Beshalach- Measurement related activities

"This is the thing that Hashem has commanded, 'Gather from it, for every man according to what he eats- an omer per person- according to the number of your souls, everyone according to whomever is in his tent shall you take." ~Shemot 16:16

"They measured in an omer and whoever took more had nothing extra and whoever took less was not lacking; everyone according to what he eats had they gathered." ~Shemot 16:18

"It happened on the sixth day that they gathered double bread, two omers for each; and all the princes of the assembly came and they told Moshe." ~Shemot 16:22 

Rashi on Shemot 16:22 explains that the Israelites actually collected the same amount when they went out on Fridays, but when they came home to measure how much they had, they found that they had 2 omers.

"The omer is a tenth of an ephah." ~Shemot 16:36

Rashi on Shemot 16:36 explains- 
1 ephah = 3 se'in
1 se'ah = 6 kabin
1 kav = 4 lugin
1 log = 6 beitzim
1 beitzah = the volume of 1 egg
Therefore, a tenth of an ephah is 43 beitzim & 1/5 of a beitzah

(Chart from Artscroll Series, The Sapirstein Edition; Student size edition; Vol 2- Shemos/Exodus pg 198)

Activity Connections:
Activity ideas in increasing conceptual difficulty:

• [Pre-activity 1] Younger students begin learning about measurement by playing with standard and non-standard measurements. Young students may begin by playing around with how many paper-clips long different items are, how many blocks tall each of their friends is (using standard size blocks), or how many classroom cups of water or rice it takes to fill a container. By playing around with these non-standard measurements, they get a sense of how to compare lengths of items without needing to worry about how to read rulers or measuring cups. Following-up on a non-standard activity, students can have a discussion about how you might be able to talk about your non-standard activities with students in other classes. What if their paperclips/blocks/cups are a different size? How could you express your measurements to those students? This helps illustrate for them where standard measurements are needed.
• [Pre-activity 2] Younger students can also have an opportunity to investigate different containers or items that have been pre-measured with standard units for them to compare quart/liter/gallon, inch/foot/yard- or metric equivalents. 
• Connecting to this week's parsha, a pre-measured container could also show the estimated volume of an omer (based on our understanding of the measurement- see below), so that students can see how much food each Israelite was apportioned for each day in the desert.
• Using a pre-measured "Omer Container" students could play with the measurement and see how it compares to our current day measurements. Using investigation- which is bigger- an omer or a quart? an omer or a gallon? How many of the smaller measure does it take to make up the larger measure? Can you express the smaller measure as a fractional amount of the larger measure? These are two different ways of expressing the same equivalency based on the measure you use for a frame of reference (ie 2 cups = 1 pint; 1/2 pint = 1 cup).
• Older students can work on calculating conversions based on the information provided by Rashi in 16:36. How do the different Biblical measures compare to each other? A modern day understanding of the volume of a beitzah is indicated here as 2.53 fl. oz. (or 75 ml). Using this understanding of beitzah, what would each of the other volume measurements be in Rashi's explanation? Ultimately, working through the calculations, what would the volume of an omer have actually been? What would that look like in current day measures? What was the volume of food that the Israelites had for each day's sustenance? What did the volume collected on Fridays (in preparation for Shabbat) look like?

Bo- Using Unit Conversions in Problem Solving

"The Children of Israel journeyed from Rameses to Succoth, about six hundred thousand on foot, the men, aside from the children." ~Shemot 12:37

Rashi here explains that the distance from Rameses to Succoth was 120 mils, and that Hashem made their travel there as if in an instant.

To gain a better understanding of this explanation in today's measurements, 1 mil = 2,000 amot, and 1 amah is estimated between 18-24 inches. Based on this estimate, 1 mil would be anywhere from 3,000-4,000 ft.

Rashi also explains that the estimated number that is given here is men from twenty years and older, possibly draft age, which would be men 20-60 years old.

As I was reading this, I wondered how much distance a group of this size would cover when they're traveling as a whole group (ie, how much land would the whole group cover at once?). If Hashem hadn't brought them immediately through a miracle, would the first person have reached Succoth before the last people left Rameses?

