"And you shall see from among the entire people, men of means, G-d-fearing people, men of truth, people who despise money, and you shall appoint them leaders of thousands, leaders of hundreds, leaders of fifties, and leaders of tens." ~Shemot 18;21
"Moshe heeded the voice of his father-in-law, and did everything that he had said. Moshe chose men of accomplishment from among all Israel and appointed them heads of the people, leaders of thousands, leaders of hundreds, leaders of fifties, and leaders of tens." ~Shemot 18;24
Rashi on 18;21--
Leaders of Thousands- They were six hundred of these leaders for six-hundred-thousand adult male Israelites.
Leaders of Hundreds- Of these there were six thousand
Leaders of Fifty- twelve thousand
And Leaders of Tens- sixty thousand
From Parshat Bo (Shemot 12;37) we know "The Children of Israel journeyed from Ramses to Succoth, about six hundred thousand on foot, the men, aside from the children"
In this week's parsha, we learn about Moshe trying to arbitrate on issues for the Jewish people. We hear of how he sits from morning until evening just hearing cases for people. When Yitro sees all the time that Moshe dedicates to arbitration, Yitro makes a suggestion to appoint levels of leaders to help relieve some of Moshe's burden; a suggestion which Moshe heeds.
In Rashi's commentary on the suggested levels of leaders, he does some basic division for us to let us know how many leaders there were at each level. This division calculation is important for us to understand that the leaders were not separated from the whole group of Jewish people, but rather, when calculating how many leaders are needed, the leaders themselves are included in the number of Jewish people.
Definitions:
Division: The process of separating a group or item into a number of smaller, equal-sized groups or pieces. With smaller numbers, it's easy to model the actual division with manipulatives (pieces that can be laid out and moved around to model a mathematical situation) or draw a diagram to aid in calculation until students become proficient with basic math facts.
Dividend: The whole group that is being divided
Divisor: The number of groups into which you are separating the dividend
Quotient: The answer to a division calculation
When we write basic division problems, we write dividend ÷ divisor. When we write division as a fraction, the dividend goes on top (numerator) and the divisor goes on the bottom (denominator). Many students find the algorithm set-up for long-division confusing, because when we set-up the division box, the dividend goes inside the box and the divisor goes on the outside.
When we need to start dividing very large numbers- in the tens, hundreds, thousands, etc.- the numbers become too big to model, and students begin to rely more on algorithmic notations (those big long division bars that we all remember) to keep track of their calculations. The other option is to draw representative diagrams to help students feel more comfortable with their number sense and help them connect meaning to the division of large numbers. For example, students can draw cubes to represent numbers of thousands, squares to represent hundreds, lines to represent tens, and dots to represent ones. In this way, they can visualize the breakdown of the numbers into equal groups. For students who have difficulty switching to the algorithm, working through calculations side-by-side visually and with the algorithm can help them to connect the concept.
Division trick: When dividing with large numbers that end in zeros (are multiples of tens, hundreds, thousands, etc.), you can remove any number of zeros from the end of both dividend and divisor as long as you can remove the same number of zeros from both. You can then do your division and your quotient will be the same as it would be for the larger number. For example, 100 ÷ 20: we can take one zero away from each number, divide 10 ÷ 2 = 5, so, 100 ÷ 20 = 5.
Connection to Parsha:
So, if we know how to divide, we can check on Rashi's calculations and make sure that all the numbers match up.
For 600,000 adult males, we have:
--Leaders of thousands- 600,000 ÷ 1,000 = 600 (same as 600 ÷ 1, if we remove the 3 zeros at the end of both numbers)
--Leaders of hundreds- 600,000 ÷ 100 = 6,000 (same as 6,000 ÷ 1)
--Leaders of fifties- 600,000 ÷ 50 = 12,000 (same as 60,000 ÷ 5)
--Leaders of tens- 600,000 ÷ 10 = 60,000 (same as 60,000 ÷ 1)
So, here we have a total of 78,600 leaders total for the whole group.
If the numbers of leaders had not been included in the number of the whole group, the numbers would have been much different. How different? Let's assume that the leaders from each previous tier are not included in the next lower breakdown.
600,000 - 600 (leaders of thousands) = 599,400 ÷ 100 = 5,994
600,000 - 6,594 (leaders of thousands + leaders of hundreds) = 593,406 ÷ 50 = 11,868.12 (let's assume 11,868)
600,000 - 18,462 (leaders of thousands + leaders of hundreds + leaders of fifties)= 581,538 ÷ 10 = 58,153.8 (let's assume 58,153)
So, all the number of leaders here would have been 76,615 (leaders of thousands + leaders of hundreds + leaders of fifties + leaders of tens) for the whole group.
