Using Semi-Log Graphs for the Beginner.

 Adapted from:

https://www.sampletemplates.com/business-templates/semilog-graph-paper.html

The semi-log graph paper shown below has a linear scale along the horizontal axis and a logarithmic scale along the vertical axis; thus it is called a "semi-log" graph.  The vertical scale on this piece of graph paper has three “decades”.  The values start at 1 and increase until 10, then increase in multiples of 10 until 100.  The set beyond 10²=100 increases by units of 100 until 1000, and so forth.  You’ll notice that the lines are not linearly spaced.  Notice how the vertical axis is labeled in the graph below.

In your previous graphing experience, graphs of data are plotted on paper that has linear scales on both the horizontal and vertical axes.  However, this is a different kind of scale that is very different. This exercise will help you to see some of the reasons for this.

And now, your personal exploration to increase comprehension:

1. Make a table of values for the function F(x) = 2^x    with values of x from x = 0 to x = 9.  Plot the graph of this function on the semi-log graph paper.  Use caution when plotting the F(x) =y values since the vertical scale is not linear.  Draw a smooth curve through the points and label the equation of the function on the graph. Do the same for G(x)=3^x.

2. Make a table of values for the function f(x) = 2x with values of x from x = 0 to x = 9.   Plot the graph of this function on the semi-log graph paper.  Again, use caution when plotting the f(x) = y values since the vertical scale is not linear.  Draw a smooth curve through the points and label the equation of the function on the graph. Do the same for g(x)=3x.

3. Briefly describe the differences between the graphs on a semi-log paper and on the rectangular graph paper you’ve previously used.  What are some reasons why you might want to graph data on semi-log paper?  What are some advantages and disadvantages? How could a graph that has a logarithmic scale be misleading?

TPC: Logarithmic Graphs: Log Regressions. Due Feb 27

In your last blog, you were asked to try to graph a set of numbers on a standard unit piece of graph paper. The values became large quickly and it was impossible to put all the values on the graph at the same time.  Some of you discovered the log graphing tool; some of you remember the blog from last year.  Here's a new twist. We will use data on distance from the sun to help us see the usefulness of logs. 

This is an example from your text: Larson, Precalculus (2007).  

The data below is each planet's distance from the sun in Astronomical Units ( 1 AU is the distance from the sun to our Earth) and the length of time it takes to orbit the sun in terms of our year.

Find a LINEAR equation that expresses the relationship between x and y. Notice that the table seems to suggest that you need to put "x" in L1 and "y" in L3.  Please do that.  Start by scatterplotting the following data. What’s the scatterplot look like? Then Stat-calc-linreg L1 vs L3. You'll see the line fits the data pretty well, but that the data appears to be more on a concave up curve, like an exponential function. None of the points fall exactly on the line. 

Take the natural log of each x and each y; put "lnx" in L2 and "lny" in L4. (You can do this quickly by placed the cursor at the top of L2, and typing "lnL1" as a formula then the down arrow; then put the cursor at the top of L4, then type "lnL3" then a down arrow. )  Then scatterplot lnx versus lny (in other words, scatterplot L2 vs. L4). What do you observe?  Now either run a linear regression of the lnx and lny data or choose any two points to find the slope. 

Explain what you discovered in your blog; take photos or snips of your calculator image to help you explain. 

TPC: How To Graph Large Numbers: Semi-Log Graphs.Due _________

In your previous blog, you wrote about the fast spread of information on the internet using unrealistically conservative models: y = 3^x and y=10^x. You let values of x become as large as 15; at that point the numbers became so large that you couldn't graph them on the graphing paper.  Some folks scaled their y axis so that they could include the larger numbers (great job!), but then it was impossible to see the level of growth for the first several hours.  You watched Sal Khan name and discuss these exponential functions in his video posted in my last blog.

So basically, on your graphs, when you show what happens for the larger values of x, you can't see what's going on for the smaller values of x (like the y intercept or the y values when x is 1 or 2 or 3....).  If you have your axis settings so that you can see what happens for the smaller values of x, you can't even include the coordinates for the larger values of x on your graph.  These functions simply increase too rapidly to show everything clearly.

What to do?

You're not the first set of folks to be concerned about this and to be frustrated with the graphs.  So we have a different way of graphing.  We use log graphs.  I have log graph paper copied for you already to use.  You'll be able to show all values of the exponential functions above on these graphs.  It's actually called "semi-log" graph paper because one side is spaced constantly (like you are used to) and the second side is spaced differently -- explained below.

When you look at this paper, notice the spacing of the vertical lines.  These, obviously, measure the "x" values.  The spacing is even; the physical distance on the graph is a constant one unit.

