Wyoming Math 8.NS.1d

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. Convert repeating decimals to fractions.

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 recognize that rational decimals end or eventually repeat, while irrational decimals do neither. They convert repeating decimals to fractions and use decimal estimates for irrational numbers in measurements.
What Mastery Looks Like: A student can explain why 0.75 and 0.1666... are rational but √2 is irrational. They convert 0.1666... to 1/6 by shifting and subtracting decimals, then use 1.41 to estimate a unit square's diagonal.
Common Misconceptions: Students may call 0.333... irrational because it never ends. They may miss the repeating 6 in 0.1666... or write √2 = 1.41 instead of √2 ≈ 1.41.
How to Assess It: Exit ticket: Convert 0.1666... to a fraction using shifted copies, then state whether √2 is rational and estimate it to two decimal places.

Lesson moves

Ways to Teach It

Draw a 1-by-1 square on grid paper, measure its diagonal, and compare the measurement with √2.
Write why 0.1666... is rational but √2 is irrational, even though both decimals go on forever.
Match cards showing 0.333..., 0.1666..., and 0.2727... with fraction cards, then check each match by shifting and subtracting decimals.
Measure a jar lid's diameter, write π times that measurement, and use 3.14 to estimate the ribbon needed around it.

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