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.