To begin to figure this out, we have 2 calculations to consider:
1) What would be today's equivalent of the distance they needed to travel?
2) What would the full population estimate be for the group?

Distance Traveled:
Based on Rashi's information above, they traveled 120 mils, where each mil is anywhere from 3,000-4,000 ft. For an easier calculation, you can choose either the lower estimate or the upper estimate. I'm going to work through both calculations and offer a range.
To calculate, we will multiply the distance in feet for 1 mil times the number of mils that was their travel distance (120).
For the lower estimate, that means 3,000 (ft) x 120 (mils) = 360,000 ft
For the upper estimate, that means 4,000 (ft) x 120 (mils) = 480,000 ft

So, the distance from Rameses to Succoth was anywhere from 360,000 - 480,000 ft.

Another level of unit conversion to consider:
If we know that 5,280 ft = 1 mile, we can convert this range into miles. Do we multiply or divide here? This question will stump many students. Let's think, we are trying to convert a number of feet into miles. We know that every group of 5,280 ft together make 1 mile. We are taking big chunks of feet and changing them each out for 1 mile. To make this conversion, we are finding out how many chunks of 5,280 ft can be fit into, say, 360,000 ft. This means division.

So, 360,000 ft ÷ 5,280 ft = approximately 68.18 miles 
and 480,000 ft ÷ 5,280 ft = approximately 90.91 miles

So, the distance from Rameses to Succoth was anywhere from 68.18 - 90.91 miles.

Population Estimate:
If we use the estimate that there were approximately 600,000 men between the ages of 20-60, it's probably a fair estimate to say that this is 1/3 of the male population (men younger than 20 and men older than 60 making up the other 2/3 of the male population). We can also say that the male population is only half of the total population, assuming that the population was roughly 1/2 male and 1/2 female. 

To calculate how many men there were, we can multiply 600,000 x 3 = 1,800,000 men.
To calculate the total population, we can double this number: 1,800,000 x 2 = 3,600,000 total estimated population. 

So, we can safely estimate that there were more than 3.5 million people who left Egypt and traveled from Rameses to Succoth.

Final Calculations:
Although the numbers are larger to calculate with, I'd like to use the distance measured in feet to finalize our calculations. One important aspect of understanding the information and numbers that you're working with is recognizing which numbers make the most sense to use for a calculation. If we want to estimate how much space a person needs, it's easier to estimate this number using a measurement of feet rather than miles. 
If we estimate, conservatively, that each person needs 2 feet of space, then with a distance of 360,000 ft, you could have a line of 360,000 ÷ 2 = 180,000 people, at most.

With the same conservative estimate, then with a distance of 480,000 ft, you could have a line of 480,000 ÷ 2 = 240,000 people, at most.

So, with little space in front or back of them, you could have a line of between 180,000 - 240,000 people reaching all the way from Rameses to Succoth. 

But we have an estimate of 3,600,000 people!?
3,600,000 (population) ÷ 180,000 (people in each line) = 20 lines of people 
or
3,600,000 (population) ÷ 240,000 (people in each line) = 15 lines of people 

So, without consideration for all the animals and possessions that the Israelites left with, if they lined up almost front to back with each other, there would have been 15-20 lines of people spanning the entire distance from Rameses to Succoth.

An Extra Look:
In Judaism, we often speak of a person's "daled amot" or 4 amot of personal space. With this measurement in mind, we could use the distance from Rameses to Succoth and calculate how much space each person would have needed if we allowed each person 4 amot.

Let's go back to the first equivalency above: 1 mil = 2,000 amot
We know that there were 120 mil, so that would be 
120 x 2,000 = 240,000 amot as the distance between Rameses and Succoth.

If we want to allow 4 amot each to 3,600,000 people, we need to calculate 
3,600,000 x 4 = 14,400,000 amot needed.

Ok, so now we need to figure out how many times the distance from Rameses to Succoth would be needed in order to accommodate all the space needed by the population.

