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: 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: Optional activity
In the final phase of the drawing project, students reflect on and revise their work. Students who chose the same paper option confer in small groups to analyze and compare their floor plans. They discuss their decisions, evaluate the accuracy of their drawings, and then revise them as needed.
After revision, students debrief as a class and discuss how the choice of scale, units, and paper affected the drawing process and the floor plans they created.
Launch
Arrange students who use the same type and size of paper into small groups. Give them 8–10 minutes to share and explain their drawings. Display and read aloud questions such as the following. Ask students to use them to guide their discussion.
What scale did you use? How did you decide on the scale?
Do the scaled measurements in each drawing seem accurate? Do they represent actual measurements correctly?
Did the scale seem appropriate for the chosen paper? Why or why not?
What was the first thing you drew in your drawing? Why?
How did you decide on the objects to show in your drawing?
What aspects of your drawings are different?
How could each floor plan be revised to better represent the classroom?
Student Facing
Trade floor plans with another student who used the same paper size as you. Discuss your observations and thinking.
Trade floor plans with another student who used a different paper size than you. Discuss your observations and thinking.
Based on your discussions, record ideas for how your floor plan could be improved.
Student Response
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Activity Synthesis
Before debriefing as a class, give students 4–5 minutes of quiet time to reflect. Ask them to write down ideas for revising their floor plan and strategies for creating accurate scale drawings based on their conversation.
Though much of the discussion will take place within the groups, debrief as a class so students can see floor plans created at a variety of scales and on different paper types or sizes. Display a range of scale drawings for all to see and discuss the following questions. (Alternatively, consider posting all students’ work for a gallery walk and ask students to reflect on these questions.)
What are the differences in these drawings?
How did different scales impact the final drawing?
How did the size of paper impact the choice of scale?
What choices were really important when creating the scale drawing?
Would these choices be the same if you were doing a different room in the school? Or some other building?
Representing, Conversing: MLR7 Compare and Connect.
Use this routine to prepare students for the whole-class discussion. Invite students to quietly circulate and read at least 2 other posters or visual displays in the room prior to the whole-class discussion. Give students quiet think time to consider what is the same and what is different about the visual displays of the floor plans created at a variety of scales and on different paper types or sizes. Next, ask students to find a partner to discuss what they noticed. Listen for and amplify observations that include mathematical language and reasoning about how the choice of scale, units, and paper affect the process of creating scaled drawings.
Design Principle(s): Cultivate conversation


aligns: no