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