Tzav- Statistics part 2

Once again, this week we have a significant portion of the parsha dedicated to specifics of korbanot, or sacrificial offerings. Again, rather than quoting from the parsha, I've organized some of the information related to the offerings in order to analyze.

Pie Chart/Circle Graph:
Whereas the bar graphs we looked at last week compared a measured amount between different categories, a circle graph compares portion sizes of categories that together make up 100% of something. For example, out of your total budget allotment, what percentage of the budget goes to different categories (food, medical, housing costs, etc.)? Out of all after school enrollment, what percentage of students go to each of the different activities? Out of the total hours in a day, what percentage of time is spent sleeping, eating, working, etc.? The key to circle graphs is that all categories together add to 100% and create a complete circle. 

When analyzing data to convert into a circle graph, we'll need our knowledge of fractions (Parshat Vayishlach), proportions (Parshat Ki Tisa), and percentages (Parshat Vayigash) to convert the raw data into a format that will fit the circle graph.

Connection to Parsha:
Let's take a look at a sample data set from this week's parsha to walk us through the process. Let's look at what percentage of sacrifices were to be fully burned and what percentage of sacrifices were allowed to be partially eaten in some form.

We can start by making a chart to organize the information so that we see which category each sacrifice falls into.

Next, we use the chart to count up how many sacrifices are in each category out of the total number of sacrifices. According to our chart, out of the 6 total sacrifices, we have 3 sacrifices that are completely burned and 3 sacrifices that are partially burned and partially eaten.

Let's restate this as fractions- 
3/6 = burned 
3/6 = burned then eaten

If we want, we can reduce these fractions (each reduces to 1/2), but we don't have to. In this case, we will reduce them, since it will make our next calculation easier to work with. 

So, we have-
1/2 of the sacrifices are burned
1/2 of the sacrifices are burned then eaten

Now, remember that the most important part of a circle graph is that it shows a percentage out of 100%, where 100% (or all) of what you're analyzing makes up the full circle. So, if 100% of the sacrifices is all of them, we know that 1/2 of the sacrifices were burned: 100% x 1/2 = 50% and 1/2 of the sacrifices were burned then eaten: 100% x 1/2 = 50%. So now we have calculated that 50% of the sacrifices were totally burned and 50% were burned then eaten. Once we have this calculation, we can create our graph. 

While I made the graph electronically, you can also construct a circle and measure out the angle measures for each section using pencil, paper, drawing compass, and protractor. Such an activity offers a good opportunity for art/geometry integrations in math. To calculate the sections by hand, we need to first know a full circle is 360°. Since we need 1/2 of the circle for each category, we multiply 360° x 1/2 = 180°. This means that the angle measure for each section of the circle needs to be 180°. Measuring out this 180°, you'll find that you've perfectly divided the circle into two 50% sections, one for each category.

It happened that the numbers for this activity fell perfectly into two equal groups. While it may seem simplistic because of the easy numbers and category divisions, it would make a very simple way to practice going through the steps of the process and practicing drawing circles and interior angles. Once practicing with this easy example, another problem with more complex category divisions could be introduced.

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.

Pikudei- Mixed Measures

"All the gold that was used for the work- for all the labor of the Sanctuary- the offered-up gold was twenty-nine kikar and seven hundred thirty shekel, in the shekel of the Sanctuary. The silver of the accountings of the assembly, a hundred kikar, one thousand seven hundred seventy-five shekel, in the shekel of the Sanctuary." ~Shemot 38;24-25

"The offered-up copper was seventy kikar and two thousand four hundred shekel." ~Shemot 38;29

Rashi on 38;24 explains that a kikar of the Sanctuary is three thousand shekalim, and, therefore, the individual shekalim that are listed are mentioned because together they don't add up to one kikar.

What is a "mixed measure"?:
Understanding how to measure with standard measurements is a basic skill that students begin practicing with from an early age. How many feet long is the room? How many cups of water are in the pitcher? But, what happens when you don't have enough to complete an exact measurement with the measure that you're using? What if the room is between 10 and 11 feet long? What if the pitcher has between 5 and 6 cups of water? Sometimes we use fractions to indicate these middle measures. However, we also have other, smaller measurements that can be used to measure these partial portions. We have a standard equivalency that 12 inches are equivalent to 1 foot, for example. So, if my room is between 10 and 11 feet long, I can measure the "extra" length using inches. Maybe my room is 10 feet and 3 inches. This type of measurement is known as a mixed measure. To simplify a measurement out of a mixed measure, we could convert the mixed measure into the larger measurement using a mixed number (a whole number with a fraction) or decimal, or we could convert the mixed measure into the smaller measurement. With our room example, 10 ft 3 in would be 10 1/4 ft or 10.25 ft when measured only in feet, or 123 inches when measured only in inches. For younger students, the calculation for converting the mixed measure into the smaller unit (from feet and inches to just inches) is an easier one and also helps to give them a better sense of the full meaning of the number, where mixed measures and mixed numbers are often harder for them to visualize and fully understand.

Let's calculate:
So how do we convert to the smaller measure unit? Let's use our 10 ft 3 in room example. We know that 1 foot is made up of 12 inches. So, we multiply 10 x 12 (10 feet, with 12 inches in each) = 120 inches. Then, we add the extra 3 inches that were left over, so 120 + 3 = 123 inches.

What does this have to do with the parsha?:
In this week's parsha, Rashi helps us understand that the weight measurements that we're given for the gold, silver, and copper that were donated to the mishkan are mixed measures using kikar and shekalim. Rashi explains that 1 kikar = 3,000 shekalim.

The amounts that we are given in the parsha are:
Gold: 29 kikar and 730 shekel
Silver: 100 kikar and 1,775 shekel
Copper: 70 kikar and 2,400 shekel

How can we convert these to all shekalim measures?
We know that 1 kikar has 3,000 shekalim, so:
Gold: 29 x 3,000 (29 kikar, with 3,000 shekalim in each) = 87,000 + 730 shekel = 87,730 shekel
Silver: 100 x 3,000 (100 kikar, with 3,000 shekalim in each) = 300,000 + 1,775 shekel = 301,775 shekel
Copper: 70 x 3,000 (70 kikar, with 3,000 shekalim in each) = 210,000 + 2,400 shekel = 212,400 shekel

So, our final calculations after converting from mixed measures to just shekalim are:
Gold: 87,730 shekel
Silver: 301,775 shekel
Copper: 212,400 shekel

A further thought:
The silver calculation is particularly meaningful, since we are told "a beka for every head, a half-shekel in the shekel of the Sanctuary for everyone who passed the counters, from twenty years of age and up, for the six hundred three thousand, five hundred fifty." (28;26) So, our silver calculation should be accurate for 603,550 men 20 years of age and older having each given a 1/2 shekel. Does this work? 
301,775 shekel of silver x 2 (two 1/2-shekel pieces for each shekel) = 603,550 1/2-shekel pieces.
So, it seems that there isn't even estimation here, but rather the calculations work exactly!