openfreebooks-bookgenerator/content/math/counting-and-numbers/core.html

<div class="book-prose">
  <p>
    Look around you. How many birds? How many cups? You can answer by
    <strong>counting</strong> — pointing at one thing at a time until everything has been pointed at.
    This chapter uses <strong>pictures of dots</strong> to stand for real things, like marbles or
    fingers.
  </p>

  <h2 id="what-is-counting" class="book-prose__heading">What is counting?</h2>
  <p>
    To <strong>count</strong> is to touch (or point at) each object once. Say a word for each touch:
    “one, two, three…” The last word tells you <strong>how many</strong> there are.
  </p>
  <p>
    Try it on your fingers: one finger up for each thing you counted. If every object has a finger,
    the number of fingers matches the number of objects.
  </p>

  <h2 id="zero" class="book-prose__heading">Zero</h2>
  <p>
    When there is <strong>nothing</strong> to point at, we write <strong>0</strong> and say
    <strong>zero</strong>. An empty plate, no toys in the box — zero dots:
  </p>
  <p class="counting-dots counting-dots--empty" aria-label="zero dots">
    <span class="counting-dots__none">(none)</span>
  </p>
  <p>Zero comes before one when we count: first nothing, then one, then two, and so on.</p>

  <div class="book-callout" role="note">
    <p class="book-callout__label">Remember</p>
    <p>Zero means “none here,” not “forgot to count.”</p>
  </div>

  <h2 id="numbers-one-to-ten" class="book-prose__heading">Numbers 1 to 10</h2>
  <p>
    Each number has a <strong>digit</strong> (the symbol), a <strong>word</strong>, and a
    <strong>picture of dots</strong> — one dot for each thing you counted.
  </p>
  <div class="book-table-wrap">
    <table class="book-table counting-number-table">
      <thead>
        <tr>
          <th scope="col">Digit</th>
          <th scope="col">Word</th>
          <th scope="col">Dots</th>
        </tr>
      </thead>
      <tbody>
        <tr>
          <td>1</td>
          <td>one</td>
          <td><span class="counting-dots" role="img" aria-label="1 dot"><span class="counting-dots__dot"></span></span></td>
        </tr>
        <tr>
          <td>2</td>
          <td>two</td>
          <td><span class="counting-dots" role="img" aria-label="2 dots"><span class="counting-dots__dot"></span><span class="counting-dots__dot"></span></span></td>
        </tr>
        <tr>
          <td>3</td>
          <td>three</td>
          <td><span class="counting-dots" role="img" aria-label="3 dots"><span class="counting-dots__dot"></span><span class="counting-dots__dot"></span><span class="counting-dots__dot"></span></span></td>
        </tr>
        <tr>
          <td>4</td>
          <td>four</td>
          <td><span class="counting-dots" role="img" aria-label="4 dots"><span class="counting-dots__dot"></span><span class="counting-dots__dot"></span><span class="counting-dots__dot"></span><span class="counting-dots__dot"></span></span></td>
        </tr>
        <tr>
          <td>5</td>
          <td>five</td>
          <td><span class="counting-dots counting-dots--five" role="img" aria-label="5 dots"><span class="counting-dots__dot"></span><span class="counting-dots__dot"></span><span class="counting-dots__dot"></span><span class="counting-dots__dot"></span><span class="counting-dots__dot"></span></span></td>
        </tr>
        <tr>
          <td>6</td>
          <td>six</td>
          <td><span class="counting-dots" role="img" aria-label="6 dots"><span class="counting-dots__dot"></span><span class="counting-dots__dot"></span><span class="counting-dots__dot"></span><span class="counting-dots__dot"></span><span class="counting-dots__dot"></span><span class="counting-dots__dot"></span></span></td>
        </tr>
        <tr>
          <td>7</td>
          <td>seven</td>
          <td><span class="counting-dots" role="img" aria-label="7 dots"><span class="counting-dots__dot"></span><span class="counting-dots__dot"></span><span class="counting-dots__dot"></span><span class="counting-dots__dot"></span><span class="counting-dots__dot"></span><span class="counting-dots__dot"></span><span class="counting-dots__dot"></span></span></td>
        </tr>
        <tr>
          <td>8</td>
          <td>eight</td>
          <td><span class="counting-dots" role="img" aria-label="8 dots"><span class="counting-dots__dot"></span><span class="counting-dots__dot"></span><span class="counting-dots__dot"></span><span class="counting-dots__dot"></span><span class="counting-dots__dot"></span><span class="counting-dots__dot"></span><span class="counting-dots__dot"></span><span class="counting-dots__dot"></span></span></td>
        </tr>
        <tr>
          <td>9</td>
          <td>nine</td>
          <td><span class="counting-dots" role="img" aria-label="9 dots"><span class="counting-dots__dot"></span><span class="counting-dots__dot"></span><span class="counting-dots__dot"></span><span class="counting-dots__dot"></span><span class="counting-dots__dot"></span><span class="counting-dots__dot"></span><span class="counting-dots__dot"></span><span class="counting-dots__dot"></span><span class="counting-dots__dot"></span></span></td>
        </tr>
        <tr>
          <td>10</td>
          <td>ten</td>
          <td><span class="counting-dots counting-dots--ten" role="img" aria-label="10 dots"><span class="counting-dots__dot"></span><span class="counting-dots__dot"></span><span class="counting-dots__dot"></span><span class="counting-dots__dot"></span><span class="counting-dots__dot"></span><span class="counting-dots__dot"></span><span class="counting-dots__dot"></span><span class="counting-dots__dot"></span><span class="counting-dots__dot"></span><span class="counting-dots__dot"></span></span></td>
        </tr>
      </tbody>
    </table>
  </div>
  <p>Which group has <strong>more</strong> dots — the bigger picture wins:</p>
  <figure class="counting-compare" role="group" aria-label="Seven dots is more than four dots">
    <div class="counting-compare__group">
      <span class="counting-compare__digit">7</span>
      <span class="counting-dots counting-dots--compare" role="img" aria-hidden="true">
        <span class="counting-dots__dot"></span><span class="counting-dots__dot"></span><span class="counting-dots__dot"></span><span class="counting-dots__dot"></span><span class="counting-dots__dot"></span><span class="counting-dots__dot"></span><span class="counting-dots__dot"></span>
      </span>
      <span class="counting-compare__word">more</span>
    </div>
    <div class="counting-compare__group">
      <span class="counting-compare__digit">4</span>
      <span class="counting-dots counting-dots--compare" role="img" aria-hidden="true">
        <span class="counting-dots__dot"></span><span class="counting-dots__dot"></span><span class="counting-dots__dot"></span><span class="counting-dots__dot"></span>
      </span>
      <span class="counting-compare__word">fewer</span>
    </div>
  </figure>

