options: 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.

Represent and solve equations and inequalities graphically.

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

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

Solve systems of equations.

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

Interpret the structure of expressions.

Represent and solve problems involving multiplication and division.

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

Generate and analyze patterns.

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

Represent and solve problems involving addition and subtraction.

Work with addition and subtraction equations.

Understand the relationship between zeros and factors of polynomials.

Work with radicals and integer exponents.

Rewrite rational expressions.

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

Perform arithmetic operations on polynomials.

Write expressions in equivalent forms to solve problems.

Create equations that describe numbers or relationships

Multiply and divide within 100.

Write and interpret numerical expressions.

Represent and analyze quantitative relationships between dependent and independent variables.

Use the four operations with whole numbers to solve problems.

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

Analyze patterns and relationships.

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

Gain familiarity with factors and multiples.

Apply and extend previous understandings of arithmetic to algebraic expressions.

Use polynomial identities to solve problems.

Add and subtract within 20.

Solve equations and inequalities in one variable.

Reason about and solve one-variable equations and inequalities.
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


correct_option_index: 32