In this week's parsha, through chapters 20 and 21, the Jews are trying to make there way through the desert up to Israel. Along the way, they make requests to Edom, Amor, and Moab to be able to cut through their land, but their requests are denied. They are left with no choice but to travel around the outer perimeter of the Edomite, Amorite, and Moabite lands, where they are eventually able to overcome the Amorites and they end up settling in the land of the Amorites and the Moabites (which was overtaken by the Amorites). By looking at a map of their route, we can see that it appears as though their route took them much longer than if they had been able to travel in a straight path up from the Red Sea up to the Dead Sea. Is this accurate? How much farther did they travel due to their not being allowed to peacefully travel through these other lands?
We know that the Jews were at Mount Hor, where we learn of Aaron's death (Numbers 20;24-29). Let's use this stop at Mount Hor as a starting point for our calculations. So, to clarify exactly what we're investigating in order to answer our questions:
1) What is the distance between Mount Hor and Jericho (the city where Joshua first brought the Jews when they entered Israel)? I should note here that if the Jews took a more straightforward route, it's possible that their point of entry would not have been Jericho. For the purpose of calculation and comparison, however, it seemed the most logical point to use in order to maintain the same start point and end point.
2) What is the distance that was traveled instead, by going around these other nations? (For consistency, we will base our answer on the estimated route given on the map linked above.)
3) What is the difference between the distance of the straightforward route and the distance of the circuitous route?
The map that we're using conveniently gives us a scale to use to estimate the mileage. Using this scale, we can compare the distances.
Answer 1: If we draw a straight line between Mount Hor and Jericho and then measure this distance, we measure a distance of just less than 125 miles.
Answer 2: If we measure the combined distances along the estimated route given on the map, we measure a distance of approximately 312.5 miles.
Answer 3: To find the difference between the two distances, we take the longer distance (#2) and subtract from it the shorter distance (#1) to find how far the extra distance traveled was.
312.5 - 125 = 187.5
So, from our calculations, the Jews traveled an extra distance of 187.5 miles due to the circuitous route that they needed to take. Another way to look at this is that they traveled more than twice the distance that they actually needed to in order to reach their destination.
Mathematical Connection:
Using a map with scale measurements to measure distances allows students an opportunity to practice many important math skills which can be differentiated based on level:
*measuring with a ruler (either using a ruler to measure the distance and comparing it to the scale given, or using the scale measurement as a ruler replacement)
*using scale factor (a standard measure = a represented distance on the map)
*adding measured distances together- finding a successful technique for accurately measuring distances that may or may not be in a straight line (using string on the map to measure, marking locations on a marking paper to add distances, measuring with a ruler and adding measures together, etc.)
Additionally, such an activity also offers integration with and reinforcement of Geography skills.
Everyday Connection:
Have you ever needed to compare driving routes- whether just trying to get around town or when planning a road trip? Don't just rely on an app or GPS to tell you where to go. Check it out for yourself. Is your GPS really giving you the shortest route? (Of course, you'd have to take into account the speed limits on different roadways and other road restrictions that might keep you from really taking the most ideal route just based on looking at a map). How much farther do you need to travel if you choose to pass by a specific landmark or stop along the way?
Friday, June 27, 2014
Friday, June 20, 2014
Korach- Equally Shared Surface Area
"Hashem spoke to Moshe, saying, "Say to Elazar son of Aaron the pans from amid the blaze- and he should move away the fire- for they have become holy. As for the fire-pans of these sinners against their souls- they shall make them thinned-out sheets as a covering for the Altar, for they offered them before Hashem, so they became holy; they shall be for a sign to the Children of Israel." ~Bamidbar 17;1-3
In this week's parsha, we learn that Elazar is told to collect the fire-pans belonging to Korach and his followers, thin out the pans, and make a covering for the Copper Alter out of the thinned fire-pans. On reading this directive, I wondered how much area each pan actually covered once it was thinned out.
If we look back in Shemot 27;1-2, we can remember the description of the Copper Altar:
"You shall make the Mizbe'ach of shittim wood, five amot long and five amot wide- the Mizbe'ach shall be square- and three amot its height. You shall make its horns on its four corners, from it shall its horns be; and you shall cover it with copper."
Back in Parshat Terumah we looked at how to calculate surface area. Multiplying the length x width gives us the measurement of square area, telling us how much flat space is covered. Using this calculation, and knowing from Shemot that the Copper Altar is 5 amot long and 5 amot wide, we can calculate that the area of the altar was 25 square amot.
