Poisonacid River
(page 180)

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 seventh-graders at Pershing Middle School, which appears to be in Houston, Texas. I have bolded and blued an especially wonderful observation about 30-60-90 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 e-mail.

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


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 means-extremes 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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