  <h2 id="patterns" class="book-prose__heading">Patterns</h2>
  <p>
    A <strong>pattern</strong> is when something repeats or grows the same way each time. You can
    often guess what comes next if you look at the pictures.
  </p>

  <h3 id="pattern-one-more" class="book-prose__heading">One more dot each time</h3>
  <p>Each step adds exactly one dot:</p>
  <figure class="counting-pattern-board">
    <ol class="counting-pattern-ladder">
      <li class="counting-pattern-step">
        <span class="counting-pattern-step__digit">1</span>
        <span class="counting-dots" role="img" aria-label="1 dot"><span class="counting-dots__dot"></span></span>
      </li>
      <li class="counting-pattern-step">
        <span class="counting-pattern-step__digit">2</span>
        <span class="counting-dots" role="img" aria-label="2 dots"><span class="counting-dots__dot"></span><span class="counting-dots__dot"></span></span>
      </li>
      <li class="counting-pattern-step">
        <span class="counting-pattern-step__digit">3</span>
        <span class="counting-dots" role="img" aria-label="3 dots"><span class="counting-dots__dot"></span><span class="counting-dots__dot"></span><span class="counting-dots__dot"></span></span>
      </li>
      <li class="counting-pattern-step">
        <span class="counting-pattern-step__digit">4</span>
        <span class="counting-dots" role="img" aria-label="4 dots"><span class="counting-dots__dot"></span><span class="counting-dots__dot"></span><span class="counting-dots__dot"></span><span class="counting-dots__dot"></span></span>
      </li>
      <li class="counting-pattern-step">
        <span class="counting-pattern-step__digit">5</span>
        <span class="counting-dots counting-dots--five" role="img" aria-label="5 dots"><span class="counting-dots__dot"></span><span class="counting-dots__dot"></span><span class="counting-dots__dot"></span><span class="counting-dots__dot"></span><span class="counting-dots__dot"></span></span>
      </li>
    </ol>
    <figcaption class="counting-pattern-board__caption">
      The pictures keep growing: 6 dots, then 7, all the way to 10 — one new dot every time.
    </figcaption>
  </figure>

