In this lesson, we explore some fascinating stories from the news and history (and the future) about number encodings in computers. These stories should serve to illuminate how the kinds of decisions we have been making about number encodings are the same kinds of things that real scientists in the world have to worry about, sometimes with disastrous consequences. While this lesson has the possibility of running long, it is meant only as a short excursion into real-world application and should be limited to one class period.

Students will be able to:

- Discover the different ways number systems have been constructed and used throughout history.
- Examine real-world issues related to number encodings in computers.

Determining the number of bits that will be used in each binary number can have a profound impact on computing systems we rely upon. Many older operating systems and protocols for encoding time made use of 32-bit numbers. As computational power and time have both progressed, standards have migrated to larger binary numbers, usually 64-bit, in order to accommodate changing demands of these systems. Moving all members of the computing community over to these new standards can take some time, however, as many users continue to expect to receive and send information using the older 32-bit protocol.

Students will spend the majority of today's lesson independently researching a topic related to real-world number systems. You will see today as part of a broader investigation into the impacts different number systems have upon the way we represent and reason about numbers.

Present the following prompt to students as a short lecture before beginning this lesson's activity.

- Over the past couple of lessons, we have been studying the binary number system, since it can be constructed using only bits, the building blocks of information within a computer. Throughout history, many different number systems have been used to represent numbers. Each number system has its associated pros and cons, and often there are unintended consequences that arise from how those number systems are structured. For today's activity, you will be researching one of these number systems to further contextualize the ideas we've explored for the binary number system.

If possible, place students in groups of 2 to 4 people to complete the initial research. Distribute the "Encoding Numbers in the Real World Activity Guide" resource, one copy per student. Assign or have groups choose one of the articles found below in the "Resource Articles" section.

Give students 15 to 20 minutes to conduct their initial research in groups and complete the top portion of their Activity Guide. Encourage students to stay focused on the core ideas of their topic before moving on to understanding more detailed or technical aspects of the topic.

Provide students a hard time limit when conducting research at the beginning of class and consider projecting a countdown timer. Use this constraint to encourage students to divide up their work and focus their attention on the overall topic, rather than specific technical details.

Below is a set of articles that address different numbering systems and their uses. Feel free to add additional sources or select multiple sources when assigning topics.

- The 2038 Problem
- The Y2K Problem
- Polynesian Binary
- 12 Numbering Systems
- Barcodes
- Ancient Number Systems
- Roman Numeral Arithmetic
- Jacquard Loom
- 60 Minutes in an Hour
- 32 Vs. 64 Bits in Computers

After researching, have students find three classmates who researched a different article and exchange the information they learned. They should record the key points from each topic in the space provided at the bottom of their Activity Guides.

During the Jigsaw, consider providing students with 5- to 8-minute periods during which they can exchange information with a classmate. As before, their job is to cover the key points and make connections to their study of binary numbers.

Conclude the lesson with a sense-making discussion summarizing the broad themes running through this activity. The key points are that the binary number system, like any number system, has benefits while also imposing limits upon how we represent information. Give particular focus to at least one topic in which selecting a certain width binary number created unforeseen challenges later on. Some conversation starters are listed below.

- Does the way you represent numbers make a difference in the way you think about numbers?
- How might the way you choose to represent numbers in a computer go wrong? What potential problems might arise from the way you choose to represent numbers?
- Of all the number systems in the world, why do you think modern computers have settled on binary? Why didn't someone build a computer that used base 10?
- List a few rules number encoding systems must follow in order to be useful.
- Why do you think some number encoding systems are more successful than others?

**Reflection:**Choose one of the topics you most enjoyed learning about today. Describe what new information you learned and how it relates to the way we create and use number systems.

Ask students to conduct further research into a topic they found compelling. Use this activity to develop skills students will need to employ for the Explore Performance Task.

**CSTA K-12 Computer Science Standards:**CI.L2:2**CSTA K-12 Computer Science Standards:**CL.L2:3**CSTA K-12 Computer Science Standards:**CT.L2:7, CT.L2:8, CT.L2:14, CT.L3A:6**Computer Science Principles:**2.1.1 (A, B, C, D, E, F, G)**Computer Science Principles:**2.1.2 (A, B, D, E, F)