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 problem or activity shown below enable students to learn the full intent of the following concept or skill?

shot_info: fl_problem_001392_neg_0

8
standard_description: Correctly name shapes regardless of their orientations or overall size.
problem_activity: Warm-up
The mathematical purpose of this activity is for students to reason about the area under a normal curve using what they know about composition and decomposition of areas.
If students are using a table of values with
\(z\)
-scores for this lesson, this work will be essential to finding the desired areas in many cases.
Launch
Remind students that the phrase “the area under the curve” means the area between the
\(x\)
-axis and the curve.
Student Facing
The images show a normal curve with mean of 40 and standard deviation of 2.
The area under the curve to the left of 39 is 0.3085.
[IMG]
The area under the curve to the left of 43 is 0.9332.
[IMG]
Since it is a normal curve, we know that the total area under the curve is 1. Use the given areas to find the areas in question. Explain your reasoning for each.
Find the area under the curve to the right of 39.
Find the area under the curve between 39 and 43.
Find the area under the curve to the left of 40.
Find the area under the curve between 39 and 40.
Student Response
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Activity Synthesis
The purpose of this discussion is to make sure students can use the given areas to compose or decompose them to find additional areas. For each question, ask a student to explain how they reasoned about their answer. Highlight places where students used the symmetry of the distribution or cases where they could add or subtract known areas to find new areas.


aligns: no