About This Chapter
Standard: Recognize and represent proportional relationships between quantities. (CCSS.Math.Content.7.RP.A.2)
Standard: 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. (CCSS.Math.Content.7.RP.A.2.A)
Standard: Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. (CCSS.Math.Content.7.RP.A.2.B)
Standard: 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. (CCSS.Math.Content.7.RP.A.2.C)
Standard: 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. (CCSS.Math.Content.7.RP.A.2.D)
About This Chapter
Working with students on the material in this chapter will help ensure they understand proportions theoretically, visually, and functionally. As comprehension of proportions builds, students will be prepared for higher-level mathematics where proportional relationships become increasingly important. It also provides them with a background to use in weighing proportions in daily life, a critical component of moderating behavior and consumption. The lessons in this chapter discuss:
- Proportional relationships between quantities
- The constant of proportionality and how to identify it
- Representations of proportional relationships with equations
- Proportional relationship graphs
Mastery of these lessons will be evident as students demonstrate the capacity to translate real-life examples of proportional relationships into equations and graphs. They will display comprehension and synthesis of the lesson content as they manipulate models and functions to explore different aspects of proportionality and attempt to tell a story of the relationship between its elements.
How to Use These Lessons in Your Classroom
If you're having a hard time coming up with interactive ways to incorporate proportions in general, and these videos in particular, in your classroom instruction, here are some ideas to help get you started.
How much sugar is in that?
Putting a mathematical twist on the ubiquitous health class display showing how much sugar is in our soft drinks and candy bars, have every student collect data on their sugar intake in a day or, preferably, an entire week. Then they can chart the data, comparing the amount of sugar they consumed to established standards of health. Break students into small groups to research healthy alternatives to their common diet to reduce sugar intake below the standards. Visual aids can be developed using glass vials or plastic tubes with volume markings and a bag of sugar. Keep in mind that thinner vessels project more exaggerated proportional relationships.
Identifying good deals
Is it better to buy a 15-oz can of beans at $2.89 or a 7.5-oz can at $1.83? Help students understand constant of proportionality through price comparisons of grocery story items. First watch the video Identifying the Constant of Proportionality together in class. Provide a brief example of a price comparison, then assign the homework of visiting a grocery store and comparing like products across brands and volumes, calculating unit rates of at least three different products. Have students show which, if any, of their product comparisons exhibited a proportional relationship--that is, that the unit rate across two or more volumes of a product was equal. For added practice in depicting proportions visually, have students graph or chart their results.
Provide examples of utility bills which charge a rate by volume/quantity of use (e.g. water, gas, or electricity), being sure to mark out personal data if you provide real examples. Have students identify unit rates, volume/quantity measures, and total prices and write the equation representing that data. Help students understand budgeting by giving a maximum price for a given utility and have them determine how much of the utility they can use before breaking their budget. Provide more depth to the lesson by identifying proportion of utility use of various objects, activities, or appliances. Just how much does leaving the faucet running while brushing your teeth cost each year? Why does it cost so much to have the A/C set to low temperatures?
1. Proportional Relationships Between Two Quantities
When two values always maintain the same ratio, forming the same fraction when you divide them, they have a proportional relationship. In this lesson, you can learn about proportional relationships between two quantities.
2. Identifying the Constant of Proportionality
Constants of proportionality might sound like something complicated, but chances are, you've encountered them before. In this lesson, we'll explore the constant of proportionality and how to find it.
3. Representing Proportional Relationships by Equations
After you read this lesson, you'll see just how easy it is to find an equation that represents your proportional relationship. You'll learn the formula you can use to find your equation for any proportion you have.
4. Graphing Proportional Relationships
Two values are proportional if they maintain the same ratio and their graph will always be a straight line that goes through the origin (0,0). In this lesson we'll explore how to draw proportional graphs and use them to solve desperate situations!
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Other chapters within the Common Core Math Grade 7 - Ratios & Proportional Relationships: Standards course