few_shots: Example:
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?


Standard options: 
A. Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.
B. Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
C. Fluently multiply multi-digit whole numbers using the standard algorithm.


Your response:
Thought: 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.
Answer: C

Now, assign the following problem or activity to one or more relevant standards it aligns with.
shot_info: fl_problem_000674

5
options: Interpret expressions that represent a quantity in terms of its context.

Use the structure of an expression to identify ways to rewrite it. For example, see x^(4) – y^(4 )as (x^(2))^(2 )– (y^(2))^(2), thus recognizing it as a difference of squares that can be factored as (x^(2 )– y^(2))(x^(2 )+ y^(2)).
problem_activity: Activity
Earlier, students noticed that when an expression
\(kx+m\)
is squared and written in standard form
\(ax^2+bx+c\)
, the value of
\(b\)
is
\(2km\)
and the value of
\(c\)
is
\(m^2\)
. In this activity, they use these observations to complete expressions to make them perfect squares and then write them as squared factors. Then, they apply this skill to solve quadratic equations.
Launch
To reiterate the connections between
\((kx + m)^2\)
and its equivalent expression of the form
\(ax^2+bx+c\)
, display these two expressions for all to see and ask students:
“If the two expressions are perfect squares and are equivalent, how is
\(a\)
related to
\(k\)
?” (
\(a\)
is
\(k^2\)
.)
“How is
\(c\)
related to
\(m\)
?” (
\(c\)
is
\(m^2\)
.)
“How is
\(b\)
related to
\(k\)
and
\(m\)
?” (
\(b\)
is
\(2km\)
.)
Tell students they will use these insights to complete some squares. Consider keeping students in groups of 2.
Representation: Internalize Comprehension.
Use color and annotations to illustrate student thinking. As students share their reasoning about the connections between
\((kx+m)^2\)
and
\(ax^2+ bx+c\)
, scribe their thinking on a visible display. Invite students to suggest language or diagrams to include that will support their understanding. Keep the display visible as students continue to work.
Supports accessibility for: Visual-spatial processing; Conceptual processing
Student Facing
Find the value of
\(c\)
to make each expression in the left column a perfect square in standard form. Then, write an equivalent expression in the form of squared factors. In the last row, write your own pair of equivalent expressions.
| standard form                       \((ax^2+bx+c)\) | squared factors                       \((kx+m)^2\) |
|-----------------------------------------------------|----------------------------------------------------|
|                  \(100x^2+80x+c\)                   |                                                    |
|                   \(36x^2-60x+c\)                   |                                                    |
|                   \(25x^2+40x+c\)                   |                                                    |
|                  \(0.25x^2-14x+c\)                  |                                                    |
|                                                     |                                                    |
Solve each equation by completing the square:
\(25x^2 + 40x = \text-12\)
\(36x^2 - 60x + 10 = \text-6\)
Student Response
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Anticipated Misconceptions
If students struggle to follow the generalized relationships between
\((kx + m)^2\)
and its equivalent expression of the form
\(ax^2+bx+c\)
(as discussed in the launch), consider revisiting a concrete example from an earlier activity—for example,
\((5x-2)^2\)
. Ask students to relate each term in the squared factor to the terms of its expression in standard form,
\(25x^2 - 20x + 4\)
.
If further scaffolding is helpful, consider using an example with an additional variable (for instance,
\((3x+n)^2\)
and its counterpart in standard form,
\(9x^2 + 2(3)(n)x + n^2\)
) before generalizing the equivalence of the two forms in entirely abstract terms.
Activity Synthesis
Ask students to write the standard-form expression they invented on a piece of scrap paper, except without the constant term. (From the blank row of the table.) Invite them to switch papers with a partner, and complete the square and write each other’s expressions in factored form. If they get stuck, encourage them to talk with their partner to work together and fix any mistakes.
If time permits, invite students to share their responses and strategies. Discuss questions such as:
“How did you know what value to use for
\(c\)
in, say,
\(100x^2 + 80x + c\)
?” (
\(k^2=100\)
, so
\(k\)
is either 10 or -10. If
\(k\)
is 10, then
\(2(10)m=80\)
and
\(m=4\)
. We know that
\(c=m^2\)
, so
\(c\)
is
\(4^2\)
or 16. If
\(k\)
is -10,then
\(2(\text-10) m=80\)
and
\(m= \text-4\)
. Squaring -4 still gives 16.)
“How did you know whether
\(c\)
would be positive or negative?” (
\(c\)
is some number squared, so it will always be positive.)
“In the equation
\(36x^2 - 60x + 10 = \text-6\)
, the expression on the left already has 10 as a constant term but it is not a perfect square. How do we make it a perfect square?” (25 would make it a perfect square, so we can either add 15 to each side of the equation, or first subtract 10 from each side and then add 25.)
As students explain their solution methods on the second question, record and display their reasoning for all to see, or display a worked solution such as:
\(\begin{align} 36x^2 - 60x + 10 &= \text-6\\ 36x^2 - 60x + 10 + 15 &= \text-6 + 15\\ 36x^2 - 60x + 25 &= 9\\ (6x-5)^2 &=9 \end{align}\)
\( \)
\(\begin {align} 6x-5 =3 \quad &\text{or} \quad 6x-5 =\text-3\\ 6x = 8 \quad &\text{or} \quad 6x = 2\\ x = \frac86 =\frac43 \quad & \text{or} \quad x = \frac13 \end{align}\)


correct_option_index: 1