options: Use the four operations with whole numbers to solve problems.

Represent and solve problems involving addition and subtraction.

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

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

Use properties of operations to generate equivalent expressions.

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

Solve systems of equations.

Apply and extend previous understandings of arithmetic to algebraic expressions.

Generate and analyze patterns.

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

Create equations that describe numbers or relationships

Understand the relationship between zeros and factors of polynomials.

Represent and analyze quantitative relationships between dependent and independent variables.

Work with addition and subtraction equations.

Analyze patterns and relationships.

Multiply and divide within 100.

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

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

Gain familiarity with factors and multiples.

Solve equations and inequalities in one variable.

Add and subtract within 20.

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

Interpret the structure of expressions.

Reason about and solve one-variable equations and inequalities.

Rewrite rational expressions.

Represent and solve problems involving multiplication and division.

Write and interpret numerical expressions.

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

Use polynomial identities to solve problems.

Perform arithmetic operations on polynomials.

Represent and solve equations and inequalities graphically.

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

Work with radicals and integer exponents.

Write expressions in equivalent forms to solve problems.
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: 19

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