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: 
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 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_neg_0

HS_2
standard_description: Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression 3 (2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6 (4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y.
problem_activity: Activity
One purpose of this activity is to practice seeing the total length of a segment as the sum of its pieces, and using the whole side length of a rectangle to express its area. Another is to generate examples of equivalent expressions, and understand why they are equivalent based on an understanding of area and the distributive property.
Monitor for students who write different but equivalent expressions to represent the area of a rectangle. For example,
\(2(t+5+t)\)
,
\(2t+10+2t\)
, and
\(4t+10\)
for figure B.
Student Facing
Complete the table with the length, width, and area of each rectangle.
[IMG]
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}, "_value": "rectangle"}, {"_attributes": {"scope": "col"}, "_value": "length (units)"}, {"_attributes": {"scope": "col"}, "_value": "width (units)"}, {"_attributes": {"scope": "col"}, "_value": "area (square units)"}]}]}], "tbody": [{"tr": [{"td": [{"_value": "A"}, {"span": [{"_attributes": {"class": ["math", "math-repaired"]}, "_value": "\\(a+4\\)"}]}, {}, {}]}, {"td": [{"_value": "B"}, {}, {"_value": "2"}, {}]}, {"td": [{"_value": "C"}, {}, {}, {}]}, {"td": [{"_value": "D"}, {}, {}, {}]}, {"td": [{"_value": "E"}, {}, {}, {}]}]}]}]}
Student Response
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Activity Synthesis
Invite previously-selected students to share different but equivalent expressions for length and area. (Reassure students that it doesn’t matter which side is considered the length or the width.) If any students express confusion over a response, encourage the student who came up with the response to explain. For example, a student might wonder why
\(a+a+a\)
can be expressed as
\(3a\)
.
Display the table from the task statement for all to see, and write any equivalent expressions that are mentioned next to each other. For example, for the area of figure A, you might write
\(5(a+4)\)
,
\((4+a) \boldcdot 5\)
, and
\(5a+20\)
. Show in the figure how you can see
\(5(a+4)\)
(the product of one side and the other whole side) and
\(5a+20\)
(the area of each smaller rectangle, added up). Emphasize that all of these expressions correctly represent the area of the whole figure, and highlight use of the distributive property where it occurs.


aligns: yes