"They shall make an Aron of shittim wood, two and a half amot its length; an amah and a half its width; and an amah and a half its height. You shall cover it with pure gold, from inside and from outside you shall cover it, and you shall make on it a golden diadem all around...You shall make a lid of pure gold, two and a half amot its length, and an amah and a half its width" ~Shemot 25;10-11, 17
In Parshat Terumah, we are essentially given the blueprints for building the mishkan, or traveling tabernacle, that the Jewish people built while traveling in the dessert. This parsha is filled with so many mathematical possibilities, but I'd like to focus on a follow-up topic from back in Parshat Vayeishev.
When investigating with 3-dimensional figures, the major calculations that students investigate are volume and surface area. Volume, as discussed in Parshat Vayeishev, is the space, in cubic units, that an object occupies or the amount of space inside the shell of a shape that can be filled. We learned that the calculation for the volume of a rectangular prism (or box shape) is to multiply the length x width x height of the prism.
Surface area is the calculation of the 2-dimensional area that covers the entire shape. For a rectangular prism, that would be the total area of all 6 sides of the prism added together. While volume is measured in cubic units, surface area is calculated in square units (the difference between a 3-D measurement for volume and a 2-D measurement for area). Lateral surface area, a variant of surface area, is the calculation of the area that covers the middle section of the prism, but does not include the top and the bottom. As a reminder, the area of a rectangular 2-dimensional shape is calculated by multiplying the length x width.
An example that I like to use to help understand the meaning of volume, surface area, and lateral surface area:
If I have a rectangular shaped room...
...volume is the amount of space inside the room if I filled it completely from floor to ceiling and wall to wall.
...surface area is the amount of wall space that I would cover if I wanted to paint the entire room- all four walls, the floor, and the ceiling.
...lateral surface area is the amount of wall space that I would cover if I wanted to paint or wallpaper just around the four walls.
Connection to the parsha:
Given the information that we are given for building the mishkan, can we calculate volume, surface area, and lateral surface area? If so, what are each of these three measurements?
We are given the 3 necessary dimensions for accurately building the mishkan, so we do have enough information to make our calculations.
Length- 2.5 amot
Width- 1.5 amot
Height- 1.5 amot
We also know that the mishkan had a lid made for it as well, with a length of 2.5 amot and a width of 1.5 amot, which matches perfectly to fit right on top of the mishkan and close it to create a closed rectangular prism.
Volume- how much space was inside the mishkan?
2.5 x 1.5 x 1.5 = 5.625 cubic amot
Surface area- the parsha states that the mishkan was to be covered in gold, both the box and the lid. So, how much gold was needed to cover the mishkan on all sides?
The area of the top and bottom of the mishkan was 2.5 x 1.5 = 3.75 square amot for each (or 3.75 x 2 = 7.5 square amot total for those two sides).
The area of the front and back of the mishkan were 2.5 x 1.5 = 3.75 square amot for each (or 3.75 x 2 = 7.5 square amot total for those two sides).
The area of the left and right sides of the mishkan were 1.5 x 1.5 = 2.25 square amot for each (or 2.25 x 2 = 4.5 square amot total for those two sides).
When you add the square area of all six sides together, you get 7.5 + 7.5 + 4.5 = 19.5 square amot of gold to cover the entire mishkan.
Lateral surface area would be the same calculation as above, but we would leave out the area of the top and bottom, which would leave us with 7.5 + 4.5 = 12 square amot. (Note that another way to calculate lateral surface area is to calculate the perimeter, or length around the outside of the base shape, and multiply that length times the height of the prism.)
An extra question- The parsha states that the bottom of the mishkan was made out of shittim wood before being covered in gold, but the top is just made of gold. How much wood was necessary to build the wooden framework of the mishkan?
Answer- This is a combination of the calculation for surface area and lateral surface area. Really, the calculation is for lateral surface area with the addition of just the bottom area (but not the top area). From our lateral surface area calculations, we know that the area of the four sides is 12 square amot. We also know that the area of just the bottom part was 3.75 square amot. So, we add 12 + 3.75 = 15.75 square amot of shittim wood for the base framework of the mishkan.
