Let's look at an example that
we like to call THE SECRET, MESSAGE, WHISPER, CHAIN.
Did you ever play that game
when you were a kid where you whisper something into someone's ear and
then they whisper the message to the person beside them and then that
person whispers the message to the person beside them and so on until
you get to the end of the chain?
We recently played this game
with six people. We recorded the time it took to reach the end of the
chain for only one person being told, then for two people passing it on,
and then for three, four, five, and six. It took longer each time we added
a person, and we thought that maybe an equation could describe the process,
so we did a regression.
It turns out we were right.
You can describe this process with an equation. Look at the steps below
to see what we did. Then give the activities on the next few days a try.
First we gathered
all the data and put it into a little table of ordered pairs.
Always be sure to label your table so you know what the numbers
of persons told the secret
to reach the end of chain
we got some graph
paper and set up a grid that would fit all of our data points
with a bit of room left over. Then we gave the graph a TITLE
and we described the numbers on the x and y axes, this is called labeling,
so that anyone reading our graph would know exactly what the points
meant. Then we graphed all six data points.
Sometimes data points
will be scattered all over a graph, and they don't seem to follow
any pattern, these are called random points. But in this case, we
noticed that the points had some order to them. They looked like
they were trying to form a line. This made us believe that there
was probably an equation that would describe this whisper chain
situation. If we could only find the best
experimented a bit using the least
squares regression tool, until we found a line that seemed to
describe the pattern of the data best. In general if you are doing
this manually we recommend a ruler or a piece of uncooked spaghetti.
Slide the spaghetti string or ruler around until you feel you have
the same number of data points above the line as you have below
it. Often you can find a line that actually passes through two,
or three, or more of your data points.
call this line the "line of best fit". When found
with a piece of spaghetti or ruler it is at best still an estimate,
but it's usually good enough to generate a reasonable equation.
Below is a picture of the line we chose.
Two people finding a line of best fit may not always come up with
the exact same line, but they should be reasonably close.
Next we looked carefully
at the line and selected two points. We circled and labeled
them. You can pick any two points on the line, but it's best to
pick two points that are on a grid intersection or ones for which
you know the coordinates from data points.
Our line passed through
three data points, so we chose two of them, (3,9) and (5,14).
Now we were ready to
get that equation. All we needed was the slope and the y-intercept,
so we started with the slope.
The first point was (3,9)
and the second was (5,14), so we plugged these values into the slope
and we got
Next we substituted
this slope into the "slope/intercept" form of the
equation, y = sx
+ i, remembering that the
spot where the letter "m" sits is where the slope of the
So now our equation looked
like this, y = 2.5x + i.
The last thing we
needed was the value for the y-intercept, which is always stored
in the variable "i".
To find it, we had to
select either one of our two points in Step 4, and then substitute
the x and y coordinates from this point for x
and y in the equation in Step 6.
We used (3,9) because
we thought it looked easier, and when we substituted 3 for x
and 9 for y the equation looked like this:
9 = 2.5(3) + i, solving
for b looked like this
9 = 7.5 + b
9 +(-7.5) = 7.5 + (-7.5)
1.5 = i
So the slope is
2.5 and the y-intercept is 1.5. Which means the equation
for this situation is,y
= 2.5x + 1.5
where x stands for the number of people in the chain and y
stands for the time it takes to tell the secret to all those people.
Now that we have the equation,
we can play "WHAT IF". We can determine how long it will take
for any number of people to run the Secret Message Whisper Chain. Or if
we know the time allowed, we can determine how many people can be in the
Try to use the equation
to answer the following two questions.
- How long
will it take 100 people to pass the message?
- How many
people could hear the message in 1 hour?
HINT: The equation is calibrated in seconds, so you
will need to enter how many seconds are in an hour to get the right
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