few_shots: 
Example:
Problem/activity:
Problem 1
The table below contains single equations in a single variable, single equations in two variables, and pairs of simultaneous equations in two variables. Check the appropriate box to show whether there is no solution, exactly 1 solution, or if there is an infinite number of solutions.
|                                                          | **           No solution          ** | **           Exactly 1 solution          ** | **           Infinite number of solutions          ** |
|----------------------------------------------------------|--------------------------------------|---------------------------------------------|-------------------------------------------------------|
|                       $${9=4x+7}$$                       |                                      |                                             |                                                       |
|                       $${y=3y+5}$$                       |                                      |                                             |                                                       |
|                    $${y={x\over2}}$$                     |                                      |                                             |                                                       |
|                     $${3x+6y-9=0}$$                      |                                      |                                             |                                                       |
|    $${x=y}$$                        $${1.25x=1.25y}$$    |                                      |                                             |                                                       |
| $${9.9=6x+8y}$$                        $${9x=2.5y-8.8}$$ |                                      |                                             |                                                       |
| $${11x-2y=1.5}$$                        $${11x-2y=2.5}$$ |                                      |                                             |                                                       |
Problem 2
Kim has a small container and a large container as shown.
[IMG]
It takes 16 of the small containers to fill the large container. Three small containers leave 1.95 gallons of space in the big container.
What is the size of each of the two containers?


Concept/skill: 
Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.

Answer: 
No. Completing neither of these problems requires replacing one equation by the sum of that equation and a multiple of the other equation.

Now, does the following problem or activity enable students to learn the full intent of the following concept or skill?

shot_info: fl_problem_001392_neg_0

8
standard_description: Prove the Pythagorean identity sin^(2)(θ) + cos^(2)(θ) = 1 and use it to find sin(θ), cos(θ),or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.
problem_activity: Optional activity
In this activity, students use the areas of composite shapes from the previous activity to reason about the area of each tangram shape. Students may have recognized previously that the area of one small triangle is
\(\frac12\)
square unit, the area of one medium triangle is 1 square unit, and the area of one large triangle is 2 square units. Here they practice articulating how they know that these observations are true (MP3). The explanations could be written in words, or as clearly-labeled illustrations that support their answers.
As partners discuss, look for two ways of thinking about the area of each assigned triangle: by
composing
copies of the triangle into a square or a larger triangle, or by
decomposing
the triangle or the unit square into smaller pieces and
rearranging
the pieces. Identify at least one student who uses each approach.
Digital
Launch
Arrange students in groups of 2. Assign the first and second questions to one partner and the second and third questions to the other partner. Give each group access to the geometry toolkits and the same set of tangram pieces as used in the earlier activity.
Give students 3–4 minutes of quiet time to find the areas of their assigned triangles and to construct their explanations, followed by a few minutes to share their responses with their partner. Tell students that as one partner explains, the other should listen carefully and either agree or disagree with the explanation. They should then come to an agreement about the answers and explanations.
Classrooms using the digital activities can use an applet to assist in determining the areas of the triangles.
Speaking, Writing: MLR8 Discussion Supports.
Use this routine when students compare areas of triangles and squares to support the use of mathematical language. As students share their responses with their partner, circulate and encourage listeners to push speakers to use the language “compose,” “decompose,” or “rearrange” in their explanations. Look for students who name the square or larger triangles “composite figures” and amplify this language. Encourage students to borrow words and phrases from each other and to use this language in their written responses.
Design Principle(s): Cultivate conversation; Optimize output (for explanation)
Student Facing
Recall that the area of the square you saw earlier is 1 square unit. Complete each statement and explain your reasoning.
The area of the small triangle is ____________ square units. I know this because . . .
The area of the medium triangle is ____________ square units. I know this because . . .
The area of the large triangle is ____________ square units. I know this because . . .
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
Arrange students in groups of 2. Assign the first and second questions to one partner and the second and third questions to the other partner. Give each group access to the geometry toolkits and the same set of tangram pieces as used in the earlier activity.
Give students 3–4 minutes of quiet time to find the areas of their assigned triangles and to construct their explanations, followed by a few minutes to share their responses with their partner. Tell students that as one partner explains, the other should listen carefully and either agree or disagree with the explanation. They should then come to an agreement about the answers and explanations.
Classrooms using the digital activities can use an applet to assist in determining the areas of the triangles.
Speaking, Writing: MLR8 Discussion Supports.
Use this routine when students compare areas of triangles and squares to support the use of mathematical language. As students share their responses with their partner, circulate and encourage listeners to push speakers to use the language “compose,” “decompose,” or “rearrange” in their explanations. Look for students who name the square or larger triangles “composite figures” and amplify this language. Encourage students to borrow words and phrases from each other and to use this language in their written responses.
Design Principle(s): Cultivate conversation; Optimize output (for explanation)
Student Facing
Recall that the area of the square you saw earlier is 1 square unit. Complete each statement and explain your reasoning.
The area of the small triangle is _________ square units. I know this because . . .
The area of the medium triangle is _________ square units. I know this because . . .
The area of the large triangle is _________ square units. I know this because . . .
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
If students initially have trouble determining the areas of the shapes, ask how they reasoned about areas in the previous activity. Have samples of composed and decomposed shapes that form one square unit available for students to reference.
Activity Synthesis
After partners shared and agreed on the correct areas and explanations, discuss with the class:
“Did you and your partner use the same strategy to find the area of each triangle?”
“How were your explanations similar? How were they different?”
Select two previously identified students to share their explanations: one who reasoned in terms of
composing
copies of their assigned triangle into another shape, and one who reasoned in terms of
decomposing
their triangle or the unit square into smaller pieces and
rearranging
them. If these approaches are not brought up by students, be sure to make them explicit at the end of the lesson.


aligns: no