# The Learning Principles Behind Early Puzzles of Wuzzit Trouble Game

*Wuzzit Trouble *encourages the development of holistic mathematical thinking and problem-solving abilities as opposed to mastery of discrete facts or specific procedures. The purpose of the introductory puzzles (levels 1-1 to 2-10) is for players to become familiar with how the game works and to practice basic math skills (addition and multiplication) required to solve the puzzles.

The first twenty-six puzzles feature a single small gear, called a cog. After that, they progress to two cogs, then three, and finally four. Puzzles with two or more cogs require (and hence develop) sophisticated mathematical reasoning ability. (*Wuzzit Trouble* as a whole has over two trillion possible solution paths. So trial-and-error is not a feasible approach for more advanced puzzles.)

**Level 1-1.** This is a demo level. The goal is to for players to experience the game as they discover how to navigate and use the tools. The only math skill required to achieve a maximum score is to know that 2 x 5 = 10.

**Levels 1-2 to 1-6.** In these single-cog puzzles, 5 is the counting unit. You will notice 5 teeth on the cog that players turn. Since most players are familiar with counting by 5s, they can focus their attention on using a circular scale, as opposed to the more traditional linear number line.

To maximize their score, some of the puzzles at these levels require players to collect more than one key. The most effective strategy is to recognize multiples of 5. If instead they choose to skip count, they’ll have to tally their skip counts and use multiples to collect the keys. That’s a lot more steps than they need!

Figuring out the necessary multiples provides the groundwork for understanding multiplicative equations. Doing this in the context of 5 means that the players are working with a number they are very familiar with, or “landmark” number.

Puzzles 1-4 and 1-5 require moving the cog in the negative, or counterclockwise, direction to earn three stars. This is a reminder that the familiar number line is *just one way* to represent positive and negative.

Puzzle 1-6 requires a combination of positive and negative turns to earn three stars.

**Levels 1-7 to 1-9.** These puzzles have 10 as the counting unit (10 teeth), which provides variation, while still allowing skip counting as a feasible strategy.

**Levels 1-10 to 2-1.** These puzzles progress to cogs with different numbers of teeth: 6, 4, 8, 7, and 9 in that order. Each appears several times in succession. The changes in multiples, or tooth counts, are designed to push players to abandon strategies they may develop based on particular numbers. Rather than simply count by 5s and 10s, players are challenged to explore more general principles, leading to an understanding of multiplication.

**Levels 2-2 to 2-10**. Puzzles with two cogs appear for the first time at these levels, beginning with landmark numbers: 3, 5 that they have encountered in previous single-cog puzzles. With two cogs, skip counting would require keeping a tally of turns, and become even more impractical.

For variation, puzzles with a single 12-tooth cog are mixed in with the two-cog puzzles. For many players, multiplying by 12 is a challenge, but by now the single-cog puzzle structure is familiar. Going back and forth between arithmetically challenging, but familiar one-cog puzzles, and arithmetically easy but novel, two-cog puzzles requires different strategies.

Puzzle 2-6 has cogs with 3 and 5 teeth. Puzzle 2-9 has cogs with 3 and 8 teeth. Puzzle 2-10 has cogs with 4 and 7 teeth.

At these levels, it becomes increasingly necessary for players to use multiplication facts rather than repeated addition, and moreover to use them fluently.

**NOTE:** To solve puzzles beyond level 2-10, multiplication fluency is necessary. At these levels, players juggle two, three, or four cogs to apply algorithmic solutions to obtain optimal solutions.

**TEACHER NOTES**

Later puzzles assume an accumulation of learned relationships about and between numbers, and a developing fluency in the use of the scale. Players become adept at looking at the distance between scale marks, without having to count round. Players also become adept at using negative numbers, and combinations of positive and negative numbers, even when the markings in a negative direction are not what you would find on a number line. The concept of negativity has to override the symbolic information and this is an important step in mathematics because it is dependent on structure and not on numbers.

As a result of playing the game, students will gain fluency in multiplication facts and in interpreting “distance between” even when negative numbers are involved. They may gain an understanding of what can be achieved with pairs of numbers that do/do not have factors in common, and they are likely to realize that structure and relationships on the dial are sometimes more important than individual numbers.

Students also use problem-solving skills and number fluency as they move from familiar to less familiar numbers and from simple positive movements to multi-step positive and negative movements.