What you'll notice that is different is the spacing on the horizontal lines, the "y" values.  Start with the darkened horizontal lines that are evenly spaced.  You'll see the first one, about an inch and a half up the page labeled as 10^1 (sometimes it's simply labeled as "1" in semi-log paper) then the second darker line, about 3 inches up the page labeled 10^2 and so forth.  These integers represent powers of 10.  What is 10^0?   I've marked some of these values for you. Label the ones I did not label.

Now focus on the horizontal lines between 10^0=1 and 10^1=10. You'll see that as you move your eyes up the page, these lines become closer together.  The first line represents the value 1, the second 2, and so forth. You're counting by ones.

Now focus on the lines between 10^1 and 10^2.  These lines also become closer together as you move up the page.  The distance between each line is now 10 units.  So you can count by tens between the "1" (which represents 10^1) and the "2" (which represents 10^2). Count the lines: 20,30,40,50... up to the darkened line which represents 100.  I've labeled some for you.

The pattern continues. Between the 10^2 and the 10^3, each line represents an increase of 100.  I've labeled some of these for you. Between the 10^2 and the 10^3, you count by hundreds.



To the right is a graph of three points. This graph paper uses a "10" instead of what our paper labels a "10^1."  It also uses "100" when our graph paper labels a "10^2."   The coordinates of these three points to the right are:
(1,10), (2,40) and (3,1). Match the coordinates to each point on the graph to the right.







To the right is a second image of points on a log graph. In your blog, identify the coordinates of those three points. If you are having trouble, the answers are given 11:19 in the YouTube video linked below. If you think you've got it, then do check your answers by watching the video linked below.







Continue to label the magnitudes represented by each horizontal line on your graph paper.  Here's a YouTube that will help. Watch that video now. It's about 12 minutes.  At 7 minutes, the measurement on the graph will start at "1" like we are in our graph.  This video will help you verify that you've labeled your horizontal lines properly (notice the paper I've given you is only different in how it starts.)

Then refer to your charts for y = 3^x and y = 10^x.  Plot those points on the log paper graph that I've given you. You'll be able to graph all the points for your charts.  You'll also see an interesting and simple relationship.  See if that relationship holds for another exponential function. (You can chose any base, but y = 2^x works just fine, too.) So you've now graphed THREE exponential functions on log paper.

For your blog, you will (a) turn in your three graphs on the same semi-log paper.  (b)  Write in your blog your observations about what these exponential functions graphed like on log paper. (c) Find something new you learned about this kind of graphing that was in the youtube video.  There's lots of gems in this video; pull out at least three ideas from the video to put in your blog. One could be the rhyme, another could be the proportionality issue, another could be how decimals are represented....your choice.  You also could discuss why you think these exponential functions graphed as they did on semi-log paper.

GA2: Pascal's Triangle. Due Friday February 17. 8 am.

One of the coolest and most useful pieces of mathematics that you learn in high school math is what we call Pascal's Triangle.  I've copied a few rows of the triangle below from Wikipedia:

 If you don't know how to form the values in each successive row of this triangular shape, you'll want to look that up and explain it in your blog. (Google "Pascal's Triangle)

A couple of the interesting things include the following.

Find the sums of each row.  Then compare these numbers.  What pattern is represented?

The powers of 11 are represented in this triangle; how so?  Use the triangle to find the seventh power of 11.  Explain how you've done this.  

Triangular numbers, square numbers, Fibbonacci numbers, prime numbers.....all appear in interesting patterns in Pascal's triangle. This interesting site will give you a simple introduction.

There's also some amazing visual patterns that appear in Pascal's triangle.  Explore those and report back what you find.  (I know there are some artsy folks out there -- this part's for you.) 

In your blog this time, write something about Pascal's triangle.  You can be expansive.  Perhaps you'd like to write about how the Chinese people knew and used this triangle long before Pascal was born.  Perhaps you'd like to write about Blaise Pascal himself: when did he live and what influences did he have? Of course it would also be wonderful for you to find something new and cool about the number patterns Pascal's Triangle. 

Have fun.  And. If you find some good piece of math humor, feel free to include that, too.

GA2 and TPC: Spread of Information: Monica Lewinsky Due Feb 21 or 22 (TPC) Or March 1 GA2.

Monica Lewinsky gave an important TED talk. 

Watch it. It's about 25 minutes long. There are many important messages here; please listen with your eyes opened.  Ms Lewinsky is a brilliant and thoughtful woman with impressive speaking skills and a life experience from which we can learn much about our culture, about the era of electronic communications and about the torture of public humiliation.

