"[Aaron] shall take the two he-goats and stand them before Hashem, at the entrance of the Tent of Meeting. Aaron shall place lots upon the two he-goats: one lot 'to Hashem' and one lot 'to Azazel.' Aaron shall bring near the he-goat designated by lot to Hashem, and he shall make it a sin-offering. And the he-goat designated by lot to Azazel shall be stood alive before Hashem, to atone upon it, to send it to Azazel to the wilderness." ~Vayikra 16:7-10
Rashi on 16:8 explains what is meant by placing lots on the goats. He explains that one goat was placed to Aaron's right and the other to Aaron's left. He had a lottery box with the two lots (presumably these were labels or markers of some sort)- one lot indicated to Hashem, and one lot indicated to Azazel. Aaron placed both his hands into the box and picked out one lot in each hand. The lot in his right hand labeled the goat on his right, and the lot in his left hand labeled the goat on his left.
Based on this explanation, we understand that until the lots were chosen by Aaron, either goat could be the one going to Hashem and either goat could be the one going to Azazel. This leads us to the concept of probability.
I wrote about probability once before, in my post last year on Parshat Beshalach. There, I explained probability as follows:
Definition- the chance that something will happen (merriam-webster.com)
Probability is expressed as a fraction between 0 and 1, where 0 means that there is no chance for the event to happen and 1 means that the event will definitely happen. For clarification, chance and probability are the same, however, chance is the number expressed as a percentage and probability is the number expressed as a fraction.
In my previous post, I explained how to calculate compound probabilities. Here, I would like to look at explaining simple probability for younger students.
We can think about this as simply as possible through some basic examples:
Case #1:
If you have 2 marbles in a bag- one blue and one red- and you want to pick out one marble, you can not say that you will certainly pick the blue or certainly pick the red. You have an equal chance of picking out one or the other.
How do we show this mathematically?
We can express the probability as a fraction where the denominator tells us how many total options we have, and the numerator tells us how many opportunities we have for picking the item that we want.
In this case, we have:
*2 marbles (our denominator) and
*if we prefer the red marble, we have 1 of those in the bag (our numerator)
Therefore, the probability of picking the red is 1/2.
This could be expressed as 50% chance of picking the red.
If we wanted to pick the blue marble, there are still the same 2 marbles in the bag and only 1 of the blue in the bag, so the probability of picking the blue is also 1/2.
Parsha Connection Interjection:
With this understanding, we see that each goat has a 50% chance of being sent to Hashem or to Azazel. They have equal chance of being sent to for either purpose.
More on Simple Probability:
As long as there are the same number of each type of item, then the probability of picking any of the items will be the same as the probability of picking any of the other items.
Case #2:
If you have 12 marbles in a bag, with 6 red and 6 blue, then the probability of picking 1 marble of either color would be 6/12, or still 1/2 (when the fraction is reduced).
Case #3:
What if you had 12 marbles in 4 different colors?
If you had 3 marbles of each color- blue, red, yellow, and green- then the probability of picking one in any of those 4 colors would be 3/12, or 1/4 when reduced.
Case #4:
What happens if we don't have the same number of each color?
Let's say we had 10 marbles:
*5 green
*2 red
*2 blue
*1 yellow.
What would be the probability of picking each color when picking 1 marble?
Green- 5/10 (reduces to 1/2)
Red- 2/10 (reduces to 1/5)
Blue- 2/10 (reduces to 1/5)
Yellow- 1/10
So, we have the best chance of picking a green, a smaller, but equal chance of picking red or blue, and it is the least likely that we'll pick yellow.
Activities to Test Probability: (Theoretical vs. Experimental Probability)
There are many similar ways of testing probability:
*flipping a coin (chance of heads or tails)
*rolling a die (chance of rolling numbers 1-6, or 6 colors, or 6 other labels)
*picking items unseen from a group (it's important that the objects aren't distinguishable by size, or touch)
Theoretical Probability is what the probability is for picking an item based on calculation.
Experimental Probability is what the probability looks like as you actually test picking item.
