These problems are embedded in a story. That gives them a different flavor. It allows the students to solve along with you, but it also allows the characters in the story to suggest ideas.
I have two daughters, Kiana and Sarah. I gave them each a piece of chocolate.
[Pass out two rectangles. One is 1x11; the other is 3x4. I measured in inches.]
Kiana complains, "It's not fair. Sarah got more." Sarah whines "It's not fair. Kiana got more." So I have a problem. Who got more? Or if they got the same amount, how can I convince them that the got the same amount?
Kiana points out the Sarah's is longer. But I show her that a long skinny piece of chocolate is not necessarily bigger than a shorter but fatter piece of chocolate.
[Show them two rectangles. I did 1x20 and 12x12, measured in centimeters.]
Kiana agrees that that the longest rectangle is not always the biggest. But then Sarah says that Kiana's is bigger because it is fatter. Dad shows Sarah these two rectangles. (6x20 and 8x8)
Sarah agrees that the fattest is not always the biggest. But then, which is the biggest?
[There is a very subtle point here -- what does it mean to say that one rectangle is bigger than another? From an adult perspective, we are talking about area. So this exercise forces them to think in terms of area, with explicitly mentioning it.]
Dad says, "Girls, can't you just see that the two are the same size?" Kiana says, "I can see that Sarah's is bigger." Sarah says, "I can see that Kiana's is bigger." Dad rips out some of his hair. One the neighbors, Miss Adams, has been listening. She says that you can't judge size just by looking. She says, look at these two lines.
[Do the Muller-Lyer illusion. Have them measure the two lines. One looks longer, but they are the same size.]
[I don't know whether students will think about measuring perimeter. If they do, have them measure the perimeter of a 1x20 rectangle and a 8x8 rectangle. The 1x20 measures as more, but it should be obviously smaller.]
Dad says, "I read in a book that when mathematicians have a hard problem they can't solve, they try to simplify the problem. Then maybe they get ideas for how to solve the hard problem. Now here's a simple problem. Which square is bigger?
[Two squares, one larger than the other. E.g. 10x10 and 8x8. How can you know which square is larger?]
Another neighbor, Mr. Aldridge, says this. "Is this a simpler problem? Which piece of string is longer?"
[Two strings, one slightly longer than the other. Anything about someting extra is okay. However, circular answers, like the bigger one is bigger, are not good. Prod them if their answer is circular.]
Miss Adams says, "Is this a simpler problem? I sent my friend this letter with kisses and hugs. Did I give more hugs or kisses to my friend?"
XO XO XO XO XO XO XO XO XO XO XO XO XO XO XO XO XO XO XO X
Are there more kisses or hugs? More importantly, how can you know that there are more kisses than hugs?
Finally, everyone but Dad says "I got it." Dad still doesn't get it. What do you think everyone but Dad understands? Dad says, "Explain it to me."
Everyone starts to speak at once. Everyone has different words for the same idea. Maybe you will like what Sarah said. "Dad, there are two ways to know which is bigger." (Mr. Aldridge thinks there is only one way, but everyone else agrees.) "One way is to put a number on each of them and see which has the bigger number. The other way is to see which one has something left over." Miss Adams calls it the extra method -- see which one has something extra. Kiana is trying to fit one inside the other.
Dad tries to see how this works for each of the problems. Can you help him?
Now Dad has to figure out which square is bigger. Dad says, "I don't know how to put a number on bigness." So I will use the leftover method." Kiana says, "Good thinking, Dad." Dad puts the squares together to see which has something left over. What happens?
Dad whines "There's something left over on each of the pieces of chocolate. What can I do?" Kiana says, "Dad, you can cut chocolate into smaller pieces." Can you cut up one piece of chocolate so that the parts fit inside the other piece of chocolate?"
[Give them squares to compare by cutting. Use different colors for the squares, so they can see what belongs to each square even after they are cut up.]
So far, this is mostly an exercise in measuring length. It forces the students to think about a concept that is so simple it is difficult to think about. That is something mathematicians do. Some students will like this and some will not (in my experience).
The question came up in my discussion with my students whether counting and measuring were the same thing. My students said they were different. I asked them to say why they were different. They listed several things, and I showed that they were the same. For example, when you count, you get an answer like three, but when you measure, you get an answer like 3 inches. I said that when you count apples, you get an answer like 3 apples. Then I showed how, if you wanted to measure a long line, you could divide it into feet and count the feet. This is not to say that counting and measuring are exactly the same. The point is just that they are similar, and it is important to see the similarity.
Comparing Shapes by Cutting
I suspect that the comparison of shapes by cutting is very important. It should develop good skill in the manipulation of areas. That skill is a foundation for calculating areas. So hopefully they can confidently cut and compare in an efficient way.
This starts with comparing rectangles. But it can get into comparing a triangle to a rectangle. Other possible shapes are a rectangle with a rectangular hole or a rectangular addition. This practice can be spread out.
There can be other variations in the XO problem too. If the X's and O's are not naturally paired up, you can draw a line connecting each X and O into a pair. As a transition, they could be given a sheet with disorganized X's and O's that are paired up. A similar problem is that a room contains 43 people and one dog. Are there more arms or legs in the room?
What if I had two hourglasses and I wanted to know which one lasted longer?
For Young Children
A talented five-year old could solve the problem of comparing two different strings, and the problem of explaining why there were more X's than O's. (Conversely, an adult I gave this problem to had a strong tendency to circular answers.)
Put down two strings (or sticks) on a piece of paper, with flanking lines that make the Muller-Lyer illusion. Ask which one looks longer. Now reverse the strings and ask which string is longer. This seems to teach something. But if a child doesn't get it, I don't think you can teach anything.
This story continues with putting numbers on the bigness of the square.