Lesson 1
Matching up to Data
1.1: Notice and Wonder: Cooling Down (5 minutes)
Warm-up
The purpose of this warm-up is to allow students to consider a graph of data and the units in preparation for informally fitting functions to sets of data, which is a focus of the lesson. 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).
The specific data shown here is used in the following activity, where students decide which of two given functions best fits the data, and in a future lesson, where students transform a given equation so that the corresponding graph fits the data.
While students may notice and wonder many things about the graph, the shape of the data in the context is the important discussion point, making use of the structure of the graph and relating it to the graphs of functions they have seen in previous lessons (MP7).
Launch
Display the graph for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice and wonder with their partner, followed by a whole-class discussion.
Student Facing
What do you notice? What do you wonder?
Student Response
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Activity Synthesis
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 graph. After all responses have been recorded without commentary or editing, ask students, “Is there anything on this list that you are wondering about?” Encourage students to respectfully disagree, ask for clarification, or point out contradicting information.
If the general shape of the data or a possible situation that the data represents does not come up during the conversation, ask students to discuss these ideas.
1.2: Which Function? (15 minutes)
Activity
Building on the work in the warm-up, the purpose of this activity is for students to determine which of the given functions is a better fit for the data. For this activity, students should use any language that makes sense to them to describe why a function is or is not a good fit and how they would change the function to be a better fit for the data. Since both functions offer reasonably good fits for the data, students have the opportunity to make an argument about why they think a particular function is a better fit (MP3).
Monitor for students using different explanations of what makes a function fit the data, such as by focusing on the general shape, the accuracy for individual points, or the average error for all the points.
This activity works best when each student has access to devices that can run the Desmos applet because students will benefit from seeing the relationship in a dynamic way. If students don't have individual access, projecting the applet will be helpful during the synthesis.
Launch
Design Principle(s): Support sense-making; Optimize output (for explanation)
Student Facing
A bottle of soda water is left outside on a cold day. The scatter plot shows the temperature \(T\), in degrees Fahrenheit, of the bottle \(h\) hours after it was left outside. Here are 2 functions you can use to model the temperature as a function of time:
\(f(h) = 45 + \frac{20}{h+0.5}\)
\(g(h) = 45 + 33(0.5)^{h+0.5}\)
- Which function better fits the shape of the data? Explain your reasoning.
- Use the applet to zoom out on the graphs. Does this change your opinion about which function fits better?
- Where do you see the 45 in the expression for each function on the graph?
- For the function you thought didn’t fit the shape of the data as well, how would you change it to fit better?
Student Response
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Launch
Design Principle(s): Support sense-making; Optimize output (for explanation)
Student Facing
A bottle of soda water is left outside on a cold day. The scatter plot shows the temperature \(T\), in degrees Fahrenheit, of the bottle \(h\) hours after it was left outside. Here are 2 functions you can use to model the temperature as a function of time:
\(f(h) = 45 + \frac{20}{h+0.5}\)
\(g(h) = 45 + 33(0.5)^{h+0.5}\)
- Which function better fits the shape of the data? Explain your reasoning.
- Where do you see the 45 in the expression for each function on the graph?
- For the function you thought didn’t fit the shape of the data as well, how would you change it to fit better?
Student Response
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Student Facing
Are you ready for more?
Consider the function \(a\) given by \(a(h) = \frac{2}{3}(h-6)^2 + 46\).
- Explain how the equation defining \(a\) is related to the temperature data.
- How well does \(a\) model the data compared to \(f\) or \(g\)? Explain your reasoning.
Student Response
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Anticipated Misconceptions
In order to show the temperature trend better, the first tick mark on the temperature axis represents 45 degrees, even though each successive tick mark only represents an additional 5 degrees. If students are confused that the first tick mark does not represent 5 degrees, remind them that since the range of this function does not include any numbers less than 45, it is convenient to start the range values at 45.
Activity Synthesis
| \(h\) (hours) | \(T\) (°F) |
|---|---|
| 0.03 | 69 |
| 0.12 | 67.8 |
| 0.22 | 67.4 |
| 0.3 | 66.3 |
| 0.93 | 59.9 |
| 1.02 | 59.1 |
| 1.28 | 57.5 |
| 1.55 | 57.3 |
| 2.17 | 55 |
| 2.77 | 52.3 |
| 5.7 | 47.1 |
Use graphing technology, such as the applet in the digital version of this activity, to project the data given here along with the two functions \(f(h) = 45+\frac{20}{h+0.5}\) and \(g(h) = 45 + (0.5)^{h+0.5}\).
