Following Mathematics software are excellent for use in Math classrooms.
#1 GeoGebra
#2 Graph
#3 C.a.R
Kamis, 26 November 2009
Senin, 23 November 2009
Net Result
I really enjoy designing nets (2-D plans that fold up into 3-D objects), and I love designing them in dynamic geometry, where you can design all nets. If I ever got the time to do math research again, I can see going in that direction.
I designed these four sketches for my geometry class, which is working on a project to design their own package with a few constraints. Each sketch is available as a dynamic webpage or the geogebra file. Here's 5 nets that were made with the sketches, in a printable pdf format. Geogebra actually has very nice priniting controls, so if you're interested in designing your own, choose that option. Remember you can install it, or run it from your browser at geogebra.org.
General tetrahedron: webpage or geogebra file
Square pyramid: webpage or geogebra file
Convex oblique pentagonal prism: webpage or geogebra file
I was very disappointed that the above sketch, though intuitive, couldn't make concave prisms. This next sketch is the answer, but would be a muddle to try to figure out how it was made. The net is pretty though, for designing solids, and I'm proud of it as work.
General Oblique Pentagonal Prism: webpage or geogebra file
Note that you can use the pentagonal prism nets to make quadrilateral and triangular prism nets by making some of the base vertices collinear.
Let me know what you think, and send me your dynamic geometry design challenges!
PS> I also wrote up a memoir for my class of how I made the pyramids (which really sounds egotistical; reminds me of a Tom Lehrer line about "even the Pharoahs, had to import, Hebrew braseros" Listen at the link. If you do, check out Lobachevsky, a great math song.) Sorry for the ramble - I'm tired. Here's the memoir.
I designed these four sketches for my geometry class, which is working on a project to design their own package with a few constraints. Each sketch is available as a dynamic webpage or the geogebra file. Here's 5 nets that were made with the sketches, in a printable pdf format. Geogebra actually has very nice priniting controls, so if you're interested in designing your own, choose that option. Remember you can install it, or run it from your browser at geogebra.org.
General tetrahedron: webpage or geogebra file
Square pyramid: webpage or geogebra file
Convex oblique pentagonal prism: webpage or geogebra fileI was very disappointed that the above sketch, though intuitive, couldn't make concave prisms. This next sketch is the answer, but would be a muddle to try to figure out how it was made. The net is pretty though, for designing solids, and I'm proud of it as work.
General Oblique Pentagonal Prism: webpage or geogebra fileNote that you can use the pentagonal prism nets to make quadrilateral and triangular prism nets by making some of the base vertices collinear.
Let me know what you think, and send me your dynamic geometry design challenges!
PS> I also wrote up a memoir for my class of how I made the pyramids (which really sounds egotistical; reminds me of a Tom Lehrer line about "even the Pharoahs, had to import, Hebrew braseros" Listen at the link. If you do, check out Lobachevsky, a great math song.) Sorry for the ramble - I'm tired. Here's the memoir.
Sabtu, 21 November 2009
Math Teachers at Play 20
The new Math Teachers at Play Carnival is up at Denise's Let's Play Math! blog. Besides the interesting puzzle she starts with, and the hilarious math FAIL images, there were a few posts I found especially interesting:
- Ashley Allain's post on computing fractions using Scratch, a new, kid-friendly programming language.
- Rachel Lynnette documents her use of a free symmetry game that looks promising.
- I'll be using this cone problem on a test, I think. Pretty elegant. Wonder if my students read this?
Rabu, 18 November 2009
Constructivism and WatchKnow
Two interesting web links to pass on:
1) Interesting essay found by Michael Paul Goldenberg (no relation) at Education Notes Online: a teacher and parent commenting on constructivism and the math wars in NYC. There is some really interesting math happening in the city classrooms, in particular the Math in the City project, and the essay is a good read. Check it out: The Math Wars Revisited: Lisa, Why Doth I Love Thee....
2) New web resource: one of the founders of Wikipedia has started a collection of educational videos that is rapidly growing and becoming comprehensive. It's called WatchKnow. Everything from Carl Sagan addressing the shape of space-time (I was just talking about that with my students...) to a geometric solids story starring Sniffer Bob (which is what we were discussing...). The rating system will become really helpful as the site matures, I think.
1) Interesting essay found by Michael Paul Goldenberg (no relation) at Education Notes Online: a teacher and parent commenting on constructivism and the math wars in NYC. There is some really interesting math happening in the city classrooms, in particular the Math in the City project, and the essay is a good read. Check it out: The Math Wars Revisited: Lisa, Why Doth I Love Thee....
2) New web resource: one of the founders of Wikipedia has started a collection of educational videos that is rapidly growing and becoming comprehensive. It's called WatchKnow. Everything from Carl Sagan addressing the shape of space-time (I was just talking about that with my students...) to a geometric solids story starring Sniffer Bob (which is what we were discussing...). The rating system will become really helpful as the site matures, I think.
Jumat, 13 November 2009
Pick On Someone Your Own Size
Definitely the longest game name I've ever used. But the fourth graders who piloted the game have decided, and who am I to argue.
Fourth grade students in Mrs. Bruckbauer’s class described this as a math mystery game, because you’re trying to find out what the other player’s hiding.
Materials: game sheet, calculator (if needed for checking), scrap paper for making your plays.
