Haazinu- Looking for patterns

Parshat Haazinu is made up entirely of a single Chapter- Perek 32- towards the end of the book of Devarim. The majority of the parsha is a song written by Moshe. If you look at the parsha directly in the Torah scroll, the section of the song is written completely in two divided columns. Below is a picture that I took of this section from a tikun, or book that people use to practice the cantillation and pronunciation of the words. It is written exactly as it appears in the Torah scroll, but I took pictures of the practice side rather than the actual Torah side. (Please note that because these pictures contain G-d's name written out in them, please do not print them out or print them out and discard them.) 

The reading is in the two larger print righthand columns on each page. The pictures only include the sections with the song, not    the entire parsha.

If you've ever studied poetry in school, you'll be familiar with patterns of rhyme, beat, measure, line fragments, word placements, etc. that writers use to enhance the aesthetics of their poetry. Reading through the parsha as it is written in the Torah, do you find any patterns? Lyrical patterns- beat or measure to the pattern of the words? Visual patterns- patterns in the placement of the words or repetitions of words in the scroll? Is there a pattern in the number of words in the lines, or the way the words are broken up into columns or lines throughout the song?

Younger students can listen to the parsha being read and listen for beat or measure. Older students can investigate this on their own, or in partners. Students of all ages can look at patterns in numbers of words per line, or looking for any alliteration of letters that may appear. Younger students could handle smaller sections and maybe look at the whole as a class, while older students can look at the entire text as in partners. 

Please note that I have not investigated these questions myself, yet, but rather this is a proposed investigation. 

Ki Tavo- Leveled activities for division

"Moshe commanded the people on that day saying, 'These shall stand to bless the people on Mount Gerizim, when you have crossed the Jordan: Shimon and Levi and Judah and Issachar and Joseph and Benjamin. And these shall stand for the curse on Mount Ebal: Reuven, Gad and Asher, and Zevulun, Dan and Naphtali...'"~Devarim 27:11-13

Some activity suggestions:

• Younger students can talk about dividing the 12 tribes evenly between the two mountains. How many should be on each mountain? Connect the separation to division of 12 by 2-- will have 6 on each mountain. 
• Pattern finding: If you cross-reference with charts of children from each mother (Rachel, Leah, Bilha, and Zilpah) and the birth order of the 12 sons, can you find any connection between the siblings who were grouped together on each mountain?
• For Older students- if we take the most recent census information- from Bamidbar 26:1-51, we have a listing of the numbers of men in each tribe. Taking these numbers, was the division of people equal between the two mountains? Did one mountain have more people and one have fewer people? Could there have been a different way to divide the tribes that would have made the division more equatable between the two mountains?

Behar/Bechukotai- Graphs, Patterns, and Calculation Activities

"Hashem spoke to Moshe, saying: Speak to the Children of Israel and say to them: When a man will express a vow to Hashem regarding the valuation of souls, the valuation of a male shall be: for someone twenty years of age to sixty years of age, the valuation shall be fifty silver shekels, of the sacred shekel. If she is female, the valuation shall be thirty shekels. And if from five years of age to twenty years of age, the valuation of a male shall be twenty shekels and of a female ten shekels. And if from one month of age to five years of age, the valuation of a male shall be five silver shekels; and for a female, the valuation shall be three silver shekels. And if from sixty years of age and up, if for a male, the valuation shall be fifteen shekels; and for a female, ten shekels. But if he is destitute for the valuation, then he should have him stand before the Kohen, and the Kohen should set his valuation; according to what the hand of the person who makes the vow can attain should the Kohen set his evaluation." ~Vayikra 27:1-8

In this section of this week's parsha, we learn about the standard valuation for donations that are vowed to the Beit HaMikdash on behalf of family members. We have the standard valuations, and we are also told that if someone makes a vow but doesn't have the funds to meet the standard valuations, the Kohen can make a determination of an appropriate valuation for that person. This week I would like to look at some activities for students of varying levels based on the standard valuations that are listed.

To begin, let's organize the information (older students could be asked to chart this information for themselves by identifying and organizing the information in the passage):

Note that this could also be organized into two charts- one for males and one for females. This breakdown might be clearer for some students when trying to separate information on males and females in follow-up activities.

