few_shots: Example 1:
Problem/activity:
Task
Materials
One 10-frame per pair of students (see example below)
[IMG]
Counters, 10 per pair of students
Decks of number cards 0-5 (with 4 of each number) or two 0-5 dice
Setup
This activity is done in pairs. The students sit side by side with 10-frame and deck of cards or dice. If they have cards, the cards are shuffled and placed face down in front of them.
Action
Player 1 flips a card or rolls the die. They place that many counters on the 10-frame. Player 2 flips a card or rolls the die. They place that many more counters on the 10-frame next to the counters already on the ten frame. The students count the total together.
A recording sheet can be added when the students have been introduced to equations and number sentences. For those students who can handle higher numbers, a deck of 0-10 cards and two 10-frames can be used.
This game can also be played alone.


Standard options: 
A. Represent addition and subtraction with objects, fingers, mental images, drawings (Drawings need not show details, but should show the mathematics in the problem. (This applies wherever drawings are mentioned in the Standards.)), sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations.
B. For any number from 1 to 9, find the number that makes 10 when added to the given number, e.g., by using objects or drawings, and record the answer with a drawing or equation.
C. Decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by using objects or drawings, and record each decomposition by a drawing or equation (e.g., 5 = 2 + 3 and 5 = 4 + 1).
D. Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem.
E. Fluently add and subtract within 5.


Your response:
Thought: This task involves representing addition with objects such as number cards and dice.
Answer: A

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
$$-\frac{1}{4}$$


Standard options: 
A. Solve real-world and mathematical problems involving the four operations with rational numbers. (Computations with rational numbers extend the rules for manipulating fractions to complex fractions.)
B. Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.
C. Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.


Your response:
Thought: This problem involves understanding fractions and division to divide rational numbers, such as 1, -4, 4, and -1. This problem also involves an understanding of multiplication: if p and q are integers, then -(p/q) = (-p)/q = p/(-q).
Answer: B

Example 3:
Problem/activity:
Find the general solution for:
$${\mathrm{cos}\left({\pi\over2}+x\right)=\mathrm{tan}{\pi\over4}}$$


Standard options: 
A. Prove the Pythagorean identity sin^(2)(θ) + cos^(2)(θ) = 1 and use it to find sin(θ), cos(θ),or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.
B. Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.


Your response:
Thought: An initial step when solving for x in this problem requires the use of the addition formula for cosine on the left side of the equation. Solving this problem does not require use of the Pythagorean identity.
Answer: B

Now, assign the following problem or activity to one or more relevant standards it aligns with.
shot_info: im_task_001477

K

fl_problem_001097

7

fl_problem_001894

HS_0
options: Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.

Use the relation i^(2) = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.

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
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


correct_option_index: 1