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: Compare two three-digit numbers based on meanings of the hundreds, tens, and ones digits, using >, =, and < symbols to record the results of comparisons.
problem_activity: Narrative
The purpose of this activity is for students to use diagrams to reason about products of a whole number and a non-unit fraction with diagrams, building on their work with diagrams that represent products of a whole number an a unit fraction. They begin to generalize that the number of shaded parts in a diagram that represents
\(n \times \frac{a}{b}\)
is
\(n \times a\)
and to explain that generalization (MP8).
Launch
Groups of 2
“Let's represent some other products of a whole number and a fraction and find their values.”
Activity
“Take a few quiet minutes to work on the activity. Afterwards, share your responses with your partner.”
5–7 minutes: independent work
2–3 minutes: partner discussion
Monitor for the strategies students use to reason about the last two problems.
Identify students who reason visually (using diagrams), additively, and multiplicatively to share in the synthesis.
Student Facing
This diagram represents
\(\frac{2}{5}\)
.
[IMG]
Show how you would use or adjust the diagram to represent
\(4 \times \frac{2}{5}\)
.
What is the value of the shaded parts in your diagram?
This diagram represents
\(\frac{5}{8}\)
.
[IMG]
Show how you would use or adjust the diagram to represent
\(3 \times \frac{5}{8}\)
.
What is the value of the shaded parts in your diagram?
Find the value of each expression. Draw a diagram if you find it helpful. Be prepared to explain your reasoning.
\(2 \times \frac {1}{6}\)
\(2 \times \frac{4}{6}\)
\(2 \times \frac {5}{6}\)
\(4 \times \frac{5}{6}\)
Mai said that to multiply any fraction by a whole number, she would multiply the whole number and the numerator of the fraction and keep the same denominator. Do you agree with Mai? Explain your reasoning.
Student Response
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Activity Synthesis
Discuss the four multiplication expressions in the third problem.
Select 1–2 students who might have drawn diagrams for all expressions.
“How does your diagram show the value of
\(2 \times \frac{4}{6}\)
?” (There are 2 groups of
\(\frac{4}{6}\)
, so there are 8 sixths shaded, which is
\(\frac{8}{6}\)
.)
Select 1–2 students who drew a diagram for some expressions and reason numerically for others.
“Why did you choose to draw a diagram for some expressions and to do something else for others?” (After drawing the first two diagrams, I realized that I'd have to draw a lot of groups or parts, so I thought about the numbers instead.)
Select 1–2 students who reasoned about all expressions numerically.
“How did you find the value of the expressions without drawing diagrams at all?” (I saw a pattern in earlier problems, that we can multiply the whole number and the numerator of the fraction and keep the denominator.)
Discuss the last problem in the lesson synthesis.


aligns: no