standard_description: Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments.
problem_activity: Activity
The mathematical goal of this activity is for students to use the area under a normal curve to find the proportion of values in certain intervals. While the values in this activity are chosen to highlight the usual areas within 1, 2, and 3 standard deviations of the mean, it is not essential for students in this course to memorize those values.
Digital
Launch
Remind students that the phrase “area under the curve” means the area between the
\(x\)
-axis and the curve. Demonstrate how to use graphing technology to display the normal curve from the previous activity (a normal curve with mean of 40 and standard deviation of 2) and to find the area under the curve for values less than 39. See
ggbm.at/w9tzezej
.
If you are using the GeoGebra Probability Calculator, available in the toolkit or at
geogebra.org/classic#probability
, select Normal distribution and customize it by adjusting the mean and standard deviation. Choose the distribution you want to calculate from the three buttons labeled as shown below. Adjust the interval limits by either replacing their values in the formula, or by dragging them in the graph.﻿ The area under the curve is given by the calculated probability.
It may be helpful for students to open two GeoGebra windows—one for each of the situations in the problem statement.
[IMG]
Calculate the probability of a distribution.
[IMG]
Calculate the probability of a left-sided distribution.
[IMG]
Calculate the probability of a right-sided distribution. ﻿
[IMG]
Toggle between the graphs of the probability density function and the cumulative distribution function.
Action and Expression: Provide Access for Physical Action.
Provide access to tools and assistive technologies, such as a graphing calculator or graphing software. Some students may benefit from a checklist or list of steps to be able to use the calculator or software.
Supports accessibility for: Organization; Conceptual processing; Attention
Reading, Conversing, Writing: MLR5 Co-Craft Questions.
Use this routine to provide students with an opportunity to produce the language of mathematical questions. Display only the context of the situation, including the statistics about lifespans of two different types of lightbulbs, without revealing the questions that follow. Give students 1–2 minutes to write their own mathematical questions about the situation. Invite students to compare their questions before revealing the remainder of the question. Listen for and amplify any questions that involve estimates of proportion. This will build student understanding of the language used to describe characteristics and applications of the normal curve.
Design Principle(s): Maximize meta-awareness; Support sense-making
Student Facing
The life span of light bulbs is approximately normally distributed. Some statistics about life spans of two different types of light bulbs are listed.
LED bulbs: mean = 2,300 days, standard deviation = 230 days
incandescent bulbs: mean = 100 days, standard deviation = 10 days
To estimate the proportion of bulbs that burn out in a certain interval of time, use technology to find the area under the normal curve and above the appropriate interval.
Estimate the proportion of LED bulbs that are expected to burn out before getting within 1 standard deviation of the mean (before 2,070 days).
Estimate the proportion of incandescent bulbs that are expected to burn out before getting within 1 standard deviation of the mean (before 90 days).
Estimate the proportion of LED bulbs that are expected to burn out after getting more than 1 standard deviation greater than the mean (after 2,530 days).
Estimate the proportion of incandescent bulbs that are expected to burn out after getting more than 1 standard deviation greater than the mean (after 110 days).
Estimate the proportion of LED bulbs that are expected to burn out in the interval between 1 standard deviation less than the mean and 1 standard deviation greater than the mean (between 2,070 and 2,530 days).
Estimate the proportion of incandescent bulbs expected to burn out in the interval between 1 standard deviation less than the mean and 1 standard deviation greater than the mean (between 90 and 110 days).
Estimate the proportion of LED bulbs that are expected to burn out in the interval between 2 standard deviations less than the mean and 2 standard deviations greater than the mean (between 1,840 and 2,760 days).
Estimate the proportion of LED bulbs that are expected to burn out in the interval between 1,900 days and 2,100 days.
Estimate the proportion of incandescent bulbs that are expected to burn out in the interval between 107 and 118 days.
