Ratios, Rates, and Unit Rates in the Real World

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Objective

Students will be able to demonstrate their understanding of ratios, rates, and unit rates in real world context using a double number line and/or a tape diagram.

Big Idea

Ratios, Ratios, and More Ratios: Further development of ratio understanding.

Curriculum Reinforcer

5 minutes

The Curriculum Reinforcer is basically a quick three to five question quiz containing information from previously taught standards that will allow you to determine areas in which your students may need to review. When I create my Curriculum Reinforcers, I make sure to mix up the types of problems. I use multiple choice, open-ended questions, as well as questions that require constructed responses. I may provide a mixture all in one day or, I may have all multiple choice one day and a mixture the next. It all depends on what I am looking for my students to be able to do. Often times, I also mix the standards however, there are times when I don't. I mix them so that I can see how well my students are fairing in more than one standard. In those instances when I don't mix up the standards, I am trying to see if the students are able to understand the standard when it is presented in a variety of ways.

Engagement

5 minutes

During this engagement activity, I will open up with a PowerPoint slide that contains two problems. The first problem will ask students to change a part to part ratio to a part to whole ratio. The second problem will ask students to change a part to whole ratio to a part to part ratio. These two problems are to ensure that students understand the additive relationship between the two parts and the whole which allow us to manipulate the ratio in a manner that is consistent with the rules of ratio and proportion but, will allow us to solve problems based upon the information provided in the problem. (MP7)

 

The students will also practice finding and writing the ratio in simplest form using a group of different shapes.

Instruction & Teacher Modeling

10 minutes

The purpose of this lesson is to ensure the understanding of a proportional relationships and how proportional relationships can be used to solve many different types of problems.

 

During this lesson, the students and I will review the meaning of ratios. Then, I will demonstrate how to set up proportions and solve a scenario using the previously taught concepts of ratio, proportionality, and equivalency. I will show the students how to solve these types of problems using an organizer called a flow map, which is a four step plan to solving proportional relationships. The four steps are as follows:

·    Step 1: Figure out the ratio necessary for solving the problem.

·    Step 2: Set up the proportion ensuring to match up the units.

·    Step 3: Solve the proportion using any previously taught method to solving proportions during this step (i.e double number line, picture pattern, equivalent ratios, or a table).

·    Step 4: The students are to provide a complete answer in sentence format ensuring that they include the  unit.

 

During the instructional portion of this lesson, I will be sure to stress the following points:

 

The importance of the manner in which a proportion is set up (the units of the quantities must be in the same orientation on both sides of the equal sign…i.e. If in the scenario the ratio is birds to butterflies, then the units in this case are birds and butterflies therefore, when the proportion is set up, if you place the birds in the numerator and the butterflies in the denominator then, on the other side of the equal sign, birds should be in the numerator and butterflies should be in the denominator just as in the ratio on the other side of the equal sign… I tell my students, “Be sure that you have matched apples to apples and oranges to oranges.”

 

To model exactly how to solve a ratio and proportion problem using a flow map, I will complete two examples before giving the students an opportunity to try solving these types of problems on their own.

The examples are as follows:

  1. Out of 30 Students surveyed, 17 have a dog. Based on these results, predict how many of the 300 students in the school have a dog. (170 students)
  2. If one out of12 students at a school share a locker, how many share a locker in a school of 456 students? (38 students)

 

The Mathematical Practice Standards that are evident in the section of this lesson are MP1, MP2, MP4, MP6, and MP7

Try It Out

10 minutes

The students will now complete two problems for practice purposes. These problems will provide students with the opportunity to practice setting up and solving proportions using real life scenarios.

The problems that the students will try are as follows:

1.       Sybrina jogged 2 miles in 30 minutes. At this rate, how far would she jog in 90 minutes? At what rate did she jog each hour? (6 miles; 4 miles per hour)

2.       A survey found that 12 out of every 15 people in the United States prefer eating at a restaurant over cooking at home. If 400 people selected eating at a restaurant on the survey, how many people took the survey? (500 people)

Selected students will be chosen to provide their answer and the method in which they arrived to that answer.

Using dry erase paddles, the students will write their answer to the “Rate Yourself” section. Those students who answered “NO” will be pulled to be a part of a facilitated group for independent practice.


 

Independent Exploration

20 minutes

To explore the concepts presented in this lesson, my students will complete the worksheet attached to this section of this lesson.

 

This worksheet requires students to solve problems involving proportional relationships using a flow map. The flow map is meant to help the students to break the problems down into workable parts. Basically, it forces the students to “make sense of problems” and thereby assisting them in “persevering to solve the problem.” (MP1)

 

Furthermore, the students will be reasoning abstractly (MP2) as they create coherent representations of the problems at hand, Modeling with Mathematics (MP4) as they use the flow maps to model the relationships evident in the problem, Attending to precision (MP6) as they are careful about specifying units and calculate accurately and efficiently, and Looking for and making use of structure (MP7) as they discern the pattern and structure present in setting up a proportion.

Closing Summary

20 minutes

Teacher and students will go over the worksheet as a whole group. The method in which this will happen is as follows: Students will be given the opportunity to share their solutions with their group members. Each group will be provided with a piece of chart paper where they will write down the answer to each of the problems presented in the worksheet. The students will also write down the method they used to solve these problems and at least one thing that they have learned so far about solving problems involving proportional relationships. Each group will present their chart paper. The class will compare solutions and methods. The class will discuss those solutions, methods, and what they as a class have learned so far.

Ticket Out the Door:

Students will be presented with the following word problem to solve:

  • A car traveling at a certain speed will travel 76 feet per second. How many yards will the car travel in 120 seconds if it maintains the same speed? (Answer 3,040 yards)

 

Students will place their solutions on a sticky note and place that sticky note on the parking lot on the way out the door.