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: Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.
problem_activity: Activity
The purpose of this activity is for students to use the structure of the volume formula for cones to calculate missing dimensions of a cone given other dimensions. Students are given the image of a generic cone with marked dimensions for the radius, diameter, and height to help their reasoning about the different rows in the table.
While completing the table, students work with approximations and exact values of
\(\pi\)
as well as statements that require reasoning about squared values. The final row of the table asks students to find missing dimensions given an expression representing volume that uses letters to represent the height and the radius. This requires students to manipulate expressions consisting only of variables representing dimensions.
Encourage students to make use of work done in some rows to help find missing information in other rows. Identify students who use this strategy, and ask them to share during the discussion.
Launch
Give students 6–8 minutes of quiet work time, followed by a whole-class discussion.
Action and Expression: Internalize Executive Functions.
Chunk this task into more manageable parts to support students who benefit from support with organization and problem solving. For example ask students to complete two lines of the table at a time and then assess for accuracy and comprehension.
Supports accessibility for: Organization; Attention
Student Facing
[IMG]
Each row of the table has some information about a particular cone. Complete the table with the missing dimensions.
| diameter (units) | radius (units)  | area of the base (square units) | height (units) | volume of cone (cubic units) |
|------------------|-----------------|---------------------------------|----------------|------------------------------|
|                  |        4        |                                 |       3        |                              |
|                  | \(\frac{1}{3}\) |                                 |       6        |                              |
|                  |                 |           \(144\pi\)            |  \(\frac14\)   |                              |
|        20        |                 |                                 |                |          \(200\pi\)          |
|                  |                 |                                 |       12       |          \(64\pi\)           |
|                  |                 |                                 |       3        |             3.14             |
Student Response
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Student Facing
Are you ready for more?
A
frustum
is the result of taking a cone and slicing off a smaller cone using a cut parallel to the base.
[IMG]
Find a formula for the volume of a frustum, including deciding which quantities you are going to include in your formula.
Student Response
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click here to register or sign in
for free access to Extension Student Response.
Anticipated Misconceptions
Students might try to quickly fill in the missing dimensions without the proper calculations. Encourage students to use the volume of a cone equation and the given dimensions to figure out the unknown dimensions.
Activity Synthesis
Select previously identified students to share the strategies they used to fill in the missing information. Ask students:
“Which information, in your opinion, was the hardest to calculate?”
“If you had to pick two pieces of information given in the table which information would you want? Why?”
Select a few rows of the table, and ask students how they might find the volume of a cylinder with the same radius and height as the cone. (Multiply by 3)
Make sure students understand that when working with the volume formula for either a cylinder or cone, if they know two out of three for radius, height and volume, they can always calculate the third.
Speaking: MLR8 Discussion Supports.
Use this routine to support whole-class discussion. For each strategy that is shared, ask students to restate and/or revoice what they heard using mathematical language. Consider providing students time to restate what they hear to a partner, before selecting one or two students to share with the class. Ask the original speaker if their peer was accurately able to restate their thinking. Call students' attention to any words or phrases that helped to clarify the original statement. This will help students to produce and make sense of the language needed to communicate their own ideas when calculating missing dimensions of a cone.
Design Principle(s): Support sense-making; Optimize output (for explanation)


correct_option_index: 0