Roots inquiry
The prompt
Mathematical inquiry processes: Verify; test other cases; create alternative representations; reason. Conceptual field of inquiry: Square and cube roots; laws of exponents; graphs of functions with roots; differentiation.
The prompt was inspired by Adam Hrankowski's blog post A Deceptively Simple Math Question. The inquiry that develops from it can encompass multiple areas of mathematics (involving numerical, algebraic and graphical representations) at different levels of complexity (ranging from square and cube roots to differentiation).
The questions and observations at the start of the inquiry depend on the students' prior knowledge. If they are not familiar with square and cube roots, then the questions will focus on those. Even if they are, the first root of one often arouses curiosity and leads to discussion.
Students who have greater prior knowledge might start by verifying that the inequality in the prompt is true, giving 1 < 1.4142 < 1.4422 (accurate to 4 decimal places). The teacher might use this phase as an opportunity to familiarise students with the meaning of calculator keys.
Sequence
Seeing the first, square and cube roots of, respectively, one, two and three as the start of a sequence (the nth root of n) opens up a line of inquiry in which the inequality is extended. Is the fourth root of four greater than the cube root of three? Curiosity is heightened when students find that it is not. The fifth root of five is lower again.
At this point, a graphical representation might be helpful to understand what is happening. However, the teacher will have to prepare the ground by introducing exponents to represent roots. The nth root of n can be mapped on a graph using the equation y = x1/x.
The graph shows that the maximum point occurs between x = 2 and x = 3, after which the graph descends slowly.
Introducing e
The graph has opened yet another line of inquiry. Students have sought the x and y values at the maximum point. This can be done by calculating the y values for the tenths between two and three, then for the hundredths between 2.7 and 2.8, and then for the thousandths between 2.71. and 2.72, and so on. As students find the tenths and hundredths, the teacher should take the opportunity to introduce Euler's number (e). Alternatively, for advanced students, the teacher can differentiate y = x1/x (see Adam's blog) and find the maximum point lies at (e,e1/e).
Extending the inquiry
How could students inquire further by changing the prompt? One interesting suggestion that has arisen in classroom inquiry is to change the sequence to the the (n + 1)th root of n - that is, the square root of one, the cube root of two and so on (see below).
Creativity
Adam finishes his post with this sentiment: "Creativity is not on the math curriculum of most high schools I have seen. Mathematics is taught as a technical skill. Precious little time is devoted to actually struggling with a problem. Yet, it's that struggle that answers the persistent question: 'Will we ever actually use this in the Real World?'"
January 2022
Using laws of exponents
By using the law that states (xm )n = xmn, we can show that the inequality is true. Changing the denominator of the indices to 12 (the lowest common multiple of 2, 3, and 4) enables the class to compare the quantities. The table has sparked more discussion when students notice that the square root of two and the fourth root of four are equal. We can use the same law to show this to be true: 41/4 = (22 )1/4 = 22/4 = 21/2.
We can insert e1/e into the table above, using the same law to show that it is greater than both the square root of two and the cube root of three. (The last line has been rounded to three decimal places.) In the past, this approach has convinced students who have remained sceptical when presented with the graph only. The line, they say, seems flat in the critical region.
When students use the same approach for the sequence of the (n + 1)th root of n, the maximum value occurs between x = 3 and x = 4. This time the denominator of the exponent is 60 - that is, the lowest common multiple of 2, 3, 4, and 5. Students can practice the technique by extending the tables (for example, the sixth root of five) or by exploring the (n + 2)th root of n.
