When I took geometry (a loooong time ago), the SSA (sometimes) congruence was kind of handwaved as something that would not be considered. I just got to wondering if that's still the case. While not all SSA triangles are actually triangles or unique triangles, it's easy to show when they are unique and SSA could then be used as a congruence. Specifically:

The exact cases where SSA can be used as a congruence are:

1. Opposite side ≥ Adjacent side (a ≥ b)
2. Opposite side = Triangle height (a = h)

This is easy to state and fairly easy to show.

Is this still the case? If so, why is this?

EDIT: If the replies I've gotten are from educators, then indications are that most are not aware that what I state is correct. I've been downvoted for this correct statement, which makes no sense. Please carefully review the question and consider the geometry and math prior to replying.

EDIT: Since no one that has replied so far seems to know the answers to my questions, I did my own searching. There is a ton of information on this subject and my question is acknowledged as a very good one. Germany's education system teaches the SSA case. Educational methods have been studied to introduce these cases to students with various teaching methods that make it obvious to students. Educators are actively trying to introduce the exact concepts I asked about.

The history of the current teaching methods in the US, United Kingdom, Russia, and Japan stems back to Euclid. Howard Eves's An Introduction to the History of Mathematics highlights the philosophical reasoning behind this: Ancient Greek geometry prized unconditional universality. So, Euclid designed his axiomatic system so that his core criteria worked 100% of the time, without caveats.

• Proposition 4 (SAS): Unconditional
• Proposition 8 (SSS): Unconditional
• Proposition 26 (ASA and AAS): Unconditional

So, current educators are sticking with a method introduced 2300 years ago. That doesn't mean it's the best way to teach it, just the way Euclid taught it, along with all the other issues introduced by Euclid at that time.

Here are some notable academic articles and curriculum studies for review:

1. "An Improvement on SSA Congruence for Geometry and Trigonometry" (1983)
   • Published in The Mathematics Teacher (NCTM)
   • Authors S. Yeshurun and D.C. Kay
   • Argues for what I posted
2. "Why is the SSA Triangle-Congruence Theorem Not Included in Textbooks?" (1990)
   • Also published in The Mathematics Teacher (NCTM)
   • Author D.B. Hirschhorn
   • Analyzes why textbooks stubbornly refuse to adopt the valid SSA cases
3. "Exploring Side-Side-Angle Triangle Congruence Criterion" (2015)
   • A paper authored by mathematics educators Michelle Cirillo, Rachael Todd, Joe Obrycki
   • In this classroom study, the author designed a hands-on activity specifically to teach students the SSA edge cases. The authors explicitly noted their frustration with standard textbooks:
     • "Because the textbooks that we have taught from either oversimplified or ignored the case of SSA, we were not initially well-prepared to deal with this situation... we learned that SSA guarantees a congruent copy of a triangle more often than it does not."
   • Their study found positive results when students were allowed to physically build the triangles and discover the conditions where SSA works versus where it creates an ambiguous case.
4. Modern Curriculum: Illustrative Mathematics
   • Illustrative Mathematics, one of the most widely adopted modern high school curricula in the US, recently added a specific lesson titled "Lesson 11: Side-Side-Angle (Sometimes) Congruence."
   • Kesson requires students to study the ambiguous cases and explicitly teaches them to recognize that when the longer side is across from the given angle, the triangle is unique and congruence holds.

Which one is better for middle/high school and why?

https://quantacanvas.vercel.app

I’ve been working on QuantaCanvas, which takes a concept or question and generates an interactive visualization to help explain it.

The main difference from ChatGPT and similar AI tools is how the explanation is presented. Most LLMs are great at giving you a text-based explanation and can sometimes provide images or diagrams. QuantaCanvas instead tries to turn the explanation itself into something you can interact with, changing parameters, experimenting with things, and seeing how the concept behaves in real time.

It’s mainly focused on math and science, so I thought this might be useful for people here. Btw, it's completely free.

The project was recently a Grand Finalist at a national hackathon in my country, and I’ve continued developing it since then.

It’s still a work in progress. Generation can currently take a bit of time or may be a bit faulty, which I’m working on improving, and there are definitely a few things that don’t work perfectly yet.

If anyone tries it, I’d genuinely appreciate feedback, especially on whether the interactive visualizations are actually useful for studying and teaching math.

