Va'eira- Charting Information

"These are the heads of their fathers' houses: The sons of Reuven the firstborn of Israel: Hanoch and Pallu, Hezron and Carmi; these are the families of Reuven. The sons of Shimon: Jemuel, Jamin, Ohad, Jachin, Zohar; and Shaul the son of a Canaanite woman; these are the families of Shimon. These are the names of the sons of Levi in order of their birth: Gershon, Kehat, and Merari; the years of Levi's life were a hundred and thirty-seven years. The sons of Gershon: Livni and Shimi, according to their families. The sons of Kehat: Amram, Izhar, Hebron, and Uziel; the years of Kehat's life were a hundred and thirty-three years. The sons of Merari: Mahli and Mushi; these were the Levite families, in order of their birth. Amram took Jochebed his aunt as a wife and she bore him Aaron and Moshe; the years of Amram's life were a hundred and thirty seven years. The sons of Izhar: Korah, Nefeg, and Zicri. The sons of Uziel: Mishael, Elzafan, and Sithri. Aaron took Elisheba daughter of Aminadab, sister of Nahshon, as a wife; and she bore him Nadab and Abihu, Elazar and Ithamar. The sons of Korah: Assir, Elkana, and Abiasaf; these were the Korahite families. Elazar son of Aaron took for himself from the daughters of Putiel as a wife, and she bore Pinhas; these were the leaders of the fathers of the Levites, according to their families." ~Shemot 6;14-25

In this week's parsha, we are given background information on the families of Moshe and Aaron, so that we can see exactly how they are directly related to Jacob (aka Israel). We are given a snippet of their family tree. The Torah doesn't draw us a family tree with maps of branches and how it all interconnects. However, we can use the descriptions that we're given to convert the information into a chart so that we can have a visual representation of the information to clearly show the relationships that are mentioned here.

When transferring information from word form into a visual form (chart, graph, table), there a few basic steps to follow:
1) Take a look at the general information type to determine what form of chart or graph will best organize the information at hand.
2) Break down the words into small, manageable sections, stopping each time new information is introduced.
3) With each new piece of information, you appropriately record the information into your chart or graph.
4) When all of the information is transferred, take a look at the chart or graph all together to make sure all the individual pieces are connected to each other as they are supposed to be.

Let's apply this process to our information here about Moshe and Aaron:
1) Since this information is related to genealogy, a standard genealogical family tree chart is appropriate here.

2) & 3) Break down each section to be charted and chart the information given. Note that as you go through charting the individual sections, you need to maintain enough information in each chart so that you can remember how to piece the whole thing together when you're done.

"The sons of Reuven the firstborn of Israel: Hanoch and Pallu, Hezron and Carmi; these are the families of Reuven." ~Shemot 6;14

"The sons of Shimon: Jemuel, Jamin, Ohad, Jachin, Zohar; and Shaul the son of a Canaanite woman; these are the families of Shimon." ~Shemot 6;15

"These are the names of the sons of Levi in order of their birth: Gershon, Kehat, and Merari; the years of Levi's life were a hundred and thirty-seven years." ~Shemot 6;16

"The sons of Gershon: Livni and Shimi, according to their families. The sons of Kehat: Amram, Izhar, Hebron, and Uziel; the years of Kehat's life were a hundred and thirty-three years. The sons of Merari: Mahli and Mushi; these were the Levite families, in order of their birth." ~Shemot 6;17-19

"Amram took Jochebed his aunt as a wife and she bore him Aaron and Moshe; the years of Amram's life were a hundred and thirty seven years. The sons of Izhar: Korah, Nefeg, and Zicri. The sons of Uziel: Mishael, Elzafan, and Sithri." ~Shemot 20-22

"Aaron took Elisheba daughter of Aminadab, sister of Nahshon, as a wife; and she bore him Nadab and Abihu, Elazar and Ithamar." ~Shemot 6;23

"The sons of Korah: Assir, Elkana, and Abiasaf; these were the Korahite families." ~Shemot 6;24

"Elazar son of Aaron took for himself from the daughters of Putiel as a wife, and she bore Pinhas; these were the leaders of the fathers of the Levites, according to their families." ~Shemot 6;25

4) Now that we've charted all of the individual pieces of the family tree, we can piece all of them back together to see how they all fit together.

Note: When doing this kind of activity with younger students, you'll probably want to offer them a pre-organized chart for them to use to fill in the information as they work through the sections. For older students, you can allow them to try choosing an organizational method to start from scratch. While there is usually more than one option for what type of chart or graph is appropriate, there are charts that won't work, and you'll have to help them determine whether their chosen organizer is appropriate or not. While some students are able to organize information freely from a younger age, others will need pre-prepared charts and organizers up into Middle and High School.

