Talk:Ambiguous Case

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Contents

Response to Checklist

This page is a helper page. I therefore used the Checklist for writing helper pages.

NOTE: This response to the checklist just addresses the content part of the page up through Teaching Materials. This week I hope to be working closely with Chris, Ann, Diana, etc. on the actual Teaching Materials section. This is the site's first page to include such materials, so one goal is to make this a sort of an experiment. Anna/Chris/Steve, I ask that you only review the content part of this page for right now.

Messages for Future

I think that the real value of this page is the Teaching Materials section that is going to happen. Other than that, I think that the content of the page itself is a solid, thorough description of the ambiguous case and I don't see much that could be added content-wise.

References

All images were made by me and the applet was made by Reza.

Quality of prose and page structuring

  • This page is set up so that the top part and description outlines the rest of the page and introduces the topic thoroughly.
  • The main goal of this page is to set up a geometric perspective for a topic grounded in trigonometry.
  • The sections are strategically set up so that the most difficult scenarios are at the end. Each section individually is also easy to follow and as clear and spelled out as possible.

Integration of Images and Text

The images, table, and applet are all necessary and useful parts of this page. They are clear and used in the description of the content.

Links to other Pages

All of the pages that link to it are listed.

Examples, Calculations, Applications, Proofs

Examples, visuals, and proofs are given throughout the page. This page is really at a basic level discussing a geometric proof for the ambiguous case--why there are different solutions for different side lengths.

Mathematical Accuracy and precision of language

The math is clearly explained when necessary. It has been checked for errors time and time again.

Layout

The paragraphs are short and the page effectively includes bold phrases and several balloons.

The page has been tested in different window sizes and has been tested to reduce white spaces and appropriately fit the titles and pictures.

I've gotten a lot of feedback on this page and I've really taken the time to sit down with this and make sure it is clear and concise. I'm really excited to get to work on this page as a teaching material. I think there's a lot of potential in this page for that reason.

Teaching Material Comments

Comments from Steve Weimar, the Math Forum 6/28

  • It appears to be a demonstration rather than an activity for thinking and investigating. I imagine that is the current standard or expectation for these pages. I wonder if we considered activities where we, for instance, have students use materials (spaghetti) and ask them to investigate the triangles that can be made given certain conditions (what is known). Then offer the board setup for students to come up and share their thinking.?
I had the word "demo" or "demonstration" on the page one or two times but changed it for consistency to "activity". What is up there now is more of a demo than an activity, and I think it's feasible to modify the demonstration to become an activity. Does anyone have any thoughts???? Richard 6/29
Chris 7/1 Per my e-mail, you might offer two prospective lessons from which a teacher could choose. The first would be your present lesson which is more of a demonstration. The second would be more of a student exploration activity based on the Geometer's Sketchpad lesson Triangle Congruence.


  • tape on chalk boards can be unstable/not adhere enough. I wonder if there is a type to recommend that both sticks and doesn't leave tape on the board. Chris, other methods you have used for attaching string to a board? Possibly attaching something at the top? Many schools use whiteboards and smartboards. Might be worth also describing an alternate method for those with software, etc.

Ideally, this page will ultimately include an applet. maybe magnets would work too? Richard 6/29
Chris 7/1 Magnets work well. I think the teacher can be trusted to devise something that works in this case.
Gene 18:52, 30 June 2011 (UTC) Richard, I want to encourage you very strongly to learn enough Geometer's Sketchpad to do this applet, and perhaps other things as well. I see other useful possibilities for interactive diagrams, and this would likely be a very valuable tool for you to have.
Gene, do we have anyone who knows GSP well enough? I've sort of come to a point in my exploring where I can't seem to get much farther. Richard 7/5


  • AnnaP 6/26 You also want to provide some suggestions for discussion questions. Let me (or Chris) know if you want more help creating these. You could also suggest to have students do this on poster board to be used later on in the class to remind kids about the discussion.
Diana 6/27/11 10:57 This section would feel much more fleshed-out and useful to teachers if you included something like an "extensions" section where you gave ideas of what to do with the activity. That is, ways to lead it into a discussion or an experiment or other geometrically ambiguous scenarios. Whatever you come up with, it would feel more like a "finished" section if it gave an indication of where to go from here.
Diana and I discussed that this demo/activity needs some sort of context, but also needs to maintain its accessibility to be inserted into different lesson plans. Richard 6/29


  • Conversation with Ann 6/29:
  • Use Suzanne Alejandre's lesson plans as a potential example (Math Forum) http://mathforum.org/alejandre/index.html
  • Goal: Make this become a more interactive activity. Make students ask the questions.
  • Context: Don't necessarily write a lesson plan, but you could. Draw from past experience. Different activities help different people
Sometimes a lecture/demo helps more advanced math students more, and activity helps more struggling students more.
  • Ideally the page would have an applet, but if there can be no applet, perhaps a video will be useful.


