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: Optional activity
This challenging activity examines how measurement errors behave when 3 quantities are multiplied (versus 2 quantities in the previous activity). In other words, if I have measurements
\(a\)
,
\(b\)
, and
\(c\)
each with a maximum error of 5%, what percent error can
\(a \boldcdot b \boldcdot c\)
have? The arithmetic and algebraic demands of this task are high because students take a product of
three
quantities that each have a maximum percent error of 5%.
Launch
Arrange students in groups of 2. Provide access to calculators. Make sure students realize that the first question gives the measured values, not the actual values, for each dimension.
Give students 10 minutes to discuss with their partners, followed by whole-class discussion.
Conversing: MLR8 Discussion Supports.
Before students determine the maximum measurement error for the volume of a prism, invite pairs to make a prediction and justification. Use a sentence frame such as: “In a prism, if each dimension has a 5% maximum measurement error, we predict the volume’s maximum percentage error is ___ because….”. This will help students use the mathematical language of justifications to begin reasoning about the measurement error for volume.
Design Principle(s): Cultivate conversation; Support sense-making
Student Facing
The length, width, and height of a rectangular prism were measured to be 10 cm, 12 cm, and 25 cm. Assuming that these measurements are accurate to the nearest cm, what is the largest percent error possible for:
each of the dimensions?
the volume of the prism?
If the length, width, and height of a right rectangular prism have a maximum percent error of 1%, what is the largest percent error possible for the volume of the prism?
Student Response
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Anticipated Misconceptions
Students may think that the maximum error possible for the volume is 1% because the length, width, and height are within 1% of the actual values. Encourage these students to make calculations of the biggest and smallest possible length, width, and height. Then ask them to make calculations for the biggest and smallest possible volumes.
Activity Synthesis
Some discussion points include:
The first problem gives measurements and errors (but no percent error), while the second problem gives no measurements but does give the percent error. This makes the calculations notably different for the two problems.
In the first problem, we are given measurements and the possible size of error. We need to find the greatest percent error and, as we have seen in other cases, this happens for the smallest possible value of the measurement. If we were to find the percent error of each measurement, we would find that the error for the volume is a
larger
percent error than for any of the individual measurements.
In the second problem, we are given the maximum possible percent error but no measurements, and we need to find the largest possible error for the volume, that is, for the product of the three unknown measurements. The greatest percent error possible for the volume occurs when the measured value is as large as possible.
A unifying feature in these two problems is that we notice that the largest percent error occurs when the actual measurements are
smaller
than the measured values, as much smaller as possible.
Representation: Internalize Comprehension.
Use color and annotations to illustrate connections between representations. As students share their diagrams and reasoning, use color and annotations to scribe their thinking on a display of each problem so that it is visible for all students.
Supports accessibility for: Visual-spatial processing; Conceptual processing


aligns: no