Lesson 7

Areas under a Normal Curve

Lesson Narrative

The mathematical purpose of the lesson is to find areas under a normal curve and interpret the area as a proportion of values in the interval using a variety of different contexts. The work of this lesson connects to previous work because students applied the concepts of mean and standard deviation to data modeled using a normal distribution and were introduced to the area under a normal curve. The work of this lesson connects to upcoming work because students will compare models to data collected from simulations. When students find the areas under a normal curve and interpret the proportion of values in certain intervals for a given context, they are looking for and making use of structure (MP7).

Students can perform the work in this lesson using a table and \(z\)-scores; however, the activities in this lesson work best when each student has access to technology because it may take too long to do otherwise. If students are using technology to find areas under normal curves, technology will give the exact area without needing calculations that would be required when using the tables.

Learning Goals

Teacher Facing

  • Determine areas under a normal curve.
  • Interpret (orally and in writing) the proportion of values in certain intervals under a normal curve in a variety of different contexts.

Student Facing

  • Let’s use the normal distribution to estimate the proportion of data values falling within given intervals.

Required Materials

Required Preparation

Be prepared to display a GeoGebra applet for all to see during the activity Life of Lights.

Learning Targets

Student Facing

  • I can use the mean and standard deviation of a normally distributed data set to estimate intervals when given a proportion.
  • I can use the mean and standard deviation of a normally distributed data set to estimate proportions.

CCSS Standards

Addressing

Glossary Entries

  • normal distribution

    mean = 10. standard deviation = 1

    A bell-shaped distribution.

    mean = 10. standard deviation = 2

    A bell-shaped distribution.

    mean = 8. standard deviation = 2

    A bell-shaped distribution.

    A specific distribution in statistics whose graph is symmetric and bell-shaped, has an area of 1 between the \(x\)-axis and the graph, and has the \(x\)-axis as a horizontal asymptote. 

  • relative frequency histogram
    Histogram. 

    A histogram where the height of each bar is the fraction of the entire data set that falls into the corresponding interval (that is, it is the relative frequency with which the data values fall into that interval).