(PG. 1 OF 2) Tutoring/Intervention Centers

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1 Constant of Proportionality (PG. 1 OF 2 SE Purpose Students identify constant of proportionality and connect it to unit rate. They will represent constant of proportionality through tables, graphs, algebraic equations, and verbal descriptions. Whole Class Teacher-facilitated Small Groups Small Student Groups Tutoring/Intervention Centers Journal Anchor Chart Setting Up For Instruction Make 1 copy of Soil Cleanup Make 1 copy of Soil Cleanup Journal (PG. 96 for each student. Other materials: Trail mix with raisins: 1 fun-size bag for each group, to represent soil mixtures (Have extra for the students to enjoy afterwards! Napkins Calculators How-To Guide CONSTANT OF PROPORTIONALITY THROUGH TABLES, GRAPHS, VERBAL DESCRIPTIONS, AND ALGEBRAIC RELATIONSHIPS AND THEIR CONNECTION TO UNIT RATE Goal: Calculate constant of proportionality by making the connection between unit rate and constant rate of change; represent in tables, graphs, equations, and verbal descriptions 1. Have a group discussion about environmental engineering. Ask students what they know about a career in environmental engineering. Discuss what it is, what they might do, and what salaries they might expect to receive. Here are a couple of links to use as resources., watch?v=k2epvauedci. In this activity, students will be environmental engineers exploring the idea of using clay to assist with contaminated soil. 2. Pass out the bagged samples of soil (trail mix and napkins. Tell students not to eat it! One way to keep them from eating it is to tell them that you ve been using the same trail mix for a few years and that it has germs. The raisins represent the contaminant. Have student separate out the contaminant from their sample and count how many parts of contaminant to 3. Hand out Soil Cleanup and have students work together to solve the problems. You may want to read the Soil Cleanup introduction and directions together. Hand out Soil Cleanup Journal. Students should work independently to solve the problems. Back to the Table of Contents Table of Standards

2 (PG. 2 OF 2 Thought Extenders What are the 2 variables? What is constant of proportionality similar to? What does constant of proportionality look like on a graph? Where does the table show constant of proportionality? What do you know from the problem? What do you need to know to solve the problem? Do you know how to use the constant of proportionality to write an equation? How can the equation be used? Constant of Proportionality Breakdown The constant of proportionality is the constant ratio between two proportional quantities. Let's take a detailed look at the constant of proportionality. Only proportional relationships have a constant of proportionality. For example, in the equation y = 2x, 2 is the constant of proportionality. In this equation no matter the value of x, the value of y is twice as much, so y x = 2 and the slope, 1 Change in y would also result in the number 2 since for every one change in x there is twice the change in y. Change in x In the equation y = 2x y = 2x Change in y even though the = 2, it is not proportional because of the additive component. Change in x The constant of proportionality is a rational number. 7 th grade students know the word constant. But they likely have not previously encountered it in mathematics. The constant of proportionality is a ratio between y and x. It is also called k. The constant of proportionality represents direct variation, where the x value is directly proportional to the y value. It is shown in the equation y = kx. A direct variation equation contains only a multiplicative component. A common relationship that expresses direct variation is d = rt. The rate (r is the constant of proportionality, as in constant rate of speed. When you solve this equation for the rate, you get r = d t this case is d t. When graphing y = kx, x is the independent variable and y is the dependent variable.. So, another way to say the constant of proportionality in The independent variable, x, describes the input value in a relationship. This is the same x in the ordered pair (x, y. The dependent variable, y, describes the output value in a relationship. This the same y in the ordered pair (x, y. The constant of proportionality can never be zero. In 7 th grade, students must also be able to locate and explain the constant of proportionality as seen in a vertical or horizontal table, a verbal description, a graph, or in an equation

3 ANSWER KEY (PG. 1 OF 3 There are hundreds of thousands of contaminated soil and water sites in the US alone. As an engineer, with many products today is that, while they re good at cleaning up a particular type of contaminant, they method of modifying clay to take some contaminants out of the soil and encourage the growth of natural microorganisms, like bacteria, to help eat some of the contaminants. For the clay to work, you have to know the proportionality between the contaminants and the soil mixture. You also need to identify the constant of proportionality. Directions: Separate the contaminants (raisins from the other parts of the soil (everything else in the trail mix. Use the data to solve the problems below. 1. Complete the chart below, comparing the parts of the soil that are contaminants (raisins to total parts. Contaminants (x Process Total (y 2. Scientists often use scaled models in their research. If your soil sample doubled, what would the contaminants total ratio be (assuming the contamination was proportional? Complete the table below with your original entry and the doubled entry. Contaminants (x Process Total (y 8 8( 8 3. Suppose the soil sample was 10 times the original amount. Complete the table below, with the original, doubled, and ten times amounts. Contaminants (x Process Total (y 8 8(

