standard_description: Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 5^(1/3) to be the cube root of 5 because we want (5^(1/3))^(3) = 5(^(1/3))^(3) to hold, so (5^(1/3))^(3) must equal 5.
problem_activity: Activity
Students solved equations of the form
\(x+p = q\)
and
\(px=q\)
in grade 6, but the equations only involved positive values. This activity bridges their understanding of a solution to an equation as a value that makes the equation true with their understanding of operations involving negative numbers from this unit. This activity builds on the work students have done in this lesson evaluating expressions at different values.
Monitor for students who:
take an arithmetic approach by substituting in values and evaluating. (Does
\(\text-2 \boldcdot (\text-4.5) = \text-9\)
?)
take an algebraic approach by writing an equivalent equation using an inverse operation (If
\(\text-2 \boldcdot x=\text-9\)
, then
\(x=\text-9 \div \text-2\)
.)
Digital
Launch
Allow students 5 minutes quiet work time followed by whole class discussion.
The digital version has an applet that allows students to see how many correct answers they have at any time.
Action and Expression: Internalize Executive Functions.
To support development of organizational skills, check in with students within the first 2-3 minutes of work time. Check to make sure students matched the correct solution to the first equation and can explain their reasoning with an arithmetic or algebraic approach.
Supports accessibility for: Memory; Organization
Student Facing
Match each equation to a value that makes it true by dragging the answer to the corresponding equation. Be prepared to explain your reasoning.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Print
Launch
Allow students 5 minutes quiet work time followed by whole class discussion.
The digital version has an applet that allows students to see how many correct answers they have at any time.
Action and Expression: Internalize Executive Functions.
To support development of organizational skills, check in with students within the first 2-3 minutes of work time. Check to make sure students matched the correct solution to the first equation and can explain their reasoning with an arithmetic or algebraic approach.
Supports accessibility for: Memory; Organization
Student Facing
Match each equation to its solution.
\(\frac12 x=\text-5\)
\(\text-2x=\text-9\)
\(\text-\frac12 x=\frac14\)
\(\text-2x=7\)
\(x+\text-2 = \text-6.5\)
\(\text-2+x=\frac12\)
\(x = \text-4.5\)
\(x = \text-\frac12\)
\(x = \text-10\)
\(x = 4.5\)
\(x = 2 \frac12\)
\(x=\text-3.5\)
Be prepared to explain your reasoning.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Activity Synthesis
Select an equation for which there is a student who took an arithmetic approach and a student who took an algebraic approach. Ask students to share their reasoning for why a solution is correct. Sequence arithmetic approaches before algebraic approaches. An example of an arithmetic approach: “I know that -3.5 is the solution to
\(\text-2x=7\)
, because I know that
\(\text-2 \boldcdot (\text-3.5)=7\)
. An example of an algebraic approach: “If
\(\text-2x=7\)
, then I know that
\(x=7 \div \text-2\)
.) Record their work, side by side, for all to see. If there are no equations for which students took both approaches, present both approaches anyway. Tell students that either approach is valid, but that in the next unit they will see some more complicated equations for which one approach might be simpler than the other.
Representing, Conversing, Listening: MLR7 Compare and Connect.
Use this routine to support whole-class discussion. Ask students, “What is the same and what is different?” about how they matched the expressions. Connect strategies by showing the different ways operations are used in each approach (e.g., arithmetic method by substituting in values and evaluating; algebraic method by using an inverse operation). Use gestures, and color on the display to highlight these connections. This helps students use mathematical language as they reason about their strategies to evaluate equivalent equations.
Design Principle(s): Maximize meta-awareness; Support sense-making


aligns: no