Math Problems for 3rd grade - Part 1

1. How many digits are all together in all numbers from 1 to 100

2. One wheel of the cart is 3 times larger than the other. The big wheel has made over 1000 rotations on the path. How many rotations has the smaller wheel made?

3. A person answers questions with "yes" or "no". He has the right not to say truth only once. With how many questions you can guess what number he/she thinks of, if that number is from 1 to 4?

4. There are 8 coins. One of them is fake and it is lighter. There are pan scales. In how many weighings you can find fake coin?

5. There are two cans of water: large and small. The large can has 5 times more water than the small can. The small can has 8 oz less water than the large can. How much water is in each can?

6. The length of the room is 20 ft. The snail moves from one side of the room to another. During the day the snail moves 2 ft towards the end of the room, but during the night the snail moves back 1 ft. In how many days will the snail reach the end of the room?

7. What is the missing number? (587 + 213 + 111) x ? = 0

8. How many rooks can be arranged on a chessboard such that none of them threaten the other?
Examples of correct answers:


















9. Two tourists were making sandwiches. Third tourist came, and the first two tourists gave him something to eat: the first gave him 3 sandwiches, and the second 2 sandwiches. The third tourist paid them $10. How do the first two tourists need to divide money among themselves?

10. Grandfather is 56 years old and his granddaughter is 14. In how many years grandfather's age will be twice as much as his granddaughter's?

Problems for summer time - Part 1

1. 5 friends and handshakes

5 friends exchanged 4 handshakes with each other. How many handshakes they did?

Answer: 40

2. Fake coin

There are 9 coins and one of them is fake. We know that the fake coin is lighter than others. How to determine which coin is fake with 2 attempts?

Answer: Divide 9 coins into 3 sets with 3 coins in each set. Compare 2 sets. If one set is lighter than the other then it means that the fake coin in the lighter set. If those two sets weight is the same then the fake coin in the third set. Now we know that the fake coin among 3 coins. Select 2 coins from those 3 coins and compare them. If one is lighter then that would be the fake coin. If they are equal then the third coin is fake.

3. Continue the sequence

Here is the sequence: 2, 20, 40, 400, 800. Could you please continue it?

Answer: The second number is 10 times more then the first. The third number is twice as much as second, the fourth is 10 times more than the third, the fifth is twice as much as the fourth
2, 20, 40, 400, 800, 8000, 16000, ...

4. Interesting square

Imagine the square 3 x 3 (9 cells all together). Put numbers from 1 to 9 in such a way that one number should be in one cell, a number is not repeated, sum of the numbers that reside on the same level horizontally, vertically, and diagonally is the same.

Answer:
One of the answers
2 7 6
9 5 1
4 3 8

5. Cleaning windows

6 workers clean 6 windows within 6 minutes. What is the minimal number of workers needed to clean 100 windows within 100 minutes?

Answer: 6

6. Treasure hunters

2 treasure hunters found treasure. They don't have any means to measure that. How to divide the treasure between those two treasure hunters in most fair fashion?

Answer:
One treasure hunter divides treasure in the way he/she thinks fair and the other one chooses the part he/she likes.

7. Saturday

September 1 2001 was Saturday. What day of the week was October 1 2001?

Answer: Monday

8. Kids in circle

There is some number of kids stay in circle. We know that the third person on the left from James is the same person who is 11th on the left from James. How many are in the circle?

Answer: 8 or 4 or 2

9. Third grade

On Tuesday there are 4 classes in the third grade: Math, Writing, Reading, and Arts. How many different combinations (in what order they go) exist?

Answer: 24

10. Cats competition

2 cats participate in running competition. They need to run first from point A to point B and then back from point B to point A. The first cat is running with the same speed the whole distance. The second cat is running twice as fast than the first one from point A to point B , but from point B to point A the first cat is running twice as fast as the second cat.
Who will finish first?

Answer: First cat

Proposal for Math lesson on May 21 2010

1. How many apples were on the table?

3 brothers asked their mother to leave apples on the dinner table and went to sleep. The eldest brother woke up, found apples on the table, ate 1/3 of them and went back to bed. The middle brother woke up, found apples on the table, ate 1/3 of them and went back to bed. The youngest brother woke up, found apples on the table, ate 1/3 of them. Then he woke up two brothers and offered them to finish eating 24 apples that were at the table.
How many apples were on the table originally?

2. Magic quadrant
Imagine a quadrant 5x5 (5 columns & 5 rows). Each number 1, 2, 3, 4, 5 is repeated 5 times. In each row has only one occurrence of each number and in each column has one occurrence of each number. Draw this quadrant.

3. Stamps
Jane purchased 3 stamps: one from the USA, second from France, and third from Italy.
The sum of the stamps without USA stamp is $13. The sum of the stamps without Italian stamp is $11. The sum of all stamps without French stamp is $12.
How much does each stamp cost?

4. Garages
This problem is similar to the 'shelves' problem we had a couple of months ago.
There are two huge garages. The number of cars in both garages is equal to 22. First garage has 8 cars less than the second garage.
How many cars are in each garage?

Proposal for Math class on May 14, 2010

1. How many are three digit numbers where all digits are odd and no digit repeats in the same number?

2. Where the elevators stops more?
There is a 10 stories building. The entrance to the building on the first floor. 1 person lives on the first floor, 2 people on the second, ..., 10 people live on the 10th floor.
Where (on what floor) the elevators stops more?

3. What was the day of the week, March 1 1996?
We know that February 1 1996 was Thursday. What day of the week was March 1 1996?

4. How many black cats who like salmon?
There are 15 cats, 8 of them are black and 8 of them like to eat salmon. How many cats are black and like to eat salmon.