The next step is to figure out how big a section from the 25 square amot came from each fire-pan. To find this out, we need to divide the area by the number of fire-pans. There were 250 men with Korach, which means that Elazar had to thin out 250 fire-pans in order to cover 25 square amot. So, we divide 25 square amot ÷ 250 pans = 1/10 square amah per fire-pan. In other words, each thinned out fire-pan covered an area of .1 square amot.
Everyday Connections:
For our reference, how big is that? An Amah is estimated between 18-24 inches. We can repeat our calculations with these two estimated lengths to have an idea of a range of the size of the fire-pans.
Assuming 1 Amah = 18 in
The length and width of the altar were each 90 in
90 x 90 = 8,100 square inches for the area of the top of the altar
8100 ÷ 250 = 32.4 square inches per fire-pan
Assuming 1 Amah = 24 in
The length and width of the altar were each 120 in
120 x 120 = 14,400 square inches for the area of the top of the altar
14,400 ÷ 250 = 57.6 square inches per fire-pan
So, from these calculations, each fire-pan was able to be thinned out to between 32.4-57.6 square inches. This means that each fire pan was only able to be thinned out to between 5.5-7.5 inches along each side if they were flattened into perfect squares. So, the final thinned out pieces were smaller than the bottom of an 8 x 8 baking pan. Imagine how small they must have been before they were thinned out!
An extra thought- What if the covering that Hashem required was supposed to cover the entire altar rather than just the top? Using surface area calculations from Parshat Terumah, how large would the surface area have been? How much bigger would each fire-pan have been with these new calculations?
In this week's parsha, we learn that Elazar is told to collect the fire-pans belonging to Korach and his followers, thin out the pans, and make a covering for the Copper Alter out of the thinned fire-pans. On reading this directive, I wondered how much area each pan actually covered once it was thinned out.
If we look back in Shemot 27;1-2, we can remember the description of the Copper Altar:
"You shall make the Mizbe'ach of shittim wood, five amot long and five amot wide- the Mizbe'ach shall be square- and three amot its height. You shall make its horns on its four corners, from it shall its horns be; and you shall cover it with copper."
Back in Parshat Terumah we looked at how to calculate surface area. Multiplying the length x width gives us the measurement of square area, telling us how much flat space is covered. Using this calculation, and knowing from Shemot that the Copper Altar is 5 amot long and 5 amot wide, we can calculate that the area of the altar was 25 square amot.
The next step is to figure out how big a section from the 25 square amot came from each fire-pan. To find this out, we need to divide the area by the number of fire-pans. There were 250 men with Korach, which means that Elazar had to thin out 250 fire-pans in order to cover 25 square amot. So, we divide 25 square amot ÷ 250 pans = 1/10 square amah per fire-pan. In other words, each thinned out fire-pan covered an area of .1 square amot.
Everyday Connections:
For our reference, how big is that? An Amah is estimated between 18-24 inches. We can repeat our calculations with these two estimated lengths to have an idea of a range of the size of the fire-pans.
Assuming 1 Amah = 18 in
The length and width of the altar were each 90 in
90 x 90 = 8,100 square inches for the area of the top of the altar
8100 ÷ 250 = 32.4 square inches per fire-pan
Assuming 1 Amah = 24 in
The length and width of the altar were each 120 in
120 x 120 = 14,400 square inches for the area of the top of the altar
14,400 ÷ 250 = 57.6 square inches per fire-pan
So, from these calculations, each fire-pan was able to be thinned out to between 32.4-57.6 square inches. This means that each fire pan was only able to be thinned out to between 5.5-7.5 inches along each side if they were flattened into perfect squares. So, the final thinned out pieces were smaller than the bottom of an 8 x 8 baking pan. Imagine how small they must have been before they were thinned out!
An extra thought- What if the covering that Hashem required was supposed to cover the entire altar rather than just the top? Using surface area calculations from Parshat Terumah, how large would the surface area have been? How much bigger would each fire-pan have been with these new calculations?
