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Primary Type: Formative Assessment Status: Published This is a resource from CPALMS (www.cpalms.org) where all educators go for bright ideas! Resource ID#: 62822 Area Under the Normal Curve 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 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 page 1 of 3 Can you draw a curve and label it with the mean and values within one and two standard deviations of the mean? Where would 24 and 32 be located? 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 a standard deviation from the mean. Review the basic properties of a density curve and the specific properties of the normal curve. Show the student how to locate the mean, as well as points within one, two, and three standard deviations of the mean on the normal curve. Model graphing and labeling a normal curve given the mean and standard deviation of a set of normally distributed data. Ask the student to find the probability of an outcome that is one, two, or three standard deviations from the mean using the 68-95-99.7 rule. Consider using the MFAS task Label a Normal Curve (S-ID.1.4). Review with the student how to find the z-score associated with a particular value and what it represents in terms of distance from the mean. Then review how to use zscores and a standard normal distribution table or graphing technology to find the percentage of data within a given number of standard deviations of the mean. Provide the student with the mean and standard deviation of another set of normally distributed data such as N(35.6, 2.7), and ask the student to find the proportion of data greater than a given value, less than a given value, and 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 zscore. 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 mean or the µ value? What is the standard deviation or the s value? 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 terms of distance from the mean. Be sure the student understands what each variable in the formula represents. Then review how to use z-scores and a standard normal distribution table or graphing technology to find the percentage of data within a given number of standard deviations of the mean. Provide the student with the mean and standard deviation of another set of normally distributed data such as N(35.6, 2.7), and ask the student to find the 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), Should We Send Out a Certificate? (http://www.illustrativemathematics.org/illustrations/1218) or Do You Fit In This Car? (http://www.illustrativemathematics.org/illustrations/1020). Consider also using the MFAS tasks Probability of Your Next Texting Thread (S-ID.1.4) or Range of Texting Thread (S-ID.1.4). 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 the proportion of scores between 24 and 32. 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 scores between 24 and 32? Instructional Implications Instruct the student in how to use the standard normal distribution table and a graphing calculator or other available technology to find the proportion of data greater than a z-score, less than a z-score, or between two z-scores. Guide the student through several examples and then provide the student with additional practice using both approaches. Consider using the NCTM lessons SAT Scores (http://www.illustrativemathematics.org/illustrations/216), Should We Send Out a Certificate? (http://www.illustrativemathematics.org/illustrations/1218) or Do You Fit In This Car? (http://www.illustrativemathematics.org/illustrations/1020). Consider also using the MFAS tasks Probability of Your Next Texting Thread (S-ID.1.4) or Range of Texting Thread (S-ID.1.4). Got It Misconception/Error page 2 of 3 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 proportions, and subtracts the proportions to find the proportion of scores between 24 and 32 (0.276 or 27.6%). The student then uses either the raw data or the z-scores to calculate the probability that a score is between 24 and 32 using a function such as normal 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 under the curve? 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), Should We Send Out a Certificate? (http://www.illustrativemathematics.org/illustrations/1218) or Do You Fit In This Car? (http://www.illustrativemathematics.org/illustrations/1020). Consider also using the MFAS tasks Probability of Your Next Texting Thread (S-ID.1.4) or Range of Texting Thread (S-ID.1.4). ACCOMMODATIONS & RECOMMENDATIONS Special Materials Needed: 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 Access Privileges: Public 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. ★ page 3 of 3
