V'Zot HaBracha- Estimating based on Non-Standard units

This Shabbat, being Chol HaMoed Sukkot, we will be reading a portion from parshat Ki Tisa. However, since next week (on Monday in Israel and Tuesday outside of Israel) we read V'Zot HaBracha and Bereishit in the same week, I felt it would be appropriate to appropriate V'Zot HaBracha for this week.

"And of Naphtali he said: Naphtali, fulfilled of desire, and filled with the blessing of Hashem; take possession of the sea and south." ~Devarim 33:23

Rashi on 33:23:
Take possession- Rashi explains that The Sea of Kinnereth (or Sea of Galilee) were Naphtali's portion, and at the Southern section of the sea, he took an additional section that was the width of a length of fishing net rope so that they would be able to fish by spreading out their nets and snares.

Baba Kamma 81b further explains that while the land of Naphtali was actually to the north and west of the Sea, the tribe was also given a strip at the south of the Sea that was wide enough to allow for their finishing nets to be pulled.

Parsha/Math Investigation:
Based on the usage of non-standard units to indicate the southern section of the Sea that was portioned to Naphtali, how can we estimate a section that could have realistically been given to them?

Using current information on the Sea of Galilee (source: Wikipedia.org), we know that the Galilee is 13 km (or 8.1 mi) at it's widest section. It's narrower sections lead from the Upper Jordan River and lead down into the Lower Jordan River. In our case of Naphtali's land portion, we are looking at the southern section of the Sea, so we are looking at the section that leads down into the Lower Jordan River. Using even just basic visual estimation, the narrowest southern section is approximately 1/6 of the widest section of the Sea. That would mean that, even at its narrowest, it's still about 2 km wide, or a little over a mile wide. If their section of inheritance from the Sea was only wide enough for fishing ropes, then we have to understand that they didn't receive a full section from one shore to the opposite shore. Rather, it had to be a section with a border somewhere between the two shores. Without knowing too much about standard lengths of fishing nets, based on some quick research, my best guess would be that the nets themselves wouldn't be more than a couple hundred yards at absolute most, which would seem to necessitate only needing to grant access or propriety to about 1/10 of a mile or approximately 1/10 of the distance out from the western shoreline on the southern section of the Sea.

More extensive researching of specific maps of the Sea of Galilee and historical fishing nets and techniques could offer information for a closer estimate to the exact Sea area that was given to Naphtali.

Related Math Investigations:

• Younger students could use different items as base units for non-standard measurement around the classroom-
• How many paper-clips long is the table?
• How many blocks tall (using identical blocks) is the bookshelf?
• Older students could measure things around the room using non-standard units, but then also compare these measurements to standard measurements.
• For older students, an idea for the classroom that is similar to the concept above from the parsha might be: 
• How many papers long is the table?
• How many inches/feet/meters long is the table?
• If I need to work on one side of the table and have space for my paper, how many inches/feet/meters of space do I need to allow myself on my end of the table?
• A step above for even older students might be to consider how much space would need to be allotted to individuals in order to have space to work at the table, and then consider, based on their estimates, how many students could effectively share the table and be able to get their work done without bumping into each other or invading each other's work space.

V'Zot HaBracha- Math Related Grammar

This Shabbat, being Chol HaMoed Sukkot, we will be reading a portion from parshat Ki Tisa. However, since next week we read V'Zot HaBracha and Bereishit in the same week, I felt it would be appropriate to appropriate V'Zot HaBracha for this week, and continue with Beresheit next week.

"And He became King over Yeshurun when the numbers of the nation are gathered- the tribes of Israel in unity." ~Devarim 33;5 

Rashi on 33;5 explains that it says "numbers of the nation" referring to when the nation gathers and is unified, they are worthy of being individually counted and being blessed by Hashem.

Mathematical Grammar:
If you are a person who is aware of proper grammar, you may have noticed that there are certain words that many people misuse on a regular basis. If the grocery store sign "12 items or less" has ever bothered you, then you have an awareness of mathematical grammar. To be accurate, the sign should read "12 items or fewer". What's the difference? There are certain words that relate specifically to countable quantities, while their counterpart words, which are often mistakenly used, actually relate to non-countable quantities (although there are exceptions where one word can be used for both).

What does this mean?

Consider asking, "Do you want more or less soup?" as opposed to "Do you want more or fewer pretzels?" Less applies to soup- a non-countable item, while fewer applies to pretzels- a countable item. If you were to switch and use fewer with the soup question or less with pretzels, it doesn't sound nearly as smooth, because the wording isn't right.

Another such word pair example is many and much. Would you ask "How many soup would you like?" or would you ask "How much soup would you like?". Consider the pretzels- "How many pretzels?" or "How much pretzels?". Again, many applies to the countable item, while much  refers to the non-countable quantity.

Parsha Connection:
If the Torah were talking about the nation- it would be referred to as a single unit- "when the nation is gathered". Rather, we are told of when the "numbers of the nation are gathered"- clearly referring to countable individuals- because when the nation is in perfect unity, each individual within the nation is worthy of being counted and blessed.

Think about referring to a change in size of the nation. We would speak of more or fewer people, but when referring to the nation as a whole, we would speak of the nation being larger or smaller. Again, more/fewer people refer to a countable entity, while larger/smaller nation is a non-countable quantity.