1. How many friends exchanged pictures.
When leaving the summer camp friends exchanged pictures. Each friend gave each other 1 picture. How many friends exchanged pictures if we know that all together they used 6 pictures.
2. How many digits '7'
Joan writes out each of the first 100 counting numbers. How many times did she write the digit 7?
3. How many months in a single year have 28 days?
4. How many candles left.
3 candles were lit. 2 faded. How many candles left.
5. Find the sum of the digits of the first 20 odd numbers
6. Who says '1'?
This problem can be solved by all students in math class sitting in a circle and doing counting.
Five students (Amy, Beth, Corey, Diego, Emily) sit in that order in a circle, counting down to 1. Amy starts by saying, “34”. Then Beth says, “33”, and so on. They continue around the circle to count down by ones. Who says, “1”?
7. How long is the fence?
Ten poles are placed 6 meters apart in a straight line. A fence runs from the first pole to the last pole. How long is the fence?
Monday, March 29, 2010
Friday, March 26, 2010
Twins
03/26/2010
Here is the problem that we did not have time to complete today. We asked children to take it as their homework. Next time when we meet I would like kids to explain how to resolve the problem.
In the second grade there are 4 pairs of twins among other children. On Halloween party each child in that class brought dad and mom and they are all in the gym. No other people are in the gym. There are 85 people all together. How many kids in that second grade?
If you help solving this problem then you can use the following directions. It is how I started to explain to my group of kids.
First I asked them a question: We have 4 pairs of twins, how many children is it? 4 x 2 = 8. Correct.
How many parents each pair of twins have? 2: mom and dad.
How many parents those 8 kids have? 4 pair x 2 = 8. Correct.
How many children and parents for those 4 pairs of twins. 8 children + 8 parents = 16 people.
We know that there are 85 people in the gym. It includes children who are twins and their parents AND children who are not twins and their parents.
We already know that there are 16 people (twins and their parents). How many people you need to add to 16 in order to get 85? This way we will find number of children who are not twins and their parents.
16 + ? = 85, where '?' is number of children who are not twins and their parents.
? = 69 Correct. So now we know that number of children who are not twins and their parents is equal to 69.
It is where our group stopped.
To continue the thought process we can do the following.
There 69 people who are children (who are not twins) and their parents. We know that each such child has 2 parents. So we have (1 child + 2 parents) x ? = 69. In other words we have a family of 3 people (1 child + 2 parents). How many such families we have in order to get to 69 people? The answer is 23.
So we have 23 families with 1 child in each of them . So how many children we have in those families all together? The answer is 23.
So we have 4 pair of twins = 8 children + 23 children (who are not twins) = 31.
It means that number of children in the class is 31.
Let's now verify that we solved the problem correctly.
4 pairs of twins => 4x2 children + 8 parents + 23 (children who are not twins) + 23 x 2 (parents for children who are not twins) = 8 + 8 + 23 + 46 = 85. So we got 85 people all together.
Here is the problem that we did not have time to complete today. We asked children to take it as their homework. Next time when we meet I would like kids to explain how to resolve the problem.
In the second grade there are 4 pairs of twins among other children. On Halloween party each child in that class brought dad and mom and they are all in the gym. No other people are in the gym. There are 85 people all together. How many kids in that second grade?
If you help solving this problem then you can use the following directions. It is how I started to explain to my group of kids.
First I asked them a question: We have 4 pairs of twins, how many children is it? 4 x 2 = 8. Correct.
How many parents each pair of twins have? 2: mom and dad.
How many parents those 8 kids have? 4 pair x 2 = 8. Correct.
How many children and parents for those 4 pairs of twins. 8 children + 8 parents = 16 people.
We know that there are 85 people in the gym. It includes children who are twins and their parents AND children who are not twins and their parents.
We already know that there are 16 people (twins and their parents). How many people you need to add to 16 in order to get 85? This way we will find number of children who are not twins and their parents.
16 + ? = 85, where '?' is number of children who are not twins and their parents.
? = 69 Correct. So now we know that number of children who are not twins and their parents is equal to 69.
It is where our group stopped.
To continue the thought process we can do the following.
There 69 people who are children (who are not twins) and their parents. We know that each such child has 2 parents. So we have (1 child + 2 parents) x ? = 69. In other words we have a family of 3 people (1 child + 2 parents). How many such families we have in order to get to 69 people? The answer is 23.