Thursday, June 12, 2014
Shelach- More Unit Conversions
"[The spies] arrived at the Valley of Eshcol and cut from there a vine with one cluster of grapes, and they carried it on a pole, by two, and of the pomegranates and of the figs." ~Bamidbar 13;23
Rashi on 13;23 says:
And they carried it on a pole, by two: From the implication of that which has been said, "And they carried it on a pole," am I not aware that it is carried by two people? Why does the Torah say "by two"? It means that it was carried by two poles, not by two people. How was this done? Eight of the spies took a cluster; one took a fig; and one a pomegranate. Joshua and Caleb did not take anything, for with [the spies'] entire being, i.e., they had no goal in bringing the fruit other than that they intended to spread slander against the Land of Israel, by saying, "Just as its fruit is unusual in its size, so are its people unusual." And if you wish to know how much the burden of one of them was, i.e., how much one of them was able to carry, you can go and derive this from the stones which they set up in Gilgal. They lifted for themselves, each individual, one stone from the Jordan onto his shoulder, and set it up at Gilgal. Our Rabbis weighed [these stones]. The weight of each one was forty se'ah. And we learned that the load that a person can raise onto his own shoulder is nothing but a third of the load of the entire load that can be raised when they assist him to raise.
Before I begin, I want to make note that I first got this idea from 2 of my students who used this topic as the basis for their Math 'n Torah Fair project.
Back in Parshat Chayei Sarah we converted between two Biblical units to be able to compare weights that were measured with different Biblical measurements. What if we want to figure out, here, the weight of each fruit? Or what about how much weight each person was carrying? Again, we'll need to look to unit conversion to find out answer. In order to figure out these weights, we'll need to know the equivalent weight of a se'ah.
A se'ah is usually used for measuring volume. However, the reference which Rashi is using here is from Gemara Sotah (34a), where they chose to use the weight measure of se'ah. Most modern day estimates for the weight of a se'ah are between 2.25 gallons and 4 gallons. For our purposes, let's use the 2.25 gallon reference, as this will tell us the smallest possible estimate for the weight of the fruit. 1 gallon of water is just over 8 lbs, so the weight of the fruit, if we use the weight of water, would be 2.25 x 8 = 18 lbs.
Let's think about:
1) What was the weight of each fruit?
2) How much weight was each person carrying?
To begin, let's organize the information that we have from Rashi.
*The fig was carried by one person
*The pomegranate was carried by one person
*The cluster of grapes was carried by 8 people
According to Rashi, each of the spies could carry a weight of 40 se'ah. This would mean that the spies carrying the fig and pomegranate were each carrying 40 se'ah x 18 lbs each, which is 720 lbs.
Now, for the grapes, Rashi says that the load that a person carries on his own is only 1/3 of the load that a person can carry with assistance. By this reasoning, 40 se'ah was only 1/3 of what each spy could carry; in other words, each spy carrying the grapes would be able to carry 40 x 3= 120 se'ah. This is the same as 120 x 18 = 2,160 lbs.
By this reasoning, the 8 spies carrying the grapes together should have been able to carry a total of:
120 x 8 = 960 se'ah or
2,160 x 8 = 17,280 lbs
So, by our calculations the weights were:
pomegranate = 40 se'ah = minimum of 720 lbs
fig = 40 se'ah = minimum of 720 lbs
cluster of grapes = 960 se'ah = minimum of 17,280 lbs
The parsha does say that the fruit was big...
To give a little perspective,
*Arnold Schwarzenegger is recorded as deadlifting 710 lbs. This would mean that each of the spies was approximately as strong as Arnold Schwarzenegger.
*A large elephant is approximately 15,000 lbs, which means that, based on the estimates here, 8 Arnold Schwarzeneggers could possibly carry 1 reasonably patient large elephant.
Everyday Connection:
Let's test the premise- The idea that each spy carrying with a group was able to carry 3x the amount that they could carry individually is an interesting one. I wonder how this would test out in everyday life. How much weight can you lift on your own? How much weight can you lift with one other person? Two other people? Is there are pattern that emerges in the ratio of [# of people] to [weight lifted]?
Rashi on 13;23 says:
And they carried it on a pole, by two: From the implication of that which has been said, "And they carried it on a pole," am I not aware that it is carried by two people? Why does the Torah say "by two"? It means that it was carried by two poles, not by two people. How was this done? Eight of the spies took a cluster; one took a fig; and one a pomegranate. Joshua and Caleb did not take anything, for with [the spies'] entire being, i.e., they had no goal in bringing the fruit other than that they intended to spread slander against the Land of Israel, by saying, "Just as its fruit is unusual in its size, so are its people unusual." And if you wish to know how much the burden of one of them was, i.e., how much one of them was able to carry, you can go and derive this from the stones which they set up in Gilgal. They lifted for themselves, each individual, one stone from the Jordan onto his shoulder, and set it up at Gilgal. Our Rabbis weighed [these stones]. The weight of each one was forty se'ah. And we learned that the load that a person can raise onto his own shoulder is nothing but a third of the load of the entire load that can be raised when they assist him to raise.
