Using Ratios to Solve Problems

Today's Curriculum Reinforcer deals with concepts from our Number Sense Unit. I will continue to spiral problems from this unit into our warmup activities as we dig deeper into ratios and proportions. Today's practice problems focus on identifying the Least Common Multiple (LCM) of two numbers.

During today's engagement activity, we will discuss what we have learned so far about ratios. My plan is to cover the following topics:

• How can we use ratios to describe a pattern?
• How do we decide which quantity goes first when we write a comparison statement using a ratio?
• Why are there so many different ways to write a ratio?
• Why do we simplify ratios? When we do, what information is lost?

 The Games At Recess task used in this section of the lesson is from the Illustrative Mathematics website. 

After discussing ratios describing the data from the Games at Recess task, we will discuss how ratios can also be used to help us analyze patterns. I will also use this PowerPoint presentation to lead students through this discussion. As we work, we will look at both part-to-part and part-to-whole comparisons. 

There are two instructional tasks embedded in this PowerPoint presentation.

1. The first requires my students to create their own pattern using cubes. Once they create a pattern, they are to analyze the pattern and write down the different ratios that can be used to describe the pattern.
2. The second task asks students to identify different ratios that can be used to describe a given scenario. 

Both of these tasks help us to appreciate how ratios can be used to model situations where it makes sense to employ a multiplicative comparison (MP4). 

Examples from the Demonstration:

• Janice has apples and oranges. The ratio of apples to amount of fruit Jill has is 3:5. If Jill has 6 apples, how many oranges does she have?

• The ratio of boys to girls in the classroom is 2:3 if there are 30 boys and girls, how many girls are there?

I will model these examples step by step asking strategic questions to ensure understanding. Here are some of the questions that I will use:

• What is the pattern described in this problem?
• What type of ratio is needed to solve this problem? Part-to-part or part-to-whole?
• Does the ratio need to be changed in order to solve this problem? If so, to what?
• Can we use a double number line to think about this problem?
• How else might we be able to solve this problem?

Next I will give my students a chance to apply the ideas that we have discussed so far in the lesson (MP1). I will give my students this word problem to solve.

At the local Animal Shelter, the ratio of cats-to-dogs is typically 3-to-5. If there are currently a total of 48 cats and dogs all together, how many dogs would you expect to find at the shelter?

In order to successfully solve this problem, they will need to figure out the following:

• What is the pattern described in this problem?
• What type of ratio is needed to solve this problem?
• Does the ratio need to be changed in order to solve this problem? If so, to what?

As my students complete their exploration of the problem, I will ask them to write a sentence explaining their answer to this problem.

Since my students are likely to face different types of ratio problems on testing, I decided to break our Independent Practice into two mini practice sessions. As we begin each task I will announce the amount of time during which I expect students to complete the problems. 

1. Activity 1 (5 minutes):  My students will solve 5 problems that will help them to review simple ratios. They will also have two problems where they will be asked to change the ratio.
2. Activity 2 (10 minutes):  My students will solve 2 problems where they will be given a scenario in the form of word problem. These scenarios mirror those presented in the instructional portion of this lesson.

As students work I will be circulating and asking them the following questions: 

• What is the pattern described in this problem?
• What is the given ratio?
• Does this ratio need to change? Why or Why not?
• If the ratio needs to change, what should it be changed to?
• Can you simplify this ratio? If so, then simplify.
• What is the problem asking you to do?
• Find the solution to this problem using a double number line?
• Can you see another way that this problem can be solved? If so, solve this problem in that manner and present it to the class when given the opportunity.

To bring today's lesson to a close, I will first ask selected students to present their answers from the second set of practice problems to the group using the document camera. I will have chosen examples that do a particularly good job organizing their work and describing their answer (MP3 & MP6). After a few well-presented examples, we will finish with a discussion of today's Topic of the Day (TOTD). Today we will discuss why it is important to identify whether we are working with a part-to-part ratio or a part-to-whole ratio.