options: Apply and extend previous understandings of arithmetic to algebraic expressions.

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

Generate and analyze patterns.

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

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

Write expressions in equivalent forms to solve problems.

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

Add and subtract within 20.

Use polynomial identities to solve problems.

Solve equations and inequalities in one variable.

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

Multiply and divide within 100.

Work with addition and subtraction equations.

Represent and solve equations and inequalities graphically.

Represent and solve problems involving addition and subtraction.

Represent and analyze quantitative relationships between dependent and independent variables.

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

Solve systems of equations.

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

Gain familiarity with factors and multiples.

Rewrite rational expressions.

Use properties of operations to generate equivalent expressions.

Reason about and solve one-variable equations and inequalities.

Write and interpret numerical expressions.

Use the four operations with whole numbers to solve problems.

Represent and solve problems involving multiplication and division.

Analyze patterns and relationships.

Understand the relationship between zeros and factors of polynomials.

Perform arithmetic operations on polynomials.

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

Create equations that describe numbers or relationships

Work with radicals and integer exponents.

Interpret the structure of expressions.

Understand properties of multiplication and the relationship between multiplication and division.
problem_activity: Activity
The purpose of this activity is for students to use rational approximations of irrational numbers to place both rational and irrational numbers on a number line, and to reinforce the definition of a cube root as a solution to the equation of the form
\(x^3=a\)
. This is also the first time that students have thought about negative cube roots.
Launch
No access to calculators. Students in groups of 2. Two minutes of quiet work time, followed by partner then whole-class discussion.
Action and Expression: Develop Expression and Communication.
Maintain a display of important terms and vocabulary. During the launch take time to review the following term with a number line and sample calculations from previous activities that students will need to access for this activity: cube root.
Supports accessibility for: Memory; Language
Student Facing
The numbers
\(x\)
,
\(y\)
, and
\(z\)
are positive, and:
\(\displaystyle x^3= 5\)
\(\displaystyle y^3= 27\)
\(\displaystyle z^3= 700\)
[IMG]
Plot
\(x\)
,
\(y\)
, and
\(z\)
on the number line. Be prepared to share your reasoning with the class.
Plot
\(\text- \sqrt[3]{2}\)
on the number line.
Student Response
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Student Facing
Are you ready for more?
Diego knows that
\(8^2=64\)
and that
\(4^3=64\)
. He says that this means the following are all true:
\(\sqrt{64}=8\)
\(\sqrt[3]{64}=4\)
\(\sqrt{\text -64}=\text-8\)
\(\sqrt[3]{\text -64}=\text -4\)
Is he correct? Explain how you know.
Student Response
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for free access to Extension Student Response.
Activity Synthesis
Display the number line from the activity for all to see. Select groups to share how they chose to place values onto the number line. Place the values on the displayed number line as groups share, and after each placement poll the class to ask if students used the same reasoning or different reasoning. If any students used different reasoning, invite them to share with the class.
Conclude the discussion by asking students to share how they placed -
\(\sqrt[3]{2}\)
on the number line.
Representing, Conversing, Listening: MLR7 Compare and Connect.
As students prepare their work before the discussion, look for approaches that favor the algebraic definition, and approaches that use the geometric interpretation. Call students’ attention to the different ways students describe how they operated on numbers cubed to determine the approximate cube roots, and to the different ways these operations are made visible in each representation (e.g., using the fact that
\(\sqrt[3]{a}\)
is a solution to the equation
\(x^3 = a\)
, and finding rational approximations of the irrational numbers to plot the cube roots on the number line). Emphasize language used to make sense of strategies reinforcing the definition of a cube root as a solution to the equation of the form
\(x^3 = a\)
.
Design Principle(s): Maximize meta-awareness; Support sense-making


correct_option_index: 31