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


Concept/skill: 
Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.

Answer: 
No. This problem does not directly teach how side ratios in right triangles lead to definitions of trigonometric ratios. Instead, it involves solving a trigonometric function.

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: 
Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.

Answer: 
Yes. This task asks students to draw a ray that bisects an angle. The act of drawing circles and rays may involve compass and straightedge or dynamic geometric software.

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_neg_0

HS_0

im_task_000870_pos_0

HS_1

im_task_001160_pos_0

HS_2
standard_description: Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results.
problem_activity: Optional activity
To close the unit, students practice using the methods they have learned. Students begin by collecting data in small groups and then analyzing the data for the entire class. They create a scatter plot and two-way tables to look for patterns in the data (MP2). Although it is expected there is a very weak, if any, association in the data, the discussion asks students to think about what it would mean if there were a positive or negative association in context.
Launch
Arrange students in groups of 4. Select one student from each group to submit the data from the group when they finish recording their data. Prepare a table for all to see that is large enough to include the data from all of the students in the class.
Tell students that the same data is treated as both numerical and categorical data at different times within this activity. This is sometimes done when there is a threshold that makes sense for the context. In this case, it makes sense to look at data for which the second time is greater or less than the first time. Although the actual values may be lost when viewing the data as categorical, it can provide a different type of insight. In this example, we can more easily see if there was an improvement in reaction time for the second attempt.
Engagement: Internalize Self Regulation.
Provide a project checklist that chunks the various steps of the project into a set of manageable tasks.
Supports accessibility for: Organization; Attention
Representing, Conversing: MLR8 Discussion Supports.
Demonstrate the process for one round of the experiment, in which a group of students and staff play an example round while the rest of the class observes. This will help invite more student participation, conversations, and meta-awareness of language involving collecting and analyzing the data.
Design Principle(s): Support sense-making; Maximize meta-awareness.
Student Facing
In this activity, you’ll get two chances to guess at how long 30 seconds is, then look for an association between the two guesses of all students.
Work with a partner. Follow the instructions listed here to gather your data.
One of you will hold a stopwatch where the other person cannot see it.
The person holding the stopwatch says “go” and starts the timer.
The other person says “stop” when they think 30 seconds have passed.
The person holding the stopwatch will stop the timer, then report and record the time to the nearest second.
The person holding the stopwatch will give a second chance, repeating the experiment.
After
both
times are recorded, switch roles.
Record the group data in this table. When you finish, a group member should give the data to the teacher.
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "row"}, "_value": "name"}, {"_attributes": {"scope": "col"}, "_value": "time 1"}, {"_attributes": {"scope": "col"}, "_value": "time 2"}]}]}], "tbody": [{"tr": [{"th": [{"_attributes": {"scope": "row"}}], "td": [{}, {}]}, {"th": [{"_attributes": {"scope": "row"}}], "td": [{}, {}]}, {"th": [{"_attributes": {"scope": "row"}}], "td": [{}, {}]}, {"th": [{"_attributes": {"scope": "row"}}], "td": [{}, {}]}]}]}]}
Look at your data. Comparing Time 1 to Time 2, do you think there is a positive association, a negative association, or no association? Discuss your thinking with your group.
What are some ways you could organize and represent the entire class's data?
Make a scatter plot of the entire class’s data and look for patterns. Identify any outliers and the type of any association you observe.
[IMG]
Draw two lines on your scatter plot: a vertical line and a horizontal line, each representing 30 seconds for one trial. Use the table for the class’s data to complete this two-way table.
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}}, {"_attributes": {"scope": "col"}, "_value": "time 2 < 30 sec"}, {"_attributes": {"scope": "col"}, "_value": "time 2 = 30 sec"}, {"_attributes": {"scope": "col"}, "_value": "time 2 > 30 sec"}, {"_attributes": {"scope": "col"}, "_value": "total"}]}]}], "tbody": [{"tr": [{"th": [{"_value": "time 1 < 30 sec"}], "td": [{}, {}, {}, {}]}, {"th": [{"_value": "time 1 = 30 sec"}], "td": [{}, {}, {}, {}]}, {"th": [{"_value": "time 1 > 30 sec"}], "td": [{}, {}, {}, {}]}, {"th": [{"_value": "total"}], "td": [{}, {}, {}, {}]}]}]}]}
Use the two-way table to decide whether there is an association between Time 1 and Time 2. Explain how you know.
Student Response
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Activity Synthesis
The purpose of the discussion is to solidify understanding of the strategies and tools from the unit. Specifically, how to organize and analyze categorical data to determine if there is an association between two variables.
Some questions for discussion:
“What representation (data table, scatter plot, two-way table, or something else) helped the most in determining any relationships between time 1 and time 2?”
“If someone's time was slower on the second attempt, how would that appear on the scatter plot?” (The point would be above and to the left of the line
\(y = x\)
, since the value for the
\(y\)
-coordinate would be greater than the value for the
\(x\)
-coordinate like the point
\((1.2, 1.4)\)
.)
“If there was a positive association between the variables, what would that mean?” (In general, the slower the first time, the slower the second time. The faster the first time, the faster the second time as well.)
“If there was a negative association between the variables, what would that mean?” (In general, the slower the first time, the faster the second time. The faster the first time, the slower the second time.)
"If the class showed a tendency to overcompensate by going from a time that was longer than 30 seconds to one that was shorter or the other way around, how would that appear in the two-way table?" (The numbers in the table would be greatest on the bottom left and top right of the table with smaller numbers elsewhere.)


aligns: no