So we have 23 families with 1 child in each of them . So how many children we have in those families all together? The answer is 23.
So we have 4 pair of twins = 8 children + 23 children (who are not twins) = 31.
It means that number of children in the class is 31.
Let's now verify that we solved the problem correctly.
4 pairs of twins => 4x2 children + 8 parents + 23 (children who are not twins) + 23 x 2 (parents for children who are not twins) = 8 + 8 + 23 + 46 = 85. So we got 85 people all together.
How to make a glass half full?
03/26/2010
There are three glasses. The first glass can contain maximum 8 oz of water, the second 5 oz, and the third 3 oz. The first glass is full with water. How to make this glass half full?
This problem is similar to what we solved the other day with two jars (How to fill a jar with water) . See here.
For each sub-group I brought 3 cups : the first representing 8 oz, the second 5 oz, and the third 3 oz.
Kids went to the water fountain and filled in 8 oz cups full with water. Then we started thinking what we should do in order to make 8 oz cup filled with 4 oz of water. Somebody suggested to put 3 oz into 3 oz cup from the largest cup.
Kids thought about various alternatives: fill in 5 oz cup from 8 oz cup, pour 3 oz to 5 oz from 3 oz cup . I hinted that it might be a path to the actual solution. So they did it and continued discussing the next steps.
At that time I hinted them that if necessary they can pour some water out (not necessarily into another cup). Children continued deliberating. Somebody suggested fill 3 oz cup from 8 oz cup again. So at this point we had 2 oz in 8 oz cup.
We continued the discussion. With my hints they poured water from 3 oz cup into 5 oz to make 5 oz cup full. Then I told them that they can spill out 1 oz remaining in 3 oz cup.
So after that step they had 2 oz in 8 oz cup, 5 oz in 5 oz cup, and nothing in 3 oz cup.
I told kids that they should keep 2 oz in 8 oz cup and do something with 5 oz and 3 oz cups. So they filled 3 oz cup. I asked them: How much water in 5 oz cup now? 2 oz.
How to get 4 oz in 8 oz cup now? They realized that by pouring 2 oz into 8 oz cup they would solve the problem.
After that I asked everybody to repeat how to solve the problem. So they all went through the thought process again.
There are three glasses. The first glass can contain maximum 8 oz of water, the second 5 oz, and the third 3 oz. The first glass is full with water. How to make this glass half full?
This problem is similar to what we solved the other day with two jars (How to fill a jar with water) . See here.
For each sub-group I brought 3 cups : the first representing 8 oz, the second 5 oz, and the third 3 oz.
Kids went to the water fountain and filled in 8 oz cups full with water. Then we started thinking what we should do in order to make 8 oz cup filled with 4 oz of water. Somebody suggested to put 3 oz into 3 oz cup from the largest cup.
Kids thought about various alternatives: fill in 5 oz cup from 8 oz cup, pour 3 oz to 5 oz from 3 oz cup . I hinted that it might be a path to the actual solution. So they did it and continued discussing the next steps.
At that time I hinted them that if necessary they can pour some water out (not necessarily into another cup). Children continued deliberating. Somebody suggested fill 3 oz cup from 8 oz cup again. So at this point we had 2 oz in 8 oz cup.
We continued the discussion. With my hints they poured water from 3 oz cup into 5 oz to make 5 oz cup full. Then I told them that they can spill out 1 oz remaining in 3 oz cup.
So after that step they had 2 oz in 8 oz cup, 5 oz in 5 oz cup, and nothing in 3 oz cup.
I told kids that they should keep 2 oz in 8 oz cup and do something with 5 oz and 3 oz cups. So they filled 3 oz cup. I asked them: How much water in 5 oz cup now? 2 oz.
How to get 4 oz in 8 oz cup now? They realized that by pouring 2 oz into 8 oz cup they would solve the problem.
After that I asked everybody to repeat how to solve the problem. So they all went through the thought process again.
Rainy day
03/26/2010
I gave the following problem to children, so they would have some fun. No calculations required.
There are 10 adults, 20 children, 5 dogs. They are all standing under one tiny umbrella. In what particular case they will not be wet?
My answer: No rain at that day, so nobody will be wet.
