3.1: Observing a Drone (5 minutes)
In this warm-up, students are prompted to compare function values. To do so, they need to interpret statements in function notation and connect their interpretations to the graph of the function.
Previously, students recognized that if a point with coordinates \((2,20)\) is on the graph of a function, the 2 is an input and the 20 is a corresponding output. Here, they begin to see the second value in a coordinate pair more abstractly, as \(f(2)\) when the input is 2, or as \(f(4)\) when the input is 4.
Give students 1–2 minutes of quiet think time. Follow with a whole-class discussion.
Some students might be unfamiliar with drones. If needed, give a brief explanation of what they are.
Here is a graph that represents function \(f\), which gives the height of a drone, in meters, \(t\) seconds after it leaves the ground.
Decide which function value is greater.
- \(f(0)\) or \(f(4)\)
- \(f(2)\) or \(f(5)\)
- \(f(3)\) or \(f(7)\)
- \(f(t)\) or \(f(t+1)\)
Invite students to share how they compared each pair of outputs. If not mentioned in students’ explanations, point out that for each pair, what we are comparing are the vertical values of the points on the graph at different horizontal values. Even though the statements don’t tell us the values of, say \(f(3)\) and \(f(7)\), and the vertical axis shows no scale, we can tell from the graph that the function has a greater value when \(t\) is 3 than when \(t\) is 7.
Tell students that the coordinates \((0, f(0))\) represent the starting point of the drone. Ask students, “What are the coordinates of the points when the drone starts leveling off? When it starts to descend? When it lands?” (\((2, f(2)), (5,f(5))\), and \((7,f(7))\), respectively.)
Highlight that the coordinates of each point on a graph of a function are \((x, f(x))\).
Discuss with students how they compared \(f(t)\) and \(f(t+1)\) (in the last question). Make sure students see that the \(f(t+1)\) could be less than, equal to, or greater than \(f(t)\), depending on the value of \(t\).
- If \(t\) is 2, then \(t+1\) is 3. From the graph, we can see that \(f(2)\) is equal to \(f(3)\) because the graph has the same vertical value when \(t\) is 2 and 3.
- If \(t\) is 5, then \(t+1\) is 6. From the graph, we can see that \(f(5)\) is greater than \(f(6)\) because the graph has a greater vertical value when \(t\) is 5 than when \(t\) is 6.
3.2: Smartphones (20 minutes)
In this activity, students continue to interpret statements in function notation in terms of a situation. Several things are new here, all of which provide opportunities to attend to precision (MP6) and to reason quantitatively and abstractly (MP2).
- The output of the function is measured in millions, so students need to attend carefully to the units or otherwise may misinterpret the situation (for example, saying that about 2 thousand people owned smartphones in 2017 rather than about 2 billion people).
- The input of the function is measured in “years after 2000,” so a positive input value \(t\) needs to be interpreted as year \(2000+t\), and the year 2010 to be associated with \(t=10\).
- One of the input values is negative, so students will need to reason about what this means in this situation.
Students also consolidate various pieces of information about the function, given in descriptions and function notation, in order to sketch the graph of a function. It is possible for students to sketch many different graphs through the given points, but the context suggests that, over time, the function values are increasing roughly exponentially (though students do not need to know that term at this point).
Read the opening sentence in the activity statement as a class. Ask students to identify the input and output of this function, and the units in which each variable is measured.
Arrange students in groups of 2. Give students 1–2 minutes of quiet time to read and make sense of the first two questions (without writing responses). Urge them to think about what the input and output values are in each statement (given in function notation or in words).
Give students another few minutes to share their thinking with their partner, and then, when reaching an agreement, to write their responses. Insist that they write their interpretations for statements such as \(P(17)=2,\!320\) in complete sentences and use the quantity names and units. Ask students to pause for a class discussion before they continue with the rest of the activity.
Invite students to share their responses to the first two questions. Before students complete the remaining questions, make sure they see why:
- The number of smartphone owners in the first question is not a couple of thousand people. The output is measured in millions, so a number such as 2,320 means 2,320 million or 2.32 billion people.
- The input value for year 2010 is not 2010. The input is measured in “years after year 2000,” so the input for year 2010 is 10. An input of 2010 would mean year 4010!
Supports accessibility for: Language; Social-emotional skills
The function \(P\) gives the number of people, in millions, who own a smartphone, \(t\) years after year 2000.
