Rational Approximations of Irrational Numbers

Hand out the Real Number System Reference Sheet first and keep it out during the sorts. Use the classification and comparison sheets for instruction, and save the maze, riddles and mystery picture for review.

Trap √2, √10, √55 and π between tenths, then between hundredths, by squaring decimals. The directions on the page read: Square tenths in the box to find the two tenths each square root lies between. Then square hundredths to find the two hundredths. For π, use the digits given. An answer key is included.

Yes, both. We made it for 8th grade number theory, and it works the same at home or at school. At home, a student can work through it alone and then go over it with you, which shows quickly what is solid and what needs another look. In class it works as independent practice, homework or a warm-up, and you can print one copy or a full set.

A decimal that ends, like negative 4.25, or repeats, like 0.454545…, is rational. A decimal that goes on forever without a repeating block, like 0.1010010001…, is irrational. Sort Decimals and Fractions as Rational or Irrational asks students to explain this difference.

Yes, sometimes. √2 × √2 = 2, which is rational, but √2 × √3 is irrational. Always, Sometimes or Never With Real Numbers and True or False Statements About Real Numbers test claims like this with examples and counterexamples.

Natural numbers sit inside the whole numbers, whole numbers inside the integers and integers inside the rational numbers. Irrational numbers are the real numbers that are not rational. The card sort and the set diagram sheet place each number in the smallest set it belongs to.

Write both as decimals with enough digits to see which is greater. With π about 3.14159 and 22/7 about 3.142857, 22/7 is greater. Compare Rational and Irrational Numbers practices pairs like this.