The prompt introduces the concept of rounding to classes in years 7 and 8 (grades 8 and 9). As students round square roots to the nearest whole number, they realise that two square roots round to 1, four round to 2, six round to 3, and, in general, 2n round to n (see the spreadsheet for the square roots of the numbers from one to 100). Once students have a clear understanding of the prompt's meaning, they take great delight in showing that it is true.

During the inquiry students develop fluency in rounding. However, the teacher should guide students in the degree of accuracy they use at each stage of the inquiry to avoid the following two misconceptions:

1. Truncation

When students round the square roots of numbers less than 100 to one significant figure, they find the nearest positive integer. For example, the square root of 12 appears as 3.464101615 on a calculator. As the square root (x) is in the range 2.5 ≤ x < 3.5, the square root of 12 is three to the nearest integer. However, rounding to one significant figure at the start of the inquiry has led to students truncating square roots instead of rounding. The square root of 13 (3.605551275) will also 'round' to three, they reason, because there is a three in the units column. 

2. A 'double rounding'

A misconception involving a 'double rounding' can also arise when students round square roots to one decimal place. The square root of 12, rounded to one decimal place, is 3.5. Students have then rounded for a second time, arguing that 3.5 rounds up to four.

To avoid misconceptions at the launch of the inquiry, the teacher might guide the class to round to two decimal places. In that case the square root of 12 rounds to 3.46, which is lower than 3.5 and, therefore, closer to three than four.

As students become more confident, the teacher can guide them to use other degrees of accuracy.

Question, notice, and wonder

In the initial phase of the inquiry, students attempt to make meaning of the prompt. Responses in the question, notice, and wonder phase include:

• Rounding can be to the nearest 10 or 100;

• Square roots are something to do with squares. An example of a square number is 16 beause 4 x 4 = 16.

• What is a positive integer?

• Does n stand for a number? What is 2n?

The teacher should be ready to draw out as much relevant knowledge as the class holds and draw upon students' contributions.

Structured inquiry

In a structured approach, the teacher might break the inquiry into stages: at first, students use a calculator to find the square roots of the numbers from one to 20; afterwards, they round the roots to the nearest positive integer. 

In this way, students can participate in the inquiry without having a clear understanding of the 'big picture'. The full meaning of the prompt might emerge only once the class reflects on the results (see the table below from the slides) and when the teacher orchestrates a discussion on the aim of the inquiry.