Shemot- Mathematical Language

"The king of Egypt said to the Hebrew midwives, of whom the name of the first was Shifrah and the name of the second was Puah-" ~Shemot 1;15

"[Pharaoh decreed] '...But the total of the bricks that they were making yesterday and before yesterday you shall place upon them- do not reduce it- for they are becoming lax; therefore they cry out saying, "Let us go and bring offerings to our G-d." Let the work weigh heavier upon the men and let them engage in it; and let them not engage in words of falsity.' The taskmasters of the people and it's guards went out and spoke to the people, saying, 'So said Pharaoh, I am not giving you straw. You, go, take straw for yourselves from whatever you find, for nothing is being deducted from your work.' So the people spread out through the entire land of Egypt to gather a gathering for straw. The taskmasters were pressing, saying, 'Complete your work, each day's quota on that day, as when there was straw!'..." ~Shemot 5;8-13

Math is the foundation of the universe. It exists all around us and permeates our lives, mostly on a subconscious level. How much we connect with that math is dependent on how tuned into it we are at any given time. When reading through Parshat Shemot, you could easily read through the storyline of the parsha without any thought towards mathematics. However, if you read carefully, you can actually see that the parsha includes grammatically correct, mathematical language in at least two places. I'm not talking about throwing numbers into the storyline, I'm talking about actual mathematical wording when discussing information.

Ordinal Numbers:
In our day to day lives, we have 3 different types of numbers that we use- ordinal numbers, cardinal numbers, and nominal numbers.

Ordinal Numbers are numbers that indicate the order of a group of items (eg. first, second, third,...).
Cardinal Numbers are numbers that tell us the quantity or value of something (eg. 7 apples, $4.15, 17 years)
Nominal Numbers are numbers that don't have a numerical meaning, but just have a number as a label (eg. zip codes, bank routing numbers, telephone numbers)

Parsha connection- In Shemot 1;15, we learn about the Hebrew midwives whom the king of Egypt appointed to oversee the Jewish births. When they are introduced to us, they aren't just listed as "one was Shifrah and one was Puah". Rather, we learn that the first was Shifrah and the second was Puah. For whatever reason, the Torah uses ordinal numbers to give them a specific ranking or order.


Arithmetic Vocabulary:
Just as in any other area of our lives, in order to understand mathematical data that is part of our daily lives, there is certain vocabulary that has mathematical meaning with which we must familiarize ourselves. Some basic examples are understanding that when we want more of something, we need to add to the quantity; when we want less of something, we need to subtract from the quantity. We need to understand that when equally sharing a quantity, it means to divide evenly amongst the group. We even have commonly used suffixes that indicate comparative value of items: -er, -est. For example: "I ran faster than him. She ran the fastest." These sentences both indicate comparative speeds, and we need to understand from this language who was faster than whom.

Parsha connection- In Shemot 5, we learn of the decree for reduction of materials without adjustment in the quota requirement for the Hebrew slaves in Egypt. As we read through this perek, it is filled with mathematical vocabulary that indicates to us the severity of the intense work that was expected of the slaves. Below, I will recopy the excerpt from Perek 5, but with emphasis on the mathematical vocabulary.

"[Pharaoh decreed] '...But the total of the bricks that they were making yesterday and before yesterday you shall place upon them- do not reduce it- for they are becoming lax; therefore they cry out saying, "Let us go and bring offerings to our G-d." Let the work weigh heavier upon the men and let them engage in it; and let them not engage in words of falsity.' The taskmasters of the people and it's guards went out and spoke to the people, saying, 'So said Pharaoh, I am not giving you straw. You, go, take straw for yourselves from whatever you find, for nothing is being deducted from your work.' So the people spread out through the entire land of Egypt to gather a gathering for straw. The taskmasters were pressing, saying, 'Complete your work, each day's quota on that day, as when there was straw!'..." ~Shemot 5;8-13

We all use math, all the time- even those of us who are scared of it. We just don't always process it as math. What math have you been taking for granted in your daily life?

Vayechi- Some Basic Arithmetic

"Joseph dwelt in Egypt- he and his father's household- and Joseph lived one hundred and ten years. Joseph saw three generations through Ephraim; even the sons of Machir son of Manasseh were raised on Joseph's knees." ~Bereisit 50;22-23

Subtraction/Passing of Time:
In math, when trying to calculate the distance between two numbers, there are a couple of ways that students begin to work with the concept. Some students will begin by counting up from the lower number to the higher number, while keeping track of how many "steps" or numbers they pass as they count. Some students will use the same method, but count their "steps" backwards, moving from the higher number to the lower number. Ultimately, we want students to recognize that in order to calculate the distance between two numbers, the fastest calculation method is to take the higher number and subtract from it the lower number. Although calculation and "regrouping" with time can become more complex, this process works in exactly the same way to calculate passage of time; we take the later date, time, etc and subtract from it the earlier date, time, etc. Through this calculation, we can see how much time has passed between two identified time periods.

Note that I said "take the higher number and subtract from it the lower number" instead of the more direct "subtract the lower number from the higher number". This is because a common confusion among children occurs when you say "subtract" and then put the subtrahend (second number) before the minuend (first number).

Some numbers that we know about Joseph:
--We know from back in Parshat Vayeishev (Bereishit 37;3) that Joseph was 17 yrs old when his brothers threw him into the pit and he was sold and brought down to Egypt.
--We know from back in Parshat Miketz (Bereishit 31;45-46) that he was 30 yrs when Pharoah gave Joseph Poti-phera's daughter as a wife and made him viceroy of Egypt.
--We are now told that Joseph lived to be 110 yrs and lived to see three generations (see below for specifics).

