few_shots: 
Example 1:
Concept/skill: 
Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

Problem/activity:
Find the general solution for:
$${\mathrm{cos}\left({\pi\over2}+x\right)=\mathrm{tan}{\pi\over4}}$$


Answer: 
Yes. An initial step when solving for x in this problem requires the use of the addition formula for cosine on the left side of the equation.

Example 2:
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).

Problem/activity:
Task
Solve the quadratic equation
$$ x^2 = (2x - 9)^2 $$
using as many different methods as possible.


Answer: 
No. This task does not involve an equation in two variables. There is only one variable, x.

Example 3:
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.

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 $P$.


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, determine whether the following problem or activity can enable students to understand the concepts or skills in the provided description.

shot_info: fl_problem_001894_pos_0

HS_0

im_task_000870_neg_0

HS_1

im_task_001160_pos_0

HS_2
standard_description: Determine the unknown whole number in an addition or subtraction equation relating to three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8 + ? = 11, 5 = �� – 3, 6 + 6 = ��.
problem_activity: Narrative
The purpose of this activity is for students to describe the structure of an array by identifying the number of columns, the number of objects in each column, and the total number of objects. They learn that the columns of an array go up and down. Students begin to reason about how they can use the structure of the array to find the total number of objects without counting by 1.
This activity uses
MLR8 Discussion Supports. Advances: speaking, conversing
Representation: Develop Language and Symbols. S
upport understanding of the rows and columns by inviting students to act it out. For example, use students to create the first array (3 rows of 2). Discuss the students who create the rows. Now switch to talking about columns (2 columns of 3). Discuss the difference in which students create the columns.
Supports accessibility for: Visual-Spatial Processing, Organization
Required Materials
Materials to Gather
Counters
Launch
Groups of 2
Give students counters.
“Use your counters to make an array with 3 rows with 4 in each row. Make sure the counters in each row line up with the counters in the other rows.”
Display counters in an array as shown:
[IMG]
“Arrays are made up of rows, but they also have columns that go up and down. How many columns are in this array?” (4)
30 seconds: quiet think time
Share responses.
“How many counters are in each column of the array you made?” (3)
30 seconds: quiet think time
Share responses.
Activity
“Now you will look at a few arrays to decide how many columns they have, how many in each column, and find the total number of counters. Then you will make arrays using counters and answer a few questions.”
MLR8 Discussion Supports
Display sentence frames to support students when they describe the structure of the array:
“There are _____ columns in the array.”
“There are _____ counters in each column.”
“There are _____ counters in all.”
8 minutes: partner work time
Student Facing
[IMG]
How many columns are in this array?
How many counters are in each column?
How many counters are there in all?
[IMG]
How many columns are in this array?
How many counters are in each column?
How many counters are there in all?
Use 10 counters to make 2 columns with the same number in each column.
How many counters are in each column?
How many rows are in the array?
How could you count these counters without counting by ones?
Use 15 counters to make 3 columns with the same number in each column.
How many counters are in each column?
How many rows are in the array?
How many counters are in each row?
How could you count these counters without counting by ones?
Student Response
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Activity Synthesis
Display student work for an array with 10 counters.
“Describe _____’s array.” (It has 10 counters, it has 2 columns and 5 counters in each column. It has 5 rows with 2 counters in each row.)
“How can we prove there are 10 counters in this array?” (We can skip count by 2. We can add
\(2 + 2 + 2 + 2 + 2\)
. We can add
\(5 + 5\)
.)


aligns: no