ThinkPairShare Activity
“Today 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.” Ask students to think about the meaning of the points, (0, 0), and (1, r), on the graph of a proportional relationship. Give students 2–3 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’ 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 realworld situations.
For each example, arrange students in groups of 2–3. 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’s pattern is below:
Diagram 1

Diagram 2

Diagram 3

Diagram …




…

 “What statement can you make about the number of squares in each diagram compared to the diagram number?” (The relationship is proportional.) “How may this relationship be represented in equation form?” (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:
 “What will the graph of the relationship look like? How do you know?”
 “Where will the graph cross the yaxis?”
 “What point will represent the unit rate, or constant of proportionality?”
 Show students a graph of the relationship after they have had time to contemplate the questions.
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 (−2, −6) indicate? Does interpretation of this point make sense in the context of this example? Explain.

The point (−2, −6) 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?

(4, 12)

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 xvalue of 5 corresponds to a yvalue 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 xvalue that corresponds to the yvalue of 21. That xvalue 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.
 “What may be stated regarding the relationship of her cumulative savings to the number of months that have passed?” (The relationship is proportional.)
 “How may this relationship be represented as an equation?” (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:
 “What will the graph of the relationship look like? How do you know?”
 “Where will the graph cross the yaxis?”
 “What point will represent the unit rate, or constant of proportionality?”
 Show students a graph of the relationship after they have had time to contemplate the questions.
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 $0. 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 (−1, −25) indicate? Does interpretation of this point make sense in the context of this example? Explain.

The point (−1, −25) 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?

(3, 75)

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 yvalue that corresponds to the xvalue 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 xvalue that corresponds to the yvalue of $200. That xvalue 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’s membership to a wholesale club costs $35 per year.
 “What can we say about the relationship of the cumulative yearly membership costs to the number of years that have passed?” (The relationship is proportional.)
 “How may this relationship be represented in equation form?” (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:
 “What will the graph of the relationship look like? How do you know?”
 “Where will it cross the yaxis?”
 “What point will represent the unit rate, or constant of proportionality?”
 Show students a graph of the relationship after they have had time to contemplate the questions.
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 $0. 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 (−1, −35) indicate? Does interpretation of this point make sense in the context of this example? Explain.

The point (−1, −35) 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?

(9, 315)

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 yvalue that corresponds to the xvalue 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 xvalue that corresponds to the yvalue of $280. That xvalue 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 xvalue 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?

(14, 280)

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 yvalue that corresponds to the xvalue 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 xvalue that corresponds to the yvalue of $100. That xvalue 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 xvalue 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?

(6, 24)

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 yvalue that corresponds to the xvalue 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 realworld relationship. The description should not include the unit rate. Students should ask a partner 3–4 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’s questions, students should switch roles.
The questions and answers may be uploaded as files to the class website, for review purposes.
Have students complete the Lesson 3 Exit Ticket (M733_Lesson 3 Exit Ticket and KEY.docx) at the close of the lesson to evaluate student level of understanding.
Extension:
 Routine: During the school year, have students identify the yintercept and unit rate when graphing linear equations that represent proportions. When proportional relationships are mentioned, ask them to describe the point at which the graph crosses the yaxis, as well as the point representing the unit rate.
 Small Groups: Students who need additional practice may by pulled into small groups to work on the Small Group Practice worksheet (M733_Small Group Practice and KEY.docx). Students can work on the problems together or work individually and compare answers when done.
 Expansion: Students who are prepared for a challenge beyond the requirements of the standard may be given the Expansion Work sheet (M733_Expansion Work and KEY.docx). The worksheet asks students to create and graph proportional relationship, interpret points, and consider the conceptual meaning of the unit rate.