Zack Miodownik, a teacher of mathematics in London (UK), created the prompt for his year 9 class. The students had studied Pythagoras' theorem and Zack used the prompt to introduce the class to surds. 

Students' responses from the question, notice, wonder phase include:

• The numbers on the short sides of the triangles are reversed.

• One number is three times the other one.

• As they are right-angled triangles, you can use Pythagoras' theorem: The length of the hypotenuse equals the square root of the sum of the squares of the two short sides.

• How do you square a term with a surd?

• The square root of any number squared is the number, for example √(6) x √(6) = 6.

• Why are some triangles missing, such as the one for 3√(9) and  9√(3)?

• Which is the longest short side, the base or the height?

• Do the sides form a sequence from triangle to triangle? Is it a geometric sequence?

Formal mathematical language

As students feed back, the teacher can formalise the mathematical language by defining a surd.

At this point, the teacher should point out the non-examples that arise if the triangles with base lengths 3√(9) and 4√(12) are included in the prompt. As 3√(9) = 9 and 12√(4) = 24, neither is an irrational number and, therefore, not a surd. We can speculate that the sides of the triangles in the prompt have to be surds.

Reasoning

During a discussion of the longest short side in each right-angled triangle, the teacher can start to introduce the laws of surds, such as √(ab) = √(a) x √(b), at a relevant and meaningful point in the inquiry.

In the first triangle the longer of the two short sides is 6√(2) because 6√(2) = 2 x 3 x √(2), while 2√(6) = 2 x √(3) x √(2), and 3 > √(3).

The same law arises when explaining why the triangle with base length 6√(18) is also missing from the prompt. As 6√(18) can be simplified to 18√(2), the length of a short side would not be in its simplest form. 

Thus, we conclude that permissible triangles in the context of the inquiry are those with surds that cannot be simplified. The next case after those in the prompt has base length 10√(30).

Pattern

As students square the lengths of the short sides and square root their sum, they begin to identify a pattern:  (2√(6))2 + (6√(2))2 = √(24+72) = √(96) = 4√(6). In the first case the hypotenuse is twice the length of the shortest short side. Is that true for all cases?

Structured inquiry

Zack designed a four-part sheet for his students to structure their inquiry (included in the slides). Alternatively, experienced inquirers might use the regulatory cards to decide how to proceed with the inquiry.