wendyroo Posted January 19, 2019 Share Posted January 19, 2019 DS (9) is almost done with AOPS Pre-Algebra. By and large he has cruised through it, so it took me by surprise when he started faltering a little recently as he tackled the geometry chapters (Angles, Perimeter and Area, and Right Triangles and Quadrilaterals including the Pythagorean theorem). It's not that he was struggling per se, he does understand those topics and his work did reflect that, but he certainly was not as confident in those chapters as in the previous ones. For example, in the more algebra-y chapters he was getting about 90% of the regular problems and 80% of the challenge questions correct on the first try. In the geometry chapters he was getting about 75% of the regular problems and just over half of the challenge questions correct on the first try. And several challenge problems per section completely stumped him to the point he didn't even know how to start; that only happened a couple times total in all the algebra chapters. Now that he has completed the geometry chapters, he is working through those topics on Alcumus, and he continues to have a hard time with the more challenging questions. I'm wondering if after he finished Pre-algebra in another month or so, I should have him work through something geometry-y to strengthen those skills a bit before having him dive into algebra. Otherwise it feels like we will just be letting that weakness fester for a year until it comes back to bite him when he starts geometry. But I don't have a clue what we could use. His geometry up until now has been: the Math Mammoth geometry chapters (which I always thought were weak), Hands on Geometry, and some topics in Mathematics: A Human Endeavor (a book he REALLY disliked). I feel like Keys to Geometry would be too easy, and a waste of time and money, since he is already getting most of the AOPS regular difficulty problems correct. I'd really like to find something with a similar challenge level as AOPS, but presented differently...like the geometry equivalent to Jousting Armadillos. Any ideas? Thanks. Quote Link to comment Share on other sites More sharing options...

HomeAgain Posted January 19, 2019 Share Posted January 19, 2019 Right Start G might fit the bill. Or Patty Paper Geometry - you can go through this with either open or guided investigations. There are lessons for both in the book. The only thing is you have to look at reviews to find the "right" brand of patty paper (the ones I initially bought were not quite square) There's also Dragon Box Elements for an app. We've all enjoyed that one. 1 Quote Link to comment Share on other sites More sharing options...

calbear Posted January 19, 2019 Share Posted January 19, 2019 Was going to suggest Patty Paper Geometry as well. Quote Link to comment Share on other sites More sharing options...

Momto6inIN Posted January 20, 2019 Share Posted January 20, 2019 CTC Discovering Geometry was pretty good. Quote Link to comment Share on other sites More sharing options...

LJPPKGFGSC Posted January 21, 2019 Share Posted January 21, 2019 It could be that his spatial reasoning skills are not as advanced as his mathematical / logical skills. But I think that there are a variety of strategies that he might employ in order to solve the geometric problems. Can you give an example of one that stumped him and maybe we can help you brainstorm some ideas to suggest when he is stuck on a challenging problem? Quote Link to comment Share on other sites More sharing options...

wendyroo Posted January 21, 2019 Author Share Posted January 21, 2019 20 minutes ago, LJPPKGFGSC said: It could be that his spatial reasoning skills are not as advanced as his mathematical / logical skills. But I think that there are a variety of strategies that he might employ in order to solve the geometric problems. Can you give an example of one that stumped him and maybe we can help you brainstorm some ideas to suggest when he is stuck on a challenging problem? Sure, here are a couple he had a lot of trouble with: "Find the area of a triangle whose sides have lengths 13, 14, and 15." "What is the side length of a regular polygon whose interior angles each measure 168 degrees and whose perimeter is 120 cm?" "A revolving restaurant rotates one complete revolution every 56 minutes. In the 21 minutes it takes to eat the peaches jubilee dessert, through how many degrees does the restaurant revolve?" Intellectually he knows that a complete revolution is 360 degrees, but I think his mental picture is of a protractor numbering the degrees. He kept insisting that we couldn't figure out the answer unless we knew "at what degree it started". I think both that his spatial reasoning skills are slightly weak and that he needs more practice implementing strategies when tackling geometry problems. He has no problem diving into algebraic problems and trying to puzzle his way to a solution, but often he looks at a geometry problem and just doesn't have a clue even how to start. I'm hoping to find a resource that will present the concepts in a slightly different way and give him more problems on which to practice various strategies. I think the only way he will build up his comfort and fluency will be working a wide range of challenging problems. Wendy Quote Link to comment Share on other sites More sharing options...