All together, then, by having the leaders included in the total count of Jewish people, there were a minimum of about 2,000 additional leaders who were assigned to help Moshe with arbitration of matters.
Thursday, January 16, 2014
Friday, January 10, 2014
Beshalach- Probability
"The enemy said, 'I will pursue, I will overtake, I will divide spoils; my soul shall be filled with them. I will draw my sword, my hand will impoverish them.'" ~Shemot 15;9
In this section of the parsha, Moshe and the Jewish people are singing a song of praise to Hashem after he's taken them safely out of Egypt. The specific phrase above is unique in that it is the only instance in the entire Torah of a 5-word alliteration (in the Hebrew text). The first 5 words of the pasuk all begin with the letter aleph (א). While we believe that every word, and even every letter, in the Torah was carefully chosen and is included with purpose, this alliteration still raises the question: What is the probability of having 5 words in a row beginning with the same letter?
Probability:
Definition- the chance that something will happen (merriam-webster.com)
Probability is expressed as a fraction between 0 and 1, where 0 means that there is no chance for the event to happen and 1 means that the event will definitely happen. For clarification, chance and probability are the same, however, chance is the number expressed as a percentage and probability is the number expressed as a fraction.
Let's look at a simple example:
Let's imagine a bag with 10 marbles, where 3 are red, 3 are yellow, and 4 are blue. If we want to know the probability of picking 1 red marble, we see that there are 3 red marbles out of the total 10, so the probability is 3/10. (Remember when we converted fractions to percentages back in Parshat Vayigash? We can use that here to calculate the chance of any probability)
Now, let's make it a little harder and think about how to calculate the probability of picking out a red marble from that bag twice in a row. Assuming that we pick a marble, put it back in the bag, and pick again...
The probability of picking the red marble the first time is 3/10 and the probability of picking it out the second time is also 3/10. To find the compound probability you multiply these two together so
3/10 x 3/10 = 9/100
If we didn't put back the red marble after picking the first time, then the calculation would be slightly different. The probability of picking the red marble the first time would still be 3/10, but, assuming that first marble is red like we want, and we're holding it out of the bag, the there are only 9 marbles left in the bag, and only 2 of them are still red. This means that the probability of picking a second red marble in this case is 2/9. Now, to calculate the compound probability, we still multiply the first probability times the second, so...
3/10 x 2/9 = 6/90, which can be reduced to 1/15
Connection to the parsha:
Let's think about how to apply this to our question. First of all, there are 22 letters in the Hebrew alphabet. Now, in our situation, we're talking about probability of something happening 5 times in a row, so we're looking at compound probability. Does our situation match with the marble being replaced or not replaced? Since letters can be used and reused without limit in language, our case matches with the marble being replaced.
So, we have 1 aleph out of 22 letters in the alphabet, and it appears at the start of the word 5 times in a row. So, the probability is:
1/22 x 1/22 x 1/22 x 1/22 x 1/22 which equals 1/5153632. Rounded off, we can say that the probability of this alliteration happening is 1 in 5 million.
"...אמר אויב ארדף אשיג אחלק שלל"
Transliteration: "Amar oyeiv erdof asig echaleik shallal..."In this section of the parsha, Moshe and the Jewish people are singing a song of praise to Hashem after he's taken them safely out of Egypt. The specific phrase above is unique in that it is the only instance in the entire Torah of a 5-word alliteration (in the Hebrew text). The first 5 words of the pasuk all begin with the letter aleph (א). While we believe that every word, and even every letter, in the Torah was carefully chosen and is included with purpose, this alliteration still raises the question: What is the probability of having 5 words in a row beginning with the same letter?
Probability:
Definition- the chance that something will happen (merriam-webster.com)
Probability is expressed as a fraction between 0 and 1, where 0 means that there is no chance for the event to happen and 1 means that the event will definitely happen. For clarification, chance and probability are the same, however, chance is the number expressed as a percentage and probability is the number expressed as a fraction.
Let's look at a simple example:
Let's imagine a bag with 10 marbles, where 3 are red, 3 are yellow, and 4 are blue. If we want to know the probability of picking 1 red marble, we see that there are 3 red marbles out of the total 10, so the probability is 3/10. (Remember when we converted fractions to percentages back in Parshat Vayigash? We can use that here to calculate the chance of any probability)
Now, let's make it a little harder and think about how to calculate the probability of picking out a red marble from that bag twice in a row. Assuming that we pick a marble, put it back in the bag, and pick again...
