few_shots: Example:
Problem/activity:
Find the general solution for:
$${\mathrm{cos}\left({\pi\over2}+x\right)=\mathrm{tan}{\pi\over4}}$$


Mathematical concept/skill options: 
A. Prove and apply trigonometric identities.
B. Model periodic phenomena with trigonometric functions.
C. Interpret functions that arise in applications in terms of the context.
D. Build a function that models a relationship between two quantities.
E. Define, evaluate, and compare functions.
F. Use functions to model relationships between quantities.
G. Understand the concept of a function and use function notation.
H. Analyze functions using different representations.
I. Extend the domain of trigonometric functions using the unit circle.
J. Interpret expressions for functions in terms of the situation they model.
K. Build new functions from existing functions.
L. Construct and compare linear and exponential models and solve problems.


Your response:
{
  "explanation": "This problem applies trigonometric identities, including the angle addition formula for cosine.",
  "answer": "A"
}

Now, assign the following problem or activity to one or more relevant mathematical concepts/skills it teaches.
shot_info: fl_problem_001894

HS
options: Summarize and describe distributions.

Summarize, represent, and interpret data on two categorical and quantitative variables.

Develop understanding of statistical variability.

Use the rules of probability to compute probabilities of compound events in a uniform probability model.

Interpret linear models.

Investigate patterns of association in bivariate data.

Understand independence and conditional probability and use them to interpret data.

Draw informal comparative inferences about two populations.

Understand and evaluate random processes underlying statistical experiments.

Summarize, represent, and interpret data on a single count or measurement variable.

Use random sampling to draw inferences about a population.

Investigate chance processes and develop, use, and evaluate probability models.

Make inferences and justify conclusions from sample surveys, experiments, and observational studies.

Calculate expected values and use them to solve problems.

Use probability to evaluate outcomes of decisions.
problem_activity: Activity
In this activity, students draw a bar graph and histogram, then describe the distributions shown on each display. Although the two visual displays may appear similar at first glance, there are important distinctions between the representations. Students notice differences in how we might characterize distributions in bar graphs and those in histograms, including how we describe typical values or categories. Along the way, students consolidate their understanding about categorical and numerical data.
Launch
Students will need the data on their travel methods and times, collected at the beginning of the unit. Distribute or display the data collected for these questions from the survey given earlier in the unit. Alternatively, complete the tables in the blackline master ahead of time.
Arrange students in groups of 2. Give one copy of the blackline master to each group of students. Display the data from the prior survey or the completed frequency tables for all to see or give a copy to each group of 2 students. Give students 5–6 minutes to complete the activity. Ask one partner to create a bar graph to represent the data on the class’s travel methods and the other to create a histogram to represent the data on travel times, and then answer the questions together.
Student Facing
Your teacher will provide you with some data that your class collected the other day.
Use the data to draw a histogram that shows your class’s travel times.
[IMG]
Describe the distribution of travel times. Comment on the center and spread of the data, as well as the shape and features.
Use the data on methods of travel to draw a bar graph. Include labels for the horizontal axis.
[IMG]
Describe what you learned about your class’s methods of transportation to school. Comment on any patterns you noticed.
Compare the histogram and the bar graph that you drew. How are they the same? How are they different?
Student Response
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for free access to Student Response.
Student Facing
Are you ready for more?
Use one of these suggestions (or make up your own). Research data to create a histogram. Then, describe the distribution.
Heights of 30 athletes from multiple sports
Heights of 30 athletes from the same sport
High temperatures for each day of the last month in a city you would like to visit
Prices for all the menu items at a local restaurant
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Activity Synthesis
The purpose of the discussion is for students to recognize the differences between histograms and bar graphs.
Select one student to show a completed histogram and another to show a completed bar graph. Then, solicit several observations about how the two graphical displays compare. Ask questions such as:
“How are the bar graphs and histograms alike? How are they different?”
“Can we use a bar graph to display the data on travel times? Why or why not?”
“Can we use a histogram to display the data on methods of travel? Why or why not?”
Next, select a few other students to share their descriptions of the distributions shown on each type of display. Then, ask questions such as:
“How are your descriptions of the distribution for travel methods different than those for travel times?”
“Can you talk about the shape of a distribution shown on a bar graph? Can you talk about its center and spread? Why or why not?”
Students should recognize that only the distribution of numerical data can be described in terms of shape, center, or spread. We cannot analyze these features for a distribution of a categorical data on a bar graph because a bar graph does
not
use a number line. This means the bars can be drawn anywhere, in any order, and with any kind of spacing, so shape, center, and spread would have no meaning.
Action and Expression: Internalize Executive Functions.
Provide students with a graphic organizer to record differences between histograms and bar graphs.  Consider creating a shared display and keeping it visible throughout the unit.
Supports accessibility for: Language; Organization
Representing, Conversing: MLR2 Collect and Display.
As students discuss the similarities and differences between histograms and bar graphs, record and display phrases you hear in a chart, such as a Venn diagram. Use this as a reference when highlighting or revoicing academic vocabulary from this unit (frequency, bins, numerical data, categorical data). This will help students connect unique and shared characteristics of histograms and bar graphs while reinforcing mathematical language.
Design Principle(s): Maximize meta-awareness


correct_option_index: 0

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