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:
{
  "explanation": "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:
{
  "explanation": "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:
{
  "explanation": "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: Analyze and solve pairs of simultaneous linear equations.

Solve linear equations in one variable.
problem_activity: Activity
The purpose of this activity is for students to recognize that multiplying an equation by a value results in the same solution. This will be useful when transforming equations for the elimination method of solving systems of equations in the supported Algebra lesson. Students will work in pairs and each partner is responsible for answering the questions in either column A or column B. Although each row has two different problems, they share the same answer. Ensure that students work their problems out independently and collaborate with one another when they do not arrive to the same answers. Students construct viable arguments and critique the reasoning of others (MP3) when they resolve errors by critiquing their partner's work or explaining their reasoning.
Launch
Arrange students in groups of two. In each group, ask students to decide who will work on column A and who will work on column B. If students are unfamiliar with the row game structure, demonstrate the protocol before they start working.
Student Facing
Work independently on your column. Partner A completes the questions in column A only and partner B completes the questions in column B only. Your answers in each row should match. Work on one row at a time and check if your answer matches your partner’s before moving on. If you don’t get the same answer, work together to find any mistakes.
Solve each equation.
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}, "strong": [{"_value": "row"}]}, {"_attributes": {"scope": "col"}, "_value": "column A"}, {"_attributes": {"scope": "col"}, "_value": "column B"}]}]}], "tbody": [{"tr": [{"td": [{"p": [{"strong": [{"_value": "1"}]}]}, {"p": [{"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "24745"}, "_value": "\\(3x-1=5\\)"}]}]}, {"p": [{"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "24750"}, "_value": "\\(6x-2=10\\)"}]}]}]}, {"td": [{"p": [{"strong": [{"_value": "2"}]}]}, {"p": [{"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "24757"}, "_value": "\\(4(x+1) = 3x - 12\\)"}]}]}, {"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "24764"}, "_value": "\\(4x+4 = 3(x-4)\\)"}]}]}, {"td": [{"p": [{"strong": [{"_value": "3"}]}]}, {"p": [{"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "24768"}, "_value": "\\(14x+10=4x+6\\)"}]}]}, {"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "24773"}, "_value": "\\(7x+5 = 2x+3\\)"}]}]}, {"td": [{"p": [{"strong": [{"_value": "4"}]}]}, {"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "24781"}, "_value": "\\(6x+3=33\\)"}]}, {"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "24786"}, "_value": "\\(4x+2=22\\)"}]}]}, {"td": [{"p": [{"strong": [{"_value": "5"}]}]}, {"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "24792"}, "_value": "\\(4x + 5y = 2 - 4x + 5y\\)"}]}, {"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "24799"}, "_value": "\\(4x + 9y = 2 + 9y - 4x\\)"}]}]}, {"td": [{"p": [{"strong": [{"_value": "6"}]}]}, {"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "24805"}, "_value": "\\(2x + 6y = 10\\)"}]}, {"span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "24810"}, "_value": "\\(5x + 15y = 25\\)"}]}]}]}]}]}
Student Response
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Activity Synthesis
The purpose of the discussion is to notice that equations that have been multiplied by the same value have the same solution.
Consider asking students to explain why their solutions might be the same for each row.
For each row, the two equations are equivalent. Sample responses for each row:
The equation in column B is 2 times the equation in column A.
In each column, one side of the equation is the distributed form of the same side of the equation in the other column.
The equation in column B is
\(\frac{1}{2}\)
of the equation in column A.
The equation in column B is
\(\frac{2}{3}\)
of the equation in column A.
The equation in column B has
\(4y\)
added to both sides of the equation from column A.
The equation in column B is
\(\frac{5}{2}\)
of the equation in column A.
While there may be other valid ways of connecting the equations (for example, in row 4, dividing the equation in column A by 3 and dividing the equation in column B by 2 result in the same equation), encourage students to recognize that there is a single multiple of one equation that results in the other equation.


correct_option_index: 1