# 2.4.2: Interpreting Graphs of Proportional Relationships

## Lesson

Let's read stories from the graphs of proportional relationships.

Exercise (PageIndex{1}): What Could the Graph Represent?

Here is a graph that represents a proportional relationship.

1. Invent a situation that could be represented by this graph.
2. Label the axes with the quantities in your situation.
3. Give the graph a title.
4. There is a point on the graph. What are its coordinates? What does it represent in your situation?

Exercise (PageIndex{2}): Tyler's Walk

Tyler was at the amusement park. He walked at a steady pace from the ticket booth to the bumper cars.

1. The point on the graph shows his arrival at the bumper cars. What do the coordinates of the point tell us about the situation?
2. The table representing Tyler's walk shows other values of time and distance. Complete the table. Next, plot the pairs of values on the grid.
3. What does the point ((0,0)) mean in this situation?
4. How far away from the ticket booth was Tyler after 1 second? Label the point on the graph that shows this information with its coordinates.
5. What is the constant of proportionality for the relationship between time and distance? What does it tell you about Tyler's walk? Where do you see it in the graph?
time (seconds)distance (meters)
(0)(0)
(20)(25)
(30)(37.5)
(40)(50)
(1)
Table (PageIndex{1})

Are you ready for more?

If Tyler wanted to get to the bumper cars in half the time, how would the graph representing his walk change? How would the table change? What about the constant of proportionality?

Exercise (PageIndex{3}): Seagulls Eat What?

4 seagulls ate 10 pounds of garbage. Assume this information describes a proportional relationship.

1. Plot a point that shows the number of seagulls and the amount of garbage they ate.
2. Use a straight edge to draw a line through this point and ((0,0)).
3. Plot the point ((1,k)) on the line. What is the value of (k)? What does the value of (k) tell you about this context?

### Summary

For the relationship represented in this table, (y) is proportional to (x). We can see in the table that (frac{5}{4}) is the constant of proportionality because it’s the (y) value when (x) is 1.

The equation (y=frac{5}{4}x) also represents this relationship.

(x)(y)
(4)(5)
(5)(frac{25}{4})
(8)(10)
(1)(frac{5}{4})
Table (PageIndex{2})

Here is the graph of this relationship.

If (y) represents the distance in feet that a snail crawls in (x) minutes, then the point ((4,5)) tells us that the snail can crawl 5 feet in 4 minutes.

If (y) represents the cups of yogurt and (x) represents the teaspoons of cinnamon in a recipe for fruit dip, then the point ((4,5)) tells us that you can mix 4 teaspoons of cinnamon with 5 cups of yogurt to make this fruit dip.

We can find the constant of proportionality by looking at the graph, because (frac{5}{4}) is the (y)-coordinate of the point on the graph where the (x)-coordinate is 1. This could mean the snail is traveling (frac{5}{4}) feet per minute or that the recipe calls for (1frac{1}{4}) cups of yogurt for every teaspoon of cinnamon.

In general, when (y) is proportional to (x), the corresponding constant of proportionality is the (y)-value when (x=1).

### Glossary Entries

Definition: Coordinate Plane

The coordinate plane is a system for telling where points are. For example. point (R) is located at ((3,2)) on the coordinate plane, because it is three units to the right and two units up.

Definition: Origin

The origin is the point ((0,0)) in the coordinate plane. This is where the horizontal axis and the vertical axis cross.

## Practice

Exercise (PageIndex{4})

