**Bubbling Cauldrons**

##### (Integer Partitions were first explored by Issai Schur, 1916)

This addition puzzle is one of the best ways to engage a classroom of children of wide ranging ability. It is so good that MathPickle will not give its stamp of approval on any curriculum that doesn’t include it.

After students have found the optimal answer (eight frogs in two bubbling cauldrons) you can challenge students in three directions:

1) Add a cauldron. Now the students must try to get as high as possible with three bubbling cauldrons. After a first attempt, students should be encouraged to try to get at least 20 frogs in the three cauldrons. The highest possible is 23, but students should not be told this unless they get to 22. I have never had a grade 2 student get to 23 in 45 minutes.

2) The first frog escapes! Still working with two cauldrons show that you can fit the numbers 2-12 in the cauldrons. After they solve this, say the first two frogs escape… then the first three…

3) Three frogs get added up instead of two. What is the smallest numbered frog for which either cauldron would explode if that frog was dropped into it? Don’t tell students the answer (11). Let them have a mini-competition to figure it out. Anything under fifteen it is pretty good.

Junior high algebra challenge: I believe the solution to the second of these puzzles are {2-12, 3-17, 4-22, 5-27, 6-32, 7-37, 8-42, n to 5n+2}. I have not proved this. For junior high students working on algebra, discovering this sequence is interesting. Adventurous algebra students may find it interesting to find a similar sequence for three or more cauldrons.

Drop a stone into water. It makes a sound, “glop” for a big stone, “splitch” for a small stone. Can you predict the pitch of the sound from the size of the stone? The usual teaching of mathematical theories is like a pyramid. Young people tend to become passive (if passionate) admirers of a structure built by old people, and problems they are taught to solve make them walk straight up to the peak. But what if we want to explore a natural mountain range, whose peaks are invisible among clouds, whose trails among trees are unknown? The problem of the sound of a stone falling into water is natural, so natural that every child knows the phenomenon and can wonder about it. The mathematics involved is extremely hard, so hard that it is not taught at any mathematics department in the world.

**Tadashi Tokeida**

**Standards for ****Mathematical Practice**

MathPickle puzzle and game designs engage a wide spectrum of student abilities while targeting the following Standards for Mathematical Practice:

##### MP1 Toughen up!

This is problem solving where our students develop grit and resiliency in the face of nasty, thorny problems. It is the most sought after skill for our students.

##### MP3 Work together!

This is collaborative problem solving in which students discuss their strategies to solve a problem and identify missteps in a failed solution. MathPickle recommends pairing up students for all its puzzles.

##### MP6 Be precise!

This is where our students learn to communicate using precise terminology. MathPickle encourages students not only to use the precise terms of others, but to invent and rigorously define their own terms.

##### MP7 Be observant!

One of the things that the human brain does very well is identify pattern. We sometimes do this too well and identify patterns that don't really exist.