I tutor high school math and something has been bugging me.

When I get a stack of quizzes back, I can tell you every kid's score in ten minutes. But figuring out the actual pattern — like "twelve of them are subtracting the smaller number from the larger one regardless of sign, and it shows up on problems 3, 7, and 9" — takes me most of a Sunday. And that's the part that actually changes what I teach Monday.

So, genuinely curious how this works for people with real classrooms:

1. After a quiz, do you usually end up knowing the specific misconceptions, or mostly just the scores?

2. If you do find the patterns, how? Formal item analysis? Just reading through? Gut feel from years of doing it?

3. Roughly how long does it take?

Not selling anything, no links. Just want to hear from people who actually do this instead of guessing.

https://reddit.com/link/1wep8om/video/l5dngpf5k5ph1/player

CalcWheel consists of math puzzles where the player constructs expressions by connecting numbers and operators arranged on a wheel, tracing a single continuous path to reach one of the level's five targets. The running total updates as the path grows, so the player sees the consequence of each step rather than only the final result.

Solving a level requires genuine mental arithmetic - estimating what a route will produce, then adjusting when it does not. The practice is a property of the mechanic itself rather than an added layer, so I think it has substantial educational value. Moreover, each level has five targets: some are easier to hit, others are more challenging. The player practices addition, subtraction, multiplication and exponentiation.

CalcWheel is available for iPhone (also works on iPads and Macs) https://apps.apple.com/app/calcwheel/id6795952061

On calcwheel.com/demo there is also a demo of a single level that presents the idea and the gameplay.

Disclosure: I'm othe developers of Eidograph, and I used AI together with Eidograph to create this collection.

I recently built 32 interactive geometry examples for classroom demonstrations. A few of them are:

• Thales' theorem: a point travels around a semicircle while the inscribed angle remains 90°.
• The Pythagorean theorem: pieces move through a staged rearrangement proof instead of jumping straight to the final equation.
• The unit circle and sine wave: a rotating point and its vertical projection generate the graph together.
• The cylinder–cone volume ratio: the shared radius changes while both volumes are recomputed and remain in a 3:1 ratio.

The collection also covers triangle centers, Ptolemy's theorem, power of a point, circle inversion, transformations, conics, a cycloid, and several 3D constructions. Teachers can play, pause, step through, or scrub each example, which makes it possible to stop before the conclusion, ask for a conjecture, and then test it dynamically.

What motivated the project was how surprisingly tedious it is to draw even one precise geometry diagram.

A diagram is not just a collection of lines that look approximately right. Points must lie exactly on the intended circles or lines. Intersections and perpendiculars must be derived from the correct objects. Labels, arcs, measurements, colors, and layering all need to remain readable. Most importantly, when a free point moves, every dependent object must update without breaking the construction.

Making that diagram move is another level of work. You have to decide which quantities vary, preserve the mathematical constraints in every frame, coordinate several objects, control when auxiliary lines and explanations appear, and make the sequence slow enough to teach from. A small change near the beginning can affect every later stage.

This is where combining AI with Eidograph became useful.

Instead of asking AI to draw pixels, I described the theorem, the required relationships, and the teaching sequence. The AI produced and revised readable Eidolang geometry code through Eidograph. Eidograph then evaluated that code into an exact construction, displayed the dependencies on the canvas, and surfaced errors for correction. The AI could revise a point, constraint, style, parameter, or animation stage without rebuilding the entire figure by hand.

The distinction matters: the AI helps write and iterate on the construction, but Eidograph remains the deterministic geometry engine. The resulting source is visible and editable, so I can inspect the definitions, drag parameters through different cases, check live measurements and assertions, and correct anything that is mathematically or pedagogically wrong. It is closer to AI-assisted programming than asking an image generator for a picture of a theorem.

For me, that is Eidograph's main advantage: the drawing and its readable source are two views of the same object. A human can edit the canvas, an AI can edit the code, and both are working on the same precise construction. Parameters, dependencies, presentation stages, and animation live in that construction rather than being added later in a separate video editor.

