options: 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.

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.

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.

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

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

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.

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

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.

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.
problem_activity: Task
Complete the table below. Is $\Delta x$ a constant?  If so, what constant is it? What do you notice about the 3rd column of the table?
{"table": [{"tbody": [{"tr": [{"th": [{"_value": "$x$"}, {"_value": "$f(x)=128\\cdot \\left({\\frac {1}{2}}\\right)^x$"}, {"_value": "Successive quotients"}]}, {"td": [{"_value": "0"}, {"_value": "128"}, {"_value": "----"}]}, {"td": [{"_value": "1"}, {"_value": "64"}, {"_value": "$\\frac{f(1)}{f(0)}=\\frac {64}{128} = \\frac{1}{2}$"}]}, {"td": [{"_value": "2"}, {}, {"_value": "$\\frac{f(2)}{f(1)}=$"}]}, {"td": [{"_value": "3"}, {}, {}]}, {"td": [{"_value": "4"}, {}, {}]}]}]}]}
Complete the table below. Is $\Delta x$ a constant?  If so, what constant is it? What do you notice about the 3rd column of the table?
{"table": [{"tbody": [{"tr": [{"th": [{"_value": "$x$"}, {"_value": "$f(x)=128\\cdot \\left({\\frac {1}{2}}\\right)^x$"}, {"_value": "Successive quotients"}]}, {"td": [{"_value": "0"}, {"_value": "128"}, {"_value": "---"}]}, {"td": [{"_value": "2"}, {"_value": "32"}, {"_value": "$\\frac{f(2)}{f(0)} = \\frac {32}{128} = \\frac{1}{4}$"}]}, {"td": [{"_value": "4"}, {}, {"_value": "$\\frac{f(4)}{f(2)}=$"}]}, {"td": [{"_value": "6"}, {}, {}]}, {"td": [{"_value": "8"}, {}, {}]}]}]}]}
Let $f(x)=a \cdot b^x$. Let $x_0$ be any particular $x$-value. Show that if $x_0$ is increased by a constant, $\Delta x$, the successive quotient  $$\frac{f(x_0+ \Delta x)}{f(x_0)}$$ is always the same no matter what $x_0$ is.
Is b) an example of the result of c)? Explain.


correct_option_index: 8