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

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

Interpret the structure of expressions.

Represent and solve problems involving addition and subtraction.

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

Work with radicals and integer exponents.

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

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

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

Solve equations and inequalities in one variable.

Use properties of operations to generate equivalent expressions.

Understand the relationship between zeros and factors of polynomials.

Use the four operations with whole numbers to solve problems.

Represent and analyze quantitative relationships between dependent and independent variables.

Add and subtract within 20.

Generate and analyze patterns.

Analyze patterns and relationships.

Write expressions in equivalent forms to solve problems.

Apply and extend previous understandings of arithmetic to algebraic expressions.

Create equations that describe numbers or relationships

Solve systems of equations.

Rewrite rational expressions.

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

Work with addition and subtraction equations.

Perform arithmetic operations on polynomials.

Represent and solve problems involving multiplication and division.

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

Reason about and solve one-variable equations and inequalities.

Write and interpret numerical expressions.

Use polynomial identities to solve problems.

Multiply and divide within 100.

Gain familiarity with factors and multiples.

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

Represent and solve equations and inequalities graphically.
problem_activity: Optional activity
This activity is optional because it goes beyond the depth of understanding required by the standards.
The purpose of this activity is to connect graphs of quadratic functions with solutions to related quadratic equations to tell how many solutions an equation has, and whether those solutions are real or non-real. Students have already solved two of the equations in the previous activity, and the equation they haven’t solved involves the perfect square that connects all three of the equations. Graphically,
\(y=f(x)\)
intersects the
\(x\)
-axis twice, which corresponds to the two real solutions to
\(x^2 - 8x + 13 = 0\)
. The graph
\(y=g(x)\)
intersects the
\(x\)
-axis once, which corresponds to the single real solution to
\(x^2 - 8x + 16 = 0\)
. However, the graph
\(y=h(x)\)
doesn’t intersect the
\(x\)
-axis, which reflects the fact that the complex solutions to
\(x^2 - 8x + 19 = 0\)
can’t be described with a real number line.
Launch
Provide access to graphing technology.
Student Facing
How many real solutions does each equation have? How many non-real solutions?
\(x^2 - 8x + 13 = 0\)
\(x^2 - 8x + 16 = 0\)
\(x^2 - 8x + 19 = 0\)
How do the graphs of these functions help us answer the previous question?
\(f(x) = x^2 - 8x + 13\)
\(g(x) = x^2 - 8x + 16\)
\(h(x) = x^2 - 8x + 19\)
Student Response
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for free access to Student Response.
Anticipated Misconceptions
Some students may attempt to plot the complex solutions in the coordinate plane as if it were the complex plane. Remind these students that points in the coordinate plane are pairs of real numbers
\((x,y)\)
, whereas points in the complex plane are single complex numbers. In other words, the point
\((2,4)\)
in the coordinate plane is a different kind of object than the point
\(2+4i\)
in the complex plane.
Activity Synthesis
Select students to share their solutions and discuss the connections between the graphs and the equations. Discuss the idea that if a graph of a quadratic function doesn’t cross a given horizontal line (in this case,
\(x=0\)
) this doesn’t indicate that the related equation has no solutions, but rather that it has no
real
solutions that can be graphed with real number lines.
Speaking: MLR8 Discussion Supports.
To help students produce statements about the connection between graphs of quadratic functions and solutions to related quadratic equations, provide sentence frames such as: “If _____ then _____ because…” and  “On the graph, I see _____, so I know….” Show concepts multi-modally by displaying the graphs of
\(f(x)\)
,
\(g(x)\)
, and
\(h(x)\)
.
Design Principle(s): Support sense-making


correct_option_index: 9

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