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: Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. (Sizes are compared directly or visually, not compared by measuring.) Identify triangles, quadrilaterals, pentagons, hexagons, and cubes.
problem_activity: Narrative
The purpose of this activity is for students to solve two-step problems without the scaffold of having the first step explicitly stated. Students solve in a way that makes sense to them and might use diagrams to help them make sense of the story. In the synthesis, the tape diagram is highlighted.
MLR7 Compare and Connect.
Synthesis: Invite group to prepare a visual display that shows the strategy they used to solve the story problems. Encourage students to include details that will help others interpret their thinking. For example, using labels, notes, diagrams, or drawings. Give students time to investigate each other’s work. During the whole-class discussion, ask students, “What did the representations have in common?”, “How were they different?” and “Did anyone solve the problem the same way, but would explain it differently?”
Advances: Representing, Conversing
Launch
Groups of 2
“With a partner, choose a problem from the first activity to solve. Discuss how the diagram can help you think about the problem.”
4 minutes: partner work time
Activity
“Now you will be solving a few money problems on your own.”
“Use a diagram if it helps you make sense of the story.”
10 minutes: independent work time
Monitor for students who use a diagram to represent and solve the problem about Tyler, Andre, and Noah.
Student Facing
For each problem, show your thinking. Write your final answer using the \$. Use a diagram if it helps.
Mai has \$27, Elena has \$48, and Jada has \$16. How much money do they have altogether?
Tyler has \$45, Andre has \$36, and Noah has \$28. How much less money does Tyler have than Andre and Noah combined?
Lin has \$19. Together, Lin and Han have \$45. Then Han gets \$17 more. How much money does Han have now?
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 want to represent the story with a diagram or equation but need support to start, consider asking:
“What is the story about? How could you break the problem into smaller parts?”
“How could you use a diagram or equation to represent the smaller parts?”
Activity Synthesis
Invite previously identified students to share the diagram for comparing Tyler's money to Andre's and Noah's money.
“How does this diagram represent the story?” (It shows Tyler's amount and then the combined amount of Andre and Noah. I can see that Tyler has less than Andre and Noah and that I need to add Andre's amount and Noah's amount and then subtract Tyler's amount from that.)


aligns: no