Question | |||||||
How do you think video games have changed over time? |
In 1965, computer scientist Gordon Moore predicted that computer processor speeds would double every two years. Twelve years later, Atari released the 2600 with a processor speed of 1.2 MHz. Based on Moore’s Law, how fast would you expect the processors to be in each of the consoles below? |
Write an expression to estimate how fast console processors should be in 2077, a century after the original Atari. |
We can think of the release of the Atari 2600 as the start of the new video game era, where 1977 is VG year 0. Let t represent the number of years that have passed since 1977, i.e. the “video game year.” Write an equation for the expected processor speed for a given year. Based on this, how fast would you expect the processor to be in a Nintendo Wii U, released in 2012? |
Using a site like Wikipedia, find out when each console below was released and calculate its predicted processor speed. Then find out the actual processor speeds. Have video game processor speeds followed Moore’s Law? (Note: 1 GHz = 1000 MHz. If a console uses multiple processors, use the fastest one.) | Using technology, write the equation of the exponential function that fits the actual data as closely as possible. How has the growth in processor speeds compared to the growth that Moore predicted? |
At some point it stops being possible to make processors faster; they vibrate too much and become too hot. When this happens, manufacturers can still build faster consoles; instead of installing a single processor, they install multiple. (For instance, the PS3 has one 3.2 GHz chip, while the PS4 has eight 1.6 GHz chips.) Do you think video games and other simulations will ever be so powerful that you won’t be able to distinguish them from reality? If so, what might be some of the consequences? |
Commentary | ||||||
The question is intentionally vague, and students are free to interpret it how they want. Some might sketch how enjoyment has changed over time, while others may focus on realism. The goal of the exercise is for students to realize that they can only graph what they can measure, and they can only measure what they can quantify. While it may be difficult to quantify some characteristics, it’s certainly possible to quantify others. | In Q3, students will write an equation to model speeds predicted by Moore’s Law: S = 1.2 • 2^{t/2}, where t represents the number of years that have passed since the Atari was released. Exponential growth is fundamentally about repeated multiplication, and the table helps establish that in a concrete way: 1.2 • 2 • 2… To facilitate this, we only included consoles that were released in odd-numbered years (but with different increments between them). | If we asked students to write the Moore’s Law equation now, many would write S = 1.2 • 2^{t}. The 1.2 • 2 part is correct but the exponent is not; it suggests that speeds double every year rather than every two. By asking students about the specific case of 100 years, the goal is for students to reason, “100 years means 50 doublings…so 1.2 • 2^{100/2}.” Focusing on the relationship between time passed and the number of doublings should smooth the transition to the equation in Q3. |
Now that students understand both the concept and mechanics of Moore’s Law, they should be better prepared to translate it into a general equation: S = 1.2 • 2^{t/2}, or S = 1.2 • 2^{0.5t}. |
Narratively, this is an important question; it provides students the first chance to use their models to compare predicted vs. actual console speeds. |
The equation of the curve-of-best fit through the actual data is S = 0.44 • 1.31^{t}. This suggests that processor speeds increased by 31% each year, which students may conclude is slower than the “doubling” predicted by Moore. However, while the calculator regression is in terms of annual growth, the Moore's Law equation is in terms of biennial growth. To compare predicted to actual growth, students must reinterpret Moore's Law into annual terms. | The purpose of Mathalicious lessons is to help students explore mathematical concepts, and to use them to think critically about the world. If a parent asks, “What did you learn today?”, the goal is for students to say, “I learned how video games have changed” rather than “I learned about exponential growth.” Ideally, the last question students answer should be the first thing they remember. |