Related Activity Suggestions:

• Graph It!- A good way to compare data of this nature is by graphing it. This information lends itself well to a double bar graph, with a category for each age grouping, and two bars within each category (one for males and one for females). The left side of the bar graph (y-axis) would indicate the number of shekels donated. As always, grade levels recommended are an approximation, and individualization for students is necessary to meet their actual ability levels. K-2 students could fill in a pre-made class-sized graphing chart as a class activity, with older students also copying the information onto personal graphs. Grades 3-5 students could create individual graphs on pre-printed graph templates with spacing indicated for different levels of labeling, and then graphing the information. Students in grades 6+ could create their own double-bar graph from start to finish.

• Is there a pattern?- Looking at the table above and/or the double-bar graph created in the first activity, students can look for comparisons between males and females at each age group. They can also look for comparisons between the different age groups for males and then between the different age groups for females. 
• What trends do they see over a person's lifetime? Is this consistent for both men and women? Why do they think the valuations may have been set in the way that they were?
• After making note of comparisons, older students could look for specific numerical comparisons between males and females in each age group and then across the data for males and females at different age groups. Are any of the ratios between groups consistent in any way? Rashi's note on 27:7 is also relevant to these comparisons, as he specifically points out some differences in the ways that the valuations change between the different age groups.

• Problem Solving- Processing the information through sample family valuations- Using the valuations given, students could be provided with sample problems with different family groupings to calculate the valuation for each family. Students could also create their own sample families to calculate or switch with friends to calculate. What about students' own families? What would each student's personal family valuation be if they were donating to the Beit HaMikdash? Can older students come up with an algebraic formula for calculating the valuation for any given family scenario?

Ki Tavo- Linear vs. Exponential Growth Patterns

"Then you shall call out and say before Hashem, your G-d, 'An Aramean would have destroyed my father, and he descended to Egypt and sojourned there, few in number, and there he became a nation- great, strong, and numerous...'" ~Devarim 26;5

Rashi on 26;5:
"Few in Number"- That is, with seventy souls

Linear vs. Exponential Growth:
When you look at the specific way that a pattern grows over time, there are two basic types of growth patterns. The first type of growth is called linear growth. With linear growth, each new number in the pattern is found by adding (or subtracting) a number from the last one. This pattern is called "linear" because when you put these numbers into a graph, it makes a straight line. For a linear pattern, if you know which number in the pattern you are looking for (term number), you can multiply it times the number that is added between terms to find out your missing number.

An example:
Every chair that I have has 4 legs. Let's write out a pattern to show how many legs I have based on how many chairs I have:

Term # (# of chairs):	1	2	3	4	…
# of Chair Legs:	4	8	12	16	…

What if I have 12 chairs? How many chair legs will I have? You can see above that each new step in the pattern (bottom row) can be found by adding 4 to the previous number. This tells us that we have a linear pattern. We can find how many chair legs we have by multiplying the # of chairs x 4. So, for 12 chairs, we can multiply 12 x 4 = 36; 36 chair legs.

Let's look at the graph of this pattern:

For more advanced students (pre-algebra/algebra), they can think about how to amend this pattern calculation if you have a pattern that starts with a number before the pattern begins. In other words, with our chair example, if you have 0 chairs, you have 0 legs. Patterns that begin in this way are said to be directly proportional. But, what if I have $10 in my bank account and I earn $5 for every hour that I babysit. How does this change the way that the pattern works? How can I change my calculation to use what I learned in the chair example, but also include my original $10 in my calculation? How will it change the graph of the information? How would a subtraction example look- in a pattern? in a chart? in a graph?

The second type of growth is called exponential growth. With exponential growth, each new number in the pattern is found by multiplying (or dividing) a number by the last one. This pattern is called "exponential" because the method for calculating the change in the pattern over time is by using an exponent. The graph for this type of growth increases (or decreases) more quickly and will curve as it continues on the graph. The number that you multiply (or divide by) is your base number, and the step number in the pattern will be your exponent.

An example:
A plant triples it's height every month. How tall will the plant grow in the first 12 months?

Term # (# of months):	1	2	3	4	…
Plant height in cm:	1	3	9	27	…

How can I calculate how tall the plant will be after 12 months? You can see above that each new step in the pattern (bottom row) can be found by multiplying the previous number by 3. This tells us that we have an exponential pattern. We can find how tall the plant will be by calculating 3 to the power of the month that we want. So, for 12 months, we can calculate 3¹² = 531,441 cm (or 531.441 meters). That's a bit of a ridiculous example, but it shows you how it works. You could be wrapping your house in this vine, at that rate.

