Grade 8 · Real Numbers & Exponents · MAT-08-NS-002
Understand Decimal Expansions of Rational Numbers
Represent and solve understand decimal expansions of rational numbers problems using precise mathematical notation, multiple representations, and justified reasoning.
Learning objectives
What you should be able to do
- Represent and solve understand decimal expansions of rational numbers problems using precise mathematical notation, multiple representations, and justified reasoning.
- Interpret and verify understand decimal expansions of rational numbers results in context, including units, constraints, and reasonableness.
Prerequisite check
Make sure the foundation is ready.
Grade 8 uses this prerequisite as active knowledge in a more formal representation.
Learn
Build the idea from meaning, not memorization.
Meaning and structure
Understand Decimal Expansions of Rational Numbers belongs to real numbers & exponents. Identify the governing property or relationship before choosing a procedure.
Multiple representations
Represent understand decimal expansions of rational numbers numerically and symbolically, then connect it to the Grade 8 Math Box. Tables, graphs, equations, coordinate models, diagrams, and verbal descriptions must agree.
Verification and interpretation
Check with substitution, an inverse operation, an equivalent representation, geometric invariants, estimation, or context. State whether the result is exact or approximate and preserve relevant units and constraints.
Math Box · Interactive lesson
Touch the math. Change it. See what stays true.
Use the interactive model before and after the worked examples. Change the inputs, make a prediction, then use the model to test whether your reasoning holds.
Irrational Number-Line Lab
Bracket a square root between nearby integers, then refine its decimal position.
- Model understand decimal expansions of rational numbers in the Math Box.
- Change one input and predict which quantities or invariants should change.
- Explain what the visual, equation, and numerical result say about the same relationship.
Worked examples
Twenty different ways to see the concept work.
Will the reduced form of 1/2 terminate or repeat as a decimal?
terminatingAfter simplification, the denominator has only factors 2 and/or 5.
Will the reduced form of 2/3 terminate or repeat as a decimal?
repeatingAfter simplification, the denominator retains a prime factor other than 2 or 5.
Will the reduced form of 3/4 terminate or repeat as a decimal?
terminatingAfter simplification, the denominator has only factors 2 and/or 5.
Will the reduced form of 4/5 terminate or repeat as a decimal?
terminatingAfter simplification, the denominator has only factors 2 and/or 5.
Will the reduced form of 5/6 terminate or repeat as a decimal?
repeatingAfter simplification, the denominator retains a prime factor other than 2 or 5.
Will the reduced form of 6/7 terminate or repeat as a decimal?
repeatingAfter simplification, the denominator retains a prime factor other than 2 or 5.
Will the reduced form of 7/8 terminate or repeat as a decimal?
terminatingAfter simplification, the denominator has only factors 2 and/or 5.
Will the reduced form of 8/9 terminate or repeat as a decimal?
repeatingAfter simplification, the denominator retains a prime factor other than 2 or 5.
Will the reduced form of 9/10 terminate or repeat as a decimal?
terminatingAfter simplification, the denominator has only factors 2 and/or 5.
Will the reduced form of 10/11 terminate or repeat as a decimal?
repeatingAfter simplification, the denominator retains a prime factor other than 2 or 5.
Will the reduced form of 11/12 terminate or repeat as a decimal?
repeatingAfter simplification, the denominator retains a prime factor other than 2 or 5.
Will the reduced form of 12/13 terminate or repeat as a decimal?
repeatingAfter simplification, the denominator retains a prime factor other than 2 or 5.
Will the reduced form of 13/14 terminate or repeat as a decimal?
repeatingAfter simplification, the denominator retains a prime factor other than 2 or 5.
Will the reduced form of 14/15 terminate or repeat as a decimal?
repeatingAfter simplification, the denominator retains a prime factor other than 2 or 5.
Will the reduced form of 15/16 terminate or repeat as a decimal?
terminatingAfter simplification, the denominator has only factors 2 and/or 5.
Will the reduced form of 16/17 terminate or repeat as a decimal?
repeatingAfter simplification, the denominator retains a prime factor other than 2 or 5.
Will the reduced form of 17/18 terminate or repeat as a decimal?
repeatingAfter simplification, the denominator retains a prime factor other than 2 or 5.
