Beacon Lesson Plan Library

Base It

Melanie Malone

Description

To expose students to number systems other than the decimal system and explain why we need to know these systems (binary: electronics and computers; octal and hexadecimal: flight test, computers)

Objectives

The student understands the structure of number systems other than the decimal number system.

Materials

-Information on Ada Lovelace, Grace Hopper, and Charles Babbage for historical information on computers.
-Teacher generated number system charts in bases of 10, 2, 8, and 16 or whatever base you wish to teach.
-Place Value Chart for each student (see associated file)
-Teacher created problems to change into different bases

Preparations

1. Make copies of Place Value Charts.
2. Gather information on the mathematicians.
3. Create your assessment problems and key in desired base.

Procedures

I am fortunate. My father found this wonderful clock that has the five minute hash marks but no numbers. Every year, on the first day of school, I have it set up in a system other than ours. I usually choose binary for the first nine weeks and alter it throughout the year. I have found that this is the highlight for Open House night. The students are really anxious to show their parents my "weird" clock, but they are also anxious to know how it works. I generally start introducing it about week two of school and then re-introduce it every six to nine weeks after that. Depending on which level of students I am teaching, they are ready to learn about the middle of the year. They need to understand the decimal system and exponents prior to doing this. I never take it to the fractional part of the number system in other bases. I keep it greater than or equal to 1. I also allow students to work informally with those seated close to them in the class. We wind up in a general group discussion with lots of small group work intermingled.

1. Begin the class by discussing the contributions of Ada Lovelace to the world of mathematics, especially computers. She helped Charles Babbage with the documentation on his first computer. It is said that she understood the system better than any other colleague of Babbage. Our computer language ADA is named for her. You can get wonderful information on her from the AIMS series on Mathematicians and from the NCTM book, [Women in Science and Mathematics]. You may also want to discuss Babbage here as well. This is where you can draw the correlation between the different number systems and their use in computer programming and electronics.

2. Distribute the Place Value Charts (see associated file) to students and have them write down some numbers on the base 10 chart. Show, or have a student show, what the place of each number means (348: the 3 means 3x100 or 3 x 10
2). It is important that they cross over to the idea that we can also use exponents for each place. You may want to revise the Place Value Chart to reflect that. It makes each base easier to work with and set up. It is also important that they understand that the 3 stands for 3-one hundreds and that if you have 10-one hundreds, you have to move up to the next place. We cannot have a "10" in base 10. If you havenít already, this is also a perfect place to introduce the scientific calculatorís ability to do x
y, taking any base and raising it to a power.

3. After students feel comfortable with the base 10 chart, give them a number to translate on the base 2 chart such as 110(base 2). See if any of the students can, working together, place the number on the chart properly. If so, ask them to identify the decimal equivalent. Allow students to teach other students unless you see they are growing more confused. After they feel comfortable with base 2, allow them to work with other bases.

It is important to understand that you may have to re-visit this concept throughout the year before students feel really comfortable with it. They will come to understand it as the year progresses. I might suggest you purchase a cheap, battery-operated wall clock and see if you can get a dry-erase type face put on it. It works wonderfully in my class.

Assessments

Assess the students by giving them about ten to twelve problems in other bases to convert to base 10 and then give them the same amount of problems in base 10 to change to other bases. I allow students to use the charts. Assess for correctness.

Web Links

Web supplement
Past Notable Women of Computing

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