Ms Lewinsky refers to herself as "Patient Zero" in cyberbullying. Over 20 years ago, information about her life changing event spread rapidly and internationally over the internet.  Her image, along with negative labels and sordid stories and commentaries, was spread with never-seen-before speed.  Her story was likely exaggerated through repetition (ever play "telephone" as a kid?) and carpeted the globe. Her name became household; she was recognized on the street and grocery store.   She claims, likely accurately, to be the first to be publicly humiliated in this extreme manner and at a time before "cyberbullying" was a word.  Since then, bullying on the internet has become lethal.  Take a deep breath and process that idea, google if you'd like. We will now sanitize her heavy message and look at just the math.

So if you have not already watched her TED talk, then stop reading MY blog now, click on the link above to her TED talk, and watch her full talk.  Then return to this page and continue. 

Again, let's separate ourselves from her personal humiliation and look at the spread of information on the internet.  We will start slowly, and with an unrealistically simple model to set the stage.  Our first model will minimize the spread but will still illustrate how rapidly information is passed from person to person electronically.  We'll put a name to this kind of growth.   Let's start by working by hand.  You'll need full-sized graphing paper , some regular lined paper for some work, and at least two writing implements of different colors.  (The link will take you to free graph paper; you can choose your size. Choose Cartesian Graph Paper.  Let your size be 85x11 inches. Choose millimeters. Measure in 5 cm units.  You are lucky, though, I've copied some of this paper for you to use for this exercise. )

With a ruler or straightedge, darken a horizontal line one unit upwards from the bottom of your graph paper, oriented vertically.  This will be your x axis. Each PAIR of blocks will represent an HOUR time interval.   Then darken a vertical line one unit to the left of the left side of your paper.  This will be your y axis.  Each block will represent the number of new people who receives the information.  Label your axes with words and with numbers; labeling will help you keep track.

The parameters we will set include a limitation that you all might agree is very conservative.  Let's say that each time a person receives this juicy information, they forward this information -- a simple "click" -- to only 3 different people. Let's say that folks send on information not continuously or within seconds, but once each HOUR.  (Ok, so in reality, it could be shared on any social media account where potentially HUNDREDS of people would learn the information with a single instantaneous "click," but we are working with a simpler and far more conservative model. We are starting by spreading the message slowly and to three additional people at a time.  Do you agree that we are really really slowing down the spread of information?)

At time = 0 (x=0, therefore the y intercept), have 1 person know the information.  Then after 1 hour (x = 1), that person "clicks" the mouse to send the information to 3 people (so the coordinate is (1,3) )  Then at t = 2 (2 hours later), those 3 people each send the information to 3 more people (so the coordinate is (2,9))  At 3 hours, each of those 9 people send the message to 3 more new people: the coordinate is then (3,27).  On your lined, paper, make a chart that represents hours in one column and number of new people who receive the information in the second column.  You'll want to use your calculator and likely round appropriately.  Find the numbers for up to 15 hours.  (Use scientific notation if you need to!) This is, remember, just over half a day.

When you are done with the chart, plot the points on your graph paper. You might have trouble plotting some of the points on this single graph paper. You have nearly 40 blocks on the x axis, so you can represent 18 hours on your graph; let your chart be complete for each hour up to 15 hours. Record your observations, in writing.  Here's the start of your blog.

Then find an equation that fits your data points. (in the form y = _______) Of course, Khan Academy parallels what we are doing nearly perfectly in a two minute video.  In this video, you'll learn the name of these kinds of relations. So if you have not already watched Sal Khan in the video link above, stop reading this blog now and watch the Khan Academy lesson now then return to this page to continue.  Be sure your BLOG TITLE includes the name of these relations.  (Hint: begins with "e.")

Now let's adjust a single parameter.  What if information is transferred only once each hour, but instead of only 3 people getting the information, 10 people receive the information?  (This is still pretty conservative, in light of the fact that each "click" could spread the information to hundreds of your friends.)  Re-do your chart on your paper.  Plot the new points on your graph in a different color.  Find the equation that fits your data points.  Record your observations.

By now, you likely get the idea.   What would happen if information was passed on -- more realistically -- once every 15 seconds?  What if information was passed on to 100 new people with each "click" each person makes?  How does the graph change for each adjustment?   Report your understanding in your blog. Be expansive and thoughtful and creative.   At some point, we can cease to understand the magnitude of the numbers our calculator reports: the numbers are simply too large.

On February 21 (Feb 22 for period 4),  I will collect your charts and graphs and expect your blogs to be uploaded into canvas.

Our next blog will consider a different graphing strategy so we can graph more information about the large numbers that appear in these functions.