There is a proportional relationship between the number of months a person has had a streaming movie subscription and the total amount of money they have paid for the subscription. The cost for 6 months is $47.94. The point ((6,47.94)) is shown on the graph below. 1. What is the constant of proportionality in this relationship? 2. What does the constant of proportionality tell us about the situation? 3. Add at least three more points to the graph and label them with their coordinates. 4. Write an equation that represents the relationship between (C), the total cost of the subscription, and (m), the number of months. Exercise (PageIndex{5}) The graph shows the amounts of almonds, in grams, for different amounts of oats, in cups, in a granola mix. Label the point ((1,k)) on the graph, find the value of (k), and explain its meaning. Exercise (PageIndex{6}) To make a friendship bracelet, some long strings are lined up then taking one string and tying it in a knot with each of the other strings to create a row of knots. A new string is chosen and knotted with the all the other strings to create a second row. This process is repeated until there are enough rows to make a bracelet to fit around your friend's wrist. Are the number of knots proportional to the number of rows? Explain your reasoning. (From Unit 2.3.3) Exercise (PageIndex{7}) What information do you need to know to write an equation relating two quantities that have a proportional relationship? (From Unit 2.3.3) ## Interpreting Points on the Graph of a Proportional Relationship Students will interpret points on the graph of a proportional relationship. Students will: • interpret points on a graph, which models a real-world proportional relationship. The points will include the origin and the unit rate. #### Essential Questions • How is mathematics used to quantify, compare, represent, and model numbers? • How are relationships represented mathematically? • How can expressions, equations and inequalities be used to quantify, solve, model and/or analyze mathematical situations? ### Vocabulary • Proportion: An equation of the form that states that the two ratios are equivalent. • Ratio: A comparison of two numbers by division. • Unit Rate: A rate simplified so that it has a denominator of 1. ### Duration ### Prerequisite Skills ### Materials • one copy of the Lesson 3 Exit Ticket (M-7-3-3_Lesson 3 Exit Ticket and KEY.docx) per student • copies of the Small Group Practice worksheet (M-7-3-3_Small Group Practice and KEY.docx) as needed • Copies of the Expansion Work sheet (M-7-3-3_Expansion Work and KEY.docx) as needed ### Related Unit and Lesson Plans ### Related Materials & Resources The possible inclusion of commercial websites below is not an implied endorsement of their products, which are not free, and are not required for this lesson plan. • one copy of the Lesson 3 Exit Ticket (M-7-3-3_Lesson 3 Exit Ticket and KEY.docx) per student • copies of the Small Group Practice worksheet (M-7-3-3_Small Group Practice and KEY.docx) as needed • Copies of the Expansion Work sheet (M-7-3-3_Expansion Work and KEY.docx) as needed ### Formative Assessment • Use responses from the Think-Pair-Share Activity to gauge student level of understanding of the meaning of origin and unit rate in proportional relationships. • Use the Partner Game to determine students&rsquo ability to generate a proportional relationship and interpret points on a line. • Use the Lesson 3 Exit Ticket (M-7-3-3_Lesson 3 Exit Ticket and KEY.docx) to quickly evaluate student mastery. ### Suggested Instructional Supports Scaffolding, Active Engagement, Metacognition, Modeling, Formative Assessment  W: Students will learn to interpret points on the graph of a proportional relationship in terms of the context. These points will include the unit rate and origin. H: Students will be hooked into the lesson by first brainstorming about the concepts of origin and unit rate in a proportional relationship. E: The focus of the lesson is on interpreting points on the line of a graph of a proportional relationship. Students will work in small groups to explore and examine proportional relationships in great depth as the class is led through teacher-directed examples. R: Opportunities for discussion start at the beginning of the lesson. Using the Think-Pair-Share Activity, students are given an opportunity to reflect on their understanding and revise as necessary. Students will discuss the examples in their groups and with the whole class. The Partner Game will serve as a review as it requires students to create a proportional relationship and ask and answer questions related to the appearance of the graph. E: Evaluate student level of understanding based on responses to the Lesson 3 Exit Ticket. If limited time is available, two or three questions may be selected for use on the exit Ticket instead of administering it in its entirety. T: The Extension section may be used to tailor the lesson to meet the needs of the students. The Routine section provides opportunities for lesson concept review during the course of the school year. The Small Group section includes ideas for students who may benefit from additional practice or learning opportunities. The Expansion section details options for students who are ready for a challenge beyond the requirements of the standard. O: The lesson is scaffolded so that students first focus on the conceptual meaning of points on the line of a graph of a proportional relationship. Next, students work through explicit examples of real-world contexts that represent proportional relationships. They will interpret the meaning of points on the graph, including the unit rate and origin. Following discussion of the examples, students will participate in the Partner Game and complete the final assessment. ### Instructional Procedures Think-Pair-Share Activity &ldquoToday we are going to talk about some of the points on the graph of a line. We want to figure out the significance of two points in particular and figure out what is so special about those points.