standard_description: Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, Ö3) lies on the circle centered at the origin and containing the point (0, 2).
problem_activity: Activity
Previously, students worked with ratios in which one quantity (distance run) had the same value and the other (time elapsed) did not. In the context of running, they concluded that the runners did not run at the same rate. Here, students work with two ratios in which neither quantity (number of tickets bought and money paid) has the same value, and decide if the two people in the situation bought tickets at the same rate. Students may approach the task in several ways. They may use a double number line to generate ratios that are equivalent to
\(47:3\)
, representing the prices Diego would pay for different number of tickets. Once the price for 9 tickets is determined by scaling up the first ratio, they can compare it to the amount that Andre paid for the same number of tickets.
They may remember, without drawing number lines, that multiplying two values of a ratio by the same number produces a ratio that is equivalent. They may notice that multiplying 3 tickets by 3 results in 9 tickets, making the values for one quantity in the ratios match. They can then compare the two prices for 9 tickets. Alternatively, they may divide Andre’s 9 tickets and
$
141 payment by 3 to get his price for 3 tickets.
Another approach is to calculate the price of 1 ticket, as students did in a previous lesson. To make this approach less attractive here, the numbers have been deliberately chosen so the price of a single ticket (
$
\(15\frac23\)
) is not a whole number.
As students work, monitor for those who reason correctly using the three approaches described above.
Digital
Launch
Ask students what comes to mind when they hear the term “at the
same rate
”? Ask if they can think of any examples of situations that happen at the same rate. An example is two things traveling at the same speed. Any distance traveled will have the same associated time for things traveling at the same rate. Remind students that when situations happen “at the same rate,” they can be described by ratios that are equivalent.
Give students quiet think time to complete the activity and then time to share their explanation with a partner.
If students have digital access, they can use an applet to explore the problem and justify their reasoning before sharing with a partner. If students have not used the number line applet in previous activities or need a refresher as to how to use it, demonstrate the treadmill problem with the applet.
Representation: Internalize Comprehension.
Activate background knowledge about finding ratios that are equivalent. Allow students to use calculators to ensure inclusive participation in the activity.
Supports accessibility for: Memory; Conceptual processing
Reading: MLR6 Three Reads.
This is the first time Math Language Routine 6 is suggested as a support in this course. In this routine, students are supported in reading a mathematical text, situation, or word problem three times, each with a particular focus. During the first read, students focus on comprehending the situation; during the second read, students identify quantities; during the third read, students brainstorm possible strategies to answer the question. The question to be answered does not become a focus until the third read so that students can make sense of the whole context before rushing to a solution. The purpose of this routine is to support students’ reading comprehension as they make sense of mathematical situations and information through conversation.
Design Principle(s): Support sense-making
How It Happens:
In the first read, students read the problem with the goal of comprehending the situation.
Invite a student to read the problem aloud while everyone else reads with them and then ask, “What is this situation about?”
Allow one minute to discuss with a partner, and then share with the whole class. A clear response would be: “Diego and Andre both bought tickets to a concert.”
In the second read, students analyze the mathematical structure of the story by naming quantities.
Invite students to read the problem aloud with their partner or select a different student to read to the class and then prompt students by asking, “What can be counted or measured in this situation? For now we don’t need to focus on how many or how much of anything, but what can we count in this situation?” Give students one minute of quiet think time followed by another minute to share with their partner. Quantities may include: the number of tickets Diego bought, the number of tickets Andre bought, the amount Diego paid, the amount Andre paid, the rate of dollars per ticket that Diego paid, the rate of dollars per ticket that Andre paid.
Call attention to the fact that whether we are talking about Andre or Diego, the two important quantities in this situation are: number of tickets, and amount paid in dollars.
In the third read, students brainstorm possible strategies to answer the question, “Did they pay at the same rate?”
Invite students to read the problem aloud with their partner or select a different student to read to the class. Instruct students to think of ways to approach the question without actually solving the problem.
Consider using these questions to prompt students: “How would you approach this question?,” “What strategy or method would you try first?,” and “Can you think of a different way to solve it?”
Give students one minute of quiet think time followed by another minute to discuss with their partner. Provide these sentence frames as partners discuss: “To compare the rates, I would use a double number line by….”, “One way to approach the question would be to….”
Sample responses include: “I would figure out how much each person paid for one ticket”, “I know that if I multiply 3 tickets by 3, I get 9, so I would see what happens when I multiply $47 by 3”, “I would draw a diagram to figure out how much 3 groups of three tickets would cost at Diego’s rate”, and “I would use a double number line to scale up the rate for Diego’s tickets (or scale down the rate for Andre’s tickets) to see if they are the same rate.” This will help students concentrate on making sense of the situation before rushing to a solution or method.
As partners are discussing their strategies, select 1–2 students to share their ideas with the whole class. As students are presenting ideas to the whole class, create a display that summarizes ideas about the question.
Listen for quantities that were mentioned during the second read, and take note of strategies for relating number of tickets to cost (amount paid).