Let's try out some questions that would just require basic calculation.
*How long had Joseph been living down in Egypt when he died?
To answer this, we need to subtract his age at his death minus his age when he was sold and went down to Egypt:
110 - 17 = 93; So, Joseph lived in Egypt for 93 years

*How long had Joseph been married when he died?
To answer this, we need to subtract his age at his death minus his age when he was married:
110 - 30 = 80; So, Joseph was married for 80 years

To add some complexity:
If we want to create a more complex problem for a higher level learner, we could ask them to think about some approximate ages of Joseph's generations given the information presented.

What we know here:
--Joseph's married life spanned 80 yrs (see our subtraction above)
--Joseph saw 3 generations of children through Ephraim (children, grandchildren, great-grandchildren, and great-great-grandchildren)

If all generations were spaced evenly, we could divide 80 ÷ 3 to tell us the approximate years between each generation. 80 ÷ 3 = 26.7; This would mean that every 26-27 years, a new generation was born. Let's count this up according to Joseph's comparable ages:
Joseph get's married= Joseph is 30 yrs old
Ephraim is born/Joseph becomes a father= Let's assume Joseph is 31 yrs old
Ephraim becomes a father/ Joseph becomes a grandfather= 31 + 26 = Joseph is 57 yrs old
Ephraim becomes a grandfather/ Joseph becomes a great-grandfather= 57 + 27 = Joseph is 84 yrs old
Ephraim becomes a great-grandfather/Joseph becomes a great-great-grandfather= 84 + 26 = 110

Vayigash- Percentages

"And it will be at the ingatherings that you will give a fifth to Pharaoh; the [other] four parts shall be yours, as seed for the field, and for feeding yourselves and for those who are in your household, and to feed your young ones." ~Bereishit 47;24

Back in Parshat Vayishlach, we looked at how to create ratios and fractions with a set of numbers that represent portions of a whole group. In this week's parsha, we actually have fractions given to us straight out. Working as an agent of Pharoah, throughout the course of the years of famine, Joseph purchased land from all of the landowners across Egypt in exchange for giving them food from his food storage. We are then told that the Priests were given food, and the landowners- who now all lived on land belonging to Pharoah and Joseph- were given seeds to plant for crops. They were instructed to work the fields, grow crops from the seeds that they received, and when they harvested their crops, 1/5 would be given to Pharoah, and the other 4/5 was to be kept for feeding themselves, their families, and their servants.

Percentage:
A rate or proportion per one hundred (dictionary.com)

While fractions are great for comparing parts of a group, there are times when you want to be able to compare across different groups, which can become messy and difficult if you're trying to compare fractions with different denominators. One way to create a standardization for comparison of information is to use percentages. Percentages readjust all the different groups so that for each group, the whole amount is represented by 100. Any portion of the group is then a percentage from 0-100%. All the total percentages within a group, when added together, should equal 100%.

How do we convert our information from fractions to percentages?
There are two basic ways to convert a fraction into a percentage.
1) Just as we could simplify fractions by dividing both numerator and denominator by the same divisor, we can make equivalent larger fractions by multiplying both numerator and denominator by the same factor. Since a percentage is a fraction out of 100 (per cent), then if you can easily identify how to multiply or divide your original denominator to make it 100, then you do the same to your numerator to find your percent.
Two basic examples:
--To find the percentage of 1/2, we know that 2x50=100, and then we can calculate 1x50=50. So, the percentage for 1/2 is 50/100 or 50%.
--To find the percentage of 350/500, we know that 500÷5=100, and then we can calculate 350÷5=70, so the percentage for 350/500 is 70/100 or 70%.
2) Some numbers just don't calculate nicely using the first method- try 1/8 or 1/3, for example. For these cases, you can divide the numerator divided by the denominator (eg. 1÷8 or 1÷3). This calculation will result in a decimal number. (1/8=.125 and 1/3=.33333...). These numbers multiplied times 100 (so the decimal moves 2 places to the right) are your percentages (1/8=12.5% and 1/3=33.3% -rounded off).
Intuitive connection- the first two decimal places, from left to right are tenths and hundredths; the fractions over 100 are read as "fifty hundredths" or "seventy hundredths"; when we multiply the decimal times 100, we are just changing it into the number that we would use in the fraction over 100

Using our method for calculating percentages, we can now look back to see what percentage of crops Joseph was collecting as taxes on Pharoah's behalf, and what percentage the previous landowners were allowed to keep for themselves and their families.

We are told that 1/5 was to be for Pharoah and the other 4/5 would be for the families. We lucked out here, since fifths are a fairly easy denominator to work with- 5x20=100. So, the percentages were:
Taxes for Pharoah: 1x20=20; 20/100 or 20%
Remaining crops for families: 4x20=80; 80/100 or 80%

To check our work, let's test our 100%. 20% taxes for Pharoah + 80% remaining for families should equal 100%. 20+80=100, so our math works and our calculated numbers are correct.