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: Add up to four two-digit numbers using strategies based on place value and properties of operations.

Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds.

Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.

Mentally add 10 or 100 to a given number 100–900, and mentally subtract 10 or 100 from a given number 100–900.

Explain why addition and subtraction strategies work, using place value and the properties of operations. (Explanations may be supported by drawings or objects.)
problem_activity: Narrative
The purpose of this activity is for students to compare different methods for solving problems within 100 using data presented in a bar graph. Students may use whatever method makes the most sense to them. The synthesis focuses on sharing multiple methods that students use to find the difference. Monitor for students who use methods that rely on using the bar graph to count on or count back and those that use more abstract methods, such as adding or subtracting by place value.
For example, when combining categories, some students may choose to use the graph to count on. Other students may choose to combine tens and ones with or without drawing a base-ten diagram or other representation.
Engagement: Provide Access by Recruiting Interest.
Provide choice. Invite students to choose a strategy and tool that works for them. Encourage students to use that same strategy and tool for both problems so they are not overwhelmed.
Supports accessibility for: Conceptual Processing, Organization, Attention
Required Materials
Materials to Gather
Connecting cubes in towers of 10 and singles
Required Preparation
Create towers of 10 with the connecting cubes.
Have single connecting cubes available.
Launch
Groups of 2
Display the bar graph.
“What does this graph tell us?” (students’ favorite movie snacks, students picked their favorite movie snacks)
1 minute: quiet think time
1 minuter: partner discussion
Share responses.
Give students access to connecting cubes in towers of ten and singles.
Activity
“Use the bar graph to answer the questions. Show your thinking using drawings, numbers, or words.”
“You can use the connecting cubes or any of the other representations we saw in the warm-up to help you.”
8 minutes: independent work time
“Now compare your methods with your partner. How are they similar or different?”
4 minutes: partner discussion
As students work, monitor for students who:
use the bar graph to count on or count back
use the connecting cubes or base-ten drawings to show adding or subtracting tens with tens and ones with ones
Student Facing
Use the bar graph to answer the questions.
[IMG]
What is the total number of students that chose popcorn or pretzels? Show your thinking.
How many more students chose nachos than chose popcorn? Show your thinking.
[IMG]
Student Response
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Activity Synthesis
Invite previously identified students to share the method they used to find how many more students chose nachos than chose popcorn.
As needed, record student methods using equations.
Consider asking:
“How are these methods the same? How are they different?”
“How does the method work? Why does each method find the same value?”


correct_option_index: 2