Before I begin, I want to make note that I first got this idea from 2 of my students who used this topic as the basis for their Math 'n Torah Fair project.
Back in Parshat Chayei Sarah we converted between two Biblical units to be able to compare weights that were measured with different Biblical measurements. What if we want to figure out, here, the weight of each fruit? Or what about how much weight each person was carrying? Again, we'll need to look to unit conversion to find out answer. In order to figure out these weights, we'll need to know the equivalent weight of a se'ah.
A se'ah is usually used for measuring volume. However, the reference which Rashi is using here is from Gemara Sotah (34a), where they chose to use the weight measure of se'ah. Most modern day estimates for the weight of a se'ah are between 2.25 gallons and 4 gallons. For our purposes, let's use the 2.25 gallon reference, as this will tell us the smallest possible estimate for the weight of the fruit. 1 gallon of water is just over 8 lbs, so the weight of the fruit, if we use the weight of water, would be 2.25 x 8 = 18 lbs.
Let's think about:
1) What was the weight of each fruit?
2) How much weight was each person carrying?
To begin, let's organize the information that we have from Rashi.
*The fig was carried by one person
*The pomegranate was carried by one person
*The cluster of grapes was carried by 8 people
According to Rashi, each of the spies could carry a weight of 40 se'ah. This would mean that the spies carrying the fig and pomegranate were each carrying 40 se'ah x 18 lbs each, which is 720 lbs.
Now, for the grapes, Rashi says that the load that a person carries on his own is only 1/3 of the load that a person can carry with assistance. By this reasoning, 40 se'ah was only 1/3 of what each spy could carry; in other words, each spy carrying the grapes would be able to carry 40 x 3= 120 se'ah. This is the same as 120 x 18 = 2,160 lbs.
By this reasoning, the 8 spies carrying the grapes together should have been able to carry a total of:
120 x 8 = 960 se'ah or
2,160 x 8 = 17,280 lbs
So, by our calculations the weights were:
pomegranate = 40 se'ah = minimum of 720 lbs
fig = 40 se'ah = minimum of 720 lbs
cluster of grapes = 960 se'ah = minimum of 17,280 lbs
The parsha does say that the fruit was big...
To give a little perspective,
*Arnold Schwarzenegger is recorded as deadlifting 710 lbs. This would mean that each of the spies was approximately as strong as Arnold Schwarzenegger.
*A large elephant is approximately 15,000 lbs, which means that, based on the estimates here, 8 Arnold Schwarzeneggers could possibly carry 1 reasonably patient large elephant.
Everyday Connection:
Let's test the premise- The idea that each spy carrying with a group was able to carry 3x the amount that they could carry individually is an interesting one. I wonder how this would test out in everyday life. How much weight can you lift on your own? How much weight can you lift with one other person? Two other people? Is there are pattern that emerges in the ratio of [# of people] to [weight lifted]?
Friday, June 6, 2014
Beha'alotcha- Directionality and Proprioception
"Hashem spoke to Moshe, saying, 'Speak to Aaron and say to him: When you kindle the lamps, toward the face of the Menorah shall the seven lamps cast light.'" ~Bamidbar 8;1-2
Rashi on 8;2 says:
Toward the face of the menorah. This means toward the middle lamp, which is called "the Menorah" because it is not on the branches, but rather, on the body of the Menorah, i.e., on its central shaft.
Shall the seven lamps cast light. Of the six lamps that are on the six branches, the three of them on the east, the wicks in them turn toward the middle [lamp], and similarly, the three of them on the west, the ends of their wicks turn toward the middle [lamp]. Why? So that [people] should not say "He needs its light."
proprioception- the unconscious perception of movement and spatial orientation arising from stimuli within the body itself (dictionary.com)
Direction or orientation is a concept that develops over time. Understanding of directions like up and down develops early on in children- often before children are even toddlers; babies demand "Up!" when they want to be picked up or "Down!" when they want to be let down to play with or reach something. Understanding of forward and backward also develops by a young age. Proficient use of left and right takes much longer to develop and remains difficult for many people even into adulthood. The understanding of left and right is part of proprioception. This is different from up, down, front, back, etc., which are based on a person's understanding of the outside world in relation to themselves, rather than the understanding of one part of their body in relation to another part.
