Thank you, O Respondents, for telling me HOW
you got your answers!
These are so long, here are links....
Anne, Jane, Ekaterina, Grace, and Melissa
Fei, Van, and Jessica
Dee, Kim, Daniel, and Sam
My comments about the first two
First: Fred, Varun, David, and Craig, I
downloaded your file to the wrong computer and it'll take me a while
to retrieve it. But thanks for sending it; I look forward to
including it.
Now. Here's a solution from seventhgraders
at Pershing Middle School, which appears to be in Houston, Texas. I
have bolded and blued an especially wonderful observation about
306090 triangles that, if these kids remember it until they're old
and gray (or at least in gradualte school) it will do them
well!

Explanation given by: Anne, Jane,
Ekaterina, Grace, and Melissa from Pershing Middle School in the 7th
grade Algebra class.
To find the width of the river, pick a place for Dan, by the
bank; everything else should revolve around him. Make a map scale,
preferably a simple one such as 1cm=5m and construct the rest of your
picture in accordance with the scale.
Find out what distance Mike walked from Dan by using the
equation 21/5x4.6. 21 is the number of steps Mike walked, Vicky found
that 4.6 was the length of 5 of his paces, so you must find how many
5 paces are in 21. Put Mike where he should be, by putting your
answer in accordance to your scale. Draw Charlene next to him.
Using a ruler, draw a line from Dan to the north, the line
will help you find the location of the tree later on.
Measure with a protractor, 30 degrees to the west from
Charlene's north side. Draw a straight line, so you create 30 degrees
angle from Charlene. Where the line from Dan and the line from
Charlene meet is where your tree is.
Measure the distance between Dan and the tree in the
measurement of your scale, and multiply that number by the amount of
distance your scale represents. The product is the width of the
river. Your distance from
Charlene to the tree should be twice the size from Charlene to Dan.
Pictures that go with explanation are attached to this
email.
We had trouble understanding what west of north in the clues
because we did not know what that meant at first. We thought that you
must measured 30 degrees north of Charlene and where that angle
and Dans sight meet is where the tree was. We also had
trouble at first making the map to scale because we made the river
first and then put Dan and Charlene on the sides of the river. Then
we realized we had to start with Dan and then work our way to make
Charlene, the river, and the tree all in the correct
places. We assumed that Mike and Charlene were approximately in the
same place so we just put Charlene on the map and not Mike. Answer:
The width of the poison acid river is 32.2m

WOW!
And there's a diagram in a file I haven't
been able to decode yet.
And no sooner did I get this solution than
another came in from the SAME school. It's a lot of reading, partly
because these are visual problems and the writers are describing a
drawing.
Isn't it amazing how hard it is to write
about a picture?
Does that tell you something about how
important it is to make diagrams?
But here's what's really cool:
FIRST, they say, "we do not know
trigonometry." Good. You did brilliantly (both groups) at using a
terrific strategy, namely, making a scale drawing and
measuring.
SECOND, their answer is a range of numbers.
THIRD, their range does not include the
answer from the group above.
So a question for anyone reading this: did
either group make a mistake? If so, what was it?