&rdquo Ask students to think about the meaning of the points, (0, 0), and (1, r), on the graph of a proportional relationship. Give students 2&ndash3 minutes to brainstorm some ideas about the significance of these two points. Then have students choose a partner to share their ideas. After about 5 minutes, the class may reconvene and one member from each group may share their thoughts on the meaning of the two points. By starting the lesson with a more general approach, you may determine students&rsquo level of prior knowledge related to the idea of the origin and the unit rate. After students have shared their ideas, the lesson may proceed to some specific examples involving real-world situations. For each example, arrange students in groups of 2&ndash3. Groups may work together as the whole class moves from one example to the next. Students should discuss and debate ideas with one another. Groups should contribute to the class discussion. The given questions are intended as samples and guidelines. Also, more examples may need to be shown. The first three examples include the unit rate in the statement, which describes the proportional relationship. In these examples, the graph also has the point (1, r) labeled. The next few examples provide students with a graph without any points labeled. Following the examples, students will have an opportunity to perform some individual work. Example 1: Ana creates a pattern using diagrams of squares. For the first diagram in her pattern, she draws 3 squares. For each subsequent diagram, she draws 3 more squares than shown in the previous diagram. An illustration of Ana&rsquos pattern is below: • &ldquoWhat statement can you make about the number of squares in each diagram compared to the diagram number?&rdquo (The relationship is proportional.) &ldquoHow may this relationship be represented in equation form?&rdquo (It may be written as, where y represents the number of squares in the diagram and x represents the position number of the diagram.) • Have students think about the appearance of the graph of this pattern. Ask questions similar to the following: • &ldquoWhat will the graph of the relationship look like? How do you know?&rdquo • &ldquoWhere will the graph cross the y-axis?&rdquo • &ldquoWhat point will represent the unit rate, or constant of proportionality?&rdquo A table with sample questions and correct responses is shown below: Sample Questions Sample Responses What does the origin, or point (0, 0), indicate in the context of this example? The point (0, 0) indicates that Diagram 0 would have 0 squares. In other words, there are not any squares shown in a diagram prior to Diagram 1. What point represents the unit rate? The unit rate is represented by the point (1, 3). What does the point (1, 3) indicate? State the meaning within the context of this example. The point (1, 3) is the unit rate or constant of proportionality. It indicates the number of new squares added to each diagram. Diagram 1 includes 3 squares. Thus, each subsequent diagram shows 3 additional squares. What does the point (&minus2, &minus6) indicate? Does interpretation of this point make sense in the context of this example? Explain. The point (&minus2, &minus6) would indicate that the position of negative 2 shows negative 6 squares. This does not make sense, since the diagrams start at position 1, with a positive number of squares. What does the point (3, 9) indicate? State the meaning, within the context of this example. The point (3, 9) indicates that Diagram 3 has 9 squares. What point on the graph represents the number of squares included in Diagram 4? How many squares will be included in Diagram 5? How did you determine your answer? Diagram 5 will have 15 squares because 3 times 5 equals 15. An x-value of 5 corresponds to a y-value of 15. What diagram number will have a total of 21 squares? How did you determine your answer? Diagram 7 will have 21 squares. This answer may be determined by finding the x-value that corresponds to the y-value of 21. That x-value is 7. What diagram number will have a total of 51 squares? How did you determine your answer? Diagram 17 will have 51 squares. Since the rate of change is 3, the following equation may be written: . Solving for x gives x = 17. What do the other points on the line indicate? The other points indicate all other combinations of diagram position numbers and number of squares in the diagram. Name another point found on the line of the graph. State the meaning of the point. Another point is (8, 24). This point indicates that Diagram 8 contains 24 squares. Example 2: Aubrey saves$25 per month.