Post the summary where all students can use it as a reference, and suggest that students consider using the Three Reads routine for the next activity as well.
Student Facing
Diego paid
$
47 for 3 tickets to a concert. Andre paid
$
141 for 9 tickets to a concert. Did they pay at the
same rate
? Explain your reasoning.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Print
Launch
Ask students what comes to mind when they hear the term “at the
same rate
”? Ask if they can think of any examples of situations that happen at the same rate. An example is two things traveling at the same speed. Any distance traveled will have the same associated time for things traveling at the same rate. Remind students that when situations happen “at the same rate,” they can be described by ratios that are equivalent.
Give students quiet think time to complete the activity and then time to share their explanation with a partner.
If students have digital access, they can use an applet to explore the problem and justify their reasoning before sharing with a partner. If students have not used the number line applet in previous activities or need a refresher as to how to use it, demonstrate the treadmill problem with the applet.
Representation: Internalize Comprehension.
Activate background knowledge about finding ratios that are equivalent. Allow students to use calculators to ensure inclusive participation in the activity.
Supports accessibility for: Memory; Conceptual processing
Reading: MLR6 Three Reads.
This is the first time Math Language Routine 6 is suggested as a support in this course. In this routine, students are supported in reading a mathematical text, situation, or word problem three times, each with a particular focus. During the first read, students focus on comprehending the situation; during the second read, students identify quantities; during the third read, students brainstorm possible strategies to answer the question. The question to be answered does not become a focus until the third read so that students can make sense of the whole context before rushing to a solution. The purpose of this routine is to support students’ reading comprehension as they make sense of mathematical situations and information through conversation.
Design Principle(s): Support sense-making
How It Happens:
In the first read, students read the problem with the goal of comprehending the situation.
Invite a student to read the problem aloud while everyone else reads with them and then ask, “What is this situation about?”
Allow one minute to discuss with a partner, and then share with the whole class. A clear response would be: “Diego and Andre both bought tickets to a concert.”
In the second read, students analyze the mathematical structure of the story by naming quantities.
Invite students to read the problem aloud with their partner or select a different student to read to the class and then prompt students by asking, “What can be counted or measured in this situation? For now we don’t need to focus on how many or how much of anything, but what can we count in this situation?” Give students one minute of quiet think time followed by another minute to share with their partner. Quantities may include: the number of tickets Diego bought, the number of tickets Andre bought, the amount Diego paid, the amount Andre paid, the rate of dollars per ticket that Diego paid, the rate of dollars per ticket that Andre paid.
Call attention to the fact that whether we are talking about Andre or Diego, the two important quantities in this situation are: number of tickets, and amount paid in dollars.
In the third read, students brainstorm possible strategies to answer the question, “Did they pay at the same rate?”
Invite students to read the problem aloud with their partner or select a different student to read to the class. Instruct students to think of ways to approach the question without actually solving the problem.
Consider using these questions to prompt students: “How would you approach this question?,” “What strategy or method would you try first?,” and “Can you think of a different way to solve it?”
Give students one minute of quiet think time followed by another minute to discuss with their partner. Provide these sentence frames as partners discuss: “To compare the rates, I would use a double number line by….”, “One way to approach the question would be to….”
Sample responses include: “I would figure out how much each person paid for one ticket”, “I know that if I multiply 3 tickets by 3, I get 9, so I would see what happens when I multiply $47 by 3”, “I would draw a diagram to figure out how much 3 groups of three tickets would cost at Diego’s rate”, and “I would use a double number line to scale up the rate for Diego’s tickets (or scale down the rate for Andre’s tickets) to see if they are the same rate.” This will help students concentrate on making sense of the situation before rushing to a solution or method.
As partners are discussing their strategies, select 1–2 students to share their ideas with the whole class. As students are presenting ideas to the whole class, create a display that summarizes ideas about the question.
Listen for quantities that were mentioned during the second read, and take note of strategies for relating number of tickets to cost (amount paid).
Post the summary where all students can use it as a reference, and suggest that students consider using the Three Reads routine for the next activity as well.
Student Facing
Diego paid
$
47 for 3 tickets to a concert. Andre paid
$
141 for 9 tickets to a concert. Did they pay at the
same rate
? Explain your reasoning.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Activity Synthesis
The main strategy to highlight here is one that could tell us what Diego would pay for 9 tickets if he paid at the same rate as he did for 3 tickets. For 9 tickets, he would have paid
$
141, which is what Andre paid for 9 tickets and which tells us that they paid at the same rate.
Select 2–3 students to present their work to the class in the order of their methods:
Correct use of a double number line to show that the given ratios are equivalent.
Correct use of multiplication (or division) without using a double number line.
(Optional) Correct use of unit price (i.e., by finding out
\(141\div9\)
and
\(47\div3\)
). Though it's a less efficient approach here, the outcome also shows that the two people paid at the same rate.
Recap that using equivalent ratios to make one of the corresponding quantities the same can help us compare the other quantity and tell whether the situations involve the same rate.


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