"[Joseph's brothers] were seated before [Joseph], the firstborn according to his seniority and the youngest according to his youth..." ~Bereishit 43;33
"[The man in charge of Joseph's house] searched; he began with the oldest and ended with the youngest; and the goblet was found in Benjamin's saddlebag." ~Bereishit 44;12
In this week's parsha, we can see two instances where Joseph's brothers were organized according to their age. Putting items in order, or sequencing, is a critical math skill that is used by students of all ages (preschool to adulthood), both in school and in life.
Sequencing:
Sequencing means putting items into a specific order or arrangement.
The skill of putting any group of items or information into a designated order is an important one. Children can begin working on this skill at a very early age. By the time they reach Kindergarten and 1st grade, children should be able to put objects in a designated order, and they can work on designating attributes to objects by which to order them. For example, the same group of students could be placed together and first asked to put themselves in height order and then asked to rearrange themselves according to their birthdays.
Connection to Parsha:
In Bereishit 43;33, Rashi explains that Joseph arranged his brothers to sit first by mother, and then in birth order within their groups. So, they were seated in the following order:
Reuven, Shimon, Levi, Yehuda, Issachar, Zevulun [children of Leah], Dan, Naphtali [children of Bilhah], Gad, Asher [children of Zilpah], and Binyamin sat with Joseph [children of Rachel/"without a mother" to quote Rashi here].
In Bereishit 44;12, Rashi explains that they were searched by oldest to youngest so that they wouldn't sense that the messenger already knew where the goblet was packed. According to this understanding, at this time they were searched in the following order:
Reuven, Shimon, Levi, Yehuda, Dan, Naphtali, Gad, Asher, Issachar, Zevulun, and Binyamin.
Sequencing Activites for the young and old:
For younger students, sequencing activities could be:
--putting dates into a calendar (cut and paste)
--putting counting numbers in order
--setting out a pattern with items or shapes and having them continue the pattern
--asking students to put a group of people in order or put themselves, as a group, in order by height, birthday, day of the month that they were born, etc. A game of asking them to sequence a group of people based on a non-physical attribute would best be accomplished with the people holding cards so that the child has something concrete to look at while putting the people in order
--creating a pattern of shapes, numbers, or information and asking a child to identify the pattern is also a good thought-provoking exercise for students.
For older students, there are still very relevant sequencing activities. Such activities could be:
--writing a timeline of events
--expanding on a number line- ordering integers (positive and negative numbers), fractions, decimals, or any mixture of rational numbers
--looking to identify patterns in numbers or shapes (this is a skill that they will use in all subjects as they learn- patterns in poetry tempo or lines, patterns in a science experiment, patterns in social history, etc)
--asking students as a group to organize themselves in a particular sequence based on a physical or non-physical attribute (this is often used as an ice-breaker activity for new groups of people first meeting each other at conventions and seminars).
For an extra challenge: There is a variation where students have an index card with a piece of information on their forehead. The students do not know what their own cards say, only what they see on everyone else's cards. Using this information and speaking to others in the group, they need to work together to put themselves in proper sequence for the information on the cards (without explicitly stating what is written on other people's cards).
What sequences can you find in your lives?
Thursday, November 28, 2013
Thursday, November 21, 2013
Vayeishev- Volume
"Joseph, at the age of seventeen years, was a shepherd with his brothers by the flock, and he was a youth with the sons of Bilhah and the sons of Zilpah, his father's wives; and Joseph would bring evil reports of them to their father." ~Bereishit 37;3
"Then [Joseph's brothers] took [Joseph], and cast him into the pit; and the pit was empty, no water was in it" ~Bereishit 37;24
This week we learn that Joseph was 17 when his brothers had enough of his reporting on their behavior and they decided to throw him into a pit before, eventually, pulling him out and selling him off to a passing caravan. So our question is, what would be a reasonable estimated minimum size for a pit from which a 17 year old man would be unable to get out?
Volume: the amount of space, measured in cubic units, that an object or substance occupies (ref dictionary.com)
For our purposes, when we think about volume, it's going to be the amount of empty space (technically, air filled space; this happened in Canaan, not in the vacuum of Space) inside the dug out pit. Volume, being a measurement of amount of space that is occupied or contained in an area, is calculated by multiplying the length x width x height (Let's keep it to simple shapes, for right now). This calculation works nicely with a space that is a box or a cube, but not, say, a cylinder. A more general calculation rule, which will also work for cylinders and some other shapes, is to calculate the area of the bottom shape and then multiply the bottom area times the height of the object. For a cylinder, this bottom shape would be the circle. Area of a circle = (radius)2 x π
**radius is the distance from the center of the circle to any point along the edge or circumference of the circle. From the exact center, this distance will be the same for any point along the circumference.
