More Chocolate

This activity came to mind while searching for ways to motivate the activity Discovering Pythagorean Triples which will be a natural consequence of this one.

The basic idea is to ask the viewer to compare areas of plane regions by looking at their shapes. The activity is rich in potential ideas and has many ramifications that have not been completely explored, so the description that follows must be regarded as a draft version.

At present we show two square regions that are almost equal in area, and the viewer is asked to choose the one with larger area. It is formulated as a game in which a correct choice increases the user's score, and an incorrect choice decreases the score.

We intend to generalize the activity by showing two rectangles, one of which obviously is much larger in area than the other. These rectangles represent chocolate bars of equal thickness. The viewer will be asked:

Which one would you choose as having more chocolate?

The player chooses one of the rectangles, and the computer responds with something like:

Obviously, you like chocolate! Or, Don't you like chocolate?

The player can click to see more examples:

An applet will produce a sequence of examples of pairs of rectangles to be compared in area, in which the aspect ratios change: for example, one rectangle is long and thin, the other nearly a square. Each clicked choice results in a response: Yes!, if the user selects the larger one, or No!, if the user selects the smaller one.

As the figures change with more or less random values for the areas, the areas gradually become more nearly equal to each other and it becomes increasingly difficult to see visually which is the larger area. Now, after each response of Yes! or No!, a visual clue appears near the bottom of the screen, for example, a small rectangle whose area is the difference in areas of the two chocolate bars. This small rectangle appears on the same side of the screen as the larger area so the player can actually see by how much the areas differ.

These examples show that it's not always easy to compare the size of chocolate bars just by looking at them. At the next stage, the bars are made up of many small squares of equal size, and the viewer could compare the chocolate bars by actually counting to see which one contains more squares. Many examples are shown of rectangles made up of large numbers of smaller squares, for which counting would be tedious and difficult. The viewer is asked:

Can you see an easier way to find the total number of squares in a chocolate bar without counting all of them?

A tutorial will be developed for comparing areas of two rectangles geometrically by repeated overlapping of one rectangle over another.

At this stage we have an applet for comparing areas of two rectangles that are nearly squares.

Three Square Chocolate Bars

Next we show three square chocolate bars, two of which are on the left half of the screen, and one on the right. Each bar is divided into unit squares of a convenient size.

As before, the user is asked to select:

Which has more chocolate, the two bars on the left, or the single one on the right?

Again, after a choice is made, the difference in areas is revealed visually as a collection of unit squares of chocolate on the same side of the screen as the larger area.

This can be made into a game in which the player gains squares of chocolate when correct, and loses squares of chocolate when wrong. Another visual clue can reveal the cumulative gain or loss after many examples are shown.

There should also be a response button in the center of the screen that says Equal! Extra bonus chocolate is awarded when the player correctly chooses Equal! with a corresponding penalty if this choice is incorrect.

Equality of areas will seem rare and surprising, and it is natural to ask when this can happen. This question leads naturally to the activity Discovering Pythagorean triples, which can be visited at this point if the user wishes to do so.

The activity can also be extended to involve more than three square chocolate bars. Among other interesting results, this will lead to the famous theorem that every positive integer is the sum of 1, 2, 3 or 4 squares.

Triangular chocolate bars

The activity can also be generalized to chocolate bars of other shapes, for example, triangles.

We have seen that the comparison game for squares leads naturally to the concept of Pythagorean triples. Similarly, we can construct chocolate bars made of unit squares arranged on a triangular grid, which leads naturally to the concept of triangular numbers, and we can play the same kind of comparison game-instead of Pythagorean triples of squares we could relate several triangular numbers. For example, the triangular number 21 is 15 + 6, the sum of two triangular numbers. Also, the triangular number 10 is 6 + 3 + 1, the sum of three triangular numbers. This illustrates the famous theorem that every number is the sum of 1, 2 or 3 triangular numbers.