few_shots: 
Example:
Concept/skill: 
Fluently multiply multi-digit whole numbers using the standard algorithm.

Problem/activity:
Problem 1
Solve using the standard algorithm. If you get stuck, use an area model and/or the partial products algorithm to help. Then assess the reasonableness of your answer.
a.
$$5,803\times46$$
b.
$$63\times2,116$$
Problem 2
Dunkin' introduced a croissant doughnut in 2014. In the first three months of its debut, they sold 7,870 dozen of them per day. How many croissant doughnuts were sold in the first three months?


Answer: 
Yes. This set of problems involves multi-digit whole numbers because the numbers in each problem are four and two digit integers, and the mathematical calculations involved in both problems focus on multiplication. The first problem explicitly asks the student to solve its subproblems using the standard algorithm.

Now, determine whether the following problem or activity can enable students to understand the concepts or skills in the provided description.

shot_info: fl_problem_000674_pos_0

5
standard_description: Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions.
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
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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


aligns: no