  • xd 7/6 one little thing in the teaching material. you might want to change 'ASS' to 'SSA' ? otherwise, i think the page is great


  • Additional comment from Steve W e-mail 7/5:Btw, I wonder if there isn't a mismatch here between the reader for whom the chalk-board exercise is necessary to grasp the first part and the reader who benefits from the Law of Sines explanation? I wonder if we want to catch a less developed reader and drop them into the teaching example before they encounter the trig formulas?


  • Conversation with Diana 7/7: Pictures in lesson are in a different order than instructions.


  • Notes on Suzanne Alejandre Lessons 7/6
  • Puts up lessons/demos/activities/all different types of resources
  • A lot of the lessons use both a technological and a simpler hands o demonstration. This makes a particular lesson or demo accessible for all different types of teachers.
  • Her Lessons are really focused on problem solving, having the students do the problem solving
  • Some ideas for ambiguous case
  • an applet will provide the technological interactive alternative like
  • The teaching materials section can really be about the hands on activity that promotes inquiry based learning



  • Conversation with Chris 7/13: talked about getting rid of demo and replacing it with hands on activity, but Chris suggests to keep demo, it will allow teachers to take different things from the site
  • Need to change the one solution oblique picture...

Page Comments

  • So I was thinking about it today, and none of these cases seem to work when the fixed angle is obtuse. In this case, there is only one solution or no solution. Same goes for a given right angle. I guess this is a section I should write up? Richard 6/30
Chris 7/1 I wouldn't bother. It doesn't come up in anything I've ever read about Ambiguous Case and it's fairly intuitive as to why it's not ambiguous once you start working with it.
Abram, 7/5/11:It is probably important to note somewhere that we are only addressing in this page the case where the given angle is acute.
Mentioned that A is an acute angle for all of the scenarios in the top general description. Richard 19:44, 12 July 2011 (UTC) 7/12


  • This appears to be fairly remote from one of the featured images. Is that true? Are we also working on lessons that directly connect to the images?
This is a helper page and not an image page, though I was thinking: should there be some sort of "main image" for this page? Richard 6/29


  • In the solution discussions one of the One Solution situations is not there and I wondered if that was on purpose. You do address it in your Teaching Materials Instructions in the paragraph that begins "First".
AnnaP6/26 I want to echo two of Chris's points that he made. Your activity is well laid out, but you do present a case that isn't exactly covered elsewhere. This case isn't truly a "Angle-Side-Side" solution since one of the angles becomes an exterior angle. It's an interesting case to explore, but make sure you explain it well in the main page.

Chris 6/26 My main suggestion involves the "solution" in which the swinging side length is greater than the fixed side length. This scenario is not discussed in the main section itself but is presented in the Ambiguous Case Activity. You then include a "Sample Picture" of that scenario as one of the pair of two solutions. While it is a scenario for what can be done with the string, it is not actually a possible solution since one of the two solutions does not include the fixed angle. It makes sense to allude to it (I'd do so both the main section and the teaching material) because you can extend the string to that length and make a triangle out of it, but it is also important to note that it is not a valid solution because of the fixed angle.
This is also a comment from Prof Maurer that is on my list of things to do. Right now, there are pictures for the activity that include this case in the "one solution" section.Richard6/29
I (at least partially) explained this case in a fourth scenario section. Richard 7/5


  • The language is sometimes difficult to follow when describing the situation: " the upper point of the other side " partly because the points are labeled, etc. There may be other ways without using labeled points, but the current language is sometimes hard to follow.
This mirrors a comment from Prof Maurer. He was thinking I should try and label the sides/vertices, but he also sees why I can't label an undefined vertex. I think I'm going to have to pick a phrase for each part of the triangle and stick with it. Staying extra-consistent for the purposes of this page??? Richard6/29

Abram, 7/5/11: In the "no solution" scenario, you have this really nice sentence: "In the picture below, no matter how the orange side swings, it will never touch the base of the triangle. This triangle will never be complete." You might be able to get rid of some of the clunky language by mirroring that style. For instance, in the second paragraph of the page, "This means that this third side can be positioned..." could be replaced with "you can swing this third side side on its "hinge" to any position where it ends somewhere along the dotted base".
Addressed this comment. Richard 7/13


