options: Statistics & Probability: students learn to analyze and produce data distributions, and build understanding of univariate and bivariate data. They learn to interpret data, make statistical inferences, justify conclusions, understand rules of probability, and use probability to make decisions.

Number & Operations in Base Ten: students learn to work with the base-ten system and build place value understanding.

Operations & Algebra: students learn to solve problems using algebraic thinking and operations such as addition, subtraction, multiplication, and division. They may learn to identify and explain arithmetic patterns, evaluate or manipulate numerical expressions, and reason with equations and inequalities.

Ratios & Proportional Relationships: students learn to recognize, describe, represent, and reason with ratios, rates, proportional relationships, and percent.

Counting & Cardinality: students learn to know number names and the count sequence, count to tell the number of objects, and compare numbers.

Functions: students learn to define, use, and evaluate functions to model relationships between quantities.

Number & Operations - Fractions: students learn to understand fractions as numbers and may work with them using addition, subtraction, multiplication, and division.

Number Systems and Quantity: students learn to understand the complex number system and reason quantitatively.

Modeling: students learn to choose and use appropriate mathematics and statistics to analyze real-world empirical situations, improve decisions, and report on their conclusions and the reasoning behind them.

Measurement & Data: students learn to work with data and measure attributes such as time, money, length, area, and volume. They may learn to compare measurements to operations and convert between different units of measure.

Geometry: students learn to classify geometric figures by their properties, and understand and compare the relationships between them. Students develop and use formulas to compute lengths, areas, and volumes, and they use transformations to generate new shapes and compare existing ones. Geometry can be studied with and without coordinates.
problem_activity: Activity
The mathematical purpose of this activity is to introduce students to curves representing normal distributions. Students compare normal curves with different means and standard deviations to determine how those statistics affect the curve. In particular, ensure students understand that normal curves are entirely defined by the mean and standard deviation, that the mean defines the center of the distribution, and the standard deviation affects the spread of the distribution.
Student Facing
These curves represent
normal distributions
with different means and standard deviations. What do you notice?
mean = 10. standard deviation = 1
[IMG]
mean = 10. standard deviation = 2
[IMG]
mean = 8. standard deviation = 2
[IMG]
mean = 12. standard deviation = 1
[IMG]
mean = 10. standard deviation = 0.8
[IMG]
Student Response
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Student Facing
Are you ready for more?
The equation for the curve for a normal distribution is
\(f(x) = \frac{1}{\sigma \sqrt{2\pi}} \boldcdot e^{\frac{-(x-\mu)^2}{2\sigma^2}}\)
where
\(\sigma\)
is the standard deviation,
\(\mu\)
is the mean, and
\(e\)
is a particular number close to 2.718.
Notice the part of the function that is
\((x - \mu)\)
. Using your understanding of transformations of functions, how does changing
\(\mu\)
affect the graph of the function?
Student Response
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Activity Synthesis
The important things for students to notice are:
The curves are entirely defined by the mean and standard deviation.
The curve is symmetric around the mean.
The standard deviation affects the width and height of the curve. A larger standard deviation is wider and shorter. A smaller standard deviation is thinner and taller.
Here are some questions for discussion:
“Why do you think the normal curve for a larger standard deviation is wider and shorter than one with a smaller standard deviation?” (We would expect a larger standard deviation to mean that data is more spread out, so it makes sense that is wider. Since the area must remain 1 under the curve, it has to get shorter to get wider.)
“What is the shape of the normal curve?” (Bell-shaped and symmetric)
“Where is the line of symmetry for each curve?” (
\(x = 10\)
,
\(x = 8\)
,
\(x = 12\)
, and
\(x = 10\)
, respectively)
Tell students,
“Although it is difficult to see in these graphs, the
\(x\)
-axis is a horizontal asymptote for normal curves. As you follow the curve to the right (or left), the curve gets closer and closer to the
\(x\)
-axis without ever actually reaching it.”
“Normal curves are bell-shaped and are most reliably used to model bell-shaped or approximately bell-shaped distributions.”
“The area under the normal curve (between the curve and the
\(x\)
-axis) is always 1. When the standard deviation increases, the curve spreads out to the right and left. To maintain the same area, though, the height of the peak must decrease.”


correct_option_index: 0