Nasso- Are the numbers useful?

From Parshat Bamidbar:

"...these are the Gershonite families- their counted ones according to the number of every male, from one month of age and up; their counted ones [were] seven thousand, five hundred." ~Bamidbar 3;21-22

"...these were the Kohathite families- the number of every male from one month of age and up [was] eight thousand, six hundred; the guardians of the charge of the sanctity." ~Bamidbar 3;27-28

"...these were the Merarite families- their counted ones according to the number of every male from one month of age and up [was] six thousand, two hundred." ~Bamidbar 3;33-34

From Parshat Nasso:
"From thirty years of age and up, until fifty years of age, everyone who comes to the legion for the work in the Tent of Meeting. Their counted ones according to their families were two thousand, seven hundred and fifty. These are the counted ones of the Kohathite families, all who work in the Tent of Meeting, whom Moshe and Aaron counted, at the word of Hashem, under the authority of Moshe." ~Bamidbar 4;35-37

"[F]rom thirty years of age and up, until fifty years of age, everyone who comes to the legion for the work in the Tent of Meeting. Their counted ones according to their families, according to their fathers' house were two thousand, six hundred and thirty. These are the counted ones of the families of the sons of Gershon, all who work in the Tent of Meeting, whom Moshe and Aaron counted, at the word of Hashem." ~Bamidbar 4;39-41

"[F]rom thirty years of age and up, until fifty years of age, everyone who comes to the legion, for the work in the Tent of Meeting. Their counted ones according to their families were three thousand, two hundred. These were the counted ones of the families of the sons of Merari, whom Moshe and Aaron counted, at the word of Hashem, through Moshe." ~Bamidbar 4;43-45

Math Connection:
Oftentimes, when children are learning a new skill and they see numbers in a problem, they just plug the numbers into the memorized algorithm that they were taught and they assume that the number that comes out of their calculation is the answer to whatever question is asked of them. Real life does not always work this way; just because the numbers are in front of you doesn't mean that the chosen calculation will answer your questions. Additionally, just because you have numbers in front of you, it doesn't mean that the information you want can be found in the information that you are given. It's important to always keep in mind the meaning of the numbers that you have and the meaning of the calculation or operation that you choose to employ in your problem solving process. 

Parsha Connection:
In last week's parsha, we are given the census numbers from the Levite males ages one month and older. The numbers from that census are (in the order listed in the parsha):

Gershon: 7,500 males

Kohath: 8,600 males
Merari: 6,200 males

In this week's parsha, we are given the census numbers from the all Levite males ages 30-50 who were to work in Holy service. The numbers from that census are (in the order listed in the parsha):
Kohath: 2,750 males
Gershon: 2,630 males
Merari: 3,200 males

Given these two sets of census numbers, we can compare the numbers from each category. It's important, though, to keep in mind exactly what information we are gaining from our comparison. 

If we know that there were 7,500 Gershonite males from one month and older, and 2,630 males ages 30-50, we can calculate that there's a difference of 7,500-2,630, or 4,870. From here, we know that there are 4,870 males who were not between the ages of 30-50. But, can we figure out how many of those 4,870 are younger than 30yo and/or how many of them are older than 50yo? 

We can make the same calculations for Kohath and Merari.
Kohath: 8,600-2,750 = 5,850
Merari: 6,200-3,200 = 3,000

And from these calculations, we know that there were 5,850 Kohathite males, and 3,000 Merarite males who were not between the ages of 30-50. But, again, we have no way of knowing how many of those 5,850 or 3,000 were between one month and 30 years, and how many were 50 years or older. 

Everyday Connection:
If the weather report tells you that Tuesday's morning temperature is 57° F and that the temperature rose 15°over the course of the day, no matter what calculation you use, you can't use the information in front of you to find out what the temperature was on Wednesday. A student who is asked to find the temperature by the end of the day Tuesday, would be able to add 57+15= 72°F. However, if a student is just given these two numbers and asked whether or not they know Wednesday's temperature, they might just assume that they should be adding the two numbers to find the sum. However, the information given does not tell us how much the temperature might have fallen or risen overnight from Tuesday evening until Wednesday morning. Thoughtfulness and understanding of both your information and your calculations are critical to finding meaning in the numbers that you use in every situation.

