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: Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.
problem_activity: Activity
This activity allows students to see how changing the target number in a guessing game changes the absolute guessing errors and changes the scatter plot of the data.
Here, students see that they are still finding the difference between the guesses and a number, that the absolute guessing errors are still all positive, and that the data points still seem to form a V above the horizontal axis. What is different is that the position of the V shape has now shifted horizontally and that the number of points forming the two pieces of the V may have changed.
The work here prepares students to later see the absolute value function in terms of finding the distance between input values and 0.
The activity is written so that students could perform the calculations and graphing individually and by hand. To make more time for reasoning and analysis, however, use of statistical and graphing technology for computation and plotting is recommended, as is splitting up the work among students.
Launch
Tell students that suppose there had been a mistake in the reported number of items in the jar, and that their job is to find out how the absolute guessing errors and the scatter plot would change once they find out the corrected number of items.
Consider giving one half of the class one value for the actual number of objects, and giving the other half of the class another value so that they could observe the general behavior of the function. (For example, give “50” to half the class and “40” to the other half.)
Students could follow the same process as earlier: calculating the absolute errors using the new "actual" number, recording them in a table, and plotting the data points on a coordinate plane. If doing so by hand, ask students to use Table B and the second coordinate plane on the handout given earlier. Keep students in groups of 2–4 so they could split up the calculations. Provide access to calculators.
Alternatively, give students the option of using technology (statistical or graphing tool) to calculate the absolute errors and to create the scatter plot. Another option is to arrange for the calculations to be split up among students and collected in a shared table or spreadsheet, and then create a class scatter plot, either by hand or using technology, displayed for all to see.
Student Facing
Earlier, you guessed the number of objects in a container and then your teacher told you the actual number.
Suppose your teacher made a mistake about the number of objects in the jar and would like to correct it. The actual number of objects in the jar is
\(\underline{\hspace{0.5in}}\)
.
Find the new absolute guessing errors based on this new information. Record the errors in Table B of the handout (or elsewhere as directed by your teacher).
Make 1–2 observations about the new set of absolute guessing errors.
Predict how the scatter plot would change given the new actual number of objects. (Would it have the same shape as in the first scatter plot? If so, what would be different about it? If not, what would it look like?)
Use technology to plot the points and test your prediction.
Can you write a rule for finding the output (absolute guessing error) given the input (a guess)?
Student Response
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Activity Synthesis
Invite students to share their observations of the new data set (or data sets, if two new "actual" numbers were given to students).
If time is limited, focus the discussion on the features of the new scatter plot, and on whether the relationship between the guesses and the absolute guessing errors form a function.
Display a completed scatter plot for all to see. Ask questions such as:
"How is this scatter plot like the first scatter plot we saw earlier?" (The points form two straight lines that seem to converge at a point, forming a V.)
"How is this scatter plot different than the first one?" (The points seem to have been shifted horizontally. Earlier, the two sets of points—each forming a line—seemed to meet at
\((47,0)\)
. Now they see to meet at
\((50,0)\)
.)
Students may also note that there were more points (or fewer points, depending on the data) that represent underestimates (or overestimates) in the new scatter plot than in the first one.
Conversing, Writing: MLR2 Collect and Display.
Before the whole-class discussion, invite students to share with their partner their observations of the new data set and compare them with what they saw in the previous activity. Listen for and collect vocabulary and phrases students use to describe similarities and differences between the scatter plots on a visual display. Include words and phrases such as “form a V,” “no negative numbers,” and “shift to the right” on the display for all to see. Remind students to borrow language from the display as needed. This will help students discuss the features of a scatter plot using appropriate mathematical language.
Design Principle(s): Support sense-making
Engagement: Develop Effort and Persistence.
Encourage and support opportunities for peer interactions. Invite students to talk with their partner about their observations of the new data set before writing them down. Display sentence frames to support students when they explain their strategy. For example “One thing that is the same is . . .”, “One thing that is different is . . . .”
Supports accessibility for: Language; Social-emotional skills


aligns: no