Wyoming Math 8.NS.1b

Math8th GradeThe Number System

The Standard

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. Explore the real number system and its appropriate usage in real-world situations. Understand that all real numbers have a decimal expansion.

2023 Wyoming Mathematics Content and Performance Standards · Know that there are numbers that are not rational, and approximate them by rational numbers.

Teacher's field guide

What This Standard Means

What Students Need to Do: Students divide fractions to see why their decimals end or repeat. They turn repeating decimals into fractions. They recognize that every real number has a decimal expansion, even if it never ends.
What Mastery Looks Like: A student can divide 1 by 6 to get 0.1666... and point to the repeated remainder. They can turn 0.1666... back into 1/6. They can identify √2 as irrational and use 1.41 as an estimate, not an exact value.
Common Misconceptions: Students may call a terminating decimal irrational because its digits do not visibly repeat. Some think 0.1666... cannot equal a fraction because it never ends. Others assume that a few digits on a calculator prove a number is irrational.
How to Assess It: Exit ticket: Divide 1 by 6, circle the repeated remainder, and convert 0.1666... back to a fraction.

Lesson moves

Ways to Teach It

Give pairs base-ten blocks to divide 1 whole by 6, trading for tenths and hundredths and recording each remainder.
Ask: Can a calculator showing 1.4142135 prove that √2 repeats? Write two sentences explaining your answer.
Deal cards showing 0.25, 0.272727..., 0.1666..., and √2; partners sort them and convert the repeating decimals to fractions.
Measure a round lid's diameter, estimate its circumference using 3.14, and explain why the result is an approximation.

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