Bamidbar- Reading, Comparing, and Calculating with large numbers

Note that the section of quotes from this week's parsha is quite long in order to show the context for Levite and firstborn census numbers. As always, I organize all of the information for you beginning below the quotes. If your time is short, feel free to scroll past the citations.

"Hashem spoke to Moshe in the Wilderness of Sinai, saying, 'Count the sons of Levi according to their fathers' house, according to their families, every male from one month of age and up shall you count them.' Moshe counted them according to the word of Hashem, as he had been commanded.

These were the sons of Levi, by their names: Gershon, Kohath, and Merari. These were the names of the sons of Gershon according to their families: Livni and Shimei. The sons of Kohath according to their families: Amram and Izhar, Hevron and Uzziel. The sons of Merari according to their families: Mahli and Mushi. These are the families of the Levites, according to their fathers' house.

For Gershon- the Livnite family and the Shimeite family; these are the Gershonite families- their counted ones according to the number of every male, from one month of age and up: their counted ones [were] seven thousand, five hundred." ~Bamidbar 3:14-22

"And for Kohath- the Amramite family, the Izharite family, the Hevronite family, and the Uzzielite family; these were the Kohathite families- the number of every male from one month of age and up [was] eight thousand, six hundred; the guardians of the charge of the sanctity." ~Bamidbar 3:27-28

"For Merari- the Mahlite family and the Mushite family; these were the Merarite families- their counted ones according to the number of every male from one month of age and up [was] six thousand, two hundred." ~Bamidbar 3:33-34

"All the counted ones of the Levites, which Moshe and Aaron counted by the word of Hashem according to their families, every male from one month of age and up, were twenty-two thousand.

Hashem said to Moshe, 'Count every firstborn male of the Children of Israel from one month of age and up, and take a census of their names. You shall take the Levites for Me- I am Hashem- in place of every firstborn of the Children of Israel, and the animals of the Levites in place of every firstborn of the animals of the Children of Israel.' Moshe counted- as Hashem had commanded him- every firstborn of the Children of Israel. Every firstborn male according to the number of their names, from one month of age and up, according to their countings, was twenty-two thousand, two hundred and seventy-three.

Hashem spoke to Moshe, saying, 'Take the Levites in place of every firstborn of the Children of Israel, and the animals of the Levites in place of their animals, and the Levites shall be Mine, I am Hashem. And as for those redeemed of [the firstborn], the two hundred and seventy-three of the firstborn of the Children of Israel who are in excess over the Levites, you shall take five shekels each according to the head count, in the sacred shekel shall you take; the shekel is twenty geirah. You shall give the money to Aaron and his sons, of those redeemed in excess among them.'" ~Bamidbar 3:39-48

In this week's parsha, we learn of multiple censuses that were taken in the desert. Last year, I looked at the census information for the 12 Israelite tribes. This census was taken for men from the age of 20 years old and older. In my post last year, I analyzed that information from a basic statistical averages viewpoint- exploring the mean, median, and mode for the data set of the census. 

This year, I would like to look at the census information for the Levites. Since we're dealing with a lot of information here, let's first focus and pull out the numerical information that we're trying to look at. The census information for the Levites is first broken down into smaller census numbers from the individual Levite families, and then summed up at the end. Then, following that information, there is another census of the firstborn Israelites. We will look at all of these sets of information.

Let's begin by weeding through the information above to organize the pertinent information that we need:

Gershon- 7,500 males from 1 month of age and older

Kohath- 8,600 males from 1 month of age and older

Merari- 6,200 males from 1 month of age and older

Then, the total sum listed for all Levites is 22,000 males from 1 month of age and older

Stop for a minute and test out this calculation. 

• Does 7,500 + 8,600 + 6,200 = 22,000? In fact, it actually adds to 22,300. 

So what's going on here? Why does the Torah omit 300 Levites in it's calculation? Rashi on 3:39 deals with this exact question. He explains that the total number of Levites was needed in order to redeem the firstborn Israelites. For every 1 Levite, 1 firstborn Israelite would be redeemed. Rashi says that the 300 Levites who are not included in the total sum were actually firstborns, themselves, and therefore would not count towards being able to redeem other Israelite firstborns, since they would be redeeming themselves.

