Vayikra- Understanding Fractions through Perspective

"Hashem spoke to Moshe, saying: If a person will commit a misuse, and sins unintentionally against Hashem's holies, he shall bring his guilt-offering to Hashem, an unblemished ram from the flock, with a value of silver shekels, according to the sacred shekel, for a guilt-offering. For what he has deprived the Sanctuary he shall make restitution, and add a fifth to it, and he shall give it to the Kohen; then the Kohen shall provide him atonement with the ram of the guilt-offering and it shall be forgiven him." ~Vayikra 5:14-16

In this section of this week's parsha, we learn about a person who is required to bring a monetary restitution for an unintentional sin of misusing an item in the Tabernacle. His restitution, in addition to bringing a sacrifice of a certain value, is to also repay the value of the damages done, plus 1/5 of the value.

In the Sapirstein Edition of the Artscroll Vayikra (Torah with Rashi's Commentary), there is a footnote on Rashi 5:16 ("For what he has deprived the sanctuary he shall make restitution") that explains that when the Torah speaks of 1/5, it is really referring to 1/5 of the final total. In other words, the amount that was added is 1/5 of the final total after adding in that additional piece. 

What does this mean when thinking of fractions as we know them? If we want to have a section be 1/5 of the total, it means that before that piece was added, there were only 4 of those identically sized pieces. In other words, the original whole was cut into 4 quarters. Then, another 1 piece of equal size to the other 4 pieces is added on. Now, the final added piece is 1/5 of the entire new whole.

Why might this be so confusing to understand?
Think of an optical illusion type of artwork that you may have seen- maybe this well known image where, from one angle, you see an old woman, and from a different angle you see a young woman. It is all a matter of perspective. If you look at one set of features in a certain way, then you can see the old woman; if you focus on a different set of features slightly differently, then you see the young woman. 

The fraction situation that we have in our parsha this week can also be confusing to learners nowadays because of perspective. When we teach and think about fractions nowadays, in most cases, we think about our original value as one whole unit. If someone told us to add 1/5, we would assume that we needed to break our original unit into fifths (ie 5 even pieces) and then add on 1 additional piece of the same size. Alternatively, in the Torah and rabbinical writings through the Talmudic period, the entire unit was determined after the final additional amount was added on. Therefore, the 1/5 that was added on was 1/5 of the final unit, and in order to determine the size of that unit, the smaller segment actually needed to be broken into quarters.

Let's look at an example to see how this would differ, practically:

For ease of calculation, let's take a starting value of $100 restitution that's owed. I will break down the case, for the same starting value, first with our current day perspective, and then with the Torah perspective.

Current Day Perspective:
Starting value owed: $100
We need to add 1/5 of the starting value: 1/5 of $100 is $20 (100 ÷ 5 = 20).
The total value to be paid is the original value + 1/5, or $100 + $20 = $120

Torah Perspective:
Starting value owed: $100
We need to add an amount in order to have that amount be 1/5 of the final value. Remember, I noted above that if you have 4 pieces of equal size, and add one more that is the same size, now you will have 5 pieces of equal size and that last piece that you added will be 1/5 of the total final value. 
So, we need to first figure out 1/4 of the starting value: 1/4 of $100 is $25 (100 ÷ 4 = 25).
When we add $100 + $25 = $125. We can check to confirm that $25 is 1/5 of $125 (125 ÷ 5 = 25).
Since this calculation does work out, we have confirmed that the total value to be paid is the $125.

If we compare these two sample calculations, we can see that, according to our calculations, the Torah Perspective actually turns out to be more costly for the sinner than the Current Day Perspective does, and the calculation process is slightly more involved (although maybe not if that's how you're used to calculating!).

Some questions to keep you thinking about fractions:
Does this perspective difference in the understanding of 1/5 always result in the Torah calculation being more costly than the current day calculation? Does it change if you use a smaller original value? a larger original value? What if you used different fractions and compared them in current day vs. Torah perspective? 

[Note that with an understanding of what a fraction is and represents, these questions are developmentally appropriate for students in younger elementary grades and upwards. The approach to finding the answers would vary based on other skill levels and developmental understanding of the concepts needed to think about the questions and make sense of them.]

Vayikra- Statistics part 1

About the Parsha:
In Parshat Vayikra we have 5 chapters that just list different types of sacrificial offerings that can be brought. Rather than quoting sections of the parsha, I've reduced the information into the summary below:

olah-offering-- male cattle
olah-offering--male sheep or goat
olah-offering--turtledoves or young doves
meal-offering--fine flour, oil, frankincense

meal-offering--first fruits

peace-offering--male or female cattle

peace-offering-- male or female from flock (sheep or goat)

Kohen's sin-offering-- young male cattle

Community sin-offering-- young bull (male cattle)

Ruler sin-offering-- male goat

individual sin-offering-- female goat or sheep

guilt-offering--female sheep or goat

guilt-offering-- two turtledoves or two young doves (if can't afford sheep/goat)

guilt-offering-- flour (if can't afford doves)

guilt-offering-- ram

What is a graph?
A graph is a visual representation of information in order to easily compare information in a given situation. There are many types of graphs- line graphs, bar graphs, pie charts, histograms, venn diagrams- each type of graph is suited to a certain type of information and a certain format. A pie chart, for example, gives information about the division of categories within a total group, or sectional percentages out of 100%.

For the next few weeks, I'd like to look at different types of graphs and statistical analysis of information that's given in the parshiot.

For this week, we'll look at bar graphs. Bar graphs are graphs that are used to compare categories. A vertical bar graph has the names of the categories on the bottom of the graph, and the left side of the graph has the number scale to show how much is measured in each category. A horizontal bar graph can display the same exact information, but the graph is turned sideways- the names of categories are on the left of the graph, and the number scale is along the bottom of the graph.

It's interesting to look at different ways that the same information can be graphed- each graph bringing out different aspects of the information.

Let's first make a graph showing how many options there are for each type of offering. First we'll organize our data:

*olah-offering: (3 options)
--cattle
--sheep/goat
--doves
*meal-offering: (2 options)
--fine flour mixture
--first fruits mixture
*peace-offering: (2 options)
--cattle
--sheep/goat
*sin-offering: (4 options)
--Kohen (cattle)
--community (cattle)
--ruler (goat/sheep)
--individual (goat/sheep)
*guilt-offering: (4 options)
--sheep/goat
--doves
--flour
--ram

Now that we have our data organized the way we want it, we can graph it (chart on top, graph underneath):

Number of Options for Sacrifice Types:

Now, what if we make a graph of how many must be male animals, must be female animals, could be either male or female, or are grain/fruit?

Again, let's first organize our data:

*male animals: (8 options)

--olah-offering (x3)
--sin-offerings (x3)
--guilt-offering (x2)
*female animals: (2 options)
--sin-offering (x1)
--guilt-offering (x1)
*male or female: (2 options)
--peace-offering (x2)
*grain/fruit: (3 options)
--meal-offering (x2)
--guilt-offering (x1)

Now that we have our data organized the way we want it, we can graph it (chart on top, graph underneath):

Number of Sacrifices by Animal Type:

So, here we have two very different graphs, both based on the same exact data set, but focusing on different aspects of the information; one looks at which sacrifices have the most options for what can be offered for that type of sacrifice, and the other looks at how many sacrifice types each animal can be used for. 

Some good questions for testing for understanding of graphs might be: Which category has the most options? Which category has the least? Are there any that have the same amount? How much difference is there between two given categories? These questions can be tailored for each specific graph that you're reading.