Dear Tim:
We are a group of seventh grade, Algebra 1 students from
Pershing Middle School, Houston Independent School District. We are
from Houston, Texas. Our names are Fei, Van, and Jessica. By using
proportions, we think we can find approximately the width of
Poisonacid River. (We do not know trigonometry.) And we would like to
describe to you how we got the approximate answer in the following
paragraph.
By reading the clues, we were more and more confused. So we
decided we would draw a picture to represent what's going on.
A line is drawn from west to east to represent the one bank of
the river. We did that because one of the clues said, "Where they
are, the Poisonacid River runs east and west." So we made the left
and the right of the paper east and west.
The next clue we read was "Dan stands directly across from a
tall cedar tree they see on the opposite bank. He uses his compass.
The tree is due north." In this clue it told us where Dan is
standing; across from the cedar tree. However, we do not know where
exact the cedar tree is, so we decided we would keep this little note
in our mind, and come back to it.
The next clue we read was "Charlene is standing on the south
bank of the Poisonacid River helping her friends figure out how wide
it is." So we plot a point on the line we drew earlier, and named it
point C, which makes that line the south bank of the river, that's
where Charlene is standing. So that's the south side of the picture.
If the south is the bottom part of the picture, the north is going to
be the top part of the picture. So east would be on the right of
north (which is right portion of the picture,) and west would be on
the left of north, (which is the left portion of the picture.) The
clue also said "When Charlene uses her compass, she sees that the
cedar tree opposite Dan is 30 degrees west of north from her. That
is, the angle between north and the direct to the tree is 30
degrees." This clue sort of confused us a little bit, we did not
understand what 30 degrees west of north meant. By using the bottom
part of this clue, we figured out in awhile what it meant.
So we put our protractor on the south bank of the river, the
hole of our protractor was on top of point C, (because that's how far
away the cedar tree is from Charlene, not the south bank,) and made
sure that the 90 degrees angle was pointing north, and we marked 60
degrees on the left of 90 degrees, and named it point T. Since it
said 30 degrees west of north, we thought it meant 30 degrees left of
north, because west means left. Now we know where the cedar tree
stands, now we also know where the north bank of the river is. So we
drew a line that goes through point T, representing the north bank of
the river.
Then our group went back to the first clue, where it said Dan
was standing directly across from the cedar tree. We put our ruler
where point T was, to drew a straight line that touches both banks of
the river, and named the end point that touched south bank point D,
because Dan is standing directly across on the opposite bank of the
cedar tree. Since the cedar tree is on the north bank, Dan must be
standing on the south bank. Now we know where Charlene, Dan and the
cedar tree is.
The next clue said "Mike paces 21 steps from where Dan is
standing to where Charlene is standing." This clue told us that Dan
and Charlene are standing on the same side of the bank, however from
the diagram we drew, we already know that.
The next clue said "Vicky measured 5 of Mike's paces with a
five meter tape. Altogether, they were 4.6 meters." In this clue, it
only tells you the distance of five of Mike's paces. We needed to
know the distance of all Mike's paces. We divided 5 into 21, we got 4
and reminder 1. So if there is 4 set of 4.6m, if you multiply them
together you get 18.4. There is one left over, so we used 4.6 divide
by 5, to find out the distance of one of Mike's paces, we got 0.92m.
Then we added 18.4 and 4.6 and got 19.32m. Since 19.32m is the actual
distance that's between Dan and Charlene, we decided we would use
proportions to solve this problem.
We took out our ruler to measure the distance between Dan and
Charlene, we got 3.9cm. So we put the distance on paper over the
actual distance, we got something that look like this: 3.9cm/19.32m.
Since we need to find the actual distance of the river, we used
proportions to find the answer. Since we don't know the
line(represents the width) that touches the south and north bank of
the river, we named it X. And we took out our ruler to measure the
width of the river on paper. We got 6.7cm. By using the exact
proportion we used last time, I got 6.7cm/Xm. We did not convert the
meters to centimeters because it doesn't matter, whatever we get for
X it has to be meters. If we converted it, whatever we get has to be
centimeters. As long as the top of both proportions have the same
unit of measurement, and the bottom of both proportions have the same
unit of measurement, we won't get messed up. So we got a proportion
that looks like this: 3.9cm/19.32m = 6.7cm/Xm.
Then we used 19.32 to multiply by 6.7 and got 129.444, next
divided by 3.9, we got approximately 33.19m, which is the
meansextremes property. So the width of the river is 33.19 meters.
However, one of our group members did this problem many times, see if
she gets the same thing. However, she got very close measurements.
From about 33 meters to 35 meters. So we think the width of
Poisonacid River is about 33 meters to 35 meters.
Sincerely yours,
Fei, Van & Jessica

But wait!
There's more! Yet another group sent this elegant explanation. I
think this is a great example of choosing well what to include in the
writeup. This does not mean that a more exhaustive writeup (like the
previous one) is bad, but this is shorter without sacrificing any of
the important math. A teacher can tell that you understand the
problem.

Solved By:
Dee Prohorenko (age 13, grade 8)
Kim Angwin (age 13, grade 8)
Daniel Cheng (age 13, grade 8)
Sam Mo (age 13, grade 7)
In: Mrs. Buan's 7th and 8th grade algebra class at Pershing
Middle School (Houston Independent School District)
To find the distance from Dan to the tree we first read the
given clues. Then we sketched out a visual aid so we could see the
information. This sketch was not to scale, but after rereading the
clues and looking the sketch over we quickly set to making one that
was to scale.
We first placed point D (Dan) on the paper. Then, using the
fact that there were 21 steps between point D and point C (Charlene)
and that every five steps equaled 4.6 meters, we calculated that the
distance between them was 19.32 meters.
After using this calculation we drew a three inch line between
them. Therefore, making 3 inches equal to 19.32 meters. Then, knowing
that point D would be turned into a right angle (angle D) but not
knowing how far north (towards the tree) to draw it, we drew a long
line in the northern direction.
After rereading that Charlene had discovered the tree opposite
Dan was 30 degrees west of north from her, we knew point T (the tree)
would be turned into a 30 degree angle (angle T). Since a triangle
contains 180 degrees, point C would be a 60 degree angle (angle C).
After measuring drawing and remeasuring, we agreed that the triangle
was to scale and we would be able to set up a ratio to determine the
distance from Dan to the tree.
We compared line CD to line DT and set up the ratio 19.32 to 3
equals x (the unknown distance) to 5. Five being five inches (the
length of line DT) and three being three inches (the length of line
CD). After solving the problem fist in metrics, we then checked them
in both the metric and customary systems. Our answer which checked
out properly both times is 32.2 meters.
Thank you for your time,
Dee, Kim, Daniel, and Sam

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