Sunday, May 12, 2013

Lesson 19- Everything Math

·         By overlapping two circles, you can make three different areas . How many different areas can you make with two squares?

I gave each of the kids a piece of paper and let them play around with the problem for a few minutes, and the kids found a way of making nine different areas by overlapping two squares.

·         Do a very easy Sudoku puzzle (below)

I made a big mistake here. I should’ve brought Sudoku puzzles that are 4x4, or instead, Sudoku puzzles that have pictures instead of numbers. I didn’t realize that these kids are still very little! They can’t solve Sudoku puzzles this hard (especially that some of the kids didn’t know how to do Sudoku puzzles) unless they get a lot of help from somebody else.

·         Play Mastermind

This is a little too hard for them for now. It’s possible to play this game with the kids, but they need a lot of hints and explanations during the game, which is fine, but don’t expect them to be able to figure it out on their own.

·         10 adult people are trying to crowd under one small umbrella, but nobody gets wet. How is this possible?

“They can stand on each other’s shoulders and the top person can hold the umbrella” – THIS IS THE SOLUTION THAT ALL LITTLE KIDS OFFER AT FIRST (even I solved it this way when I was solving it when I was 5 or 6). I don’t know what is so obvious about that solution, but all little kids seem to think that it’s the best way to solve this problem. Later, a few kids realized that the 10 people could be in a house or under something- that’s a good solution. Then they couldn’t think of any more solutions, so I asked “Where/when can you get wet?” and they said “in the rain”. So, then I gave them a minute or so to think, and then they realized “wait, you never said that it was raining outside!”


- What should the next picture in the pattern be?

I first let them look at it for a little bit and try to figure out what the pattern is. Then after seeing that they weren’t getting anywhere I covered half of each of the terms, and asked them what it looked like. Then, when they realized the pattern, I asked them to continue the pattern for the next 2 terms.

·         Which of these shapes does this cat not have

a)      Rectangles b) Squares c) Circles d) Triangles

They didn’t understand this problem, so to make it easier, I rounded the tips of the cats ears too, and then the saw that there weren’t any triangles in the cat.







Lesson 17- Basic Proofs

Everybody needs a mouth:

If we didn’t have a mouth, we wouldn’t be able to eat.

If we weren’t able to eat, we wouldn’t get energy.

If we didn’t have any energy, we would die.

Therefore, everybody needs a mouth.

Does this make sense? Is it correct?

Actually, for this problem, I asked the kids to prove why we need a mouth instead of telling them myself. It’s always better to have the kids doing something or figuring something out instead of just sitting there and listening (or in some cases, not listening).

How can you prove that you breathe with your nose and not your fingers?

They couldn’t quite figure out how to prove it at first, but then they closed their noses and said that they couldn’t breathe, and then they made fists and squeezed their fingers and they could still breathe.

Circular Reasoning- Do these make sense?


“This is a dinosaur bone!”

“How do you know?”

“Because it was stuck in this billion year old sand.”

“How do you know the sand was a billion years old?”

“Because there was a dinosaur bone inside it.”

At first, they thought that this made perfect sense, but then I drew them a diagram, and they realized that the reasoning was going in circles.


“I am not a monster. I don’t even know what a monster is.”

“How do you know that you are not a monster if you don’t know what a monster is?”

“Because if I was a monster, I would know.”

I prompted them with a few hints which were actually questions AFTER I LET THEM THINK A LITTLE BIT ON THEIR OWN. I asked them questions like “Can somebody know that they are something (or in this case, not something) if they don’t even know what that thing is?” Soon they realized that was I was saying was correct. I gave them a couple more variations of this problem after they solved just to make sure that they understood. For example, I said “If you were an apple, and somebody came up to you and said, “you are an apple” and you said “ I am not an apple. I don’t even know what an apple is”, would that make sense?”


In a rectangular room, put 8 chairs around the walls so that there are 3 chairs at each wall.

