08/08/2022

Historical ideas of time and angle

Mathematics Education to Develop Students Agency
Part III: Measurement
Time, Angle and Others (rate and ratio)
Please pose

Ancients’ astronomy and Navigation
4000 years ago: Mesopotamia Lunisolar
CalendarTask 2 after
Task 1 asfollows: One month on Luna is around 30 (29.5) days, thus a month
became 30 days or 29 days. Then, a year, 12 month, is 354 days+α(Leap month) which provided base 60 system. A year on Sun
365 days + β. Around 360 days.

Task 1I eat lunch from12:00 to 12:30.
How long did Ieat?
Conversion ofmeasuring
quantity

Longitude isgiven by time✡

 They used to refer the tables for calculation.
 Divisors of 60:

Task 2
12 equal parts ofthe circumference

Why 12 o’clock?
Task 1
6 equal parts of thecircumference

60 = 22 × 31 × 51
60, 30, 15,12, 6, 5, 4, 3, 2, 1

2 equal parts of …: No problem

Prime number: 2, 3, 5, 7, …
 Universe followed God? It follows Luna & Sun and Mathematics

Calculatorof Universityof California

Navigation:
Marine Chronometer4 equal parts of …: No problem
Mathematical Problems!
5 equal parts of …
7 equal parts of …
Trisector of angle
Why 1 hour = 60 min, &1 min = 60 seconds?

ISODA Masami, Prof/PhD
Director of CRICEDFirst decimal place

1

Second decimal place

University of Tsukuba, Japan

Time is introduced by the schedule of the day.

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Time and duration

English usage of time: How English is complicated language to learn Math.?
 Ten forty-five Quoter to eleven•Fifteen before eleven, also, Fifteen minutes to eleven, but not Fifteen minutes to eleven hours?

 Three quoter past 10 but 10.30 is a half past 10 or half to 11
 10.45 hours? However, 09.00-09.40 HRS. (NY) 10.45 hours or 9:45 hours?
Is it 10 hours, 27 minutes and 0 seconds or 10 hours 45 minutes?

In English
 10 o’clock 45 minutes is wrong. Why ten forty-five.
 If we see hours as first decimal place on the day, it will be reasonable: it is myunderstanding. I could not find this explanation on the web.
Why o’clock is limited to use? Because it express ‘not by sundial.’

 O’clock denominate only time, not duration. Hours, minutes and seconds are used bothtime and duration.

Difficulty for children: For reading the clock, short and long hands 12 numbers show how many times on count by 5.
 Even though almost 11 o’clock on the shorthand, we have to read smaller number: It is not thesame as other measurement.

 Duration is defined by the difference of times: Time cannot add but subtraction of time isduration and the representation of time and duration are the same.
 On the operation, we have to convert based by the base 60, 12 (odd or even) and 24 aswell as 7 days, week, month and year. Years and days are base 10 system.
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Ratio for different quantity: Rate
Produce new quantity by per unit quantity.

Rotation, Angle, Angle of rotation
unit of angles: 1°，right angle, π，…

5

6

All these arethe contentof learning!!

Time is introduced by theschedule of the day.

This lecture use SEA-BES: CCRLS to explain objectives

Appreciation(6)
Review Part III, L4

These terminology explain the process. Thus, if you can imagine
concrete materials for each class, you are able to develop it inyour classroom. That’s why we have been discussing materials.

Reflection(5)

Acquisition

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