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.


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.


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.


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: 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.)

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.

Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.
problem_activity: Activity
It is common to use positive numbers to represent credit and negative numbers to represent debts on a bill. This task introduces students to this convention and asks them to solve addition and subtraction questions in that context. Note that whether a number should be positive or negative is often a choice, which means one must be very clear about explaining the interpretation of a signed number in a particular context (MP6).
For the second question, monitor for students who find the amount each week and sum those, and students who sum the value of the electricity used and the value of the electricity generated separately, and then find the sum of those.
Launch
Arrange students in groups of 2. Give students 4 minutes of quiet work time followed by partner discussion. Follow with a whole-class discussion.
Representation: Access for Perception.
Read Han’s electricity bill situation aloud. Students who both listen to and read the information will benefit from extra processing time. Check in with students to see if they have any questions about the context of the situation.
Supports accessibility for: Language
Reading: MLR6 Three Reads.
Use this routine to support reading comprehension of this word problem, without solving it for students. In the first read, read the problem with the goal of comprehending the situation (e.g., This problem is about a house with solar panels that generate energy.). In the second read, ask students to analyze the mathematical quantities (e.g., used
$
83.56 worth of electricity, generated
$
6.75 worth of electricity, current charges are
$
83.56, Solar Credit is -
$
6.75 and the amount due is
$
74.81). In the third read, ask students to brainstorm possible strategies to answer the questions. This helps students connect the language in the word problem and the reasoning needed to solve the problem, while still keeping the intended level of cognitive demand in the task.
Design Principle(s): Support sense-making​
Student Facing
Han's family got a solar panel. Each month they get a credit to their account for the electricity that is generated by the solar panel. The credit they receive varies based on how sunny it is.
[IMG]
| Current charges:                         $                        83.56                         Solar Credit: -                         $                        6.75                         Amount due:                         $                        76.81 |
|------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|
Here is their electricity bill from January.
In January they used
$
83.56 worth of electricity and generated
$
6.75 worth of electricity.
In July they were traveling away from home and only used
$
19.24 worth of electricity. Their solar panel generated
$
22.75 worth of electricity. What was their amount due in July?
The table shows the value of the electricity they used and the value of the electricity they generated each week for a month. What amount is due for this month?
|        | used (                               $                              ) | generated (                               $                              ) |
|--------|-----------------------------------------------------------------------|----------------------------------------------------------------------------|
| week 1 |                                                                       |                                                                            |
| week 2 |                                                                       |                                                                            |
| week 3 |                                                                       |                                                                            |
| week 4 |                                                                       |                                                                            |
What is the difference between the value of the electricity generated in week 1 and week 2? Between week 2 and week 3?
Student Response
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Student Facing
Are you ready for more?
While most rooms in any building are all at the same level of air pressure, hospitals make use of "positive pressure rooms" and "negative pressure rooms."  What do you think it means to have negative pressure in this setting?  What could be some uses of these rooms?
Student Response
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Activity Synthesis
Ask one or more students to share their answer to the first question and resolve any discrepancies.
Ask selected students to share their reasoning for the second questions. Discuss the relative merits of different approaches to solving the problem.
Finish by going over the solution to the third question. Point out that the bill will reflect a negative number in the amount due section, but we can interpret this to mean that the family receives a credit, and it will be applied to their next bill.


correct_option_index: 0