Here are some answers from kids, nobody gave my answer though :)
They are inside of a submarine.
The umbrella is not really tiny, it is huge.
They are in the huge tent.
I gave the following problem to children, so they would have some fun. No calculations required.
There are 10 adults, 20 children, 5 dogs. They are all standing under one tiny umbrella. In what particular case they will not be wet?
My answer: No rain at that day, so nobody will be wet.
Here are some answers from kids, nobody gave my answer though :)
They are inside of a submarine.
The umbrella is not really tiny, it is huge.
They are in the huge tent.
How much cake is left for tomorrow?
03/26/2010
A cake is cut into 15 equal parts. Michael ate 1/3 of the cake, Jane 2/5 of the cake, and Liz ate 1/3 from what Jane ate. The rest of the cake is left for tomorrow. What part of the cake is left for tomorrow?
This problem was relatively easy for kids in our math group.
First, kids found how many pieces Michael ate. What is 1/3 form 15? 5.
Then they figured out how much cake Jane ate. For some kids it was difficult to find 2/5. So I posed the question: What is 1/5 of the cake? 1/5 from 15 is 3. Correct.
If we know that 1/5 is 3 what is 2/5? 2 x 3 = 6. Correct.
Liz ate 1/3 from 6 (what Jane ate). 2. Correct
So now we can calculate how many pieces Michael, Jane and Liz ate together. 5 (for Michael) + 6 (for Jane) + 2 (for Liz) = 13.
How many pieces left for tomorrow if we know that Michael, Jane and Liz ate 13 pieces? 15-13 = 2.
What part of the cake is left for tomorrow? 2/15. Correct!
A cake is cut into 15 equal parts. Michael ate 1/3 of the cake, Jane 2/5 of the cake, and Liz ate 1/3 from what Jane ate. The rest of the cake is left for tomorrow. What part of the cake is left for tomorrow?
This problem was relatively easy for kids in our math group.
First, kids found how many pieces Michael ate. What is 1/3 form 15? 5.
Then they figured out how much cake Jane ate. For some kids it was difficult to find 2/5. So I posed the question: What is 1/5 of the cake? 1/5 from 15 is 3. Correct.
If we know that 1/5 is 3 what is 2/5? 2 x 3 = 6. Correct.
Liz ate 1/3 from 6 (what Jane ate). 2. Correct
So now we can calculate how many pieces Michael, Jane and Liz ate together. 5 (for Michael) + 6 (for Jane) + 2 (for Liz) = 13.
How many pieces left for tomorrow if we know that Michael, Jane and Liz ate 13 pieces? 15-13 = 2.
What part of the cake is left for tomorrow? 2/15. Correct!
Tuesday, March 23, 2010
Proposal for Math Lesson on March 26
Father and his sons
Father is 41 years old. He has 3 sons. The first one is 13, the second is 10, and the third is 6 years old. In how many years the father's age will be equal to sum of his sons' ages?
How much cake is left for tomorrow?
A cake is cut into 15 equal parts. Michael ate 1/3 of the cake, Jane 2/5 of the cake, and Liz ate 1/3 from what Jane ate. The rest of the cake is left for tomorrow. What part of the cake is left for tomorrow?
Rainy day
There are 10 adults, 20 children, 5 dogs. They are all standing under one tiny umbrella. In what particular case they will not be wet?
How to make a glass half full?
There are three glasses. The first glass can contain maximum 8 oz of water, the second 5 oz, and the third 3 oz. The first glass is full with water. How to make this glass half full?
Twins
In the second grade there are 4 pairs of twins among other children. On Halloween party each child in that class brought dad and mom and they are all in the gym. No other people are in the gym. There are 85 people all together. How many kids in that second grade?
Horses
A pair of horses ran 20 miles. How many miles did each horse run?
Father is 41 years old. He has 3 sons. The first one is 13, the second is 10, and the third is 6 years old. In how many years the father's age will be equal to sum of his sons' ages?
How much cake is left for tomorrow?
A cake is cut into 15 equal parts. Michael ate 1/3 of the cake, Jane 2/5 of the cake, and Liz ate 1/3 from what Jane ate. The rest of the cake is left for tomorrow. What part of the cake is left for tomorrow?
