Irrational Approximations

The Common Core State Standards are filled with number sense. Students engaged in the Common Core aligned curriculum will have a deep understanding of rational and irrational numbers...including a rational approximation of an irrational number (CCSS 8.NS.A.2).

How would you find the rational approximation of this number?

Ummm...without a calculator. ;)

Here's how we approach this one in my classroom...Well. Let's start with square roots that are rational. We know the square root of 20 is somewhere between the square root of 16 and the square root of 25...so something between 4 and 5. Can you be more specific? Hmmm...the list begins to form...
root 16 = 4
root 17 = ?
root 18 = ?
root 19 = ?
root 20 = ?
......................about 4.5
root 21 = ?
root 22 = ?
root 23 = ?
root 24 = ?
root 25 = 5

Since we have 9 numbers to cover an increase of 1 unit, each increase will be a little more than 0.1. Why? Because 1/9 > 1/10. So we can approximate that root 20 is approximately 4.4. {Ish! At this point we discuss if the "little more" is "enough more" to justify rounding.}

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But what about simplest radical form?!? We've searched high and low. No sign of "simplest radical form" in our copy of the CCSS document. (If you find it...please put me out of my misery!)

Could we teach our students to simplify radicals and then in turn use that to find the rational approximation of an irrational number? In simplest radical form, the square root of 20 is exactly 2 times the square root of 5. Hmmm...the square root of 5 is some number between the square root of 4 and the square root of 9...between 2 and 3.

root 4 = 2
root 5 = ?
root 6 = ?
..................about 2.5
root 7 = ?
root 8 = ?
root 9 = 3

Since we have 5 numbers to cover an increase of 1 unit, each increase will be more than 0.1. Why? Because 1/5 > 1/10 (some students will automatically go to the fact that 1/5 = 0.2). So we can approximate that root 5 is approximately 2.2 and therefore 2(2.2) = 4.4.

Bottom line. We reach a valid approximation either way. I think it's important for our students to discover both paths...and even more important for them to critique which approach they prefer and under what circumstances they would select that particular method. I'm ready for a viable argument. How about you?!?

Cooperative practice would be crucial for students to have perspective to support their argument.

Quiz-Quiz-Trade or Inside-Outside Circle

This topic has a natural connection to Pythagorean Theorem application (CCSS 8.G.B.7). When students are determining the length of an unknown side in a right triangle, they should be able to state their solution as both an exact and approximate number. And how do we teach students to solve using the Pythagorean Theorem when they don't know the first thing about a quadratic equation? That will be for another post...after we finish investigating rational and irrational numbers...stay tuned!