14,400,000 amot (population) ÷ 240,000 amot (distance) = 60 lines of people.

Another way to look at the same calculation is to follow the same calculation process that we did above for feet.
The distance was 240,000 amot ÷ 4 amot for each person = 60,000 people would fill a line reaching from Rameses to Succoth.

Our population of 3,600,000 ÷ 60,000 people in each line = 60 lines of people.

We just confirmed the same answer through 2 different calculation approaches!

So, again, if each person had 4 amot of space to travel (not accounting for animals or possessions), they would have had 60 lines of people lined all the way from Rameses to Succoth. 

Va'eira- Does the timeline work?

For this week's post, I'd like to take a couple of different explanations regarding the course of the plagues that were brought on the Egyptians and see if these explanations actually piece together to work on a proposed overall timeline for the course of the plagues.

Regarding the plague of blood that was brought on the Egyptians:

"The seven-day period was filled after Hashem struck the river." ~Shemot 7:25

Rashi on this verse and Shemot Rabbah 9:12 explain that each plague spanned an entire month of time. This time incorporates both the time during which the Egyptians were plagued with each curse and the time during which they were warned or Pharoah was given time to change his ways. One interpretation explains that there was 1 week of plague and 3 weeks of warning/repentance time, while another interpretation says that there were 3 weeks of plague and 1 week of warning/repentance time. Either way, the full timespan for each plague was 1 month.

Regarding the plague of lice that was brought on the Egyptians:

"So they did: Aaron stretched out his hand with his staff and struck the dust of the land, and the lice-infestation was on man and beast; all the dust of the land became lice, throughout the land of Egypt." ~Shemot 8:13

The Midrash Says on the book of Shemot explains that once the plague of lice was inflicted on the Egyptians, their was no useable ground left for the Israelites to perform any of the slave labor that they had been forced to do in the past. They could not make bricks, because the land was unusable; they could not be made to sweep the streets, because the dust had all become lice. The Torah Anthology Me'Am Lo'ez Book 4 and The Midrash Says on this passage explain that even if someone dug 1 Amah (~24 inches) down into the ground, they still would only find lice- there was no dirt to be found at all. 

As The Midrash Says continues on through the plague of Wild Beasts, it explains that the Egyptians tried to make the Israelites continue working by making them carry their children far distances, but the beasts would come and take the children from the hands of the Israelites. In this way, the Israelites escaped another attempted form of labor.

In Ein Yaakov: The Ethical and Inspirational Teachings of the Talmud, compiled by R' Yaakov Ibn Chaviv, it explains that the Jews were redeemed from their hard work  6 months before their exodus from Egypt. 

Based on what we learn above, that 

• each plague spanned (in some form) an entire month
• the Israelites were unable to be forced into slave labor after the plague of lice ruined all of the dirt in Egypt
• the Israelites were unable to be forced into slave labor for the 6 months preceding their exodus

does this timeline fit together?

Let's establish a timeline that lines up with the series of plagues:

Month 1: Blood

Month 2: Frogs

Month 3: Lice **

Month 4: Wild Beasts 

Month 5: Pestilence

Month 6: Boils

Month 7: Hailstones

Month 8: Locusts

Month 9: Darkness

[Month] 10: Death of the Firstborn-- however, this was not actually a month, as there was no further time for repentance offered. The plague occurred overnight, and then the Israelites were sent out.

If we look through the timeline in this way, we can count backwards through the months. Keeping in mind that the final plague was brought at the beginning of what would have been the 10th month, there is no timespan there to actually count (see this post for a review on tracking time). By this count, the plague of lice ended exactly 6 months before the exodus. 

An additional component: Thinking back to the explanation of R' Yaakov Ibn Chaviv, he explains that the Israelites were freed from work on Rosh Hashanah and freed from Egypt on Pesach. Based on our method for tracking time, does this work for counting the Jewish months from Rosh Hashanah to Pesach? Are there 6 months between the two holidays? (Remember, to count complete months!)