Friday, January 31, 2014
Thursday, January 23, 2014
Mishpatim- Calculating Interest
"When you will lend money to My people, to the poor person who is with you, do not act toward him as a creditor; do not place interest upon him." ~Shemot 22;24
Rashi on 22;24--
Interest (root word is the same as "snake")- This means increase, for it is like the bite of a snake, for a snake bites a small wound in one's foot and [the victim] does not feel it, but suddenly it causes puffiness and swelling up to the crown of his head. So it is with interest. He does not feel it and is not aware of it, until the interest accumulates and causes him a loss of much money.
Definitions:
Principal: An original sum of money
Interest (or Simple Interest): A calculated percentage value of the principal amount
Compound Interest: A method of calculating interest, where the previous interest is added to the principal amount for future interest calculations
In my experience, interest is a particularly confusing concept for students. I believe that this stems from the somewhat complex aspects of the calculation. In order to calculate interest, you need to first convert the percentage (interest rate) into a decimal or fraction in order to make the calculation. Once you've made the calculation, you have only actually calculated the interest amount. When using interest, the interest is paid on top of the original principal amount. So, in order to calculate the full amount, you need to then add the interest amount together with the original principal amount. The combination of a calculation requiring conversion within a two-step calculation problem, tends to lead to a lot of confusion for many students. Breaking down the steps into a logical progression and making each step meaningful to students can help them make sense of the process, rather than forcing them to memorize a seemingly complex rote process.
Connection to Parsha:
Rashi explains that we are commanded not to charge interest on a loan because the interest builds up without the borrower realizing and suddenly the borrower owes a great amount of money to the lender. Let's use an example to understand Rashi's explanation of how interest accrues and how it can surreptitiously overtake a borrower.
Let's assume that I borrow $1,000 from a friend of mine, and he charges me just 1% interest per week on my loan. Since the parsha doesn't specify simple or compound interest, let's see how they compare side-by-side over the course of 3 months or a 13 week period.
Since we are calculating 1%, we change it into a fraction or decimal (1/100 or .01) in order to calculate with the number (See if you can use the information from Parshat Vayigash to work backwards to create the fractions/decimals for calculation with a percent). Then, we multiply the fraction or decimal times the principal amount. Once we have the weekly interest, we add the interest to the principal to see what is owed each week that the loan goes unpaid. For Simple Interest, we just have $10 interest that adds to the total each week; for Compound Interest, we need to recalculate 1% of each week's new total to calculate the new interest for the following week, and the new interest is added to the previous sum that was owed.
Rashi on 22;24--
Interest (root word is the same as "snake")- This means increase, for it is like the bite of a snake, for a snake bites a small wound in one's foot and [the victim] does not feel it, but suddenly it causes puffiness and swelling up to the crown of his head. So it is with interest. He does not feel it and is not aware of it, until the interest accumulates and causes him a loss of much money.
Definitions:
Principal: An original sum of money
Interest (or Simple Interest): A calculated percentage value of the principal amount
Compound Interest: A method of calculating interest, where the previous interest is added to the principal amount for future interest calculations
In my experience, interest is a particularly confusing concept for students. I believe that this stems from the somewhat complex aspects of the calculation. In order to calculate interest, you need to first convert the percentage (interest rate) into a decimal or fraction in order to make the calculation. Once you've made the calculation, you have only actually calculated the interest amount. When using interest, the interest is paid on top of the original principal amount. So, in order to calculate the full amount, you need to then add the interest amount together with the original principal amount. The combination of a calculation requiring conversion within a two-step calculation problem, tends to lead to a lot of confusion for many students. Breaking down the steps into a logical progression and making each step meaningful to students can help them make sense of the process, rather than forcing them to memorize a seemingly complex rote process.
Connection to Parsha:
Rashi explains that we are commanded not to charge interest on a loan because the interest builds up without the borrower realizing and suddenly the borrower owes a great amount of money to the lender. Let's use an example to understand Rashi's explanation of how interest accrues and how it can surreptitiously overtake a borrower.
