# Byte-Size Modeling:

## A Follow Up

Written by

Ron Lancaster,

Associate Professor,

University of Toronto

June 2021

Written as a follow up to our May post on bite-size modeling.

I recently developed the following problems for my pre-service students to use in their next practicum. I shared the problems with several classroom teachers who assigned them to their students. The feedback I received was positive and teachers particularly liked the real world connection.

In addition to sharing the problems, I will provide comments to make explicit my thinking in developing these problems.

Problem 1

Larger units of a byte have different definitions depending on the systems being used. Some systems are based (bad pun) on powers of 10 and others on powers of 2. Table 1 contains the names of the first eight sizes for both systems.

Question 1

(a) Figure 1 shows how the percentage difference between the decimal and binary systems grows with increasing storage size. How were these percentages calculated? Are they all correct? Determine the percentage difference for the following storage sizes not listed in figure 1.

(i) zettabyte and zebibyte

(ii) yottabyte and yobibyte

(b) In Figure 1, let x and y represent the quantities on the horizontal and vertical scales respectively. Use the data given in the graph to fill in the values of x and y in table 2.

(c) Determine a mathematical model for the data in table 2. Use your model to fact check the text given below.

While the numerical difference between the decimal and binary interpretations is relatively small for the kilobyte (about 2% smaller than the kibibyte), the systems deviate significantly as units grow larger (the relative deviation grows by 2.4% for each three orders of magnitude). For example, a power-of-10-based yottabyte is about 17% smaller than power-of-2-based yobibyte.

Source: https://en.wikipedia.org/wiki/Byte

(d) In parts (a) and (c), the percentage difference between the yottabyte and yobibyte has been calculated in two different ways. Are the answers the same for both methods? If not, why not?

(e) Use your mathematical model to determine the metric storage capacity in decimal units that is 50% less than its equivalent in binary units.

When I came across the data in Table 1, I was surprised to learn that there are still two different systems in use. I figured by now that there would be a dominant system. Students need to see things like this where competing ideas co-exist even though it leads to some confusion.

In Question 1(a), I resisted the urge many teachers have of telling too much - instead of explaining how the percentages were calculated, I asked students to figure this out. I continued this push not to tell in part (b) where students have to pull information from a graph. If students are familiar with logarithms, they will have to decide what base to use in order to proceed. If several different bases are used by students they can compare and contrast the mathematical models obtained in part (c) using these bases. Students not familiar with logs can log on and learn about them without having to wait until they arise in a future course.

The intention of these questions is to convey a sense of purpose to finding the mathematical model by having students act as fact checkers in Question 1(c).

Finally, I designed the questions in parts (a), (c) and (d) so that students calculate a quantity in two different ways and check if the results are consistent with each other.

These questions will not take a huge byte out of class time. Assign them to your students and see for yourself the power of these powers.