options: Represent and solve equations and inequalities graphically.

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

Write expressions in equivalent forms to solve problems.

Multiply and divide within 100.

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

Add and subtract within 20.

Generate and analyze patterns.

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

Reason about and solve one-variable equations and inequalities.

Solve systems of equations.

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

Write and interpret numerical expressions.

Use properties of operations to generate equivalent expressions.

Apply and extend previous understandings of arithmetic to algebraic expressions.

Work with radicals and integer exponents.

Gain familiarity with factors and multiples.

Solve equations and inequalities in one variable.

Analyze patterns and relationships.

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

Understand the relationship between zeros and factors of polynomials.

Interpret the structure of expressions.

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

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

Perform arithmetic operations on polynomials.

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

Represent and solve problems involving addition and subtraction.

Represent and analyze quantitative relationships between dependent and independent variables.

Work with addition and subtraction equations.

Rewrite rational expressions.

Represent and solve problems involving multiplication and division.

Use the four operations with whole numbers to solve problems.

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

Create equations that describe numbers or relationships

Use polynomial identities to solve problems.
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


correct_option_index: 1