We did a few easier examples before this one. For example, I asked them to put 12 chairs around a square room so that there would be an equal number of chairs at each side. Then I asked them to figure out a strategy for placing an equal number of chairs at each wall to add up to a certain number (the solution was to keep adding one chair to each wall in a circle until there is enough chairs). Then, after a few tries, most of the kids figured out the answer to the problem and THE KIDS THAT FIGURED IT OUT ALREADY EXPLAINED IT TO THE KIDS THAT DIDN’T UNDERSTAND.


a)         How many people are in the house

b)         How many poker chips I am holding

c)         How many poker chips are in this bag

d)         How many people live on this street

e)         How many apples equal the weight of you

The answers varied wildly. Some people said that there were a thousand poker chips in the bag, and some said that there were 5 (there were actually about 50 of them). Also, before we did “how many apples equal the weight of you” we did “how many apples equal the weight of a cat” because that was an easier variation.

General Comments:

I was planning to fit in a lot more things in this lesson, but we spent a lot of time on each problem (which is actually a good thing, because it’s better to have all of the kids completely understand only a couple of problems instead of having the kids have no clue what’s going on in many problems).

Lesson 18- Math


Kristina, Katya, and Max are standing in a line. They all are holding either a blue or green bouncy ball. Two people are holding a green bouncy ball, and one is holding a blue bouncy ball. We also know that some of these people always lie and some only tell the truth.  Kristina says, “I have a blue bouncy ball.” Katya says, “I have a blue bouncy ball”. Max says, “I have a green bouncy ball.” Kristina says, “Katya is lying.” Katya says, “Kristina is lying.” Max says, “Katya is telling the truth.” Who is holding which color bouncy ball? Who are the truth tellers, and who are the liars?

For this problem, I drew a diagram on the white board which showed which person was holding which color bouncy ball. Also, the diagram had arrows pointing from one person to the other with Xs and checkmarks at the ends (depending on whether that specific person was saying that the other was a liar or a truth teller). We checked the possibilities for who could’ve been the liar or the truth teller, and the diagram helped a lot. This problem wasn’t too hard, but it wasn’t very easy either, so it was a good warm up.

Handshake Puzzle:

If there are: 1, 2, 3, 4, 5, or 6 people in a room, and every person shakes hands with every other person, how many handshakes will happen? Can you find any patterns?

ACT IT OUT. Ask a few kids to come to the front of the room, and have them shake each other’s hands, and count how many handshakes would happen in all. We recorded our results in a table, and we also recorded how many handshakes

a)      The person that shook everyone’s hands first

b)      The person that shook everyone’s hands second

c)      The person that shook everyone’s hands third, etc.

They found a few different patterns while solving this problem.


Dima has a string. He is holding the ends of the string with his hands. How can Dima tie a knot in the string without switching the ends of the string with different hands?

I brought a scarf so that each child would get a chance to try doing this problem. After each child had a turn and nobody figured out yet, I gave them the hint that your hands can be crossed in any way when you start, as long as your hands are holding the ends of the scarf, and that they stay there for the whole time while solving the problem. After about 5-10 minutes of them trying to solve it (with me giving more and more hints every time), I showed them how to do it. Then I let a few kids try repeating what I did to show them that it was actually possible (they thought that I was cheating, because it was obviously impossible).

Toothpick puzzle:

There are 4 squares joined together to make a bigger square (out of toothpicks). How can you move just two toothpicks to make seven squares?

I brought popsicle sticks (toothpicks are too sharp and small) and they solved the problem using them. I let each child try, and they figured out after a few minutes that the popsicle sticks could be crossed over each other.

Try making a magic square (3x3)

I drew a 3x3 square on the white board, and gave them two numbers to start with (in random places in the square). They tried to solve it from there, but many of the times it ended up not working (obviously, since I just placed the numbers in randomly). Then, I let them try and figure out on their own how to start (and finish) the puzzle. This is actually a pretty hard puzzle.

