few_shots: Example 1:
Problem/activity:
Task
Materials
One 10-frame per pair of students (see example below)
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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: Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., p^(2)). For example, by truncating the decimal expansion of Ö2, show that Ö2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations.

Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.
problem_activity: Activity
In this activity, students use what they know about finding the area of a square from a given side length to think about the converse problem: how do you find the side length of a square with a given area? Students solve this problem in general in the next lesson, but this lesson sets them up for that work.
First, they work with squares whose side lengths lie along grid lines, so both the side lengths and areas can be found simply by counting length or area units.
They also see a “tilted” square whose area can be determined by counting square units or by decomposing and rearranging the square or a related figure. In this case, the side lengths are not apparent, so students have to reason that since the area is the same as one of the earlier squares, the side lengths must also be the same. They should check that this appears to be true by tracing one of the squares on tracing paper and seeing that it does, in fact, line up with the other.
The last question asks them to use the insights from the previous questions, tracing paper, or a ruler to estimate the side lengths of some other tilted squares (MP5). In the next lesson, they will see other techniques for estimating those side lengths as well.
Monitor for students who use these strategies for the fourth question:
reason that the side lengths are between 5 and 6 because the areas are between 25 and 36 (MP3)
measure with tracing paper or a ruler
Launch
Provide access to tracing paper. Students in groups of 2. 3 minutes of quiet work time followed by a partner discussion on the first three problems. Pause students and make sure everyone agrees on the answers for the first three questions. Students continue with the fourth question, then a whole-class discussion.
Engagement: Develop Effort and Persistence.
Encourage and support opportunities for peer interactions. Invite students to talk about their ideas with a partner before writing them down. Display sentence frames to support students when they explain their strategy. For example, “I noticed _____ so I…”, “By comparing triangles I…”, or “Another strategy could be _____ because….”
Supports accessibility for: Language; Social-emotional skills
Conversing, Writing: MLR5 Co-Craft Questions.
Before revealing the questions in this activity, display the diagram of the three squares and invite students to write possible mathematical questions about the diagram. Ask students to compare the questions they generated with a partner before sharing questions with the whole class. Listen for and amplify questions about the side lengths and area of each square. If no student asks about the side lengths or area of each square, ask students to adapt a question to align with the learning goals of this lesson. Then reveal and ask students to work on the actual questions of the task. This routine will help develop students’ meta-awareness of language as they generate questions about the side lengths and areas of squares in the coordinate plane.
Design Principle(s): Maximize meta-awareness
Student Facing
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What is the side length of square A? What is its area?
What is the side length of square C? What is its area?
What is the area of square B? What is its side length? (Use tracing paper to check your answer to this.)
Find the areas of squares D, E, and F. Which of these squares must have a side length that is greater than 5 but less than 6? Explain how you know.
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Student Response
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Activity Synthesis
A key take-away from this activity is that if the area of a square is in between the areas of two other squares, then the side lengths are also in between, which in the case of the squares in this task can be reinforced by estimating with tracing paper.
Select previously identified students to share their solutions. Sequence with a reasoning strategy followed by a measuring strategy using a ruler or tracing paper. Use the measuring strategy to verify the reasoning strategy.


correct_option_index: 0