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: 
Solve quadratic equations in one variable.

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


Answer: 
Yes. This task shows one quadratic equation with one variable, x.

Example 3:
Concept/skill: 
Construct a tangent line from a point outside a given circle to the circle.

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: 
No. This task does not involve drawing lines that are tangent to circles. The only line that may be drawn is the ray between points P and B.

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_pos_0

HS_1

im_task_001160_neg_0

HS_2
standard_description: Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.
problem_activity: Activity
The purpose of this activity is for students to give precise descriptions of translations, rotations, and reflections. By the end of the previous lesson, students have identified and sketched these transformations from written directions, however they have not used this more precise language themselves to give descriptions of the three motions. The images in this activity are given on grids to allow and to encourage students to describe the transformation in terms of specific points, lines or angles. Describing the moves precisely and clearly will require students to engage in MP6.
There are four different transformation cards students use in this activity: 1A, 1B, 2A, and 2B. Each card has the original image and the image under a transformation. Students are arranged in groups of 2 and each one gets a different transformation card: some pairs are given cards 1A and 1B while other pairs get 2A and 2B. Each A card is a translation, while the B cards show either a rotation or reflection.
As students are describing their transformations and sketching their images under transformation, monitor for students using precise descriptions to their partner in terms of specific points, lines, or angles.
Digital
Launch
Introduce the word
transformation:
This is a word for a move (like the reflection in the warm-up) or for a sequence of moves. It's a mathematical word which will be used in place of "move," which has been used up to this point.
Arrange students in groups of 2. Display the original image that each student also has in front of them. Give each student one of the four transformation cards and tracing paper—make sure students know
not
to show their card to their partner! Tell students they will sketch a transformation based on directions given to them by a partner.
Partners have Cards 1A and 1B or Cards 2A and 2B. Tell students in the first round, those with the A cards give a precise description of the transformation displayed on their card to their partner. Their partners may use tracing paper to produce the image under transformation on the grid with the original image. In digital classrooms, students have access to applets they can use to transform the figure. When the sketch is complete, the student describing the transformation reveals their card, and together, students decide if the sketch is correct. In the second round, the roles are reversed. The students with the B cards describe their transformation while their partner sketches.
The student describing the transformation is allowed to repeat, revise, or add any important information as their partner sketches, however, they are not allowed to tell them to fix anything until they are finished. The student sketching should not speak, just sketch. (This is to encourage the describer to use mathematical language.) Remind students to use the geometric language for describing reflections, rotations, and translations that was used in the previous lesson.
Representation: Develop Language and Symbols.
Display or provide charts with symbols and meanings. For example, display the terms translation, rotation, and reflection and their respective meanings “slide,” “turn,” and “mirror,” accompanied by a visual example depicting the transformation. Remind students that they can refer to the display when giving directions to their partner.
Supports accessibility for: Conceptual processing; Memory
Listening, Speaking: Math Language Routine 2 Collect and Display.
This is the first time Math Language Routine 2 is suggested as a support in this course. In this routine, the teacher circulates and listens to student talk while jotting down words, phrases, drawings, or writing students use. The language collected is displayed visually for the whole class to use throughout the lesson and unit. Generally, the display contains different examples of students using features of the disciplinary language functions, such as interpreting, justifying, or comparing. The purpose of this routine is to capture a variety of students’ words and phrases in a display that students can refer to, build on, or make connections with during future discussions, and to increase students’ awareness of language used in mathematics conversations.
Design Principle(s): Optimize output (for explanation); Maximize meta-awareness
How It Happens:
As students describe the transformation of triangle
\(ABC\)
to their partner, listen for and collect vocabulary and phrases students use to describe the moves. Focus on capturing students using geometric language for describing reflections, rotations, and translations.
If the speaker is stuck, consider asking these questions: “How did point
\(A\)
transform to
\(A’\)
?”, “Choose one of the points, lines, or angles and describe how it changed.”, and “Overall, does it look like the new triangle is a translation, rotation, or reflection of the original?”
If using the applet, check for precision and labels as students place the new image under the transformation.
Write students’ words on a visual display. Divide the display into 3 sections. Group language about Cards 1A and 2A on the left side of the display, language about Card 1B in the middle, and language about Card 2B on the right side.
Record all language (whether precise, ambiguous, correct, or incorrect) in the appropriate column. as described by the students.
Arrange students in groups of 2, and invite partners to discuss which words or phrases stand out to them. Prompt students by asking, “Are there any words or phrases that stand out to you or don’t belong in a specific column? Why?” Again, circulate around the room, collecting any additional words, phrases, and sketches onto the display.
Students should notice that the left side consists of language describing translations, the middle consists of language describing reflections, and the right side consists of language describing rotations.
Select 3–4 groups to share their ideas with the class. Invite students to demonstrate their reasoning with the applet or tracing paper and be sure to modify the display accordingly.
Use this discussion to clarify, revise, and improve how ideas are communicated and represented. If students are still using vague words (e.g., move, flip, mirror image, etc.), reinforce the precise geometric terms (e.g., transformation, translation, rotation, reflection, etc.). Ask students, “Is there another way we can say this?” or “Can someone help clarify this language?”