Rainy day
There are 10 adults, 20 children, 5 dogs. They are all standing under one tiny umbrella. In what particular case they will not be wet?
How to make a glass half full?
There are three glasses. The first glass can contain maximum 8 oz of water, the second 5 oz, and the third 3 oz. The first glass is full with water. How to make this glass half full?
Twins
In the second grade there are 4 pairs of twins among other children. On Halloween party each child in that class brought dad and mom and they are all in the gym. No other people are in the gym. There are 85 people all together. How many kids in that second grade?
Horses
A pair of horses ran 20 miles. How many miles did each horse run?
Tuesday, March 16, 2010
Proposal for Math lesson on March 19
How many apples a farmer needs to pick up?
A farmer picked up 16 apples. This is 1/4 from all apples he planned to pick up. How many more apples a farmer needs to pick up.
How much candies a woman purchased?
A woman bought 30 lbs of candies. 1/3 of all candies are made with cherry, 2/5 of all candies are with strawberry, the rest are with raspberry. How much candies with raspberry she purchased?
What is the prices for a cake?
A person earned $2,000. From this amount he spent 1% on purchasing a cake. What is the price for a cake?
How many people are in the movie theater?
There are many people are in the movie theater. 1% from those people is 7 people. How many people are in the movie theater?
How a sheet of paper can be divided?
Take a sheet of paper and draw 3 straight lines. What is the maximum number of parts that sheet of paper can be divide by those 3 lines?
How many two digits numbers where all digits are even?
A farmer picked up 16 apples. This is 1/4 from all apples he planned to pick up. How many more apples a farmer needs to pick up.
How much candies a woman purchased?
A woman bought 30 lbs of candies. 1/3 of all candies are made with cherry, 2/5 of all candies are with strawberry, the rest are with raspberry. How much candies with raspberry she purchased?
What is the prices for a cake?
A person earned $2,000. From this amount he spent 1% on purchasing a cake. What is the price for a cake?
How many people are in the movie theater?
There are many people are in the movie theater. 1% from those people is 7 people. How many people are in the movie theater?
How a sheet of paper can be divided?
Take a sheet of paper and draw 3 straight lines. What is the maximum number of parts that sheet of paper can be divide by those 3 lines?
How many two digits numbers where all digits are even?
Friday, March 12, 2010
How to fill a jar with water?
3/12/2010
First of all, thanks to Smita who helped to teach today's class.
Wow! Today our 2nd grade math class solved the very difficult problem. I am still amazed that they did it.
There are 2 empty jars. The first jar can be filled with 5 ounces of water and the second jar with 3 ounces.
How do you fill the largest jar with 1 ounce of water? You can fill a jar or empty it, if necessary, and use any amount of water.
Before going to the class I asked several adults to solve the problem. Some adults found a solution within several minutes others spent hours. Before reading how to solve the problem try yourself. When preparing for the class I did not realize that the problem can be solved in more than one way.
We divided the class in two groups, so Smita worked with one and I with another. I brought 2 pairs of glasses. Each group were given 2 glasses: one glass represented the largest jar (5 oz) and another the smallest one (3 oz).
Kids started with running to the water fountain, filling glasses with water and pouring water from one glass to another. Some kids suggested to fill in the largest glass with 5 oz, pour water into the smallest glass leaving 2 oz in the largest one. However they realized that they could not get precisely 1 oz. Kids tried several ways, but nothing worked. They thought hard for about 10 minutes. Then we asked them to stop and we switched to other problems. We told them that this will be their homework...
Because kids finished all the problems that we prepared for them fast we had some time to work at 'jar' problem again.
Smita guided her group through the thinking process. I believe that their solution was the following.
Kids made 5 oz glass full . Then they poured 3 oz from the largest into the smallest glass. So they had 2 oz left in the largest glass. Because kids could see through the glass they took a marker and marked the level for 2 oz on the glass. Then they emptied the largest glass, poured 3 oz from the smallest to the largest. Afterward kids poured water from the largest into smallest glass up to the mark they put earlier. This way they ensured that they had only 1 oz in the smallest glass. Then the largest glass was emptied and they poured 1 oz from 3 oz glass into 5 oz glass. The problem is solved!