• &ldquoWhat may be stated regarding the relationship of her cumulative savings to the number of months that have passed?&rdquo (The relationship is proportional.)
• &ldquoHow may this relationship be represented as an equation?&rdquo (It may be written as, where y represents the cumulative savings and x represents the number of months that have passed.)
• Have students think about the appearance of the graph of this pattern. Ask questions similar to the following:
• &ldquoWhat will the graph of the relationship look like? How do you know?&rdquo
• &ldquoWhere will the graph cross the y-axis?&rdquo
• &ldquoWhat point will represent the unit rate, or constant of proportionality?&rdquo

A table with sample questions and correct responses is shown below:

Sample Questions

Sample Responses

What does the origin, or point (0, 0) indicate in the context of this example?

The point (0, 0) indicates that after 0 months, her cumulative savings is . In other words, she does not make an initial deposit prior to month 1. You may also discern from this point that Month 1 does not include any savings, in addition to the unit rate amount.

What point represents the unit rate?

The unit rate is represented by the point (1, 25).

What does the point (1, 25) indicate? State the meaning within the context of this example.

The point (1, 25) is the unit rate or constant of proportionality. It indicates the amount of savings per month. After 1 month, she has saved $25. Thus, for each subsequent month, she saves an additional$25.

What does the point (&minus1, &minus25) indicate? Does interpretation of this point make sense in the context of this example? Explain.

The point (&minus1, &minus25) would indicate that after negative 1 month, she has saved negative 25 dollars. This does not make sense, in the context of the example, since you cannot have a negative number of months.

What point on the graph represents her cumulative savings after the third month?

What does the point (4, 100) indicate? State the meaning within the context of this example.

The point (4, 100) indicates that she has saved $100 after 4 months. How much money will she have saved after 6 months? How did you determine your answer? She will have saved$150 after 6 months. The y-value that corresponds to the x-value of 6 is 150.

After how many months will she have saved a total of $200? How did you determine your answer? After 8 months, she will have saved$200. This answer may be determined by finding the x-value that corresponds to the y-value of $200. That x-value is 8. After how many months will she have saved a total of$350? How did you determine your answer?

After 14 months, she will have saved $350. Since the rate of change is$25, the following equation may be written: . Solving for x gives x = 14.

What do the other points on the line indicate?

The other points indicate all other combinations of number of months and cumulative savings amounts.

Name another point on the line of the graph. State the meaning of the point.

Another point on the graph is (11, 275). She will have saved $275 after 11 months. Example 3: Monique&rsquos membership to a wholesale club costs$35 per year.

• &ldquoWhat can we say about the relationship of the cumulative yearly membership costs to the number of years that have passed?&rdquo (The relationship is proportional.)
• &ldquoHow may this relationship be represented in equation form?&rdquo (It may be written as, where y represents the cumulative membership costs and x represents the number of years that have passed.)
• Have students think about the appearance of the graph of this pattern. Ask questions similar to the following:
• &ldquoWhat will the graph of the relationship look like? How do you know?&rdquo
• &ldquoWhere will it cross the y-axis?&rdquo
• &ldquoWhat point will represent the unit rate, or constant of proportionality?&rdquo

A table with sample questions and correct responses is shown below:

Sample Questions

Sample Responses

What does the origin, or point (0, 0), indicate in the context of this example?

The point (0, 0) indicates that after 0 years, her cumulative membership costs equal . In other words, she does not pay any membership amount prior to Year 1.

What point represents the unit rate?

The unit rate is represented by the point (1, 35).

What does the point (1, 35) indicate? State the meaning, within the context of this example.

The point (1, 35) is the unit rate or constant of proportionality. It indicates the cost of membership per year. After 1 year, she has paid $35 in membership costs. Thus, for each subsequent year, she will pay an additional$35.

What does the point (&minus1, &minus35) indicate? Does interpretation of this point make sense in the context of this example? Explain.

The point (&minus1, &minus35) would indicate that after negative 1 year, she has paid negative 35 dollars. This does not make sense, in the context of the example, since you cannot have a negative number of years.

What point on the graph represents her cumulative membership costs, after the 9 th year?

What does the point (5, 175) indicate? State the meaning within the context of this example.

The point (5, 175) indicates that she will have paid a total of $175 after 5 years. How much will she have paid in membership costs after 7 years? How did you determine your answer? She will have paid$245. The y-value that corresponds to the x-value of 7 is 245.