How to Play: Each player comes up with 3 numbers that add up to 1000. When both players are set, they fill in the fight boxes from largest to smallest. You get a point when you have the greater number.
For example, 800, 100, 100 and 500, 300, 200; player 1 gets 1 point and player 2 gets 2 points. It’s okay to have two of your numbers be the same. If your number is equal to an opponent's, no points.
Player1 Player 2
800 > 500
100 < 300
100 < 200
1 point 2 points
You play for five rounds, and the person with the most points wins.
PDF of the game.
Teaching with the game: I started with a guessing game. I picked a number between 0 and 1000, and they tried to guess it, using only my more or less answer as clues. I picked 673, and the students took turns guessing. I wrote down on the resulting comparison. using x for my number, like x>89. They guessed: 89, 587, 1000, 998, 823, 650, 657, 760, 700, 699, 689, 666, 680, 676 and 673. (I wrote 673, then x then filled in the =, to great rejoicing, x=673.) Just refreshing use of the comparison relation, and seeing how they were with ordering numbers in the hundreds. They really enjoyed this, and I think that could be turned into a lesson of its own.
Then I played me vs. the students, as I often like to teach games. It solidifies rules and shares some strategies. My triples were (900,99,1), (500, 250, 250), (501, 498, 1), (350, 350, 300), (501, 301, 198). As the students played later I saw them using several variations on these, and also the idea of modifying a previous guess.
The game was really engaging, and students took their guesses seriously. I watched at first, and a couple hadn't really gotten the adds to 1000 idea, and several had to be reminded to put their guesses in order. I went back and forth on that in design, but it really worked well for game play.
Rules they wanted to add were having 1/2 point or 1 point each for ties, and playing an extra set if it ends in a tie. The ties were nice for getting the students to modify their guesses up or down a few ticks.
The math involved encourages students to think about sums to nice numbers, as well as how to partition 1000. I shared with students how they could pick two numbers and do 1000-sum to find the third. We had calculators for them to quickly check their opponents picks, but mostly they did it in their head. Several students, not too familiar with the calculators, really enjoyed playing with them to try different numbers.
Game analysis: it's a little more interesting than you might think, kind of a nuanced Roshambo. (Rock-Paper-Scissors) The three basic strategies are big-small-small, medium-medium-small, and third-third-third. Of course, you can win by a point, and that's where some nuance and psychology comes in. Much like Rock-Paper-Scissors.
Let me know if you give this a try and what you think! Thanks to Denise at Let's Play Math! for catching the missing instruction.
Pick on Someone Your Own Size
A Math Mystery Game
A Math Mystery Game
Fourth grade students in Mrs. Bruckbauer’s class described this as a math mystery game, because you’re trying to find out what the other player’s hiding.
Materials: game sheet, calculator (if needed for checking), scrap paper for making your plays.
How to Play: Each player comes up with 3 numbers that add up to 1000. When both players are set, they fill in the fight boxes from largest to smallest. You get a point when you have the greater number.
For example, 800, 100, 100 and 500, 300, 200; player 1 gets 1 point and player 2 gets 2 points. It’s okay to have two of your numbers be the same. If your number is equal to an opponent's, no points.
Player1 Player 2
800 > 500
100 < 300
100 < 200
1 point 2 points
You play for five rounds, and the person with the most points wins.
Player 1 Score ....................................................................... Player 2 Score
PDF of the game.
Teaching with the game: I started with a guessing game. I picked a number between 0 and 1000, and they tried to guess it, using only my more or less answer as clues. I picked 673, and the students took turns guessing. I wrote down on the resulting comparison. using x for my number, like x>89. They guessed: 89, 587, 1000, 998, 823, 650, 657, 760, 700, 699, 689, 666, 680, 676 and 673. (I wrote 673, then x then filled in the =, to great rejoicing, x=673.) Just refreshing use of the comparison relation, and seeing how they were with ordering numbers in the hundreds. They really enjoyed this, and I think that could be turned into a lesson of its own.
Then I played me vs. the students, as I often like to teach games. It solidifies rules and shares some strategies. My triples were (900,99,1), (500, 250, 250), (501, 498, 1), (350, 350, 300), (501, 301, 198). As the students played later I saw them using several variations on these, and also the idea of modifying a previous guess.
The game was really engaging, and students took their guesses seriously. I watched at first, and a couple hadn't really gotten the adds to 1000 idea, and several had to be reminded to put their guesses in order. I went back and forth on that in design, but it really worked well for game play.
Rules they wanted to add were having 1/2 point or 1 point each for ties, and playing an extra set if it ends in a tie. The ties were nice for getting the students to modify their guesses up or down a few ticks.
The math involved encourages students to think about sums to nice numbers, as well as how to partition 1000. I shared with students how they could pick two numbers and do 1000-sum to find the third. We had calculators for them to quickly check their opponents picks, but mostly they did it in their head. Several students, not too familiar with the calculators, really enjoyed playing with them to try different numbers.
Game analysis: it's a little more interesting than you might think, kind of a nuanced Roshambo. (Rock-Paper-Scissors) The three basic strategies are big-small-small, medium-medium-small, and third-third-third. Of course, you can win by a point, and that's where some nuance and psychology comes in. Much like Rock-Paper-Scissors.
Let me know if you give this a try and what you think! Thanks to Denise at Let's Play Math! for catching the missing instruction.