My group solved the problem by accident :) One kid suggested to make 3 oz glass full, pour water from that glass into 5 oz glass, and then repeat it one more time. He did not realize that he was so close to solving the problem. Then one girl said: Stop! If we put 3 oz of water from the smallest glass into the largest and repeat the same thing up to the point until 5 oz glass is full, then we would have exactly 1 oz in the smallest glass and we can put that 1 oz into the largest one afterward. The solution is found!
After the class one boy came to me and said: It was awesome!
First of all, thanks to Smita who helped to teach today's class.
Wow! Today our 2nd grade math class solved the very difficult problem. I am still amazed that they did it.
There are 2 empty jars. The first jar can be filled with 5 ounces of water and the second jar with 3 ounces.
How do you fill the largest jar with 1 ounce of water? You can fill a jar or empty it, if necessary, and use any amount of water.
Before going to the class I asked several adults to solve the problem. Some adults found a solution within several minutes others spent hours. Before reading how to solve the problem try yourself. When preparing for the class I did not realize that the problem can be solved in more than one way.
We divided the class in two groups, so Smita worked with one and I with another. I brought 2 pairs of glasses. Each group were given 2 glasses: one glass represented the largest jar (5 oz) and another the smallest one (3 oz).
Kids started with running to the water fountain, filling glasses with water and pouring water from one glass to another. Some kids suggested to fill in the largest glass with 5 oz, pour water into the smallest glass leaving 2 oz in the largest one. However they realized that they could not get precisely 1 oz. Kids tried several ways, but nothing worked. They thought hard for about 10 minutes. Then we asked them to stop and we switched to other problems. We told them that this will be their homework...
Because kids finished all the problems that we prepared for them fast we had some time to work at 'jar' problem again.
Smita guided her group through the thinking process. I believe that their solution was the following.
Kids made 5 oz glass full . Then they poured 3 oz from the largest into the smallest glass. So they had 2 oz left in the largest glass. Because kids could see through the glass they took a marker and marked the level for 2 oz on the glass. Then they emptied the largest glass, poured 3 oz from the smallest to the largest. Afterward kids poured water from the largest into smallest glass up to the mark they put earlier. This way they ensured that they had only 1 oz in the smallest glass. Then the largest glass was emptied and they poured 1 oz from 3 oz glass into 5 oz glass. The problem is solved!
My group solved the problem by accident :) One kid suggested to make 3 oz glass full, pour water from that glass into 5 oz glass, and then repeat it one more time. He did not realize that he was so close to solving the problem. Then one girl said: Stop! If we put 3 oz of water from the smallest glass into the largest and repeat the same thing up to the point until 5 oz glass is full, then we would have exactly 1 oz in the smallest glass and we can put that 1 oz into the largest one afterward. The solution is found!
After the class one boy came to me and said: It was awesome!
What is the weight of the apple?
3/12/2010
I knew that 2nd graders study fractions this month, so I decided to give them 'fraction' problem.
The weight of the apple is 400 grams. What is the weight of 1/5, 3/10, 3/4 of that apple?
To determine the weight of 1/5, kids drew an apple and divided it into 5 parts. However it proved to be a very complicated task for them.
Then we started with more simple problem. Let's say the weight is not 400, but 100 grams. What is the weight of 1/2? Kids replied: 50 grams. Correct! What is the weight of 1/4? 25 grams, kids replied. Correct! What is the weight of 3/4? 75. Correct!
Now back to the original problem, the weight is 400 grams. What is the weight of 1/2? 200 grams. Correct! What is the weight of 1/4? 100 grams. Correct! What is the weight of 1/5? This was still difficult for kids.
I drew a circle representing an apple. Divided circle with 5 equal parts. Each part is 1/5. I marked each part with ?. We have ? + ? + ? + ? + ? = 400 grams => 5 x ? = 400 grams. What number do you need to take 5 times to get 400? Kids replied: 80. Correct! What is 2/5 then? Kids replied: 160. Correct! What is 3/5? 240. Correct!
I knew that 2nd graders study fractions this month, so I decided to give them 'fraction' problem.
The weight of the apple is 400 grams. What is the weight of 1/5, 3/10, 3/4 of that apple?
To determine the weight of 1/5, kids drew an apple and divided it into 5 parts. However it proved to be a very complicated task for them.