Let's look on further- We are told that the firstborn Israelite males from 1 month of age and older totaled 22,273. 

So, if we replace one-for-one each firstborn Israelite with a non-firstborn Levite, we have 22,273 firstborns & 22,000 Levites available for redeeming:

22,273 - 22,000 = 273

Now, it seems that we have 273 firstborns who do not have a Levite to redeem them. If we continue reading, though, we see that Hashem has told Moshe that there are, in fact, 273 firstborns "in excess" of the Levites, and these 273 should be redeemed by collecting 5 shekels per person for each of them and that money will go to the Kohanim. 

Some other thoughts & activities:

• It my be interesting to compare the census for the Levites with the census of the rest of the Israelites that's listed in the first part of the parsha. In making a comparison of the numbers, it would be important to keep in mind that the census for the 12 Israelite tribes was taken from 20 years and older, while the Levite census was taken from 1 month and older. Even more so, what interesting findings do you see when comparing the numbers of the two groups?
• An interesting discussion topic may be to think about whether there is a way to compare the information given for the general Israelite census to the firstborn Israelite census to figure out any further information regarding numbers of Israelites between certain ages? Is there enough information given to create comparisons of smaller age groups- either from each tribe or from the whole group?
• For younger students, it might be interesting to just think about place value and reading large numbers. Students could have a matching activity in which they have to identify the numbers of the censuses and match them to the appropriate tribes. Students could also talk about how to think about comparing which numbers are larger and which numbers are smaller- place values and how you can use place values to compare numbers to each other to figure out which is larger and which is smaller. Students who catch on quickly to this concept could think about ordering the numbers from smallest to largest or largest to smallest.
• Follow-up activity- values are given for how many shekels were collected for the 273 "excess" firstborns who did not have Levite redeemers (Bamidbar 3:50). Older students could check the calculations for the values to see if it matches the amount that they calculate should have been donated. We are also told (Bamidbar 3:47) that a shekel was worth 20 geirah. Students can calculate a conversion of the value to also figure out the value of the donation in geirahs. 

Tazria/Metzora- Investigating Patterns in Doubling

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?

Vayeishev- Timeline and basic arithmetic related activity ideas

"Then Jacob rent his garments and placed sackcloth on his loins; he mourned for his son many days." ~Bereishit 37;34

This week we learn of when Joseph's brothers sold him and implied to Jacob, their father, that he was attacked and killed by a wild animal. Upon seeing Joseph's bloody tunic and surmising his death, we are told that Jacob mourned for "many days". Rashi  here explains that "many days" actually means that he mourned for Joseph the entire time that Joseph was missing to him- which actually calculated to be 22 years.

Activity Connections:
Activity ideas in increasing conceptual difficulty:

• The most basic calculation connection would be to begin with the statement 1) in Bereishit 37;2 (at the start of this week's parsha) that Joseph was 17 years old at the time of the story of his sale and 2) in Bereishit 41;46 (in next week's parsha- Miketz) that Joseph was 30 years old at the time that he first was called to come before Pharoah. Younger students can just calculate the passage of time between Joseph's sale and his appearance before Pharoah. Before learning subtraction, this could be drawn on a time line for students where they can physically count the years. Children who are a little older can count up from 17 to 30 or subtract 30-17.
• The next step would be to look at the calculation that is offered by Rashi regarding the 22 years that pass before Jacob sees Joseph again. Children can take the calculation from the activity above and pull out (from Parshat Miketz) the additional years that passed- 7 plentiful years followed by 2 years of famine before Jacob travels to Egypt. Students would then add the 3 numbers together to confirm that the time that passed was truly 22 years.
• A larger activity, which could be leveled in many ways depending on the age and abilities of students, would be to pull out the information from larger segments of Sefer Bereishit and place the information onto a more comprehensive timeline. Students could either create timelines for individual sections, or a more comprehensive timeline that incorporates multiple people's storylines for students to see the crossover of time between the different stories that we are given throughout Sefer Bereishit. Along with Rashi's interpretations and explanations throughout the sefer, students can use their own calculations for comparison of information and calculating where everything that we learn would actually fall on a grand-scale timeline.

