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: 
Analyze and solve pairs of simultaneous linear equations.

Answer: 
Yes. The last three rows of the table in the first problem involves solving pairs of simultaneous linear equations. The second word problem can also be represented as two simultaneous linear equations.

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_pos_0

8
standard_description: Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane. For example, given the rule “Add 3” and the starting number 0, and given the rule “Add 6” and the starting number 0, generate terms in the resulting sequences, and observe that the terms in one sequence are twice the corresponding terms in the other sequence. Explain informally why this is so.
problem_activity: Activity
The purpose of this activity is for students to solve two related quadratic equations by completing the square. The process of solving will be nearly identical, with the difference being that one will involve square roots of a negative number and the other will not. It is not necessary for students to prove a general statement about all monic quadratic equations, but it is important for students to compare and understand why some have real solutions and others have non-real solutions.
Launch
Representation: Internalize Comprehension.
Activate or supply background knowledge about completing the square. Allow students to use calculators to ensure inclusive participation in the activity.
Supports accessibility for: Memory; Conceptual processing
Student Facing
Solve these equations by completing the square.
\(x^2 - 8x + 13 = 0\)
\(x^2 - 8x + 19 = 0\)
Student Response
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Student Facing
Are you ready for more?
For which values of
\(a\)
does the equation
\(x^2 - 8x + a = 0\)
have two real solutions? One real solution? No real solutions? Explain your reasoning.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Activity Synthesis
Select students to share their solutions. Record and display their thinking for all to see. If not brought up during the discussion, ask students to clarify why one equation has real solutions while the other has complex solutions.
Display these three equations for all to see:
\(x^2+14x+44=0\)
\(x^2+14x+49=0\)
\(x^2+14x+54=0\)
Ask students, “How many solutions do these equations have? Which, if any, of these equations has solutions that involve imaginary numbers?” (
\(x^2+14x+44\)
has two real solutions,
\(x^2+14x+49\)
has one real solution, and
\(x^2+14x+54\)
has two non-real solutions. Compare each left hand side to the perfect square
\((x+7)^2\)
, which is equal to
\(x^2+14x+49\)
. When completing the square, the first equation will have
\((x+7)^2 = 5\)
, the second equation will have
\((x+7)^2 = 0\)
, and the third equation will have
\((x+7)^2 = \text-5\)
.)
Conversing: MLR2 Collect and Display.
During the discussion, listen for and collect the language students use to compare characteristics of equations that have real and complex solutions. Call students’ attention to phrases such as “the right hand side is negative”, “factored form”, or “replace with the imaginary number
\(i\)
.” Write the students’ words and phrases on a visual display and update it throughout the remainder of the lesson. Remind students to borrow language from the display as needed. This will help students use mathematical language to explain solutions to quadratic equations.
Design Principle(s): Maximize meta-awareness


aligns: no