Saturday, May 4, 2013

Lesson 14- Logic Puzzles

This lesson includes a few problems from: The Moscow Puzzles book, and from Julia Brodsky (Art of Inquiry math circle)

General comments: the kids seemed to love this lesson- they paid attention, they didn’t fool around, and they seemed to enjoy the problems. Two of the parents stayed in the room, and made sure that the kids were behaving themselves. The kids were very focused.


A boy went to the dentist to have his cavity filled. The boy was the dentist’s son, but the dentist was not his father. How can this be?

“Oh, that’s impossible! Hmmm… Wait! Maybe it was his mom!” Everybody was paying attention and thinking.


*Sorry, I couldn't get the image into the post. I will add it to the post later.

You may touch and move only one glass to get the following:

*Sorry, I couldn't get the image into the post. I will add it to the post later.

This problem can be presented in a couple different ways; either there can be actual paper/plastic cups used and toys or something similar put inside them, or you could draw it on a whiteboard (which is how I did it). The kids were focused, and in their seats, and after trying different strategies of switching around the cups, they realized that water can also be poured from one cup to another. After their discovery, they soon figured out the solution to the problem.


Paper clip problem

If you drop a paper clip, it will land flat on its side. What can you do so when you drop it, it will land balanced on edge?

The kids seemed to enjoy solving a problem that had science incorporated into it. I was just trying to have the kids solve a problem where they have to figure out a creative way of doing something, or of using random materials to complete a task. They thought of many ways of solving this problem; stick it into a crack, squeeze it between two books/boxes, putting two magnets on the sides of it, and multiple other ideas.

The bathroom scale problem

What happens if you stand on a bathroom scale and lift one foot up? What will the scale read? What if you put one foot on one scale, and another foot on another scale?

This took some explanation, but the kids realized that their foot wouldn’t be disappearing anywhere- it would still be there, it’s just that it was in a different place. I explained to them that the weight still “flows” into the scale- it’s just that it all goes through one foot. They seemed to understand, and to check, I asked one of the kids to clarify why a person’s weight stays the same on a scale when they lift one foot up. When I asked them whether the weight would still be the same if they put a foot on a different scale, and they said that they weight would be about equal on both scales. Maybe, if there aren’t too many kids, after some exploring and thinking, the kids could actually try standing on a  scale and lifting their foot up and checking if there would be a difference- just a suggestion.

Discuss: If halves are equal, then wholes are equal. 

Example 1: If half-length of one road is equal to half-length of another road, then road lengths are equal. Right?

Example 2: Consider a glass below. Half empty glass is the same as half full glass. Does it mean that an empty glass is a full glass?

“Hahaha! No! The second example isn’t right! Probably the rule doesn’t work for everything…”


Two camels

Two camels in the middle of the desert were facing in opposite directions.  One was facing east and another west. How could they manage to see each other without walking, turning around or even moving their heads?

We acted it out, because the kids had trouble imagining this in their head- they soon realized that looking in different directions didn’t necessarily have to mean looking away from each other. 


Moving Checkers:

6 checkers lay in a row, alternating in color (black, white, black, white, black, etc.). There are four empty spaces on the left. Checkers can only be moved in pairs together with an adjacent checker. Move the checkers in such a way that all of the white checkers end up on the left, with the black ones following them.

The kids loved this exercise- and to my amazement, figured it out after only a few tries (we went around in a circle, and every child got to move a pair of poker chips (that is what we used instead of checkers) on their turn. They even re-did the activity multiple times because they wanted to.


If you have two colors, how many combinations can you find of coloring in three empty circles? Now try it with four empty circles? What about if you have three colors? How many combinations for three empty circles then? Try to get the most possible number of combinations.

They really liked finding the combinations, and watching the pace at which the number of combinations grew.

Coloring Activity

Using 3/4/5 different colors, color the picture so that neighboring parts differ by color.

