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: 
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).

Answer: 
No. This task does not involve an equation in two variables. There is only 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 following problem or activity 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_neg_0

HS_1

im_task_001160_pos_0

HS_2
standard_description: Solve real-world and mathematical problems involving the four operations with rational numbers. (Computations with rational numbers extend the rules for manipulating fractions to complex fractions.)
problem_activity: Warm-up
This warm-up prompts students to attend to precision in measurements, which will be important in upcoming work.
Launch
Arrange students in groups of 2. Give students 1 minute of quiet think time to estimate the size of their own foot in centimeters or inches, and a moment to share their estimate with a partner. Then, ask them to complete the task.
Student Facing
If a student uses a ruler like this to measure the length of their foot, which choices would be appropriate measurements? Select
all
that apply. Be prepared to explain your reasoning.
[IMG]
\(9\frac14\)
inches
\(9\frac{5}{64}\)
inches
23.47659 centimeters
23.5 centimeters
23.48 centimeters
Here is a scale drawing of an average seventh-grade student's foot next to a scale drawing of a foot belonging to the person with the largest feet in the world. Estimate the length of the larger foot.
[IMG]
Student Response
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for free access to Student Response.
Anticipated Misconceptions
Some students may say the large foot is about
\(3\frac12\)
inches or about 9 centimeters long, because they assume the ruler shown in the first question is at the same scale as the feet shown in the second question. Explain that the images are drawn at different scales.
Activity Synthesis
Select a few students to share the measurements they think would be appropriate based on the given ruler. Consider displaying the picture of the ruler for all to see and recording students' responses on it. After each response, poll the class on whether they agree or disagree.
If students consider B, C, or E to be an appropriate measurement, ask them to share how to get such a level of precision on the ruler. Make sure students understand that reporting measurements to the nearest
\(\frac{1}{64}\)
of an inch or to the hundred-thousandths of a centimeter would not be appropriate (i.e., show that the ruler does not allow for these levels of precision).
Choice E of 23.48 cm may merit specific attention. With the ruler, it is possible to
guess
that the hundredths place is an 8. This may even be correct. The problem with reporting the measurement in this way is that someone who sees this might misinterpret it and imagine that an extremely accurate measuring device was used to measure the foot, rather than this ruler. The way a measurement is reported reflects how the measurement was taken.
Next, invite students to share their estimates for the length of the large foot. Since it is difficult to measure the length of these feet very precisely, these measurements should not be reported with a high level of precision; the nearest centimeter would be appropriate.


aligns: no