Then we started with more simple problem. Let's say the weight is not 400, but 100 grams. What is the weight of 1/2? Kids replied: 50 grams. Correct! What is the weight of 1/4? 25 grams, kids replied. Correct! What is the weight of 3/4? 75. Correct!
Now back to the original problem, the weight is 400 grams. What is the weight of 1/2? 200 grams. Correct! What is the weight of 1/4? 100 grams. Correct! What is the weight of 1/5? This was still difficult for kids.
I drew a circle representing an apple. Divided circle with 5 equal parts. Each part is 1/5. I marked each part with ?. We have ? + ? + ? + ? + ? = 400 grams => 5 x ? = 400 grams. What number do you need to take 5 times to get 400? Kids replied: 80. Correct! What is 2/5 then? Kids replied: 160. Correct! What is 3/5? 240. Correct!
What is the price of the book?
3/12/2010
This is another fraction problem we did today.
We know that 1/5 of the book's price is $2. What is the price of the book?
Kids solved the problem within several seconds. The answer is 10. Too easy!
This is another fraction problem we did today.
We know that 1/5 of the book's price is $2. What is the price of the book?
Kids solved the problem within several seconds. The answer is 10. Too easy!
How to get from Boston to New York?
3/12/2010
You can reach Hartford from Boston by bus and by train. You can reach New York City from Hartford by bus, train, and plane. In how many different ways you can reach New York City from Boston if you know that you shall stop in Hartford? Please note that you can change transportation in Hartford (e.g. switching from bus to train).
I asked kids to draw 3 points on the white board representing Boston, Hartford, and New York City. They connected Boston -Hartford and Hartford - New York. Then to help them visualizing they drew bus and train next to Boston - Hartford line and bus, train, and plane next to Hartford - New York line. Then they started counting all combinations: bus - bus, bus - train, etc. One girl wrote 'bus' and 'train' on the left side of her white board and 'bus', 'train', and 'plane' on the right side. Then she connected all possible combinations.
The answer is 6.
Afterward I showed to them that instead of just counting all combinations one by one they could just multiple 2 by 3, where 2 is how many different ways to get from Boston to Hartford and 3 is how many ways to get from Hartford to New York.
You can reach Hartford from Boston by bus and by train. You can reach New York City from Hartford by bus, train, and plane. In how many different ways you can reach New York City from Boston if you know that you shall stop in Hartford? Please note that you can change transportation in Hartford (e.g. switching from bus to train).
I asked kids to draw 3 points on the white board representing Boston, Hartford, and New York City. They connected Boston -Hartford and Hartford - New York. Then to help them visualizing they drew bus and train next to Boston - Hartford line and bus, train, and plane next to Hartford - New York line. Then they started counting all combinations: bus - bus, bus - train, etc. One girl wrote 'bus' and 'train' on the left side of her white board and 'bus', 'train', and 'plane' on the right side. Then she connected all possible combinations.
The answer is 6.
Afterward I showed to them that instead of just counting all combinations one by one they could just multiple 2 by 3, where 2 is how many different ways to get from Boston to Hartford and 3 is how many ways to get from Hartford to New York.
Wednesday, March 10, 2010
Proposal for Math Lesson on March 12
Here is the proposed problem for next Friday.
How to fill a jar with water?
There are 2 empty jars. The first jar can be filled with 5 gallons of water
and the second jar with 3 gallons.
How do you fill the largest jar with 1 gallon of water?
What is the weight of the apple?
The weight of the apple is 400 grams. What is the weight of 1/5, 3/10, 3/4 of that apple?
What is the price of the book?
We know that 1/5 of the book's price is $2. What is the price of the book?
How to get from Boston to New York?
You can reach Hartford from Boston by bus and by train. You can reach New York City from Hartford by bus, train, and plane. In how many different ways you can reach New York City from Boston if you know that you shall stop in Hartford? Please note that you can change transportation in Hartford (e.g. switching from bus to train).
How to fill a jar with water?
There are 2 empty jars. The first jar can be filled with 5 gallons of water
and the second jar with 3 gallons.
How do you fill the largest jar with 1 gallon of water?
What is the weight of the apple?
The weight of the apple is 400 grams. What is the weight of 1/5, 3/10, 3/4 of that apple?
What is the price of the book?
We know that 1/5 of the book's price is $2. What is the price of the book?
How to get from Boston to New York?
