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
This task helps to transition students from thinking about “how many groups?” to “what fraction of a group?”.
Students compare different lengths of ropes and express their relative lengths in multiplicative terms. Rope B and C are 5 and
\(2\frac12\)
times as long as rope A, respectively, but rope D is shorter than rope A, so then we say that it is
\(\frac34\)
times as long as rope A. They see that it is possible that the answer to a “how many groups?” question is a number less than 1 when the given amount is smaller than the size of a group.
As students work, notice how they go about making multiplicative comparisons. Select students who write clear and concise questions for the equations in the last problem so they can share later.
Digital
Launch
Arrange students in groups of 2. Give students 4–5 minutes of quiet think time and then a couple of minutes to compare their responses with a partner and discuss any disagreements. Clarify that rope A is 4 units long.
Students using the digital materials can use the applet at
ggbm.at/kZUgANCC
to compare the segments. The segments can be moved by dragging endpoints with open circles. The yellow “pins” can help students keep track of the groups.
Student Facing
The segments in the applet represent 4 different lengths of rope. Compare one rope to another, moving the rope by dragging the open circle at one endpoint. You can use the yellow pins to mark off lengths.
Complete each sentence comparing the lengths of the ropes. Then, use the measurements shown on the grid to write a multiplication equation and a division equation for each comparison.
Rope B is _______ times as long as rope A.
Rope C is _______ times as long as rope A.
Rope D is _______ times as long as rope A.
Each equation can be used to answer a question about Ropes C and D. What could each question be?
\({?} \boldcdot 3=9\)
and
\(9 \div 3={?}\)
\({?} \boldcdot 9=3\)
and
\(3 \div 9= {?}\)
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
Arrange students in groups of 2. Give students 4–5 minutes of quiet think time and then a couple of minutes to compare their responses with a partner and discuss any disagreements. Clarify that rope A is 4 units long.
Students using the digital materials can use the applet at
ggbm.at/kZUgANCC
to compare the segments. The segments can be moved by dragging endpoints with open circles. The yellow “pins” can help students keep track of the groups.
Student Facing
Here is a diagram that shows four ropes of different lengths.
[IMG]
Complete each sentence comparing the lengths of the ropes. Then, use the measurements shown on the grid to write a multiplication equation and a division equation for each comparison.
Rope B is _______ times as long as Rope A.
Rope C is _______ times as long as Rope A.
Rope D is _______ times as long as Rope A.
Each equation can be used to answer a question about Ropes C and D. What could each question be?
\({?} \boldcdot 3=9\)
and
\(9 \div 3={?}\)
\({?} \boldcdot 9=3\)
and
\(3 \div 9= {?}\)
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Anticipated Misconceptions
Some students might associate the wrong lengths with the ropes or confuse the order of comparison (e.g., comparing A to C instead of C to A). Encourage them to put the length of each rope next to the diagram and attend more closely to the ropes being compared.
Activity Synthesis
Display the solutions to the first set of problems for all to see. Give students a minute to check their answers and ask questions. Then, focus class discussion on two key ideas:
The connection between “how many groups?” questions and “how many times as long?” questions. Ask students how these two types of questions are similar and different. Make sure students see that both have the structure of
\(? \boldcdot a =b\)
, where
\(a\)
is the size of 1 group (or the unit we are using for comparison), and
\(b\)
is a given number.
The language commonly used when referring to a situation in which the number of groups is less than 1 whole. Explain that we have seen equal-sized groups where the number of groups is greater than 1, but some situations involve a part of 1 group. So instead of saying “the number of groups” or asking “how many groups?,” we would ask “what fraction of a group?” or “what part of a group?”. For example, in the case of rope D, where the answer is less than 1, we can ask, “What fraction of rope A is rope D?”
Ask 1–2 previously identified students to share the question they wrote for the last pair of equations (
\(? \boldcdot 9=3\)
and
\(3 \div 9= ?\)
). Make sure students see that this pair of equations represent a situation with a fractional group
(i.e., rope D is shorter than rope C, so the length of rope D is a fraction of that of rope C).
Representation: Internalize Comprehension.
Use color and annotations to illustrate connections between representations. As students describe their reasoning about “how many groups?” and “how many times as long?”, 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
Speaking: MLR8 Discussion Supports.
As students compare and contrast these two types of division questions, provide a sentence frame such as: “Something these two types of questions have in common is . . .” and "A difference between these two types of questions is . . . ." This will help students produce and make sense of the language needed to communicate their ideas about the relationship between multiplication and division equations.
Design Principle(s): Support sense-making; Optimize output (for comparison)


aligns: no