  <h3 id="pattern-even" class="book-prose__heading">Two dots at a time (even numbers)</h3>
  <p>
    <strong>Even</strong> counts split into pairs with nothing left over. We jump two dots at a time:
  </p>
  <figure class="counting-pattern-board">
    <ol class="counting-pattern-ladder counting-pattern-ladder--pairs">
      <li class="counting-pattern-step">
        <span class="counting-pattern-step__digit">0</span>
        <span class="counting-dots counting-dots--empty" role="img" aria-label="0 dots"><span class="counting-dots__none">none</span></span>
      </li>
      <li class="counting-pattern-step">
        <span class="counting-pattern-step__digit">2</span>
        <span class="counting-dots counting-dots--pair" role="img" aria-label="2 dots in a pair">
          <span class="counting-dots__dot"></span><span class="counting-dots__dot"></span>
        </span>
      </li>
      <li class="counting-pattern-step">
        <span class="counting-pattern-step__digit">4</span>
        <span class="counting-dots counting-dots--pair" role="img" aria-label="4 dots, two pairs">
          <span class="counting-dots__pair"><span class="counting-dots__dot"></span><span class="counting-dots__dot"></span></span>
          <span class="counting-dots__pair"><span class="counting-dots__dot"></span><span class="counting-dots__dot"></span></span>
        </span>
      </li>
      <li class="counting-pattern-step">
        <span class="counting-pattern-step__digit">6</span>
        <span class="counting-dots counting-dots--pair" role="img" aria-label="6 dots, three pairs">
          <span class="counting-dots__pair"><span class="counting-dots__dot"></span><span class="counting-dots__dot"></span></span>
          <span class="counting-dots__pair"><span class="counting-dots__dot"></span><span class="counting-dots__dot"></span></span>
          <span class="counting-dots__pair"><span class="counting-dots__dot"></span><span class="counting-dots__dot"></span></span>
        </span>
      </li>
      <li class="counting-pattern-step">
        <span class="counting-pattern-step__digit">8</span>
        <span class="counting-dots counting-dots--pair" role="img" aria-label="8 dots, four pairs">
          <span class="counting-dots__pair"><span class="counting-dots__dot"></span><span class="counting-dots__dot"></span></span>
          <span class="counting-dots__pair"><span class="counting-dots__dot"></span><span class="counting-dots__dot"></span></span>
          <span class="counting-dots__pair"><span class="counting-dots__dot"></span><span class="counting-dots__dot"></span></span>
          <span class="counting-dots__pair"><span class="counting-dots__dot"></span><span class="counting-dots__dot"></span></span>
        </span>
      </li>
      <li class="counting-pattern-step">
        <span class="counting-pattern-step__digit">10</span>
        <span class="counting-dots counting-dots--pair counting-dots--ten-pairs" role="img" aria-label="10 dots, five pairs">
          <span class="counting-dots__pair"><span class="counting-dots__dot"></span><span class="counting-dots__dot"></span></span>
          <span class="counting-dots__pair"><span class="counting-dots__dot"></span><span class="counting-dots__dot"></span></span>
          <span class="counting-dots__pair"><span class="counting-dots__dot"></span><span class="counting-dots__dot"></span></span>
          <span class="counting-dots__pair"><span class="counting-dots__dot"></span><span class="counting-dots__dot"></span></span>
          <span class="counting-dots__pair"><span class="counting-dots__dot"></span><span class="counting-dots__dot"></span></span>
        </span>
      </li>
    </ol>
    <figcaption class="counting-pattern-board__caption">
      Each picture has one more <em>pair</em> of dots than the picture before. Every dot has a buddy.
    </figcaption>
  </figure>