Senin, 09 November 2009
Permen No 75 Tentang Ujian Nasional
Tepatnya tanggal 13 Oktober 2009, di akhir masa jabatannya sebagai Mentri Pendidikan Nasional pada kabinet Indonesia Bersatu I Bambang Sudibyo mengeluarkan peraturan mentri no 75 tentang Ujian Nasional. Ada beberapa hal yang agak sedikit berbeda mengenai pelaksanaan Ujian Nasional 2010 dengan Ujian Nasional tahu 2009, antara lain :1. WaktuUjian Nasional tahun ini akan dilaksanakan pada minggu
Minggu, 08 November 2009
Jangan Pernah Menyerah dengan Matematika
Windy adalah salah satu siswa yang terkenal di kelas matematika selama duduk di bangku SMA. Di kelas sewaktu sekolah, memang ada beberapa siswa yang sangat terkenal dan dikenal oleh guru dan teman-temannya. Biasanya siswa tersebut pinter, jagoan di kelas atau sebaliknya- siswa yang ekstrem satunya, “ngga bisa apa-apa” atau “katrok” bangetlah pinjam istilahnya Tukul Arwana. Dan sayangnya, Windy masuk kategori yang ekstrem kedua, seringkali dianggap ‘nothing’ dalam pelajaran matematika. Setiap habis ulangan bisa dipastikan dia salah satu siswa yang harus mengikuti kelas remedial. Sehingga julukan “Mr. Remidi” melekat padanya.
Apakah Windy, diam dan menyerah begitu saja? Bagi kebanyakan siswa ketika dirinya dianggap “nothing” oleh kebanyakan teman dan gurunya dia biasanya akan menyerah, diam dan tidak melakukan apa-apa, cenderung memandang rendah dirinya sendiri, nggak percaya diri dan lebih percaya kalau dirinya memang nggak bisa apa-apa. Tapi hal itu ternyata tidak untuk Windy! Dia mengakui memang dirinya lemah dalam pelajaran matematika. Dia menyadari dia kurang dalam pelajaran ini. Tapi dia masih punya dua hal, pertama, keyakinan bahwa matematika bisa dipelajarinya, kedua dia memiliki ketekunan. Mungkin dua hal itu, yang tidak dimiliki oleh banyak orang yang menyerah terhadap matematika. Dengan dua hal itu Windy bertahan untuk terus belajar matematika.
Windy yang SD dan SMPnya ditempuh di Papua dan ketika menginjak SMA melanjutkan studi di Yogyakarta itu, memang menyadari benar kemampuannya. Dia tidak malu mengakui bahwa dirinya butuh lebih banyak waktu untuk memahami suatu materi matematika dibanding teman-temannya yang lain. Mungkin sekali dijelaskan, teman-temannya sudah paham. Tapi windy butuh dua atau tiga kali penjelasan untuk materi yang sama. Untunglah, guru matematikanya seorang pastor yang memiliki kesabaran dua atau tiga kali lebih banyak daripada guru matematika biasa dan mau membuka kelas matrikulasi untuk siswa-siswa yang tertinggal dalam pelajaran matematika.
Untuk mengejar ketertinggalannya Windy selalu mengikuti kelas matrikulasi matematika yang diadakan sore hari di sekolahnya. Selain itu, dia juga menyediakan waktu khusus untuk belajar matematika sendiri, entah untuk mengulangi materi atau mengerjakan PR. Jadi bisa dipastikan, Windy adalah salah satu siswa di kelas dan mungkin di sekolahnya yang selalu mengerjakan PR matematika. Meski juga bisa dipastikan dari pekerjaan itu banyak yang salah juga. Tapi dari mengerjakan PR itu, dia sedikit-sedikit tahu kesalahan yang dibuatnya, bisa belajar dari kesalahan-kesalahan yang dibuatnya tersebut dan memperbaiki kesalahan itu lewat mengerjakan ulang PR yang salah itu.
Windy memang “Mr Remidi”, setiap kali ulangan dia memang boleh dikata nilainya hampir selalu dibawah standar ketuntasan belajar. Tapi nilai hasil remidi yang diikutinya pasti selalu diatas tujuh puluh, tak jarang mendekati nilai sempurna. Dan jangan heran, meski jagoan “remidi”, Windy lulus SMA dari jurusan IPA. Kini ia sedang menyelesaikan tugas akhirnya di jurusan Teknik Informatika di salah satu PTS cukup terkenal di Yogyakarta. “Jangan pernah menyerah dengan matematika, meski sesulit apapun. Dia akan memberikan pada kita sesuatu yang memang pantas untuk diperjuangkan. Cara berpikir, bernalar yang benar, berpikir secara logis dan sistematis.” Ungkapnya penuh keyakinan. “Kamu boleh tidak percaya, tapi saya sudah membuktikannya!” Percaya deh…
Apakah Windy, diam dan menyerah begitu saja? Bagi kebanyakan siswa ketika dirinya dianggap “nothing” oleh kebanyakan teman dan gurunya dia biasanya akan menyerah, diam dan tidak melakukan apa-apa, cenderung memandang rendah dirinya sendiri, nggak percaya diri dan lebih percaya kalau dirinya memang nggak bisa apa-apa. Tapi hal itu ternyata tidak untuk Windy! Dia mengakui memang dirinya lemah dalam pelajaran matematika. Dia menyadari dia kurang dalam pelajaran ini. Tapi dia masih punya dua hal, pertama, keyakinan bahwa matematika bisa dipelajarinya, kedua dia memiliki ketekunan. Mungkin dua hal itu, yang tidak dimiliki oleh banyak orang yang menyerah terhadap matematika. Dengan dua hal itu Windy bertahan untuk terus belajar matematika.