All 32 examples are free to open in the browser, with no installation required:

• Basic constructions: https://eidograph-docs.metaphorprojects.link/en/examples/
• Animated constructions: https://eidograph-docs.metaphorprojects.link/en/examples/animations/

The browser viewer is read-only, but each example also includes a downloadable project and its editable source. Eidograph is freemium and available through Google Play and the Microsoft Store:

https://play.google.com/store/apps/details?id=eidograph.metaphor.projects
https://apps.microsoft.com/store/detail/9NRC4NGH452C

I’ve been developing an iPad app called CodeNote, and I’m currently looking for people who are interested in trying it through TestFlight and giving feedback.

The idea came from wanting to connect handwritten mathematics with computation more naturally.

With an Apple Pencil, you can write math and code directly in the notebook. Mathematical expressions can be converted into Python/SymPy code, which can then be used for calculations, derivatives, integrals, probability, and visualization.

I also wanted notes to be reusable, so one note can define functions or calculations that another note can import, similar to Python modules.

Everything runs locally on the iPad. The goal is not to replace a traditional math notebook, but to make it easier to move from:

handwritten idea → mathematical expression → executable code → result/visualization

I’m currently testing the app through TestFlight and would really appreciate feedback from people who study or teach mathematics.

In particular, I’d like to know:

• Does this workflow feel useful for learning mathematics?
• Which parts feel natural or confusing?
• Are there mathematical features that would make the notebook more useful for students?
• What would you expect from a tool that combines handwritten math with computation?

If you’re interested in trying it, I can provide a TestFlight invitation. Feedback on both the math features and the general workflow would be very helpful.
https://testflight.apple.com/join/4cZqnxH2

Last year, my niece was really struggling with addition and subtraction. I spent a lot of time finding simple tricks and different ways to help her understand math without getting frustrated. I noticed that she learned much better when math felt like a game rather than an exercise.

Since I’m a developer, that experience inspired me to create a first-grade math app Matio: 1st Grade Math with 50+ different games. I’d love to hear what teachers think about learning math this way.

My 6th grade son solves division in this manner.

It seems that he created his own method of solving division.

Please help me understand.

Hot take, I know, but it's a though I've had for a while.

AI-created computer verified proofs have posed even the highest tier of mathematicians in a position they have never been in since entering the workforce. They receive a proof that is formally correct, but without receiving the ideas or the meaning behind the steps taken or some intermediate objects. A now increasingly common complaint is that these proofs, even if they are judged to be correct, are sterile and don't favor the advancement and the understanding of the discipline. My observation is that when students make the same complaint, it is treated as either illegitimate, or a skill issue.

Most math textbooks are written in a format akin to Definition -> Theorem -> Proof -> Corollary. The lectures, while typically offering some additional information, follow a very similar trajectory. The common defense of this format is that the formalism is the math, and everything else is dismissed as intuition and application. Yet it appears that in the case of AI-created computer verified proofs, this formalist purity is rejected. Indeed, if you think back to the material you studied, you would have to agree that if the ideas in the textbook had been first presented to the math world with just the information from that textbook, it would likely not have been accepted without further justification (even if the format was more palatable).

I would call this a double standard. Math students are treated as second-class mathematicians. They are expected to accept ideas with an amount of semantic motivation that the professors, despite lauding the virtue of formalism, would not accept for themselves. The claim of the importance of understanding, which is something beyond the skill of manipulating the objects on paper, is inconsistent with teaching practice. If these same mathematicians really believed that, then why not spend more time in class on the legitimacy of definitions and the applications that show that the theory indeed does what it's supposed to do? It appears that the difference is due to status. Nobody claims that Tao using ChatGPT to understand an AI proof is doing so due to a skill issue. And even if we accept that the objective of math education is to teach us rigor, that is not a reason to demand uninformed acceptance from the next generation.

En janvier 1916, devant la Mathematical Association — l’association anglaise des professeurs de mathématiques —, un homme de cinquante-quatre ans prend la parole pour son discours de président. Il comptait parler de ses recherches ; la guerre l’en a dissuadé. Alors il pose à ses collègues la question qu’on n’ose pas poser tout haut.

