Sunday, February 17, 2008
Sunday, February 10, 2008
春燕和她先生在亚非学院
昨天春燕和她的先生和我想去大英博物馆。我们下午两点半左右去，可是大英博物馆的外面有太多人。我们没进里面。昨天天气很好，所以我们走路去 Warren Street， 可是墨西哥饭馆儿下午三点关门。 所以我们走路回 Holborn。我们去日本饭馆儿。春燕先生说：《多吃点！多吃点！》然后他买午饭。这么中国人！
在中国，中国人回家过春节。春燕和她先生没回家过春节，所以他们昨天给家里打电话。春燕问我《你今天给你家打电话马？》她想我也是中国人。
Yesterday, Chunyan and her husband and I wanted to go to the British Museum for Chinese New Year. We went around 2:30, but there were too many people outside. We didn't go in. The weather was very nice yesterday, so we walked to Warren Street, but the Mexican restaurant closed at 3. So we walked back to Holborn. We went to a Japanese restaurant. Chunyan's husband said, "Eat more! Eat more!". Then he paid for lunch. So Chinese!
In China, people go home to see their families for Chinese New Year. Chunyan and her husband did not go to China for Chinese New Year, so yesterday they called their families. Chunyan asked me, "Did you call your family today?" She thought I was also Chinese.
Tuesday, February 05, 2008
看电影
昨天晚上我和春燕一起去电影院看电影。以后我们去他家吃晚饭。她做晚饭了。 晚饭很好吃。她做得很好。
她住在 Finchley Road 地铁站附近。她住在国王学校学生房子。一个我的学生也住在那里。他认识我，可是我不认识他。
我昨天晚上十一点半回家了。每天早上我差一刻六点左右起床。所以今天我很累。
Last night, Chunyan and I went to the movie theatre to see a film. Afterwards, we went to her house for dinner. She made dinner. It was delicious. She is a good cook.
She lives near Finchley Road tube station. She lives in a King's College residence hall. One of my students also lives there. He recognized me, but I didn't recognize him.
Last night I got home at 11:30. Every morning I get up around 5:45. So today I am very tired.
她住在 Finchley Road 地铁站附近。她住在国王学校学生房子。一个我的学生也住在那里。他认识我，可是我不认识他。
我昨天晚上十一点半回家了。每天早上我差一刻六点左右起床。所以今天我很累。
Last night, Chunyan and I went to the movie theatre to see a film. Afterwards, we went to her house for dinner. She made dinner. It was delicious. She is a good cook.
She lives near Finchley Road tube station. She lives in a King's College residence hall. One of my students also lives there. He recognized me, but I didn't recognize him.
Last night I got home at 11:30. Every morning I get up around 5:45. So today I am very tired.
Monday, January 21, 2008
Divisibility tests
One night while I was home over the holidays, I ended up helping Emma with her homework. First some spelling homework and then maths. She was confused about some problems on the distributive property and she also had to draw three rectangles with area 24, without using 1 x 24. She didn't have any sort of systematic method; she just cast about for integers that divide 24. She had come up with 6 x 4 and 3 x 8 but couldn't find a third rectangle. So I taught her about prime numbers and prime factorisation.
Somehow this led us to divisibility tests. She knew the one for 2 (and I assume she knows the similar tests for 5 and 10). Most people know these tests. My dad was making nachos (for some reason we were doing her homework on the kitchen floor) and overheard me teaching her the divisibility test for 4. He got all indignant: "I didn't know that! Why don't they teach us cool things like that in school?" and ran into the other room, from where we heard his shouts to Kathi, "Did you know that?!?"
The divisibility test for 4 has the same justification as the test for 2 (or, for that matter, for 5 or 10). For 2, you can write your (positive, for the sake of argument) lengthy integer as 10X + Y where Y is less than 10 (and greater than or equal to 0). Then, if Y is divisible by 2 (so Y = 2Y') we have
10X + Y = 2(5X) + 2Y' = 2(5X + Y')
so the big integer is also divisible by 2. (Note that this works because of the distributive property mentioned earlier.) And, likewise, if Y is not even then the big long integer is also not even. For 4, you instead write your integer as 100X + Y, where Y is now less than 100. Since 4 divides 100, you only have to check whether 4 divides Y (so now you have to check whether 4 divides a 2 digit number, which is more work than the even/odd test, but considerably less work than trying to divide 4 into, say, a 7 digit long number).