Windy yang SD dan SMPnya ditempuh di Papua dan ketika menginjak SMA melanjutkan studi di Yogyakarta itu, memang menyadari benar kemampuannya. Dia tidak malu mengakui bahwa dirinya butuh lebih banyak waktu untuk memahami suatu materi matematika dibanding teman-temannya yang lain. Mungkin sekali dijelaskan, teman-temannya sudah paham. Tapi windy butuh dua atau tiga kali penjelasan untuk materi yang sama. Untunglah, guru matematikanya seorang pastor yang memiliki kesabaran dua atau tiga kali lebih banyak daripada guru matematika biasa dan mau membuka kelas matrikulasi untuk siswa-siswa yang tertinggal dalam pelajaran matematika.
Untuk mengejar ketertinggalannya Windy selalu mengikuti kelas matrikulasi matematika yang diadakan sore hari di sekolahnya. Selain itu, dia juga menyediakan waktu khusus untuk belajar matematika sendiri, entah untuk mengulangi materi atau mengerjakan PR. Jadi bisa dipastikan, Windy adalah salah satu siswa di kelas dan mungkin di sekolahnya yang selalu mengerjakan PR matematika. Meski juga bisa dipastikan dari pekerjaan itu banyak yang salah juga. Tapi dari mengerjakan PR itu, dia sedikit-sedikit tahu kesalahan yang dibuatnya, bisa belajar dari kesalahan-kesalahan yang dibuatnya tersebut dan memperbaiki kesalahan itu lewat mengerjakan ulang PR yang salah itu.
Windy memang “Mr Remidi”, setiap kali ulangan dia memang boleh dikata nilainya hampir selalu dibawah standar ketuntasan belajar. Tapi nilai hasil remidi yang diikutinya pasti selalu diatas tujuh puluh, tak jarang mendekati nilai sempurna. Dan jangan heran, meski jagoan “remidi”, Windy lulus SMA dari jurusan IPA. Kini ia sedang menyelesaikan tugas akhirnya di jurusan Teknik Informatika di salah satu PTS cukup terkenal di Yogyakarta. “Jangan pernah menyerah dengan matematika, meski sesulit apapun. Dia akan memberikan pada kita sesuatu yang memang pantas untuk diperjuangkan. Cara berpikir, bernalar yang benar, berpikir secara logis dan sistematis.” Ungkapnya penuh keyakinan. “Kamu boleh tidak percaya, tapi saya sudah membuktikannya!” Percaya deh…
Berbagi Matematika
Pak Yanto, begitu dia sering disapa, mengakui suka dengan matematika sejak kecil. Dan jika ditanya “mengapa suka matematika?” hingga sekarang dia selalu mengatakan tidak tahu alasannya. “Suka saja!”, begitu ia selalu menjawab. Sewaktu SD matematika adalah pelajaran favoritnya. PR yang selalu dikerjakannya adalah PR matematika. Bahkan ketika tidak ada PR pun dia selalu memaksa ibunya untuk membuatkan soal PR matematika.
Namun, kesukaannya terhadap matematika sempat meredup ketika SMP. Nilai matematikanya tidak cukup memuaskan buat dirinya, meskipun jika dibanding teman-temannya yang lain sebenarnya lebih dari cukup. “Waktu di SMP, guru matematikanya tidak menyenangkan.” kilahnya. Namun demikian, dia tetap belajar matematika meski standar seperti teman-teman lainnya, tidak seperti waktu di SD dulu.
Sewaktu SMA minatnya pada matematika kembali tumbuh. Di sekolah dia dikenal sebagai jago matematika oleh teman-temannya. Setiap pagi, dia selalu menjadi orang yang paling ditunggu di kelasnya. Teman-temannya sudah banyak yang antri untuk melihat dan mencontoh PR atau tugas matematikanya. Kadang malah ada yang pagi-pagi sudah nyamperi kerumahnya. Resikonya, karena “digilir” banyak temannya, kertas pekerjaannya jadi lecek dan tak jarang sobek. Untuk itu kadang dia harus buat tugas yang dikumpul rangkap dua. Maka jangan heran, kalau dia sering terpilih mewakili sekolah untuk mengikuti lomba matematika meski selama SMA itu tak satupun kejuaraan yang dimenanginya. Tapi itu tidak membuatnya surut menyukai matematika. Sebaliknya, hal itu membawanya pada kesadaran bahwa dia belajar matematika untuk tahu, mengerti dan paham tentang konsep matematika dan kalau bisa membagi apa yang diketahuinya itu pada teman-teman lain yang mau belajar matematika, bukan untuk menjadi juara lomba matematika, apalagi sekedar mendapatkan nilai matematika yang baik. Kesadaran baru itu semakin membawanya pada rasa suka yang lebih besar terhadap matematika.
