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: Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.

Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.
problem_activity: Activity
In this activity, students see that taking the square root of each side of an equation doesn’t necessarily preserve solutions like other algebraic moves they have used in the past.
Launch
Arrange students in groups of 2. Encourage students to discuss their responses with a partner as they work. If they disagree with their partner, they should work to reach agreement.
Writing, Listening, Conversing: MLR1 Stronger and Clearer Each Time.
Use this routine to help students improve their written responses for Priya’s question. Give students time to meet with 2–3 partners to share and receive feedback on their responses. Display feedback prompts that will help students strengthen their ideas and clarify their language. For example, “How do the graphs show that there are two solutions?”, “What is the definition of a square root?”, and “What happened when Tyler took the square root of each side?” Invite students to go back and revise or refine their written responses based on the feedback from peers. This will help students justify why the equation
\(x^2-1=3\)
must have two solutions.
Design Principle(s): Optimize output (for justification); Cultivate conversation
Student Facing
Tyler was solving this equation:
\(\displaystyle x^2 - 1 = 3\)
He said, “I can add 1 to each side of the equation and it doesn’t change the equation. I get
\(x^2 = 4\)
.”
Priya said, “It does change the equation. It just doesn’t change the solutions!” Then she showed these two graphs.
Figure A
[IMG]
Figure B
[IMG]
How can you see the solution to the equation
\(x^2 - 1 = 3\)
in Figure A?
How can you see the solution to the equation
\(x^2 = 4\)
in Figure B?
Use the graphs to explain why the equations have the same solutions.
Tyler said, “Now I can take the square root of each side to get the solution to
\(x^2 = 4\)
. The square root of
\(x^2\)
is
\(x\)
. The square root of 4 is 2.” He wrote:
\(\begin{align}x^2 &= 4\\ \sqrt{x^2} &= \sqrt{4} \\ x &= 2 \end{align}\)
Priya said, “But the graphs show that there are
two
solutions!” What went wrong?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Activity Synthesis
Invite students to share their responses. Discuss the connection between the graphs and the solutions to the equations. The important idea is that the operation of taking the square root results in a positive number, but there are two square roots of every non-zero number, one positive and one negative. This means that when Tyler takes the square root of both sides of the equation, he does not get an equivalent equation. To find all possible solutions, he should remember that
\(x^2\)
has two roots,
\(x\)
and
\(\text-x\)
, and that 4 has two square roots, 2 and -2. This gives him the pair of equations
\(x=2\)
and
\(x=\text-2\)
.
Representation: Internalize Comprehension.
Use color-coding and annotations to highlight connections between representations in a problem. As students share their reasoning about the solutions to the equations, scribe their thinking on a visible display. Use the same color to highlight the connection between the graphs and the solutions to the equations.
Supports accessibility for: Visual-spatial processing; Conceptual processing


correct_option_index: 1