  • there are explanations where I wondered whether a student reading it would grasp the issue and the "proof": "the height can always be determined because the furthest vertex from the base is known" In general I wonder whether we should be exposing some of the questions that led to the exposition that follows and encouraging the reader to notice and wonder first before we explain: "what are the possible cases for this situation? what can we use to define the cases? If we use the height, can we always determine it?" Then : "the height is the perpendicular distance from the vertex to the line containing the opposite side. In this case we know both the vertex and the line containing the opposite side, so we can always determine the height."
  • Prof Maurer suggested different ways to organize the page. This may be the way that I choose: introduce height first and include the scenarios in the description, then answer the questions when I discuss each scenario.
  • In regards to the teaching material, this perspective may be extremely useful in making discussion questions or something Richard6/29

Abram, 7/5/11: It seems like what Steve W. is getting at is not about the order, but about framing the question and the observations more clearly. The style of this site hasn't seemed to include the kinds of priming questions Steve describes (though that's not been an official decision, and it is an effective way to frame things). But either way, you can write things like, "As you can see in Image x, we can draw the height of the triangle even though we don't know the length of the base. Below, we will see how we can determine the number of solutions by comparing the known length of the swinging third side to this known height of the triangle." Sentences like this indicate to the reader what you are establishing, reminds them what you are ultimately trying to do, and signals how this current fact is related to the end goal.
Addressed this comment. Let me know if you think I should do more. Richard 7/13


  • the initial trig formula for determining the height could be more clearly motivated (and maybe later in the text), and at some point we could discuss the nature of the sin function and why it can generate equivalent values at different angles.
Asked for clarification 7/5
E-mail response from Steve W 7/5

I guess there are a couple of different ideas in that comment. Sorry. The first is probably mostly about whether the (average K-12 teacher and student) reader has enough help to know what the text is doing and to follow the development or changes. How would you label the text if you were presenting an outline? Right now there is nothing in the text organization or formatting in the top section that helps a beginner follow the shifts:

SSA Postulate: what we can't know

Why is that? Explanation/Informal proof

What can be known from SSA?

How can one figure out which case it is?

I think the trig is being used to answer this last question but I am not sure many readers would know whether you are still proving the postulate or shifted to describing a procedure for what you can do in this situation. It could be that some sort of labeling of the paragraphs or even just posing questions that the subsequent text answers might help. And, it might be useful for some readers to read a sentence that points out why trig is needed/used in such situations.

The second point was a kind of free association that if we are going to the trouble of introducing trig here, would it be useful to at least pose a thinking question about the sin function and how the one solution-two solution outcome is reflected in its cyclic nature?

Had a conversation with Abram on 7/13 about shifting the focus of the actual content of the page to be more geometrically based rather than trigonometric. Added a table and hid the height section to make it more clear and based in geometry. Richard 7/14


  • in that trig exposition you use the word "values" differently than I would. We tend to say, when working with pre-college students, that there are quantities and values. There are variables to represent the quantities and the values are usually numbers calculated for those quantities. I might be inclined to say that you are using trig to express the quantity "height" and to calculate its value in specific situations. Then I might say that you are writing the expression using the variables used in your diagram.
Changed to variables. Richard 7/13


  • in the solutions discussions you don't label the "a" side length on the diagram which would help interpret it when used in the narrative.
Changed the labels. Richard 7/13


  • "just solve for the first triangle normally" I think you mean something like "using the Law of Sines" where you write "normally".
This content was directly taken from my Law of sines page. I'm going to have to add some more to make it be able to stand on its own legs.Richard6/29

Changed the text. Richard 7/13


  • Richard: The reader can clearly understand SSA ambiguous case from reading this page and both student and teacher alike will find the teaching activity most useful in mastering the concept. I plan to use your page with my 9th grade Geometry students next year and demonstrate the concept using the Ambiguous Case Activity.
Strong points of the page:
1. The text sections are short, clearly written, and accessible and alternate with strong visuals that clearly illustrate the concept.
2. Using 30˚for your angle and 10 for your fixed length and are strong choices so that the side length for one solution is half of the fixed length (hypotenuse) and 4, 5, and 6 can be the numbers used for the various cases.
3. The teaching activity uses easily accessible materials that take very little time and effort to set up yet demonstrate the concept clearly. It's interesting that, since you are cutting the string a bit at a time, the order of solutions presented in the activity is opposite from those presented in the main section. I think that it's fine to have a different order, it's just interesting that the physical constraints of the activity make the opposite order so much more sensible.


