few_shots: 
Example 1:
Problem/activity:
Find the general solution for:
$${\mathrm{cos}\left({\pi\over2}+x\right)=\mathrm{tan}{\pi\over4}}$$


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

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:
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_pos_0

HS_0

im_task_000870_pos_0

HS_1

im_task_001160_pos_0

HS_2
standard_description: Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.
problem_activity: Narrative
The purpose of this activity is for students to find the unknown addend in an equation in a way that makes sense to them and compare their methods. In the launch, students are introduced to base-ten blocks and compare them to connecting cubes. During the launch, students should be given time to observe the image and touch the connecting cubes and base-ten blocks.
Students may find the unknown addend using any method that makes sense to them. Monitor and select students with the following methods to share in the synthesis:
count on or count back using connecting cubes or base-ten blocks
use base-ten blocks to show combining or separating tens and ones
use base-ten drawings to show combining or separating tens and ones
Students have the opportunity in the activity and the activity synthesis to consider the available tools and make a choice that best helps them find the unknown addend (MP5). To support student reflection on the utility of each tool, provide each group with towers of ten connecting cubes, but not enough to represent the numbers in the equation without needing to create new towers of ten.
MLR8 Discussion Supports.
When groups compare methods, invite students to take turns sharing their responses. Ask students to restate what they heard using precise mathematical language and their own words. Display the sentence frame: “I heard you say . . . .” Original speakers can agree or clarify for their partner.
Advances: Listening, Speaking
Required Materials
Materials to Gather
Base-ten blocks
Connecting cubes
Required Preparation
Each group of 2 needs 90
–100 connecting cubes, but
no more than 3
–
5 towers of 10 cubes should be included in their collection. Break apart any extra towers for this activity.
Launch
Groups of 2
Give each group towers of 10, single connecting cubes, and base-ten blocks.
Display the image.
“Each group has some connecting cubes and some base-ten blocks.”
“What is the same and what is different between these tools?” (They both are cubes or towers of cubes. The connecting cubes are in towers of 10, single cubes, and some are in towers of different sizes. The blocks are only in tens and ones. The blocks in tens do not come apart.)
1 minute: quiet think time
1 minute: partner discussion
Share responses.
Activity
“Work together to find the number that makes the equation true. You can use the connecting cubes, base-ten blocks, or other representations to help find the number or show your thinking. Be prepared to explain your thinking.”
6 minutes: partner work time
As students work, consider asking:
“Why did you choose this tool?”
“How did you find the number that makes the equation true?”
“What is another way you could use this tool to find the unknown number?”
“Now compare your method with another group. How are your methods the same? How are they different?”
2 minutes: group discussion
Student Facing
[IMG]
What is the same and what is different between these tools?
Find the number that makes the equation true. Show your thinking using the cubes, blocks, or drawings.
\(41 + \underline{\phantom{\hspace{1.05cm}}}=84\)
Student Response
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for free access to Student Response.
Activity Synthesis
Invite previously identified groups to share their methods in the given order.
Consider asking:
“Why did you choose this tool to help you find the number?”
“How are the methods the same? How are they different?” (Some methods are the same, they just used different tools to show it. Some methods used the same tool, but are different because one group added tens and ones to find the unknown number, but another group took away tens and ones to find the unknown number.)


aligns: yes