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: Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. For example, estimate the population of the United States as 3 × 108 and the population of the world as 7 × 109, and determine that the world population is more than 20 times larger.
problem_activity: Warm-up
Students reason about the relationship between distance, rate, and time to solve a problem. The purpose is to reactivate what students know about constant speed contexts, where constant speed is represented by a set of equivalent ratios associating distance traveled and elapsed time. In the longer activity that follows, students represent a constant speed context using a table, equations, and graphs.
As students work, watch for different representations used (particularly tables) as well as for students who calculate each person's speed in miles per hour or each person's pace in hours per mile.
Launch
Consider starting off by having students close their books or devices, and display the following situation. Ask students, "What do you notice?" "What do you wonder?"
Lin and Jada each leave school at 3 p.m. to walk to the library. They each walk at a steady rate. When do they arrive?
Give them 1 minute to think of at least one thing they notice and one thing they wonder.
Students might notice that they leave at 3 p.m., that they walk at a steady rate (also called a constant speed), and that there is not enough information given to answer the question.
Students might wonder many things, but in order to answer the question, they would need to know:
How fast do they each walk?
How far away is the library?
Ask students to open their books or devices and use the additional information to solve the problem by any method they choose. If desired, remind students of tools that may be appropriate including double number lines or tables of equivalent ratios.
Student Facing
Lin and Jada each walk at a steady rate from school to the library. Lin can walk 13 miles in 5 hours, and Jada can walk 25 miles in 10 hours. They each leave school at 3:00 and walk
\(3\frac14\)
miles to the library. What time do they each arrive?
Student Response
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Activity Synthesis
Invite students who used different representations and lines of reasoning to share. If no student mentions it, demonstrate how to represent on person's trip using a table of equivalent ratios with columns representing distance and time. Ask a student to explain how to use the table to reason about the distance traveled in 1 hour and the time it takes to travel 1 mile.
One way to reason is to notice that Lin can walk 26 miles in 10 hours, so she walks slightly faster than Jada (who can complete 25 miles in 10 hours) and should arrive a bit sooner. Both of these ways of reasoning are in preparation for the following activity.


aligns: no