few_shots: 
Example 1:
Problem/activity:
You’re mixing ingredients for cookies. The recipe says to combine 6 tablespoons, or ¾ stick, of butter with 1 cup of sugar. You accidentally mix in a whole stick of butter (8 tablespoons) with the cup of sugar. How can you fix this?


Concept/skill: 
Recognize and represent proportional relationships between quantities.

Answer: 
No. Usually problems that teach recognizing and representing proportional relationships between quantities involve deciding whether two quantities are in a proportional relationship or identifying the constant of proportionality, and then representing proportional relationships by equations or explaining proportional relationship using coordinate graphs.

Example 2:
Problem/activity:
For each pair of numbers, determine if the values are equivalent. Explain your reasoning for each one.
a.
$$-0.25$$
and
$$\frac{1}{-4}$$
b.
$$-\left ( \frac{1}{4} \right )$$
and
$$\frac{-1}{-4}$$
c.
$$\frac{-1}{4}$$
and
$$-\frac{1}{4}$$


Concept/skill: 
Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.

Answer: 
No. The expressions shown in this problem do not involve multiplying fractions with other fractions, nor whole numbers with fractions.

Example 3:
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_000972_neg_0

6

fl_problem_001097_neg_0

7

fl_problem_001392_neg_0

8
standard_description: Extend polynomial identities to the complex numbers. For example, rewrite x^(2) + 4 as (x + 2i)(x – 2i).
problem_activity: Activity
The purpose of this activity is to introduce the idea that sums of complex numbers can be rewritten as a single complex number in the form
\(a+bi\)
, where
\(a\)
and
\(b\)
are real numbers. In previous lessons, students have used the complex plane to visualize complex numbers, and now they continue that work using the complex plane to visualize why the sum of two complex numbers is another complex number.
Launch
Display the image from the task statement 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 1 minute to discuss the things they notice with their partner, followed by a brief whole-class discussion. 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.
Arrange students in groups of 2 and give them a few minutes of work time for the activity, encouraging students to discuss each problem and, if there is disagreement, to work to reach agreement. Follow up with a whole-class discussion.
Conversing, Writing: MLR5 Co-Craft Questions.
Use this routine to spark students’ curiosity about the sums of complex numbers. Display only the image and ask students to write down possible mathematical questions that could be asked about the situation. Invite students to compare their questions before revealing the activity’s questions. Listen for and amplify any questions involving how parts of the expression are represented on the diagram. This will help students create the language of mathematical questions before feeling pressure to produce solutions.
Design Principle(s): Maximize meta-awareness; Support sense-making
Student Facing
This diagram represents
\((2 + 3i) + (\text- 8 - 8i)\)
.
[IMG]
How do you see
\(2 + 3i\)
represented?
How do you see
\(\text- 8 - 8i\)
represented?
What complex number does
\(A\)
represent?
Add “like terms” in the expression
\((2 + 3i) + (\text- 8 - 8i)\)
. What do you get?
Write these sums and differences in the form
\(a+bi\)
, where
\(a\)
and
\(b\)
are real numbers.
\((\text- 3 + 2i) + (4 - 5i)\)
(Check your work by drawing a diagram.)
\((\text- 37 - 45i) + (11 + 81i)\)
\((\text- 3 + 2i) - (4 - 5i)\)
\((\text- 37 - 45i) - (11 + 81i)\)
Student Response
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Activity Synthesis
Select students to share their arithmetic results and connect them to arrows in the complex plane. It is important to discuss that doing operations with complex numbers results in another complex number. In order to see this, it is helpful to write the result in the form
\(a+bi\)
, where
\(a\)
and
\(b\)
are real numbers. When a complex number is written this way,
\(a\)
is called the
real part
of the number, and
\(bi\)
is called the
imaginary part
.
If time allows, ask students, “Is
\((2 + 3i) + (\text- 8 - 8i)\)
the same as
\((\text- 8 - 8i) + (2 + 3i)\)
?” (Yes. The diagrams will look different, but, for example, going right 2 then left 8 is the same as going left 8 and right 2.)
Representation: Internalize Comprehension.
Use color-coding and annotations to highlight connections between representations in a problem. For example, highlight connects between representations by showing the expressions
\(2 + 3i\)
and using one color to connect it to the arrows in the complex plane. Use a second color to highlight the connection between the expression
\(\text-8 - 8i\)
and the arrows in the complex plane.
Supports accessibility for: Visual-spatial processing


aligns: no