few_shots: 
Example 1:
Concept/skill: 
Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

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


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:
Concept/skill: 
Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).

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


Answer: 
No. This task does not involve an equation in two variables. There is only one variable, x.

Example 3:
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.

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 $P$.


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, determine whether the following problem or activity can enable students to understand the concepts or skills in the provided description.

shot_info: fl_problem_001894_pos_0

HS_0

im_task_000870_neg_0

HS_1

im_task_001160_pos_0

HS_2
standard_description: Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.
problem_activity: Activity
In previous lessons, students examined the estimation of the mean and median for populations using data from a sample. In this activity, students apply similar reasoning to estimating the proportion of a population that matches certain characteristics. Students collect a sample of 20 reaction times and compute the fraction of responses in their sample that are in a given range. Then, in the discussion, students compare their estimations to the known population proportion and use the class's proportions to gauge the accuracy of their estimate (MP6).
Launch
Arrange students in groups of 2. Distribute bags of slips cut from the blackline master.
Tell students that, in statistics, a
proportion
is a number between 0 and 1 that represents the fraction of the data that fits into the desired category. For example, with the data: {Yes, Yes, Yes, No, Maybe} the proportion of “Yes” answers is
\(\frac{3}{5}\)
or 0.6.
Introduce the context: All 120 seniors at a high school were asked to click a button as soon as they noticed a box change color and the response time was recorded in seconds. These 120 response times represent the population for this activity. Their responses are written on the slips of paper in the bag.
When selecting a sample of 20, each value does not need to be replaced before taking the next one.
Allow students 10 minutes of partner work time followed by a whole-class discussion.
Action and Expression: Internalize Executive Functions.
Chunk this task into more manageable parts. After students have solved the first 2-3 problems, check-in with either select groups of students or the whole class. Invite students to share the strategies they have used so far as well as any questions they have before continuing.
Supports accessibility for: Organization; Attention
Student Facing
The track coach at a high school needs a student whose reaction time is less than 0.4 seconds to help out at track meets. All the twelfth graders in the school measured their reaction times. Your teacher will give you a bag of papers that list their results.
Work with your partner to select a random sample of 20 reaction times, and record them in the table.
|   |   |   |   |   |   |   |   |   |   |
|---|---|---|---|---|---|---|---|---|---|
|   |   |   |   |   |   |   |   |   |   |
What
proportion
of your sample is less than 0.4 seconds?
Estimate the proportion of all twelfth graders at this school who have a reaction time of less than 0.4 seconds. Explain your reasoning.
There are 120 twelfth graders at this school. Estimate how many of them have a reaction time of less than 0.4 seconds.
Suppose another group in your class comes up with a different estimate than yours for the previous question.
What is another estimate that would be
reasonable
?
What is an estimate you would consider
unreasonable
?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Anticipated Misconceptions
The reason a proportion is between 0 and 1 is because we are specifically looking for the proportion of a desired category out of the whole group. This is a “part out of whole” fraction, which will be a number between 0 and 1.
Activity Synthesis
The purpose of the discussion is for students to see how multiple sample proportions can help revise their estimates and give an idea of how accurate the individual estimates from samples might be.
Ask the groups to share the proportion from their sample that had fast reaction times and display the results for all to see.
Some questions for discussion:
“Using the class’s data, how accurate do you think your group's estimate is? Explain or show your reasoning.” (Students should mention the variability of the proportions from the samples influencing the accuracy of the estimate.)
“The actual proportion for this population is 0.5. How close was your estimate? Explain why your estimate was not exactly the same.” (Each sample might be slightly different since they do not include all of the values, but they should be close.)
“If each group had 40 reaction times in their samples instead of 20, do you think the estimate would be more or less accurate?” (The estimate should be more accurate since there is more information available.)
Representing, Speaking: MLR7 Compare and Connect.
Invite students to create a visual display of their work to share with other students. Displays should include a representation of the sample reaction times, the proportion of fast reaction times, and an estimate for the number of twelfth-graders with fast reaction times. As students investigate each other’s work, ask students to compare their samples and proportion of fast reaction times. Listen for and amplify the language students use to justify the accuracy of their estimate for the proportion of fast reaction times using the definition of a random sample. This will help students compare their sample proportions in order to see how accurate their population estimates might be.
Design Principle(s): Optimize output (for comparison); Cultivate conversation


aligns: no