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: 
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 following problem or activity 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_neg_0

HS_2
standard_description: Solve two-step word problems using the four operations. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding. (This standard is limited to problems posed with whole numbers and having whole number answers; students should know how to perform operations in the conventional order when there are no parentheses to specify a particular order (Order of Operations).)
problem_activity: Narrative
The purpose of an Estimation Exploration is to practice the skill of estimating a reasonable answer based on experience and known information. In the synthesis, discuss what students know about Estimation Explorations and what they need to think about to create one like this example.
Launch
Groups of 2
Display the image.
“What is an estimate that’s too high? Too low? About right?”
1 minute: quiet think time
Activity
“Discuss your thinking with your partner.”
1 minute: partner discussion
Record responses.
Student Facing
What is the length of this earthworm?
[IMG]
Record an estimate that is:
|                    too low                     |                  about right                   |                    too high                    |
|------------------------------------------------|------------------------------------------------|------------------------------------------------|
| \(\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}\) | \(\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}\) | \(\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}\) |
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
“What do you know about Estimation Explorations?” (It's not about finding the exact answer. We make an estimate that is too low, too high, and about right. The estimate could be the value of an expression, the number of items in an image, or a measurement.)
Consider asking: “What would you have to think about if you were going to design an Estimation Exploration like this one?” (There is something in the image that gives a clue about the general length of the object. Choosing an object that is not an exact length in whole inches.)
Record and display responses for all to see.


aligns: no