We can use the information given in this week's parsha and next week's parsha to analyze the passage of time that Rashi refers to on Bereishit 37;24. Creating a timeline helps students formulate an understanding of the chronology of events, which provides them a framework for the both the logical progression of events and the greater picture of how multiple events fit together with each other. In addition to the sequential understanding that is gained, students also practice basic addition and subtraction by calculating the time lapses between different events. Students also work on identifying which information needs to be added or subtracted in order to find missing information, which is a critical problem solving skill. Finally, when the completed timeline has been pieced together, students can look to compare and contrast information to gain a deeper insight that they may not have picked up on in their original, single-story understanding of the events.

Bereishit- Place Value and Regrouping

Addition and Place Values:
When younger children begin learning to add (and subtract) multi-digit numbers, the algorithms alone for these calculations can be confusing. As a starting point, children first need to understand the meaning behind what each digit in a number represents. As they expand their understanding, it is helpful for children to recognize the pattern of every three place values from the decimal point moving towards the left. Larger decimal place values come later, but, again, understanding the pattern of place value will help students pick up the larger picture of what's happening with the numbers.

Basic place values to the millions:

The biggest hurdle for students to overcome is that each place value can't have more than 9 of it's kind, and every time you reach a group of 10, you have reached the next place value and need to "regroup" your numbers.

*10 ones = 1 ten and 0 ones [10]
*10 tens = 1 hundred and 0 tens and 0 ones [100]
*10 hundreds = 1 thousand and 0 hundreds and 0 tens and 0 ones [1,000]
*10 thousands = 1 ten-thousand and 0 thousands and 0 hundreds and 0 tens and 0 ones [10,000]
*10 ten-thousands = 1 hundred-thousand and 0 ten-thousands and 0 thousands and 0 hundreds and 0 tens and 0 ones [100,000]
*10 hundred-thousands = 1 million and 0 hundred-thousands and 0 ten-thousands and 0 thousands and 0 hundreds and 0 tens and 0 ones [1,000,000]

It helps for children to visualize this concept using manipulatives such as base-10 blocks, which have models for ones, tens, hundreds, and thousands. Paper clips are a good make-shift manipulative, since students can clip and unclip into and out of groups of 10, 100, etc. as they make sense of the numbers.

One key that is critical for students to remember is that they need to line up the place values- you can only add ones with ones, tens with tens, hundreds with hundreds, etc. (and then regroup as needed).

Parsha Connection:
Chapter 5 of this week's parsha gives details of the direct lineage from Adam to Noah. We are given information on names and ages for this direct line of descendants. We are given enough information about each person to provide 9 sample addition problems, where we can check the numbers that we are given in the parsha.

As a first step, let's organize the individual pieces of information that we are given. As we organize information for each person, we can check the math.

1st generation:
"This is the account of the descendants of Adam- on the day of G-d's creating of Man, He made him in the likeness of G-d. He created them male and female. He blessed them and called their name Man on the day they were created- when Adam had lived one hundred and thirty years, he begot in his likeness and his image, and he named him Seth. And the days of Adam after begetting Seth were eight hundred years, and he begot sons and daughters. All the days that Adam lived were nine hundred and thirty years; and he died." ~Bereishit 5;1-5

Adam was 130 years old when Seth was born.
Adam lived 800 years after Seth was born.
Adam died at 930 years old

  130
+800 
  930    so this works

2nd generation:
"Seth lived one hundred and five years and begot Enosh. And Seth lived eight hundred and seven years after begetting Enosh, and he begot sons and daughters. All the days of Seth were nine hundred and twelve years; and he died." ~Bereishit 5;6-8

Seth was 105 years old when Enosh was born.
Seth lived 807 years after Enosh was born.
Seth died at 912 years old

  105
+807
  912    so this works

The above example offers regrouping in the ones (to tens) digits.

3rd generation:
"Enosh lived ninety years, and begot Kenan. And Enosh lived eight hundred and fifteen years after begetting Kenan, and he begot sons and daughters. All the days of Enosh were nine hundred and five years; and he died." ~Bereishit 5;9-11

Enosh was 90 years old when Kenan was born.
Enosh lived 815 years after Kenan was born.
Enosh died at 905 years old

   90
+815
  905    so this works

The above example offers a sample with different sized numbers (one tens and one hundreds), which offers a good check on understanding lining up the place values. Also, we have regrouping in the tens (to hundreds) digits here.

