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.
Wednesday, October 8, 2014
Thursday, October 2, 2014
Yom Kippur- Shofar Blasts and Equivalent Fractions
The parsha that we read on Yom Kippur comes from a section of Acharei Mot. My previous post from Acharei Mot covers the topic of the two goats that were brought for atonement by the Kohen Gadol on behalf of the Jewish people on Yom Kippur.
As one feature characteristic of this time of year is the blowing of the shofar, I thought I would look at the shofar blowing this week. The month of Elul leading up to Rosh Hashana, then the first 10 days of Tishrei- the days of atonement between Rosh Hashana and Yom Kippur- we listen to the shofar. Throughout Elul, we hear the shofar blown once each day. On Rosh Hashana, we hear 100 blasts of the shofar on both days. On Yom Kippur, we hear a final shofar blast at the end of the day.
Equivalent Fractions:
Equivalent fractions are different ways of writing the same fractional piece of a whole item (or group). They are two or more fractions that represent the same amount of an item but are written with different numerators and denominators. We can find these fractions by breaking the whole item into more or fewer pieces. The important thing to look for is that when you cut it into more pieces, a certain number of those pieces has to exactly create a larger piece.
For example, [Case 1] if I have a pizza and I cut it into two pieces and eat one of the pieces, then I've eaten 1/2 of the pizza. Now, [Case 2] if I had cut the pizza into 4 pieces and eaten 2 of the pieces, then I would have eaten 2/4 pieces, which is actually the same amount of pizza as in Case 1. [Case 3] If I had cut the pizza into 6 pieces and eaten 3 of the pieces, then I would have eaten 3/6, which is, again, actually the same amount of pizza as in the first two cases. But now, what if I had cut the pizza into 3 pieces, or 5 pieces? In these two situations, we could not take a number of pieces to make them exactly 1/2 of the pizza. So, we can make equivalent fractions with 1/2 using fourths and sixths, but we can't find equivalent fractions using thirds or fifths.
Shofar Connection:
From the time that young children begin learning about the shofar blasts, they are taught that there are 3 different types of blasts: tekiah, shevarim, and teruah. They learn that tekiah is one long blast, shevarim is 3 shorter blasts, and teruah is 9 staccato blasts.
(artwork credit to my preschool daughter)
If we think about these blasts as fractions, we can consider that tekiah is one whole blast (one single unit), the 3 parts of the shevarim blast are each 1/3 of the length of the tekiah, and the 9 parts of the teruah blast are each 1/9 the length of the tekiah. If we look further, we can also compare that each group of 3 blasts of teruah is the same as 1 of the shevarim sections. In other words, we can see that 1/3 of the shevarim blast is equivalent to 3/9 of the teruah blast.
In theory, this is all nice, but how do we quantify the blasts? What measurements do we use? Are they actually equivalent fractions? I checked with my husband on this- our local shofar blower (ba'al tokeah)- to get an answer. Below is a purely theoretical example of how it should work with the blasts measured in seconds.
Note: I'm purposely using numbers here that are slightly larger than in practice to make calculations easier)
If Tekiah is set to 9 seconds, then Shevarim is also set to 9 seconds for the whole blast (so the total is equivalent to tekiah), which means that each 1/3 section must be 3 seconds to be considered kosher.
Teruah would also be set to 9 seconds for the whole blast (so the total is equivalent to tekiah and shevarim), which means that each 1/9 section must be 1 second (and 3 sections together equal 3 seconds to be equivalent to a 1/3 shevarim section) to be considered kosher.
These would work for both "tekiah-shevarim-tekiah" and "tekiah-teruah-tekiah" blasts. Since the middle section cannot be longer than the tekiah itself, we would need to adjust for the "tekiah-shevarim/teruah-tekiah". For these blasts, the tekiah would be set to 12 seconds, and the shevarim/teruah together would need to be 12 seconds, so shevarim would be 6 seconds (2 seconds for each short blast), and teruah would be 6 seconds (6/9 or 2/3 of a second for each of the staccato blasts).
Now, in practice, when my husband blows shofar, our Rabbi watches with a stop watch to make sure that the shofar blasts are, in fact, fractional pieces of each other. What does this look like in practice?