The probability of picking the red marble the first time is 3/10 and the probability of picking it out the second time is also 3/10. To find the compound probability you multiply these two together so
3/10 x 3/10 = 9/100
If we didn't put back the red marble after picking the first time, then the calculation would be slightly different. The probability of picking the red marble the first time would still be 3/10, but, assuming that first marble is red like we want, and we're holding it out of the bag, the there are only 9 marbles left in the bag, and only 2 of them are still red. This means that the probability of picking a second red marble in this case is 2/9. Now, to calculate the compound probability, we still multiply the first probability times the second, so...
3/10 x 2/9 = 6/90, which can be reduced to 1/15
Connection to the parsha:
Let's think about how to apply this to our question. First of all, there are 22 letters in the Hebrew alphabet. Now, in our situation, we're talking about probability of something happening 5 times in a row, so we're looking at compound probability. Does our situation match with the marble being replaced or not replaced? Since letters can be used and reused without limit in language, our case matches with the marble being replaced.
So, we have 1 aleph out of 22 letters in the alphabet, and it appears at the start of the word 5 times in a row. So, the probability is:
1/22 x 1/22 x 1/22 x 1/22 x 1/22 which equals 1/5153632. Rounded off, we can say that the probability of this alliteration happening is 1 in 5 million.
Thursday, January 2, 2014
Bo- Finding the midpoint
"And Moshe said, 'So said Hashem, 'At midnight I shall go out in the midst of Egypt. Every firstborn in the land of Egypt shall die, from the firstborn of Pharaoh who sits on his throne to the firstborn of the slave-woman who is behind the millstone and all the firstborn of the animal." ~Shemot 11;4-5
"It was at midnight, and Hashem smote every firstborn in the land of Egypt, from the firstborn of Pharaoh sitting on his throne to the firsborn of the captive who was in the dungeon, and every firstborn animal." ~Shemot 12;29
In his commentary on 11;4 above, Rashi explains that "[Midnight] means when the night is divided". This is understood to mean when the night is divided exactly in half, into two equal portions. In other words, the exact time in the middle of the night when the nighttime that has passed and the nighttime that remains is exactly the same.
Midpoint:
In math, the midpoint is the exact point at which half is before the point and half is after the point. There are different areas of math where midpoint is needed.
*In Geometry, we can measure the midpoint of lines or sides of shapes (exactly halfway along a line or side).
*In Statistics, given a set of data, there is often a need to calculate the midpoint between two numbers.
Calculating to find a midpoint is the same as trying to find the mean or arithmetic average of two numbers. We can find the midpoint by adding the two numbers together and dividing the sum by 2.
*So, if you want to locate the midpoint of a 10 ft wall, think about the measuring tape along the side of the wall: the measuring tape starts at 0 (the beginning of the wall) and ends at 10 ft (where the wall ends). You would add 0 + 10 = 10, then divide 10 ÷ 2 = 5. So, the 5 ft mark would be the midpoint of the wall.
*If you want to find the midpoint between any two numbers, say 10 and 70, you would add 10 + 70 = 80, then divide 80 ÷ 2 = 40. So, the midpoint between 10 and 70 is 40.
Connection to the parsha:
In this week's parsha, we learn that Hashem promises that he will kill the non-Jewish firstborns (humans and animals) at the exact midpoint of the night- the exact midpoint between sunset and sunrise. This is actually a calculation that we still use nowadays when calculating times according to Jewish law. Let's go through a sample calculation for finding "midnight", according to the Torah understanding of the term (ie. the midpoint of the night).
Let's take some data for sunset 1/2/2014 and sunrise 1/3/2014 in Boston, MA. The midpoint between these two times, should be exact midnight.
--sunset 1/2/2014 is 4:23 pm
--sunrise 1/3/2014 is 7:14 am
*First, we need to calculate how many hours there are between sunset and sunrise (ie how many hours of darkness). This time difference is 14 hrs and 51 min.
*Next, we divide this time in half. 14 hrs and 51 min divided in half is 7 hrs and 25.5 min.
*Finally, we can either add the 7 hrs and 25.5 min to 4:23 am or subtract it from 7:14 pm (get it? because it's exactly halfway between the two times, you can get to it either way). Since it's usually more intuitive to add rather than subtract, especially when working with time, let's add 7 hrs and 25.5 min onto 4:23. This give us 11:48:30 pm (that's hrs:min:sec)
For fun, let's make a connection directly to the parsha; let's calculate midnight in Egypt in the month of April, to get an idea for what time the parsha is referring to.
