Primary Type: Formative Assessment
Status: Published
This is a resource from CPALMS (www.cpalms.org) where all educators go for bright ideas!
Resource ID#: 56089
Comparing Irrational Numbers
Students are asked to estimate the value of several irrational numbers using a calculator and order them on a number line.
Subject(s): Mathematics
Grade Level(s): 8
Intended Audience: Educators
Freely Available: Yes
Keywords: MFAS, irrational numbers, estimation, square root
Resource Collection: MFAS Formative Assessments
ATTACHMENTS
MFAS_ComparingIrrationalNumbers_Worksheet.docx
FORMATIVE ASSESSMENT TASK
Instructions for Implementing the Task
This task can be implemented individually, with small groups, or with the whole class.
1. The teacher asks the student to complete the problem on the Comparing Irrational Numbers worksheet.
2. The teacher asks follow-up questions, as needed.
Note: The teacher may need to remind the student to show an estimate for each irrational number on his or her paper.
TASK RUBRIC
Getting Started
Misconception/Error
The student is unable to correctly estimate values of irrational numbers using a calculator.
Examples of Student Work at this Level
The student:
Attempts an estimation strategy that does not involve a calculator.
page 1 of 4 Disregards the order of operations conventions when using a calculator and incorrectly calculates
as 5 or 5.4.
Finds an interval bounded by two consecutive whole numbers in which the irrational number falls and then guesses the value of an estimate.
Makes several errors including calculating
as
.
Questions Eliciting Thinking
How might you use your calculator to estimate these values?
How would you apply the order of operations rules to calculating
What is
? Is
the same as 9
? Is that what you “asked” your calculator to compute?
2?
Instructional Implications
Provide instruction on and practice using a calculator to evaluate numerical expressions involving fractions and square roots. Review the definitions of rational number and
irrational number. Discuss with the student the need to estimate irrational numbers in order to graph and order them. Ask the student to approximate irrational numbers by
rounding to different decimal places and to consider the number of decimal places needed in order to adequately compare two values.
Provide the student with additional opportunities to find rational approximations of irrational numbers using a calculator.
Moving Forward
Misconception/Error
The student makes errors in ordering or graphing the set of numbers.
Examples of Student Work at this Level
The student correctly estimates each irrational number but:
Graphs
and
in the same location or in reverse order.
Graphs numbers in the correct order but locates several incorrectly on the number line.
Questions Eliciting Thinking
Do
and
have the same value? Which is larger? How can you tell?
page 2 of 4 By what interval is this number line scaled? How can you tell if numbers are graphed in the right places on this number line?
Instructional Implications
Assist the student in identifying the scale on the number line and then correctly graphing each value. Provide additional sets of irrational numbers for the student to
estimate and graph. Discuss with the student considerations in determining a scale that is appropriate for graphing a given set of numbers. Provide opportunities for
independent practice in scaling and graphing irrational numbers on a number line.
Almost There
Misconception/Error
The student does not demonstrate an understanding of the difference between an irrational number and its rational approximation.
Examples of Student Work at this Level
The student labels the graphed points on the number line with the approximations rather than the original numbers to be graphed.
Upon questioning, the student does not appear to understand that the value produced by the calculator is a rational approximation of the irrational number.
Questions Eliciting Thinking
What kind of number is
? What kind of number is 1.41? Are these numbers equal?
Could you have ordered the original numbers without first approximating them? What was the point of using the calculator?
What numbers were you asked to graph – the original irrational numbers or their decimal approximations?
Instructional Implications
Review the definitions of rational number and irrational number. Discuss with the student the need to estimate irrational numbers in order to graph and order them. Ask the
student to label his or her graphed points with the corresponding irrational numbers. Provide the student with additional opportunities to find rational approximations of
irrational numbers in real-world contexts and in the context of graphing.
Got It
Misconception/Error
The student provides complete and correct responses to all components of the task.
Examples of Student Work at this Level
The student approximates each irrational number
The positions of
and
and accurately graphs each number on the number line.
are clearly differentiated and the student labels the graphed points with their corresponding irrational representations.
Questions Eliciting Thinking
What is the difference between an irrational number like
and a number like 3.14 or
Is it possible to find a fraction that represents the exact value of
?
? Why or why not?
Could you have ordered the original numbers without first approximating them?
Instructional Implications
Ask the student to compare a common decimal approximation (e.g., 3.14) to the common fraction approximation
of
to determine which is closer to its actual value
and by about how much. Ask the student to consider when one might use each of the two approximations. Challenge the student to find a fraction that is a better
approximation of
than
.
ACCOMMODATIONS & RECOMMENDATIONS
Special Materials Needed:
Comparing Irrational Numbers worksheet
Calculator with square root function
page 3 of 4 SOURCE AND ACCESS INFORMATION
Contributed by: MFAS FCRSTEM
Name of Author/Source: MFAS FCRSTEM
District/Organization of Contributor(s): Okaloosa
Is this Resource freely Available? Yes
Access Privileges: Public
License: CPALMS License - no distribution - non commercial
Related Standards
Name
MAFS.8.NS.1.2:
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., π²). For example, by truncating the decimal
expansion of √2, show that √2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get
better approximations.
page 4 of 4