Haazinu- One to One Correspondence

"When the Supreme One gave nations their portion, when He separated the children of man, He set the borders of peoples according to the number of the Children of Israel." ~Devarim 32;8

Rashi on 32;8 makes reference to 70 languages which correspond to the 70 nations enumerated in Bereishit chapter 10, which developed from the descendants of Noah.

One to One Correspondence:
Have you ever heard a young child count pictures in a book and watched as they accurately count up to 13, only to see that there are actually only 5 pictures on the page? Have you watched a small child try to count items in front of them, trying to figure out which ones were already counted?

One to One Correspondence is one of the first concepts that young children need to understand in order to understand what numbers actually mean. One to one correspondence is the idea that each counting number correlates to one item that is being counted. The way to assign a number to an entire group is to have each item represented by a single number; we can then say that the size of the group is that of the largest counting number used to count the items. If I have 12 marbles, for example, I would be able to know this by counting each individual marble once, and identifying that when I've counted the last marble, the number that I used to count it was number 12. Since I ran out of marbles at number 12, that means that I have 12 marbles in my group.

Once children know the proper order of the counting numbers, the biggest confusion for them when learning to count is how to keep track of what they have already counted. The best strategy for teaching tracking while you're counting is to move the counted items to the side, out of the way, so that they can easily identify which items have already been counted and which items still need to be counted. So how can you help children practice counting when they are given pictures or drawings of items? The best thing to do in that situation is to line up or organize the drawings on the paper in such a way that children can methodically move through each section of the picture in a way that they can keep track of what has already been counted. Organized rows and columns work very well in this type of situation. The fancy books with dynamic types of designs tend to be more confusing and leave children less successful in their counting.

Parsha Connection:
When Rashi references the 70 languages which correspond to the 70 nations that came from the descendants of Noah, this is a perfect example of one to one correspondence. In this case, we have 70 descendants- each of those 70 descendants developed into their own nation, resulting in 70 nations (one descendant for each nation)- and each of the 70 nations had its own language, resulting in 70 languages (one nation for each language).

Another, year-long exercise in one to one correspondence is matching each parsha (or double-parsha) with each shabbat. There is a specific order to the parshiot readings, and there are certain parshiot that are specifically read at certain times. For example, Devarim is always read the week before Tisha B'Av, and Va'etchanan is always read the week  the week after Tisha B'Av; V'Zot Ha'Bracha is always read on Simchat Torah. Using these placeholders in the calendar, we can go through the year of parshiot and match them up to the shabbatot in which they will be read. This process of matching one to one is an exercise in one to one correspondence.

Everyday Connection:
The mere fact that we are each able to count items successfully on a daily basis for day-to-day activities shows that we have each internalized and successfully rely on our understanding of one-to-one correspondence. Interestingly, as we grow and our brains develop, we can start to automatically, accurately identify the quantities of small groups of items. Visually organizing items makes it easier for our brains to automatically, accurately identify the quantities of larger groups of items.

For a different example, musicians who play the harp need to identify the notes of the strings. As identification, the C note strings are all red, and the F note strings are all blue. Using these placeholders, musicians are able to identify which strings are which notes and what their finger placements need to be- 2 examples of one to one correspondence- matching strings to notes and matching fingers to strings.

What examples are in your daily life?

Nitzavim-Vayeilech- Statistical Overview of the Torah

This week we have the only double-parsha of this year's cycle of reading of the complete Torah. Due to this year being a leap year, we had an extra month in the calendar, so these two parshiot are the only 2 that need to be combined in order to get through reading the entire Torah within one Jewish calendar year.

In this week's double parsha, we read of Moshe bringing together all of the Jewish people, leaders of all levels and laypeople alike. He then reviews with them the promises that Hashem made to Abraham, Isaac, and Jacob; he reviews how Hashem saved the Jewish people from slavery in Egypt; and he reviews that if they follow the commandments, they will be blessed, and if they don't, they will be cursed. Then, Moshe discusses that he has lived his complete life, and he will pass on leadership to Yehoshua ben Nun to bring the Jewish people into the land of Israel.

The content of this double-parsha struck me as the beginning of the conclusion of the first 5 books of the Torah. As we approach the end of the cycle, I thought it would be interesting to look at some statistics related to all the parshiot.

Statistics:
If we look back over the math concepts that have been touched on this year, we looked at statistics back in Parshat Bamidbar. There, we talked about the purpose of statistics for comparing data in order to learn patterns and trends about the situation you are looking at. We also looked at using statistics to create graphs in Parshat Vayikra and Parshat Tzav, where we used the graphs to help us look for trends and comparisons of information.

Below I've complied data related to the 5 Books of Torah and each of the individual parshiot. (Note: These statistics were compiled using a combination of information found on Wikipedia.org and torahtots.com.)

If we look at the charts, we can see that there are:
12 parshiot in Bereishit
11 parshiot in Shemot
10 parshiot in Vayikra
10 parshiot in Bamidbar
11 parshiot in Devarim

For some basic statistics, while we see that the parshiot are approximately evenly divided between the 5 books, Vayikra and Bamidbar have the least number of parshiot (10 each), and Bereishit has the greatest number (12). The number of parshiot in each of the 5 books ranges from 10-12 (so there is a range of 2--> 12 - 10 = 2). If we wanted to find out how many parshiot, on average, each book has, we would find the mean- add the 5 numbers together and divide by 5.