Bamidbar- Basic Statistics: Mean, Median, and Mode

"Their counted ones, for the tribe of Reuven: forty-six thousand, five hundred." ~Bamidbar 1;20
"Their counted ones, for the tribe of Shimon: fifty-nine thousand, three hundred." ~Bamidbar 1;23
"Their counted ones, for the tribe of Gad: forty-five thousands, six hundred and fifty." ~Bamidbar 1;25
"Their counted ones, for the tribe of Judah: seventy-four thousand, six hundred." ~Bamidbar 1;27
"Their counted ones, for the tribe of Issachar: fifty-four thousand, four hundred." ~Bamidbar 1;29
"Their counted ones, for the tribe of Zevulun: fifty-seven thousand, four hundred." ~Bamidbar 1;31
"Their counted ones, for the tribe of Ephraim: forty thousand, five hundred." ~Bamidbar 1;33
"Their counted ones, for the tribe of Menasheh: thirty-two thousand, two hundred." ~Bamidbar 1;35
"Their counted ones, for the tribe of Binyamin: thirty-five thousand, four hundred." ~Bamidbar 1;37
"Their counted ones, for the tribe of Dan: sixty-two thousand, seven hundred." ~Bamidbar 1;39
"Their counted ones, for the tribe of Asher: forty-one thousand, five hundred." ~Bamidbar 1;41
"Their counted ones, for the tribe of Naftali: fifty-three thousand, four hundred." ~Bamidbar 1;43

"...these were all the counted ones of the Children of Israel, according to their fathers' house, from twenty years of age and up, everyone who goes out to the army in Israel; All their counted ones were six hundred and three thousand, five hundred and fifty." ~Bamidbar 1;45-46

This week's parsha begins with Hashem telling Moshe to take a census of all the men in Israel from the age of 20 and older. Hashem appoints one person from each tribe as a prince to help Moshe take the census.

What is Statistics?
In simple terms, statistics is the mathematical study of data (numbers in a certain situation) in order to provide information about the situation. For example, if I have test scores from a class of students, I could use statistics to find out information about how the whole class performed on the test. Or, if I want to know information about the general height of students in the class, I can use statistics to analyze the heights of the students in the class. The data for these examples would be the list of all the test scores, or the list of each students' height.

What could we do with these sets of data? There are 3 different calculations that are known as the statistical averages: 
1) mean 
2) median
3) mode

Each of these averages can be more or less meaningful in given situations. 
*Mean is calculated by adding all of the numbers in your data set together and dividing by the number of numbers that were added together. So, if I have 5 numbers in my data set, I would add them together and divide that sum by 5. That quotient would be the mean of my data set. (This is the classic "average", or arithmetic average, that we learn about in Elementary school)

*Median is identified by finding the exact middle number of the data set. The first step is to put all the numbers in numerical order. Next, identify the middle number- for younger children, it's easier to methodically cross off numbers alternating between the top number that's left and the bottom number that's left until they're just left with the middle number (or two numbers). If there is an odd amount in the data set, then the middle number is the median; if there is an even amount, then the number that is exactly halfway between the two middle data points is the median. 

*Mode is identified by finding the number that appears the most times in the data set. If every number appears evenly in the data set, then there is no mode. There can also be more than one mode in a data set.

One other important term that is usually introduced with these 3 averages is range. Range is the distance spanned by the numbers in a data set. Range is calculated by taking the highest number in the set minus the smallest number in the set. The resulting difference is the range.

Parsha Connection:
We can take the census numbers from the 12 tribes and calculate the range and averages of the number of men aged 20 and older. Let's start by rewriting our data.

Reuven-          46,500
Shimon-          59,300
Gad-               45,650
Judah-             74,600
Issachar-          54,400
Zevulun-          57,400
Ephraim-          40,500
Menasheh-       32,200
Binyamin-        35,400
Dan-                 62,700
Asher-              41,500
Naftali-             53,400

Now, let's put those numbers in numerical order:

Menasheh-       32,200
Binyamin-        35,400
Ephraim-          40,500
Asher-              41,500
Gad-                45,650
Reuven-           46,500
Naftali-            53,400
Issachar-          54,400
Zevulun-          57,400
Shimon-          59,300
Dan-                62,700
Judah-             74,600

Before doing any calculations, we can look at the newly organized data and see that Menasheh had the smallest number of 20+ men, and Judah had the most. Now let's do those statistical calculations.

Range:
Most of the census numbers range between 32,000-60,000, while Judah had much more. The actual range of census numbers is
74,600 - 32,200 = 42,400

Mode:
We see that each tribe had a unique number of men, so there is no mode to this data.

Median:
Since there are 12 tribes here, the median will be exactly between the 6th and 7th tribes; between Reuven & Naftali. So how do we find the midpoint between 46,500 and 53,400? We can add them together and divide by 2.
46,500 + 53,400 = 99,900
99,900  ÷ 2 = 49,950
So, the median number of men in each tribe was 49,950.

Mean:
We start by adding all 12 numbers together. When added together, the sum is 603,550 (notice that this is the exact number that the parsha gives us for the total census for the 12 tribes together). Next we divide 603,550  ÷  12 = 50,295.8
So, the mean number of men in each tribe was 50,295.8 (or approximately 50,296 men)

Everyday Connection:
What are some hot topics in your house that you can use for data collection? How much time do your children spend, on average, working on homework each night? Are they getting enough sleep; what's the average number of hours of sleep that they get each night? 

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? 