Will the reduced form of 18/19 terminate or repeat as a decimal?
repeatingAfter simplification, the denominator retains a prime factor other than 2 or 5.
Will the reduced form of 19/20 terminate or repeat as a decimal?
terminatingAfter simplification, the denominator has only factors 2 and/or 5.
Will the reduced form of 20/21 terminate or repeat as a decimal?
repeatingAfter simplification, the denominator retains a prime factor other than 2 or 5.
Practice
Guided practice with hints
01Will the reduced form of 21/22 terminate or repeat as a decimal?
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: repeating
02Will the reduced form of 22/23 terminate or repeat as a decimal?
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: repeating
03Will the reduced form of 23/24 terminate or repeat as a decimal?
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: repeating
04Will the reduced form of 24/25 terminate or repeat as a decimal?
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: terminating
05Will the reduced form of 25/26 terminate or repeat as a decimal?
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: repeating
06Will the reduced form of 26/27 terminate or repeat as a decimal?
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: repeating
Practice
Independent practice
01Will the reduced form of 27/28 terminate or repeat as a decimal?
Answer: repeating
02Will the reduced form of 28/29 terminate or repeat as a decimal?
Answer: repeating
03Will the reduced form of 29/30 terminate or repeat as a decimal?
Answer: repeating
04Will the reduced form of 30/31 terminate or repeat as a decimal?
Answer: repeating
05Will the reduced form of 31/32 terminate or repeat as a decimal?
Answer: terminating
06Will the reduced form of 32/33 terminate or repeat as a decimal?
Answer: repeating
07Will the reduced form of 33/34 terminate or repeat as a decimal?
Answer: repeating
08Will the reduced form of 34/35 terminate or repeat as a decimal?
Answer: repeating
Common mistakes
Learn to catch the error, not just the answer.
Repeating nonterminating decimals are rational.
Product and quotient exponent rules require the same base and multiplication or division.
Applications & challenge
Use the idea beyond a single exercise.
Applications
- Represent extremely large and small measurements with scientific notation.
- Approximate roots and irrational quantities when decimal estimates are needed.
Challenge problems
Will the reduced form of 35/36 terminate or repeat as a decimal?
repeating
Will the reduced form of 36/37 terminate or repeat as a decimal?
repeating
Will the reduced form of 37/38 terminate or repeat as a decimal?
repeating
Will the reduced form of 38/39 terminate or repeat as a decimal?
repeating
Flashcards
Retrieve it from memory.
Flashcards
Card 1 of 4
Game mode
Grade 8 Algebra & Modeling Sprint
Solve understand decimal expansions of rational numbers problems and justify each result with a valid verification method.
Practice
Mastery check
01Will the reduced form of 39/40 terminate or repeat as a decimal?
Answer: terminating
02Will the reduced form of 40/41 terminate or repeat as a decimal?
Answer: repeating
03Will the reduced form of 41/42 terminate or repeat as a decimal?
Answer: repeating
04Will the reduced form of 42/43 terminate or repeat as a decimal?
Answer: repeating
05Will the reduced form of 43/44 terminate or repeat as a decimal?
Answer: repeating
06Will the reduced form of 44/45 terminate or repeat as a decimal?
Answer: repeating
07Will the reduced form of 45/46 terminate or repeat as a decimal?
Answer: repeating
08Will the reduced form of 46/47 terminate or repeat as a decimal?
Answer: repeating
Terminology
Words to know
- rational number
- A real number expressible as a ratio of integers with nonzero denominator.
- irrational number
- A real number that cannot be written as a ratio of integers.
- exponent
- A number indicating repeated multiplication by a base.
- scientific notation
- A representation a × 10^n where 1 ≤ |a| < 10 and n is an integer.
Continue learning
Connected concepts
Curriculum references
Standards and instructional references
California Mathematics Framework (2023)
Grade 8 instructional guidance, mathematical practices, modeling, and progression toward high-school mathematics.Common Core State Standards for Mathematics — Grade 8
Grade-level expectations for real numbers, equations, functions, geometry, and statistics.Principles to Actions: Ensuring Mathematical Success for All
Research-informed emphasis on reasoning, representations, discourse, productive struggle, and conceptual understanding.