4th generation:
"Kenan lived seventy years, and begot Mahalalel. And Kenan lived eight hundred and forty years after begetting Mahalalel, and he begot sons and daughters. All the days of Kenan were nine hundred and ten years; and he died." ~Bereishit 5;11-14

Kenan was 70 years old when Mahalalel was born.
Kenan lived 840 years after Mahalalel was born.
Kenan died at 910 years old

   70
+840
  910    so this works

Again, here we have different sized numbers and regrouping in the tens (to hundreds) digits.

5th generation:
"Mahalalel lived sixty-five years, and begot Jared. And Mahalalel lived eight hundred and thirty years after begetting Jared, and he begot sons and daughters. All the days of Mahalalal were eight hundred and ninety-five years; and he died."  ~Bereishit 5;15-17

Mahalalel was 65 years old when Jared was born.
Mahalalel lived 830 years after Jared was born.
Mahalalel died at 895 years old

   65
+830
  895    so this works

Again, here we have different sized numbers.

6th generation:

"Jared lived one hundred and sixty-two years, and begot Enoch. And Jared lived eight hundred years after begetting Enoch and he begot sons and daughters. All the days of Jared came to nine hundred and sixty-two years; and he died." ~Bereishit 5;18-20

Jared was 162 years old when Enoch was born.
Jared lived 800 years after Enoch was born.
Jared died at 962 years old

  162
+800
  962    so this works

7th generation:
"Enoch lived sixty-five years, and begot Methuselah. Enoch walked with G-d for three hundred years after begetting Mehuselah; and he begot sons and daughters. All the days of Enoch were three hundred and sixty-five years. And Enoch walked with G-d; then he was no more, for G-d had taken him." ~Bereishit 5;21-24

Enoch was 65 years old when Methuselah was born.
Enoch lived 300 years after Methuselah was born.
Enoch died at 365 years old

   65
+300
  365    so this works

8th generation:

"Methuselah lived one hundred and eighty-seven years and begot Lamech. And Methuselah lived seven hundred and eighty-two years after begetting Lamech, and he begot sons and daughters. All the days of Methuselah were nine hundred and sixty-nine years; and he died." ~Bereishit 5;25-27

Methuselah was 187 years old when Lamech was born.
Methuselah lived 782 years after Lamech was born.
Methuselah died at 969 years old

  187
+782
  969    so this works

Again, here we have regrouping in the tens (to hundreds) digits.

9th generation:
"Lamech lived one hundred and eighty-two years, and he begot a son. And he called his name Noah, saying, 'This one will bring us ease from our work and from the toil of our hands, from the ground which Hashem had cursed.' Lamech lived five hundred and ninety-five years after begetting Noah, and he begot sons and daughters. All the days of Lamech were seven hundred and seventy-seven years; and he died." ~Bereishit 5;28-31

Lamech was 182 years old when Noah was born.
Lamech lived 595 years after Noah was born.
Lamech died at 777 years old

  182
+595
  777    so this works

Again, here we have regrouping in the tens (to hundreds) digits.

10th generation:
"When Noah was five hundred years old, Noah begot Shem, Ham, and Japheth." ~Bereishit 5;32

Noah was 500 years old when Shem, Ham, and Japheth were born.

Follow-up Activities:
If you take this base information and use it to try to place the births and deaths in a timeline, students can calculate how old each person was at the births of successive descendants, as well as how old each of the descendants was at the time of death of their ancestors. These calculations will offer students more, similar calculation practice.

For an added exercise, having students identify which numbers to use for these different comparative calculations offers practice with problem solving and identifying important information.

