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

Apply and extend previous understandings of arithmetic to algebraic expressions.

Write and interpret numerical expressions.

Work with radicals and integer exponents.

Represent and solve equations and inequalities graphically.

Generate and analyze patterns.

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

Represent and analyze quantitative relationships between dependent and independent variables.

Rewrite rational expressions.

Multiply and divide within 100.

Understand the relationship between zeros and factors of polynomials.

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

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

Solve equations and inequalities in one variable.

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

Add and subtract within 20.

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

Write expressions in equivalent forms to solve problems.

Represent and solve problems involving addition and subtraction.

Represent and solve problems involving multiplication and division.

Use properties of operations to generate equivalent expressions.

Work with addition and subtraction equations.

Solve systems of equations.

Reason about and solve one-variable equations and inequalities.

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

Perform arithmetic operations on polynomials.

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

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

Analyze patterns and relationships.

Gain familiarity with factors and multiples.

Use polynomial identities to solve problems.

Interpret the structure of expressions.

Create equations that describe numbers or relationships

Solve real-life and mathematical problems using numerical and algebraic expressions and equations.
problem_activity: Warm-up
In this activity, students have an opportunity to notice and make use of structure (MP7) in order to identify a point where the graphs of two given functions intersect. The work here leads directly into the next activity in which students use algebraic methods to identify all points of intersection between two polynomials.
Launch
Display one problem at a time. Give students quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all problems displayed throughout the talk. Follow with a whole-class discussion.
Representation: Internalize Comprehension.
To support working memory, provide students with sticky notes or mini whiteboards.
Supports accessibility for: Memory; Organization
Student Facing
Mentally identify a point where the graphs of the two functions intersect, if one exists.
\(f(x)=x\)
and
\(g(x)=3\)
\(j(x)=(x+3)(x-3)\)
and
\(k(x)=0\)
\(m(x)=(x+3)(x-3)\)
and
\(n(x)=(x-3)\)
\(p(x)=(x+5)(x-5)\)
and
\(q(x)=(x+3)(x-3)\)
Student Response
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Activity Synthesis
Ask students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking:
“Who can restate
\(\underline{\hspace{.5in}}\)
’s reasoning in a different way?”
“Did anyone have the same strategy but would explain it differently?”
“Did anyone solve the problem in a different way?”
“Does anyone want to add on to
\(\underline{\hspace{.5in}}\)
’s strategy?”
“Do you agree or disagree? Why?”
If not brought up during the discussion, remind students that these are called systems of equations. Ask, “What are some ways you have solved systems of equations in the past?” (Graphing, substitution, elimination.) Tell students that in the next activity, they are going to think about how to use substitution to solve systems of equations in which at least one of the equations is a quadratic.
Speaking: MLR8 Discussion Supports.
Display sentence frames to support students when they explain their strategy. For example, “First, I _____ because . . . .” or “I noticed _____ so I . . . .” Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class.
Design Principle(s): Optimize output (for explanation)


correct_option_index: 22