Pada waktu SMA itulah, ia juga berinisiatif mengajari teman-temannya belajar matematika seminggu dua kali untuk persiapan ujian nasional di sekolahnya. Dan hasilnya, semua teman satu kelasnya lulus, meski dia sendiri hanya memperoleh nilai 6,25 untuk pelajaran matematika.
Kecintaannya terhadap matematika membuatnya memilih jurusan pendidikan matematika di salah satu PTS terkemuka di Yogyakarta, sebagai pintu masuknya menjadi guru untuk berbagi matematika dengan banyak orang. Kini ia pun menjadi guru matematika dan berbagi matematika dengan murid-muridnya.
Namun, kesukaannya terhadap matematika sempat meredup ketika SMP. Nilai matematikanya tidak cukup memuaskan buat dirinya, meskipun jika dibanding teman-temannya yang lain sebenarnya lebih dari cukup. “Waktu di SMP, guru matematikanya tidak menyenangkan.” kilahnya. Namun demikian, dia tetap belajar matematika meski standar seperti teman-teman lainnya, tidak seperti waktu di SD dulu.
Sewaktu SMA minatnya pada matematika kembali tumbuh. Di sekolah dia dikenal sebagai jago matematika oleh teman-temannya. Setiap pagi, dia selalu menjadi orang yang paling ditunggu di kelasnya. Teman-temannya sudah banyak yang antri untuk melihat dan mencontoh PR atau tugas matematikanya. Kadang malah ada yang pagi-pagi sudah nyamperi kerumahnya. Resikonya, karena “digilir” banyak temannya, kertas pekerjaannya jadi lecek dan tak jarang sobek. Untuk itu kadang dia harus buat tugas yang dikumpul rangkap dua. Maka jangan heran, kalau dia sering terpilih mewakili sekolah untuk mengikuti lomba matematika meski selama SMA itu tak satupun kejuaraan yang dimenanginya. Tapi itu tidak membuatnya surut menyukai matematika. Sebaliknya, hal itu membawanya pada kesadaran bahwa dia belajar matematika untuk tahu, mengerti dan paham tentang konsep matematika dan kalau bisa membagi apa yang diketahuinya itu pada teman-teman lain yang mau belajar matematika, bukan untuk menjadi juara lomba matematika, apalagi sekedar mendapatkan nilai matematika yang baik. Kesadaran baru itu semakin membawanya pada rasa suka yang lebih besar terhadap matematika.
Pada waktu SMA itulah, ia juga berinisiatif mengajari teman-temannya belajar matematika seminggu dua kali untuk persiapan ujian nasional di sekolahnya. Dan hasilnya, semua teman satu kelasnya lulus, meski dia sendiri hanya memperoleh nilai 6,25 untuk pelajaran matematika.
Kecintaannya terhadap matematika membuatnya memilih jurusan pendidikan matematika di salah satu PTS terkemuka di Yogyakarta, sebagai pintu masuknya menjadi guru untuk berbagi matematika dengan banyak orang. Kini ia pun menjadi guru matematika dan berbagi matematika dengan murid-muridnya.
Jumat, 06 November 2009
Golden Math Cartoon & Math Teachers at Play 19
I had to post this. I'm trying to help edit a collection of extension activities for Geometry being put together by teams of students under Char Beckmann for the Michigan Council of Teachers of Mathematics. (The Sum More books - more as they become available.) One of the activities involves cutting up a cartoon and then trying to arrange it in order to make the idea of arranging arguments in order in a proof. Long ramble, here's the cartoon: (Click for full size)
Their take on a pi joke. Pretty good for 9 and 10, eh? I inked it, and was told I boffed Fiddlestick's mouth in the last frame. They did a great job putting in clues as to what comes next.
Math Teachers at Play 19 (back on numbering) came out last Friday. My contribution was Area Block. The carnival is moving to a monthly schedule, which I think will be good. What was most fascinating to me:
Math Teachers at Play 19 (back on numbering) came out last Friday. My contribution was Area Block. The carnival is moving to a monthly schedule, which I think will be good. What was most fascinating to me:
- Maria Anderson, just up the road in Muskegon, is doing amazing work with technology in a math for elementary teachers class. I've got to meet her sometime!
- Jason Dyer's post on a reading experiment was interesting. (I wonder if it has to do with this cognitive miser issue. (Not a MTAP link.)
- Kendra at the Pumpkin Patch had a cute, quick sum game.
Selasa, 03 November 2009
Mencari beda dalam deret aritmetika
Salah satu bentuk soal matematika yang sering muncul dalam materi deret adalah mencari beda dalam deret aritmetika. Sebagaimana telah diketahui sebuah beda dalam deret aritmetika dapat dicari dengan aturan mencari selisih antara sebuah suku dengan suku sebelumnya, selain itu beda dalam deret aritmetika ( yang berhubungan denga Sn ) juga bisa dicari menggunakan aturan Un = Sn - Sn-1.Berikut adalah
Kamis, 29 Oktober 2009
An Ad?
While Math Hombre is traditionally ad free, I do owe a link to my daughter Ysabela. In exchange for her review of Babymouse: Dragonslayer, the excellent math/fantasy graphic novel by Jennifer Holm, I promised to plug her market craft.
Yzzy's Sticky Silverware has recycled silverware affixed with two powerful magnets and artistic embellishments to make a beautiful and practical refrigerator decoration. A steal at $1.25 decorated ($1 plain), you might pay up to $5 for a similar product at a craft show. Check it out! Mailing is at cost.