After how many years will her cumulative membership costs total $280? How did you determine your answer? After 8 years, her cumulative membership costs will total$280. This answer may be determined by finding the x-value that corresponds to the y-value of $280. That x-value is 8. After how many years will her cumulative membership costs total$525? How did you determine your answer?

After 15 months, she will have paid $525, in membership costs. Since the rate of change is$35, the following equation may be written: . Solving for x gives x = 15.

What do the other points on the line indicate?

The other points indicate all other combinations of number of years and cumulative membership costs.

Name another point found on the line of the graph. State the meaning of the point.

Another point on the line is (20, 700). After 20 years, she will have paid $700 in membership costs. Example 4: On a map, a certain number of miles is represented by a certain number of inches. The number of actual miles and the inches on the map form a proportional relationship. In other words, the number of actual miles varies directly with the number of inches shown on the map. A graph is shown below: A table with sample questions and correct responses is shown below: Sample Questions Sample Responses What does the origin, or point (0, 0), indicate in the context of this example? The point (0, 0) indicates 0 inches represents 0 actual miles. What point represents the unit rate? State the meaning of the point within the context of this example. The unit rate is represented by the point (1, 20). This means that 1 inch, on the map, represents 20 actual miles. Thus, for every additional inch, 20 more miles are added to the distance. Will a point with a negative x-value make sense in the context of this example? Explain. No, it will not, because you cannot have negative inches. What point on the graph represents the number of miles represented by 14 inches? What does the point (10, 200) indicate? State the meaning, within the context of this example. The point (10, 200) indicates that 200 miles are represented by 10 inches on the map. How many miles are represented by 17 inches? How did you determine your answer? There are 340 miles represented by 17 inches. The y-value that corresponds to the x-value of 17 is 340. How many inches will represent 100 miles? How did you determine your answer? 5 inches will represent 100 miles. This answer may be determined by finding the x-value that corresponds to the y-value of$100. That x-value is 5.

How many inches will represent a total distance of 440 miles? How did you determine your answer?

22 inches will represent 440 miles. Since the rate of change is 20, the following equation may be written as. Solving for x gives x = 22.

What do the other points on the line indicate?

The other points indicate all other combinations of number of inches on the map and number of actual miles.

Name another point found on the line of the graph. State the meaning of the point.

Another point on the line is (18, 360). 18 inches represents 360 miles.

Example 5: The dimensions of two similar isosceles triangles form a proportional

relationship. The graph below represents the base lengths of the two triangles (in centimeters):

A table with sample questions and correct responses is shown below:

Sample Questions

Correct Responses

What does the origin, or point (0, 0), indicate in the context of this example?

The point (0, 0) indicates a base length of 0 cm on the smaller triangle corresponds to a base length of 0 cm on the larger triangle.

What point represents the unit rate? State the meaning of the point within the context of this example.

The unit rate is represented by the point (1, 4). This means that a base length of 1 cm on the smaller triangle would result in a base length of 4 cm on the larger triangle. Thus, for every additional cm of length on the smaller triangle, the base length of the larger triangle increases by 4 cm.

Will a point with a negative x-value make sense in the context of this example? Explain.

No, it will not, because you cannot have negative base lengths of triangles.

What point on the graph represents the base length of the larger triangle, given that the smaller triangle has a base length of 6 cm?

What does the point (3, 12) indicate? State the meaning within the context of this example.

The point (3, 12) indicates that a base length of 3 cm on the smaller triangle will result in a base length of 12 cm on the larger triangle.

How long will the base of the larger triangle be, given the base of the smaller triangle is 5 cm? How did you determine your answer?

The base of the larger triangle will be 20 cm. The y-value that corresponds to the x-value of 5 is 20.

How long will the base of the smaller triangle be, given that the base of the larger triangle is 44 cm? How did you determine your answer?

The base of the smaller triangle will be 11 cm. Since the rate of change is 4, the following equation may be written: . Solving for x gives x = 11.

What do the other points on the line indicate?

The other points indicate all other combinations of lengths of bases of the two similar triangles.

Name another point found on the line of the graph. State the meaning of the point.

