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 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_pos_0

HS_2
standard_description: Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.
problem_activity: Narrative
The purpose of this activity is for students to measure lengths using a ruler that is marked with half inches and quarter inches. Building on their understanding of equivalent fractions, students recognize that lengths that line up with a half-inch mark can be read as one-half of an inch or two-fourths of an inch. They also recognize that lengths that are whole numbers of inches can be expressed as fractions.
MLR8 Discussion Supports.
Synthesis: Provide students with the opportunity to rehearse what they will say with a partner before they share with the whole class.
Advances: Speaking
Required Materials
Materials to Gather
Materials from a previous activity
Rulers (inches)
Required Preparation
Each student needs a ruler marked with half inches and quarter inches from the warm-up.
Launch
Groups of 2
Display the problem.
“Think about how Kiran and Jada record the length of the worm. Use the ruler to explain how both of their measurements make sense.”
1 minute: quiet think time
2–3 minutes: partner discussion
Monitor for students that say they both make sense because
\(4\frac{1}{2}\)
is equivalent to
\(4\frac{2}{4}\)
and select them to share their responses.
Make sure each student has a ruler marked with half inches and quarter inches from the previous activity.
Activity
“Now, work with your partner to measure the length of some worms. Some lengths can be recorded in more than one way. Be ready to explain how you recorded each length.”
3–5 minutes: partner work time
Student Facing
Kiran and Jada are discussing the length of a worm.
[IMG]
[IMG]
Kiran says that the worm is
\(4\frac{2}{4}\)
inches long.
Jada says that the worm is
\(4\frac{1}{2}\)
inches long.
Use the ruler to explain how both of their measurements are correct.
Measure the length of the following worms.
A
[IMG]
B
[IMG]
C
[IMG]
D
[IMG]
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Advancing Student Thinking
If students start measuring from the edge of their ruler instead of from 0, consider asking:
“In the image for the first problem, how is the ruler lined up to the worm?”
“Why does it make sense to not line up the left end of the ruler with the left end of the worm?”
Activity Synthesis
Invite students to share how they recorded their measurements and why they chose to record them in the way they did.
“Let’s think more about the first worm you measured. What are all the ways that we could record its length?” (4 inches.
\(\frac{8}{2}\)
inches.
\(\frac{16}{4}\)
inches.)
“What are all the ways that we could record the length of the last worm?” (
\(3\frac{1}{2}\)
inches.
\(3\frac{2}{4}\)
inches.
\(\frac{7}{2}\)
inches.
\(\frac{14}{4}\)
inches.)
“We can use different numbers to describe the same length because the numbers are equivalent.”


aligns: no