Va'etchanan- Gematria and Simple Addition

"I implored Hashem at that time, saying, 'My Lord, Hashem, You had begun to show Your servant Your greatness and Your strong hand, for what power is there in the heavens or on the earth that can perform according to Your deeds and according to Your mighty acts? Please let me cross and see the good land that is on the other side of the Jordan, this good mountain and the Lebanon.' But Hashem became angry with me because of you, and He did not listen to me; Hashem said to me, 'It is much for you! Do not continue to speak to Me further about this matter...'" ~Devarim 3;23-26

"When you beget children and grandchildren and you will have been long in the land, you will grow corrupt and make a carved image of the likeness of any thing, and you will do evil in the eyes of Hashem, your G-d, to anger Him." ~Devarim 4;25

Rashi on Devarim 4;25:
"And you will have been long"- He hinted to them that they would be exiled from [the land] at the end of eight hundred and fifty-two years, like the numerical value of the word v'noshantem (and you will have been long in the land). But [G-d] advanced the date and exiled them at the end of eight hundred and fifty years. He advanced the exile two years ahead of the number represented by v'noshantem so that "you will surely perish" should not be fulfilled in them. This is the meaning of that which is said, "Hashem hurried regarding the evil and brought it upon us for Hashem, our G-d, is charitable; that is, He acted charitably with us for He hurried to bring [the exile] two years ahead of time.

Gematria & Simple Addition:
For young students who are trying to improve their simple addition skills, practice is critical. While a student might be able to handle the process of addition, every opportunity for practice that a student has will contribute to his fluency and speed in mental calculation. Gematria is the Jewish concept that every letter of the Hebrew alphabet has a numerical value assigned to it. Using these values, it is possible to add the value of each letter in a word, phrase, sentence, etc. to find a deeper meaning between the meaning of the word or phrase and the numerical value assigned to it through Gematria. This Gematria chart gives the values assigned to each hebrew letter. There are places throughout Judaism where you can find Gematria based interpretations for things- throughout the Torah, related to holidays, etc. Younger students can calculate the Gematria of their name or of individual words and phrases. I've had older students search for Gematria patterns in words related to specific holidays. 

Parsha Connection:
In this week's parsha, alone, we have at least two instances of significant Gematria that is explained. The very first word of the parsha- "va'etchanan"- is discussed in Midrash Rabbah. In the beginning storyline of th e parsha, Moshe is praying to Hashem to please allow him to enter the land of Israel along with the Jewish people, and Hashem subsequently angers towards Moshe insisting that Moshe stop this line of prayer. The explanation is given that the Gematria value of the the word va'etchanan (the word meaning Moshe prayed) is 515. Let's check this:
vav- 6
aleph- 1
taph- 400
chet- 8
nun- 50
nun- 50

6 + 1 = 7 + 400 = 407 + 8 = 415 + 50 = 465 + 50 = 515

Midrash Rabbah explains that Moshe offered 515 prayers (the Gematria value of va'etchanan) to Hashem to plead to be allowed to lead the Jewish people into Israel. If Hashem had not cut him off, but had allowed him to pray for the 516th time, it is explained, then Hashem would have had to allow Moshe to enter Israel with the Jewish people.

The second instance of Gematria significance in the parsha come in 4;25, where we are told of the Jewish people being settled and living for many generations in Israel, and then they will anger Hashem. Rashi here uses the Gematria value of the word "v'noshantem", which means "you will have been long in the land". Rashi explains that the Gematria value of the word is 852- let's check this:
vav- 6
nun- 50
vav- 6
shin- 300
nun- 50
taph- 400
mem- 40

6 + 50 = 56 + 6 = 62 + 300 = 362 + 50 = 412 + 400 = 812 + 40 = 852

According to Rashi, this means that the Jewish people should have been settled in the land for 852 years before they were exiled. However, when looking at the timeline of Jewish History, the Jews were actually only settled for 850 years before they were exiled. Why this discrepancy? Rashi explains that if they had been settled for the full 852 years, then they would have strayed so far that their behavior would have caused Hashem to not only exile them, but they would have also been destroyed. In an act of mercy, Hashem caused the Jewish people to be exiled 2 years early so that when they were exiled, they did not perish as well.

