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?

Bechukotai- Finding a rule...or not?

"I will provide peace in the land, and you will lie down with none to frighten you; I will cause wild beasts to withdraw from the land, and a sword will not cross your land. You will pursue your enemies; and they will fall before you by the sword. Five from among you will pursue a hundred, and a hundred from among you will pursue ten thousand; and your enemies will fall before you by the sword." ~Vayikra 26;6-8

Rashi on 26;8 says:
Five [from among you will pursue] a hundred, and a hundred from among you [will pursue] ten thousand. Is this the correct calculation? Should it not have rather said, "and a hundred from among you will pursue two thousand?" But, you cannot compare, a few who perform the commandments of the Torah to many who perform the commandments of the Torah.

This week, rather than teaching a new concept, I want to focus on using some preexisting knowledge to think about mathematical brainstorming. Sometimes, more than teaching new information, it's just as important to allow students the opportunity to think and learn how they best organize information and think about how numbers work together- using their number sense to work through real applications of what they know. 

In this week's parsha, we learn that when the Jews are to enter the land of Israel, they will be protected and supported by Hashem. When pursuing their enemies, we are told that 5 people could overcome 100 enemies, and 100 people could overcome 10,000 enemies.

Rashi's commentary picks up on a big mathematical question with these numbers:
Back in Parshat Ki Tisa we looked at maintaining proportions, or keeping fractions of a group the same even if we make a larger group or smaller group. In this week's parsha, Rashi points out that the numbers in the Torah here are not proportional. If...

5 people can overcome 100 enemies, and 
5 x 20 = 100, then 
100 people should be able to overcome 2,000 enemies (100 x 20 = 2,000).

Rashi explains that the numbers do not work like this in our case, because the strength of many righteous people is increased more than just proportionally. So if we look at the strength of 100 people, maybe it should be an exponential relationship? This is when the number being used is multiplied by itself. We see that 100 x 100 = 10,000.

So did we find the rule for how we can figure out the strength of a group of any size? Unfortunately, no, because it needs to work in both cases. Since 5 x 5 = 25 (not 100), then this rule only works for the second set of numbers, not the first.

How can we try to find a rule here? This is where brainstorming and playing around with numbers comes in. We lay out what we know:

5 people can overcome 100 enemies (this is x20).
100 people can overcome 10,000 enemies (this is x100)

Is there are rule to be found where for 5 people we would multiply x20, and for 100 people we would multiply x100? It's possible that breaking down the numbers into factors (e.g. 4x5- for 20; 20x5 for 100) will help us see a more intricate relationship that isn't visible with the larger numbers.

I've been playing around with the numbers this week to see what relationships I can find between the two comparative situations. I'm still playing with it. Sometimes there isn't enough information to definitively form a rule, and sometimes there just isn't a straightforward relationship. Can you find a relationship or rule that would fit both of these situations?