Behar- Different Bases

"You shall count for yourself seven sabbaths of years, seven years seven times; and the days of the seven sabbaths of years shall be for you forty-nine years. You shall sound a broken blast on the shofar, in the seventh month, on the tenth of the month; on Yom Kippur you shall sound the shofar throughout your land. You shall sanctify the fiftieth year and you shall proclaim freedom throughout the land for all its inhabitants; it is a yovel year for you, and you shall return, each man to his ancestral heritage, and you shall return, each man to his family." ~Vayikra 25;8-10

Base-10:
Our counting system is a base-10 system. This means that our basic counting structure is based on collecting groups of 10. Our place values in our numbers (ones, tens, hundreds, etc.) are essentially the way we keep track of our groups of 10 as we count. We have the numbers 0-9. When we are counting and reach the number 10, we are basically saying that we have 1 group of 10; as we keep counting, 20 means that we have 2 groups of 10. Continuing on, 25, for example, means 2 groups of 10 with 5 individual pieces. If we look closely at our place values, we find:

ones= digits 0-9 (the parts of a group of 10) [this can also be looked at as the number of individual parts multiplied by 10⁰]
tens= how many groups of 10 [also the number of groups multiplied by 10¹]
hundreds= how many 10 groups of 10 [also the number of groups multiplied by 10²]
thousands= how many 100 groups of 10 [also the number of groups multiplied by 10³]
etc.

So, thinking about it in this way, the number 532,749 means that we have:
9 singles
4 groups of 10
7 groups of 10 tens
2 groups of 100 tens
3 groups of 1,000 tens
5 groups of 10,000 tens

Other Bases:
The concept of other bases means that rather than bundling groups of 10, we bundle groups of other numbers. Clocks, for example, work on base-12. Every 12 hours we complete a full "bundle". Thinking in base-12, the numbers 0-9, 10 (represented as T), and 11 (represented as E) would go in the "ones" column, and the number 10 would actually mean that you have 1 full bundle of 12.

Some examples in base-12:
*the number 20 would mean that you have 2 full bundles of 12 (or 24 in our base-10 system)
*the number 25 would mean that you have 2 full bundles of 12, plus 5 more from a partial bundle (or 24+5, or 29 in our base-10 system)
*a trickier one- the number 3E would mean that you have 3 full bundles of 12, plus 11 more from a partial bundle (or 36+11, or 47 in our base-10 system)

What does each place value mean in the base-12 system?
"Ones"= "digits" 0-12 [or the number of individual pieces multiplied by  12⁰]
"Tens" (or Twelves)= how many groups of 12 [or the number of groups multiplied by 12¹]
"Hundreds" (or 144's)= how many 12 groups of 12 [or the number of groups multiplied by 12²]
"Thousands" (or 1,728's)= how many 144 groups of 12 [or the number of groups multiplied by 12³]

Binary, or base-2, is often associated with computers. In addition to our 12 hour clock system, base-24 could also be used when dealing with hours.

Parsha Connection:
In this weeks parsha, we read about the shmita year, which occurs every 7 years once the Jewish people entered into the land of Israel. We then learn that after 7 cycles of 7 years there is a yovel year. Here we see a system for counting in groups of 7- basically we are calculating in base-7.

How can we use our base notation to keep track of the years for shmita and yovel?
Let's think about this. First of all, let's set-up what each place value is representing.
"Ones"= digits 0-6 [or the number of individual pieces multiplied by 7⁰]
"Tens" (or Sevens)= how many groups of 7 [or the number of groups multiplied by 7¹]
"Hundreds" (or 49's)= how many 7 groups of 7 [or the number of groups multiplied by 7²]

"Thousands" (or 343's)= how many 49 groups of 7 [or the number of groups multiplied by 7³]

Let's try to map out the years for shmita and yovel using base-7 notation:

Regular Counting year	Base-7 notation	Special Year
Year 1	1
Year 2	2
Year 3	3
Year 4	4
Year 5	5
Year 6	6
Year 7	10	Shmita
Year 8	11
Year 9	12
Year 10	13
Year 11	14
Year 12	15
Year 13	16
Year 14	20	Shmita
Year 15	21
Year 16	22
Year 17	23
Year 18	24
Year 19	25
Year 20	26
Year 21	30	Shmita
Year 22	31
Year 23	32
Year 24	33
Year 25	34
Year 26	35
Year 27	36
Year 28	40	Shmita
Year 29	41
Year 30	42
Year 31	43
Year 32	44
Year 33	45
Year 34	46
Year 35	50	Shmita
Year 36	51
Year 37	52
Year 38	53
Year 39	54
Year 40	55
Year 41	56
Year 42	60	Shmita
Year 43	61
Year 44	62
Year 45	63
Year 46	64
Year 47	65
Year 48	66
Year 49	70	Shmita
Year 50	71	Yovel
Year 51	72

What patterns do you notice here?

If you continue the pattern, you'll notice a slight shift over time. Can you find a pattern in the way that the first pattern shifts?