The market scales puzzle & connections with ancient Egyptian mathematics: Homework for Monday Sept 28

 Here's a puzzle that connects with the principles of ancient Egyptian multiplication in interesting ways. There's a story connected with this puzzle that I'll tell you in class -- but here's the puzzle:

A market vendor sells dried cooking herbs in whole-number amounts from 1 to 40 grams. The vendor has an old-fashioned two pan weigh scale, and has exactly four weights of different amounts that allows them to weigh out any of these amounts of herbs -- without using the herbs or any other object as an auxiliary weight.

• Make sure you understand how a two-pan scale works...the weights can be placed on either or both of the pans.
  
  
• What must the values of the four weights be? Why?
  
  
• What if it was a one-pan scale -- i.e., a scale where you could only put the weights on one of the pans, rather than both of them? What five weights would you need to weigh up to, say, 31grams?
  
  
• How could you extend this puzzle to help your students understand the mathematics more deeply?
  
  
• How does this puzzle connect with ideas about number theory and bases you are already familiar with?
  
  
  

Please post your blog on this problem next  Monday September 28 by 9 AM.

Enjoy!

Dave Wagner: two-pan market scales, Bhutan. Used with permission.

 

Ancient Egyptian mathematics: Numeration

  

Value	1	10	100	1,000	10,000	100,000	1 million, or many
Hieroglyph
Description	Single stroke	Cattle hobble	Coil of rope	Water lily (also called lotus)	Bent finger	Tadpole	Heh[3]

(From Wikimedia)

What differences do you notice between ancient Mesopotamian/ Babylonian numeration and ancient Egyptian numeration systems?

(And if you are familiar with Roman numerals -- which we will visit later -- what similarities and differences do you notice?)

What affordances and constraints do you notice for the Egyptian system? For the Babylonian system? 

The 'Russian Peasant Multiplication' -- closely related to Ancient Egyptian mathematics and binary number systems

  Here is a fascinating Numberphile video from 2020 that shows the 'Russian peasant method' for multiplication. How is this the same and different from Ancient Egyptian multiplication? Why are they equivalent?

Numberphile Video (approx 5:00) < https://youtu.be/HJ_PP5rqLg0>

 

Examples of Ancient Egyptian multiplication and division

Here are examples of the Ancient Egyptian method of multiplication and division, by doubling and halving. They depend on the fact (clearly known in that culture) that all the whole numbers can be constructed from sums the powers of 2 (including 2^0, which equals 1).

Ancient Egyptians also used only unit fractions in the form 1/x, with the exception, for some reason, of 2/3 — something we will explore further. 

Take a good look at the division example. Would this technique work with all whole number divisors, or only even ones? Could this technique be used with other unit fractions?

                                                             Multiplying 22 by 7




                                                       Dividing 43 by 8

**EDIT: Blog post due Wed Sept 23: Did Mesopotamian scribes have algebra? (And how can we recognize 'rhetorical' and 'syncopated' algebra?)

Here is the link to an explanation of Babylonia algebra to help understand Babylonian word problems. (Note that this is the sheet we were using in class as a handout!)

https://drive.google.com/file/d/0B00n89L6TX5gZ01BQndwczRQaEE/view?usp=drivesdk&resourcekey=0-arSdOZ88pkPwDfajhNbQMg

You'll see that the form of the problem is different from the algebra we are used to -- and there is no use of compact alphanumeric symbols for variables. Instead, concrete words and images are used.

 

(**Note that algebraic relationships stated in words are known as 'rhetorical algebra'; algebraic relationships stated with some abbreviations but not a systematic symbolism are known as 'syncopated algebra'. It's interesting to think how this relates to our students' initial learning of algebra...!)

Questions to consider:

• How could one state a general mathematical principle in a time before the development of algebra and algebraic notation?

• Is mathematics all about generalization and abstraction?

•Thinking about various areas of mathematical knowledge -- number theory, geometries, calculus, graph theory, etc. etc. -- how could you imagine stating general or abstract relationships without algebra?

• Do our students go through similar stages of rhetorical and syncopated algebra as they move toward learning and doing  symbolic algebra?

**EDIT (note change of date): Reading for Monday 28: More on Babylonian word problems

  

Our reading comes from a chapter on the history of word problems from my book, A man left Albuquerque heading east: Word problems as genre in mathematics education. This section is mostly about Babylonian word problems -- we may read the rest of the chapter as we consider the history of mathematics teaching and learning alongside the history of mathematics in itself.

Please post your response by  **EDIT: Mon September 28 at 9 AM.

A few additional resources: timelines and Babylonian word problem

  Here are some resources from Dr. Irene Percival (from SFU) including a timeline of overlaps between different cultures we're looking at and their mathematics, and an example of a quadratic Babylonian word problem!

https://drive.google.com/file/d/1EArxj8x8ihpA16iJwrHVLQOKELIdfyYX/view?usp=drivesdk