Kerileanne99 Posted August 4, 2014 Share Posted August 4, 2014 This question could probably be answered in the general forum, but I didn't want to risk the crickets. Our school today was indefinitely canceled as dd disappeared down the rabbit hole😄 We have started ending our math lessons on a 'stumper' problem, trying to work on the idea that not all math will come easily and shouldn't...and she asked for a whole day devoted to them. I have been pulling from a wide variety of materials, and today I gave her a warm up problem. One of those typical 'what comes next' pattern problems. This one was a word problem, about the growth of a frog. Said froggy is growing by squares, so you are given 1, 4, 9, and 16 under the pictures of each progressively larger frog. She did not immediately see that they were squares so decided to write them on her page. She found the difference between the squares, recognized that they were sequential odd numbers and predicted that the next should be 25. Once she wrote 25, she immediately recognized that they were all squares and excitedly predicted how big froggy would be for the next week. Great. I was thrilled that she figured it out 'her way' as I honestly would rather her spend the next couple of years on this kind of math exploration and play rather than keep advancing! This problem intrigued her and she has been busy for 2 HOURS making lists of square numbers and finding the difference to see if the pattern holds, and just informed me that she is going to try the cubes next. All fun. The problem is that she wants to know the WHY! Uh... I don't even know what you call it. I have a decent math background. I am comfortable teaching her for some time, knowing that we will probably outsource at some point. I can teach any algorithm I will need to...but I never was inclined to play with numbers and their patterns/relationships. We read a lot of kiddie math biographies and books, so I am learning too. 1) what do I call this, how can I explain the why of it? and she wants to know who (as in mathematician) studied the patterns so she can read about them:) 2) where can I find more like this to pique her interest, especially when I need some time to myself, lol? She has been quiet as a mouse...although she IS rolling around the living room floor on and every surface as she writes. I did find a great NRICH list of projects I think we can do that will thrill her, but mommy needs a bit of education first! Quote Link to comment Share on other sites More sharing options...
Kerileanne99 Posted August 4, 2014 Author Share Posted August 4, 2014 And can I just say that someday she is going to hate me for this as she doesn't have a calculator😄 I am not planning on one for a very long time. So she is now up to 132^2, and has had to work them out by hand on a dry erase as mommy occasionally giggles maniacally from the kitchen. She knows them up to 15^2, and can sometimes catch a break when she recognizes that she can figure them out from what she knows...but mostly just by hand:) How long do you think I can garner from the cubes?! He he... Quote Link to comment Share on other sites More sharing options...
EKS Posted August 4, 2014 Share Posted August 4, 2014 Someone else who actually knows something about this is sure to chime in here, but I believe that the why that you're looking for is actually what mathematical proofs do. A proof would show that the pattern holds indefinitely by showing that it is generally true. So, my sort-of proof for this would be that you have a particular case of a difference of squares x^2-y^2 in that y=x+1. So you can rewrite x^2-y^2 as x^2-(x+1)^2. If you factor that you get (x+x+1)(x-(x+1)) which simplifies to 2x+1. And 2x+1 defines the pattern in a general way. ETA: There are some sign errors in the above, which is why I really should refrain from posting about the details of math... Quote Link to comment Share on other sites More sharing options...
quark Posted August 4, 2014 Share Posted August 4, 2014 Explanation by DS: ---------------------------- Dear Ma'am, Hello, this is Quark's son. Suppose you have a square with dimensions of n, with n2 unit squares. We now remove a square from the corner of this square. The new square has dimensions n-1, with (n-1)2 unit squares. What remains is an L shape with a total of 2n-1 unit squares, where 2n-1 is always an odd number (please see attached graphic). n2 - (n-1)2 = 2n-1 and (n+1)2-n2=2n+1, and so on, creating sequential odd numbers. For cubes, the difference of two consecutive cubes is 3n2-3n+1, which factors to 3[n(n-1)]+1. Perhaps your daughter would like to show that 3[n(n-1)]+1 is always odd. This blog has nice explanations in general for many math concepts. Example: http://mathwithbaddrawings.com/2014/01/13/undiscovered-math/ Quote Link to comment Share on other sites More sharing options...
lewelma Posted August 4, 2014 Share Posted August 4, 2014 Oh Quark :001_wub: Quote Link to comment Share on other sites More sharing options...
