standard_description: Prove the Pythagorean identity sin^(2)(θ) + cos^(2)(θ) = 1 and use it to find sin(θ), cos(θ),or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.
problem_activity: Optional activity
This optional activity gives students an additional opportunity to apply what they know about key characteristics of an exponential function and use it to model real-world data. Unlike previous activities, the data have not been adjusted to perfectly fit an exponential model. Students will need to make estimates and think strategically about what model to choose. Because the years chosen for data points are close together, students may use repeated multiplication to answer the questions rather than writing and evaluating an expression that models the exponential function.
The activity also allows students to practice articulating their reasoning process. The info gap structure requires students to determine what information is necessary to solve a problem and then to ask for the information. This may take several rounds of discussion, as their first requests may not yield the necessary information (MP1). It also allows students to refine the language they use and ask increasingly more precise questions until they get the information they need (MP6).
Here is the text of the cards for reference and planning:
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In Problem Card 1, answers will vary based on the data points and the strategy students use to find the growth factor. Strategies to watch for include:
Estimate that the phone sales are roughly doubling each year and use that to estimate future sales
Calculate a growth factor using the data from two consecutive years
Find the average of the growth factors between 2008 and 2009 and between 2009 and 2010
For the second problem, the third approach will not arise because only two sales numbers are available on the data card.
Launch
Tell students they will continue to work with interpreting exponential relationships. Ask students “How many of you have a smartphone?” Give students a brief overview on the first appearance of smartphones and when they became widely popular (around 2007). Explain that annual records of smartphone sales are available starting from 2008 until now.
Explain the info gap structure, and consider demonstrating the protocol if students are unfamiliar with it.
Arrange students in groups of 2. In each group, distribute a problem card to one student and a data card to the other student. After reviewing students' work on the first problem, give them the cards for a second problem and instruct partners to switch roles.
Conversing:
This activity uses
MLR4 Information Gap
to give students a purpose for discussing information necessary to solve problems involving equivalent ratios. Consider providing questions or question starters for students who need a starting point such as: “Can you tell me . . .”, and “Why do you need to know . . .?”
Design Principle(s): Cultivate Conversation
Engagement: Develop Effort and Persistence.
Display or provide students with a physical copy of the written directions. Check for understanding by inviting students to rephrase directions in their own words. Keep the display of directions visible throughout the activity.
Supports accessibility for: Memory; Organization
Student Facing
Your teacher will give you either a problem card or a data card. Do not show or read your card to your partner.
If your teacher gives you the data card:
Silently read the information on your card.
Ask your partner “What specific information do you need?” and wait for your partner to ask for information. Only give information that is on your card. (Do not figure out anything for your partner!)
Before telling your partner the information, ask “Why do you need to know (that piece of information)?”
Read the problem card, and solve the problem independently.
Share the data card, and discuss your reasoning.
If your teacher gives you the problem card:
Silently read your card and think about what information you need to answer the question.
Ask your partner for the specific information that you need.
Explain to your partner how you are using the information to solve the problem.
When you have enough information, share the problem card with your partner, and solve the problem independently.
Read the data card, and discuss your reasoning.
Pause here so your teacher can review your work. Ask your teacher for a new set of cards and repeat the activity, trading roles with your partner.
Student Response
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Activity Synthesis
Invite students to share different ways they estimated a growth factor for smartphone sales, including:
using a rough estimate of 2
using sales data from two consecutive years (2008 and 2009, or 2009 and 2010)
using sales data from all three years and taking an average of growth factors for consecutive years
The first of these methods is efficient and allows for quick calculations. It is not as accurate, however, as the other estimates and may lead to a less reliable prediction. In this case, it leads to an overestimate of phone sales since the actual growth factors are less than 2.
Consider showing a model and a graph generated using a calculator or graphing technology. Based on this model, about 125 million of that brand of smartphones were sold in 2012.
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Next, focus the whole-class discussion on the limitations of a real-world model. In this context, market saturation limits the growth of the cell phone sales. Discuss questions such as:
“According to your model, how many smartphones were sold worldwide in 2018?” (About 5 billion.)
“The world population in 2018 is about 7.6 billion. Is the number of smartphones sold based on your model realistic? Why or why not?” (No.  Allowing for children, the elderly, and underdeveloped countries, 66% of the population buying one brand of smartphone is not realistic.)
“Could the sales of smartphones continue to grow exponentially? Why or why not?” (No. There is a finite number of people who will buy a particular brand of smartphone. The data should level off, with surges when new models come out.)


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