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: Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
problem_activity: Optional activity
The purpose of this task is for students to write a recursive definition for a sequence that represents a mathematical context and to create other representations of the sequence.
Monitor for groups who create their recursive definitions in different ways to share during the whole-class discussion. For example, some students may first create a table showing step number and the associated values while others may draw additional steps.
Allow students to use graph paper to sketch their graphs if needed. Making graphing technology available gives students an opportunity to choose appropriate tools strategically (MP5).
Launch
Display the image for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time, and then ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the image. After all responses have been recorded without commentary or editing, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to respectfully disagree, ask for clarification, or point out contradicting information.
If not brought up by students, ask “How would you describe the total number of small squares in Step 3 compared to the total number of small squares in Step 2?” (There are 9, or
\(3^2\)
, more squares in Step 4 than in Step 3.)
Arrange students in groups of 2. Encourage them to check in with their partner frequently as they work through the task.
Action and Expression: Internalize Executive Functions.
Provide students with grid or graph paper to organize their work with representations of geometric and arithmetic sequences.
Supports accessibility for: Language; Organization
Student Facing
Here is a pattern where the number of small squares increases with each new step.
[IMG]
Write a recursive definition for the total number of small squares
\(S(n)\)
in Step
\(n\)
.
Sketch a graph of
\(S\)
that shows Steps 1 to 7.
Is this sequence geometric, arithmetic, or neither? Be prepared to explain how you know.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Student Facing
Are you ready for more?
Start with a circle. If you make 1 cut, you have 2 pieces. If you make 2 cuts, you can have a maximum of 4 pieces. If you make 3 cuts, you can have a
maximum
of 7 pieces.
Draw a picture to show how 3 cuts can give 7 pieces.
Find the maximum number of pieces you can get from 4 cuts.
From 5 cuts.
Can you find a function that gives the maximum number of pieces from
\(n\)
cuts?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Anticipated Misconceptions
Students may not be sure where to begin with the graph since no axes are provided in the task statement. Encourage these students to first figure out what values they need to plot before drawing, scaling, and labeling their axes.
Activity Synthesis
The goal of this discussion is for students to share how they reasoned about a recursive definition for
\(S\)
and how they created their graph for Steps 1 to 7. Invite previously identified groups to share how they created their definitions, making sure to display for all to see any additional representations they used to help their thinking. After these students have shared, ask "Did anyone use a different strategy for writing their definition?" and invite any new students to share their thinking.
Conclude the discussion by reviewing graphing strategies. Select 2–4 students to share how they created their sketch of
\(S\)
. In particular, focus on how the scale on each axis was chosen.


aligns: yes