**π- read pi- is a constant number that, for our purposes, can be rounded to 3.14)
Since we don't have records of Joseph's size and no Biblical growth charts remain intact for us to be able to determine the size of the average Biblical 17 year old male, we will make our calculations using the assumption that they were the same size as we are today. Any variations from this assumption will cause the answer to be scaled proportionally.
Height- If we use a growth chart, we can see that, nowadays, an average 17 year old man would be approximately 70 inches tall (about 5 ft 10 in).
Width- Shoulder to shoulder would be the widest part of the body. I'm having trouble finding a shoulder to shoulder measurement from a reliable source, but an average men's suit is approximately 18.25 inches across at the shoulders. Let's assume, given that Joseph was out working as a shepherd, that he was broader than that- let's say 20 inches.
Length- Just to establish an easier set of numbers, let's just say that, whether circular or square, he would have fit into the pit in any direction in which he was put, so the length would also need to be more than 20 inches for him to fit inside.
Okay, so if we assume that this pit was a square at the bottom, and he had to fit deep inside the pit in order to be stuck and not pull himself out, and it had to be wide enough for him to actually fit in when they threw him down, we could estimate the dimensions to be:
Height- 120 inches. Human arm span is typically equivalent to a person's height, which is 70 for Joseph. We already assumed his shoulder span to be 20 inches. If we subtract his shoulder span from his arm span we will get the distance from his shoulders to his fingertips (70 - 20 = 50). Divide that by 2 and we get the distance from each shoulder to fingertips. (50/2 = 25). The average man's fingers are approximately 4 inches, and if we subtract that from his shoulder to fingertip length, we get the distance from his shoulders to knuckles - just enough for him to grab on the the edge of the pit and pull himself up, assuming sufficient upper body strength. That number is 25 - 4 = 21 inches. Most online sources of head size only give dimensions for head circumference, which is relevant for hat sizes but not the distance from shoulder to head. Based upon observation we can assume that when a person's arms are outstretched above their head, the head reaches halfway up the arms. This would reach 10.5 inches on Josephs arms, let's make it 11 to simplify the numbers. 21 inches of shoulder to knuckle length - 11 inches of shoulder to top of head height = 10 inches that his arms reach above his head. Add that to his height: 70 + 10 = 80 inches is the height of his knuckles with his arms outstretched. The average vertical leap for an NBA player is 28 inches. Adding this to his outstretched knuckle height and we get 80 + 28 = 108 inches, or 9 feet. That means that if Joseph had explosive jumping power and a cliff hanger's upper body strength, he could hoist himself out of a pit 9 feet deep. There are records of NBA players with vertical jumps ranging from 40 inches to 60 inches, but we're going to stick with the average for this and say that a 10 foot pit (120 inches) is sufficiently deep enough for Joseph to be unable to rescue himself.
Width and length- 40 inches in each direction (that would give him a little less than an extra foot on either side of him to move around inside the pit)
Now for the calculation:
in inches: 40 x 40 x 120 = 192,000 cubic inches
in feet: 3 1/3 x 3 1/3 x 10= approximately 111 cubic feet
If we wanted to make it more challenging, we could try the calculation assuming that the pit was a cylinder. In that case, we could still use the 120 in or 10 ft for the height, and we would use 20 in or 1 2/3 ft for the radius (if we assume that it's 40 inches all the way across, then 20 inches would be from the center to the side of the pit).
The calculation:
in inches: (20)2 x 3.14 x 120 = 150,720 cubic inches
in feet: (1 2/3)2 x 3.14 x 10 = approximately 87 cubic feet
If you compare the volume of the square pit to the volume of the cylindrical pit, you can see how much extra space is cut out when you have a comparable sized circle cut into a square. That extra corner space really adds up!
So, how big would your pit be?