The first thing to keep in mind is that practical situations are variable, and it's better to overestimate the length of a blast in order to make sure that it's kosher. Basically, when you have a kosher blast, you want to be able to count it. In order to do this, the procedure is basically as follows:
Since we can't predict the exact time in seconds that the middle blasts will be (shevarim, teruah, and shevarim/teruah), we overestimate on the time length for the first tekiah to make sure that the total length of that middle blast is always about the same length as the tekiah, but never longer than the tekiah. Since, for the final tekiah, you know exactly how long the middle blasts took, the final tekiah is timed to be exactly as long as the middle section. Also, since holding 100 shofar blasts for 10-12 seconds is extremely wearing, the blasts are actually shorter (and do not use quick, simple numbers), which makes a lot of the calculations involve fractions of seconds.
So, in real life, the blasts look something like this:
For "tekiah-shevarim-tekiah", the first tekiah is overestimated at about 5 seconds, the goal for shevarim is to have 3 1-1/2 second sections, making the whole shevarim 4-1/2 seconds long, and the final tekiah is 4-1/2 seconds.
For "tekiah-teruah-tekiah", the first tekiah is overestimated at about 6 seconds, the goal for the teruah is to have 9 sections that are just over 1/2 second each (5/9 of a second), making the whole teruah 5 seconds long, and the final tekiah is 5 seconds.
For "tekiah-shevarim/teruah-tekiah", the first tekiah is overestimated at about 9 seconds, the goal for the shevarim/teruah is to have 3 1-1/3 second blasts for the shevarim (4 seconds total for that part) and 9 sections that are just under 1/2 second each (4/9 of a second) for the teruah (4 seconds total), making the total shevarim/teruah 8 seconds long, and the final tekiah is 8 seconds long.
What does this mean about the tekiah gedolah (the long tekiah at the end of the 100 blasts)? My husband tries to make that one 12 seconds. Why 12 seconds? Because this would be equivalent to 1 estimated tekiah length from the shevarim/teruah set, plus another 1/3 or another 3 seconds, since that additional fractional amount is enough to make it noticeably longer than the other blasts.
Best wishes for a G'mar Chatima Tova- a sealed inscription for a year filled with only good things.
Tuesday, September 23, 2014
Haazinu- One to One Correspondence
"When the Supreme One gave nations their portion, when He separated the children of man, He set the borders of peoples according to the number of the Children of Israel." ~Devarim 32;8
Rashi on 32;8 makes reference to 70 languages which correspond to the 70 nations enumerated in Bereishit chapter 10, which developed from the descendants of Noah.
One to One Correspondence:
Have you ever heard a young child count pictures in a book and watched as they accurately count up to 13, only to see that there are actually only 5 pictures on the page? Have you watched a small child try to count items in front of them, trying to figure out which ones were already counted?
One to One Correspondence is one of the first concepts that young children need to understand in order to understand what numbers actually mean. One to one correspondence is the idea that each counting number correlates to one item that is being counted. The way to assign a number to an entire group is to have each item represented by a single number; we can then say that the size of the group is that of the largest counting number used to count the items. If I have 12 marbles, for example, I would be able to know this by counting each individual marble once, and identifying that when I've counted the last marble, the number that I used to count it was number 12. Since I ran out of marbles at number 12, that means that I have 12 marbles in my group.
Once children know the proper order of the counting numbers, the biggest confusion for them when learning to count is how to keep track of what they have already counted. The best strategy for teaching tracking while you're counting is to move the counted items to the side, out of the way, so that they can easily identify which items have already been counted and which items still need to be counted. So how can you help children practice counting when they are given pictures or drawings of items? The best thing to do in that situation is to line up or organize the drawings on the paper in such a way that children can methodically move through each section of the picture in a way that they can keep track of what has already been counted. Organized rows and columns work very well in this type of situation. The fancy books with dynamic types of designs tend to be more confusing and leave children less successful in their counting.
Parsha Connection:
When Rashi references the 70 languages which correspond to the 70 nations that came from the descendants of Noah, this is a perfect example of one to one correspondence. In this case, we have 70 descendants- each of those 70 descendants developed into their own nation, resulting in 70 nations (one descendant for each nation)- and each of the 70 nations had its own language, resulting in 70 languages (one nation for each language).