Everyday Connection:
Gematria offers one way of integrating interesting, deeper insights into Judaic studies concepts while practicing simple addition and even offering opportunities for students to look for patterns within a given topic. While I find these added spiritual connections interesting and insightful, each person has their own subjects, ideas, or activities that they connect with. In what ways can you find basic math skills, patterning skills, etc in your everyday activities to help offer skills review and practice to the math students in your life? One person might naturally calculate all around them- calculating time until an alarm should go off, combining any types of numbers that may appear in their surroundings -street signs, page numbers, etc; someone else might need to search out problem solving scenarios within their daily activities to find meaningful calculation practice- if I can read a page every 3 minutes, how long will it take me to finish this book? how many pages do I have left to read? if I want to finish the book and walk my dog for 10 minutes before dinner time, do I have enough time to do both if I start right now?

Balak- Mathematical Insights Using Percentages and Other Basic Math

"These are the counted ones that Moses, Aaron, and the princes of Israel counted- twelve men, one man for his father's house, were they- these were all the counted ones of the Children of Israel, according to their fathers' house, from twenty years of age and up, everyone who goes out to the army in Israel: All their counted ones were six hundred and three thousand, five hundred and fifty. The Levites according to their fathers' tribe were not counted among them." ~Bamidbar 1;44-47 (Parshat Bamidbar)

In this week's parsha, following the plague as punishment for Israel becoming attached to Baal-peor:
"Hashem said to Moses, 'Take all the leaders of the people, and hang [the idol worshipers] before Hashem  opposite the sun- and the flaring wrath of Hashem will withdraw from Israel.' Moses said to the judges of Israel, 'Let each man kill his men who were attached to Baal-peor.'" ~Bamidbar 25;4-5
"Those who died in the plague were twenty-four thousand." ~Bamidbar 25;9

Rashi on Bamidbar 25;5 says:

Let each man kill his men: Each and every one of the judges of Israel would kill two and the judges of Israel were seven myriads (seventy thousand), and eight thousand as stated in Sanhedrin.

At first, Rashi's commentary confused me. At the end of the parsha, we are explicitly told that 24,000 died as the result of the plague. Rashi here is stating that 78,000 judges each killed 2 people. The numbers alone don't make sense- how did 78,000 judges kill only 24,000 people (comparing numbers- good lower elementary skill)? If we do the basic multiplication (middle elementary skill), to calculate Rashi's understanding that each judge killed 2 sinners, then 

78,000 Judges x 2 sinners killed = 156,000 men killed.

How does this make sense? I did a little more research, and it seems that Gemara Sanhedrin 82a talks about the Israelite man and the Midianite woman being Zimri son of Salu (from the tribe of Shimon) and Cozbi daughter of Zur (Midianite princess). In the gemara, it says that Zimri brought with him 24,000 Israelites from his tribe, which is what led him to sin (that's some peer pressure!). While it's not stated explicitly, my understanding from this explanation would then be that the 24,000 that we are told of in Bamidbar 25;9 are just the 24,000 Israelites from the tribe of Shimon, and that the additional sinners who were killed by the judges actually did number 156,000.

Now that we have some numbers to go on, let's work with some calculations:

*Let's start by figuring out how many people actually died all together as a result of this situation: (Good practice of long addition algorithm for younger students- remember to line up place values)

24,000 + 156,000 = 180,000 total Israelites who died 

*If we go based on these two accountings, how many Jews were left in the general population (non-Levite) after the plague? (Good practice of long subtraction algorithm for younger students- remember to line up place values)

603,550 - 180,000 = 423,550 total non-Levite Israelites who were left

*What percentage of the general population of Jews were killed in the plague? (Good calculation practice for Middle School)

Following our methods from Parshat Vayigash:
180,000/603,550 = 0.29823544031, or approximately 30%
This means that approximately 70% of the general population was left after the plague was finished. Let's check and see if those numbers are consistent:
423,550/603550 = 0.70176455968, or approximately 70%

An interesting follow-up activity:
If we are assuming that the 24,000 mentioned in Bamidbar 25;9 were all from Shimon, how would our percentage calculations look if we just took the census information for Shimon from back in Bamidbar and compared what percentage of Shimon died as a result of the plague. When making these calculations, we could also consider that Zimri was from Shimon and was killed, but not accounted for in the 24,000. Also, the Judges from Shimon each killed 2 people, so the Israelites from Shimon who were killed by the Shimonite judges would also need to be added into the Shimonite death toll for an accurate calculation.