Shemot- Adding up the years

"They embittered their lives with hard work, with mortar and with bricks, and with every labor of the field; all their labors that they performed with them was crushing labor." ~Shemot 1:14

  וימררו את חייהם בעבדה קשה..."  ~שמות א:י"ד"

Note: When clicking the link above to view the text on sefaria.org, you always have the option of viewing the text in English, Hebrew, or linear translation. This week, I've quoted the beginning of the phrase in Hebrew, as well, since my topic for this week pertains to the special notations that indicate cantillation for reading the passage (these notations are referred to as "trop"). The trop can be seen on the hebrew and linear translation views on the sefaria.org link.

In previous posts, one ongoing activity option has been to use the timing and referential ages, time spans, etc. to calculate an ongoing timeline. Based on the information provided in the entirety of the whole book of Bereishit and up through the beginning of the book of Shemot (just to bring us to-date), it is possible to create a timeline from the Creation of man through to the enslavement of the Israelites in Egypt.

If we look back at Bereishit 15:13, Avraham is told that his descendants will be slaves in a land that is not their own, and they will be afflicted for 400 years. If you put the events up until this week's parsha on a timeline, you will find the discrepancy that the number of years that the Israelites were actually in Egypt do not add to 400. Different commentaries make sense of this discrepancy in different ways. Rashi offers one explanation with a calculation of a different sum of years along with mathematical proof that it's not possible that the Israelites could have been in Egypt for 400 years based on relevant numbers that we know. His explanation can be seen here.

The Vilna Gaon offered a different type of explanation. His explanation was based on Gematria. Gematria, which I explained in more detail last year for Parshat Vaetchanan, is a system of establishing numerical values of words based on assigned values for each letter. A quick reference chart for letter values can be seen here.

The steps to the Vilna Gaon's explanation are worked out below:

Here is where I refer back to the Hebrew quote above. The first 3 words, which translate to "They embittered their lives", have the two trop signs called "kadma" (קדמא) and "azla" (אזלא). The meaning of the Aramaic words "kadma" and "azla" is "get up and go".

If you check the math of the timeline for the actual number of years that the Israelites were in Egypt (from the time Yaakov moved his family to Egypt until the Jews fled Egypt), you will find that they were there for 210 years. If you calculate the discrepancy between what Avraham was told would happen and what actually happened, you will find that the discrepancy (400 years minus 210years) is 190 years.

Now, let's check the gematria for קדמא ואזלא (kadma and azla- get up and go) :
ק -100
ד -4
מ -40
א -1
total= 145 (remember to line up your place values before adding!)

ו -6
א -1
ז -7
ל -30
א -1
total= 45 (remember to line up your place values before adding!)

Let's add those 2 together: 145 + 45 = 190.

Based on the value of 190 found here, the meaning of the words ("get up and go"), and the fact that they are found specifically on the words referring to the Egyptians embittering the lives of the Israelites, the Vilna Goan used all of this information to explain the mathematical discrepancy in the following way:
Avraham was told that his descendants would be strangers in a land and they will serve the inhabitants of the land. The Egyptians, however, took it a step further (as we see here in Shemot 1:14) and they specifically sought to embitter the lives of the Israelites. Due to the extra burden that the Egyptians put on the Israelites, G-d made it possible for the Israelites to leave Egypt early- "Get up and go!". How early did he let them go? By exactly 190 years (the value of "get up and go").

Students of varying levels can either identify this question as part of a larger timeline that they are following and tracking, or they can work through just the calculations that present to identify the question. Looking at and working through various commentators explanations of the discrepancy with their mathematical calculations is an activity that students of varying skill levels can handle. Younger students could look at a timeline provided by a teacher to look at the calculations, while older students can calculate the different time spans offered by the commentators and double-check the calculations that the commentators offer.

Vayechi- Proprioception and Mental Imaging related activity ideas

"Joseph took the two of them- Ephraim with his right [hand], to Israel's left, and Menashe with his left, to Israel's right- and he drew close to him." ~Genesis 48:13

Rashi on 48:13 explains:
Ephraim at his right, to Israel's left- One who comes toward his friend, his right is opposite his friend's left. [Joseph] intended that the firstborn should be properly prepared for the blessing.