LJPPKGFGSC Posted January 21, 2019 Share Posted January 21, 2019 I might give an incomplete response, since I am in the middle of dinner. But I will post what I can and then come back later. I do have the AOPS Pre-Algebra book, but we are getting new floors this week and it is buried somewhere. So I am not looking in the book now, just giving you some impressions.... The area of a triangle problem … This just requires you to plug in the side lengths into the formula. I think that they derive and explain the formula for using the side lengths to find the area. But I can't remember if that is in PreAlgebra or Geometry. If he is having a hard time understanding the Pythagorean Theorem, that would explain why it might not have stuck with him. The side length of a regular polygon …. Does he understand the theorems about interior and exterior angles of a polygon? if it was a polygon with less sides, I might ask him to draw 168 degree angles until it meets up again. But that would not work with this problem - the angles are too huge! So, did he really understand those theorems? Maybe you can find a video so that he can see someone manipulating the shapes to explain the formula for interior and exterior angles. The revolving restaurant … I would make a model. Draw two equal circles. Let one be the unmoving foundation of the restaurant. Take the other the circle and cut it out. Draw the table on both (the base will retain the original location.) Place the cut out circle on top of the foundation and rotate it. Maybe that will help him to see the rotation and get that the starting point is the original location of the table. Count to 21 and see where it is located, then keep counting to 56 to get back to where the table started. (Clearly, you would not be rotating as consistently as the mechanisms of the restaurant, but he should get the idea.) You might find a video of the view from a rotating restaurant. Maybe he doesn't even understand the idea of the rotating restaurant? Sometimes nonroutine problems are all more the context than the math. But in general, I think lots of kids need more life experience to be good at spatial reasoning. As a math teacher, I notice that most kids are have a clear preference for algebraic manipulation or spatial reasoning. That is why Geometry can be easier for some students, they can just see that it makes sense. But if kids just can't see it, they need to learn to use models that will help them imagine what is happening in a problem. So I would look for activities that will develop his sense of how things move in the world. And while you do that, play tons of games at home that require spatial reasoning skills. 1 Quote Link to comment Share on other sites More sharing options...

wendyroo Posted January 22, 2019 Author Share Posted January 22, 2019 15 minutes ago, LJPPKGFGSC said: I might give an incomplete response, since I am in the middle of dinner. But I will post what I can and then come back later. I do have the AOPS Pre-Algebra book, but we are getting new floors this week and it is buried somewhere. So I am not looking in the book now, just giving you some impressions.... The area of a triangle problem … This just requires you to plug in the side lengths into the formula. I think that they derive and explain the formula for using the side lengths to find the area. But I can't remember if that is in PreAlgebra or Geometry. If he is having a hard time understanding the Pythagorean Theorem, that would explain why it might not have stuck with him. The side length of a regular polygon …. Does he understand the theorems about interior and exterior angles of a polygon? if it was a polygon with less sides, I might ask him to draw 168 degree angles until it meets up again. But that would not work with this problem - the angles are too huge! So, did he really understand those theorems? Maybe you can find a video so that he can see someone manipulating the shapes to explain the formula for interior and exterior angles. The revolving restaurant … I would make a model. Draw two equal circles. Let one be the unmoving foundation of the restaurant. Take the other the circle and cut it out. Draw the table on both (the base will retain the original location.) Place the cut out circle on top of the foundation and rotate it. Maybe that will help him to see the rotation and get that the starting point is the original location of the table. Count to 21 and see where it is located, then keep counting to 56 to get back to where the table started. (Clearly, you would not be rotating as consistently as the mechanisms of the restaurant, but he should get the idea.) You might find a video of the view from a rotating restaurant. Maybe he doesn't even understand the idea of the rotating restaurant? Sometimes nonroutine problems are all more the context than the math. But in general, I think lots of kids need more life experience to be good at spatial reasoning. As a math teacher, I notice that most kids are have a clear preference for algebraic manipulation or spatial reasoning. That is why Geometry can be easier for some students, they can just see that it makes sense. But if kids just can't see it, they need to learn to use models that will help them imagine what is happening in a problem. So I would look for activities that will develop his sense of how things move in the world. And while you do that, play tons of games at home that require spatial reasoning skills. He seems to know the formulas and theorems, and be able to use them properly in simple cases, but not intuitively see how to apply them in more complex problems. For example, looking for the area of the triangle with sides 13, 14, and 15, he immediately got stuck because he didn't know the height of the triangle and therefore was at an impasse for finding its area. His mind just beat against a brick wall because he knew that the triangle did not have a right angle, and he knew that in general he lacked the tools to find the area of a non-right triangle if he was not given the altitude. My mind proceeded along the same path as his, but then concluded that since pretty much all we had to work with was Pythagoras and the area formula, that we had better draw in an altitude to create some right angles just to let us start playing around with it. He was perfectly willing to go along with that plan, but wasn't able to run with it and realize that we could now creatively apply Pythagoras to figure out the altitude and bases and therefore the areas of each of the smaller triangles. Wendy Quote Link to comment Share on other sites More sharing options...