Another, year-long exercise in one to one correspondence is matching each parsha (or double-parsha) with each shabbat. There is a specific order to the parshiot readings, and there are certain parshiot that are specifically read at certain times. For example, Devarim is always read the week before Tisha B'Av, and Va'etchanan is always read the week the week after Tisha B'Av; V'Zot Ha'Bracha is always read on Simchat Torah. Using these placeholders in the calendar, we can go through the year of parshiot and match them up to the shabbatot in which they will be read. This process of matching one to one is an exercise in one to one correspondence.
Everyday Connection:
The mere fact that we are each able to count items successfully on a daily basis for day-to-day activities shows that we have each internalized and successfully rely on our understanding of one-to-one correspondence. Interestingly, as we grow and our brains develop, we can start to automatically, accurately identify the quantities of small groups of items. Visually organizing items makes it easier for our brains to automatically, accurately identify the quantities of larger groups of items.
For a different example, musicians who play the harp need to identify the notes of the strings. As identification, the C note strings are all red, and the F note strings are all blue. Using these placeholders, musicians are able to identify which strings are which notes and what their finger placements need to be- 2 examples of one to one correspondence- matching strings to notes and matching fingers to strings.
What examples are in your daily life?
Rashi on 32;8 makes reference to 70 languages which correspond to the 70 nations enumerated in Bereishit chapter 10, which developed from the descendants of Noah.
One to One Correspondence:
Have you ever heard a young child count pictures in a book and watched as they accurately count up to 13, only to see that there are actually only 5 pictures on the page? Have you watched a small child try to count items in front of them, trying to figure out which ones were already counted?
One to One Correspondence is one of the first concepts that young children need to understand in order to understand what numbers actually mean. One to one correspondence is the idea that each counting number correlates to one item that is being counted. The way to assign a number to an entire group is to have each item represented by a single number; we can then say that the size of the group is that of the largest counting number used to count the items. If I have 12 marbles, for example, I would be able to know this by counting each individual marble once, and identifying that when I've counted the last marble, the number that I used to count it was number 12. Since I ran out of marbles at number 12, that means that I have 12 marbles in my group.
Once children know the proper order of the counting numbers, the biggest confusion for them when learning to count is how to keep track of what they have already counted. The best strategy for teaching tracking while you're counting is to move the counted items to the side, out of the way, so that they can easily identify which items have already been counted and which items still need to be counted. So how can you help children practice counting when they are given pictures or drawings of items? The best thing to do in that situation is to line up or organize the drawings on the paper in such a way that children can methodically move through each section of the picture in a way that they can keep track of what has already been counted. Organized rows and columns work very well in this type of situation. The fancy books with dynamic types of designs tend to be more confusing and leave children less successful in their counting.
Parsha Connection:
When Rashi references the 70 languages which correspond to the 70 nations that came from the descendants of Noah, this is a perfect example of one to one correspondence. In this case, we have 70 descendants- each of those 70 descendants developed into their own nation, resulting in 70 nations (one descendant for each nation)- and each of the 70 nations had its own language, resulting in 70 languages (one nation for each language).
Another, year-long exercise in one to one correspondence is matching each parsha (or double-parsha) with each shabbat. There is a specific order to the parshiot readings, and there are certain parshiot that are specifically read at certain times. For example, Devarim is always read the week before Tisha B'Av, and Va'etchanan is always read the week the week after Tisha B'Av; V'Zot Ha'Bracha is always read on Simchat Torah. Using these placeholders in the calendar, we can go through the year of parshiot and match them up to the shabbatot in which they will be read. This process of matching one to one is an exercise in one to one correspondence.
Everyday Connection:
The mere fact that we are each able to count items successfully on a daily basis for day-to-day activities shows that we have each internalized and successfully rely on our understanding of one-to-one correspondence. Interestingly, as we grow and our brains develop, we can start to automatically, accurately identify the quantities of small groups of items. Visually organizing items makes it easier for our brains to automatically, accurately identify the quantities of larger groups of items.
For a different example, musicians who play the harp need to identify the notes of the strings. As identification, the C note strings are all red, and the F note strings are all blue. Using these placeholders, musicians are able to identify which strings are which notes and what their finger placements need to be- 2 examples of one to one correspondence- matching strings to notes and matching fingers to strings.
What examples are in your daily life?