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: Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠0, and use rate language in the context of a ratio relationship. For example, “This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar.” “We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger.” (Expectations for unit rates in this grade are limited to non-complex fractions.)

Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, “The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak.” “For every vote candidate A received, candidate C received nearly three votes.”

Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.
problem_activity: Hummingbird feeders can be filled with a simple sugar water mixture. Two different recipes are shown below.
|                  |   Water (cups)   | Sugar (cups) | $$2$$ | $$\frac{1}{2}$$ | $$4$$ | $$1$$ | $$16$$ | $$4$$ | $$28$$ | $$7$$ | $$32$$ | $$8$$ |   | Water (cups) | Sugar (cups) | $$2\frac{2}{3}$$ | $$1$$ | $$4$$ | $$1\frac{1}{2}$$ | $$8$$ | $$3$$ | $$24$$ | $$9$$ | $$32$$ | $$12$$ |
|------------------|------------------|--------------|-------|-----------------|-------|-------|--------|-------|--------|-------|--------|-------|---|--------------|--------------|------------------|-------|-------|------------------|-------|-------|--------|-------|--------|--------|
|   Water (cups)   |   Sugar (cups)   |              |       |                 |       |       |        |       |        |       |        |       |   |              |              |                  |       |       |                  |       |       |        |       |        |        |
|      $$2$$       | $$\frac{1}{2}$$  |              |       |                 |       |       |        |       |        |       |        |       |   |              |              |                  |       |       |                  |       |       |        |       |        |        |
|      $$4$$       |      $$1$$       |              |       |                 |       |       |        |       |        |       |        |       |   |              |              |                  |       |       |                  |       |       |        |       |        |        |
|      $$16$$      |      $$4$$       |              |       |                 |       |       |        |       |        |       |        |       |   |              |              |                  |       |       |                  |       |       |        |       |        |        |
|      $$28$$      |      $$7$$       |              |       |                 |       |       |        |       |        |       |        |       |   |              |              |                  |       |       |                  |       |       |        |       |        |        |
|      $$32$$      |      $$8$$       |              |       |                 |       |       |        |       |        |       |        |       |   |              |              |                  |       |       |                  |       |       |        |       |        |        |
|   Water (cups)   |   Sugar (cups)   |              |       |                 |       |       |        |       |        |       |        |       |   |              |              |                  |       |       |                  |       |       |        |       |        |        |
| $$2\frac{2}{3}$$ |      $$1$$       |              |       |                 |       |       |        |       |        |       |        |       |   |              |              |                  |       |       |                  |       |       |        |       |        |        |
|      $$4$$       | $$1\frac{1}{2}$$ |              |       |                 |       |       |        |       |        |       |        |       |   |              |              |                  |       |       |                  |       |       |        |       |        |        |
|      $$8$$       |      $$3$$       |              |       |                 |       |       |        |       |        |       |        |       |   |              |              |                  |       |       |                  |       |       |        |       |        |        |
|      $$24$$      |      $$9$$       |              |       |                 |       |       |        |       |        |       |        |       |   |              |              |                  |       |       |                  |       |       |        |       |        |        |
|      $$32$$      |      $$12$$      |              |       |                 |       |       |        |       |        |       |        |       |   |              |              |                  |       |       |                  |       |       |        |       |        |        |
Compare the two different recipes. Which recipe is more sugary? Justify your response.


correct_option_index: 2