Parshat Noach - Using populations to model exponential versus linear growth

At the 2009 NCTM Boston Regional Conference, I came upon the work of an organization called Population Connection. They were advocating teaching about population growth in math classrooms - to further their ultimate goal of population stabilization. Cheekily, I attended their session and got lots of great ideas about how to educate our students about our ultimate goal of growing the Jewish population, so as to better fulfill our mission of bettering God's world.

I debuted this worksheet the next day. (It was, fortuitously, Parshat Noach that Shabbat.) Since the conference was a Thursday, and my Windows Explorer tells me I finished the worksheet at 7:06 am the next morning, you will not be surprised to note that the first draft of the worksheet is a bit rough. I'll discuss some of the potential areas of improvement below.

What does this week's parsha teach us?
וַיְבָ֣רֶךְ אֱלֹהִ֔ים אֶת־נֹ֖חַ וְאֶת־בָּנָ֑יו וַיֹּ֧אמֶר לָהֶ֛ם פְּר֥וּ וּרְב֖וּ וּמִלְא֥וּ אֶת־הָאָֽרֶץ׃
God blessed Noah and his sons, and said to them, “Be fertile and increase, and fill the earth. (Breisheet 9:1, fulltext and translation courtesy of Sefaria.org.)

The math scenarios (Gemara Yevamot 62a)

"תניא רבי נתן אומר ב"ש אומרים

שני זכרים ושתי נקבות ובה"א

זכר ונקבה"

“We learn in a Beraita: Rabbi Natan says: [Each person is commanded to bear a certain number of children.]

            Beit Shammai says: Two boys and two girls.

            Beit Hillel says: A boy and a girl.”

If Noach and his kids left the ark and immediately began to repopulate the earth, how would the earth's population be affected by using the Beit Hillel strategy as opposed to the Beit Shammai strategy?

The first time my students attempted this answer key, they found it quite challenging. Many teachers who like to develop understanding will use an exploration of exponential growth to introduce it to 8th graders (or perhaps, strong 7th graders.) The difficulty of this worksheet is based on your students' understanding of how exponential growth works. If they have not been exposed to exponential growth yet, they will need to approach the problem by calculating each row independently. It is unlikely that they will manage to find the pattern in terms of an exponential function unless they are very experienced with exponential growth.

First I'll walk through the Torah ideas behind the answer key, and then I'll suggest an adaptation to improve matters. To come up with the numbers, I made some assumptions. They are explained below.

What do we assume?
Noach had 3 sons, and they each had wives. (See Breisheet 6:10, 18).

וַיּ֥וֹלֶד נֹ֖חַ שְׁלֹשָׁ֣ה בָנִ֑ים אֶת־שֵׁ֖ם אֶת־חָ֥ם וְאֶת־יָֽפֶת׃Noah begot three sons: Shem, Ham, and Japheth.....
וַהֲקִמֹתִ֥י אֶת־בְּרִיתִ֖י אִתָּ֑ךְ וּבָאתָ֙ אֶל־הַתֵּבָ֔ה אַתָּ֕ה וּבָנֶ֛יךָ וְאִשְׁתְּךָ֥ וּנְשֵֽׁי־בָנֶ֖יךָ אִתָּֽךְ׃But I will establish My covenant with you, and you shall enter the ark, with your sons, your wife, and your sons’ wives. After the flood, these three sons were in charge of repopulating the world. We are assuming Noach and his wife had no more kids. שְׁלֹשָׁ֥ה אֵ֖לֶּה בְּנֵי־נֹ֑חַ וּמֵאֵ֖לֶּה נָֽפְצָ֥ה כָל־הָאָֽרֶץ׃These three were the sons of Noah, and from these the whole world branched out.

20

...וַֽיְחִי־נֹ֖חַ אַחַ֣ר הַמַּבּ֑וּל שְׁלֹ֤שׁ מֵאוֹת֙ שָׁנָ֔ה וַֽחֲמִשִּׁ֖ים שָׁנָֽה׃Noah lived after the Flood 350 years.

29

וַיִּֽהְיוּ֙ כָּל־יְמֵי־נֹ֔חַ תְּשַׁ֤ע מֵאוֹת֙ שָׁנָ֔ה וַחֲמִשִּׁ֖ים שָׁנָ֑ה וַיָּמֹֽת׃ (פ)And all the days of Noah came to 950 years; then he died.
(See Breisheet 9:19, 28-29. The Torah tells us explicit genealogies in these chapters, beginning with these verses. If Noach had had additional children, it is reasonable to expect they would have been enumarated in perek 10 with his other descendents.)

If Noach himself did not procreate after the flood, the base of 'people who are procreating' after the flood is 6. If each has only two children (obviously, not the case as enumerated in Breisheet 10), then we add 6 new children per generation, removing those who "age out" of the table.

Your students will discover quickly that Beit Hillel's strategy leads only to population replacement, not to population growth.

Challenges and opportunities:

Students had a lot of trouble calculating the boxes as Beit Shammai's numbers got larger. The sheet, which is portrait, not landscape, does not have a lot of space. It is beyond most middle school students to write an exponential growth function that models the total in the right-hand column.

It occurred to me belatedly that this was a perfect opportunity to create a spreadsheet activity. (Former colleague Steven Steinsaltz suggested I tell students to work down, not across. Going down the columns is easy, because the population just doubles every time, once the pesky 2* from Noach and his wife 'falls off' the table.)

Once you have the numbers for generation 6, you are set.

If you set up a Sheet with header columns like in the handout, and input the numbers we came up with for each generation, you can teach students how to use the SUM function to come up with the totals.

Careful not to include the generation number in the total population!

Fill in the generations column by typing 6, 7, 8 and then using everyone's favorite little blue box on the bottom right corner of the cell (henceforth, 'the magic blue box') to fill down.

Then we get this:

Then you can show them how to use a formula for doubling in column B.

And use the magic blue box to drag the B formula (not A!) all the way to column G.

For H, grab the total in row 2 and drag down to row 3. Then you can drag down B3:H3 all at once - the spreadsheet can accommodate different formulas for different columns.

The whole spreadsheet I worked with is here. Make yourself a copy if you want to tinker.

I highlighted the total in Generation 10....
Because your students will have been told before that there are 10 generations from Noach to Avraham, I highlighted this number. Ask your students if they think it's too high ... or too low.

Extensions:
How many kids did the generations after Noach actually have? Are these males or males and females? (See Breishit perek 10, here.)
Could we attempt to estimate how many are in generation 3, and from that make new predictions?
(I.e., change both the initial inputs and ALSO change the rate at which the population grows?) Note that to do this, we need to either make assumptions about how many males and females are born, or find rabbinic sources that attempt to answer this question and use their model.

Conclusions: 
Every so often there are conversations about demographics in the Jewish community. How many kids do Israeli Jews have? How many kids do Charedi Jews have? How many kids do modern Orthodox families have? How will having 3 kids instead of 4 per family impact schools, political clout, consumer options, housing, etc. etc? I think these conversations are fascinating but need to be grounded in a proper understanding of the math.

My new Rav, Rabbi Uri Topolosky, pointed out in a d'var Torah that the pasuk uses the language of blessing instead of commandment: "וַיְבָ֣רֶךְ אֱלֹהִ֔ים אֶת־נֹ֖חַ" (Breisheet 9:1 again.)

The fact that it's a blessing reminds us that many people struggle to conceive and that families in our era can't always have as many children as they might want. But the ability to raise the next generation is a divine blessing (as well as a divine obligation). Through our children (and our fruitful multiplication), we make a lasting impression on the world. Weighty stuff for math class!

Shabbat shalom,
Aliza