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: Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression 3 (2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6 (4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y.

Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them). For example, the expressions y + y + y and 3y are equivalent because they name the same number regardless of which number y stands for.

Write, read, and evaluate expressions in which letters stand for numbers.

Write and evaluate numerical expressions involving whole-number exponents.
problem_activity: Activity
The purpose of this activity is to give students an opportunity to practice checking whether a particular value is a solution to an equation, and to recall properties of operations and equality that preserve the solution set of an equation.
This will be useful when students consider when and why equations have the same solution in the associated Algebra 1 lesson.
Monitor for students who:
check the value of
\(x\)
in only one expression and reason about the second expression
identify properties by name such as commutative, distributive, and associative
identify properties informally such as “changing the order doesn’t matter” or “you can add first and then multiply or multiply first and then add” or “you can multiply three or more terms in any order” or “you can add the same thing to each side”
manipulate the expressions
graph one or both expressions
Making graphing technology available gives students an opportunity to choose appropriate tools strategically.
Student Facing
For each pair of equations, decide whether the given value of
\(x\)
is a solution to one or both equations:
Is
\(x = 2\)
a solution to:
\(x(2 + 3) = 10\)
\(2x + 3x = 10\)
Is
\(x = 3\)
a solution to:
\(x - 4 = 1\)
\(4 - x = 1\)
Is
\(x = \text{-}2\)
a solution to:
\(7x = \text{-}14\)
\(x \boldcdot 14 = \text{-}28\)
Is
\(x = \text{-}1\)
a solution to:
\(x + 3 = 2\)
\(3 + x = 2\)
Is
\(x = \text{-5}\)
a solution to:
\(3 - x = 8\)
\(5 - x = 10\)
Is
\(x = (8 + 1)+3\)
a solution to:
\(\frac{12}{2} = \frac12(x)\)
\(18 = 2x\)
Is
\(x = 2\)
a solution to:
\(\frac{12}{x} = 6\)
\(6x = 12\)
Is
\(x = \frac{10}{3}\)
a solution to:
\(\text{-}1 + 3x = 9\)
\(9 = 3x - 1\)
Is
\(x = \frac12\)
a solution to:
\(5(x + 1) = \frac{15}{2}\)
\(5x + 1 = \frac{15}{2}\)
Student Response
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Activity Synthesis
Ask students to talk to their neighbor about some things they can look for, to see if two equations are likely to have the same solution or different solutions. If needed, draw students’ attention to which equations had the same solution and which had different solutions, perhaps by displaying them for all to see, grouped by whether the equations were equivalent or not.
Call on students to share their thinking and make an informal list displayed for all to see. It’s fine if students do not name all possible properties, or only describe them informally. They will have more opportunities to encounter them formally in their Algebra 1 course.
Remind students that
equivalent
equations have the same solution set. Use the language of
equivalent
interchangeably with “have the same solution,” and monitor for students to pick up on this vocabulary.


correct_option_index: 2