Some of the math involved was: figuring out the unit cost from the total materials cost, and also balances, profit margin, the length of the total wire based on the wire per silverware, (which was a pretty good upper el measurement problem). What is her break even point? That's the next calculation!
Yzzy's Sticky Silverware has recycled silverware affixed with two powerful magnets and artistic embellishments to make a beautiful and practical refrigerator decoration. A steal at $1.25 decorated ($1 plain), you might pay up to $5 for a similar product at a craft show. Check it out! Mailing is at cost.
Some of the math involved was: figuring out the unit cost from the total materials cost, and also balances, profit margin, the length of the total wire based on the wire per silverware, (which was a pretty good upper el measurement problem). What is her break even point? That's the next calculation!
Minggu, 25 Oktober 2009
Area Block
New game! Blokus meets Nim, this game works on area and strategy. Tested with fourth and fifth graders, but flexible upwards with sophistication in area computation techniques. Get it as a pdf here: AreaBlock.pdf
Materials: Game Board, 2 different color pens, pencils, crayons or markers.
How to play: Players take turns making a single shape on the board that has an area of 10 or less. The game is done when the board is filled, and the player with the most squares covered wins. The table is for recording how many squares you cover each turn.
The first player to go can only color up to 8 squares on their first shape. (Otherwise it would always be best to go first.) After that the limit is always 10.
The squares colored in to make your shape need to share a side, not just touch at a corner. (The shape you make has to be a polygon.) A player can even use slanted lines – as long as they can figure out the area for their shape! When you are making a new shape, it does not have to touch your old – you can put it anywhere there’s room.
After you make or shade in your shape, record the area of your block – you don’t want to miss any points. The next player then colors a polygon of up to 10 squares of area in a different color.
When the board is completely filled, the game is done. Total up your squares and see who won!
Notice you can check if all the squares are counted by adding both scores – you should get 100.
Whoever lost gets to choose the next time if they want to go first or second.
Variations:
• The board can be changed to have holes or blocks or be a different shape. As long as there are 100 squares. After trying these boards, make your own.
• The game can be played with dice. Roll 2 dice, and you can fill in the total rolled.
• Advanced players could try where their score is not the area – but the perimeter. (You might want to try that with no slanted sides.)
Board 0:
Board 1:
Board 2:
Board 3:
Board 4:
It's pretty fun! I haven't been able to spot a degenerate strategy yet. Give it a try, and let me know what you think.
Materials: Game Board, 2 different color pens, pencils, crayons or markers.
How to play: Players take turns making a single shape on the board that has an area of 10 or less. The game is done when the board is filled, and the player with the most squares covered wins. The table is for recording how many squares you cover each turn.
The first player to go can only color up to 8 squares on their first shape. (Otherwise it would always be best to go first.) After that the limit is always 10.
The squares colored in to make your shape need to share a side, not just touch at a corner. (The shape you make has to be a polygon.) A player can even use slanted lines – as long as they can figure out the area for their shape! When you are making a new shape, it does not have to touch your old – you can put it anywhere there’s room.
When the board is completely filled, the game is done. Total up your squares and see who won!
Notice you can check if all the squares are counted by adding both scores – you should get 100.
Whoever lost gets to choose the next time if they want to go first or second.
Variations:
• The board can be changed to have holes or blocks or be a different shape. As long as there are 100 squares. After trying these boards, make your own.
• The game can be played with dice. Roll 2 dice, and you can fill in the total rolled.
• Advanced players could try where their score is not the area – but the perimeter. (You might want to try that with no slanted sides.)
Board 0:
Board 1:
Board 2:
Board 3:
Board 4:
It's pretty fun! I haven't been able to spot a degenerate strategy yet. Give it a try, and let me know what you think.
Sabtu, 24 Oktober 2009
Group Matematika SMA
Mungkin sebagian besar kita sudah memiliki akun di salah satu jejaring sosial paling banyak anggotanya di seluruh dunia, Facebook. Ya, sebagian besar kita sepertiya lebih betah untuk berlama - lama duduk di depan PC kita hanya untuk sekedar mengupdate status ataupun membalas di wall teman kita. Nah ngga ada salahnya kan kalau sebagian waktu anda ketika mengutak - atik fb anda mampir ke group
Selasa, 13 Oktober 2009
GeoGebra: Triangle Tuning
What can we deduce from the side lengths of a triangle?
This is a preservice teacher activity, easily adaptable to middle or high school use. GeoGebra is a free dynamic geometry program available as an online applet or downloadable program, from geogebra.org.
Objective: TLW explore triangle properties relating type and side length.
Schema Activation: What are the 7 triangle types? Fill them in in the ‘Type’ column below. You'll fill in the other columns as you go through the activity.
.....Type ....................... Examples [Like (3,5,5) ] .......................What do you notice?
1.
2.
3.
4.
5.
6.
7.
Focus: One of the reasons dynamic geometry is so powerful is the support it allows the teacher (or curriculum designer) to give students for finding examples. Lots of examples. As we’ve discussed in class, the natural way we reason is to go from lots of specific examples to the general.
