Balak- Quantity vs. Quality

"The officers of Moav arose and came to Balak and said, 'Balaam has refused to go with us.'

Balak kept on sending officers- more, and higher ranking than these. They came to Balaam and said to him, 'So said Balak son of Zippor, "Do not refrain from going to me, for I will honor you very much, and everything that you say to me I shall do; so go now and imprecate this people for me."'

Balaam answered and said to the servants of Balak, 'If Balak were to give me his houseful of silver and gold, I am unable to transgress the word of Hashem, my G-d, to do anything small or great. And now, you, too, please stay here for the night, and I will know what Hashem will add in speaking with me.'" ~Bamidbar 22:14-19

Quantitative vs. Qualitative:
Quantitative data is information that can be counted or measured.
Qualitative data is information that describes something. 

Parsha Connection:
In this week's parsha, Balak is trying to convince Balaam to invoke a curse upon the Children of Israel. When the first set of elders were sent from Moav, they were unsuccessful in convincing Balaam to curse the Israelites. Without giving up, we learn that Balak sent more, higher ranking officers. We aren't told how many officers were sent each time, nor are we told their specific rankings. Rather, we are given a qualitative description of the officers. We know that the second group was larger and of higher standing than the first group.

In an effort to convince Balaam, Balak sends a message that he will honor Balaam "very much". Again, Balak's offer was a qualitative one- he will honor Balaam "very much"; there is no indication of how he will honor Balaam, or what he will offer him, we just know that it will be a lot of honor.

Balaam's response, however, is a quantitative one. He responds that even if Balak gave over his entire houseful of silver and gold, Balaam is not willing to go against Hashem. Although we don't know how much silver and gold was contained in Balak's house, it is a quantifiable, calculable amount. 

Academic Connection:
Sometimes the difference between quantity and quality is important, and other times it's not critical. For example, if I have 2 sizes of containers to put away toys, the specific sizes probably don't matter- I can look and see that one size is too small, but the other size will work well. If I have half a gallon of water left and want to transfer it to a smaller container, it would be important to know the exact sizes of the two containers so that I will know if the water will fit in one or the other of the containers.

Suggested Activity:

• Younger students could have scenarios set up for them- either as part of a whole class discussion or individual stations for students to go through in partners or individually. Students could have 2 sample situations presented to them where they need to determine "Quantity or Quality?" Sample situations might include:
• Case 1: Your friend's bag of cookies is bigger than your bag of cookies. Case 2: We need 7 more cookies in order to have enough for everyone in the class.
• Case 1: The cookies have black and white frosting on top. Case 2: There are 5 black and white cookies in each package.
• Case 1: Our class collected $50 in tzedaka money. Case 2: Our class collected more money for tzedaka than we did last year.
• Case 1: My house has 3 bedrooms. Case 2: My house is blue.
• Case 1: Your friend has fewer siblings than you. Case 2: Together, you and your friend have 5 siblings.
• After working through some examples, older students could create their own examples and test each other. Another activity might be for students to create 2 short writing samples, one with only qualitative descriptions and the other with only quantitative descriptions. As a class, sharing writing samples, students can think about the differences they notice between the two types of writing samples.
• Note that qualitative information can sometimes be turned into quantitative information, and this might be confusing to students. For example, if I have more cookies than you, that's qualitative. However, we could each count our cookies and then describe the difference using actual numbers, which would be quantitative. Other qualitative information cannot be explained quantitatively- for example, my house being blue.

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.