The prompt launches a wide-ranging inquiry that connects different mathematical concepts in number and geometry. Students are intrigued by the idea that the length of a line segment can be an irrational number. They wonder how the side of the square in the prompt, for example, can be √(10) if the square root of 10 "goes on forever".

The statement in the prompt is deliberately ambiguous. Does it mean that the areas of the five rectangles in the diagram are 10 cm2? Or does it mean that you can draw only five rectangles with an area of 10 cm2 on a grid? The use of the phrase 'on a grid' (rather than 'on the grid') suggests the statement is making a general point about any grid.

Orientation to the prompt

In the orientation phase of the inquiry, the teacher must ensure that the class understands the meaning of 'on a grid'. Each vertex of a rectangle lies on a point where lines of the grid intersect. Although a rectangle of length 4 cm and width 2.5 cm has the required area, it cannot satisfy the condition of being 'on a grid'.

In the question, notice, and wonder phase, students have made the following responses to the prompt:

• How do we work out the area of the tilted rectangles?

• There is a 10 x 1 rectangle. Does a 1 x 10 rectangle count as a different one?

• The areas of three rectangles area definitely 10 cm2 - 10 x 1, 5 x 2, and the long diagonal rectangle which has four whole squares and 12 halves.

• Are there any more rectangles with the same area?

• The area of the square is 9 cm2 - 3 x 3.

• Is there a pattern? 

Explaining the prompt

The length and width of each rectangle in the prompt can be expressed as square roots:

√(100) x √(1), √(50) x √(2), √(25) x √(4), √(20) x √(5), and √(10) x √(10).

For each rectangle, the radicands are a factor pair of 100 and the product of  length and width is √(100). 

It is possible to represent all the square roots as line segments on a grid because the radicands are either a square number or the sum of two squares. An example of a square number is 25: √(25) can be drawn on the grid as a line segment of length five. An example of the sum of two squares is 20: As 22 + 42 = 20, √(20) can be drawn as the hypotenuse of a right-angled triangle using Pythagoras' theorem. The lengths of the short sides of the triangle would be two and four.

As there are five factor pairs of 100 and as it is possible to represent each square root as a line segment on a grid and as each pair of line segments are perpendicular, it follows that there are five and only five ways to draw a rectangle with an area of 10 cm2 on a grid. 

Surds

Even though√(20) is a surd (or irrational root), it can be represented as a line segment on a two-dimensional grid. However, that is not the case for other surds. (See a list in the slides of the lengths from one to 50 you can and cannot draw.)

If we consider rectangles with an area of 15 cm2, for example, then √(3) x √(75) = √(225) = 15. However, neither three nor 75 are the sum of two two squares and, therefore, it is not possible to draw line segments with lengths √(3) and √(75) on a grid or, it follows, a rectangle with those dimensions.

It is possible, however, to draw both line segments in three dimensions - √(3) is the diagonal between opposite vertices in a 1 x 1 x 1 cube and √(75) is the diagonal in a 5 x 5 x 5 cube. Making connections to three-dimensional representations is one possible extension of the inquiry (see Lines of inquiry below).

Alternative prompt

The alternative prompt, containing four rectangles with an area of 8 cm2, is a more accessible entry point to the inquiry. As the rectangles contain only whole or half squares, students find it easier to verify the truth of the statement. Once they have, the teacher gives the side lengths as square roots: √(1) x √(64), √(2) x √(32), √(4) x √(16), and √(8) x √(8). Students can then use the same approach as they explore another case.