Comparing Ratios

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Objective

SWBAT use ratio tables to compare ratios.

Big Idea

Any entry within one ratio table for one term can me compared with any entry from the ratio table for the other term.

Think About It

7 minutes

Students work in partners in the Think About It problem.  After 3 minutes of work time, I bring the class together and ask them who thinks Ms. Thody has the spicier chili, and who thinks Ms. Rey has the spicier chili.

The majority of kids will raise their hand for Ms. Rey having the spicier chili, because she uses the higher number of tablespoons in the word problem.  Students don't think about the ratios when they are working to solve this problem.  

I tell students that I am about to prove to them three different times that the spicier chili belongs to Ms. Thody.

Intro to New Material

20 minutes

I use the Think About It problem for the start of the New Material section.  At the bottom of this page, I create ratio tables for each teacher, including a total column.  I then tell students that I am going to prove to them three times that Ms. Thody has the spicier chili.

First, I ask students to find a number that shows up on both total columns.  When the total is 45, Ms. Thody uses 18 tbsp of chili powder, and Ms. Rey uses 15 tbsp of chili powder.  Usually, there are choruses of 'oooh' in the classroom.

Then, I tell students I am going to prove my point a second time.  I ask students to find a number of tbsps of beans that appears on both tables (6, for example).  I ask students questions that gets us to the conclusion that with the same amount of beans, Ms. Thody uses more chili power.

Finally, students identify a number that shows up in the chili powder column.  We talk about diluting the chili powder with beans and make conclusions about the spiciness of the chilis.  

(If this last point is confusing kids, I ask them if anyone likes KoolAid.  I have them imagine that I have 2 packets of KoolAid.  I mix one packet with 2 c of water and one packet with 10 c of water.  I have students turn and talk with their partners about the difference between the two mixes)

Once we've looked at three different comparison points, I move into the notes in the Intro to New Material section.  The first page lists the steps we followed to complete the chili problem.  I use my notes to show students a horizontal ratio table; I want them to be comfortable with both formats.  I show students how to write a complete response to the questions asks (answer the question, identify where in the ratio table the reader should look, and then draw a conclusion).  The exit ticket samples in this lesson show two student responses.  

I guide students through the second problem, using this Exemplar.  

Partner Practice

15 minutes

Students work in pairs on the Partner Practice problem set.  As they work, I circulate around the classroom.  I am looking for:

  • Are students correctly creating ratio tables from a given relationship?
  • Are students identifying appropriate numbers to compare?
  • Are students annotating the problems and ratio tables?
  • Are students correctly analyzing the problem to find out what ratio they are looking for?
  • Are students completely answering the question asked, including using justification?

I ask students:

  • What is the problem asking you to look for?  How do you know?
  • How do you know which ratios you can compare?
  • Are there other comparison points you could have used?
  • What conclusions can you draw from your ratios?
  • How will you prove your claim?

 

After 10 minutes of work time, I bring the class back together for a conversation.  We look at the second problem of partner practice and identify which paint is redder.  I ask students to share the strategies they used to find the comparison point.

I then have students look at the check for understanding problem. These two ratio tables provide a number of possible comparison points.  I cold call students to identify pairs of terms we could use to compare.  

 

Independent Practice

15 minutes

Students work on the Independent Practice problem set.  

There are only three problems in the Independent Practice problem set, but each offers different challenges for kids.

In the first problem, students may decide to add a total column or extend the first mixture table.  There are multiple pathways to a solution.

In the second problem, students need to fill in a ratio table with decimals.  The arithmetic may be slightly difficult for students here (or, they'll at least need to slow down their work to get the computation done).

The third problem requires students to create their own ratio tables before being able to make comparisons. 

Closing and Exit Ticket

10 minutes

At the end of independent work time, the class comes back together for a discussion.  I ask for a student who feels very confident about his/her work on problem 1 to share his/her work on the document camera.  First, we look at the ratio tables.  Some students may have decided to add a total column, some may have extended the Mixture A table, and some may have used 6 pints of white paint as the comparison point.  I make sure we discuss all three strategies, as all are valid.  We then look at the written response and students give positive and critical feedback.

Students work on the Exit Ticket independently to close the lesson.  Two student samples are here:  Exit Ticket Sampe A and Exit Ticket Sample B.