Operations with Rational Numbers – Equivalent Forms & Repeating Decimals

Learning Goals

Benchmarks & Learning Targets

• Benchmark: MA.7.NSO.1.2 - Part of CRM 1.1 – Operations with Rational Numbers, Lesson 1.

• Learning Targets

  • Rewrite rational numbers in different but equivalent forms (fraction, decimal, percent), including repeating decimals.

  • Explore and generalize patterns in fractions whose decimal form is repeating.

Mathematical Thinking & Reasoning (MTR 5)

• “Use patterns and structure to help understand and connect mathematical concepts.”

• You will connect prior 6th-grade strategies to 7th-grade content.

• Our goal is to help you build connections between what you already know and new, more complex ideas.

Key Vocabulary

• Rational number: any real number that can be written as $\frac{a}{b}$ where $a$ and $b$ are integers and $b \neq 0$.

• Terminating decimal: a decimal representation that comes to an end (e.g., $0.25$).

• Repeating decimal: a decimal representation that extends infinitely but repeats a pattern; denoted with a bar, e.g., $0.\overline{6}$.

• Percentage: a ratio whose denominator is $100$.

Common Misconceptions & Technology Tips

• Remember to add the repeating bar over the repeating digits (e.g., $0.\overline{3}$ not $0.333$).

• Be aware that a rounded value from calculators might not be exact.

• Different calculators round repeating decimals differently, so we’ll discuss this in class.

Materials & Technology You'll Need

• Your Student Handout (modified from Math Nation).

• Approved online scientific calculator: You can use this link: https://bit.ly/ScientificCalc2526 (This is the calculator available on 7th-grade assessments & EOC).

• Avoid using non-approved calculators in class, as they may have different features.

• Look for the calculator icons on slides; they indicate when the calculator is required.

Activities & Practice

• Warm-Up (5 min) – “Gone Fishing”

  • Convert data on fish caught to fraction, decimal, and percent.

  • Scenario: 9 bluegills + 1 yellow perch in 2 hours.

  • Example: Portion bluegills = $\frac{9}{10}=0.9=90\%$.

  • Prompts: Identify the place value of 9 (tenths). Define “percentage.”

• Refresher (10 min, Teacher-Led)

  • Review 6th-grade methods for rewriting rational numbers.

  • Rational numbers have three interchangeable forms:

    1. Fraction ➡ divide numerator by denominator.

    2. Decimal ➡ read place value to write percent.

    3. Percent ➡ multiply decimal by $100$.

  • We’ll do a matching activity: connect phrases & strategies; e.g., “turn $\frac{3}{4}$ into a percent” ↔ “multiply by $100$”.

  • Sample table (answers will be shown):

    • $0.155 = 15.5\% = \frac{155}{1000} \text{ simplified}$.

    • $0.28 = 28\% = \frac{28}{100}=\frac{7}{25}$.

    • $0.375 = 37.5\% = \frac{3}{8}$.

  • Reflection questions: When does a calculator help? (decimal → percent/fraction via built-in convert). When is it limited? (exact repeating representation).

• Guided Instruction (20 min, Teacher-Led)

  • Definition revisited: A repeating decimal repeats infinitely but with a predictable sequence.

  • Calculator identification tip: A display like 0.3333333 with many 3’s, rather than a finite mix, often signals repetition.
    ### Table Exploration (Slides 18–21)

  • Convert a set of fractions to decimals; classify each as terminating or repeating:

  • Example conversions:

    • $\frac{2}{5}=0.4$ (terminating).

    • $\frac{2}{3}=0.\overline{6}$ (repeating).

    • $\frac{8}{7}=1.142857\,\overline{142857}$ (repeating).

  • You will circle values >1 and rewrite them as mixed numbers, e.g., $\frac{8}{7}=1\,\frac{1}{7}$.
    ### Key Talking Points

  • If the simplified denominator has ONLY $2$ and/or $5$ as prime factors ➡ the decimal terminates.

  • Any other prime factor in the denominator ➡ the decimal repeats.

  • Calculators may show a last digit of 1 due to rounding errors (e.g., $0.1428571$).
    ### Highlighted Example – $\frac{85}{99}$

  • Long division pattern: $0.858585…$

  • Bar notation: $0.\overline{85}$.

  • Reason digits repeat: the denominator generates a remainder cycle of length 2.
    ### Observed Patterns

  • Denominators like $9, 11, 99, 999,\ldots$ produce repeating blocks equal in length to the number of 9’s/11’s.

  • Fractions >1 with repeating decimals still keep the repeating block in their decimal part (e.g., $4\,0.\overline{3}$).

• Try It (5 min, Student-Led)

  • Compute decimal equivalents and then circle those that are repeating.

  • Discussion prompt: “Why does the calculator sometimes end with ‘1’?”

  • Connect this to rounding versus the true infinite pattern.

• Your Turn! (5 min, Student-Led)

  • Complete a table: write decimal equivalents of given fractions.

  • Fractions provided: $\frac{5}{6}, \frac{4}{15}, \frac{1}{80}, \frac{1}{1}, \frac{9}{8}, \frac{152}{333}, \frac{2}{13}, \frac{2}{32}$.

  • Sample answers (will be shown):

    • $\frac{5}{6}=0.83\overline{3}$

    • $\frac{4}{15}=0.26\overline{6}$

    • $\frac{1}{80}=0.0125$ (terminating).

    • $\frac{9}{8}=1.125$ (terminating, mixed $1\,\frac{1}{8}$).

    • $\frac{152}{333}=0.456456…=0.\overline{456}$.

    • $\frac{2}{13}=0.\overline{153846}$.

    • $\frac{2}{32}=0.0625$.

Academic Discourse Cards

• Purpose: These cards help you structure conversations, justify your thinking, and understand others' reasoning.

• Cards include prompts for:

  • Asking clarifying questions.

  • Agreeing/disagreeing with justification.

  • Connecting to prior knowledge.

• How we'll use them: We'll prepare, use them during discussions, and reflect afterward.

Real-World & Ethical Connections

• Data interpretation: Converting between forms is crucial for understanding statistics, interest rates, and probabilities in real life.

• Technology literacy: Understanding calculator limitations helps prevent the spread of incorrect, rounded information.

Summary of Core Mathematical Ideas

• Every rational number can be represented exactly as a fraction; its decimal may terminate or repeat infinitely.

• The determining factor for whether a decimal terminates is the prime factors of the simplified denominator (only $2$ or $5$).

• Repeating decimals can be denoted concisely with a bar and converted back to fractions using a specific algebraic method:

  • Example: Let $x = 0.\overline{3}$

  • Then, $10x = 3.\overline{3}$

  • Subtracting the first equation from the second: $10x - x = 3.\overline{3} - 0.\overline{3}$

  • This gives: $9x = 3$

  • So, $x = \frac{3}{9} = \frac{1}{3}$ (simplified).

• Exploring