You can reach Hartford from Boston by bus and by train. You can reach New York City from Hartford by bus, train, and plane. In how many different ways you can reach New York City from Boston if you know that you shall stop in Hartford? Please note that you can change transportation in Hartford (e.g. switching from bus to train).
Friday, March 5, 2010
How many games teams play in soccer tournament?
3/5/2010
This problem is very similar to 'handshakes' problem.
I had 2 groups of children. First group had 3 kids and the second had 4 kids. Each child represented a soccer team. All those teams are playing in soccer tournament. In first group there are 3 teams and in the second there are 4. Teams play against another 3 games. How many games played in each group?
To solve this I reminded children about 'handshake' problem. Each handshake represents a game.
Kids did handshakes and calculated total number of games. Kids in both groups calculated number of games correctly by just counting total number of handshakes among each other.
Then I asked them to step back. I reminded them how we could solve similar problems by applying arithmetic progression: 3 handshakes between 2 kids, 6 if add third kid, 9 when fourth kid is added. So we have 3+6+9=18.
After that I asked: What if we add 5th team? We would have 3+6+9+12=30.
I reminded them how we can calculate the sum of arithmetic progression. Add the first and the last numbers (3+12=15) and add the second and second from the last numbers(6+9=15). Then add those results: 15+15=30.
This problem is very similar to 'handshakes' problem.
I had 2 groups of children. First group had 3 kids and the second had 4 kids. Each child represented a soccer team. All those teams are playing in soccer tournament. In first group there are 3 teams and in the second there are 4. Teams play against another 3 games. How many games played in each group?
To solve this I reminded children about 'handshake' problem. Each handshake represents a game.
Kids did handshakes and calculated total number of games. Kids in both groups calculated number of games correctly by just counting total number of handshakes among each other.
Then I asked them to step back. I reminded them how we could solve similar problems by applying arithmetic progression: 3 handshakes between 2 kids, 6 if add third kid, 9 when fourth kid is added. So we have 3+6+9=18.
After that I asked: What if we add 5th team? We would have 3+6+9+12=30.
I reminded them how we can calculate the sum of arithmetic progression. Add the first and the last numbers (3+12=15) and add the second and second from the last numbers(6+9=15). Then add those results: 15+15=30.
How many cats are on the farm?
3/5/2010
Today we had an interesting problem to solve and kids had a lot of fun.
I divided math group into two subgroups and asked each subgroup to solve the problem independently.
Here is the problem.
There are cats and chickens on the farm. A cat has 5 claws on one leg and a chicken has 4 claws on one leg. There are 104 claws all together. How many cats are on the farm?
One group gave me an answer right away: 5 cats and 1 chicken with only one leg :)
Because the answer was wrong kids continued solving it.
In a little while one group gave me the correct answer: 4 cats, but when I asked them how many chickens they told me 24. I guess all those 24 chickens had 1 leg and 1 claw :) Kids understood what they did wrong and gave me the correct answer: 3 chickens. The second group solved that problem just after the first one.
Then I told children that there is another solution. After a while both groups gave me the correct answer: 2 cats and 8 chicken.
After that I told them that this problem can have another answer. They could not solve it for a while and then I gave them a hint that number of cats can be any including 0,1,7, etc. They tried 0 and it worked. 0 cats and 13 chicken.
Today we had an interesting problem to solve and kids had a lot of fun.
I divided math group into two subgroups and asked each subgroup to solve the problem independently.
Here is the problem.
There are cats and chickens on the farm. A cat has 5 claws on one leg and a chicken has 4 claws on one leg. There are 104 claws all together. How many cats are on the farm?
One group gave me an answer right away: 5 cats and 1 chicken with only one leg :)
Because the answer was wrong kids continued solving it.
In a little while one group gave me the correct answer: 4 cats, but when I asked them how many chickens they told me 24. I guess all those 24 chickens had 1 leg and 1 claw :) Kids understood what they did wrong and gave me the correct answer: 3 chickens. The second group solved that problem just after the first one.
Then I told children that there is another solution. After a while both groups gave me the correct answer: 2 cats and 8 chicken.
After that I told them that this problem can have another answer. They could not solve it for a while and then I gave them a hint that number of cats can be any including 0,1,7, etc. They tried 0 and it worked. 0 cats and 13 chicken.
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