This year Pesach begins on April 10, so let's take data for sunset on 4/10/2014 and sunrise 4/11/2014 in Cairo, Egypt.
--sunset 4/10/2014 is 6:19 pm
--sunrise 4/11/2014 is 5:33 am
* First, we calculate how many hours between sunset and sunrise. This time difference is 11 hrs and 14 min.
*Next, we divide this time in half. 11 hrs and 14 min divided in half is 5.5 hrs and 7 min, in other words, 5 hrs and 37 min.
*Finally, we add 5 hrs and 37 min to 6:19 pm. This gives us 11:56 pm.
Whatever date and time you choose- any season, any region- it should always approximate midnight as we know it (12 am). Test it out and see what you find!
"It was at midnight, and Hashem smote every firstborn in the land of Egypt, from the firstborn of Pharaoh sitting on his throne to the firsborn of the captive who was in the dungeon, and every firstborn animal." ~Shemot 12;29
In his commentary on 11;4 above, Rashi explains that "[Midnight] means when the night is divided". This is understood to mean when the night is divided exactly in half, into two equal portions. In other words, the exact time in the middle of the night when the nighttime that has passed and the nighttime that remains is exactly the same.
Midpoint:
In math, the midpoint is the exact point at which half is before the point and half is after the point. There are different areas of math where midpoint is needed.
*In Geometry, we can measure the midpoint of lines or sides of shapes (exactly halfway along a line or side).
*In Statistics, given a set of data, there is often a need to calculate the midpoint between two numbers.
Calculating to find a midpoint is the same as trying to find the mean or arithmetic average of two numbers. We can find the midpoint by adding the two numbers together and dividing the sum by 2.
*So, if you want to locate the midpoint of a 10 ft wall, think about the measuring tape along the side of the wall: the measuring tape starts at 0 (the beginning of the wall) and ends at 10 ft (where the wall ends). You would add 0 + 10 = 10, then divide 10 ÷ 2 = 5. So, the 5 ft mark would be the midpoint of the wall.
*If you want to find the midpoint between any two numbers, say 10 and 70, you would add 10 + 70 = 80, then divide 80 ÷ 2 = 40. So, the midpoint between 10 and 70 is 40.
Connection to the parsha:
In this week's parsha, we learn that Hashem promises that he will kill the non-Jewish firstborns (humans and animals) at the exact midpoint of the night- the exact midpoint between sunset and sunrise. This is actually a calculation that we still use nowadays when calculating times according to Jewish law. Let's go through a sample calculation for finding "midnight", according to the Torah understanding of the term (ie. the midpoint of the night).
Let's take some data for sunset 1/2/2014 and sunrise 1/3/2014 in Boston, MA. The midpoint between these two times, should be exact midnight.
--sunset 1/2/2014 is 4:23 pm
--sunrise 1/3/2014 is 7:14 am
*First, we need to calculate how many hours there are between sunset and sunrise (ie how many hours of darkness). This time difference is 14 hrs and 51 min.
*Next, we divide this time in half. 14 hrs and 51 min divided in half is 7 hrs and 25.5 min.
*Finally, we can either add the 7 hrs and 25.5 min to 4:23 am or subtract it from 7:14 pm (get it? because it's exactly halfway between the two times, you can get to it either way). Since it's usually more intuitive to add rather than subtract, especially when working with time, let's add 7 hrs and 25.5 min onto 4:23. This give us 11:48:30 pm (that's hrs:min:sec)
For fun, let's make a connection directly to the parsha; let's calculate midnight in Egypt in the month of April, to get an idea for what time the parsha is referring to.
This year Pesach begins on April 10, so let's take data for sunset on 4/10/2014 and sunrise 4/11/2014 in Cairo, Egypt.
--sunset 4/10/2014 is 6:19 pm
--sunrise 4/11/2014 is 5:33 am
* First, we calculate how many hours between sunset and sunrise. This time difference is 11 hrs and 14 min.
*Next, we divide this time in half. 11 hrs and 14 min divided in half is 5.5 hrs and 7 min, in other words, 5 hrs and 37 min.
*Finally, we add 5 hrs and 37 min to 6:19 pm. This gives us 11:56 pm.
Whatever date and time you choose- any season, any region- it should always approximate midnight as we know it (12 am). Test it out and see what you find!