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: 
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_pos_0

HS_1

im_task_001160_pos_0

HS_2
standard_description: Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators. For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12. (In general, a/b + c/d = (ad + bc)/bd.)
problem_activity: Narrative
The goal of this activity is to use the distributive property to explain why multiplying a number by a fraction greater than one increases the size of the number while multiplying by a fraction less than one decreases the size of the number. Expressions are particularly useful here because they show explicitly how the size of the number relates to the product. For example writing
\(\frac{3}{5}\)
as
\(1 - \frac{2}{5}\)
and then multiplying by
\(\frac{4}{7}\)
gives:
\(\displaystyle \left(1 - \frac{2}{5}\right) \times \frac{4}{7} = \frac{4}{7} - \left(\frac{2}{5} \times \frac{4}{7}\right)\)
The revealing part of this calculation is that the structure of the right hand side shows that it is less than
\(\frac{4}{7}\)
without calculating the exact value (MP7). It must be less than
\(\frac{4}{7}\)
because it is
\(\frac{4}{7}\)
minus some other number.
Engagement: Internalize Self-Regulation.
Provide students an opportunity to self-assess and reflect on their own progress. For example, provide students with questions that relate to the size of the factors for them to reflect on once the activity is complete.
Supports accessibility for: Conceptual Processing, Attention, Memory
Launch
Groups of 2
Activity
1–2 minutes: quiet think time
8–10 minutes: partner work time
Monitor for students who use the expressions in the first problem to make the comparisons and then generalize about what happens when you multiply a number by any fraction greater than 1 or less than 1.
Student Facing
Rewrite each expression as a sum or difference of 2 products.
\(\left(1 - \frac{2}{5}\right) \times \frac{4}{7}\)
\(\left(1 + \frac{1}{5}\right)\times \frac{4}{7}\)
\(\left(1 - \frac{3}{8}\right)\times \frac{4}{7}\)
\(\left(1 + \frac{1}{8}\right)\times \frac{4}{7}\)
Fill in each blank with
\(<\)
or
\(>\)
to make the inequality true.
\(\left(1 - \frac{2}{5}\right)\times \frac{4}{7} \,\underline{\hspace{0.9cm}} \,\frac{4}{7}\)
\(\left(1 + \frac{1}{5}\right)\times \frac{4}{7} \,\underline{\hspace{0.9cm}}\, \frac{4}{7}\)
\(\left(1 - \frac{3}{8}\right)\times \frac{4}{7} \,\underline{\hspace{0.9cm}} \,\frac{4}{7}\)
\(\left(1 + \frac{1}{8}\right)\times \frac{4}{7} \,\underline{\hspace{0.9cm}} \,\frac{4}{7}\)
Describe the value of the product when
\(\frac{4}{7}\)
is multiplied by a fraction greater than 1. Explain your reasoning.
Describe the value of the product when
\(\frac{4}{7}\)
is multiplied by a fraction less than 1. Explain your reasoning.
Student Response
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Activity Synthesis
Invite students to share their expressions for the products in the first problem.
Display the equation:
\(\left(1 - \frac{2}{5}\right)\times \frac{4}{7} = \frac{4}{7} - \left(\frac{2}{5} \times \frac{4}{7}\right)\)
“How can you see that the value of the expression is less than
\(\frac{4}{7}\)
?” (It’s
\(\frac{4}{7}\)
minus something.)
“Does this reasoning also work for
\(\left(1 - \frac{3}{8}\right)\times \frac{4}{7}\)
?” (Yes, it’s again
\(\frac{4}{7}\)
minus some other number.)
“Will this reasoning work whenever you multiply a number less than 1 by
\(\frac{4}{7}\)
?” (Yes, I’ll always get
\(\frac{4}{7}\)
minus an amount so that’s less than
\(\frac{4}{7}\)
.)


aligns: no