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: Know there is a complex number i such that i^(2) = –1, and every complex number has the form a + bi with a and b real.
problem_activity: Activity
In the previous activity, students found that negative real numbers have two imaginary square roots. In future lessons, students will use the quadratic formula to find complex solutions, which involves interpreting expressions like
\(\sqrt{\text-36}\)
. In this activity, students learn and apply the convention that for any positive real number
\(a\)
,
\(\sqrt{\text-a}=i\sqrt{a}\)
.
Launch
Explain to students that sometimes negative numbers end up inside of the square root symbol as a result of the steps used to solve equations. In the previous activity, we saw that negative numbers have two square roots, one on the positive imaginary axis, and one on the negative imaginary axis. By convention, the
\(\sqrt{}\)
symbol with a negative number inside refers to the square root on the positive imaginary axis. For example,
\(\sqrt{\text-4}=2i\)
.
Student Facing
Write these imaginary numbers using the number
\(i\)
.
\(\sqrt{\text- 36}\)
\(\sqrt{\text- 10}\)
\(\text- \sqrt{\text- 100}\)
\(\text- \sqrt{\text- 17}\)
Student Response
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Anticipated Misconceptions
Students may need to work through some intermediate steps in order to express the square roots in terms of
\(i\)
. For example, students could break a square root apart like this:
\(\sqrt{\text-36} = \sqrt{\text-1}\boldcdot \sqrt{36} = i \boldcdot 6 = 6i\)
.
Activity Synthesis
Select students to share their responses and encourage students to show that their answers make sense by squaring. Mention that for answers that involve
\(i\)
multiplied by a radical, it is customary to write the
\(i\)
first so it doesn’t look like it might be part of the radicand. For example,
\(\sqrt{\text-15}\)
is usually written as
\(i\sqrt{15}\)
rather than
\(\sqrt{15}i\)
.
Engagement: Develop Effort and Persistence.
Encourage and support opportunities for peer interactions. Prior to the whole-class discussion, invite students to share their work with a partner. Display sentence frames to support student conversation such as: “First, I _____ because . . .”, “I noticed _____ so I . . .”, “Why did you . . .?”, “I agree/disagree because . . . .”
Supports accessibility for: Language; Social-emotional skills


aligns: yes