options: 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.

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.

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

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

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.

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.

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.

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

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.

Counting & Cardinality: students learn to know number names and the count sequence, count to tell the number of objects, and compare numbers.
problem_activity: Warm-up
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
\(x\)
-axis and the curve.
Student Facing
The 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.
[IMG]
The area under the curve to the left of 43 is 0.9332.
[IMG]
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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Activity Synthesis
The purpose of this discussion is to make sure students can use the given areas to compose or decompose them 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.


correct_option_index: 5