standard_description: Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., p^(2)). For example, by truncating the decimal expansion of Ö2, show that Ö2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations.
problem_activity: Activity
This activity connects the ideas of roots, rational exponents, graphs of exponential functions, and decimal approximations. Students graph
\(y=2^x\)
for a few integer values of
\(x\)
, approximate a smooth curve passing through those points, and use that curve to estimate the value of
\(2^x\)
for various rational inputs. Students then test these estimates using their understanding of the connection between rational exponents and roots.
Launch
Arrange students in groups of 2–3. Encourage them to compare their answers with their group after each question.
Since the first problem in this activity was designed to be completed without technology, ask students to put away any devices until after they complete it.
Representing, Conversing: MLR7 Compare and Connect.
Use this routine to prepare students for the whole class discussion about the graph of
\(y=2^x\)
. After students complete the table and sketch the graph of
\(y=2^x\)
, invite them to create a visual display of their work. Then ask students to quietly circulate and observe at least two other visual displays in the room. Give students quiet think time to consider what is the same and what is different about their tables and graphs. Next, ask students to find a partner to discuss what they noticed. Listen for and amplify the language students use to compare and contrast the estimates for
\(2^x\)
for various values of
\(x\)
between 0 and 2.
Design Principle(s): Cultivate conversation
Engagement: Develop Effort and Persistence.
Encourage and support opportunities for peer collaboration. When students share their work with a partner, display sentence frames to support conversation such as: “First, I _____ because. . .”, “How did you get . . .?”, “That could/couldn’t be true because . . . .”
Supports accessibility for: Language; Social-emotional skills
Student Facing
Complete the table as much as you can without using a calculator. (You should be able to fill in three spaces.)
{"table": [{"_attributes": {"align": "center", "border": "1"}, "tbody": [{"tr": [{"th": [{"_attributes": {"scope": "row"}, "span": [{"_attributes": {"class": ["math"], "data-png-file-id": "50"}, "_value": "\\(x\\)"}]}], "td": [{"_value": "0"}, {"span": [{"_attributes": {"class": ["math"], "data-png-file-id": "14"}, "_value": "\\(\\frac13\\)"}]}, {"span": [{"_attributes": {"class": ["math"], "data-png-file-id": "781"}, "_value": "\\(\\frac23\\)"}]}, {"_value": "1"}, {"span": [{"_attributes": {"class": ["math"], "data-png-file-id": "4325"}, "_value": "\\(\\frac43\\)"}]}, {"span": [{"_attributes": {"class": ["math"], "data-png-file-id": "11218"}, "_value": "\\(\\frac53\\)"}]}, {"_value": "2"}]}, {"th": [{"_attributes": {"scope": "row"}, "span": [{"_attributes": {"class": ["math"], "data-png-file-id": "1139"}, "_value": "\\(2^x\\)"}], "_value": "(using exponents)"}], "td": [{"span": [{"_attributes": {"class": ["math"], "data-png-file-id": "2323"}, "_value": "\\(2^0\\)"}]}, {"span": [{"_attributes": {"class": ["math"], "data-png-file-id": "2115"}, "_value": "\\(2^{\\frac13}\\)"}]}, {"span": [{"_attributes": {"class": ["math"], "data-png-file-id": "2132"}, "_value": "\\(2^{\\frac23}\\)"}]}, {"span": [{"_attributes": {"class": ["math"], "data-png-file-id": "1930"}, "_value": "\\(2^1\\)"}]}, {"span": [{"_attributes": {"class": ["math"], "data-png-file-id": "25624"}, "_value": "\\(2^{\\frac43}\\)"}]}, {"span": [{"_attributes": {"class": ["math"], "data-png-file-id": "25635"}, "_value": "\\(2^{\\frac53}\\)"}]}, {"span": [{"_attributes": {"class": ["math"], "data-png-file-id": "1131"}, "_value": "\\(2^2\\)"}]}]}, {"th": [{"_attributes": {"scope": "row"}, "span": [{"_attributes": {"class": ["math"], "data-png-file-id": "1139"}, "_value": "\\(2^x\\)"}], "_value": "(decimal approximation)"}], "td": [{}, {}, {}, {}, {}, {}, {}]}]}]}]}
Plot the points that you filled in.
[IMG]
Connect the points as smoothly as you can.
Use this graph of
\(y=2^x\)
to estimate the value of the other powers in the table, and write your estimates in the table.
Let’s investigate
\(2^{\frac13}\)
:
Write
\(2^{\frac13}\)
using radical notation.
What is
\(\left( 2^{\frac13}\right)^3\)
?
Raise your estimate from the table of
\(2^{\frac13}\)
to the third power. What should it be? How close did you get?
Let’s investigate
\(2^{\frac23}\)
:
Write
\(2^{\frac23}\)
using radical notation.
What is the value of
\(\left( 2^{\frac23}\right)^3\)
?
Raise your estimate from the table of
\(2^{\frac23}\)
to the third power. What should it be? How close did you get?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Student Facing
Are you ready for more?
Answer these questions using the fact that
\((1.26)^3 = 2.000376\)
.
Explain why
\(\sqrt[3]{2}\)
is very close to
\(1.26\)
. Is it larger or smaller than
\(1.26\)
?
Is it possible to write
\(\sqrt[3]{2}\)
exactly with a finite decimal expansion? Explain how you know.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Activity Synthesis
Display several student graphs of
\(y=2^x\)
for all to see and invite students to share how close their estimates for the cubes of
\(2^{\frac13}\)
and
\(2^{\frac23}\)
were. The important takeaway from this activity is that whether we write the number as
\(2^{\frac23}\)
or
\(\sqrt[3]{2^2}\)
, the expression is just a number and its value can be approximated.


aligns: no