What does each equation tell us about smartphone ownership?
Use function notation to represent each statement.
- In 2010, the number of people who owned a smartphone was 296,600,000.
- In 2015, about 1.86 billion people owned a smartphone.
Mai is curious about the value of \(t\) in \(P(t) = 1,\!000\).
- What would the value of \(t\) tell Mai about the situation?
- Is 4 a possible value of \(t\) here?
- Use the information you have so far to sketch a graph of the function.
Are you ready for more?
What can you say about the value or values of \(t\) when \(P(t) = 1,\!000\)?
Some students may struggle to express a number like 296,600,000 in millions, or they make think that a quantity like “2,320 million” doesn’t quite make sense. Ask them to write these quantities as numerals: 1 million, 10 million, 100 million, and so on, and then use their list to help figure out how to say 296,600,000 in millions or to write 2,320 million as a numeral.
Students may think they do not have enough information to sketch a graph of \(P\). Encourage them to read through the activity to identify points that must be part of the graph of \(P\), but not written in \((x,y)\) form.
Select students to share their responses and sketches of the graph of \(P\).
Make sure students can interpret a statement such as \(P(15) = 1,\!860\) and \(P(t)=1,\!000\) in terms of the situation and articulate it completely. For instance, students might say, “The output is 1,860 when the input is 15,” or “When the input is \(y\) years, the output is 1,000 million,” or some other variation that doesn’t convey the quantities fully. Push them to refine their interpretation so that it is clear that \(P(15) = 1,\!860\) means “1.86 billion people owned a smartphone in the year 2015” and \(P(t)=1,\!000\) means “A billion people owned a smartphone \(t\) years after 2000.”
When discussing possible graphs of \(P\), acknowledge that a graph of \(P\) could be drawn in various ways. The information we have is limited to the four input-output pairs, so what happens between the points is up for interpretation. But it would make sense, based on the context, for the graph to show very little change before year 2010 and then a rapid increase afterward.
If possible, draw students’ attention to the idea that modeling a relationship with a function involves making choices about the units used. Discuss questions such as:
- “The output of \(P\) is defined in terms of ‘millions of people’ instead of individual persons. What might be a reason for this?” (If the unit is “persons,” the scale would show very long numbers with 9 or 10 digits each, which would be much harder to read and might lead to mistakes. Using “in thousands” or “in millions” strategically makes it possible to use simpler numbers. It makes it easier to draw attention to important characteristics of a graph.)
- “The input is defined in ‘years after 2000.’ Could we have instead used calendar years, such as 2002?” (Yes)
- “What might be a reason to choose ‘years after 2000’ as the unit?” (It allows us to write smaller numbers. Another reason is that sometimes knowing a duration is more useful than knowing a particular point in time at which something happens. For instance, in the drone activity, knowing the number of seconds that had passed before the drone landed was more useful than knowing that it landed at, say, 2:45 p.m. In this activity, the year 2000 might be significant in the development of smartphones, so measuring time with that starting point might be useful.)
3.3: Boiling Water (10 minutes)
Previously, students learned that each point on the graph of a function \(f\) is of the form \((t, f(t))\) for input \(t\) and corresponding output \(f(t)\). They analyzed and plotted input-output pairs in which both values were known.
In this activity, students reason about unknown output values by relating them to values that are known (by interpreting inequalities such as \(W(5) > W(2)\)), using a graph and a context to support their reasoning. The work here prompts students to reason quantitatively and abstractly (MP2).
Students take turns explaining their interpretation of statements in function notation and making sense of their partner’s interpretation, discussing their differences if they disagree. In so doing, students practice constructing logical arguments and critiquing the reasoning of others (MP3).
Identify students who can correctly interpret statements such as \(W(15) > W(30)\) in terms of the situation and in relation to the graph of the function. Also, look for partners whose graphs are very different but are both correct. Invite them to share their responses during the whole-class discussion.
Keep students in groups of 2. Ask students to take turns explaining to their partner the meaning of each statement in the first question. The partner’s job is to listen and make sure they agree with the interpretation. If they don’t agree, the partners discuss until they come to an agreement. Based on their shared interpretation of the statements, partners then sketch their own graph of the function.
Supports accessibility for: Visual-spatial processing
The function \(W\) gives the temperature, in degrees Fahrenheit, of a pot of water on a stove, \(t\) minutes after the stove is turned on.