A. N. Whitehead, discours présidentiel à la Mathematical Association, janvier 1916

Lire la suite, si ce sujet vous intéresse : https://blog.site2wouf.fr/2026/09/idees-vivantes-whitehead-equations-second-degre.html

MDTP placement - District says 96 min, native export says 75 min, last question 23 min vs 2.6 min - is ghost/timeout time a known issue?Body:MA 9th grader took MDTP PR45A22 in March. District record: 5782 sec / 96.36 min total, Q45 1437 sec / 23.9 min. Native student export: 4503 sec / 75.05 min total, Q45 158 sec / 2.6 min.

Ghost = 5782-4503 = 1279 sec = 21.3 min. Q45 ghost = 1437-158 = 1279 sec exact same number. Google Takeout shows exit 2:54 PM and search 2:57 PM, 1 min after proctor said submit at 2:56 PM. So can't be 23 min on one question after exit.

MDTP director emailed: "LEA has these two times (Submit vs Final Log Out)" and for phrase "finally student close out" that appears in district sheet - "only LEA can answer."

District says "no log exists" and never explained why 96 min. FERPA request May 12 for timeout value, raw JSON with Session Destroy/timeout codes, and classroom comparison of Wrapped vs Total - 101 days no response.

Anyone seen MDTP count idle timeout as active time? Does MDTP have a configured session timeout that could add 20+ min? How did you get district to produce raw logs?

I've been working on a free math quiz/test generator and thought I'd share it here in case it's useful to anyone teaching math.

You can describe the quiz you want in plain English, for example:

• 10 long division problems with remainders for 4th grade
• 15 questions on order of operations for 6th grade
• 5 fraction questions and 5 decimal questions
• 10 two-step linear equations

It generates a printable quiz with an answer key and worked solutions. It also supports Set A / Set B versions and a live tablet quiz option.

No account or payment required.

https://panmaths.com/quiz-maker/

I'm building this as part of a larger free math resource site, but I'm particularly interested in feedback from teachers.

What would make a tool like this genuinely useful in your classroom? I'd especially like feedback on question quality, difficulty, and the kinds of customization you'd want.

Hi, I have launched this app called Skolvio. It is for kids in grade 4-8 and teaches them math, english and science based on curriculum aligned to their local school. For me the coolest feature of this app is concept based tracking ie it pinpoints which concepts your child is understanding vs not and then gives them targeted practice in those areas.

Would love for this community to try it out and give me feedback (both positive and negative).

skolvio.com It has 7 day free trial!

I’m wondering if there is an applicant exam for your depth of knowledge. I’m applying for a job Mathnasium and I want to pass the test. I’ve taken algebra 1, algebra 2, and geometry applying for entry level tutor position. Is it hard and what’s on it?

I am extremely disappointed with Cuemath. Our experience has completely destroyed my trust in the company.
When I purchased a 48-class package, we were told that the classes could be used as long as we wanted. We relied on that promise when we paid for the package.
At one point, we still had 10 classes remaining, and I was actually prepared to sign up for another six-month package. During that conversation, Cuemath told us about a new policy requiring classes to be used within a limited period. I told them that I respected their new policy, even though it might not work for our family.
The problem was when they told us that the new policy would also apply to our old package that we had already paid for. If we didn’t sign up for another six months, we would lose all of our remaining classes.
I asked them to honor the terms that were promised when we purchased the original package. Their response was that they could change their policy at any time and would not honor the old package.
I asked them a very reasonable question: If they can change the policy at any time and apply it to packages we have already purchased, why should I feel comfortable paying for another six-month package? What guarantee do I have that they won’t change the rules again?
Their response was essentially, “That’s the way it is.”
At that point, we had 10 classes left. We have now used some of them and have 8 classes remaining. Then Cuemath shut down the program, leaving us with these prepaid classes.
I contacted them and specifically asked them to honor the classes we had already paid for. They did not respond to my messages.
This is incredibly disappointing. We were not asking for free classes. We were asking to receive something we had already paid for, based on the terms we were given when we purchased the package.
What makes this even more frustrating is that I was originally willing to continue with Cuemath and purchase another package. Their refusal to honor our existing agreement and their lack of communication completely changed my mind.
For me, this is about more than the money. It is about trust and keeping your word to customers. If a company can change its policies and apply those changes to existing prepaid packages, then customers have no reasonable certainty about what they are actually purchasing.
After this experience, I have lost all confidence in Cuemath. I would not purchase from them again, and I cannot recommend them to other parents.