daijobu Posted January 22, 2019 Share Posted January 22, 2019 If it were me I wouldn't sweat it so much. He's going to have a full year of algebra before he's scheduled to tackle geometry, so he'll have plenty of time to gain some maturity between now and then. And if he needs additional time, you can delay geometry by doing counting & probability and/or number theory before then. I think one of the hardest things about solving these geometry problems is drawing a line that isn't currently there. Like drawing the altitude of a triangle, or extending a line segment that ends abruptly. It takes experience to be able to do that, and it's certainly an advanced skill. 1 Quote Link to comment Share on other sites More sharing options...

Kathy in Richmond Posted January 22, 2019 Share Posted January 22, 2019 9 hours ago, daijobu said: If it were me I wouldn't sweat it so much. He's going to have a full year of algebra before he's scheduled to tackle geometry, so he'll have plenty of time to gain some maturity between now and then. And if he needs additional time, you can delay geometry by doing counting & probability and/or number theory before then. I think one of the hardest things about solving these geometry problems is drawing a line that isn't currently there. Like drawing the altitude of a triangle, or extending a line segment that ends abruptly. It takes experience to be able to do that, and it's certainly an advanced skill. I agree with everything here. My first thought was to recommend Patty Paper geometry, but after reading the examples of problems that gave him a hard time, I think he's beyond that. It sounds like he can already do the basics, but geometry tends to be difficult when you have a problem that integrates several ideas. It's like having a big toolbox & needing to know which tools to pull out for a given job. Sometimes only lots of experience in problem solving will do the trick. You might want to take a look at the original AoPS problem solving Vol 1. It sounds like he might be ready to tackle some of it. I've always thought that the geometry chapters were brilliantly done. In Ch 11.9, for instance, the authors give hints on how to tackle geometry problems that have you stumped...One of them is daijobu's hint of "drawing a line that isn't currently in the diagram." The text states that this is the "most difficult to master," and goes on to give more helps and problem solving practice. I still refer back to that chapter often in my teaching & learning. Your son might also enjoy going through the geometry problems in old AMC 8s and MathCounts (school handbook, past competition) papers. If you use apps with him, there's a cool MC trainer app (from AoPS), too. It's not strictly geometry, though. 1 Quote Link to comment Share on other sites More sharing options...

wendyroo Posted January 22, 2019 Author Share Posted January 22, 2019 Thank you all for the support and ideas. I am certainly not fretting about this issue, but I am looking for opportunities for him to gradually strengthen his skills so that he is fully prepared when he hits geometry. Since I do think his spatial skills are a weakness, I would like to make sure his "pre-geometry" basics and strategies are rock solid so that he doesn't find AOPS geometry too overwhelming and frustrating. I figure an ounce of prevention is worth a pound of cure. To start, I think I am going to have the 9 year old and his 7 year old brother do some work with Patty Paper Geometry. They are just now finishing up AOPS Pre-A and Math Mammoth 4 respectively, and I think both of them could use a bit of a break to change focus and play around with some hands-on math. I am a big fan of over-teaching the basics from a variety of perspectives, so hopefully Patty Paper will get them thinking about the foundational concepts of geometry differently. After that, I think I will look at the AoPS problem solving Vol 1. If DS and I were to sit together, read, and do a couple problem in one geometry section a week, we could cover all the geometry chapters, and add all those problem solving strategies to our toolboxes, before he tackles AOPS Geometry. Quote Link to comment Share on other sites More sharing options...

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