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: Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R.

Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.

Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.
problem_activity: Activity
This activity is designed to give students opportunities to use their understandings from this unit to perform mathematical modeling.
The trail mix context is familiar from the previous activity, but students are challenged to choose quantities, determine how to represent them, interpret and reason about them, and use the model they create to make choices. It also enables students to reflect on their model and revise it as needed (MP4).
Students are likely to want to use graphing technology, as the nutritional information involves decimals and the inequalities written would be inconvenient to graph by hand. This is an opportunity for students to choose tools strategically (MP5).
No time estimate is given here because the time would depend on decisions about the research students do and on the expectations for collaboration and presentation.
Launch
Arrange students in groups of 2–4. Provide access to Desmos or other graphing technology.
Explain the expectations for researching nutritional values, for collaboration with group members, and for presentation of student work. (If each group is presenting one response, provide each group with tools for creating a visual display. If each student is presenting a response, give each student tools for creating a visual display.)
Engagement: Develop Effort and Persistence.
Provide prompts, reminders, guides, rubrics, or checklists that focus on increasing the length of on-task orientation in the face of distractions. For each question, allot a specific amount of work time and display expectations. Display a countdown timer, along with a bulleted list of what teams or individuals are to produce to complete a given step. For instance, during question 1, display a 5 minute countdown timer and the list: “You’re finished if you. . . (1) Have your two ingredients (2) Have the nutrition information ready.”
Supports accessibility for: Attention; Social-emotional skills
Student Facing
It's time to design your own trail mix!
Choose two ingredients that you like to eat. (You can choose from the ingredients in the previous activity, or you can look up nutrition information for other ingredients.)
Think about the constraints for your trail mix. What do you want to be true about its calories, protein, sugar, fat, or fiber?
Write inequalities to represent your constraints. Then, graph the inequalities.
[IMG]
Is it possible to make trail mix that meets all your constraints using your ingredients? If not, make changes to your constraints or your ingredients and record them here.
Write a possible combination of ingredients for your trail mix.
Pause here so your teacher can review your work and give further instructions for displaying your work.
Student Response
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Activity Synthesis
Select groups to share their visual displays. Encourage students to ask questions about the mathematical thinking or design approach that went into creating the display. Here are questions for discussion, if not already mentioned by students:
What constraints did every group use?
How do the graphs of the various mixes compare?
Did anyone have to revise or change their model in order to come up with a solution they could use?
How did you use the graph to choose a recipe for your mix?
Representing, Conversing: MLR7 Compare and Connect.
Use this routine to prepare students for the whole-class discussion. At the appropriate time, invite groups to create a visual display of their work. Students should consider what types of details (annotations, notes, diagrams, arrows, etc.) to include on their displays that will help communicate their reasoning. Begin the whole-class discussion by selecting and arranging 2–4 displays for all to see. Give students 2–3 minutes of quiet think time to interpret the displays before inviting the authors to present their work.
Design Principle(s): Optimize output; Cultivate conversation


correct_option_index: 1