Properties and Rational Numbers as Decimals

Lesson 2 of 2: Efficient Computation and Decimal Forms

In this lesson:

• Use properties to multiply and divide rationals efficiently
• Convert any rational number to a decimal
• Identify terminating vs. repeating decimals

What You Will Learn Today

By the end of this lesson, you should be able to:

4. Apply the equivalence when dividing
5. Use properties of operations to multiply and divide rationals efficiently
6. Convert a fraction to a decimal; identify terminating vs. repeating

One Negative Fraction Has Three Forms

These all represent the same value — negative three-fourths:

Verify with decimals: All three equal −0.75

The negative sign can live in three places:

• In front of the fraction
• In the numerator
• In the denominator

Three-Form Equivalence Shown on a Number Line

All three forms land on the same point: — one sign only.

Watch Out: Both Signs Cancel

Misconception: ?

That's positive three-fourths — not the same!

Remember: . Negating both cancels them.

Using Associativity to Regroup and Simplify

Problem:

Sign first: 2 negatives → even → positive

Regroup strategically:

Distributing a Negative Coefficient Carefully

Problem:

Verify: Let : . Check: ✓

Cancel Common Factors Before You Multiply

Problem: — sign: pos (neg × neg)

Cancel 14÷7=2 and 15÷5=3:

Multiplying first gives , then simplify — more steps, same answer.

Check: Use the Associative Property

Simplify using regrouping.

• Sign: _____ (negative × positive = ?)
• Regroup: Write 18 as , then cancel the 6

What is the result?

Answer:

• Sign: neg × pos = negative
• Regroup:

Compare: working left to right gives , then negate → −15. Same answer, more steps.

Converting Any Fraction to a Decimal

Every rational number converts via long division.

The decimal must either:

• Terminate — remainder reaches 0
• Repeat — remainder cycles

In , only possible remainders (0 to ) → eventually one must repeat.

Long Division Example: Terminating Decimal

: divide 7 by 8, bringing down zeros:

Long Division Example: Single-Digit Repeating

Convert — same remainder after every step:

Long Division Example: Block Repeating Decimal

: remainder cycles after 2 steps:

Bar Notation: Reading and Writing

The bar covers the repeating block — not the whole decimal.

Why Every Decimal Terminates or Repeats

In , remainders range from 0 to — only possibilities.

After at most steps, a remainder must repeat → decimal cycles.

• Remainder reaches 0 → terminates
• Remainder repeats (≠ 0) → repeats

No other outcome is possible.

Check: Convert a Fraction to Decimal

Convert to a decimal. Does it terminate or repeat?

Use long division:

• Divide 20 by 9 → quotient digit and remainder
• Does the remainder repeat?

Write the decimal in proper notation before advancing.

Answer: (Repeating)

The remainder 2 repeats immediately → single digit 2 repeats forever.

Lesson 2 Summary: Key Ideas to Remember

• Three forms: — one sign only
• Regroup/cancel before multiplying — more efficient
• Long division: every fraction terminates or repeats
• Bar notation: , ,

Watch out: Negating both top and bottom gives a positive
Watch out: Repeated remainder = repeating decimal

Next Up: All Four Operations Together

Coming up: All four operations with rational numbers

You can now:

• Add and subtract rational numbers (7.NS.A.1)
• Multiply and divide rational numbers (7.NS.A.2)

Next lesson: Apply all four operations together in real-world and mathematical problems — multi-step situations, rates, percentages, and expressions with rational coefficients.