Year 7 Mathematics · AC9M7N04

Equivalent Rational Numbers

We are learning to find equivalent representations of rational numbers and represent rational numbers on a number line

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Learning goalsSay it simply

We are learning to find equivalent representations of rational numbers and represent rational numbers on a number line.

A rational number can be written as a fraction of integers with a non-zero denominator. Equivalent fractions, decimals and percentages occupy the same point on a number line.

Conversion preserves value: 7/20 becomes 35/100, then 0.35 and 35%. A common representation makes comparison reliable when forms differ.

Signed rational numbers require attention to direction. Among negative values, a number with greater absolute value lies farther left and is therefore smaller.

Success criteria

  • I can convert among fraction, decimal and percentage representations.
  • I can justify that two rational representations are equivalent.
  • I can locate and order positive and negative rational numbers on a number line.
Key conceptTeach from the board

Use these models to connect the mathematical idea to values, diagrams, coordinates, graphs or structure before moving to symbolic calculation.

Follow each example from representation and setup through calculation/reasoning, interpretation and an independent check.

Example 1

Connect equivalent rational forms

  1. Begin with 7/20.
  2. Multiply numerator and denominator by 5 to obtain 35/100.
  3. Read 35/100 as decimal 0.35.
  4. Convert hundredths to 35%.

Final answer: 7/20 = 0.35 = 35%.

Check: Calculate 7 ÷ 20 = 0.35.

Example 2

Order signed rational numbers

  1. Convert −3/4 to −0.75.
  2. Keep −0.6 as a decimal.
  3. Convert 20% to 0.20.
  4. Order on a number line: −0.75 < −0.6 < 0.20.

Final answer: −3/4 < −0.6 < 20%.

Check: Values farther left on the number line are smaller.

Example 3

Application problem 1

Problem: Order −0.6, −5/8, 55% and 0.58 from least to greatest. Convert to a common representation and justify the order.

  1. Plan: Represent the information first, then calculate, interpret and independently check the result.
  2. Work: −5/8 = −0.625, so −0.625 < −0.6 < 0.55 < 0.58. Therefore −5/8, −0.6, 55%, 0.58.
  3. Interpret: This synthesis problem combines the chapter's core representation, calculation and reasoning skills.

Final answer: −5/8 = −0.625, so −0.625 < −0.6 < 0.55 < 0.58. Therefore −5/8, −0.6, 55%, 0.58.

Check: This synthesis problem combines the chapter's core representation, calculation and reasoning skills.

Example 4

Application problem 2

Problem: Order 7/8, 0.86 and 87% and state the gap between greatest and least.

  1. Plan: Write 7/8 and 87% as decimals.
  2. Work: 0.86 < 87% < 7/8; the gap is 0.875−0.86=0.015.
  3. Interpret: Convert each value to a decimal.

Final answer: 0.86 < 87% < 7/8; the gap is 0.875−0.86=0.015.

Check: Convert each value to a decimal.

Clean visual examplesOne-page board

Clean one-page examples

AC9M7N04 - Equivalent Rational Numbers
Example 1

Example 1 Connect equivalent rational forms Begin with 7/20. Multiply numerator and denominator by 5 to obtain 35/100. Read 35/100 as decimal 0.35. Convert hundredths to 35%. Final answer: 7/20 = 0.35 = 35%. Check: Calculate 7 ÷ 20 = 0.35.

Example 2

Example 2 Order signed rational numbers Convert −3/4 to −0.75. Keep −0.6 as a decimal. Convert 20% to 0.20. Order on a number line: −0.75 &lt; −0.6 &lt; 0.20. Final answer: −3/4 &lt; −0.6 &lt; 20%. Check: Values farther left on the number line are smaller.

Example 3

Example 3 Application problem 1 Problem: Order −0.6, −5/8, 55% and 0.58 from least to greatest. Convert to a common representation and justify the order. Plan: Represent the information first, then calculate, interpret and independently check the result. Work: −5/8 = −0.625, so −0.625 &lt; −0.6 &lt; 0.55 &lt; 0.58. Therefore −5/8, −0.6, 55%, 0.58. Interpret: This synthesis problem combines the chapter's core representation, calculation and reasoning skills. Final answer: −5/8 = −0.625, so −0.625 &lt; −0.6 &lt; 0.55 &lt; 0.58. Therefore −5/8, −0.6, 55%, 0.58. Check: This synthesis problem combines the chapter's core representation, calculation and reasoning skills.

Example 4

Example 4 Application problem 2 Problem: Order 7/8, 0.86 and 87% and state the gap between greatest and least. Plan: Write 7/8 and 87% as decimals. Work: 0.86 &lt; 87% &lt; 7/8; the gap is 0.875−0.86=0.015. Interpret: Convert each value to a decimal. Final answer: 0.86 &lt; 87% &lt; 7/8; the gap is 0.875−0.86=0.015. Check: Convert each value to a decimal.

