options: Create equations that describe numbers or relationships

Work with radicals and integer exponents.

Perform arithmetic operations on polynomials.

Represent and solve problems involving multiplication and division.

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

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

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

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

Use properties of operations to generate equivalent expressions.

Write and interpret numerical expressions.

Use polynomial identities to solve problems.

Represent and solve equations and inequalities graphically.

Add and subtract within 20.

Solve systems of equations.

Multiply and divide within 100.

Solve equations and inequalities in one variable.

Rewrite rational expressions.

Represent and analyze quantitative relationships between dependent and independent variables.

Apply and extend previous understandings of arithmetic to algebraic expressions.

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

Interpret the structure of expressions.

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

Use the four operations with whole numbers to solve problems.

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

Work with addition and subtraction equations.

Understand the relationship between zeros and factors of polynomials.

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

Write expressions in equivalent forms to solve problems.

Gain familiarity with factors and multiples.

Represent and solve problems involving addition and subtraction.

Reason about and solve one-variable equations and inequalities.

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

Generate and analyze patterns.

Analyze patterns and relationships.
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: 1