standard_description: Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.
problem_activity: Activity
The purpose of this activity is for students to use the four operations on rational numbers solve a problem about water in a tank. The activity presents another example where negative time is used; this time to describe before a sensor starts working. Students examine the change as a separate column before using the starting point to model the draining of the tank.
Students who see that they are doing the same computations over and over and see that the structure of the expression is the same every time are expressing regularity in repeated reasoning (MP8). Monitor for different explanations for the last question.
Launch
Arrange students in groups of 2. Give students 4 minutes of quiet work time, followed by partner and whole-class discussion.
Action and Expression: Internalize Executive Functions.
Chunk this task into more manageable parts. For example, after students have completed the first 2-3 rows of the first table, check-in with either select groups of students or the whole class. Invite students to share how they have applied generalizations from previous lessons about using the four operations on rational numbers so far.
Supports accessibility for: Conceptual processing; Organization; Memory
Student Facing
A tank of water is being drained. Due to a problem, the sensor does not start working until some time into the draining process. The sensor starts its recording at time zero when there are 770 liters in the tank.
Given that the drain empties the tank at a constant rate of 14 liters per minute, complete the table:
| time after sensor                       starts (minutes) | change in                       water (liters) |         expression         | water in the                       tank (liters) |
|----------------------------------------------------------|------------------------------------------------|----------------------------|--------------------------------------------------|
|                            0                             |                       0                        |  \(770 + (0) (\text-14)\)  |                       770                        |
|                            1                             |                      -14                       | \(770 + (1)  (\text-14) \) |                       756                        |
|                            5                             |                      -70                       |                            |                                                  |
|                            10                            |                                                |                            |                                                  |
Later, someone wants to use the data to find out how long the tank had been draining before the sensor started. Complete this table:
| time after sensor                       starts (minutes) | change in                       water (liters) |            expression            | water in the                       tank (liters) |
|----------------------------------------------------------|------------------------------------------------|----------------------------------|--------------------------------------------------|
|                            1                             |                      -14                       |    \(770 + (1) (\text-14) \)     |                       756                        |
|                            0                             |                       0                        |    \(770 + (0)  (\text-14) \)    |                       770                        |
|                            -1                            |                       14                       | \(770 + (\text-1)  (\text-14) \) |                       784                        |
|                            -2                            |                       28                       |                                  |                                                  |
|                            -3                            |                                                |                                  |                                                  |
|                            -4                            |                                                |                                  |                                                  |
|                            -5                            |                                                |                                  |                                                  |
If the sensor started working 15 minutes into the tank draining, how much was in the tank to begin with?
Student Response
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Anticipated Misconceptions
The last question might be hard for students because they have had the table to support calculating the answers.  Ask students how can they use the previous entries in the table to help them calcualte the answer?  What is something they see changing in the expressions in the table that would change in this question?
Activity Synthesis
Make sure students have filled out the tables appropriately. Select students to share their reasoning for the last one.
Speaking, Representing: MLR7 Compare and Connect.
Use this routine when students share their strategies for completing the table. Ask students to consider what is the same and what is different about the structure of each expression. Draw students’ attention to the connection between representations (e.g., “Where do you see the change in water in your expression?”, "How are do repeated computations appear in the expression?"). These exchanges strengthen students’ mathematical language use and reasoning with different representations.
Design Principle(s): Support sense-making; Maximize meta-awareness


aligns: no