Curriculum examplesCopied content

The content description and elaborations below show the curriculum ideas taught in this unit. Items marked as teaching context support lesson planning.

  • Content description: find equivalent representations of rational numbers and represent rational numbers on a number line
  • E1: investigating equivalence of fractions using common multiples and a fraction wall, diagrams or a number line to show that a fraction such as \frac23 is equivalent to \frac46 and \frac69 and therefore \frac23<\frac56
  • E2: expressing a fraction in simplest form using common divisors
  • E3: applying and explaining the equivalence between fraction, decimal and percentage representations of rational numbers; for example, 16\%, 0.16, \frac{16}{100} and \frac4{25}, using manipulatives, number lines or diagrams
  • E4: representing positive and negative fractions and mixed numerals on various intervals of the real number line, including intervals that are not symmetrical about zero
Questions and answersWith answers

Attempt each problem before opening Check answer. The set moves from core understanding to application and synthesis.

  1. 1. Convert 3/4 to a decimal and percentage.

    Check answer

    Answer: 3/4 = 0.75 = 75%.

    Hint: Divide 3 by 4, then multiply the decimal by 100.

    Why: Equivalent forms name the same value.

  2. 2. Simplify 42/56.

    Check answer

    Answer: 42/56 = 3/4.

    Hint: Find the highest common factor.

    Why: Dividing both terms by 14 preserves value.

  3. 3. Write 0.6 as a fraction and percentage.

    Check answer

    Answer: 0.6 = 3/5 = 60%.

    Hint: Write 0.6 as 6/10.

    Why: Six tenths simplifies to three fifths.

  4. 4. Order −0.5, −2/3 and 25% from least to greatest.

    Check answer

    Answer: −2/3 < −0.5 < 25%.

    Hint: Convert all three to decimals.

    Why: Converting the three rational numbers to equivalent decimals reveals their number-line order.

  5. 5. Place 5/8 between two tenths on a number line.

    Check answer

    Answer: 5/8 = 0.625, so it lies between 0.6 and 0.7.

    Hint: Calculate 5 ÷ 8.

    Why: A decimal locates the fraction precisely.

  6. 6. Which is greater: 7/12 or 58%?

    Check answer

    Answer: 7/12 ≈ 58.33%, so 7/12 is greater.

    Hint: Convert 7/12 to a percentage.

    Why: A common representation permits comparison.

  7. 7. Correct the claim that −3/5 is greater than −1/2 because 3/5 > 1/2.

    Check answer

    Answer: −3/5 = −0.6, which is less than −0.5 = −1/2.

    Hint: Plot both values left of zero.

    Why: Negatives reverse the order of positive magnitudes.

  8. 8. A tank is 0.65 full. Express this as a simplified fraction and percentage.

    Check answer

    Answer: Dividing by 100 gives 0.65=65/100; simplifying by 5 gives 13/20, and multiplying the decimal by 100 gives 65%.

    Hint: Simplify 65/100 by 5.

    Why: All forms describe the same portion of the tank.

  9. 9. Order 7/8, 0.86 and 87% and state the gap between greatest and least.

    Check answer

    Answer: 0.86 < 87% < 7/8; the gap is 0.875−0.86=0.015.

    Hint: Write 7/8 and 87% as decimals.

    Why: Convert each value to a decimal.

  10. 10. Order −0.6, −5/8, 55% and 0.58 from least to greatest. Convert to a common representation and justify the order.

    Check answer

    Answer: −5/8 = −0.625, so −0.625 < −0.6 < 0.55 < 0.58. Therefore −5/8, −0.6, 55%, 0.58.

    Hint: Represent the information first, then calculate, interpret and independently check the result.

    Why: This synthesis problem combines the chapter's core representation, calculation and reasoning skills.

Practice and reviewReady for practice

Common mistake: Percent sign removed without dividing by 100.

Correction: 60%=0.60, not 60.

Common mistake: Negative order reversed.

Correction: -0.8 is less than -0.3.

Common mistake: Equivalent numerator changed alone.

Correction: Scale numerator and denominator together.

Curriculum alignmentStart here

We are learning to find equivalent representations of rational numbers and represent rational numbers on a number line.

A rational number can be written as a fraction of integers with a non-zero denominator. Equivalent fractions, decimals and percentages occupy the same point on a number line.

Conversion preserves value: 7/20 becomes 35/100, then 0.35 and 35%. A common representation makes comparison reliable when forms differ.

Signed rational numbers require attention to direction. Among negative values, a number with greater absolute value lies farther left and is therefore smaller.

Success criteria

  • I can convert among fraction, decimal and percentage representations.
  • I can justify that two rational representations are equivalent.
  • I can locate and order positive and negative rational numbers on a number line.
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