Activity:
1) Open the sketch TriangleBySide.ggb. (Online at faculty.gvsu.edu/goldenj/TriangleBySide.html)
a. Collect at least 2 examples of each type of triangle. Where possible, try not to have the triangles be similar, where all the sides are multiples of another triangle.
b. Which were hardest to find? Was it something to do with the type or how you were looking?
2) Open the sketch PythagoreanData.ggb. (Online at faculty.gvsu.edu/goldenj/PythagoreanData.html)
a. Look at your right triangle examples on this triangle. How do you think ancient mathematicians noticed this cool pattern with the squares
b. Check your other examples on this sketch. For each, record whether the sum of the areas of the smaller squares is <, =, or > the area of the large square.
c. What pattern(s) do you notice?
d. Why do you think your pattern(s) might be true?
3) Go to http://teachers.henrico.k12.va.us/math/GeoGebra_Site/pythagoras/pythaPrblm1.html (link on Blackboard) for the Ladder sketch. Answer the questions there.
a. How high will he get if he places the ladder 3 m off the wall? Drag the ladder point and find a solution. Sketch the solution with all its lengths (s, r, h) on paper.
b. Now, calculate the solution of task (a) on paper. Which lengths are given, which are sought? Do you get the same solution?
c. At what distance of the wall should Pythagoras place his ladder in order to reach the window 4.50 m above the ground? Sketch the solution with all its lengths (s, r, h) on paper.
d. Now, calculate the solution of task (c) on paper. Compare your solution to your sketch?
e. As a teacher, what do you notice about the difference between doing the tasks in sketch and on the paper?
4) Open the sketch AdjustableLadder.ggb (Online at faculty.gvsu.edu/goldenj/AdjustableLadder.html)
a. What young Pythagoras might not have known is that a safe ladder ratio is 4:1 for the height to distance from the wall. What length ladder does he need to reach the window safely? Is there one solution or more?
b. How would you solve this problem on paper?
Reflection:
a. What confirmed, new or deepened understanding did you develop regarding triangles through these activities?
b. How did the dynamic environment affect your understanding?
EDIT:
Be sure to look at the comments. Scott Farrar has some notes on using this with 9th grade geometry, and has added a complimentary activity.
This is a preservice teacher activity, easily adaptable to middle or high school use. GeoGebra is a free dynamic geometry program available as an online applet or downloadable program, from geogebra.org.
Objective: TLW explore triangle properties relating type and side length.
Schema Activation: What are the 7 triangle types? Fill them in in the ‘Type’ column below. You'll fill in the other columns as you go through the activity.
.....Type ....................... Examples [Like (3,5,5) ] .......................What do you notice?
1.
2.
3.
4.
5.
6.
7.
Focus: One of the reasons dynamic geometry is so powerful is the support it allows the teacher (or curriculum designer) to give students for finding examples. Lots of examples. As we’ve discussed in class, the natural way we reason is to go from lots of specific examples to the general.
Activity:
1) Open the sketch TriangleBySide.ggb. (Online at faculty.gvsu.edu/goldenj/TriangleBySide.html)
b. Which were hardest to find? Was it something to do with the type or how you were looking?
2) Open the sketch PythagoreanData.ggb. (Online at faculty.gvsu.edu/goldenj/PythagoreanData.html)
b. Check your other examples on this sketch. For each, record whether the sum of the areas of the smaller squares is <, =, or > the area of the large square.
c. What pattern(s) do you notice?
d. Why do you think your pattern(s) might be true?
3) Go to http://teachers.henrico.k12.va.us/math/GeoGebra_Site/pythagoras/pythaPrblm1.html (link on Blackboard) for the Ladder sketch. Answer the questions there.
a. How high will he get if he places the ladder 3 m off the wall? Drag the ladder point and find a solution. Sketch the solution with all its lengths (s, r, h) on paper.
b. Now, calculate the solution of task (a) on paper. Which lengths are given, which are sought? Do you get the same solution?
c. At what distance of the wall should Pythagoras place his ladder in order to reach the window 4.50 m above the ground? Sketch the solution with all its lengths (s, r, h) on paper.
d. Now, calculate the solution of task (c) on paper. Compare your solution to your sketch?
e. As a teacher, what do you notice about the difference between doing the tasks in sketch and on the paper?
4) Open the sketch AdjustableLadder.ggb (Online at faculty.gvsu.edu/goldenj/AdjustableLadder.html)
a. What young Pythagoras might not have known is that a safe ladder ratio is 4:1 for the height to distance from the wall. What length ladder does he need to reach the window safely? Is there one solution or more?
b. How would you solve this problem on paper?
Reflection:
a. What confirmed, new or deepened understanding did you develop regarding triangles through these activities?
b. How did the dynamic environment affect your understanding?
EDIT:
Be sure to look at the comments. Scott Farrar has some notes on using this with 9th grade geometry, and has added a complimentary activity.
Math Resources
Moodle
Moodle Math documentation
http://docs.moodle.org/en/Mathematics
Pretty technical information, but accurate and detailed.
Book about teaching math with Moodle
http://www.packtpub.com/moodle-1-9-math/book
Content
http://illuminations.nctm.org/
NCTM’s free lesson and resource center, with many online applets. Note the Calculation Nation link leads to some excellent 1 and 2 player games, but require a free login account. (Guest access to try it.)
http://ohiorc.org/for/math/
An Ohio Resource Center with links to many topics by grade and content. Some high quality resources.
http://nrich.maths.org
From the British equivalent of the NCTM. Once you get past how they say ‘maths,’ there’s an excellent supply of interesting problems, some clever student solutions, and open challenges. Also features some interactive applets.