Students can chart their probability findings by setting up a bar graph on large graph paper (eg 1in x 1in squares) or large chart graph paper on an easel. If they are tracking the probability of heads vs tails on a coin, they would have a column for each result, and flip the coin multiple times. Each time they flip, they color in 1 square on the chart above the corresponding picture (heads or tails). Different students can test this individually and then compare their results. If you collate all the graphs from the class onto a single graph, students should start to see that the more trials they have, the closer experimental probability comes to looking like theoretical probability.
Thinking about the difference between theoretical and experimental probability- Just because you flip a coin 100 times doesn't mean that you will have exactly 50 heads results and 50 tails results. Some people might have 50-50, while another might have 48-52, and a third person could have 40-60. But if after 4 flips you have 3 heads and 1 tails, you could still end up with 50 heads and 50 tails after 100 flips.
In Vayikra 12:2-5 we learn about the Biblical periods of impurity after a woman gives birth. We learn that after giving birth to a son, she is impure for one week (7 days) and she may not touch holy items or partake in eating the terumah for an additional 33 days. At the end of the 40 day period (7 + 33 = 40), she brings sacrifices. After giving birth to a daughter, she is impure for 2 weeks (14 days) and she may not touch holy items or partake in eating the terumah for an additional 66 days. At the end of the 80 day period (14 + 66 = 80), she brings sacrifices.
Doubling Numbers & The Sums of Numbers:
If we look at the timeline for a woman's post-delivery impurity and offering of sacrifices, we can see that the timeline after the delivery of a daughter is double the timeline after the delivery of a son. Let's take a minute to break it down piece by piece:
Post delivery impurity:
Son- 1 week (7 days)
Daughter- 2 weeks (14 days)
Additional time before bringing sacrifices:
Son- 33 days
Daughter- 66 days
Total time post-delivery until sacrifices are brought:
Son- 7 days + 33 days = 40 days
Daughter- 14 days + 66 days = 80 days
As a teacher, I have seen students many times find a pattern in the way numbers work together and automatically assume that the pattern will apply to all numbers. Sometimes their assumption is correct and they have found an accurate pattern in number theory; other times students stumble upon a neat trick that works with specific types of numbers but cannot be extrapolated to other numbers. Without thinking about and testing their pattern, students may inadvertently apply a neat trick as a greater application in number theory.
With this in mind, I propose the following investigation for students:
If you double two numbers, will the sum of the doubled numbers always be double the sum of the original numbers? (Note that this investigation is applicable for students of any age who aren't yet certain of the answer and explanation of why.)
Students can test this by picking a variety of different numbers and checking what happens when they double the numbers and compare the sums of different numbers- original and then doubled. Students can work in partners or individually and compare their findings with their classmates. It's important for students to consider that, while they can't test every number, if enough numbers follow their pattern, they can feel confident that their pattern holds true for most numbers.
With enough testing, students should develop the basic premise of the theorem:
If a + b = c, then 2a + 2b = 2c
Through their testing, at least some students (at least by grade 5+) should develop enough of a thought process about their testing to begin to explain why this is true.
Some follow-up investigations:
Math related-
*Does this hold true when you add more addends to your original addition statement? i.e. if you add three numbers, will the sum of double those 3 numbers be double the original sum? 4? 5? etc. Why?
*Does this hold true for subtracting numbers?
*Does this hold true for multiplication? division?
*Does this hold true for negative numbers? fractions & decimals?
The more scenarios and number sets that they test, the stronger theory they can develop for use in their future work.
Math & Parsha related-
*Specifically related to this week's parsha, when you convert 33 days and 66 days into time measured in weeks & days, will they still appear to be doubled at first glance? Can you explain your findings? Based on your findings, do you think there's a reason that the Torah may have listed the time periods in the way that they are listed?