"Then [Joseph's brothers] took [Joseph], and cast him into the pit; and the pit was empty, no water was in it" ~Bereishit 37;24
This week we learn that Joseph was 17 when his brothers had enough of his reporting on their behavior and they decided to throw him into a pit before, eventually, pulling him out and selling him off to a passing caravan. So our question is, what would be a reasonable estimated minimum size for a pit from which a 17 year old man would be unable to get out?
Volume: the amount of space, measured in cubic units, that an object or substance occupies (ref dictionary.com)
For our purposes, when we think about volume, it's going to be the amount of empty space (technically, air filled space; this happened in Canaan, not in the vacuum of Space) inside the dug out pit. Volume, being a measurement of amount of space that is occupied or contained in an area, is calculated by multiplying the length x width x height (Let's keep it to simple shapes, for right now). This calculation works nicely with a space that is a box or a cube, but not, say, a cylinder. A more general calculation rule, which will also work for cylinders and some other shapes, is to calculate the area of the bottom shape and then multiply the bottom area times the height of the object. For a cylinder, this bottom shape would be the circle. Area of a circle = (radius)2 x π
**radius is the distance from the center of the circle to any point along the edge or circumference of the circle. From the exact center, this distance will be the same for any point along the circumference.
**π- read pi- is a constant number that, for our purposes, can be rounded to 3.14)
Since we don't have records of Joseph's size and no Biblical growth charts remain intact for us to be able to determine the size of the average Biblical 17 year old male, we will make our calculations using the assumption that they were the same size as we are today. Any variations from this assumption will cause the answer to be scaled proportionally.
Height- If we use a growth chart, we can see that, nowadays, an average 17 year old man would be approximately 70 inches tall (about 5 ft 10 in).
Width- Shoulder to shoulder would be the widest part of the body. I'm having trouble finding a shoulder to shoulder measurement from a reliable source, but an average men's suit is approximately 18.25 inches across at the shoulders. Let's assume, given that Joseph was out working as a shepherd, that he was broader than that- let's say 20 inches.
Length- Just to establish an easier set of numbers, let's just say that, whether circular or square, he would have fit into the pit in any direction in which he was put, so the length would also need to be more than 20 inches for him to fit inside.
Okay, so if we assume that this pit was a square at the bottom, and he had to fit deep inside the pit in order to be stuck and not pull himself out, and it had to be wide enough for him to actually fit in when they threw him down, we could estimate the dimensions to be:
Height- 120 inches. Human arm span is typically equivalent to a person's height, which is 70 for Joseph. We already assumed his shoulder span to be 20 inches. If we subtract his shoulder span from his arm span we will get the distance from his shoulders to his fingertips (70 - 20 = 50). Divide that by 2 and we get the distance from each shoulder to fingertips. (50/2 = 25). The average man's fingers are approximately 4 inches, and if we subtract that from his shoulder to fingertip length, we get the distance from his shoulders to knuckles - just enough for him to grab on the the edge of the pit and pull himself up, assuming sufficient upper body strength. That number is 25 - 4 = 21 inches. Most online sources of head size only give dimensions for head circumference, which is relevant for hat sizes but not the distance from shoulder to head. Based upon observation we can assume that when a person's arms are outstretched above their head, the head reaches halfway up the arms. This would reach 10.5 inches on Josephs arms, let's make it 11 to simplify the numbers. 21 inches of shoulder to knuckle length - 11 inches of shoulder to top of head height = 10 inches that his arms reach above his head. Add that to his height: 70 + 10 = 80 inches is the height of his knuckles with his arms outstretched. The average vertical leap for an NBA player is 28 inches. Adding this to his outstretched knuckle height and we get 80 + 28 = 108 inches, or 9 feet. That means that if Joseph had explosive jumping power and a cliff hanger's upper body strength, he could hoist himself out of a pit 9 feet deep. There are records of NBA players with vertical jumps ranging from 40 inches to 60 inches, but we're going to stick with the average for this and say that a 10 foot pit (120 inches) is sufficiently deep enough for Joseph to be unable to rescue himself.
Width and length- 40 inches in each direction (that would give him a little less than an extra foot on either side of him to move around inside the pit)
Now for the calculation:
in inches: 40 x 40 x 120 = 192,000 cubic inches
in feet: 3 1/3 x 3 1/3 x 10= approximately 111 cubic feet
If we wanted to make it more challenging, we could try the calculation assuming that the pit was a cylinder. In that case, we could still use the 120 in or 10 ft for the height, and we would use 20 in or 1 2/3 ft for the radius (if we assume that it's 40 inches all the way across, then 20 inches would be from the center to the side of the pit).
