options: Use properties of operations to generate equivalent expressions.

Represent and solve problems involving addition and subtraction.

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

Apply and extend previous understandings of arithmetic to algebraic expressions.

Gain familiarity with factors and multiples.

Generate and analyze patterns.

Use polynomial identities to solve problems.

Write and interpret numerical expressions.

Interpret the structure of expressions.

Write expressions in equivalent forms to solve problems.

Represent and solve problems involving multiplication and division.

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

Work with addition and subtraction equations.

Analyze patterns and relationships.

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.

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

Understand the relationship between zeros and factors of polynomials.

Work with radicals and integer exponents.

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

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

Rewrite rational expressions.

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

Represent and analyze quantitative relationships between dependent and independent variables.

Represent and solve equations and inequalities graphically.

Multiply and divide within 100.

Reason about and solve one-variable equations and inequalities.

Add and subtract within 20.

Use the four operations with whole numbers to solve problems.

Solve equations and inequalities in one variable.

Solve systems of equations.

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

Create equations that describe numbers or relationships

Perform arithmetic operations on polynomials.
problem_activity: Warm-up
This warm-up begins where students left off at the end of the previous lesson, considering how to multiply strategically to “clear the denominators” in a rational equation. The purpose of this warm-up is to elicit the idea that when multiplying to clear denominators, we only need to multiply by the least common denominator shared by the different terms in the expression. This idea will be used in the following activities as students solve rational equations and investigate extra solutions that sometimes arise in the solving process.
While students may notice and wonder many things about these images, the multiplication by
\(x(x+2)\)
and not
\(x(x+2)^2\)
(the result of multiplying together all the denominators) is the important discussion point.
Launch
Display the partially solved equation for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice and wonder with their partner, followed by a whole-class discussion.
Student Facing
What do you notice? What do you wonder?
\(\displaystyle \begin{align} \frac{3}{x(x-2)} &= \frac{2x+1}{x-2} \\ \frac{3}{x(x-2)} \boldcdot x(x-2) &= \frac{2x+1}{x-2} \boldcdot x(x-2) \\ 3 &= 2x^2 + x \\ 0 &= 2x^2+x-3 \\ \end{align}\)
Student Response
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Activity Synthesis
Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the image. After all responses have been recorded without commentary or editing, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to respectfully disagree, ask for clarification, or point out contradicting information. If students do not recall or have not heard the phrase “clearing the denominators” for this type of multiplying, tell them it is a useful description for the first few steps shown.
If time allows, have students finish solving for
\(x\)
by either using the quadratic formula or factoring (-1.5, 1).


correct_option_index: 15