5. What is the sum of all numbers from 101 to 200?

Treasure chests

The gnome separated his treasure into 3 different chests, each different
color, each standing by the wall. One chest had precious stones, one chest
had gold coins, and one chest had magic books. He remembers that red chest
is to the right of the one with the stone. And books are to the right of
the red chest. What chest contains books, if green chest is to the left of
the blue chest?

This is a problem on logic and the kids did it very successfully.
There were 9 kids in my class, so I divided them in 3 groups. Each group chose what they represent. One group was Precious stones, another was Gold coins, and the third was Magic books.
We all decided to put the 'Precious stones' team next to the wall. Kids realized that two other teams should stay on the right side from 'Precious stones'. They also found that one of those teams is red. Because 'Magic books' team on the right from red team it meant that 'Gold coins' team is red. Because green chest is to the left from blue chest it meant that 'Magic books' is blue and 'Precious stones' is green.
The problem is solved!

How many trucks and cars in the garage?

There are 17 trucks and cars in the garage.
Trucks have 6 wheels and
cars have 4 wheels. How many of each are in the garage if the total number
of wheels is 82?

This problem is very similar to the farm problem we did a couple of months ago. Surprisingly (again) all kids solved this problem very quickly :)
The answer is 10 cars and 7 trucks

What are the number in the following pattern

I gave this type of problem to the kids before.
The sequence I chose was 720, 360, 180, 90

It was a challenge for kids to determine the pattern. When I wrote those numbers on the board I made a mistake. Instead of 180, I wrote 120. So it didn't help them to find the pattern. But I could see how hard the kids were thinking :) When I gave a hint that each following number is a half from previous number. The kids were paid attention to minor details of may explanation. So when I said that 120 is a half from 360. They noticed this right away and corrected me on the spot.
I started asking what is a half of 90:45.
A half from 45: 22 1/2
A half from 22 1/2: 11/1/4
But then it was very difficult for them to get a half from 11 1/4. So I draw 11 circles and additional circle where I marked 1/4. Together we derived a half from 11: 5 1/2. Then we get a half from 1/4: 1/8. The next step was to add
5 1/2 and 1/8. It was not a trivial exercise for kids, so I posed a question: How many 1/8th in 1/2. They solved: 4/8. Then the kids realized that all they need to do next is to add 5 + 4/8 +1/8 = 5 5/8
As the next step I asked them: What is a half from 5 5/8.
We did a similar exercises as before. We got a half from 5: 2 1/2. Then we needed to make a half from 5/8. Together they got it that 5/8 is the same as 10/16 and a half from this would be 5/16.
So the result was 2 1/2 + 5/16. The next step was to add 1/2 and 5/16. I asked : how many 1/16th is 1/2. They answered 8/16. So the final answer was 2 + 8/16 + 5/16 = 2 13/16

Proposal for Math lesson May 7

1. What are the next two numbers in the following pattern
720, 360, 120, 30

2. How many trucks and cars in the garage?
There are 17 trucks and cars in the garage.
Trucks have 6 wheels and
cars have 4 wheels. How many of each are in the garage if the total number
of wheels is 82?

3. Treasure chests
The gnome separated his treasure into 3 different chests, each different
color, each standing by the wall. One chest had precious stones, one chest
had gold coins, and one chest had magic books. He remembers that red chest
is to the right of the one with the stone. And books are to the right of
the red chest. What chest contains books, if green chest is to the left of
the blue chest?

4. What is the sum of all the numbers from 101 to 200?

Proposal for Math lesson April 30

In the beginning kids will solve the last two problems from the last lesson that they did not have time.
1. $2 bill
A store owner has only $2 dollar bills. The customer who comes in has only $2 and $5 dollar bills. The items in the store cost $1, $2, $3, $4, $5, $6, $8, and $10. Prove that the customer can buy any of the items in the store and get back correct change.

2. How many children study two foreign languages
There are 25 children in the class. 17 study French, 15 study Spanish. Any student study at least 1 foreign language. How many kids study 2 foreign languages?

3. How many three digits numbers
How many are three digits numbers where all digits are 1 or 2 or 3?

Proposal for Math class on April 16

1. 4 balls

There are 4 balls (red, black, and white). There are the same number of red balls as white and black combined together. How many black balls are in the box?

2. 15 balls (red, black, and white).
There are 15, red, black and white balls in the box. There are 7 times more red balls than white balls. How many black balls are in the box?

3. Tommy and Johny

Tommy and Johnny have 15 stamps together. Tommy gave Johnny 2 stamps. How many stamps do they have now?


4. Think a number

I am thinking of a number. If you add 1, subtract 2, multiply the results by 3 and divide it by 4 you get 6. What number am I thinking of?

5. Get a new square

Look at the picture below. What is the smallest number of dots that you can add to the picture in order to get a new square?

Solution: you would need to draw dots described in the problem.







Right away you can come up with the following








But it is not optimal. Let's turn the drawing







Then you can come with the following solution.









6. Connect dots
Starting with the same 9 dots, can you connect all the dots with 4 straight lines without lifting your pencil from the paper?

7. $2 bill
A store owner has only $2 dollar bills. The customer who comes in has only $2 and $5 dollar bills. The items in the store cost $1, $2, $3, $4, $5, $6, $8, and $10. Prove that the customer can buy any of the items in the store and get back correct change.

8. How many children study two foreign languages
There are 25 children in the class. 17 study French, 15 study Spanish. Any student study at least 1 foreign language. How many kids study 2 foreign languages?

Proposal for Math lesson on Apr. 9 2010

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?

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.

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.

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.

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!

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?

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?

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!

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!

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!

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.

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 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.

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.

How many handshakes can people do?

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).

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

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.

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!