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.

Miketz- Order & Sequencing

"[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?

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?

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:

Type of Animal	# of Females (& children)	# of Males	Total #
Goats	200	20	220
Ewes/Rams	200	20	220
Camels	30	0	30
Cows/Bulls	40	10	50
Donkeys	20	10	30
Total # of Animals	490	60	550

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?