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?
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