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: 
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 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

HS_0

im_task_000870_neg_0

HS_1

im_task_001160_pos_0

HS_2
standard_description: Know relative sizes of measurement units within one system of units including km, m, cm; kg, g; lb, oz.; l, ml; hr, min, sec. Within a single system of measurement, express measurements in a larger unit in terms of a smaller unit. Record measurement equivalents in a two column table. For example, know that 1 ft is 12 times as long as 1 in. Express the length of a 4 ft snake as 48 in. Generate a conversion table for feet and inches listing the number pairs (1, 12), (2, 24), (3, 36), ...
problem_activity: Narrative
The purpose of this activity is for students to analyze a line plot that represents lengths that are measured to the nearest half inch. They make observations and write statements about the data represented in the line plot, and then generate questions that could be answered with the line plot. When students recognize how organizing data helps to read the information and to answer questions, they learn that line plots are a powerful tool to present data (MP5).
Launch
Groups of 2
Display the list and line plot.
“The list and the line plot both show the heights of seedlings. A seedling is a young plant. Where have you seen seedlings before?” (At the park. In a garden.)
“Looking at the list, can we tell the height of the shortest seedling?” (Yes,
\(\frac{1}{2}\)
inch). “What about the tallest seedling?” (Yes, 5 inches)
“Draw a quick sketch of the shortest seedling and the tallest seedling at their actual heights. Use what you know about the length of an inch.”
1 minute: independent work time
“Share your sketches with your partner.”
“What else can we tell about the seedlings from looking at the list?” (Twenty-two seedlings were measured. There are two seedlings that are 3 inches tall.)
1 minute: partner discussion
Share responses.
Activity
“Now, take a close look at the line plot. Think about what information we can gather from the line plot and what questions it can help to answer.”
“Work with your partner to complete these problems.”
7–10 minutes: partner work time
Student Facing
heights of seedlings (in inches)
\(\frac{1}{ 2}\)
1
1
\(\frac{1}{ 2}\)
\(1\frac{1}{ 2}\)
\(2\frac{1}{ 2}\)
4
\(\frac{1}{ 2}\)
3
\(1\frac{1}{ 2}\)
5
\(1\frac{1}{2}\)
\(1\frac{1}{2}\)
\(2\frac{1}{2}\)
3
\(\frac{1}{2}\)
\(2\frac{1}{2}\)
\(1\frac{1}{2}\)
1
\(1\frac{1}{2}\)
4
2
[IMG]
[IMG]
Write 3 statements about the measurements represented in the line plot.
What questions could be answered more easily with the line plot than the list? Write at least 2 questions.
Student Response
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Activity Synthesis
Display the list and the line plot.
“How is the information displayed in the line plot different from that in the list?” (In the line plot, all the measurements that are the same length are together. The measurements go from smallest to largest along the bottom. In the line plot, we don’t keep writing the numbers over and over, we would just use an x for each measurement.)
“What were some questions that could be more easily answered with the line plot than the list?” (“How many seedlings were 3 inches tall?” because we could just count the x’s at 3 instead of searching through the list. “What’s the shortest seedling?” because we can find the x on the left end of the scale. “Which seedling height was the most common?” because we can see which number has the most x’s.)


aligns: no