6.1: Notice and Wonder: The Draining Tank (5 minutes)
This warm-up prompts students to make sense of a problem before solving it by familiarizing themselves with a context and the mathematics that might be involved (MP1). In the next activity, they will hear more details about the situation, and be asked specific questions about it. When students articulate what they notice and wonder, they have an opportunity to attend to precision in the language they use (MP6). They might first propose less formal or imprecise language, and then restate their observation with more precise language in order to communicate more clearly. For example, for this prompt, they might use words like full, empty, volume, time, minutes or seconds, and rate of change.
Display the prompt for all to see. Give students 1 minute of quiet think time and ask them to be prepared to share at least one thing they notice and one thing they wonder. Give students another minute to discuss their observations and questions.
A water tank is draining at a constant rate.
What do you notice? What do you wonder?
Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the task statement. After all responses have been recorded without commentary or editing, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to respectfully disagree, ask for clarification, or point out contradicting information.
6.2: Identifying Important Points (15 minutes)
In the associated Algebra 1 lesson, students write an equation to model the distance traveled by an object moving at a constant speed. They will also identify important points on a graph representing projectile motion, and determine a reasonable domain. In this preparatory activity, they write a linear function to model a situation involving constant rate of change, practice using graphing technology to extract the coordinates of points on the graph, and determine a reasonable domain for the function based on the situation it is modeling. It is intentional that the first few entries in the table are difficult to determine using the graph—this is to encourage students to think about the information “drains at a constant rate of 2 gallons per minute.” This activity provides opportunities to attend to the meaning of quantities in the situation (MP2).
Ask students to read the stem and decide how they think the axes should be labeled, and share this with a partner. Invite a few students to share their ideas. Ensure that all students have the axes labeled correctly before proceeding with the rest of the activity.
Give students a few minutes to create the table and write a function. At that point, depending on students’ experience with graphing technology, it may be desirable to demonstrate how to set an appropriate graphing window and use the technology to extract the coordinates of the intercepts and other points on the graph.
A tank has 50 gallons of water and drains at a constant rate of 2 gallons per minute. Here is a graph representing the situation:
- Label each axis to show what it represents. Be sure to include units.
- Complete the table.
\(t\) 0 1 2 3 10 20 \(t\) \(v(t)\)
- Use the expression in terms of \(t\) from the table to write a function modeling this situation.
- Use graphing technology to graph your function. Practice setting the graphing window so that you can see both intercepts, and using graphing technology to see the coordinates of different points on your graph.
- What is a reasonable domain for this function, based on the situation it models?
Possible questions for discussion:
- “Why might it be useful to know the coordinates of intercepts of a graph that models a situation?” (The intercepts tell you the value of one quantity when the other quantity is 0. In this case, that means the volume of water in the tank when it starts draining (at 0 minutes), and how many minutes it takes the tank to empty (the time when the volume is at 0 gallons).)
- “What are some important things to keep in mind when setting a graphing window?” (You want to make sure you can see any important points on the graph, which often includes the intercepts, though it depends on the situation. Other responses might depend on the type of graphing technology used.)
6.3: Three Situations (25 minutes)
This activity is an opportunity to practice using graphing technology to determine important points on a graph and to practice writing a function to represent a situation described verbally. Students can choose to find the coordinates of the intercepts either by using the technological tool or by reasoning about the definition of the function. For example, on the graph of function \(d\), they can either use technology to find the \(y\)-coordinate when \(x\) is 0, or they can evaluate \(81 \boldcdot 3^0\).
Note that in function \(b\), the \(x\)-intercept is a very small negative number. In a few cases, students will encounter a small, negative \(x\)-intercept in the projectile motion lessons. Although intercepts like this aren’t generally meaningful in the context, they are mentioned a bit in the associated Algebra 1 lessons. So that’s the reason why such a function was included here.
The second question asks students to find the coordinates of the vertex of the graph of the quadratic function \(d\). Depending on the specific graphing technology used, they may be able to figure this out on their own, or they may need explicit instruction on how to use the technology to find the coordinates of this point.
- Create a graph of each function using graphing technology. Make a rough sketch of each graph. On each graph, label the coordinates of any intercepts.
- Function \(d\) has a maximum point. Can you find the coordinates of this point?
- Here are some situations. For each situation:
- Write an equation representing the situation. If you get stuck, consider making a table of values, thinking about what type of function it is, or thinking about the initial value and rate of change or growth factor. Be sure to explain the meaning of any variables you use.
- Sketch a graph representing each situation. Label the coordinates of any intercepts or other important points.
- A person has $128 saved, and adds $4 to their savings per week.
- A tank has 128 gallons of water, and drains at a constant rate of 4 gallons per minute.
- A patient is given 128 milligrams of a medication, and half of the medication leaves the patient’s bloodstream every hour.
Display one or more students’ work on the last question for all to see. Point out examples where students made sure to name the variables they used in their equations. If needed, provide others time to add that to their work.
- “How are the graphs of the three situations alike? How are they different?” (They all have the same \(y\)-intercept. Two of the graphs are lines and the third is the graph of an exponential function.)
- “How are the equations you wrote alike? How are they different?” (The equations all have a 128. The two linear equations either add \(4t\) or subtract \(4t\) from 128. The exponential equation has a growth factor of \(\frac12\) and the variable is the exponent.)
- “Which graphs have \(x\)-intercepts? Which graphs have a \(y\)-intercept?” (The graph of \(y=128(\frac12)^x\) does not have an \(x\)-intercept. All the graphs have the same \(y\)-intercept, \((0,128)\).)
- “What does the \(y\)-intercept of each graph tell you about the situation?” (It is the initial amount.)
- “What does the \(x\)-intercept (if there is one) tell you about the situation?” (For the graph of \(y=128-4x\), the \(x\)-intercept tells when the tank is empty. For the savings account, the \(x\)-intercept is not meaningful because it would have a negative \(x\)-coordinate and negative values for time don’t make sense in this situation.)