options: Work with equal groups of objects to gain foundations for multiplication.

Analyze patterns and relationships.

Understand the connections between proportional relationships, lines, and linear equations.

Analyze and solve linear equations and pairs of simultaneous linear equations.

Understand properties of multiplication and the relationship between multiplication and division.

Use polynomial identities to solve problems.

Understand the relationship between zeros and factors of polynomials.

Interpret the structure of expressions.

Represent and solve equations and inequalities graphically.

Generate and analyze patterns.

Create equations that describe numbers or relationships

Use properties of operations to generate equivalent expressions.

Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.

Write expressions in equivalent forms to solve problems.

Solve real-life and mathematical problems using numerical and algebraic expressions and equations.

Write and interpret numerical expressions.

Solve problems involving the four operations, and identify and explain patterns in arithmetic.

Work with addition and subtraction equations.

Represent and analyze quantitative relationships between dependent and independent variables.

Multiply and divide within 100.

Add and subtract within 20.

Gain familiarity with factors and multiples.

Rewrite rational expressions.

Understand and apply properties of operations and the relationship between addition and subtraction.

Use the four operations with whole numbers to solve problems.

Solve systems of equations.

Solve equations and inequalities in one variable.

Understand solving equations as a process of reasoning and explain the reasoning.

Reason about and solve one-variable equations and inequalities.

Perform arithmetic operations on polynomials.

Represent and solve problems involving addition and subtraction.

Work with radicals and integer exponents.

Apply and extend previous understandings of arithmetic to algebraic expressions.

Represent and solve problems involving multiplication and division.
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
Teachers with a valid work email address can
click here to register or sign in
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: 7

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