The calculation:
in inches: (20)2 x 3.14 x 120 = 150,720 cubic inches
in feet: (1 2/3)2 x 3.14 x 10 = approximately 87 cubic feet
If you compare the volume of the square pit to the volume of the cylindrical pit, you can see how much extra space is cut out when you have a comparable sized circle cut into a square. That extra corner space really adds up!
So, how big would your pit be?
Thursday, November 14, 2013
Parshat Vayishlach- Ratios and Fractions
"...then [Jacob] took, from that which had come into his hand, a tribute to Esav his brother: She-goats, two hundred, and he-goats, twenty; ewes, two hundred, and rams, twenty; nursing camels and their young, thirty; cows, forty, and bulls, ten; she-donkeys, twenty, and he-donkeys, ten." ~Bereishit 32;14-16
While my other posts addressed more mid to upper level math skills (within a K-8 range), this week's topic deals with more simplistic concepts. A nice, engaging, educational method for younger students is to introduce a concept and then give them lots and lots of opportunities to practice that concept. With this chart below, once you get the hang of what's happening, there's lots of room for organizing the information in different ways to offer lots and lots of practice with the idea of fractions and ratios.
Let's start by organizing our information into a chart:
A fraction is a number that tells you what part of a whole group you have. A fraction is written as one number over another with a bar between them (eg. 1/2). The top number, or "numerator", is the number that tells you how many pieces you have from the group; the bottom number, or "denominator", is the number that tells you how many pieces were in the group all together. So, the fraction 1/2 tells us that our group had 2 pieces all together, and we had 1 of those pieces.
A ratio is a comparison of numbers within a group. A ratio can be written as a fraction, with a colon separating the numbers that you're comparing, or writing "to" between the numbers. When written as a fraction, one number is written in the numerator spot and the other is written in the denominator spot; when written with a colon, the numbers are written next to each other with a colon separating them. So, if we have 15 marbles composed of 8 blue marbles and 7 red marbles, the ratio of blue to red is 8/7, 8:7, or 8 to 7. We can flip them around, too, to say that the ratio of red to blue is 7/8, 7:8, or 7 to 8.
Both fractions and ratios can be reduced to use smaller numbers to describe a situation. For example, if I have 8 cookies, and I ate 4 of them, I could say that I ate 4/8 of the cookies, or I could reduce that fraction to say that I ate 1/2 of the cookies. Another example- if I have 20 marbles composed of 5 purple marbles and 15 green marbles, I could say that the ratio of purple to green is 5:15, or I could reduce that to say that the ratio is 1:3. If both numbers in a fraction or ratio are divisible (can be divided) by the same number, then you can divide them to get to a reduced fraction or ratio.
Using our chart of the tribute gifts that Jacob set aside for Esav, we can use fractions and ratios to make different descriptions and comparisons of the types of animals. (Note: To avoid confusion, I'll use the colon for ratios and the bar for fractions.)
What fraction of the animals were female? 490/550, or reduced- 49/55. This means that if all the animals were divided into exactly the same groups, for every group of 55 animals, 49 of them would be female.
What was the ratio of female to male animals? 490:60, or reduced- 49:6. This means that if all the animals were divided into exactly the same groups, each group would have 49 female animals and 6 male animals. Notice that when you add the ratio numbers together, you get the total number of animals (490 females + 60 males= 550 animals) and this works for the reduced fractions and ratios, as well (49 females + 6 males= 55 animals in each group).
What do the comparisons look like for individual types of animals?
Goats:
--Fraction of females: 200/220 or 20/22 or 10/11. So, for every 11 goats, 10 were female.
--Fraction of males: 20/220 or 2/22 or 1/11. So, for every 11 goats, 1 was male.
--Ratio of females to males: 200:20 or 20:2 or 10:1. So, for every 10 female goats, there was 1 male goat.
Do you see how nicely the fractions and ratios fit together? How the numbers are consistent, even when they're reduced, so that you can make comparisons between males and females and also compare sections of the whole group to each other?
Let's try another animal. Ewes and Rams have the same ratio as Goats, and the Camels only had females and children- no males, so nothing to compare within the category.
