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.

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

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.

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.

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

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.

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

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.
problem_activity: Activity
In this activity, students see that taking the square root of each side of an equation doesn’t necessarily preserve solutions like other algebraic moves they have used in the past.
Launch
Arrange students in groups of 2. Encourage students to discuss their responses with a partner as they work. If they disagree with their partner, they should work to reach agreement.
Writing, Listening, Conversing: MLR1 Stronger and Clearer Each Time.
Use this routine to help students improve their written responses for Priya’s question. Give students time to meet with 2–3 partners to share and receive feedback on their responses. Display feedback prompts that will help students strengthen their ideas and clarify their language. For example, “How do the graphs show that there are two solutions?”, “What is the definition of a square root?”, and “What happened when Tyler took the square root of each side?” Invite students to go back and revise or refine their written responses based on the feedback from peers. This will help students justify why the equation
\(x^2-1=3\)
must have two solutions.
Design Principle(s): Optimize output (for justification); Cultivate conversation
Student Facing
Tyler was solving this equation:
\(\displaystyle x^2 - 1 = 3\)
He said, “I can add 1 to each side of the equation and it doesn’t change the equation. I get
\(x^2 = 4\)
.”
Priya said, “It does change the equation. It just doesn’t change the solutions!” Then she showed these two graphs.
Figure A
[IMG]
Figure B
[IMG]
How can you see the solution to the equation
\(x^2 - 1 = 3\)
in Figure A?
How can you see the solution to the equation
\(x^2 = 4\)
in Figure B?
Use the graphs to explain why the equations have the same solutions.
Tyler said, “Now I can take the square root of each side to get the solution to
\(x^2 = 4\)
. The square root of
\(x^2\)
is
\(x\)
. The square root of 4 is 2.” He wrote:
\(\begin{align}x^2 &= 4\\ \sqrt{x^2} &= \sqrt{4} \\ x &= 2 \end{align}\)
Priya said, “But the graphs show that there are
two
solutions!” What went wrong?
Student Response
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Activity Synthesis
Invite students to share their responses. Discuss the connection between the graphs and the solutions to the equations. The important idea is that the operation of taking the square root results in a positive number, but there are two square roots of every non-zero number, one positive and one negative. This means that when Tyler takes the square root of both sides of the equation, he does not get an equivalent equation. To find all possible solutions, he should remember that
\(x^2\)
has two roots,
\(x\)
and
\(\text-x\)
, and that 4 has two square roots, 2 and -2. This gives him the pair of equations
\(x=2\)
and
\(x=\text-2\)
.
Representation: Internalize Comprehension.
Use color-coding and annotations to highlight connections between representations in a problem. As students share their reasoning about the solutions to the equations, scribe their thinking on a visible display. Use the same color to highlight the connection between the graphs and the solutions to the equations.
Supports accessibility for: Visual-spatial processing; Conceptual processing


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