  • One other thought involves the title: Ambiguous Case could refer to many things in math. Would it make sense to specify it such as Ambiguous Case: SSA in Triangles or something like that? Chris Taranta 6.26.11
I've seen it commonly in some of the books as the "ambiguous case of the law of sines", which is why I originally had this content on the Law of Sines page. The only ambiguous case I ever remember learning was this one, and it's what comes up when you do a Google search. Maybe you could point me to a different Ambiguous case? I hesitate to make a longer title with more than one part since almost all of the other pages on the site have a short title that is to the point. I'd like this title to be the same way, but I want to also be sure that the topic of the page can definitely be known from the title. Richard 6/29
Chris 7/1 I looked into this and agree with you. Ambiguous Case is a fine title.


  • Dayo made an edit to the page to fix a grammatical error on 6/29


  • Kate 15:30, 1 July 2011 (UTC) I guess this is a more general comment, but I'm afraid of the comments above - does this page really need to be this long? None of what you have is wrong or confusing, and I like the pictures, but it seems like you could say all of these things in a lot less space.
I columnized (is that a word?) the page to make look shorter. The organization of the page was making too long. Richard 7/5
Yeah, it's definitely better. I think columnized should be a word, but the computer seems to disagree.


  • Kate 15:30, 1 July 2011 (UTC): How come there's a whole section before the table of contents? It makes the page look like it's just this section long, which it isn't.
Abram, 7/5/11: One final thought: there seems to be way too much content before the Table of Contents. Why not put it after the table of contents?
Kate showed me how to do that today. Fixed. Richard 7/6


  • Kate 15:30, 1 July 2011 (UTC): This means that this third side can be positioned in whatever way connects the upper point of the other side
I think I'd say "in any way that connects" just to be clear that it's not just one way.
Coolio! Comment Addressed Richard 7/5


  • Kate 15:30, 1 July 2011 (UTC): # If a > h, there are two solutions.
You should say If b>a>h, right? so that it's clear that this doesn't overlap with the next case.
I tried to organize so that this was an extension of the of the case with four scenarios. I feel that that organization is pretty logical. Richard 7/5
I disagree. Right now, it says "If a > h, there are two solutions. If a > b, the solution is a single non-right triangle.", which is logically inconsistent. If b > a > h, then your first statement claims that it has two solutions yet at the same time your second statement claims that it has only one. The way you actually talk about the cases later, it makes sense that the second is an extension of the first, but in this list, I think you need to be accurate. (Kate 17:48, 6 July 2011 (UTC))
Addressed this comment. Richard 7/13


  • Kate 15:30, 1 July 2011 (UTC): Kate 17:45, 1 July 2011 (UTC): Typo in the "determining Both Solutions" section- To find both triangle, just solve for the (Should be both trianges)
Thanks!!! Comment Addressed Richard 7/5


  • Abram, 7/5/11: You do this great thing right before the table of comments, which is summarizing all the possible results in one place. The two problems are (1) you don't mention the case a = b, and more importantly (2) it is buried in the height derivation. This is a summary of the whole page! Display it prominently. Even make a table of it, with headings like "Scenario" (1, 2, 3, 4), "Condition", "Number of solutions", maybe "Summary picture" (or not).
I keep ending up with these kinds of comments...oh boy. Richard 7/6


  • Your pictures are fantastic. I could imagine getting a bit confused by the first image and not realize that those three *possible* third sides, not three sides that are all there. A caption would take care of that, or would a reference to "swinging the third side in Image 1" in the text.
I talked this over with several people. The current status is always to refer to that side simply as swinging side. No matter which image we're talking about, it's always referring to the same side. Richard 7/6
Addressed this comment. Richard 7/13


  • My one other question about the first image is if it maybe makes sense to refer to the "unknown" measures as something like "unspecified" instead, to mirror the idea that the problem allows those measures to be anything that will "work".
My first inclination is to leave as is. To me, "unspecified" sounds like there are certain known possibilities/options. That sounds more fitting to describe the position of the swinging side rather than a length or a measure. (Am I making sense????) Richard 7/6


  • Another nice thing you do is saying things like "In the picture below, [nice explanation of what reader should notice]" That is great integration of images and text. It's just that the picture won't always be below, depending on the browser window size! Or it could be really far below, or... That's why we've suggested that images get anchors using the Image template, and that you write "In Image x", with "Image x" linked to that anchor.
The page is set up so that the picture will always be below. Richard 7/6


  • User:Rebecca 00:30, 8 July 2011 (UTC) I reread the page, and I think it is looking much improved! I love the addition of the demonstration and the new pictures. I do agree with Abram's most recent comment about mentioning that we're dealing with an acute angle in this page.
Comment Addressed. Richard 7/18