If we take a look into the statistics related to the number of letters in each parsha, number of words in each parsha, and number of pesukim (verses) in each parsha, we will find that:

V'Zot Habrachah has the fewest letters with 1,969 letters.
Nasso has the most letters with 8,632 letters.
Overall, the parshiot have a range of 8,632 - 1969 = 6,663

V'Zot Habrachah has the fewest words with 512 words.
Nasso has the most words with 2,264 words.
Overall, the parshiot have a range of 1,752 words.

Vayeilech (one of this week's parshiot) has the fewest verses with 30 verses.
Nasso has the most verses, with 176 verses.
Overall, the parshiot of a range of 146 verses.

In a full statistical analysis of each of these, a student would use either a line plot or stem-and-leaf plot to see the full view of how the number of letters, words, or verses fall on a number line. They would look to see if the numbers cluster in any spots (is there a pattern to be able to identify a smaller number range where most parshiot fall when looking at number of letters? number of words? number of verses?).
Students could calculate mean, median, and mode for the number of letters, words, or verses in the parshiot.
Students could graph the numbers of letters, words, or verses for on a bar graph to see how the numbers increase and decrease over the progression of the parshiot throughout the year.
Students can also look at how the other double-parshiot are paired for regular years, and how that effects changes in the numbers if we combine numbers for double-parshiot.

There are many statistical questions that a student could ask related to this data to compare the statistics of individual books and parshiot. These are just a sampling of some ideas.

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.

Ki Teitzei- Weights & Measures, Balanced Scales

"You shall not have in your pouch a stone and a stone- a large one and a small one. You shall not have in your house a measure and a measure- a large one and a small one. A perfect and honest stone shall you have, a perfect and honest measure shall you have, so that your days shall be lengthened on the land that Hashem, your G-d, gives you. For an abomination of Hashem, your G-d, are all who do this, all who act fraudulently." ~Devarim 25;13-16

Rashi on these passages explains that these stones refer to weights and measures. He also clarifies that it's not saying that you're not allowed to use different size weights. Rather, it means that you may not use two weights that are different weights but look to be the same, which would enable you to trick someone else into thinking that you are using the heavier weight when you're really using the lighter one.

Weights & Measures- Balanced Scales:
Classic scales, ones that were used before analog and digital scales with internal weight mechanisms were developed, worked by balancing two sides with each other. With an item of weight on either side, if the right side dips lower, then the item(s) on the right side are heavier; if the left side dips lower, then the item(s) on the left side are heavier; if the sides are even with each other, then the items on the two sides weigh the same amount. 

The concept of balancing a scale is also one that is commonly used now when teaching pre-algebraic and algebraic concepts. In this format, the idea of balancing equivalent combinations of numbers and variables is compared to balancing weights on a scale. If you know that the two sides of an equation are balanced, then you can perform the same operation to both sides of the balance- similar to adding or subtracting the same amount of weights on both sides of a scale- in order to isolate a variable on one side while keeping the equation balanced so the other side tells you the value of the variable.

Some examples:

If you know that-

x + 3 = 15

the "=" tells us that x + 3 is the same as (balances with) 15. Imagine that the "x" is one weight with an unknown value and the "3" is another weight with a value of 3. The "15" is a single weight with a value of 15. For younger children who need to physically manipulate to help them work through the problem, it might be an unknown weight, 3 weights with a value of 1, and 15 weights with a value of one. This set-up means that if you take away 3 from both sides, the scale (so to speak) will remain balanced. This leaves us with just our unknown, "X" weight on one side and [15 - 3 =] 12 on the other side. So, now we know that the unknown weight has a value of 12.

A more complex example:

If you know that-

5 x Y = 20

again, the "=" tells us that the 5 x Y is the same as (balances with) 20. Here, we imagine that we have 5 weights which all have the same unknown value of "Y" on one side of the scale and a weight with value of 20 (or 20 weights with a value of 1) on the other side. This set-up means that if you divide both sides into 5 equal groups, you can match-up groups of equivalent values. When we divide the "5 x Y" side by 5, we will have the 5 weights separated into 5 groups of one weight in each. With the more simplistic set-up, we can divide the 20 weights into 5 groups, and we'll have 4 weights with a value of 1 in each group. This means that 1 weight "Y" is the same as 4 weights. So, the unknown weight has a value of 4.

Parsha Connection:
In this week's parsha, we are warned against not using two weights that have the same shape and size, but have different weights. If you think about labeled weights that we use nowadays, does that mean that we're not allowed to have more than one weight? I have to choose if I'm going to measure everything with a 1 lb weight or a 5 lb weight? Rashi explains that it means that I can have weights of different weight, but I can't have, for example, a 1 lb weight and a 5 lb weight that look to be the same shape and size. You need to have weights that are clearly distinguishable from one another so that when you are weighing out items of value, it will be clear that you are in no way cheating regarding the value of the items being weighed.

Everyday Connection:
Practicing with balancing scales gives children an opportunity to manipulate the concept of equivalence. Building an understanding of being able to manipulate both sides of a scale in the same way and still maintain equivalence is a critical skill for developing algebraic thinking.

Have you every tried playing on a see-saw? If you have two people who are closer in weight, then they will balance each other and can have fun bouncing each end up and down. Sometimes, if you have one heavier person and two lighter people, you can put the two lighter people together on one side and they will approximately balance the one heavier person. If the heavier person tried to balance with just one lighter person on the other side, the heavier person will be stuck down on the ground, while his friend is stuck up in the air- not heavy enough to weigh himself down against his friend.