Another point on the line is (15, 60). A base length of 15 cm on the smaller triangle corresponds to a base length of 60 cm on the larger triangle.

Partner Game

Have students write a general description of a proportional real-world relationship. The description should not include the unit rate. Students should ask a partner 3&ndash4 questions regarding the graph of the relationship. Questions may include:

• What is the meaning of the origin?
• What is the meaning of the unit rate in the context of the example?
• What does the point (x, y) represent?
• What is another point on the graph, and what does it indicate?

After one partner has answered the other&rsquos questions, students should switch roles.

The questions and answers may be uploaded as files to the class website, for review purposes.

Watch our free video on how to Graph Proportional Relationships. This video shows how to solve problems that are on our free Graphing Proportional Relationships worksheet that you can get by submitting your email above.

Watch the free Graphing Proportional Relationships video on YouTube here: Graphing Proportional Relationships Video

Video Transcript:

This video is about graphing proportional relationships. You can get the worksheet used in this video for free by clicking on the link in the description below. In order to show you how to graph proportional relationships, we’re going to do a couple practice problems from our graphing proportional relationships worksheet.

The first problem we’re going to do on our graphing proportional relationships worksheet is number two. This problem gives us a table, a graph, and asks us to solve for the constant of proportionality. The first thing we’re going to do is we’re going to use our table to complete our proportional relationship graph on the right. The first thing we can do is we can label our axes on our graph. We know that the x-axis is going to be minutes. We know this axis is going to be minutes and we know that the y-axis is going to be feet. In order to graph this, we use the values that are given to us in our proportional table here. Our first point is going to be 0 0. We put a dot on 0 0. The second point that is given to us is 1 minute and then 2 feet. We go over to our proportional relationship graph and we go to 1 minute and then up to two feet and we put our second dot. The third point we can put on our graph is minutes is two and feet is four. We go over to two minutes and then up to four feet and we put our third dot, fourth point is going to be minutes is three and then feet is six, we put a dot there and then our last point gives us minutes is four, four minutes and then eight feet. We put our last dot there. Now that our proportion relationship has been graphed, we can find the constant of proportionality. The constant of proportionality is given as the equation k equals y divided by x, and k is the constant of proportionality, y is the y values, and x is the x values. Our x axis is minutes so we know the minutes represent the x values and the y axis is feet. We know feet are represented by the y axis. In order to solve our formula k equals y divided by x we can use any column from our table as long as we put feet in for y and minutes in for x. I’m going to use this fourth column, which gives us three minutes and six feet so our feet is six. We know that’s the y value the minutes are three, we know that’s the x value. To simplify this, we’re going to do 6 divided by 3 which is 2. Now we know that the constant of proportionality is 2 feet every minute and that’s going to be the solution.

The last problem we’re going to complete on our graphing proportional relationships worksheet is number three. The first step is to use our table to complete the proportional relationships graph. Again, we’re going to use the values from each column to plot the points on our graph here. The first step is to label our axes so we’re going to label hours on the x-axis and then miles on the y-axis in order to complete the proportion relationships graph. We’re going to use the columns from the table to plot our points. The first column gives us 0 0. That’s going to be our first point, second column gives us 1 hour is 3 miles, we put a dot there, this third column gives us 2 hours is 6 miles, and then 3 hours is nine miles and then four hours. We go to hour four and then up to 12 miles, and then five hours, and then 15 miles and then finally six hours which is 18 miles. Now that our proportional relationship has been graphed, we can find the constant of proportionality. We know that the constant of proportionality is equal to the y values divided by the x values and we know that the x values are represented by hours and the y values are represented by miles. We can take any column that we want to find the constant of proportionality. I’m going to use column number four. This column gives us miles, which is our y value as nine, this is 9 miles divided by our hours column which is 3. It’s 3 hours and then when you reduce this, you’ll get 9 divided by 3. It’s three miles for every one hour and that’s going to be our solution.

## Introduction

A proportion is an equation stating that two ratios are equivalent. A relationship that involves a collection of equivalent ratios is called a proportional situation.

In each of these situations, you can see that the relationships are proportional. For each of the data points, the ratios are equivalent.

A rate is a comparison between two quantities. When the denominator of the rate is one, it is called a unit rate.