We didn’t have time to get to this, but after the class, the kids stayed colored this (following the instructions)- I gave them the sheet.



Lesson 15- Math Mix

Heads and Legs:

 In a garden, there are five horses. On each of the first three horses, sits a rooster. On each of the last two horses, sits a fish. How many heads and legs are there all together?

“Fish and chickens ride on horses? Haha!” The kids solved the problem by making a little diagram with little circles representing heads, and lines from those circles representing the number of legs that animal has (the animals who sat on top of the horses were drawn above the horses, and the kids thought that doing that really mattered to solve the problem).  I offered the kids to draw the animals sitting on the horses in columns, with the animals under the horses, but the kids refused to do that, since it was “not following the rules of the problem”. It is good that they pay attention to the directions of the problem, but I think that I should spend some more time discussing the importance of various parts of a problem.

Fill in the blanks:
*sorry, the image was not inserted correctly. It will be posted later.

The kids figured out what had to be in each row, and then solved the problem easily. Give everybody a chance to share what they think. This didn’t seem to be too hard for the kids, but it was a good warm-up type of problem. It is good for the kids to recognize and figure out what figures and inner figures had to be in each row, column, and diagonal- don’t tell them those things! Let them think! It might not seem like it, but it is the most important part of the problem!


Is it possible to put 5 checkers in such a way that each checker touches 2 others? 3 others?  

I gave each child some poker chips (that is what we used instead of checkers), and they played around a bit. Most of them solved it after a little while, others were stuck, and were explained how to do it by the children that already figured it out, and NOT by me. We only had time to make the 5 checkers touch 2 others. Later we will try making each of the checkers touch three others.

Elevator Problem:

I know one tall building in Antarctica. Only one person lives on the first floor, 2 people live on the second floor, 3 people live on the third floor, and 4 people live on the fourth floor. There is an elevator in the building.  On what floor does the elevator stop most often?

At first, they all thought that the elevator visits the fourth floor the most often, which is the most obvious response. I let the kids think for a couple of minutes, and everybody still stuck to their answer. I gave them a hint; “is there any place in the building that all the people need to go to, that the elevator would take them to?” Soon after this, they figured out that everybody in the building needs to get to the exit (the door) at some point, and that is on the first floor. “Or they can jump out the window!” said one of the kids. I said, “Do you normally see people jumping out the window from the fourth floor?” “Fine…”

Flower colors:

Once, Sasha drew a flower. He had blue, red and white colors.  “It is not white” – said Max. “It is blue or red” – said Kristina. “It is red” – said Dima. It is known that at least one of the kids guessed wrong, and at least one guessed right.  What color is the flower?

We checked the combinations possible for who was lying and who was telling the truth, and we were in the middle of solving the problem, when the kids started to fool around, so I stopped, and moved on to the next activity. I will re-state this problem and try it again next time.

Passing Trains:

A train is going from the US to Canada, at 60 mph. Another train is heading in the opposite direction- from Canada to the US, and it is moving at the speed of 40 mph. When the two trains pass each other (meet), which one will be closer to Canada?

The kids didn’t seem to understand what the point of the problem was- “Wait… won’t they be in the same place when they meet?” OK… I guess I didn’t succeed in tricking these children… J


Boiling Eggs:

If it takes 2 minutes to boil one egg, how long will it take to boil three eggs?

“Will the eggs boil in the same pot or in different pots?” “What about you decide?”, I answered. The kids were moving in the right direction. “Um… usually eggs are boiled in the same pot, so… let’s have them boil in the same pot.” The kids were getting really close. “Wait, if they’re in the same pot, it’s not like first one will boil and then the other! They will boil at the same time, so it will also take only 2 minutes!” These kids are smart!

Play Pentagrams. Try building solid rectangles out of them.

The kids were surprisingly good at this! They built small rectangles, and a lot of them got really close to making a rectangle using all the pieces. I gave each of the kids a set of pentagrams, so they would be able to work on their own if they wanted to. Which they did.