Cows/Bulls:
--Fraction of females: 40/50 or 4/5. So, for every group of 5 cows & bulls, 4 were female.
--Fraction of males: 10/50 or 1/5. So, for every group of 5 cows & bulls, 1 was male.
--Ratio of females to males: 40:10 or 4:1. So, for every 4 cows, there was 1 bull.
Donkeys:
--Fraction of females: 20/30 or 2/3. So, for every 3 donkeys, 2 were female.
--Fraction of males: 10/30 or 1/3. So, for every 3 donkeys, 1 was male.
--Ratio of females to males: 20:10 or 2:1. So, for every 2 female donkeys, there was 1 male donkey.
There are lots of other fractions and ratios that we could look at here:
--What fraction of all animals were goats? What fraction were camels?...
--What was the ratio of goats to camels? goats to donkeys?...
--What fraction of all female animals were female goats? ewes? camels?...
--What was the ratio of female goats to female donkeys? male donkeys to bulls?...
If you're really getting into the comparisons, you can actually calculate how many different comparisons could be made between the animals, but that's getting into combinatorics- another topic for another time.
What interesting comparisons can you make? Do you see any interesting patterns in your fractions or ratios?
While my other posts addressed more mid to upper level math skills (within a K-8 range), this week's topic deals with more simplistic concepts. A nice, engaging, educational method for younger students is to introduce a concept and then give them lots and lots of opportunities to practice that concept. With this chart below, once you get the hang of what's happening, there's lots of room for organizing the information in different ways to offer lots and lots of practice with the idea of fractions and ratios.
Let's start by organizing our information into a chart:
A fraction is a number that tells you what part of a whole group you have. A fraction is written as one number over another with a bar between them (eg. 1/2). The top number, or "numerator", is the number that tells you how many pieces you have from the group; the bottom number, or "denominator", is the number that tells you how many pieces were in the group all together. So, the fraction 1/2 tells us that our group had 2 pieces all together, and we had 1 of those pieces.
A ratio is a comparison of numbers within a group. A ratio can be written as a fraction, with a colon separating the numbers that you're comparing, or writing "to" between the numbers. When written as a fraction, one number is written in the numerator spot and the other is written in the denominator spot; when written with a colon, the numbers are written next to each other with a colon separating them. So, if we have 15 marbles composed of 8 blue marbles and 7 red marbles, the ratio of blue to red is 8/7, 8:7, or 8 to 7. We can flip them around, too, to say that the ratio of red to blue is 7/8, 7:8, or 7 to 8.
Both fractions and ratios can be reduced to use smaller numbers to describe a situation. For example, if I have 8 cookies, and I ate 4 of them, I could say that I ate 4/8 of the cookies, or I could reduce that fraction to say that I ate 1/2 of the cookies. Another example- if I have 20 marbles composed of 5 purple marbles and 15 green marbles, I could say that the ratio of purple to green is 5:15, or I could reduce that to say that the ratio is 1:3. If both numbers in a fraction or ratio are divisible (can be divided) by the same number, then you can divide them to get to a reduced fraction or ratio.
Using our chart of the tribute gifts that Jacob set aside for Esav, we can use fractions and ratios to make different descriptions and comparisons of the types of animals. (Note: To avoid confusion, I'll use the colon for ratios and the bar for fractions.)
What fraction of the animals were female? 490/550, or reduced- 49/55. This means that if all the animals were divided into exactly the same groups, for every group of 55 animals, 49 of them would be female.
What was the ratio of female to male animals? 490:60, or reduced- 49:6. This means that if all the animals were divided into exactly the same groups, each group would have 49 female animals and 6 male animals. Notice that when you add the ratio numbers together, you get the total number of animals (490 females + 60 males= 550 animals) and this works for the reduced fractions and ratios, as well (49 females + 6 males= 55 animals in each group).
What do the comparisons look like for individual types of animals?
Goats:
--Fraction of females: 200/220 or 20/22 or 10/11. So, for every 11 goats, 10 were female.
--Fraction of males: 20/220 or 2/22 or 1/11. So, for every 11 goats, 1 was male.
--Ratio of females to males: 200:20 or 20:2 or 10:1. So, for every 10 female goats, there was 1 male goat.
Do you see how nicely the fractions and ratios fit together? How the numbers are consistent, even when they're reduced, so that you can make comparisons between males and females and also compare sections of the whole group to each other?