underthebridge Posted August 4, 2014 Share Posted August 4, 2014 You can see the algebraic proof above, but if you actually make models of the squares it becomes more clear. Using base 10 unit cubes, you can make a 2x2 square and place a 3x3 square beside it. You can see that the 3x3 square has 3+2 more squares (3 on one side and 2 on the top). Then place a 4x4 square next to it. You can see that it is 4+3 bigger. And so forth. So the next square is the original unsquared number plus the same number add 1, which is always odd. This is exactly what is above, but without the algebraic notation! In fact , you can give DD the blocks and let her work it out! Quote Link to comment Share on other sites More sharing options...
Kathy in Richmond Posted August 5, 2014 Share Posted August 5, 2014 That pattern and more are explored in some of the upper level Miquon Math strands; Lab Annotations includes teaching hints. I've used Miquon with early elem kids who were really math advanced & loved exploration. You both might like Marilyn Burns books at this stage: Math for Smarty Pants & The I Hate Mathematics Book I see a discussion about the square number pattern in the former volume. Kid friendly! Martin Gardner is terrific, especially his Aha Insight! and Gotcha! for elementary aged learners. She might like leafing through these, too, or reading with you. Constance Reid's From Zero To Infinity discusses basic number theory patterns for a lay audience & includes some historical material. It's written for an adult audience, but it's short & not intimidating. My kids read it on their own in mid-elementary. The Number Devil is full of math pattern explorations & is lots of fun. We read this one aloud together, stopping to try different fun ideas along the way. At a higher level, AoPS Intro to Number Theory would be the next step if you really want to learn more. Your little girl sounds perfectly delightful. Have fun exploring with her! Quote Link to comment Share on other sites More sharing options...
Kerileanne99 Posted August 5, 2014 Author Share Posted August 5, 2014 First off, before it gets lost in the post, can I just say you guys are AWESOME! Truly. Quote Link to comment Share on other sites More sharing options...
Kerileanne99 Posted August 5, 2014 Author Share Posted August 5, 2014 Someone else who actually knows something about this is sure to chime in here, but I believe that the why that you're looking for is actually what mathematical proofs do. A proof would show that the pattern holds indefinitely by showing that it is generally true. So, my sort-of proof for this would be that you have a particular case of a difference of squares x^2-y^2 in that y=x+1. So you can rewrite x^2-y^2 as x^2-(x+1)^2. If you factor that you get (x+x+1)(x-(x+1)) which simplifies to 2x+1. And 2x+1 defines the pattern in a general way. ETA: There are some sign errors in the above, which is why I really should refrain from posting about the details of math... Thanks for this:)I did figure out the signs and it was very helpful to me. I actually showed it to her this evening at dinner. Obviously, at 4 she hasn't had algebra, but she has done both Dragonbox apps and played with some Hands on Equations stuff, so I really think she enjoyed seeing a real-life probl that SHE was interested in use algebra:) Quote Link to comment Share on other sites More sharing options...
Kerileanne99 Posted August 5, 2014 Author Share Posted August 5, 2014 Explanation by DS: ---------------------------- Dear Ma'am, Hello, this is Quark's son. Suppose you have a square with dimensions of n, with n2 unit squares. We now remove a square from the corner of this square. The new square has dimensions n-1, with (n-1)2 unit squares. What remains is an L shape with a total of 2n-1 unit squares, where 2n-1 is always an odd number (please see attached graphic). n2 - (n-1)2 = 2n-1 and (n+1)2-n2=2n+1, and so on, creating sequential odd numbers. SAM_2066 (2).JPG For cubes, the difference of two consecutive cubes is 3n2-3n+1, which factors to 3[n(n-1)]+1. Perhaps your daughter would like to show that 3[n(n-1)]+1 is always odd. This blog has nice explanations in general for many math concepts. Example: http://mathwithbaddrawings.com/2014/01/13/undiscovered-math/ Dear Quark's Son: You are a gem. Thank you for taking the time to put this together and lay it out for me. Your explanation helped me put a smile on my daughter's face. I showed her your work as well, telling her that a wonderful young man was helping her. She shouted a big 'thank you' at the IPad:) Quote Link to comment Share on other sites More sharing options...
