Doodlebug Posted August 13, 2020 Share Posted August 13, 2020 We are beginning our work on geometric proofs. See DS's proof below. 1) m<SRT = m<STR; m<3 = m<4 Given 2) m<SRT - m<3 = m<1; m<STR - m<4 = m<2 Subtraction property of equality (**this is incorrect**) 3) m<SRT - m<3 = m<STR - m < 4 Substitution 4) m <1 = m<2 Substitution So far, we've only been introduced to the angle addition postulate, which is why he felt confident in step 2 of his proof (applying the inverse operation off the bat). But he doesn't have the terminology to substantiate it. I think he must begin with an angle addition postulate, m<1 + m<3 = SRT, as I don't see an "angle subtraction postulate" 😉 upcoming. My gut tells me there's a foundational principle here that's critical. But he's my first geometry student and I need to "chew" on this before I declare the way forward -- and I want to affirm the ways his thinking is right on. Thoughts??? Quote Link to comment Share on other sites More sharing options...
forty-two Posted August 13, 2020 Share Posted August 13, 2020 Given what you have, here's how I'd do it. 1) m<SRT = m<STR; m<3 = m<4 Given 2) m<SRT = m<3 + m<1; m<STR = m<4 + m<2 Addition Angle Postulate 3) m<SRT = m<4 + m<2 Substitution 4) m<3 + m<1 = m<4 + m<2 Substitution 5) m<4 + m<1 = m<4 + m<2 Substitution 6) m<1 = m<2 Addition Prop of Equality 1 Quote Link to comment Share on other sites More sharing options...
forty-two Posted August 13, 2020 Share Posted August 13, 2020 (edited) But if you wanted to keep his approach of applying the Subtraction Prop of Equality, you could do this: 1) m<SRT = m<STR; m<3 = m<4 Given 2) m<SRT = m<3 + m<1; m<STR = m<4 + m<2 Addition Angle Postulate 3) m<SRT - m<3 = m<1; m<STR - m<4 = m<2 Subtraction property of equality 4) m<SRT - m<3 = m<STR - m < 4 Substitution 5) m <1 = m<2 Substitution 37 minutes ago, Doodlebug said: I think he must begin with an angle addition postulate, m<1 + m<3 = SRT I agree. I think that his only problem, actually - not *explicitly* stating his use of the Addition Angle Postulate. Otherwise I think he was correct in how he then applied the Sub Prop of Eq to it. Edited August 13, 2020 by forty-two 1 Quote Link to comment Share on other sites More sharing options...
Doodlebug Posted August 13, 2020 Author Share Posted August 13, 2020 22 minutes ago, forty-two said: Given what you have, here's how I'd do it. 1) m<SRT = m<STR; m<3 = m<4 Given 2) m<SRT = m<3 + m<1; m<STR = m<4 + m<2 Addition Angle Postulate 3) m<SRT = m<4 + m<2 Substitution 4) m<3 + m<1 = m<4 + m<2 Substitution 5) m<4 + m<1 = m<4 + m<2 Substitution 6) m<1 = m<2 Addition Prop of Equality Thank you, forty-two! 15 minutes ago, forty-two said: But if you wanted to keep his approach of applying the Subtraction Prop of Equality, you could do this: 1) m<SRT = m<STR; m<3 = m<4 Given 2) m<SRT = m<3 + m<1; m<STR = m<4 + m<2 Addition Angle Postulate 3) m<SRT - m<3 = m<1; m<STR - m<4 = m<2 Subtraction property of equality 4) m<SRT - m<3 = m<STR - m < 4 Substitution 5) m <1 = m<2 Substitution I agree. I think that his only problem, actually - not *explicitly* stating his use of the Addition Angle Postulate. Otherwise I think he was correct in how he then applied the Sub Prop of Eq to it. This is how I proofed the problem. And now that I've had lunch, I think you're right! His only problem was not stating his use of the addition postulate -- he assumed it was "given information" because it was provided in the visual. To further complicate matters, the definition given for deductive reasoning in this chapter says that we reason using postulates/axioms, theorems, definitions, and given information. It left me scratching my head as to what can be assumed. Thank you so much for working through that. I love seeing how others come at these problems... 🙂 Doodlebug Quote Link to comment Share on other sites More sharing options...
Doodlebug Posted August 13, 2020 Author Share Posted August 13, 2020 2 hours ago, square_25 said: Oh, pfffft, fiddlesticks. I think his proof is fine and should be graded as such -- it makes sense, doesn't it? Then why make him feel like he got it wrong? I think grading things like this wrong makes kids feel like proofs are a weird magic thing. For what it's worth, literally no one does proofs in this format outside the world of high school geometry. It's extremely hard and onerous to only work from axioms, and generally, we tend to forget what the axioms are as soon as we step away from them. While figuring out what they are is interesting foundational work, I think a big part of math proficiency is feeling like you can explain math intuitively, which means you'll certainly use arguments like the above. Thank you! Of course I'm not counting it wrong... that was the point of asking a fun chewy question on a topic I'm not super confident in. (To your point of forgetting all those axioms after we step away from them.) 😄 I don't "dance" in geometry, but I work really hard to understand it and present only the ideas I can support. Our text does cover informal as well as formal logic, so we'll get a taste of both. And it's always helpful to hear from people who are better "dancers." 😉 Quote Link to comment Share on other sites More sharing options...
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