Introducing Ratios and Rates in Word Problems

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Objective

SWBAT define ratios (rates) as a comparison of two quantities by identifying ratios in problem situations.

Big Idea

students will play a cooperative game identifying ratios and exploring equivalent rate and unit rates

Do Now

10 minutes

Students enter silently according to the Daily Entrance Routine. They are reminded at the door that the Do Now is a practice quiz and that the real one will take place on tomorrow. In order to speed up the data collection and review of tough questions, all students are asked to enter their answers into Senteo clickers. A timer is set for 6 minutes and displayed on the SMART Board. As students work silently I will be walking around to monitor progress. Any students taking too long on any question may be asked to move on to the next question. The goal is to get as much data as possible about what students know. This allows me to prioritize skills students need to develop the most when reviewing the answers.

A commonly missed question in all three classes was question 5, a statement which compares the decimal 0.33 and the fraction 1/3. I predicted this would be a misconception because many of my students say 1/3 is equivalent to 0.33, rather stating it is an approximation. I wanted to make sure this essential idea was reviewed and engage the students through the results. (i.e. “let’s take care of any misunderstandings now so that you guys don’t miss this one again”).

I had a significantly tougher time getting students to complete the Do Now and enter answers. In this one class, we were not able to generate as much data and the Senteo clickers give me an immediate system of accountability. I urge students to make sure they are pushing themselves to work faster and get to each question (i.e. “without data, I don’t know what to help you with”).

Once time for this section has expired or I have answered all student questions, the Do Now sheets will be organized into binders and Class Notes will be distributed. 

Class Notes

15 minutes

Students receive Cornell Notes. I ask them to fill out the heading and the aim which is written on the white board. Red font in the sheet indicates content which students will be responsible for writing down themselves. We begin by reviewing the definition of ratio, which I have one student read out loud. Then I have another student read the example. I ask students to raise their hand, as a check for understanding, if they feel confident about the answer to this question. This is how I evaluate how many students have prior knowledge of this topic. Then I ask one student for the answer and I ask others to write it two different ways. As another check for understanding, I ask students if the ratio 1 : 2 or the fraction ½ are correct answers for this problem. This is how I check for understanding of the order in which the integers are written.

Another student is cold called to read the definition of ratio and the example. I explain to students that ratios and word problems in 7th grade will most likely include rational numbers. I ask a student to remind us about the difference between a rational number and an integer. Then I ask a student to write a ratio in the form of a fraction which includes the two quantities in the example. Students may need more scaffolding to do this (example in the video below. I will also be asking students to find equivalent ratios and explain what they mean in the context of the situation.

 

Once we have finished reviewing this example, I ask students to complete the final example in the notes on their own. A timer is displayed with 4 minutes. When the time expires I ask them to turn to a neighbor to review/complete their answers. I write the correct answer on the board. This should only take about 1 minute.

Game

20 minutes

I announce to students that we will be playing a game in teams of four today. Each student received color cards with their names on them. Before class, I prepared heterogeneous groups based on RIT scores determined MAP scores. I begin by asking students to pack up their belongings, except for a pencil. This should only take 1 minute and displaying the timer on the SMARTBoard keeps students accountable. Then, students have 2 minutes to move the classroom tables to create pods of 4. They are instructed to stand behind their chairs after this task is completed. Once this is done, students are asked to find their team members by color and take a seat at one of the pods. Two additional minutes are displayed for this direction. As students are taking a seat, I am walking around to place baskets in the middle of each pod. Each basket contains 4 paper-clipped quarter-sized sheets of paper with alphanumerical identification on the back, and a highlighter.

I introduce the game to students as the “Silent Competition”. In order to help my students feel successful in this game, I may ask lower achieving students to take the A cards, which are all “identification” tasks, lower on Bloom’s taxonomy. Since this will be a group effort, they will still have an opportunity to learn about all the other steps. Otherwise, students may choose any stack they want.

Give students the following information before they choose:

  • A cards will require the student to highlight and identify the two distinct quantities in the word problem
  • B cards will ask students to write a ratio in the form of a fraction using the highlighted in formation without reducing.
  • C cards will ask students to use B cards to find an equivalent ratio.
  • D cards will ask students to answer a question relevant to the initial word problem which uses the equivalent ratio provided by the student who completed C card.
  • Cards are handed to the following student in alphabetical order.
  • Do NOT hand cards to students unless the directions instruct you to do so (i.e. student A may not give/show her card to student C)
  • Students MAY NOT speak or provide help during each round.
  • Once student D has an answer, they must raise it in the air for me to come and check.

In order to ensure that students understand the game we will begin by completing a round out loud, the “sample” round. For this round students will be using the cards clearly labeled “sample”, which include the problem completed during the notes.

My goal with this lesson is to encourage students to work together to understand how to identify the units being compared, what equivalent ratios represent in each context, and providing exposure to the unit rate a day before teaching it directly. Through this game students are in use of MP1

Closing

10 minutes

In the last 10 minutes of class, students will receive a sheet that includes all 4 problems. They must complete the four problems using the information they gathered during the game, along with my guidance toward solutions. This is also a great resource to have available because I use it in case I lose the class in terms of overwhelming behavioral issues that do not allow enough time for the game. 

Homework is handed out at the end of class and students are dismissed.