openfreebooks-bookgenerator/content/math/properties-of-operations/supplement.html

<div class="book-prose">
  <h2 id="history" class="book-prose__heading">History</h2>
  <p>
    Long before algebra used letters, merchants and accountants rearranged piles and rows of goods.
    Indian and Islamic mathematicians in the Middle Ages wrote rules for how sums and products
    behave. The words <strong>commutative</strong>, <strong>associative</strong>, and
    <strong>distributive</strong> became standard in European algebra books in the 1800s, when
    school math settled on one notation for all.
  </p>
  <p>
    The distributive rule is ancient geometry in disguise: a large rectangle split into two smaller
    ones has total area equal to the sum of the parts. That picture is still one of the clearest
    explanations today.
  </p>

  <h2 id="checkpoints" class="book-prose__heading">Checkpoints</h2>
  <ol>
    <li>
      Which property lets you write \(7 + 4\) as \(4 + 7\) without changing the total?
    </li>
    <li>
      Is \((5 \times 2) \times 3\) the same as \(5 \times (2 \times 3)\)? Which property says so?
    </li>
    <li>
      Rewrite \(4(6 + 1)\) as a sum of two products.
    </li>
    <li>
      Why is \(8 - 3\) not commutative? Give two different answers for \(3 - 8\) and \(8 - 3\).
    </li>
    <li>
      A rectangle is 5 units tall and \((2 + 3)\) units wide. How does the distributive property
      describe its area?
    </li>
  </ol>

  <h2 id="applications" class="book-prose__heading">Applications</h2>
  <h3 id="mental-math" class="book-prose__heading">Mental math</h3>
  <p>
    Distribute to make friendly numbers: \(7 \times 98 = 7 \times (100 - 2) = 700 - 14 = 686\).
    Commute to add easier first: \(6 + 47 + 4 = (6 + 4) + 47 = 57\).
  </p>
  <h3 id="shopping" class="book-prose__heading">Shopping and sharing</h3>
  <p>
    Buying 3 packs of \((2 + 5)\) items is the same as \(3 \times 2\) items plus \(3 \times 5\)
    items. Swapping the order you scan barcodes does not change how many items you have (commute on
    addition when you merge carts).
  </p>
  <h3 id="algebra-later" class="book-prose__heading">Later algebra</h3>
  <p>
    Expanding \(a(b + c)\) in polynomials and factorising common factors both rely on the
    distributive property. You will meet them again in <strong>polynomials: expansion &amp;
    factorisation</strong>.
  </p>

  <h2 id="question-bank" class="book-prose__heading">Question bank</h2>

  <h3 id="questions-easy" class="book-prose__heading">Easy</h3>

  <details class="book-question">
    <summary class="book-question__prompt">
      Fill in the blank using a property: \(9 + 6 = 6 + \underline{\hspace{1.5em}}\).
    </summary>
    <div class="book-question__solution">
      <p><strong>Answer:</strong> \(9\) — commutative property of addition.</p>
    </div>
  </details>

  <details class="book-question">
    <summary class="book-question__prompt">Compute: \(2(4 + 3)\).</summary>
    <div class="book-question__solution">
      <p>
        <strong>Answer:</strong> \(14\). Either \(2 \times 7 = 14\), or \(2 \times 4 + 2 \times 3 = 8 +
        6 = 14\) (distributive).
      </p>
    </div>
  </details>

  <h3 id="questions-intermediate" class="book-prose__heading">Intermediate</h3>

  <details class="book-question">
    <summary class="book-question__prompt">
      Evaluate both sides and compare: \((3 + 4) + 5\) and \(3 + (4 + 5)\).
    </summary>
    <div class="book-question__solution">
      <p><strong>Answer:</strong> Both equal \(12\) — associative property of addition.</p>
    </div>
  </details>

  <details class="book-question">
    <summary class="book-question__prompt">Use distribution to find \(5 \times 102\).</summary>
    <div class="book-question__solution">
      <p><strong>Answer:</strong> \(510\). Steps: \(5 \times (100 + 2) = 500 + 10 = 510\).</p>
    </div>
  </details>

  <h3 id="questions-difficult" class="book-prose__heading">Difficult</h3>

  <details class="book-question">
    <summary class="book-question__prompt">
      Explain why \(12 \div (3 \div 2)\) is not the same as \((12 \div 3) \div 2\).
    </summary>
    <div class="book-question__solution">
      <p>
        <strong>Answer:</strong> Left: \(12 \div 1.5 = 8\). Right: \(4 \div 2 = 2\). Division is not
        associative; grouping changes the meaning.
      </p>
    </div>
  </details>

  <details class="book-question">
    <summary class="book-question__prompt">
      A garden bed is 4 m wide and \((2 + 3)\) m long. Write two ways to find its area and show they
      match.
    </summary>
    <div class="book-question__solution">
      <p>
        <strong>Answer:</strong> \(20\) m². One rectangle: \(4 \times 5 = 20\). Split length:
        \(4 \times 2 + 4 \times 3 = 8 + 12 = 20\) (distributive).
      </p>
    </div>
  </details>
</div>