Parsha Connection:
When reading the portion that tells about Aaron lighting the menorah, it struck me that the Torah doesn't give specific direction; rather, it says that the lamps should be lit "toward the face of the Menorah". Interestingly, Rashi's explanation also refers to the lights using external directions- i.e. east and west. While east and west might also be confusing, since you first need to orient yourself in relation to the cardinal directions (North, South, East, West), once you identify one direction, it's easier to figure out where the other directions are in relation to your orienting direction. Left and right would have been much harder to figure out or understand, since left and right are always based on the relationship of the specific person or the object being referred to. For example, if the Torah (or Rashi) mentioned the wicks on the left compared to the wicks on the right, how would we know if it was referring to them based on the perspective of the person lighting or the perspective of the Menorah itself?
Everyday Connection:
Differentiating left from right is always difficult. Practicing referring to the left and right of someone else is even more difficult. Putting left and right labels and then turning around in different directions to see where the left and right end up can start to help children develop an understanding of how left and right move around as an object changes location. For example, if you're talking about your right or left and a child's right or left, when you're facing each other it's very hard for a child to understand why your left and their left are diagonally across from each other. Turning to face the same direction, identifying your left hand with a label or by raising it up and then turning around while keeping it raised can help the child actually see how your left changes its location as you turn around.
One trick that many people use to remember left from right is to put their index fingers and thumbs on each hand into an "L" shape. With the back of your hand facing you, your left hand will make a proper "L", while your right hand will make a backwards "L".
Interesting note- when dealing with stage directions, to avoid confusion the director always refers to directions from the perspective of the actors standing on stage and facing the audience. This means that the director (when facing the stage) actually needs to reverse all of their innate directions in order for the actors to understand in which direction they need to move.
Rashi on 8;2 says:
Toward the face of the menorah. This means toward the middle lamp, which is called "the Menorah" because it is not on the branches, but rather, on the body of the Menorah, i.e., on its central shaft.
Shall the seven lamps cast light. Of the six lamps that are on the six branches, the three of them on the east, the wicks in them turn toward the middle [lamp], and similarly, the three of them on the west, the ends of their wicks turn toward the middle [lamp]. Why? So that [people] should not say "He needs its light."
proprioception- the unconscious perception of movement and spatial orientation arising from stimuli within the body itself (dictionary.com)
Direction or orientation is a concept that develops over time. Understanding of directions like up and down develops early on in children- often before children are even toddlers; babies demand "Up!" when they want to be picked up or "Down!" when they want to be let down to play with or reach something. Understanding of forward and backward also develops by a young age. Proficient use of left and right takes much longer to develop and remains difficult for many people even into adulthood. The understanding of left and right is part of proprioception. This is different from up, down, front, back, etc., which are based on a person's understanding of the outside world in relation to themselves, rather than the understanding of one part of their body in relation to another part.
Parsha Connection:
When reading the portion that tells about Aaron lighting the menorah, it struck me that the Torah doesn't give specific direction; rather, it says that the lamps should be lit "toward the face of the Menorah". Interestingly, Rashi's explanation also refers to the lights using external directions- i.e. east and west. While east and west might also be confusing, since you first need to orient yourself in relation to the cardinal directions (North, South, East, West), once you identify one direction, it's easier to figure out where the other directions are in relation to your orienting direction. Left and right would have been much harder to figure out or understand, since left and right are always based on the relationship of the specific person or the object being referred to. For example, if the Torah (or Rashi) mentioned the wicks on the left compared to the wicks on the right, how would we know if it was referring to them based on the perspective of the person lighting or the perspective of the Menorah itself?
Everyday Connection:
Differentiating left from right is always difficult. Practicing referring to the left and right of someone else is even more difficult. Putting left and right labels and then turning around in different directions to see where the left and right end up can start to help children develop an understanding of how left and right move around as an object changes location. For example, if you're talking about your right or left and a child's right or left, when you're facing each other it's very hard for a child to understand why your left and their left are diagonally across from each other. Turning to face the same direction, identifying your left hand with a label or by raising it up and then turning around while keeping it raised can help the child actually see how your left changes its location as you turn around.
One trick that many people use to remember left from right is to put their index fingers and thumbs on each hand into an "L" shape. With the back of your hand facing you, your left hand will make a proper "L", while your right hand will make a backwards "L".
Interesting note- when dealing with stage directions, to avoid confusion the director always refers to directions from the perspective of the actors standing on stage and facing the audience. This means that the director (when facing the stage) actually needs to reverse all of their innate directions in order for the actors to understand in which direction they need to move.