Last year in Parshat Beha'alotcha I wrote about proprioception and the difficulties that many people have (even as adults) in differentiating between their right and left. Given this issue that many people struggle with, I thought it interesting when I reread this week's parsha and noticed that:

1. the Torah itself states the distinction- Ephraim was on Joseph's right, which was Israel's left; and Menashe was on Joseph's left, which was on Israel's right
2. Rashi makes a point of also explaining this difficult concept further- when one person approaches another, the person's right is actually opposite his friend's left

Spatial awareness is important for us as we live day-to-day in our 3-dimensional world. Shapes, the space they take up, and their orientation in relation to other items in their vicinity are part of our everyday lives. 

• To run and catch the subway train as its doors are closing, I have to turn sideways to fit through
• Which direction do I need to turn the table to fit it through the doorway?

The geometric math concepts that we work on with children as they grow and develop are all meant to help them build this awareness of spatial relationships, how shapes can be used for the greatest stability in our buildings, how shapes can fit together for the best use of space in our daily lives, and how the space that we take up within our surroundings will intersect with the space that other people and objects in our surroundings take up.

Activity Connections:
Activity ideas in increasing conceptual difficulty:

• For younger students, just starting by recognizing their left and right sides is difficult enough. Using left and right in directions as much as possible where applicable will help increase their familiarity. Everyday exposure to the distinction will make a big difference in making the concept more familiar and usable for them. Labeling left and right on a board or in primary locations at home or in the classroom will help give students a quick reference when they're unsure and thinking about the concept. When students are comfortable with the alphabet, they can also use their hands to check themselves- the thumb and forefinger of the left hand form an "L" when you're looking at the back of your hand, while the same formation with your right hand will be a backwards "L". 
• As students begin to feel comfortable with their own right and left, a fun activity is to have them wear left and right labels (on their hands or near their shoulders) and see what happens to their own left/right and their friends' left/right as they stand in line behind each other, walk towards each other, and turn around in different ways. This activity will help students gain a better understanding of the concept that the parsha and Rashi are explaining this week. As students work more and more with left and right, if they do have difficulty, they will also learn to develop tricks that they find helpful for themselves.
• On a higher level, spatial conceptualization can move beyond right/left orientation towards rotations (turns), reflections (flips), and translations (slides) of 2-dimensional and 3-dimensional shapes. 
• Again, for younger students they can have actual shapes that they turn, flip, and slide and they can draw diagrams to go with their activities. Playing with shapes and looking at the way a shape appears differently in different positions helps students develop their mental imaging skills, while diagraming the changes that they find helps them make note of particulars that they may overlook when just looking at the shape.
• Students will usually find it easier to diagram the 2D shapes and then they can progress to diagraming 3D shapes. Students will begin to notice more interesting changes as they move into 3D shapes- sometimes not all of the sides are the same shape, for example. As students explore, they will find similarities and differences between different shapes and begin to categorize shapes for themselves in different ways. As they develop their understandings of the different shapes, they will be storing mental imaging for themselves that they can call on in future learning.
• Once students have formed mental images for themselves, they can think about given situations where they are provided with information about spatial relationships and they need to work out the connections between different orientations. For example:
• Remember those baby blocks that have different shapes on each side? If students are given drawings of that type of block from different angles, they should be able to figure out which shapes are on which sides of each other in order to recreate an image of a full block.
• If you diagram a building block structure, can you figure out from the diagram how many blocks you need and how to recreate the structure? Younger learners can practice by actually trying to recreate the the structure with real blocks, while more adept students can work on their mental visualization of the shape in order to figure out the information that they're asked to find.
• Try to fit two dressers (of certain dimensions) next to each other in a given space (of certain dimensions and configuration)- what will be the best way to fit them next to each other? What will the space look like if I put them side by side? What about back to back? Is there a different orientation that will fit in the space in a better way? Remember, you need to be able to open the drawers!