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Helping Students With Mathematics Difficulties Understand Ratios and Proportions

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Helping Students With Mathematics Difficulties Understand Ratios and Proportions

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Ratio and Proportional reasoning activity

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Ratio and Proportional reasoning activity

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Seventh grade students have had wide-ranging experiences with fractions and operations on fractions, but ratios and rates may be new concepts for them. This activity introduces students to the idea of rates via a familiar and tangible product: Kool-Aid. Students will add varying amounts of Kool-Aid powder to a set amount of water, thereby producing a sensory experience of rates. The Kool-Aid example also provides a base experience upon which other examples can build.

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Illinois Open Educational Resource Detail

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#IOER, Illinois Open Educational Resources, STEM, #STEM

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What is a Proportion in Math? Calculate & Solve Proportions & Equations - [6-3-3]

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Definitions:

A ratio is an ordered pair of numbers a and b, written a / b where b does not equal 0.

A proportion is an equation in which two ratios are set equal to each other.

For example, if there is 1 boy and 3 girls you could write the ratio as:

1 : 3 (for every one boy there are 3 girls)

1 / 4 are boys and 3 / 4 are girls

0.25 are boys (by dividing 1 by 4)

25% are boys (0.25 as a percentage)

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Library Guides: Math Skills Overview Guide: Understanding Ratios & Proportions

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A ratio is an ordered pair of numbers a and b, written a / b where b does not equal 0. A proportion is an equation in which two ratios are set equal to each other.

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Definitions:

A ratio is an ordered pair of numbers a and b, written a / b where b does not equal 0.

A proportion is an equation in which two ratios are set equal to each other.

For example, if there is 1 boy and 3 girls you could write the ratio as:

1 : 3 (for every one boy there are 3 girls)

1 / 4 are boys and 3 / 4 are girls

0.25 are boys (by dividing 1 by 4)

25% are boys (0.25 as a percentage)

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Library Guides: Math Skills Overview Guide: Understanding Ratios & Proportions

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A ratio is an ordered pair of numbers a and b, written a / b where b does not equal 0. A proportion is an equation in which two ratios are set equal to each other.

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standards, activities, teaching strategies 

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standards, activities, teaching strategies 

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Proportional relationships are ingrained in our everyday, they just make sense. I think that is why I really enjoy proportions and seeing a students' ability to find something that is "missing". Many students do well with proportions, there is a simple formula to teach, the steps remain the same, solve, label, etc. A piece of cake. Other students look at a proportion and all they see two dreaded fractions. Que dark scary music.

Today, I am sharing how proportional relationships go far beyond just solving for a missing number and set a foundation for success in Algebra.

7.RP.2 Recognize and represent proportional relationships between quantities.

7.RP.2A Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin.

7.RP.2B Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.

7.RP.2C Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn.

7.RP.2D Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate.

Visual Representations

These standards are heavily weighted with visual representations including tables, graphs on the coordinate plane, and diagrams. Some students will have been exposed to these multiple representations in 6th grade with standard 6.RP.3 and equivalent ratios. The goal is for students to fluently connect the table, equation, graph, verbal description, and even diagram together. When given one piece of information, students should be able to represent that same information in multiple ways. Students who can do this will have a firm foundation for 8th grade math and Algebra 1.

There are tons of great ideas and activities out there, but below are a few of my favorite "tried and true". They are all easy to incorporate and provide scaffolding for students who struggle in the math classroom.

Multiple Representations Graphic Organizer

Some students need to see everything together and that is where this multiple representations graphic organizer comes in handy. It is the perfect size for students to show their work and simultaneously is easy for students to draw on their own. I really like how the components stay the same, but the given information changes.

These are perfect for introducing the various visual representations, but also can be laminated to use in tutoring or math intervention. Once students are familiar with the graphic organizer, I would even give students butcher paper and various pieces of information. Then, they would use markers to represent the remaining information. A quick informal assessment idea or even partner work for a Friday.

Highlight the Unit Rate/Constant of Proportionality

A quick trick to help students see all the connections is to use a highlighter! Model and have students highlight the unit rate/constant of proportionality within the different representations. This is perfect for representing k in the equation and table, or meeting 7.RP.2d by labeling (1, r) on the graph.

Emphasize Vocabulary

Hopefully, students are familiar with the term unit rate and how to find it. Constant of proportionality sounds so cumbersome and challenging. Unit rate is often emphasized when we are talking about unit price (cost per ounce, etc), so I think students have trouble seeing a graph or table and also using the phrase unit rate. Either way, students need to be familiar with the verbiage and understand what a question is asking. Here is a more detailed explanation for why they are the same.

Common Misconceptions

1. A relationship is not proportional unless (0,0) is visible in the table or graph

2. Dividing x/y to find k

3. General confusion about k

4. General disconnect between k and proportions

5. Proportional relationships only involve positive numbers

6. Mixing up x and y on the table, when it is not given

Anchor Chart Ideas

Anchor charts are fabulous ways to showcase the content in a visual manner for students to reference. They can easily be created before the lesson or as you are teaching, depending on the content. The visual representations of proportional relationships are perfect for anchor charts.