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
Remind students that the phrase “area under the curve” means the area between the
\(x\)
-axis and the curve. Demonstrate how to use graphing technology to display the normal curve from the previous activity (a normal curve with mean of 40 and standard deviation of 2) and to find the area under the curve for values less than 39.
Action and Expression: Provide Access for Physical Action.
Provide access to tools and assistive technologies, such as a graphing calculator or graphing software. Some students may benefit from a checklist or list of steps to be able to use the calculator or software.
Supports accessibility for: Organization; Conceptual processing; Attention
Reading, Conversing, Writing: MLR5 Co-Craft Questions.
Use this routine to provide students with an opportunity to produce the language of mathematical questions. Display only the context of the situation, including the statistics about lifespans of two different types of lightbulbs, without revealing the questions that follow. Give students 1–2 minutes to write their own mathematical questions about the situation. Invite students to compare their questions before revealing the remainder of the question. Listen for and amplify any questions that involve estimates of proportion. This will build student understanding of the language used to describe characteristics and applications of the normal curve.
Design Principle(s): Maximize meta-awareness; Support sense-making
Student Facing
The life span of light bulbs is approximately normally distributed. Some statistics about life spans of two different types of light bulbs are listed.
LED bulbs: mean: 2,300 days, standard deviation: 230 days
incandescent bulbs: mean: 100 days, standard deviation: 10 days
To estimate the proportion of bulbs that burn out in a certain interval of time, use technology to find the area under the normal curve and above the appropriate interval.
Estimate the proportion of LED bulbs that are expected to burn out before getting within 1 standard deviation of the mean (before 2,070 days).
Estimate the proportion of incandescent bulbs that are expected to burn out before getting within 1 standard deviation of the mean (before 90 days).
Estimate the proportion of LED bulbs that are expected to burn out after getting more than 1 standard deviation greater than the mean (after 2,530 days).
Estimate the proportion of incandescent bulbs that are expected to burn out after getting more than 1 standard deviation greater than the mean (after 110 days).
Estimate the proportion of LED bulbs that are expected to burn out in the interval between 1 standard deviation less than the mean and 1 standard deviation greater than the mean (between 2,070 and 2,530 days).
Estimate the proportion of incandescent bulbs expected to burn out in the interval between 1 standard deviation less than the mean and 1 standard deviation greater than the mean (between 90 and 110 days).
Estimate the proportion of LED bulbs that are expected to burn out in the interval between 2 standard deviations less than the mean and 2 standard deviations greater than the mean (between 1,840 and 2,760 days).
Estimate the proportion of LED bulbs that are expected to burn out in the interval between 1,900 days and 2,100 days.
Estimate the proportion of incandescent bulbs that are expected to burn out in the interval between 107 and 118 days.
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
Students should be careful to use the correct mean and standard deviation for the type of bulb in the question. For students who struggle, ask them to more clearly identify the questions related to LED (or incandescent) bulbs to ensure they use the correct values in their technology.
Activity Synthesis
The goal of this discussion is for students to understand that a normal curve can be used to estimate the proportion of data that lie within a certain interval when the data is approximately normal. Ask,
“Why is it important that we are told that the life spans are approximately normally distributed?” (If the data had a different type of distribution, modeling with a normal distribution would not accurately describe the situation. Compare this to modeling quadratic data with a line.)
“Why are the answers for the first 4 questions the same?” (Since the normal curve is determined by the standard deviation and mean, the area to the left of the position one standard deviation below the mean is the same regardless of the values. Since the normal curve is symmetric, the area less than one standard deviation below the mean is equal to the area greater than one standard deviation above the mean.)
“Since all normal curves have the same areas for the regions relative to the mean and number of standard deviations away, approximately what percentage of values are within 1 standard deviation of the mean? (About 68% are within 1 standard deviation of the mean.)
“Approximately what percentage of values are within 2 standard deviations of the mean?” (About 95% are within 2 standard deviations of the mean.)
“Approximately what percentage of values are within 3 standard deviations of the mean?” (About 99.7% are within 3 standard deviations of the mean.)


aligns: no