Internet Resources for in class use
http://math.hippocampus.org/
An extensive collection of online exposition of topics from algebra and calculus. (As well as Bio, Chemistry, Physics, etc. Some support for Spanish.) Includes a voice delivering reading of the text (with a skip forward feature), and assessment along the way. Not clear what the purchase of a license adds, but everything I’ve seen was free.
http://www.imagination3.com/
GE’s interactive whiteboard, open to collaboration, optional graph paper background. Colors and some shapes are possible. You can save files.
http://www.wolframalpha.com/
Wolfram|Alpha, a free computational engine online. Amazing and worth exploring. Also some encyclopedia function. Supercedes some of the older online utilities.
http://geogebra.org
GeoGebra, a free dynamic geometry program that can be used as a web app or by downloading and installing. Some powerful algebraic applications, too.
http://www.plu.edu/~heathdj/java/
A collection of java applets, from many different mathematical content areas.
http://www.univie.ac.at/future.media/moe/onlinewerkzeuge.html
Powerful collection of online tools. One of the best online function plotters (graphing calculator) is here.
Games
http://www.funbrain.com/brain/MathBrain/MathBrain.html
K-8 math games. Not very challenging, but there are students for wom this will be just right. Connected with Poptropica, which my kids love, and features some decent problem solving.
http://gprime.net/game/
Huge collection of flash games – not all kid appropriate. Some excellent math and reasoning games, though. Try 3D logic, a maze game on 3 cube faces (http://gprime.net/game.php/3dlogic), gridlock, the classic game (http://gprime.net/game.php/gridlock) or grow island, a deductive reasoning game (http://gprime.net/game.php/growisland), and one of a family of grow games (http://www.eyezmaze.com/grow/v3/index.html).
http://www.logicmazes.com/
Very challenging puzzles by a very interesting man, Robert Abbott. Also links to some games of his creation.
Utility
http://incompetech.com/graphpaper/lined/
Free graph paper pdf generator. Very handy.
Teacher
http://www.learner.org/resources/browse.html?discipline=5
Annenberg Teacher Resource Center. Extensive collection of lessons, videos of students and teachers, and professional development materials. All free. Sortable by grade level.
Moodle Math documentation
http://docs.moodle.org/en/Mathematics
Pretty technical information, but accurate and detailed.
Book about teaching math with Moodle
http://www.packtpub.com/moodle-1-9-math/book
Content
http://illuminations.nctm.org/
NCTM’s free lesson and resource center, with many online applets. Note the Calculation Nation link leads to some excellent 1 and 2 player games, but require a free login account. (Guest access to try it.)
http://ohiorc.org/for/math/
An Ohio Resource Center with links to many topics by grade and content. Some high quality resources.
http://nrich.maths.org
From the British equivalent of the NCTM. Once you get past how they say ‘maths,’ there’s an excellent supply of interesting problems, some clever student solutions, and open challenges. Also features some interactive applets.
Internet Resources for in class use
http://math.hippocampus.org/
An extensive collection of online exposition of topics from algebra and calculus. (As well as Bio, Chemistry, Physics, etc. Some support for Spanish.) Includes a voice delivering reading of the text (with a skip forward feature), and assessment along the way. Not clear what the purchase of a license adds, but everything I’ve seen was free.
http://www.imagination3.com/
GE’s interactive whiteboard, open to collaboration, optional graph paper background. Colors and some shapes are possible. You can save files.
http://www.wolframalpha.com/
Wolfram|Alpha, a free computational engine online. Amazing and worth exploring. Also some encyclopedia function. Supercedes some of the older online utilities.
http://geogebra.org
GeoGebra, a free dynamic geometry program that can be used as a web app or by downloading and installing. Some powerful algebraic applications, too.
http://www.plu.edu/~heathdj/java/
A collection of java applets, from many different mathematical content areas.
http://www.univie.ac.at/future.media/moe/onlinewerkzeuge.html
Powerful collection of online tools. One of the best online function plotters (graphing calculator) is here.
Games
http://www.funbrain.com/brain/MathBrain/MathBrain.html
K-8 math games. Not very challenging, but there are students for wom this will be just right. Connected with Poptropica, which my kids love, and features some decent problem solving.
http://gprime.net/game/
Huge collection of flash games – not all kid appropriate. Some excellent math and reasoning games, though. Try 3D logic, a maze game on 3 cube faces (http://gprime.net/game.php/3dlogic), gridlock, the classic game (http://gprime.net/game.php/gridlock) or grow island, a deductive reasoning game (http://gprime.net/game.php/growisland), and one of a family of grow games (http://www.eyezmaze.com/grow/v3/index.html).
http://www.logicmazes.com/
Very challenging puzzles by a very interesting man, Robert Abbott. Also links to some games of his creation.
Utility
http://incompetech.com/graphpaper/lined/
Free graph paper pdf generator. Very handy.
Teacher
http://www.learner.org/resources/browse.html?discipline=5
Annenberg Teacher Resource Center. Extensive collection of lessons, videos of students and teachers, and professional development materials. All free. Sortable by grade level.
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