Download Area Under the Normal Curve

Area Under the Normal Curve
Resource ID#: 62822
Primary Type: Formative Assessment
This document was generated on CPALMS - www.cpalms.org
Students are asked to find the probability that an outcome of a normally distributed variable is
between two given values using both a Standard Normal Distribution Table and technology.
Subject(s): Mathematics
Grade Level(s): 9, 10, 11, 12
Intended Audience: Educators
Freely Available: Yes
Keywords: MFAS, normal curve, standard deviation, normal distribution, probability
Instructional Component Type(s): Formative Assessment
Resource Collection: MFAS Formative Assessments
ATTACHMENTS
MFAS_AreaUnderTheNormalCurve_Worksheet.docx
MFAS_AreaUnderTheNormalCurve_Worksheet.pdf
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 problems on the Area Under the Normal
Curve worksheet.
2. The teacher asks follow-up questions, as needed.
TASK RUBRIC
Getting Started
Misconception/Error
The student is unable to determine the probability that a score is between 24 and 32.
Examples of Student Work at this Level
The student:



Does not understand the need to calculate z-scores.
Looks up 0.24 and 0.32 on the table and then subtracts these values and states that the probability is 3.1%
Subtracts 24 from 32 and then looks up 0.08 on the table and states that the probability is 5.32%.
Questions Eliciting Thinking
Can you draw a curve and label it with the mean and values within one and two standard deviations of the mean
How do you calculate a z-score? If you knew the z-scores, would you be able to calculate the probability?
Instructional Implications
If needed, review how to use the mean and standard deviation of a distribution to calculate values at multiples of
density curve and the specific properties of the normal curve. Show the student how to locate the mean, as well a
normal curve. Model graphing and labeling a normal curve given the mean and standard deviation of a set of nor
outcome that is one, two, or three standard deviations from the mean using the 68-95-99.7 rule. Consider using t
Review with the student how to find the z-score associated with a particular value and what it represents in term
standard normal distribution table or graphing technology to find the percentage of data within a given number o
standard deviation of another set of normally distributed data such as N(35.6, 2.7), and ask the student to find th
between two given values.
Moving Forward
Misconception/Error
The student makes errors in calculating z-scores.
Examples of Student Work at this Level
The student attempts to find the z-score but incorrectly interchanges the x and µ resulting in a negative z-score.
Questions Eliciting Thinking
What is the formula for finding the z-score? What is the value to be standardized or the x value? What is the mea
Did you substitute correctly?
Instructional Implications
Review with the student how to find the z-score associated with a particular value and what it represents in term
variable in the formula represents. Then review how to use z-scores and a standard normal distribution table or g
standard deviations of the mean. Provide the student with the mean and standard deviation of another set of norm
proportion of data greater than a given value, less than a given value, and between two given values.
Consider using the NCTM lessons SAT Scores (http://www.illustrativemathematics.org/illustrations/216), Shoul
(http://www.illustrativemathematics.org/illustrations/1218) or Do You Fit In This Car? (http://www.illustrativem
Consider also using the MFAS tasks Probability of Your Next Texting Thread (S-ID.1.4) or Range of Texting Th
Almost There
Misconception/Error
The student is unable to find the probability using one or both methods.
Examples of Student Work at this Level
The student correctly calculates the z-scores, but:

Does not understand how to use either the standard normal distribution table or technology to calculate th

Can calculate the proportion of scores between 24 and 32 using one of the two methods but not both.
Questions Eliciting Thinking
Do you know how to use the table to calculate the proportion of scores between 24 and 32?
Do you know how to use a graphing calculator (or other available technology) to calculate the proportion of sco
Instructional Implications
Instruct the student in how to use the standard normal distribution table and a graphing calculator or other availa
than a z-score, or between two z-scores. Guide the student through several examples and then provide the studen
Consider using the NCTM lessons SAT Scores (http://www.illustrativemathematics.org/illustrations/216), Shoul
(http://www.illustrativemathematics.org/illustrations/1218) or Do You Fit In This Car? (http://www.illustrativem
Consider also using the MFAS tasks Probability of Your Next Texting Thread (S-ID.1.4) or Range of Texting Th
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 uses the table to calculate the z-scores of 24 (0.4286) and 32 (1.5714), determines the associated pro
between 24 and 32 (0.276 or 27.6%). The student then uses either the raw data or the z-scores to calculate the pr
cdf on a graphing calculator. The student explains how technology was used to determine the probability.
Questions Eliciting Thinking
Can you use the symmetry of the normal curve to identify two other values that will yield the exact same area un
Instructional Implications
Pair the student with an Almost There student to practice finding probabilities using technology.
Consider using the NCTM lessons SAT Scores (http://www.illustrativemathematics.org/illustrations/216), Shoul
(http://www.illustrativemathematics.org/illustrations/1218) or Do You Fit In This Car? (http://www.illustrativem
Consider also using the MFAS tasks Probability of Your Next Texting Thread (S-ID.1.4) or Range of Texting Th
ACCOMMODATIONS & RECOMMENDATIONS

Special Materials Needed:
o
o
Area Under the Normal Curve worksheet
Graphing calculator or other technology capable of statistics computations
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
License: CPALMS License - no distribution - non commercial
Related Standards
Name
MAFS.912.S-ID.1.4:
Description
Use the mean and standard deviation of a data set to fit it to a
normal distribution and to estimate population percentages.
Recognize that there are data sets for which such a procedure is
not appropriate. Use calculators, spreadsheets, and tables to
estimate areas under the normal curve. ★