In this lesson, you will extend what you know about unit rates and look for patterns in graphs of proportional relationships. You will also investigate slope, which is a special characteristic of linear relationships. In a linear relationship, the slope of the line is steepness of the graph of the line. Slope is also described as the ratio of the vertical change to the horizontal change between two points on the line.

## Warm up

In the Rose Ratios Warm Up students consider a vase that holds red and white roses. The vase contains three red roses for every two white roses. Students need to decide how many flowers might be in the vase.

I expect some students may stop at the answer 5 while others will fill a page with possibilities. Because it is productive for students to consider the nature of a complete answer to this problem, I ask my class to discuss their solutions with their math family groups. If there is an entire group who came up with 5 as the only solution, I will direct them to read the problem carefully. I will ask, "Does it say that the vase holds "exactly" three red and "exactly" two white roses?"

For students who are considering other possibilities, I will wait for them to begin to ask, "When can we stop?" I will respond, "How big do you think the vase could be?" Though it is not the primary purpose of the lesson, I want them to reason about constraints, an important habit when making sense of and determining the reasonableness of solutions (MP1).

As noted above, I expect students to be working at different levels in my classroom. I am ready to help them build their capacity for multiplicative thinking with some scaffolding questions (see Warm Up Rose Ratios Notes). During a lesson like this one, I accept all student explanations, but I try to model explanations in a way that allows students to notice the patterns and connections between additive and multiplicative thinking (MP8).

## Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways.

The constant of variation is the number that relates two variables that are directly proportional or inversely proportional to one another. But why is it called the constant of variation? This tutorial answers that question, so take a look!

#### What Does Direct Variation Look Like on a Graph?

Want to know what a direct variation looks like graphically? Basically, it's a straight line that goes through the origin. To get a better picture, check out this tutorial!

#### What's the Direct Variation or Direct Proportionality Formula?

Ever heard of two things being directly proportional? Well, a good example is speed and distance. The bigger your speed, the farther you'll go over a given time period. So as one variable goes up, the other goes up too, and that's the idea of direct proportionality. But you can express direct proportionality using equations, and that's an important thing to do in algebra. See how to do that in the tutorial!

## Testing the proportional hazard assumption in Cox models

When modeling a Cox proportional hazard model a key assumption is proportional hazards. There are a number of basic concepts for testing proportionality but the implementation of these concepts differ across statistical packages. The goal of this page is to illustrate how to test for proportionality in STATA, SAS and SPLUS using an example from Applied Survival Analysis by Hosmer and Lemeshow .

Works best for time fixed covariates with few levels. If the predictor satisfy the proportional hazard assumption then the graph of the survival function versus the survival time should results in a graph with parallel curves, similarly the graph of the log(-log(survival)) versus log of survival time graph should result in parallel lines if the predictor is proportional. This method does not work well for continuous predictor or categorical predictors that have many levels because the graph becomes to “cluttered”. Furthermore, the curves are sparse when there are fewer time points and it may be difficult to gage how close to parallel is close enough. Due to space limitations we will only show the graph for the predictor treat.

SAS It is very easy to create the graphs in SAS using proc lifetest. The plot option in the model statement lets you specify both the survival function versus time as well as the log(-log(survival) versus log(time).

STATA The sts graph command in STATA will generate the survival function versus time graph.

SPLUS The plot function applied to a survfit object will generate a graph of the survival function versus the survival time.

2. Including Time Dependent Covariates in the Cox Model

Generate the time dependent covariates by creating interactions of the predictors and a function of survival time and include in the model. If any of the time dependent covariates are significant then those predictors are not proportional.

SAS In SAS it is possible to create all the time dependent variable inside proc phreg as demonstrated. Furthermore, by using the test statement is is possibly to test all the time dependent covariates all at once.

STATA We use the tvc and the texp option in the stcox command. We list the predictors that we would like to include as interaction with log(time) in the tvc option (tvc = time varying covariates). The texp option is where we can specify the function of time that we would like used in the time dependent covariates. By using the lrtest commands it is possible to tests all the time dependent covariates together by comparing the smaller model without any time dependent covariates to the larger model that includes all the time dependent covariates.

