2/5/2010 & 2/12/2010
The following problem gave kids introduction to arithmetic progression.
We had 11 kids in math class on that day.
I asked all kids to stand up and chose two kids to come forward. Then I asked everybody how many handshakes these two kids can do (condition, they should not do more than 1 handshake among each other). Kids said: 1. Two kids did 1 handshake. Correct!
Then I chose another child to join the first two and asked the same question. Kids calculated: 3.
3 kids did 3 handshakes. Correct!
I added fourth child asking the same question. Kids calculated: 6. 4 kids did their handshakes and confirmed that the answer is 6.
We did the same by adding more and more kids to the group.
At the point when there were 8 kids I showed them that by adding 1 more child the number of handshakes = Previous Number of Handshakes + The handshakes that the new kid has to do. So in case of 8 kids we have 21 + 7 = 28.
When we have 9 kids there will be 28 (from previous time) + 8 = 36. 10 kids: 36 + 9 = 45.
Then I showed that for
2 children -> 1 handshake
3 children -> 1 + 2 handshakes
4 children -> 1+2+3 handshakes
etc.
Next, I showed them arithmetic progression
1+2+3+4+5+6+7+8+9+10
I asked them a question whether they see a pattern if you add the first and last number, 1+10, second and second from last, 2+9,etc. All those sums are equal to 11.
Then I asked how many sums like this (or pairs) there are. They calculated: 5.
So we need to add 11 five times = >11 x 5 = 55
Then I asked kids to add numbers 1+2+3+...+10 and they got 55 as well.
I showed them that this rule works for any sequence (arithmetic progression). They tried to add 2+3+4+5, 1+2+3+4+5+6, etc. using the rule above (adding first and last number and multiplying it by number of pairs).
Sunday, February 28, 2010
How old are dad and his son?
2/26/2010
This problem proved to be very easy for my second graders.
The total age for a father and his son is 40 years old.
How old will they be in 3 years?
All kids answered and explained why within seconds. 46
This problem proved to be very easy for my second graders.
The total age for a father and his son is 40 years old.
How old will they be in 3 years?
All kids answered and explained why within seconds. 46
How many books are on each shelf?
2/26/2010
Today I had 8 kids from the second grades at my advanced math lesson.
Here is the problem that I gave to them.
There are 42 book on two shelves. First shelf has 12 books more than the second one. How many books are on each shelf?
I asked kids to give me an answer writing it on their small whiteboards they brought to the class. Surprisingly they all gave me the right answer within several minutes. They just tried several numbers combinations and found the correct one.
Then I showed on the blackboard we have in the classroom how I would solve the same problem using simple equation. At each step I asked them questions to make it more interactive and ensure that they are engaged.
Second shelf has ? books.
How many books are on the first shelf?: ? + 12.
How many books all together? 42. Or you need to add number of books on the first shelf, ?, and number of books on the second shelf, ? + 12.
So we have, ? + (? + 12) = 42 =>
? + ? + 12 = 42 =>
(? + ?) = 42 - 12 =>
? + ? = 30 =>
2 x ? = 30 =>
What number you need to multiply by 2 to get 30? They answered 15.
So now we know that the second shelf has 15 books.
How many books are on the first shelf? 15 + 12 = 27 books.
Let's check the answer. 15 (number of books on the second shelf) + 27 (number of books on the second shelf) = 42 - Correct!
Next I gave them different variations of the same problem, asking one of them to come to the blackboard and solve the problem while others would do the same on their whiteboards. I asked them not to try different combinations but rather use equations we just learned.
30 books, first shelf has 4 more books than the second, 18 books with the first shelf having 2 more books, etc.
Today I had 8 kids from the second grades at my advanced math lesson.
Here is the problem that I gave to them.
There are 42 book on two shelves. First shelf has 12 books more than the second one. How many books are on each shelf?
I asked kids to give me an answer writing it on their small whiteboards they brought to the class. Surprisingly they all gave me the right answer within several minutes. They just tried several numbers combinations and found the correct one.
Then I showed on the blackboard we have in the classroom how I would solve the same problem using simple equation. At each step I asked them questions to make it more interactive and ensure that they are engaged.
Second shelf has ? books.
How many books are on the first shelf?: ? + 12.
How many books all together? 42. Or you need to add number of books on the first shelf, ?, and number of books on the second shelf, ? + 12.
So we have, ? + (? + 12) = 42 =>
? + ? + 12 = 42 =>
(? + ?) = 42 - 12 =>
? + ? = 30 =>
2 x ? = 30 =>
What number you need to multiply by 2 to get 30? They answered 15.
So now we know that the second shelf has 15 books.
How many books are on the first shelf? 15 + 12 = 27 books.
Let's check the answer. 15 (number of books on the second shelf) + 27 (number of books on the second shelf) = 42 - Correct!
Next I gave them different variations of the same problem, asking one of them to come to the blackboard and solve the problem while others would do the same on their whiteboards. I asked them not to try different combinations but rather use equations we just learned.
30 books, first shelf has 4 more books than the second, 18 books with the first shelf having 2 more books, etc.
Saturday, February 20, 2010
When will we meet?
2/12/2010
I asked children to solve the following problem.
Let's say between Farmington and West Hartford is 10 miles. One person leaves Farmington and moves towards West Hartford. Another person goes from West Hartford towards Farmington. They both take the same road. The first person goes 4 miles per hour, the second 1 mile per hour. In how many hours will they meet?
I took toys and marked each mile with a toy on the floor. I asked 2 kids to volunteer . The first child stood at one end of the 'road': Farmington. The second child stood in another end: West Hartford. I asked kids where they would be in one hour. The first child stepped to the toy marking end of 4th mile. The second child moved to the toy marking end of 1st mile (from his side). Then I asked them where they would be in two hours. Kids met at 8th mile (for first child) and 2nd mile (for second child).
In how many hours will they meet? 2. Correct!
After that I changed distance and speeds calling different children and asking the same question: When will they meet?
After we were done I showed another way to solve similar problems.
I explained that in one hour both people will move 4 + 1 miles together. 4 + 1 = 5 miles. We have 10 miles total. How many times you need to take 5 miles to get to 10? 2. Correct!
I asked children to solve the following problem.
Let's say between Farmington and West Hartford is 10 miles. One person leaves Farmington and moves towards West Hartford. Another person goes from West Hartford towards Farmington. They both take the same road. The first person goes 4 miles per hour, the second 1 mile per hour. In how many hours will they meet?
I took toys and marked each mile with a toy on the floor. I asked 2 kids to volunteer . The first child stood at one end of the 'road': Farmington. The second child stood in another end: West Hartford. I asked kids where they would be in one hour. The first child stepped to the toy marking end of 4th mile. The second child moved to the toy marking end of 1st mile (from his side). Then I asked them where they would be in two hours. Kids met at 8th mile (for first child) and 2nd mile (for second child).
In how many hours will they meet? 2. Correct!
After that I changed distance and speeds calling different children and asking the same question: When will they meet?
After we were done I showed another way to solve similar problems.
I explained that in one hour both people will move 4 + 1 miles together. 4 + 1 = 5 miles. We have 10 miles total. How many times you need to take 5 miles to get to 10? 2. Correct!