Let's try another animal. Ewes and Rams have the same ratio as Goats, and the Camels only had females and children- no males, so nothing to compare within the category.
Cows/Bulls:
--Fraction of females: 40/50 or 4/5. So, for every group of 5 cows & bulls, 4 were female.
--Fraction of males: 10/50 or 1/5. So, for every group of 5 cows & bulls, 1 was male.
--Ratio of females to males: 40:10 or 4:1. So, for every 4 cows, there was 1 bull.
Donkeys:
--Fraction of females: 20/30 or 2/3. So, for every 3 donkeys, 2 were female.
--Fraction of males: 10/30 or 1/3. So, for every 3 donkeys, 1 was male.
--Ratio of females to males: 20:10 or 2:1. So, for every 2 female donkeys, there was 1 male donkey.
There are lots of other fractions and ratios that we could look at here:
--What fraction of all animals were goats? What fraction were camels?...
--What was the ratio of goats to camels? goats to donkeys?...
--What fraction of all female animals were female goats? ewes? camels?...
--What was the ratio of female goats to female donkeys? male donkeys to bulls?...
If you're really getting into the comparisons, you can actually calculate how many different comparisons could be made between the animals, but that's getting into combinatorics- another topic for another time.
What interesting comparisons can you make? Do you see any interesting patterns in your fractions or ratios?
Thursday, November 7, 2013
Vayetzei- Rate of Speed: What's your walking speed?
"And [Lavan] put a distance of three days between himself and Jacob; and Jacob tended Lavan's flock that remained" ~Bereishit 30;36
"So [Jacob] fled- he and all he had; he arose and crossed the river and he set his direction toward Mount Gilead. It was told to Lavan on the third day that Jacob had fled. So he took his brethren with him and he chased after him a seven-days' journey, and caught up with him on Mount Gilead." ~Bereishit 31;21-23
Rashi here explains the timing and distance. To begin with, Lavan was already 3 days away from Jacob, as we are told in 30;36. Therefore, when Jacob left, it took the informer 3 days to reach Lavan with the news of Jacob's departure. Rashi then explains that "a seven-days' journey" is for us to understand that, while Jacob was 7 days ahead of Lavan (3 days to begin with, plus 3 days of travel forward while the informer was traveling in the other direction to tell Lavan, plus the day on which Lavan reached him), Lavan actually covered this distance to reach Jacob on his first day of travel- 4 days after Jacob left.
Rate: a quantity, amount, or degree of something measured per unit of something else (merriam-webster.com)
Basically, when comparing any two different measurements to each other, you have a rate. Common examples from our daily lives are miles per hour (speed), or dollars per pound (cost). When calculating with a rate, it is helpful to calculate a base rate, which tells you the amount of one measurement per one single unit of the other measurement.
If we look at how far Jacob and Lavan each traveled, and the length of time that it took each of them to travel their respective distances, then we can calculate the rate of speed that each of them was traveling.
From the text and Rashi's commentary, we know that Jacob traveled for a total of 4 days, and Lavan traveled for a total of 1 days. Now for the distance that each man traveled:
Based on the information that we have from the text and from Rashi, let's assume that the distance that they were each traveling was from the general Padan-Aram/Haran area to Mount Gilead. On a current map, we can use the map coordinates for Harran and for Ramoth-Gilead to approximate the distance that they traveled. Although it won't calculate travel distance, if we plot the coordinates onto a map and estimate the distance between the two locations, it's approximately 300 mi between the two locations.
Now, let's consider that Jacob (and his wives and children and sheep) traveled those 300 miles in 4 days, and Lavan chased after him and traveled the 300 miles in 1 day.
7-days journey- If the Torah assumes that this journey should have taken 7 days, then 300 miles in 7 days is roughly 43 miles each day. Assuming 12 hours of walking time each day, the standard travel rate would be about 3.5 m/h (miles per hour).
Jacob- 300 miles was the 7 day journey. If Jacob was only traveling for 4 days (note that he was 3 days away from Lavan's flocks), we can subtract about 120 miles from his trip (3 days x 40 miles each day). This means that he traveled 300-120= 180 miles. These 180 miles divided into 4 days means that he traveled 45 miles per day. This is equal to just under 2 m/h if he didn't stop to rest or sleep, but just walked 24 hours a day straight for the 4 days. To be more realistic, let's assume that he walked a total of 12 hours each day- to allow for sleeping at night and resting with children. 45 miles divided into 12 hours means that they were walking at a leisurely rate of 3.75 m/h.
