few_shots: 
Example 1:
Problem/activity:
Find the general solution for:
$${\mathrm{cos}\left({\pi\over2}+x\right)=\mathrm{tan}{\pi\over4}}$$


Concept/skill: 
Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

Answer: 
Yes. An initial step when solving for x in this problem requires the use of the addition formula for cosine on the left side of the equation.

Example 2:
Problem/activity:
Task
Solve the quadratic equation
$$ x^2 = (2x - 9)^2 $$
using as many different methods as possible.


Concept/skill: 
Solve quadratic equations in one variable.

Answer: 
Yes. This task shows one quadratic equation with one variable, x.

Example 3:
Problem/activity:
Task
Suppose $A$ is an angle with vertex $P$, as pictured below:
[IMG]
Draw a circle with center $P$ and with  radius $r \gt 0$.  Explain why the circle
meets each ray of angle $A$ in a single point.  Label these points $Q$ and $R$ respectively.
Draw circles with centers $Q$ and $R$ respectively and radius $r$.  These circles meet at $P$ and a second point to be labelled $B$.
Show that ray $\overrightarrow{PB}$  bisects angle $P$.


Concept/skill: 
Construct a tangent line from a point outside a given circle to the circle.

Answer: 
No. This task does not involve drawing lines that are tangent to circles. The only line that may be drawn is the ray between points P and B.

Now, does the problem or activity shown below enable students to learn the full intent of the following concept or skill?

shot_info: fl_problem_001894_pos_0

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im_task_000870_pos_0

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im_task_001160_neg_0

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standard_description: Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant.
problem_activity: Activity
The mathematical goal of this activity is for students to collect and summarize data from an experiment. Students will measure their heart rate after doing a light exercise and use the difference from their resting heart rate as data to analyze in a later activity.
Launch
Arrange students in the 2 groups created at random in the activity
Get Ready to Experiment
. Tell students, “One group will count out loud while exercising and the other group will remain silent while exercising. Everyone will take their pulse silently after the exercising is finished.” For the exercise, consider asking the students to do 30 jumping-jacks or running in place for 1 minute. The intensity should be such that it raises student heart rates quickly without making them overly sweaty. Adjust the time or level of exercise to meet the needs of your class.
Action and Expression: Internalize Executive Functions.
Provide students with a template so they can fill in the data for their heart rate and calculations. For example, include a line for each item students should record (resting heart rate, group, exercise heart rate for 10 seconds, multiply by 6 for beats per minute, exercise heart rate minus resting heart rate).
Supports accessibility for: Language; Organization
Student Facing
Does counting while exercising affect your heart rate? Let’s perform an experiment to find out.
Your teacher will help divide you into two groups and lead the exercise. If you are selected to be in the group that will count during the exercise, count out loud together with your group while you do the exercise. If you are selected to be in the group that will remain silent during the exercise, remain quiet.
Immediately following the exercise, measure your heart rate again. Count the number of beats in ten seconds and multiply the result by 6 to get your heart rate in beats per minute. Record this result. Find the difference in heart rate by subtracting your heart rate after exercise from the resting heart rate you found during the warm-up.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Activity Synthesis
Collect the differences in heart rate from each student and keep them separated by groups. Display the results for all to see. These results will be used later in the lesson.
The goal of this discussion is for students to think about how to analyze the data from the two groups. Here are some questions for discussion:
“How could you graphically represent the data?” (You could make a histogram showing the distribution of the data from each group.)
“What are some ways you could analyze the data?” (You could calculate and compare the mean and standard deviation of each data set, or compare the shapes of the two histograms.)
“How would you find a randomization distribution for this data?” (You could record all of the values in a spreadsheet then choose half of them at random to be the counting group and the other half to be the silent group. Then you would find the means of those two groups and subtract them. You would repeat this many times.)


aligns: yes