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# XBOX Xponential

## How have video game consoles changed over time?

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# XBOX Xponential

## How have video game consoles changed over time?

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How have video game consoles changed over time? In 1965, Intel co-founder Gordon Moore predicted that computer processors would double in speed every two years. Twelve years later, the first modern gaming console -- the Atari 2600 -- was released, sparking a revolution in video games that have become ever-faster and more realistic.

Students create exponential models to predict the speed of video game processors over time, compare their predictions to observed speeds, and consider the consequences as digital simulations become increasingly lifelike.

### REAL WORLD TAKEAWAYS

• Video games have become increasingly realistic over time. This may be related to speedier processors.
• Intel founder Gordon Moore predicted that processor speeds would double every two years; they have not, but they have grown exponentially (just less quickly).
• As virtual simulations become more lifelike, this may have profound implications for media and society.

### MATH OBJECTIVES

• Build and evaluate exponential functions to model a real-world prediction
• Generate and interpret an exponential regression (formally or informally) to describe a pattern in real-world data
• Compare different but equivalent expressions

This complex task is best as a culminating unit activity after students have developed formal knowledge and conceptual understanding.
Algebra 2
Exponential Functions (Adv.)
Algebra 2
Exponential Functions (Adv.)
Content Standards F.BF.1 Write a function that describes a relationship between two quantities. (a) Determine an explicit expression, a recursive process, or steps for calculation from a context. (b) Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. (c) (+) Compose functions. For example, if T(y) is the temperature in the atmosphere as a function of height, and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the temperature at the location of the weather balloon as a function of time. F.LE.2 Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). S.ID.6 Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. (a) Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models. (b) Informally assess the fit of a function by plotting and analyzing residuals. (c) Fit a linear function for a scatter plot that suggests a linear association. A.SSE.2 Use the structure of an expression to identify ways to rewrite it. For example, see x<sup>4</sup> &mdash; y<sup>4</sup> as (x<sup>2</sup>)<sup>2</sup> &mdash; (y<sup>2</sup>)<sup>2</sup>, thus recognizing it as a difference of squares that can be factored as (x<sup>2</sup> &mdash; y<sup>2</sup>)(x<sup>2</sup> + y<sup>2</sup>). A.SSE.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. (a) Factor a quadratic expression to reveal the zeros of the function it defines. (b) Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. (c) Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15<sup>t</sup> can be rewritten as (1.15<sup>1/12</sup>)<sup>12t</sup> &approx; 1.012<sup>12t</sup> to reveal the approximate equivalent monthly interest rate if the annual rate is 15%. F.IF.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.
Mathematical Practices MP.4 Model with mathematics. MP.7 Look for and make use of structure. MP.1 Make sense of problems and persevere in solving them. MP.2 Reason abstractly and quantitatively. MP.3 Construct viable arguments and critique the reasoning of others. MP.5 Use appropriate tools strategically. MP.6 Attend to precision. MP.8 Look for and express regularity in repeated reasoning.