Ideas for Struggling Students

1. Practice finding different points in the relationships that aren't listed on a table.

2. Begin with the equation and use and input-output table to create a table.

3. Use a four corners graphic organizer.

4. Match multiple representations.

5. Practice graphing with four quadrants (from 6th grade).

Hopefully, this gives you some ideas for teaching proportional relationships or even insight as to what knowledge your students are coming with. I would love to hear other great activities or ideas you have used! Feel free to share in the comments.

Be sure to check out these different concepts and activities that are included in my Proportional Relationships Unit and Proportional Relationships Activity Bundle.

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Proportional relationships are ingrained in our everyday, they just make sense. I think that is why I really enjoy proportions and seeing a students' ability to find something that is "missing". Many students do well with proportions, there is a simple formula to teach, the steps remain the same, solve, label, etc.

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The Ratio Design Challenge is an online, interactive tool designed to engage middle school students in developing an understanding of ratio and proportional reasoning while highlighting the value of math in real-world situations and careers.

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IM Key Lesson 7.2.4 Proportional Relationships and Equations - PA Adult Education Resources

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A resource for PA adult education and family literacy teachers, tutors, and administrators.

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Intro Video to Proportional Relationships

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Intro Video to Proportional Relationships

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- [Instructor] In this video, we are going to talk about proportional relationships, and these are relationships between two variables where the ratio between the variables is equivalent. Now that sounds complex or a little bit fancy. It'll hopefully seem a little bit more straightforward once we look at some examples. So let's say I'm looking at a recipe for some type of baked goods, maybe it's some type of pancakes, I've been making a lot of those lately, and we know that for a certain number of eggs how many cups of milk we need. So we have number of eggs, and then we're also going to have cups of milk. And in this recipe, we know that if we're going to use one egg, then we would use two cups of milk, and if we use three eggs, then we're gonna use six cups of milk, and if we use, let's say, 12 eggs, then we're going to use 24 cups of milk. So is this a proportional relationship where the two variables are the cups of milk and the number of eggs? Well, to test that we just have to think about the ratio between these two variables. And you can say that the ratio of the number of eggs to the cups of milk, or the ratio of the cups of milk to the number of eggs. But you just need to ensure that they are always equivalent in these scenarios. So let me make another column here, and I'm gonna think about the ratio of the eggs to the cups of milk. Well, in this first scenario one egg for two cups of milk. This second scenario is three to six. This third scenario is 12 to 24. Are these equivalent ratios? Well, to go from one to three you multiply by three, and also, to go from two to six, you multiply by three. So you multiplied both the variables by three. Similarly, if you multiply the number of eggs by four, then you multiply the number of cups of milk by four as well. So these indeed are all equivalent ratios, one to two, three to six, 12 to 24. In every scenario you have twice as much cups of milk as you have number of eggs. So this would be proportional. So check. Now what would be an example of a non-proportional relationship? We'll stay in this baked goods frame of mind. Let's say you're going to a cake store and you're curious about how much it would cost to buy a cake for different numbers of people. So let's say number of servings, number of servings in one column, and then the cost of the cake. And let me set up two columns right over here. And so let's say if you have 10 servings, the cake costs $20. If you have 20 servings, the cake costs $30. And if you have 40 servings, the cake costs $40. Pause this video and see if you can figure out whether this is a proportional relationship. If it is, why? If it isn't, why not? Alright, well let's just think about the ratios again. And here our two variables are the number of servings and the cost of cake. So if we look at the ratio of the servings, servings, to cost, in this first situation it is 10 to 20, and then it is 20 to 30, and then it is 40 to 40. And so to see if these are equivalent ratios, when we go from 10 to 20 on the number of servings, we're multiplying by two. But when we go from 20 to 30 on the cost of the cake, we aren't multiplying by two, we're multiplying by 1.5, or one and a half. And similarly, when we go from 20 to 40, we are multiplying by two again, but to go from 30 to 40 we aren't multiplying by two, we're multiplying by one and one third. By one and one third. When we multiply our servings by a given amount, we're not multiplying our cost of cake by the same amount. This tells us that this is not proportional. One way to think about proportional relationships, we already said, that the ratio between the variables will be equivalent. Another way to think about it is one variable will always be some constant times the first variable. So in our first example right over here we said the cups of milk is always two times the number of eggs. We can write that down. So cups of milk is always going to be equal to two times the number of eggs. And this number right over here, we call that the constant of proportionality. And you wouldn't be able to set up an equation like this in this scenario. It would have to be more complicated. And so a proportional relationship, the ratios are equivalent between the two variables and you can set it up with an equation like this where you have a constant of proportionality.

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Introduction to proportional relationships (video) | Khan Academy

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Proportional relationships are relationships between two variables where their ratios are equivalent. Another way to think about them is that, in a proportional relationship, one variable is always a constant value times the other. That constant is know as the "constant of proportionality".

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Ratios and Proportional Relationships - Teaching Math With Rigor

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Proportions Lesson Plan: Ratios, Proportions, and Percents | BrainPOP Educators

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This lesson accompanies the BrainPOP topic Proportions, and supports the standard of analyzing proportional relationships and using them to solve real-world and mathematical problems. Students demonstrate understanding through a variety of projects.

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Resources for teaching Proportions and Rates

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Resources for teaching Proportions and Rates