Close this conversation by posting the display in the front of the classroom for students to reference for the remainder of the lesson, and be sure to update the display throughout the remainder of the lesson.
Student Facing
Your partner will describe the image of this triangle after a certain
transformation
. Sketch it here.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Print
Launch
Introduce the word
transformation:
This is a word for a move (like the reflection in the warm-up) or for a sequence of moves. It's a mathematical word which will be used in place of "move," which has been used up to this point.
Arrange students in groups of 2. Display the original image that each student also has in front of them. Give each student one of the four transformation cards and tracing paper—make sure students know
not
to show their card to their partner! Tell students they will sketch a transformation based on directions given to them by a partner.
Partners have Cards 1A and 1B or Cards 2A and 2B. Tell students in the first round, those with the A cards give a precise description of the transformation displayed on their card to their partner. Their partners may use tracing paper to produce the image under transformation on the grid with the original image. In digital classrooms, students have access to applets they can use to transform the figure. When the sketch is complete, the student describing the transformation reveals their card, and together, students decide if the sketch is correct. In the second round, the roles are reversed. The students with the B cards describe their transformation while their partner sketches.
The student describing the transformation is allowed to repeat, revise, or add any important information as their partner sketches, however, they are not allowed to tell them to fix anything until they are finished. The student sketching should not speak, just sketch. (This is to encourage the describer to use mathematical language.) Remind students to use the geometric language for describing reflections, rotations, and translations that was used in the previous lesson.
Representation: Develop Language and Symbols.
Display or provide charts with symbols and meanings. For example, display the terms translation, rotation, and reflection and their respective meanings “slide,” “turn,” and “mirror,” accompanied by a visual example depicting the transformation. Remind students that they can refer to the display when giving directions to their partner.
Supports accessibility for: Conceptual processing; Memory
Listening, Speaking: Math Language Routine 2 Collect and Display.
This is the first time Math Language Routine 2 is suggested as a support in this course. In this routine, the teacher circulates and listens to student talk while jotting down words, phrases, drawings, or writing students use. The language collected is displayed visually for the whole class to use throughout the lesson and unit. Generally, the display contains different examples of students using features of the disciplinary language functions, such as interpreting, justifying, or comparing. The purpose of this routine is to capture a variety of students’ words and phrases in a display that students can refer to, build on, or make connections with during future discussions, and to increase students’ awareness of language used in mathematics conversations.
Design Principle(s): Optimize output (for explanation); Maximize meta-awareness
How It Happens:
As students describe the transformation of triangle
\(ABC\)
to their partner, listen for and collect vocabulary and phrases students use to describe the moves. Focus on capturing students using geometric language for describing reflections, rotations, and translations.
If the speaker is stuck, consider asking these questions: “How did point
\(A\)
transform to
\(A’\)
?”, “Choose one of the points, lines, or angles and describe how it changed.”, and “Overall, does it look like the new triangle is a translation, rotation, or reflection of the original?”
If using the applet, check for precision and labels as students place the new image under the transformation.
Write students’ words on a visual display. Divide the display into 3 sections. Group language about Cards 1A and 2A on the left side of the display, language about Card 1B in the middle, and language about Card 2B on the right side.
Record all language (whether precise, ambiguous, correct, or incorrect) in the appropriate column. as described by the students.
Arrange students in groups of 2, and invite partners to discuss which words or phrases stand out to them. Prompt students by asking, “Are there any words or phrases that stand out to you or don’t belong in a specific column? Why?” Again, circulate around the room, collecting any additional words, phrases, and sketches onto the display.
Students should notice that the left side consists of language describing translations, the middle consists of language describing reflections, and the right side consists of language describing rotations.
Select 3–4 groups to share their ideas with the class. Invite students to demonstrate their reasoning with the applet or tracing paper and be sure to modify the display accordingly.
Use this discussion to clarify, revise, and improve how ideas are communicated and represented. If students are still using vague words (e.g., move, flip, mirror image, etc.), reinforce the precise geometric terms (e.g., transformation, translation, rotation, reflection, etc.). Ask students, “Is there another way we can say this?” or “Can someone help clarify this language?”
Close this conversation by posting the display in the front of the classroom for students to reference for the remainder of the lesson, and be sure to update the display throughout the remainder of the lesson.
Student Facing
Your partner will describe the image of this triangle after a certain
transformation
. Sketch it here.
[IMG]
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Anticipated Misconceptions
Students may get stuck thinking they need to use the precise terms for the transformation in their description. Encourage these students to describe it in a way that makes sense to them and to look for things they know about the specific points, lines, or angles on their card to help them.
Activity Synthesis
Display the following questions for all to see and give groups 2 minutes to discuss:
What pieces of your partner’s description were helpful when you were sketching?
What pieces did you find difficult to explain to your partner? Point to specific examples on your cards.
When you were sketching, what questions would have been helpful to be able to ask the describer?
Ask selected students who were observed using precise descriptions and sketching based on those descriptions to explain why they used the information they did and how it was helpful in sketching. Focus on:
The direction and distance of a translation.
The center and the measure of a rotation.
The line of a reflection.
If there is time, ask students who were both using and not using tracing paper to explain their process.
Reinforce the term
transformation
as a term that encompasses translations, rotations, and reflections. Tell them that there are other types of transformations, but for now, we will focus on these three types.


aligns: no