"Moshe spoke to Aaron and to Elazar and Ithamar, his remaining sons, 'Take the meal-offering that is left from the fire-offerings of Hashem, and eat it unleavened near the Mizbe'ach; for it is that which is holy of the highest degree. You shall eat it in a holy place, for it is your portion and the portion of your sons from the fire-offerings of Hashem, for so have I been commanded. And the breast of the waving and the thigh of the raising you are to eat in a pure place, you and your sons and your daughters with you; for they have been given as your portion and the portion of your sons from the sacrifices of the peace-offerings of the Children of Israel. They are to bring the thigh of the raising and the breast of the waving upon the fire-offering fats to wave as a wave service before Hashem; and it shall be for you and your sons with you for an eternal decree, as Hashem has commanded.'" ~Vayikra 10:12-15
In the above section of this week's parsha, Moshe explains to Aaron about how the meal-offering (mincha-מנחה), the wave offering (tenupha-תנופה), and the raise offering (terumah-תרומה) are divided amongst Aaron and his children. Rashi discusses the apportioning on 10:13 and 10:14, where there is differentiation between the mincha, which is the holiest, and the tenupha and terumah, which must be eaten in a pure place.
If we look at the comparison of these different sacrifices, we see that the mincha is to be apportioned between Aaron and his sons, with no mention of daughters at all. Regarding Tenupha and terumah, however, Moshe mentions both the sons and daughters eating along with Aaron, but he mentions portions for only Aaron and his sons. Rashi explains these differences in wording to mean that the mincha should only be portioned for and eaten by Aaron and his sons- his daughters may not partake. For tenupha and terumah, the portions are again only to be divided between Aaron and his sons, but for these, when they are eating their portions, they may share their portions with their daughters and wives.
Practically speaking, what does this mean? For ease of calculating, let's think about the portions of sacrifices based on made up weights that would be divided between Aaron and his sons. Before we begin, we also need to know about the composition of Aaron's immediate family, to the best of our knowledge.
*Aaron was married to Elisheva
*Their 2 oldest sons, Nadav & Avihu, were killed at the beginning of this chapter for bringing a fire onto the altar without checking with Moshe.
*Aaron and Elisheva's son Elazar was married
*Elazar and his wife had a son Pinchas
*Aaron and Elisheva's 4th son was Ithamar
So, in sum, there were 4 Kohanim (priests)- Aaron, Elazar, Ithamar, and Pinchas. Then, there were 2 women in the immediate family- Elisheva and Elazar's wife.
Now let's look at a sample situation to understand the calculations. Let's assume that there's 8 lbs. of each sacrifice to be apportioned.
Mincha- This will be divided evenly between the 4 Kohanim, and each Kohen will have 2 lbs. (8 lbs. ÷ 4 Kohanim = 2 lbs. per Kohen). Each Kohen will also eat the full 2 lbs. on their own.
Tenupha & Terumah- This will also be divided evenly between the 4 Kohamin, and each Kohen will, again, have 2 lbs. (same calculation as above). However, they will not each have 2 lbs. of each sacrifice to eat.
*Aaron will share his 2 lbs. with Elisheva. So, if divided evenly, Aaron & Elisheva will each have 1 lb. of tenupha and 1 lb. of terumah.
*Elazar will share his 2 lbs. with his wife. So, as with Aaron, Elazar and his wife will each have 1 lb. of each of the 2 sacrifices.
*Pinchas and Ithamar having no family at this point, would each eat their entire portions of 2 lbs. of each sacrifice.
If we think about this situation related to fractions, we would say that each sacrifice was divided the same, and each Kohen received 1/4 of the sacrifice. When it came to eating the sacrifices, though, while each Kohen ate 1/4 of the mincha, this was not the case for the tenupha & terumah. Let's think about those fractions:
*Aaron had 1/4 of each, which he shared with Elisheva- What is half of 1/4? If we take our 1/4 pieces and break them each into half, we would have 8 pieces out of our original 4, so Aaron's 1/4 was the same as 2/8, and he and Elisheva each had 1/8 of each the tenupha & terumah.
*Comparably, Elazar and his wife also each had 1/8 of each of these 2 sacrifices.
*As with the portion weights sampled above, Pinchas and Ithamar each ate their entire 1/4 of each of these sacrifices.
To help students internalize these calculations (by weights and/or by fractions), we could have them personalize these calculations by applying them to their own family compositions. How much of each of the 3 sacrifices would the people in their own families receive? After they calculate for their own families, they can compare calculations with their peers, to extend their thinking again to a third sample size.