Take turns with your partner to explain the meaning of each statement in this situation. When it’s your partner’s turn, listen carefully to their interpretation. If you disagree, discuss your thinking and work to reach an agreement.
- \(W(0) = 72\)
- \(W(5) > W(2)\)
- \(W(10) = 212\)
- \(W(12) = W(10)\)
- \(W(15) > W(30)\)
- \(W(0) < W(30)\)
If all statements in the previous question represent the situation, sketch a possible graph of function \(W\).
Be prepared to show where each statement can be seen on your graph.
Some students may think there is not enough information to accurately graph the function. Assure them that this is true, but clarify that we are not after the graph, but rather a possible graph of the function based on the information we do have.
Select previously identified students to share their interpretations of the inequalities in the first question and to show their graphs. Ask them to explain how each statement is evident in their graph. Discuss questions such as:
- “Why might it be true that \(W(15) > W(30)\)?” (The heat was turned off at or after 15 minutes, or the kettle was taken off the stove.)
- “You and your partner agreed on what each statement meant. Are your graphs identical? If not, why might that be?” (Function \(W\) was not not completely defined. We have information about the temperature at some points in time and how they compare, but we don’t have all the information about all points in time.)
Design Principle(s): Support sense-making
To highlight the importance of attending to units of measurement when working with statements in function notation, display for all to see the following description and list of statements.
At noon, a news organization published a piece of breaking news on its website. The function \(Q\) gives the number of visitors, in thousands, to the website \(h\) hours after the news was published. By 2:30 p.m., the news article had been viewed by 1.6 million visitors.
\(Q(2\!:\!30) = 1.6\)
\(Q(2.5) = 1.6\)
\(Q(2.5) = 1,\!600\)
\(Q(2\!:\!30) = 1,\!600,\!000\)
\(Q(150) = 1,\!600\)
\(Q(2.5) = 1,\!600,\!000\)
Give students a moment of quiet think time to decide which statement in function notation correctly represents the visitor data at 2:30 p.m. Ask them to be prepared to explain why they believe all the other statements that are not selected do not accurately represent the situation.
Make sure students understand why \(Q(2.5)=1,\!600\) is the only one that represents the quantities in the situation.
3.4: Cool-down - Visitors in a Museum (5 minutes)
Cool-downs for this lesson are available at one of our IM Certified Partners
Student Lesson Summary
What does a statement like \(p(3)=12\) mean?
On its own, \(p(3)=12\) only tells us that when \(p\) takes 3 as its input, its output is 12.
If we know what quantities the input and output represent, however, we can learn much more about the situation that the function represents.
If function \(p\) gives the perimeter of a square whose side length is \(x\) and both measurements are in inches, then we can interpret \(p(3)=12\) to mean “a square whose side length is 3 inches has a perimeter of 12 inches.”
We can also interpret statements like \(p(x)=32\) to mean “a square with side length \(x\) has a perimeter of 32 inches,” which then allows us to reason that \(x\) must be 8 inches and to write \(p(8)=32\).
If function \(p\) gives the number of blog subscribers, in thousands, \(x\) months after a blogger started publishing online, then \(p(3)=12\) means “3 months after a blogger started publishing online, the blog has 12,000 subscribers.”
It is important to pay attention to the units of measurement when analyzing a function. Otherwise, we might mistake what is happening in the situation. If we miss that \(p(x)\) is measured in thousands, we might misinterpret \(p(x)=36\) to mean “there are 36 blog subscribers after \(x\) months,” while it actually means “there are 36,000 subscribers after \(x\) months.”
A graph of a function can likewise help us interpret statements in function notation.
Function \(f\) gives the depth, in inches, of water in a tub as a function of time, \(t\), in minutes, since the tub started being drained.
Here is a graph of \(f\).
Each point on the graph has the coordinates \((t, f(t))\), where the first value is the input of the function and the second value is the output.
\(f(2)\) represents the depth of water 2 minutes after the tub started being drained. The graph passes through \((2,5)\), so the depth of water is 5 inches when \(t= 2\). The equation \(f(2)=5\) captures this information.
\(f(0)\) gives the depth of the water when the draining began, when \(t=0\). The graph shows the depth of water to be 6 inches at that time, so we can write \(f(0)=6\).
\(f(t)= 3\) tells us that \(t\) minutes after the tub started draining, the depth of the water is 3 inches. The graph shows that this happens when when \(t\) is 6.