Ratios & Proportions

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Objective

Students will be able to demonstrate the concept of proportionality as a ratio relationship.

Big Idea

It's the same but it's different: Understanding the concept of proportionality.

Curriculum Reinforcer

5 minutes

Today's curriculum reinforcement is just a way for students to see and practice previously taught concepts. This activity in particular deals with concepts from the Number Sense Unit which deals with Division of whole numbers and fractions, calculating with decimals, factors, multiples, and the distributive property.

Engagement

5 minutes

To start out today’s lesson, I will have my students answer the question, “What does it mean to be proportional?”

I will first, give each of my students a sticky note. They are to write their answer to this question on the sticky not and place their sticky note on a chart paper labeled, “Before Instruction,” that will be hanging in the front of the classroom. However, before the students answer the question, I will provide my students with the definitions for proportional and dimension.

I will ask my students to keep these words and their meanings in mind as they are providing their answers to the question presented.

  • Proportional – Comparative relation between things or magnitudes as to size, quantity, number, etc.; ratio.
  • Dimension – measurement in length, width, and thickness

The purpose of this exercise is to get students thinking about the meaning of proportionality in a more deliberate and precise manner. We will revisit the sticky notes later on during the closing of this lesson. (MP6)

 

Instruction & Teacher Modeling

10 minutes

The purpose of today’s lesson is to deepen the understanding of ratios to include the concept of scaling up and scaling down. Students should understand that ratios are constant. They don’t change, no matter how much something is scaled up or scaled down.

To illustrate to my students this concept of scaling up and down, I will show a clip from, Honey I Shrunk the Kids. This clip is for the purpose of demonstrating the meaning of proportionality and scale. I will also show a clip from, Alice In Wonderland which, is for the same purpose. (MP2)

I will then explain that the ratio of proportionality does not change. It is constant. Then, I will ask my students, “What does constant mean?” to ensure that they understand the concept. If students have misconceptions, I will take the time to clarify at this point before moving on.

Next, I will then go on to explain that the ratio of proportionality is what ensures that an item will look the same no matter what size it is. The teacher will also ensure to teach the students that when scaling up or down in SIZE, we call the ratio of constant proportionality the SCALE FACTOR.

To probe the students for understanding the connection between what they saw in the video and this concept of proportionality, I will ask the following question concerning the clips: What item in the clip represents, by how much the dimensions of the characters would change? What is the term used to describe the change in size? Did their proportions change? What is the term used to describe the phenomenon that keeps the proportions the same even though the size has changed? (MP2)

  • Honey I Shrunk the Kids – The Machine… Scale Factor… No… Scale Factor
  • Alice In Wonderland – The cake and the drink… Scale Factor… No… Scale Factor

 

After having this discussion, I will ask my students if they can come up with any other examples of things that have been scaled down using a ratio of proportionality… A globe, a map

Next, I will demonstrate this concept of proportionality even further with the use of triangles. I will do this by using triangles with grid lines. The first set of triangles will be right triangles that are 1 x 2 and 12 by 24. The second set will have a height of 3 and a base of 4 and the other will have a height of 9 and a base of 12. During this demonstration, I will be sure to connect my students' understanding of equivalency to this concept of proportionality. See the instructional video attached to this section of this lesson see exactly what I did during this portion of this lesson. (MP4)

To further the understanding of my students, I will then model how to solve two problems that involves proportionality. For the purpose of this lesson, I will focus on using methods such as filling in a table, creating a double number line, and drawing a picture. In the next lesson, I will focus on setting up a proportion. The two problems that I will be modeling are presented at the end of the instructional video. These problems are also presented in the attached PowerPoint for your convenience. During the modeling of these problems, I will ensure to connect my students' understanding of equivalency to the concept of proportionality.

Through this demonstration, the students should be able to see that a ratio is what keeps an object proportional while still determining the manner in which something will scale up or scale down.

*** Throughout this lesson be sure to constantly refer to equivalency. If you need a good lesson that deals with equivalency as a prerequisite to this lesson, click on the link to my lesson entitled Equivalent Ratios.

 

Independent Exploration

20 minutes

To explore the concept that I have taught in today’s lesson and show mastery of the material. My students will be given the following three problems:

Problem #1: Nathan's pet snail crawls at 2.5 feet per hour. Nathan's room is 10 feet long. If Nathan's pet snail takes 2 hours to cross   of Nathan's room in 2 hours, is this rate proportional?

Problem #2: It takes 4 pints to fill one jug of water. Qasim needs to fill 8 jugs. If Qasim uses 16 quarts to fill up the 8 jugs, will all the jugs be full?

Problem #3: Given these two figures, would you say that the two of them are proportional? Why or why not?

Please click here to see the PowerPoint that will display these three problems in their entirety. The figures for Problem #3 are not able to be seen unless you view the PowerPoint.

These problems will have to be presented as a poster. Furthermore, my students will have to solve the problems using one of the methods demonstrated in today’s lesson (Each problem must use a different method) (MP7).

  1. Table
  2. Double Number Line
  3. Picture

Secondly, my students will also have to provide evidence as to how their answer is correct. Their evidence should be written as an essay (MP3)

Example of Poster:

Problem #1:

Word Problem

Problem #2:

Word Problem

Problem #3:

Word Problem

Solution to the Problem

Solution to the Problem

Short Essay to Provide Evidence

Short Essay to Provide Evidence

Short Essay to Provide Evidence

 

Closing Summary

20 minutes

Selected students will present their solutions to the three problems to the class. (MP6) This should take no more than 10 minutes. Students will bring their work up to place under the document camera so that all students can see their solution.

Each presenting students must be prepared to answer questions from the peers as well as questions that will come from me. Those students who are not presenting must be ready to critique, ask questions, and or add more information to what has already been presented. (MP3)

As a ticket out of the door, I will present my students with the same question that I presented to them as the beginning of class.

 “What does it mean to be proportional?”

Like before, they will answer the question on a colored sticky note. This time we will use a different color than before. This will allow us to compare the answers from the beginning of the lesson to the answers at the end. We will discuss the answers as a way to reflect on what we learned today.