So Emma tested some big long numbers for divisibility by 4 and after we found a few that were not divisible by 4, I had her come up with a big long number that was divisible by 4. She ended her number with 08, which I liked.
Then I taught her the divisibility test for 3. If you add up the digits of the number and that sum is divisible by 3, then the whole number is divisible by 3. Rather than give a proof, which is much cleaner and nicer if you know about modular arithmetic, I will just give an example. This also uses the distributive property Emma was working on.
635,148 = 6(100,000) + 3(10,000) + 5(1,000) + 1(100) + 4(10) + 8
=6(1+99,999) + 3(1+9,999) + 5(1+999) + 1(1+99) + 4(1+9) + 8
=6+3+5+1+4+8 + 6(99,999) + 3(9,999) + 5(999) + 1(99) + 4(9)
=(6+3+5+1+4+8) + 3[6(33,333) + 3(3,333) + 5(333) + 1(33) + 4(3)]
So, since the right hand part of the last line (the 3[6(33,333) + 3(3,333) + 5(333) + 1(33) + 4(3)] part) is divisible by 3, that means the whole number is divisible by 3 if and only if the left hand part of the last line (the (6+3+5+1+4+8) part) is also divisible by 3. Since this left hand part is simply the sum of the digits of the number, we get our divisibility test. Of course the same argument works to show that the divisibility by 9 test is analogous: an integer is divisible by 9 if and only if the sum of the digits is divisible by 9.
Somehow this led us to divisibility tests. She knew the one for 2 (and I assume she knows the similar tests for 5 and 10). Most people know these tests. My dad was making nachos (for some reason we were doing her homework on the kitchen floor) and overheard me teaching her the divisibility test for 4. He got all indignant: "I didn't know that! Why don't they teach us cool things like that in school?" and ran into the other room, from where we heard his shouts to Kathi, "Did you know that?!?"
The divisibility test for 4 has the same justification as the test for 2 (or, for that matter, for 5 or 10). For 2, you can write your (positive, for the sake of argument) lengthy integer as 10X + Y where Y is less than 10 (and greater than or equal to 0). Then, if Y is divisible by 2 (so Y = 2Y') we have
so the big integer is also divisible by 2. (Note that this works because of the distributive property mentioned earlier.) And, likewise, if Y is not even then the big long integer is also not even. For 4, you instead write your integer as 100X + Y, where Y is now less than 100. Since 4 divides 100, you only have to check whether 4 divides Y (so now you have to check whether 4 divides a 2 digit number, which is more work than the even/odd test, but considerably less work than trying to divide 4 into, say, a 7 digit long number).
So Emma tested some big long numbers for divisibility by 4 and after we found a few that were not divisible by 4, I had her come up with a big long number that was divisible by 4. She ended her number with 08, which I liked.
Then I taught her the divisibility test for 3. If you add up the digits of the number and that sum is divisible by 3, then the whole number is divisible by 3. Rather than give a proof, which is much cleaner and nicer if you know about modular arithmetic, I will just give an example. This also uses the distributive property Emma was working on.
635,148 = 6(100,000) + 3(10,000) + 5(1,000) + 1(100) + 4(10) + 8
=6(1+99,999) + 3(1+9,999) + 5(1+999) + 1(1+99) + 4(1+9) + 8
=6+3+5+1+4+8 + 6(99,999) + 3(9,999) + 5(999) + 1(99) + 4(9)
=(6+3+5+1+4+8) + 3[6(33,333) + 3(3,333) + 5(333) + 1(33) + 4(3)]
So, since the right hand part of the last line (the 3[6(33,333) + 3(3,333) + 5(333) + 1(33) + 4(3)] part) is divisible by 3, that means the whole number is divisible by 3 if and only if the left hand part of the last line (the (6+3+5+1+4+8) part) is also divisible by 3. Since this left hand part is simply the sum of the digits of the number, we get our divisibility test. Of course the same argument works to show that the divisibility by 9 test is analogous: an integer is divisible by 9 if and only if the sum of the digits is divisible by 9.
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