Lavan- He traveled 300 miles in a single day. Since we know that he was chasing after Jacob to catch him right away, and he was running with just his brothers, let's assume that they ran until they caught up to Jacob. If it took him a full 24 hours, then he was running at a rate of 12.5 m/h. If we assume that he needed to stop at nightfall, and could, at most, travel 12 hours in the day, then he was traveling at a rate of 25 m/h.
Before you scoff at these rates, we could say that they were actually traveling by camel (it does specifically say this for Jacob- Bereishit 31;17), which is able to reach speeds of 40 m/h and sustain a speed of up to 25 m/h (ref. wikipedia).
"So [Jacob] fled- he and all he had; he arose and crossed the river and he set his direction toward Mount Gilead. It was told to Lavan on the third day that Jacob had fled. So he took his brethren with him and he chased after him a seven-days' journey, and caught up with him on Mount Gilead." ~Bereishit 31;21-23
Rashi here explains the timing and distance. To begin with, Lavan was already 3 days away from Jacob, as we are told in 30;36. Therefore, when Jacob left, it took the informer 3 days to reach Lavan with the news of Jacob's departure. Rashi then explains that "a seven-days' journey" is for us to understand that, while Jacob was 7 days ahead of Lavan (3 days to begin with, plus 3 days of travel forward while the informer was traveling in the other direction to tell Lavan, plus the day on which Lavan reached him), Lavan actually covered this distance to reach Jacob on his first day of travel- 4 days after Jacob left.
Rate: a quantity, amount, or degree of something measured per unit of something else (merriam-webster.com)
Basically, when comparing any two different measurements to each other, you have a rate. Common examples from our daily lives are miles per hour (speed), or dollars per pound (cost). When calculating with a rate, it is helpful to calculate a base rate, which tells you the amount of one measurement per one single unit of the other measurement.
If we look at how far Jacob and Lavan each traveled, and the length of time that it took each of them to travel their respective distances, then we can calculate the rate of speed that each of them was traveling.
From the text and Rashi's commentary, we know that Jacob traveled for a total of 4 days, and Lavan traveled for a total of 1 days. Now for the distance that each man traveled:
Based on the information that we have from the text and from Rashi, let's assume that the distance that they were each traveling was from the general Padan-Aram/Haran area to Mount Gilead. On a current map, we can use the map coordinates for Harran and for Ramoth-Gilead to approximate the distance that they traveled. Although it won't calculate travel distance, if we plot the coordinates onto a map and estimate the distance between the two locations, it's approximately 300 mi between the two locations.
Now, let's consider that Jacob (and his wives and children and sheep) traveled those 300 miles in 4 days, and Lavan chased after him and traveled the 300 miles in 1 day.
7-days journey- If the Torah assumes that this journey should have taken 7 days, then 300 miles in 7 days is roughly 43 miles each day. Assuming 12 hours of walking time each day, the standard travel rate would be about 3.5 m/h (miles per hour).
Jacob- 300 miles was the 7 day journey. If Jacob was only traveling for 4 days (note that he was 3 days away from Lavan's flocks), we can subtract about 120 miles from his trip (3 days x 40 miles each day). This means that he traveled 300-120= 180 miles. These 180 miles divided into 4 days means that he traveled 45 miles per day. This is equal to just under 2 m/h if he didn't stop to rest or sleep, but just walked 24 hours a day straight for the 4 days. To be more realistic, let's assume that he walked a total of 12 hours each day- to allow for sleeping at night and resting with children. 45 miles divided into 12 hours means that they were walking at a leisurely rate of 3.75 m/h.
Lavan- He traveled 300 miles in a single day. Since we know that he was chasing after Jacob to catch him right away, and he was running with just his brothers, let's assume that they ran until they caught up to Jacob. If it took him a full 24 hours, then he was running at a rate of 12.5 m/h. If we assume that he needed to stop at nightfall, and could, at most, travel 12 hours in the day, then he was traveling at a rate of 25 m/h.
Before you scoff at these rates, we could say that they were actually traveling by camel (it does specifically say this for Jacob- Bereishit 31;17), which is able to reach speeds of 40 m/h and sustain a speed of up to 25 m/h (ref. wikipedia).