Let's assume that I borrow $1,000 from a friend of mine, and he charges me just 1% interest per week on my loan. Since the parsha doesn't specify simple or compound interest, let's see how they compare side-by-side over the course of 3 months or a 13 week period.
Since we are calculating 1%, we change it into a fraction or decimal (1/100 or .01) in order to calculate with the number (See if you can use the information from Parshat Vayigash to work backwards to create the fractions/decimals for calculation with a percent). Then, we multiply the fraction or decimal times the principal amount. Once we have the weekly interest, we add the interest to the principal to see what is owed each week that the loan goes unpaid. For Simple Interest, we just have $10 interest that adds to the total each week; for Compound Interest, we need to recalculate 1% of each week's new total to calculate the new interest for the following week, and the new interest is added to the previous sum that was owed.
| Simple Interest | Compound Interest | |
| Initial Loan | $1,000 | $1,000 |
| Week 1- interest | $10 | $10 |
| Week 1- total due | $1,010 | $1,010 |
| Week 2- interest | $10 | $10 |
| Week 2- total due | $1,020 | $1,020 |
| Week 3- interest | $10 | $10 |
| Week 3- total due | $1,030 | $1,030 |
| Week 4- interest | $10 | $10 |
| Week 4- total due | $1,040 | $1,041 |
| Week 5- interest | $10 | $10 |
| Week 5- total due | $1,050 | $1,051 |
| Week 6- interest | $10 | $11 |
| Week 6- total due | $1,060 | $1,062 |
| Week 7- interest | $10 | $11 |
| Week 7- total due | $1,070 | $1,072 |
| Week 8- interest | $10 | $11 |
| Week 8- total due | $1,080 | $1,083 |
| Week 9- interest | $10 | $11 |
| Week 9- total due | $1,090 | $1,094 |
| Week 10- interest | $10 | $11 |
| Week 10- total due | $1,100 | $1,105 |
| Week 11- interest | $10 | $11 |
| Week 11- total due | $1,110 | $1,116 |
| Week 12- interest | $10 | $11 |
| Week 12- total due | $1,120 | $1,127 |
| Week 13- interest | $10 | $11 |
| Week 13- total due | $1,130 | $1,138 |
You'll notice in the calculations above that the compound interest doesn't seem much different from the simple interest for almost the full first half of the loan period. This is because we used such a small interest rate.
Let's say that my friend is not as nice to me, and charges me just a little bit more, say 5% interest. Below we'll see a much bigger difference.
| Simple Interest | Compound Interest | |
| Initial Loan | $1,000 | $1,000 |
| Week 1- interest | $50 | $50 |
| Week 1- total due | $1,050 | $1,050 |
| Week 2- interest | $50 | $53 |
| Week 2- total due | $1,100 | $1,103 |
| Week 3- interest | $50 | $55 |
| Week 3- total due | $1,150 | $1,158 |
| Week 4- interest | $50 | $58 |
| Week 4- total due | $1,200 | $1,216 |
| Week 5- interest | $50 | $61 |
| Week 5- total due | $1,250 | $1,276 |
| Week 6- interest | $50 | $64 |
| Week 6- total due | $1,300 | $1,340 |
| Week 7- interest | $50 | $67 |
| Week 7- total due | $1,350 | $1,407 |
| Week 8- interest | $50 | $70 |
| Week 8- total due | $1,400 | $1,477 |
| Week 9- interest | $50 | $74 |
| Week 9- total due | $1,450 | $1,551 |
| Week 10- interest | $50 | $78 |
| Week 10- total due | $1,500 | $1,629 |
| Week 11- interest | $50 | $81 |
| Week 11- total due | $1,550 | $1,710 |
| Week 12- interest | $50 | $86 |
| Week 12- total due | $1,600 | $1,796 |
| Week 13- interest | $50 | $90 |
| Week 13- total due | $1,650 | $1,886 |
From these two examples, we can really see what Rashi is referring to when he talks about interest accumulating quickly without the borrower realizing what he has entered into.
Thursday, January 16, 2014
Yitro- Basic division with large numbers
"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.
"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.
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!