scenarios for this task.

Scenario #1Hilman has a pencil box with four pencils, twelve markers, two pens, three erasers, and twenty four crayons.

Scenario #2Jasmine has fifteen barrettes, twelve headbands, and ten hairbands in her bathroom drawer.

Then, I'll say, "I want you to write as many different ratios as they can come up with using the scenarios. As you write each ration think about the comparison that you are making."

 

Independent Exploration

20 minutes

For additional practice with ratios, I will have my students complete this activity. In this activity my students again write as many ratios as they can given two scenarios. Then, they will have to write their own scenario. Finally, they will list all of the possible ratios that can be written to represent their scenario. As they work I like to listen to the comparison statements that they make. It provides me with a good indication of how well they are getting the idea.

Problems for Independent Exploration

Scenario #1Lynn asked her classmates what type of pet they had. The results were as follows:

  • Three students said they have a guinea pig
  • Eight students said that they have a dog
  • Four students said that they have a cat
  • One student said that they have a lizard
  • Six students said that they have a bird
  • Two students said that they have a snake
  • Three students said that they have a turtle

Scenario #2Olivia collects stickers, bracelets, and seashells. So far, Olivia has collected thirty stickers, 18 bracelets, and 12 sea shells. 

Write your own scenario: On a separate sheet of paper, write your own scenario that can be described using ratios. Then, identify all of the ratios that can be used to describe the relationships in your scenario.

Closing Summary

20 minutes

As we approach the end of the lesson, I will choose one student to present their answers for Scenario 1 and Scenario 2 from the Independent Exploration. I will ask these students to explain the meaning of each comparison as they state the ratio (MP3). During the students' presentations I will ask probing (or guiding) questions as necessary. I want students to hear their peers' explanations, while also learning to use correct mathematical terminology.

After we've discussed all of the possible ratios for each scenario, I will choose several students to present the scenario that they created. I will ask these students to read their scenario. Before I do, I will ask the class to listen for the ratio comparisons that can be made using the data in the scenario. After a scenario is read, I will ask the class if they can make any ratio comparisons. I will ask the author to check the students' suggestions with his/her own list. 

With about ten minutes left in the period, we will stop discussing the students' scenarios. For today's ticket out the door I will ask my students to write a letter to an absent student explaining what they have learned during today's lesson.