Why does this work? Why is the 3 the base number and not the exponent number? Each step in the pattern is the previous step x3. So, for the 12 months, it's the same as:
1 (our starting height) x 3 x 3 x 3 x 3 x 3 x 3 x 3 x 3 x 3 x 3 x 3 x 3
This can be written more simply as 3¹²

Let's look at a graph of this pattern:

Parsha Connection:

In this weeks' parsha, Rashi points out that there were 70 people in Yaakov's family when they went down to Egypt, and over the course of their 210 years in Egypt, they became quite numerous. How can we calculate the approximate number of Jews by the time they left Egypt?

First, we can just calculate a base number. Let's assume that each generation doubled the number of people (that would mean that each couple had 4 children; 2 parents --> 4 children = double the number). If we use this as a base, it is an approximation, since some families would have more and some families would have fewer children. How many generations should we use? Regardless of people's lifespan at the time, we can assume that within 20 years, each new generation of children would begin having children of their own. This would mean that there were approximately 10 generations that began having children while in Egypt. 

So, if each generation is doubling, that means that each number in our pattern is multiplied x2 to find the next number. 

Term # (Generation #)	1	2	3	4	5	6	7	8	9	10
Number of people (doubling)	70	140	280	560	1,120	2,240	4,480	8,960	17,920	35,840

Let's think about how to make a formula for calculating this pattern (it's a little more complex than my examples above). First, when dealing with exponential "generations" of growth, you need to know that the very first generation is always labeled as 0 (not 1). We do this, because when we're calculating the growth from one generation to the next, the 1st new generation is one step up (so numbered as #1). So, when we're talking about doubling our pattern, we're talking about calculating exponentially with a base of 2 (2 for doubling), and our exponent will be the generation number (or one less than our term number).

So, the first part of our formula looks like: 

2^{(term #-1)}

That's easy enough, but where does the 70 original people come in? So, we need to multiply the whole formula x70 in order to fit that in.

70 x 2^{(term #-1)} = the number of people at the exodus from Egypt

Truthfully, our numbers above look small, based on census taken after the exodus (especially knowing that there were some people who died during the period of slavery). Let's try our calculation tripling each generation. Then, the formula would be:

70 x 3^{(term #-1)} = the number of people at the exodus from Egypt

And our chart would look like:

Term # (Generation #) 1 2 3 4 5 6 7 8 9 10

Number of people (tripling)

70

210

630

1,890

5,670

17,010

51,030

153,090

459,270

1,377,810

Knowing that some of the population died in slavery, these numbers look more realistic.

And the graphs of the double and triple populations?

You'll notice that the graphs follow a similar curve, but pay attention to the numbers on the y-axis (left side). The tripled numbers are MUCH bigger than the doubled numbers. They only look similar because the scale on the tripled graph is much bigger than the scale on the doubled graph. Also note that the curve on the tripled graph increases more steeply than on the doubled graph. What do you think would happen if we quadrupled? How would the graph look?

Everyday Connection:

Population growth is actually one of the classic examples of exponential growth. Certain bank interest calculations also grow exponentially. You will also find examples in science and statistics.

While you might not find exponential growth patterns on a regular basis, it's important to be able to identify how different patterns work and what kind of a pattern you're dealing with. How many people can you seat using a number of 8-seater tables? How much will you earn in a week at your hourly wage? How much will you have saved if you consider your savings plus a month of work hours? How much will your have in your bank account after a year of saving with interest?

Do you really want to sit and calculate each situation one step at a time? If you can identify the pattern and corresponding formula, then you will save yourself tedious calculations of all the individual steps in the pattern- you can jump right to the number that you want to find.

Basic patterning practice for younger students (as I've written of before) is all work for building the skills for them to be able to think about these higher level patterning ideas as they get older.

Re'eh- Overlapping Patterns

"At the end of seven years you shall institute a release. This is the matter of the release: Every creditor shall release his authority over what he has lent his fellow; he shall not press his fellow or his brother, for He has proclaimed a release for Hashem." ~Devarim 15;1-2

Rashi on Devarim 15;1:
Rashi here explains that one might think that each time a person makes a loan, they count seven years on that particular loan. Rather, we learn here that the seven years mentioned are referring to counting of the year according to the shmita cycle, and all loans are released every time we come to a shmita year, regardless of when they were made.

