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: Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.
problem_activity: Activity
The purpose of this activity is for students to solve two related quadratic equations by completing the square. The process of solving will be nearly identical, with the difference being that one will involve square roots of a negative number and the other will not. It is not necessary for students to prove a general statement about all monic quadratic equations, but it is important for students to compare and understand why some have real solutions and others have non-real solutions.
Launch
Representation: Internalize Comprehension.
Activate or supply background knowledge about completing the square. Allow students to use calculators to ensure inclusive participation in the activity.
Supports accessibility for: Memory; Conceptual processing
Student Facing
Solve these equations by completing the square.
\(x^2 - 8x + 13 = 0\)
\(x^2 - 8x + 19 = 0\)
Student Response
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Student Facing
Are you ready for more?
For which values of
\(a\)
does the equation
\(x^2 - 8x + a = 0\)
have two real solutions? One real solution? No real solutions? Explain your reasoning.
Student Response
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click here to register or sign in
for free access to Extension Student Response.
Activity Synthesis
Select students to share their solutions. Record and display their thinking for all to see. If not brought up during the discussion, ask students to clarify why one equation has real solutions while the other has complex solutions.
Display these three equations for all to see:
\(x^2+14x+44=0\)
\(x^2+14x+49=0\)
\(x^2+14x+54=0\)
Ask students, “How many solutions do these equations have? Which, if any, of these equations has solutions that involve imaginary numbers?” (
\(x^2+14x+44\)
has two real solutions,
\(x^2+14x+49\)
has one real solution, and
\(x^2+14x+54\)
has two non-real solutions. Compare each left hand side to the perfect square
\((x+7)^2\)
, which is equal to
\(x^2+14x+49\)
. When completing the square, the first equation will have
\((x+7)^2 = 5\)
, the second equation will have
\((x+7)^2 = 0\)
, and the third equation will have
\((x+7)^2 = \text-5\)
.)
Conversing: MLR2 Collect and Display.
During the discussion, listen for and collect the language students use to compare characteristics of equations that have real and complex solutions. Call students’ attention to phrases such as “the right hand side is negative”, “factored form”, or “replace with the imaginary number
\(i\)
.” Write the students’ words and phrases on a visual display and update it throughout the remainder of the lesson. Remind students to borrow language from the display as needed. This will help students use mathematical language to explain solutions to quadratic equations.
Design Principle(s): Maximize meta-awareness


aligns: no