options: Operations & Algebra: students learn to solve problems using algebraic thinking and operations such as addition, subtraction, multiplication, and division. They may learn to identify and explain arithmetic patterns, evaluate or manipulate numerical expressions, and reason with equations and inequalities.

Number & Operations - Fractions: students learn to understand fractions as numbers and may work with them using addition, subtraction, multiplication, and division.

Functions: students learn to define, use, and evaluate functions to model relationships between quantities.

Ratios & Proportional Relationships: students learn to recognize, describe, represent, and reason with ratios, rates, proportional relationships, and percent.

Counting & Cardinality: students learn to know number names and the count sequence, count to tell the number of objects, and compare numbers.

Number Systems and Quantity: students learn to understand the complex number system and reason quantitatively.

Modeling: students learn to choose and use appropriate mathematics and statistics to analyze real-world empirical situations, improve decisions, and report on their conclusions and the reasoning behind them.

Measurement & Data: students learn to work with data and measure attributes such as time, money, length, area, and volume. They may learn to compare measurements to operations and convert between different units of measure.

Statistics & Probability: students learn to analyze and produce data distributions, and build understanding of univariate and bivariate data. They learn to interpret data, make statistical inferences, justify conclusions, understand rules of probability, and use probability to make decisions.

Geometry: students learn to classify geometric figures by their properties, and understand and compare the relationships between them. Students develop and use formulas to compute lengths, areas, and volumes, and they use transformations to generate new shapes and compare existing ones. Geometry can be studied with and without coordinates.

Number & Operations in Base Ten: students learn to work with the base-ten system and build place value understanding.
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: 0