I'd like to use this information about a different aspect of shmita years to build on the shmita cycle information that we looked at in Parshat Behar. 

In the shmita/yovel chart in that post, you can see that every 7 years is a shmita year. At the time of the post, I made the chart based on my understanding of the text- with shmita every 7 years, and a final yovel year in the 50th year (after 7 groups of 7 years). With further research, I found a discrepancy in the understanding of how the cycle works. One explanation of the discrepancy can be seen here in an explanation on Parshat Behar.

In essence, the calculation discrepancy is as follows:
*Rabbi Yehuda understood that the yovel cycle was independent from the shmita cycle. In his view, there was a shmita cycle occurring every 7 years and a yovel cycle occurring every 50 years. The separate cycles overlapped in such a way that yovel fell in the 1st year of every 8th new shmita cycle.

*The sages understood that the yovel cycle was a final year within the larger shmita cycle. In their view, yovel was the 50th and final year of the previous shmita cycle, and the next shmita cycle would start the following year at year 1.

Here is a chart that compares the calculation implications in these differing understandings (bear with me- the chart gets lengthy in order to start to see the difference as we get into the second set of cycles):

Regular Counting year	Rabbi Yehuda	Sages
Year 1
Year 2
Year 3
Year 4
Year 5
Year 6
Year 7	Shmita	Shmita
Year 8
Year 9
Year 10
Year 11
Year 12
Year 13
Year 14	Shmita	Shmita
Year 15
Year 16
Year 17
Year 18
Year 19
Year 20
Year 21	Shmita	Shmita
Year 22
Year 23
Year 24
Year 25
Year 26
Year 27
Year 28	Shmita	Shmita
Year 29
Year 30
Year 31
Year 32
Year 33
Year 34
Year 35	Shmita	Shmita
Year 36
Year 37
Year 38
Year 39
Year 40
Year 41
Year 42	Shmita	Shmita
Year 43
Year 44
Year 45
Year 46
Year 47
Year 48
Year 49	Shmita	Shmita
Year 50	Yovel (yr 1 of new shmita cycle)	Yovel (yr 50 of old cycle)
Year 51		(yr 1 of new shmita/yovel cycle)
Year 52
Year 53
Year 54
Year 55
Year 56	Shmita
Year 57		Shmita
Year 58
Year 59
Year 60
Year 61
Year 62
Year 63	Shmita
Year 64		Shmita
Year 65
Year 66
Year 67
Year 68
Year 69
Year 70	Shmita
Year 71		Shmita

Just looking at the comparison chart, we can see that after the first full shmita/yovel cycle, the Sages' cycle falls behind Rabbi Yehuda's. For the next 7 shmita cycles, Rabbi Yehuda's shmita years fall one year earlier than the Rabbi Yehuda's. If we continue this pattern, after the next 7 shmita cycles, the Sages' shmita years will fall two years earlier. This discrepancy continues over time. Although the Sage's will fall farther and farther behind, every certain number of cycles, they will be in line again. Can you figure out how frequently that alignment will occur? To connect with counting in different bases, discussed in Parshat Behar, can you count the years in base-7 (as I started in the chart there) to help identify the alignment shift?

We can look at how this discrepancy in understanding changes the meaning of loan release that we learn of in our parsha this week. It says clearly that the loan release takes place after 7 years, which Rashi explains is in the shmita year. According to Rabbi Yehuda, the math works, and every 7 years is a shmita year, even when a yovel year follows a shmita year. According to the sages, though, when a yovel year falls out, there are actually 8 years between the shmita year before and the next shmita year. What happens if someone makes a loan during a yovel year? According to my understanding of the sages, a loan made during the yovel year would not be required to be released until the following shmita year, which would actually be 8 years. Following the sages, of the calculation were to keep to 7 years, then in the 2nd yovel cycle loans would actually be released the year before shmita, and in the 3rd yovel cycle loans would be released two years before shmita, and so on.

Although I did try to research this question, I was not able to find an answer. It seems that the Gaonim followed the Sages calculation of the yovel cycle, but at the time of exile from Israel, yovel was discontinued and the 50th yovel year was removed from the cycle. With this new calculation, the counting cycle looks like the column for Rabbi Yehuda's cycle, but with the yovel label taken out, since the new shmita cycle is bumped up a year.