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
The purpose of this activity is to help students understand that quantities measured using the same two units of measure form a set of equivalent ratios. All of the strategies and representations they have for reasoning about equivalent ratios can be used for reasoning about converting from one unit of measure to another. Any ratio
\(a:b\)
has two associated unit rates:
\(\frac{a}{b}\)
and
\(\frac{b}{a}\)
, with a particular meaning in the context. For example, since there are 8 kilometers in approximately 5 miles, there are
\(\frac85\)
kilometers in 1 mile, and there are
\(\frac58\)
of a mile in one kilometer. We want students to notice that finding “how much per 1” and reasoning with these unit rates is efficient, but to make sense of these efficient strategies by using familiar representations like double number lines and tables. In a constant speed context, students are explicitly asked to compute each unit rate, and then they are asked to solve a problem where either unit rate can be used. For the second problem, monitor for one student who uses each strategy to solve the problem:
Creating a double number line or a table to represent the association between miles and the equivalent distance in kilometers as a set of equivalent ratios. (If both of these representations are used, it is fine to include both.)
Converting 80 kilometers into 50 miles by evaluating
\(80 \boldcdot \frac58\)
(in order to compare the resulting 50 miles per hour with 75 miles per hour)
Converting 75 miles into 120 kilometers by evaluating
\(75 \boldcdot \frac85\)
(in order to compare the resulting 120 kilometers per hour with 80 kilometers per hour)
Continuing to draw connections between representations of equivalent ratios and more efficient methods will help students make sense of the more efficient methods.
Launch
Display the image from the task statement for all to see, tell students it is a traffic sign you might see while driving, and ask students to explain what it means. They will likely guess it is a speed limit sign and assume it means 80 miles per hour. If no one brings it up, tell students that this is a sign you might see while driving in Canada or another country that uses the metric system for more things than we use it for in the United States.
Give students 2 minutes of quiet work time and ask them to pause after the first question. Ensure that everyone has correct answers for the first question before proceeding with the second question. Follow with whole-class discussion.
Representation: Internalize Comprehension.
Provide appropriate reading accommodations and supports to ensure students access to written directions, word problems and other text-based content.
Supports accessibility for: Language; Conceptual processing
Student Facing
Elena and her mom are on a road trip outside the United States. Elena sees this road sign.
Elena’s mom is driving 75 miles per hour when she gets pulled over for speeding.
[IMG]
The police officer explains that 8 kilometers is approximately 5 miles.
How many kilometers are in 1 mile?
How many miles are in 1 kilometer?
If the speed limit is 80 kilometers per hour, and Elena’s mom was driving 75 miles per hour, was she speeding? By how much?
Student Response
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Anticipated Misconceptions
It is acceptable to express the answers to the first question in either fraction or decimal form. If students express uncertainty about carrying out the division of
\(5 \div 8\)
or
\(8 \div 5\)
, encourage them to express the quotient in fraction form.
Activity Synthesis
Focus discussion on different approaches to the second question. If any students with less-efficient methods were selected, have them go first in the sequence, or present one of these representations yourself. As students are presenting their work, encourage them to explain the meaning of any numbers used and the reason they decided to use particular operations. For example, if a student multiplies 80 by
\(\frac58\)
, ask them to explain what
\(\frac58\)
means in this context and why they decided to multiply 80 by it. It can be handy to have representations like double number lines or tables displayed to facilitate these explanations.
Representing, Listening, Speaking: MLR7 Compare and Connect.
Ask students to display their approaches to determine whether or not Elena’s mom was speeding. As students share their work, encourage them to explain the meaning of each quantity they use. For example, if they convert 80 miles per hour into kilometers per hour, where 80 is multiplied by
\(\frac58\)
, ask what
\(\frac58\)
means in this context and why they decide to multiply it by 80. If students used a table or a double number line, ask how these representations connect with other strategies. This will help students make sense of the various approaches to reason about equivalent ratios which can be used for reasoning about converting one unit of measure to another.
Design Principle(s): Cultivate conversation; Maximize meta-awareness


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