The teaching of irrational number concepts has been historically relegated to the final chapter of books, when conducting computations with radicals. Instead, students need to be introduced to the need for irrational numbers as the needs arise in context. The following are some of the major points and examples from my presentation for high school teachers:
RATIONALS FIRST
Students need to be comfortable with rational numbers and skills involving them prior to the introduction of irrational numbers. Examples include finding the percentage of a number (mentally and on paper), proper use of decimals and fractions, and recognition of equivalent forms of quantities (e.g., 1/4 = 0.25 = 25%, etc.).
QUADRATICS TO INTRODUCE IRRATIONALS
Allow the study of irrational numbers to flow from contextual situations. One example is to study how to solve quadratic equations by looking for x-intercepts (zeros) of the function on a graphing calculator. If students can find rational roots for an equation such as x^2 - 2x - 8 = 0, then when they encounter an equation such as x^2 - 4x + 2 = 0, they will recognize that answers need to be rounded, and they begin realize that solutions do not always work out "exactly."
PROOF OF SQR(2) [Square Root of 2] BEING IRRATIONAL
After developing the quadratic formula, students are ready to talk about how solutions that have a discriminant that is not a perfect square, the answers do not work out evenly. At this point, it is important that they be convinced that SQR(2) is irrational, and Euclid's proof from 300 B.C. is a nice way to do it. For an explanation of this proof, click HERE.
CALCULATOR EXPLORATION
Students are also interested in finding out how many decimal places their calculators "know" the values of irrational numbers, such as SQR(2). Have students calculate the value and help them conclude that by subtracting "1", the calculator will now display one more digit of the answer. If they multiply by 10 and then subtract the "4", another number is revealed, and so forth. They can continue this process until the calculator runs out of information. Since they have already proven that SQR(2) is irrational, they now know the limit of what their calculator can do in terms of approximating values of irrational numbers. They can do the same with pi, etc.
CONSTRUCTING IRRATIONALS
If students use a compass and straightedge, they can construct (a) congruent segments and (b) perpendicular lines. Using the first construction, they can begin with a ray and construct a number line from 0 to 10. The problem raised is, "How do I find exactly where SQR(5) lies on that number line?" Students discover that if they construct a right triangle (by using the perpendicular lines construction) and then, using the "unit" from their number line, construct the triangle so that the sides are of length 1 and 2. By the Pythagorean Theorem, the length of the hypotenuse is SQR(5), so they can measure it with the compass and construct that length on the number line to show its location. Similarly, students can construct irrationals such as SQR(2), SQR(10), SQR(13), SQR(20), SQR(34), and so on. They can also use their constructions to show that SQR(20) = 2 * SQR(5), which opens the gateway to discussing how to simplify radical expressions.
PAPER-FOLDING IRRATIONALS
For a nice description of how to use paper folding to generate a 30-60-90 triangle and use it to create a number line, see the article entitled "Irrational Numbers on the Number Line: Perfectly Placed," which appeared in the September 2001 Mathematics Teacher journal (Vol. 94, No. 6, page 453).
OTHER CONNECTIONS
Students can continue their exploration of irrational numbers by looking at pi, e, and the golden ratio. The golden ratio is one of my favorites, since its value is (1 + SQR(5))/2 from the quadratic formula. The Web has many good sites on this ratio, but for one of my favorites, click HERE.