options: Counting & Cardinality: students learn to know number names and the count sequence, count to tell the number of objects, and compare numbers.

Statistics & Probability: students learn to analyze and produce data distributions, and build understanding of univariate and bivariate data. They learn to interpret data, make statistical inferences, justify conclusions, understand rules of probability, and use probability to make decisions.

Measurement & Data: students learn to work with data and measure attributes such as time, money, length, area, and volume. They may learn to compare measurements to operations and convert between different units of measure.

Number Systems and Quantity: students learn to understand the complex number system and reason quantitatively.

Operations & Algebra: students learn to solve problems using algebraic thinking and operations such as addition, subtraction, multiplication, and division. They may learn to identify and explain arithmetic patterns, evaluate or manipulate numerical expressions, and reason with equations and inequalities.

Ratios & Proportional Relationships: students learn to recognize, describe, represent, and reason with ratios, rates, proportional relationships, and percent.

Modeling: students learn to choose and use appropriate mathematics and statistics to analyze real-world empirical situations, improve decisions, and report on their conclusions and the reasoning behind them.

Number & Operations in Base Ten: students learn to work with the base-ten system and build place value understanding.

Number & Operations - Fractions: students learn to understand fractions as numbers and may work with them using addition, subtraction, multiplication, and division.

Geometry: students learn to classify geometric figures by their properties, and understand and compare the relationships between them. Students develop and use formulas to compute lengths, areas, and volumes, and they use transformations to generate new shapes and compare existing ones. Geometry can be studied with and without coordinates.

Functions: students learn to define, use, and evaluate functions to model relationships between quantities.
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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for free access to Student Response.
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: 4