4 ANSWER KEY (PG. 2 OF 3 On a larger scale, suppose the sample were 200 times the original amount. Continue expanding the table including 200 times the original number of contaminants. Contaminants (x Process Total (y 8 8( ( 800 What is the constant of proportionality? How is the constant of proportionality related to the number of contaminants and the total mixture? It is the constant ratio of the total mixture to the contaminants. 6. Graph the soil sample contamination representation. y

5 ANSWER KEY (PG. 3 OF 3 7. Where is the constant of proportionality on the graph? 8. What equation represents this soil sample relationship? y =6x 9. Using the constant of proportionality, how many parts are contaminated if the total parts in the soil 8, parts are contaminated. 10. What does the constant represent in this relationship? contaminants. 11. In what situations would a contaminated soil sample be of concern to engineers? Answers will vary. Possible answers include where animals are, where people live, parks, schools, drinking water

6 SOIL CLEAN-UP JOURNAL CLEANING SOIL WITH FUNGI ANSWER KEY JOURNAL Researchers in Finland have found a way to use fungi to clean soil in a more sustainable way than traditional methods. The fungi alter, or change, the makeup of the pollutants so that they no longer cause harm. Directions: Answer the questions using the ratio given in the table. 1. The table below shows an example of the ratio between the parts of fungi and the contaminants. Use the constant of proportionality in the process. Fill in the table. Fungi part (x Process Contaminants (y 2 2( ( , What is the constant of proportionality? 2 3. What does the constant of proportionality represent? part fungi to clean the soil. Graph the relationship between the fungi and the contaminants. y What equation represents this relationship? 6. Where is the constant of proportionality in the equation? 2, or 1 value., is the constant of x

7 (PG. 1 OF 3 Name: There are hundreds of thousands of contaminated soil and water sites in the US alone. As an engineer, with many products today is that, while they re good at cleaning up a particular type of contaminant, they method of modifying clay to take some contaminants out of the soil and encourage the growth of natural microorganisms, like bacteria, to help eat some of the contaminants. For the clay to work, you have to know the proportionality between the contaminants and the soil mixture. You also need to identify the constant of proportionality. Directions: Separate the contaminants (raisins from the other parts of the soil (everything else in the trail mix. Use the data to solve the problems below. 1. Complete the chart below, comparing the parts of the soil that are contaminants (raisins to total parts. Contaminants (x Process Total (y 2. Scientists often use scaled models in their research. If your soil sample doubled, what would the contaminants total ratio be (assuming the contamination was proportional? Complete the table below with your original entry and the doubled entry. Contaminants (x Process Total (y 3. Suppose the soil sample was 10 times the original amount. Complete the table below, with the original, doubled, and ten times amounts. Contaminants (x Process Total (y

8 (PG. 2 OF 3 Name: including 200 times the original number of contaminants. Contaminants (x Process Total (y How is the constant of proportionality related to the number of contaminants and the total mixture? 6. Graph the soil sample contamination representation. y

9 (PG. 3 OF 3 Name: 7. Where is the constant of proportionality on the graph? 8. What equation represents this soil sample relationship? 9. Using the constant of proportionality, how many parts are contaminated if the total parts in the soil 10. What does the constant represent in this relationship? 11. In what situations would a contaminated soil sample be of concern to engineers?

10 SOIL CLEAN-UP JOURNAL CLEANING SOIL WITH FUNGI JOURNAL Name: Researchers in Finland have found a way to use fungi to clean soil in a more sustainable way than traditional methods. The fungi alter, or change, the makeup of the pollutants so that they no longer cause harm. Directions: Answer the questions using the ratio given in the table. 1. The table below shows an example of the ratio between the parts of fungi and the contaminants. Use the constant of proportionality in the process. Fill in the table. Fungi part (x Process Contaminants (y What is the constant of proportionality? 3. What does the constant of proportionality represent? Graph the relationship between the fungi and the contaminants. y What equation represents this relationship? 6. Where is the constant of proportionality in the equation?