Fraction of a square inquiry

The prompt

Mathematical inquiry processes: Identify and extend patterns; reason geometrically; conjecture and generalise. Conceptual field of inquiry: Fractions, equivalent fractions, algebraic expressions.

The prompt is designed to intrigue students and encourage them to explore squares split into more rows. The statement is false: half the square with three rows is shaded, but less than half is shaded when there are four rows.

Does the fraction continue to decrease as we split the square into more rows? Does the fraction decrease by the same amount each time?

In the orientation phase during which students notice, question and wonder, the teacher should model geometrical reasoning if it does not arise in the class discussion. You can show that half the first square is shaded by moving the top triangle to fill up the bottom row. If necessary, vertical lines can be added to the diagram to create smaller squares so that it is easier to convince students of the fraction shaded. Using this approach, it can be shown that three eighths of the second square is shaded.

What if we split the triangle into five rows? Does the fraction shaded follow a pattern ... half, three eighths, a quarter?

A rule for shading

When students begin to shade the squares, perhaps using the templates included in the PowerPoint (see resources below), it is important that they follow a rule in order to create a consistent sequence of designs.

With one diagonal line from top left corner to bottom right corner, they should shade the region to the left of the line in the first row, to the right in the second, to the left in the third and so on.

Odds and evens

As students collect their results, they notice that when the square is split into an odd number of rows the fraction shaded is always a half. This is not the case for an even number of rows. The fraction is different each time, although there is the same relationship between the part shaded and the total area of the square. The relationship can be expressed algebraically.

December 2022

Mathematical notes

1. An odd number of rows

For any odd number of rows, the fraction of the square shaded is a half. This can be shown by moving the shaded parts above the middle row to fill in the rows below (see diagram for five rows). Half the middle row itself is shaded.

2. An even number of rows

Less than half of the squares with an even number of rows are shaded. The fraction of each one shaded is as follows:

  • Two rows: one quarter

  • four rows: three eighths

  • six rows: five twelfths

  • eight rows: seven sixteenths

  • n rows: (2n - 1)/4n

To prove the fractions follow a pattern, students can be shown how to rearrange the shaded parts in the same way (see the PowerPoint).

A line of inquiry

One way to extend the inquiry is to add a second diagonal line from the top right corner to bottom left corner. When shading the parts, it is important to apply a rule consistently. The diagrams that follow use an outside-inside rule - that is, in the first row the parts outside the diagonals are shaded, then inside the diagonals for the second row, outside for the third and so on.

1. An even number of rows

In this case, half the squares with an even number of rows are shaded. This can be shown by moving the shaded parts above the middle line to fill in the spaces below.

2. An odd number of rows

The pattern for an odd number of rows is more difficult. At this stage of the inquiry, students benefit from drawing their own designs on squared paper. The fraction shaded in each case is as follows:

  • Three rows: 5/18

  • Five rows: 21/50

  • Seven rows: 41/98

  • Nine rows: 73/162

  • Eleven rows: 109/242.

One approach is to categorise the squares into two types depending on the design of the middle row: those with the inside shaded and those with the outside shaded.