  • Comments from Cathy Stambaugh, teacher at Strath Haven High School 7/11
1, Towards the beginning you define tan = opp/adj. I would say: tan of the reference angle =....
2. Towards the beginning you imply that you can use the law of sine when given any three elements of a triangle. I think that you need to say: ...any three elements of a triangle, no three of which are the same type of information.
I think these comments have more to do with basic trig functions page. Richard 7/18


  • Chris 7.16.11 The page is very strong. Here are some final edits:
Opening Paragraph: I wouldn't highlight vertex with a link to a definition, given that it's a standard geometric term and that you haven't done it for anything else (oblique, for example).
Comment Addressed. Richard 7/18
The table is a very good idea, though it seems large for the page.
Comment Addressed. Richard 7/18
Remove "to compute height" in the very last sentence.
Comment Addressed. Richard 7/18
First Scenario: Two "the"s in sentence 3.
Comment Addressed. Richard 7/18
In this and the other scenarios, the title (in this case, "No Solution") seems to be in no-man's land. Since h = b sinA is part of your solution, I would either have the title above that line or would have it further to the right to serve as a title for the diagram.
Third Scenario: Fixed "is are" in second to last sentence. Should this be titled "Two Oblique Solutions" for consistency?
Both solutions don't necessarily have to be oblique. Say angle C is 90 degrees... Richard 7/18
Determining Both Solutions: Put a link to "Law of Sines" in third sentence.
Comment Addressed. Richard 7/18
There is no angle labeled B to correspond to your equation.
Fourth Scenario: Should this be titled One Viable Solution? This is tricky because of the particular issues involved, but both solutions are oblique, it's just that one of them is not viable.

I tried to explain that within the paragraph that there is only one solution of two completed triangles, and not two solutions but only one that works. (I had trouble wording that...I think I make sense????? Richard 7/18


  • User:Gene 7/15 In First Scenario, you might say something for the 2nd paragraph like "In the picture below with the numbers we've chosen, no matter ..." since otherwise it's not clear where the numbers came from. This sets the stage pretty well, I believe, for the other scenarios, too.
Comment Addressed. Richard 7/18

Applet Comments

  • For the applet you suggested for this page, Reza said he would begin working on a Java applet as per the specifications listed on the S11 page - Rguo - 6/30



Older Comments

Originally, the page was a subsection of the Law of Sines page. After much discussion with Prof Maurer, Harrison, Gene, and others, it was decided to make an entirely separate page that can stand on its own. The law of sines page was too long with the ambiguous case section attached. There was enough content related to the Congruent triangles page that the ambiguous case could be a helper page for the two pages.

The following comments were (and still are) on the Law of Sines Discussion Page:

The ambiguous case

I want to make sure that this part of the page is very clear. If people could read this over that'd be great! Richard 5/24

  • I think you have an error in your first paragraph of this section. It's true that there is an unknown length and two unknown angles, but the swinging side cannot be connected to any point along the dashed side. It can be connected to one of two possible points corresponding to the angle, not to any point along the base. This might not have been what you mean, but i don't think the section or the picture is clear.
  • You should make the fixed length swinging side long enough so that it touches the base in two places in the picture I think, since this is the most common case.
  • I think you need to label the parts of this triangle in the picture as well. Its too hard to keep referring to things as the "third side" or the "base."
  • The end of this section is very clear!

Rebecca 01:43, 25 May 2011 (UTC)

First Scenario: No solution

* First sentence is very confusing. You have too many fragmented thoughts.

  • Second Scenario: One solution & Third Scenario: Two solutions sections are very clear!

Rebecca 01:44, 25 May 2011 (UTC)

edited first sentence Richard 5/25


xd 02:02, 25 May 2011 (UTC)

1. You need better transition between the previous section to this section. This should not be an independent section by itself.

I was actually thinking that this shouldn't be in the mathematical explanation section at all. If I move this and the example sections out, I think the remaining sections would leave just a mathematical explanation and the ambiguous case and the example would be more about computational aspects of the law of sines. Richard 5/25

2. I think the determinant of which kind of solution, i.e. 0, 1 or 2 is the swinging side with the fixed length instead of the height of tghe t

2. Determinant of solution -> there is no triangle to start with. So don't say "height of the triangle". Say distance between the vertex and the base line as shown in the picture.


  • I added a few words to show that the swinging side compared to the height is the determinant
  • but it ultimately is the height of the triangle. Would it be okay if I add a sentence or two to explain that?


Richard 5/25

Extra Picture

Image:Ambig_cas_act1.jpg

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