Kerileanne99 Posted August 5, 2014 Author Share Posted August 5, 2014 You can see the algebraic proof above, but if you actually make models of the squares it becomes more clear. Using base 10 unit cubes, you can make a 2x2 square and place a 3x3 square beside it. You can see that the 3x3 square has 3+2 more squares (3 on one side and 2 on the top). Then place a 4x4 square next to it. You can see that it is 4+3 bigger. And so forth. So the next square is the original unsquared number plus the same number add 1, which is always odd. This is exactly what is above, but without the algebraic notation! In fact , you can give DD the blocks and let her work it out! Ooh fun! I have a great big box of math tiles I am going to get out tomorrow and let Alex try this tomorrow. I think it will really appeal to her. I want to see if she can 'see' the algebra explanation with the tiles. This is something I think I would like to follow wherever she wants to take it/) Quote Link to comment Share on other sites More sharing options...
quark Posted August 5, 2014 Share Posted August 5, 2014 Dear Quark's Son: You are a gem. Thank you for taking the time to put this together and lay it out for me. Your explanation helped me put a smile on my daughter's face. I showed her your work as well, telling her that a wonderful young man was helping her. She shouted a big 'thank you' at the IPad:) You and your daughter made his day. :001_wub: Quote Link to comment Share on other sites More sharing options...
Kerileanne99 Posted August 5, 2014 Author Share Posted August 5, 2014 That pattern and more are explored in some of the upper level Miquon Math strands; Lab Annotations includes teaching hints. I've used Miquon with early elem kids who were really math advanced & loved exploration. You both might like Marilyn Burns books at this stage: Math for Smarty Pants & The I Hate Mathematics Book I see a discussion about the square number pattern in the former volume. Kid friendly! Martin Gardner is terrific, especially his Aha Insight! and Gotcha! for elementary aged learners. She might like leafing through these, too, or reading with you. Constance Reid's From Zero To Infinity discusses basic number theory patterns for a lay audience & includes some historical material. It's written for an adult audience, but it's short & not intimidating. My kids read it on their own in mid-elementary. The Number Devil is full of math pattern explorations & is lots of fun. We read this one aloud together, stopping to try different fun ideas along the way. At a higher level, AoPS Intro to Number Theory would be the next step if you really want to learn more.! Thanks for the resources! I actually have some of them. I pulled out Math for Smarty Pants and guess what? Guess mom has some reading to do! It was so much easier when we did all of our reading together:) I haven't seen the Constance Reid book so will have a look. It sounds perfect. So many of the explanations I find for kids run into the same problem as science- things get dumbed down so much to accomodate their ages that it becomes almost completely unrecognizable. I think this will improve as she gets a bit more math and experience under her belt:) Number Devil-next up on our reading list in the math dept. I tried it some time ago but she wasn't ready. I think she definitely is now. AoPS- we do Beast Academy now in part...she isn't quite ready always to tackle longer problems that she lacks the fine motor to tackle. Those perfectionist tendencies are just too overwhelming-for both of us. I actually think maybe the higher levels of AoPS would work for me though. Daddy suggested tonight that he and I start working through them together. Otherwise we are really going to be short-changing this kid if she continues like this in math. I am thinking we are going to have some interesting date nights! Quote Link to comment Share on other sites More sharing options...
Kerileanne99 Posted August 5, 2014 Author Share Posted August 5, 2014 Your little girl sounds perfectly delightful. Have fun exploring with her! Ah, thanks:) She is. Mostly😄 Although at age 4 it is definitely a journey and a process. Some days she blows me away, and other days the extent of her math exploration is to see how many fingers fit inside her nose:) But I wouldn't trade it for anything... Quote Link to comment Share on other sites More sharing options...
Kathy in Richmond Posted August 5, 2014 Share Posted August 5, 2014 Ah, thanks:) She is. Mostly😄 Although at age 4 it is definitely a journey and a process. Some days she blows me away, and other days the extent of her math exploration is to see how many fingers fit inside her nose:) But I wouldn't trade it for anything... Btdt!.....and it continues at age 22! :tongue_smilie: But I wouldn't trade it for anything, either. Enjoy the ride! Quote Link to comment Share on other sites More sharing options...
Dmmetler Posted August 5, 2014 Share Posted August 5, 2014 Dover books has some good number theory. Quote Link to comment Share on other sites More sharing options...
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