options: 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.

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

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.

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.

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.

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

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.

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

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 purpose of this warm-up is for students to recall how to calculate an average rate of change from two points. Students will use this skill throughout the lesson and return to the context and data table in the last activity of the lesson.
Launch
Ask students to close their books or devices. Tell students that their job is to think of at least one thing they notice and at least one thing they wonder. Display the table for all to see. Give students 1 minute of quiet think time, and then select 3–5 students to share something they notice or wonder about the table  and record and display their responses for all to see. If not pointed out, ask students, "Describe how the data is changing as the value of
\(t\)
is increasing." (The values of
\(p\)
are decreasing at a slower rate.) Ask students to reopen their books or devices and complete the activity. Follow with a whole-class discussion.
Student Facing
Let
\(p\)
be the function that gives the cost
\(p(t)\)
, in dollars, of producing 1 watt of solar energy
\(t\)
years after 1977. Here is a table showing the values of
\(p\)
from 1977 to 1987.
{"table": [{"_attributes": {"border": "1"}, "thead": [{"tr": [{"th": [{"_attributes": {"scope": "col"}, "span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "166"}, "_value": "\\(t\\)"}]}, {"_attributes": {"scope": "col"}, "span": [{"_attributes": {"class": ["math", "math-repaired"], "data-png-file-id": "13322"}, "_value": "\\(p(t)\\)"}]}]}]}], "tbody": [{"tr": [{"td": [{"_value": "0"}, {"_value": "80"}]}, {"td": [{"_value": "1"}, {"_value": "60"}]}, {"td": [{"_value": "2"}, {"_value": "45"}]}, {"td": [{"_value": "3"}, {"_value": "33.75"}]}, {"td": [{"_value": "4"}, {"_value": "25.31"}]}, {"td": [{"_value": "5"}, {"_value": "18.98"}]}, {"td": [{"_value": "6"}, {"_value": "14.24"}]}, {"td": [{"_value": "7"}, {"_value": "10.68"}]}, {"td": [{"_value": "8"}, {"_value": "8.01"}]}, {"td": [{"_value": "9"}, {"_value": "6.01"}]}, {"td": [{"_value": "10"}, {"_value": "4.51"}]}]}]}]}
Which expression best represents the average rate of change in solar cost between 1977 and 1987?
\(p(10)-p(0)\)
\(p(10)\)
\(\dfrac{p(10) - p(0)}{10-0}\)
\(\dfrac{p(10)}{p(0)}\)
Student Response
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Anticipated Misconceptions
Some students might confuse finding an average of the values in the table and finding an average rate of change. Help them see that average usually involves one unit, such as average number of cookies. Average rate of change involves comparing how one quantity changes when another quantity changes by 1, such as cost ($) per year.
Activity Synthesis
The goal of this discussion is for students to recall the meaning of average rate of change for a function and how it is calculated. Select students to describe what the value of each expression represents in this context. For example,
\(p(10)-p(0)\)
is the difference in cost for a solar cell between 1977 and 1987 while
\(\dfrac{p(10)}{p(0)}\)
could be used to identify the percent change from 1977 to 1987. An important takeaway for students is that the actual expression for the average rate of change,
\(\dfrac{p(10)-p(0)}{10-0} \approx \text-7.55\)
, tells us that the price decreased by -
$
7.55 each year on average since the expression looks at the total difference in price between the 1977 and 1987 divided by the total number of years that passed.


correct_option_index: 5

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