options: Create equations that describe numbers or relationships

Interpret the structure of expressions.

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

Use polynomial identities to solve problems.

Represent and solve equations and inequalities graphically.

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

Analyze patterns and relationships.

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

Perform arithmetic operations on polynomials.

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

Multiply and divide within 100.

Solve equations and inequalities in one variable.

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

Use properties of operations to generate equivalent expressions.

Represent and solve problems involving multiplication and division.

Reason about and solve one-variable equations and inequalities.

Gain familiarity with factors and multiples.

Represent and solve problems involving addition and subtraction.

Apply and extend previous understandings of arithmetic to algebraic expressions.

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

Add and subtract within 20.

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

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

Write expressions in equivalent forms to solve problems.

Understand the relationship between zeros and factors of polynomials.

Solve systems of equations.

Write and interpret numerical expressions.

Work with radicals and integer exponents.

Represent and analyze quantitative relationships between dependent and independent variables.

Rewrite rational expressions.

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

Generate and analyze patterns.

Use the four operations with whole numbers to solve problems.

Work with addition and subtraction equations.
problem_activity: Activity
Students have seen quadratic expressions in both standard form and factored form since the beginning of the unit. In this activity, they learn to distinguish the expressions by their forms and to refer to each form by its formal name. Refining their language about the different forms prepares students to be more precise in their thinking about the graphs of quadratic functions in later lessons (MP6).
Launch
Arrange students in groups of 2. Give students quiet work time and then time to share their work with a partner.
Writing, Conversing: MLR1 Stronger and Clearer Each Time.
Use this routine to help students improve their writing, by providing them with multiple opportunities to clarify their explanations through conversation. Give students time to meet with 1–2 partners to share their response to the final question, “Why do you think that form is called factored form?” Provide listeners with prompts for feedback that will help their partner add detail to strengthen and clarify their ideas. For example, students can ask their partner, “Can you say more about what each expression means?” or “I understand _____, but can you clarify . . .” Next, provide students with 2–3 minutes to revise their initial draft based on feedback from their peers.
Design Principle(s): Optimize output (for explanation)
Action and Expression: Internalize Executive Functions.
Provide students with a graphic organizer that can be used to brainstorm the activity. Include one column for standard form and one column for factored form. Include a space at the bottom of the organizer for student-generated notes or reminders about standard form and factored form.
Supports accessibility for: Language; Organization
Student Facing
The quadratic expression
\(x^2 + 4x + 3\)
is written in
standard form
.
Here are some other quadratic expressions. The expressions on the left are written in standard form and the expressions on the right are not.
Written in standard form:
\(x^2 – 1\)
\( x^2 + 9x\)
\(\frac12 x^2\)
\(4x^2 – 2x + 5\)
\(\text-3x^2 – x + 6\)
\(1 - x^2\)
Not written in standard form:
\((2x + 3)x\)
\((x+1)(x-1)\)
\(3(x-2)^2 +1\)
\( \text-4(x^2 + x) +7\)
\( (x+8)(\text{-}x+5)\)
What are some characteristics of expressions in standard form?
\((x+1)(x-1)\)
and
\((2x + 3)x\)
in the right column are quadratic expressions written in
factored form
. Why do you think that form is called factored form?
Student Response
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Student Facing
Are you ready for more?
What quadratic expression can be described as being both standard form and factored form? Explain how you know.
Student Response
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Activity Synthesis
Display the two columns of expressions for all to see. Solicit students’ ideas on the features of each form. Record their responses for all to see. Invite other students to express agreement or disagreement or to clarify their fellow students’ responses.
Define a quadratic expression in
standard form
explicitly as
\(ax^2 + bx + c\)
. Explain that we refer to
\(a\)
as the coefficient of the squared term
\(x^2\)
,
\(b\)
as the coefficient of the linear term
\(x\)
, and
\(c\)
as the constant term.
Ask students:
“How would you write
\((2x + 3)x\)
in standard form?” (
\(2x^2 + 3x\)
)
“The expression
\(2x^2 + 3x\)
only has two terms. Is it still in standard form?” (Yes, there is no constant term, which means
\(c\)
is 0.)
“How would you write
\(\text {-4}(x^2 + \frac14x) +7\)
in standard form?” (
\(\text {-4}x^2 - x + 7\)
)
“What are the values of the coefficients
\(a\)
and
\(b\)
?” (-4 and -1) “What about the constant term?” (7)
Then, clarify that a quadratic expression in
factored form
is a product of two factors that are each a linear expression. For example,
\((x+1)(x-1)\)
,
\((2x + 3)x\)
, and
\(x(4x)\)
all have two linear expressions for their factors. An expression with two factors that are linear expressions and a third factor that is a constant, for example:
\(2(x+2)(x-1)\)
, is also in factored form.
Tell students that each form gives us interesting or useful information about the function that the expression represents, and they will learn more about this in future lessons.


correct_option_index: 1

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