Invite previously identified students to share which function they think fits better and why. Since there is no single correct answer, attend to students’ explanations and ensure the reasons given are correct. Ask 2–3 students for ideas on how they would adjust either \(f\) or \(g\) to be a better fit.
Conclude the discussion by showing how the graphs of \(f\) and \(g\) change when the 45 is removed from the equation. If students called the 45 the vertical intercept, note that this is true for some equations, such as the \(b\) in \(y=mx+b\), but the constant term is not always the vertical intercept, as shown by the equations for \(f\) and \(g\). Tell students that a goal of this unit is to understand how to transform the graphs of functions in different ways and what different transformations mean for the corresponding expressions.
1.3: What Happened to the Graph? (15 minutes)
Activity
In this partner activity, students take turns describing transformations of a graph and sketching the transformed graph from the description. As students trade roles explaining their thinking and listening, they have opportunities to refine and use more precise language when describing transformations (MP6).
Encourage students to attend to details such as direction, distance, and shape. While students work, record words you notice students using in their descriptions, such as vertex, intercept, or maximum, for all to see to provide ideas for other students and to reference during the whole-class discussion.
Launch
Arrange students in groups of 2. Tell students that they are going to take turns. One partner will describe the transformation of graph a to graph b that they see on their handout, the other will draw the transformation based on the description. Each partner will draw 3 graphs and describe 3 transformations.
Ask students to be specific in their descriptions, but note that the goal is for their partner to draw the transformation correctly without needing to name specific points.
Distribute 2 half sheets to each group from the blackline master, 1 to each student. Remind students to keep their sheet hidden from their partner.
Design Principle(s): Maximize meta-awareness; Support sense-making
Supports accessibility for: Conceptual processing; Organization
Student Facing
Your teacher will give you a card. Take turns describing the transformation of the graph on your card for your partner to draw and drawing the transformed graph from your partner's description.
1. a.
1. b.
2. a.
2. b.
3. a.
3. b.
4. a.
4. b.
5. a.
5. b.
6. a.
6. b.
Student Response
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Anticipated Misconceptions
Some students may describe the transformations without enough detail, making it difficult for their partner to sketch the correct transformation. Emphasize that the goal is for their partner to draw the transformed graph with precision so that it matches what they see exactly.
Activity Synthesis
The purpose of this discussion is for students to describe the transformations they saw when graphing. Encourage students to use precise language such as translate, reflect, and stretch. Students will continue to refine their language around graphical transformations throughout the unit, so it is okay for students to use more informal language at this time.
Begin the discussion by inviting students to share what types of transformations they saw, displaying the graphs for all to see to help illustrate student descriptions and connecting back to the list of words recorded during the activity. Connect any words students used back to geometry vocabulary (translate and reflect). Ask, “Are any of these transformations dilations?” (No, they are only stretching in one direction.)
Lesson Synthesis
Lesson Synthesis
Display the three graphs. Invite students to describe how to transform each graph to fit the points.
Here are some questions for discussion:
- “Which graph fits the data the best?” (All three can fit perfectly, for example, by translating the parabola up 2 and right 1.)
- “Imagine a situation where each graph—linear, quadratic, and exponential—would be the best choice to model the relationship.” (Any explanation works so long as the rate of change in the scenario matches the graph chosen. For example: The exponential would be best if the situation was population growth over time. The line would be best if the situation was price per pound.)
- "When we fit a curve to data points, will the curve always go precisely through all of the points?" (Not always, but the shape of the curve should fit the general shape of the data.)
1.4: Cool-down - Translating Two Ways (5 minutes)
Cool-Down
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Student Lesson Summary
Student Facing
The data in the graph show the temperature \(T\), in degrees Fahrenheit, of a can of soda \(h\) hours after it was put into the refrigerator.
What if we want to build a function that fits this data set? One way to find a function that fits the data well is to start with a simpler function that has the same general shape as the data when graphed and transform it. What shape does this data form?
Let’s try an exponential decay function. We can get the right shape using a simpler equation like \(T=(0.7)^h\), but the graph doesn't fit where the data is. The graph of the function given by \(T = 36+ 45(0.7)^h\) isn't represented by a simple equation, but it does fit the data. (What did multiplying by 45 and adding 36 do to the graph?) In this unit we will learn how to translate, reflect, and stretch graphs to fit data.
