Unit 7, Lesson 7
Areas under a Normal Curve
Integrated Math 3
7.1
Warm-up
Instructional Routines
NoneMaterials
NoneActivity Narrative
The mathematical purpose of this activity is for students to reason about the area under a normal curve, using what they know about composition and decomposition of areas.
If students are using a table of values with z-scores for this lesson, this work will be essential to finding the desired areas in many cases.
Launch
Remind students that the phrase “the area under the curve” means the area between the -axis and the curve.
Activity
NoneThe images show a normal curve with mean of 40 and standard deviation of 2.
The area under the curve to the left of 39 is 0.3085.A bell-shaped distribution with the horizontal axis labeled from 36 to 46 by 2s. The peak of the bell is centered at 40. The vertical axis labeled from 0 to 0.20 by 0.05’s. The area under the curve is shaded through a vertical line at 39.
The area under the curve to the left of 43 is 0.9332.Graph of normal curve. Horizontal axis from 34 to 46 by 2’s. Vertical axis from 0 to point 2 by point 0 5’s. Curve peaks near point 1 2. Area under curve is shaded from 34 to 43.
Since it is a normal curve, we know that the total area under the curve is 1. Use the given areas to find the areas in question. Explain your reasoning for each.
- Find the area under the curve to the right of 39.
- Find the area under the curve between 39 and 43.
- Find the area under the curve to the left of 40.
- Find the area under the curve between 39 and 40.
Student Response
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Building on Student Thinking
Activity Synthesis
The purpose of this discussion is to make sure students can compose or decompose the given areas to find additional areas. For each question, ask a student to explain how they reasoned about their answer. Highlight places where students used the symmetry of the distribution or cases where they could add or subtract known areas to find new areas.
Student Lesson Summary
The normal distribution can be used to estimate the proportion of values expected in a certain interval by finding the area under the normal curve and between the bounds that define the interval. Since the total area under a normal curve is 1, the area within any particular interval can be interpreted as the proportion of values that are in that interval.
To find the area of a region, technology or reference tables can be used. When using technology, the system will need to know the mean and standard deviation for the data as well as the boundary values for the region.
For example, the weights of large plastic building blocks are approximately normally distributed. The mean weight is 20 grams, and the standard deviation is 0.7 gram. Let’s estimate the proportion of all plastic building blocks that weigh between 19.4 and 20.5 grams. This proportion is represented by the shaded area in the figure.
A bell-shaped distribution with the horizontal axis labeled from 18 to 22 by 0.5’s. The peak of the bell is centered at 20. The vertical axis labeled from 0 to 0.4 by 0.1’s. The area under the curve is shaded between the vertical lines of just before 19.5 and 20.5.
By adding the mean, standard deviation, and bounds defining the region to a technological tool, we find that the proportion of values in the region is 0.5668, or 56.68%. This also means that, when selecting a building block at random, the probability of selecting a block with a weight between 19.4 and 20.5 grams is approximately 0.5668.