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Samples from Pathways from the Past I

From the teacher’s manual:

This system uses place value, but it is based on sixty, rather than on ten. That is, successive groups of symbols are multiplied by increasing powers of 60. The numbers 1 to 59 are made by adding combinations of the two basic symbols.... By working with powers of 60, instead of powers of 10, students will see more clearly the essential features of any place value system.

Students may think it strange that the Babylonians thought about numbers in 60s much like we think in 10s, but we do it ourselves in very ordinary situations. For instance:

•  We think in 60s when we tell time: 60 minutes in an hour, 60 seconds in a minute.  That's 3600 seconds in an hour.

•  We measure angles with a system based on 60:  360 (= 6 × 60)  degrees in a circle, 60 minutes in a degree of arc, 60 seconds in a minute of arc.

•  Here's an example of measuring with 60s that is less well known.  For ships and planes, distance is measured in nautical miles and speed is measured in knots.  A nautical mile is the length of 1/60th of a degree of the Equator. Thus, the full 360˚ length of the Equator is 60 × 360 = 21,600 nautical miles.  A knot is one nautical mile per hour (60 minutes).

Why do we do it this way? Because of the Babylonians. These measurement customs have been handed down to us from the Babylonians of 4000 years ago!

This activity sheet introduces students to the Babylonian system in stages, starting with the numbers from 1 to 59 and progressing gradually to the powers-of-60 place values.

From the teacher’s manual:

This activity sheet focuses on numerators and denominators. To understand them, we should think of fractions primarily in terms of counting, rather than division.  This is a subtle matter of emphasis.  Think of a fraction as counting copies of a single, small enough part.  Instead of measuring out a pint and a cup of milk for a recipe, it's easier to measure three cups.  Instead of representing a fractional amount by identifying the largest single part within it and then exhausting the rest by successively smaller parts, simply look for a small part that can be counted up enough times to produce exactly the amount you want.  Two numbers then specify the total amount: the size of the unit part, and the number of times it is “counted.”

Latin writers of the Middle Ages used the terms numerator (“counter” -- how many) and denominator (“namer” -- of what size) as a convenient way to distinguish the top number of a fraction from the bottom one.  If we still spoke Latin, these terms would make much more sense to students!

The questions of this worksheet help to develop a sequence of

important ideas:

•  denominators name the size of the pieces being counted;

•  numerators count the same-size pieces;

•  larger denominators result in smaller fractions;

•  larger numerators result in larger fractions;

•  changing the denominator affects the numerator;

•  common denominators make it easy to compare fractions.

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