3. Tests and Graps Based on the Schoenfeld Residuals Testing the time dependent covariates is equivalent to testing for a non-zero slope in a generalized linear regression of the scaled Schoenfeld residuals on functions of time. A non-zero slope is an indication of a violation of the proportional hazard assumption. As with any regression it is highly recommended that you look at the graph of the regression in addition to performing the tests of non-zero slopes. There are certain types on non-proportionality that will not be detected by the tests of non-zero slopes alone but that might become obvious when looking at the graphs of the residuals such as nonlinear relationship (i.e. a quadratic fit) between the residuals and the function of time or undue influence of outliers.

SPLUS First we create the coxph object by using the coxph function. To create the plots of the Schoenfeld residuals versus log(time) create a cox.zph object by applying the cox.zph function to the cox.ph object. Then the plot function will automatically create the Schoenfeld residual plots for each of the predictors in the model including a lowess smoothing curve. The order of the residuals in the time.dep.zph object corresponds to the order in which they were entered in the coxph model. To plot one graph at a time use the bracket notation with the number corresponding to the predictor of interest. The abline function adds a reference line at y=0 to the individual plots.

STATA The tests of the non-zero slope developed by Therneau and Grambsch for SPLUS have been implemented in STATA in the stphtest command. The algorithms that STATA uses are slightly different from the algorithms used by SPLUS and therefore the results from the two programs might differ slightly. The stphtest with the detail option will perform the tests of each predictor as well as a global test. There are different functions of time available including the identity function, the log of survival time and the rank of the survival times. The stphtest command with the plot option will provide the graphs with a lowess curve. The usual graphing options can be used to include a horizontal reference line at y=0. Unlike the graphs created in SPLUS the graphs in STATA do not include 95% confidence intervals for the lowess curves which makes it more difficult to assess how much the curves may deviate from the y=0 line.

## Strategies

Knowledge of direct variation, ratios, proportions and graphing linear equations are encouraged to ensure success on this exercise.

1. A direct variation, or proportional relationship, can be represented by />.
2. The constant of proportionality, or /> from the above equation, is the slope of the line.
3. The question asked is often about a rate or a value for a certain about.
4. The labeling of the axes can be used to assist when trying to determine if a statement is true or not.

## 7.2 Introducing Proportional Relationships

IM 6–8 Math was originally developed by Open Up Resources and authored by Illustrative Mathematics®, and is copyright 2017-2019 by Open Up Resources. It is licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0). OUR's 6–8 Math Curriculum is available at https://openupresources.org/math-curriculum/.

Adaptations and updates to IM 6–8 Math are copyright 2019 by Illustrative Mathematics, and are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).

Adaptations to add additional English language learner supports are copyright 2019 by Open Up Resources, and are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).

The second set of English assessments (marked as set "B") are copyright 2019 by Open Up Resources, and are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).

Spanish translation of the "B" assessments are copyright 2020 by Illustrative Mathematics, and are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0).

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## Constant of Proportionality Worksheets

The constant of proportionality is the ratio between two variables y and x. Interpret the constant of proportionality as the slope of the linear relationship y = kx. Find the proportional relationship between x and y values to solve this set of pdf worksheets that comprise graphs, equations, and tables. Students will also learn to find the missing values in tables based on the constant of proportionality k, so derived. These printable worksheets are specially designed for students of grade 7 and grade 8. Click on the 'Free' icon to sample our worksheets.

7th grade students should use the slope of each graph to identify the constant of proportionality, k. Then, find the proportional relationship between the x and y coordinates by applying the formula y = kx.

Based on the value k, draw a straight line on the graph that passes through the origin to denote the proportional relationship y = kx. Use our answer keys to validate your responses.

8th grade students should rewrite each equation in the form of y = kx, where 'k' represents the constant of proportion. There are ten problems in each pdf worksheet.

Examine the x and y values provided in each table to find the constant of proportionality, k. Then, replace the value of k in y = kx to obtain the proportional relationship between x and y.

Each printable worksheet contains eight function tables. Using the values of x and y, determine the constant of proportionality k. Based on the constant derived, complete the table.