Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.
Official wording from the Common Core State Standards for Mathematics (© 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers). View on thecorestandards.org
Every number on the number line has a decimal expansion, but not every expansion behaves the same way. Fractions such as 3/8 or 5/11 always produce decimals that either stop (0.375) or settle into a block of digits that repeats forever (0.454545...). Numbers whose decimals never stop and never repeat, such as √2 or π, cannot be written as a ratio of two integers. Eighth graders meet the word irrational for these numbers and learn that together with the rationals they make up the real numbers.
The heart of the work is two-way conversion. Students show that long division of a fraction must eventually repeat, because there are only so many possible remainders. Going the other way, they take a repeating decimal like 0.272727... and use algebra to rewrite it as a fraction: multiply by a power of 10 that shifts one full block of repeating digits, subtract the original, and solve. That procedure is the proof that every repeating decimal is rational, which is why a number like 0.1010010001... (with a growing gap of zeros) is irrational.
A calculator showing 0.1428571429 for 1/7 tempts students to call it irrational. Point out that the digits 142857 repeat, so it is a fraction in disguise.
These are rational approximations, not the value itself. Students should say π is about 3.14, and know its decimal never ends or repeats.
√36 is 6 and √(4/9) is 2/3, both rational. Only roots of non-perfect squares, such as √5, are irrational.
For 0.454545... the block has two digits, so multiply by 100, not 10. Using 10 leaves the repeating parts misaligned and the subtraction does not clear them.
Let x = 0.272727..., where the digits 27 repeat forever. Write x as a fraction in simplest form.
Answer: 0.272727... = 3/11. Checking with division, 3 ÷ 11 = 0.2727..., which matches.
Start with a sorting activity: give pairs a set of cards (5/8, 0.333..., √49, √10, π, -4, 0.121121112...) and ask them to group by decimal behavior before introducing the vocabulary. Then let students do a few long divisions by 7 or 13 and notice the remainders cycling, which makes the repeating argument concrete rather than a rule to memorize.
Assessments often ask students to identify the irrational number in a list, or to convert a repeating decimal with a one- or two-digit block. Encourage a quick check by dividing the final fraction on a calculator, and keep the bar notation for repeating digits visible on the board so students connect 0.3 with a bar to 1/3.
Original questions written for this standard. Choose an option or type your answer, then press Check. Every question has a worked explanation.
Answer: D) √13
13 is not a perfect square, so √13 has a decimal that never ends or repeats. √49 = 7, 0.125 = 1/8 and 5/9 are all rational.
Answer: 7/9
Let x = 0.777... Then 10x = 7.777... Subtracting gives 9x = 7, so x = 7/9.
Answer: 2/11
Let x = 0.1818... Multiply by 100: 100x = 18.1818... Subtract: 99x = 18, so x = 18/99 = 2/11.
Answer: C) Because long division by 13 has only 13 possible remainders, so a remainder must come round again
Dividing by 13 can only leave remainders 0 to 12. Once a remainder repeats, the digits after it repeat too, so the decimal cannot go on without a pattern.
Answer: D) 3.14 = 314/100, so it is rational; it is only an approximation of π
3.14 terminates, so it equals 314/100 and is rational. π itself is irrational; 3.14 is just a close rational estimate.
Answer: 3 (also accepted: three)
√16 = 4, 0.5 and -3 are rational. √2 and π are irrational. That makes 3 rational numbers.
A full lesson with slides, activities and an exit ticket on rational and irrational numbers, pitched to grade 8 and editable in PowerPoint or Google Slides.
Make a lesson →A printable, differentiated worksheet on 8.NS.A.1 with an answer key, ready in about a minute.
Make a worksheet →Turn rational and irrational numbers into a quiz students answer online that marks itself, with a class summary for you.
Build a test →A rational number can be written as a fraction of two integers, and its decimal ends or repeats. An irrational number cannot be written that way, and its decimal goes on forever with no repeating block.
Yes. Using the 8.NS.A.1 method, x = 0.999... gives 10x - x = 9, so 9x = 9 and x = 1. It is a good extension question for confident students.