  <h3 id="pattern-odd" class="book-prose__heading">One extra dot (odd numbers)</h3>
  <p>
    <strong>Odd</strong> counts also grow by two dots each step, but there is always
    <strong>one dot left alone</strong> — it does not fit in a pair. Look for the single dot in orange:
  </p>
  <figure class="counting-pattern-board">
    <ol class="counting-pattern-ladder counting-pattern-ladder--odd">
      <li class="counting-pattern-step">
        <span class="counting-pattern-step__digit">1</span>
        <span class="counting-dots counting-dots--odd" role="img" aria-label="1 dot, alone">
          <span class="counting-dots__singleton"><span class="counting-dots__dot"></span></span>
        </span>
      </li>
      <li class="counting-pattern-step">
        <span class="counting-pattern-step__digit">3</span>
        <span class="counting-dots counting-dots--odd" role="img" aria-label="3 dots, one pair and one alone">
          <span class="counting-dots__pair"><span class="counting-dots__dot"></span><span class="counting-dots__dot"></span></span>
          <span class="counting-dots__singleton"><span class="counting-dots__dot"></span></span>
        </span>
      </li>
      <li class="counting-pattern-step">
        <span class="counting-pattern-step__digit">5</span>
        <span class="counting-dots counting-dots--odd" role="img" aria-label="5 dots, two pairs and one alone">
          <span class="counting-dots__pair"><span class="counting-dots__dot"></span><span class="counting-dots__dot"></span></span>
          <span class="counting-dots__pair"><span class="counting-dots__dot"></span><span class="counting-dots__dot"></span></span>
          <span class="counting-dots__singleton"><span class="counting-dots__dot"></span></span>
        </span>
      </li>
      <li class="counting-pattern-step">
        <span class="counting-pattern-step__digit">7</span>
        <span class="counting-dots counting-dots--odd" role="img" aria-label="7 dots, three pairs and one alone">
          <span class="counting-dots__pair"><span class="counting-dots__dot"></span><span class="counting-dots__dot"></span></span>
          <span class="counting-dots__pair"><span class="counting-dots__dot"></span><span class="counting-dots__dot"></span></span>
          <span class="counting-dots__pair"><span class="counting-dots__dot"></span><span class="counting-dots__dot"></span></span>
          <span class="counting-dots__singleton"><span class="counting-dots__dot"></span></span>
        </span>
      </li>
      <li class="counting-pattern-step">
        <span class="counting-pattern-step__digit">9</span>
        <span class="counting-dots counting-dots--odd" role="img" aria-label="9 dots, four pairs and one alone">
          <span class="counting-dots__pair"><span class="counting-dots__dot"></span><span class="counting-dots__dot"></span></span>
          <span class="counting-dots__pair"><span class="counting-dots__dot"></span><span class="counting-dots__dot"></span></span>
          <span class="counting-dots__pair"><span class="counting-dots__dot"></span><span class="counting-dots__dot"></span></span>
          <span class="counting-dots__pair"><span class="counting-dots__dot"></span><span class="counting-dots__dot"></span></span>
          <span class="counting-dots__singleton"><span class="counting-dots__dot"></span></span>
        </span>
      </li>
    </ol>
    <figcaption class="counting-pattern-board__caption">
      The lonely dot stays while the pairs grow: next would be <strong>11</strong> — five pairs and
      one dot by itself.
    </figcaption>
  </figure>

  <h3 id="pattern-shapes" class="book-prose__heading">Circle, square, circle, square…</h3>
  <p>Patterns can use <strong>shapes</strong>, not only dots. Watch the pictures repeat:</p>
  <figure class="counting-shape-pattern" role="group" aria-label="Repeating pattern: circle, square, circle, square, circle, square, then a question mark for what comes next">
    <ol class="counting-shape-pattern__strip">
      <li><span class="counting-shape counting-shape--circle" aria-hidden="true"></span><span class="counting-shape-pattern__label">circle</span></li>
      <li><span class="counting-shape counting-shape--square" aria-hidden="true"></span><span class="counting-shape-pattern__label">square</span></li>
      <li><span class="counting-shape counting-shape--circle" aria-hidden="true"></span><span class="counting-shape-pattern__label">circle</span></li>
      <li><span class="counting-shape counting-shape--square" aria-hidden="true"></span><span class="counting-shape-pattern__label">square</span></li>
      <li><span class="counting-shape counting-shape--circle" aria-hidden="true"></span><span class="counting-shape-pattern__label">circle</span></li>
      <li><span class="counting-shape counting-shape--square" aria-hidden="true"></span><span class="counting-shape-pattern__label">square</span></li>
      <li class="counting-shape-pattern__next">
        <span class="counting-shape counting-shape--question" aria-hidden="true">?</span>
        <span class="counting-shape-pattern__label">next?</span>
      </li>
    </ol>
    <figcaption class="counting-pattern-board__caption">
      After a square comes a <strong>circle</strong>. The shapes take turns, over and over.
    </figcaption>
  </figure>

  <h2 id="addition" class="book-prose__heading">Addition</h2>
  <p>
    <strong>Addition</strong> means putting two groups together. Three dots and four more dots make
    seven dots in one big group. We write <strong>3 + 4 = 7</strong>.
  </p>

  <div class="book-callout" role="note">
    <p class="book-callout__label">Adding</p>
    <p>Start with the first group, then place the second group beside it and count all the dots.</p>
  </div>

  <h2 id="multiplication" class="book-prose__heading">Multiplication</h2>
  <p>
    <strong>Multiplication</strong> is many <em>equal</em> groups. Three bags with two sweets each
    is the same as three pairs of dots. We write <strong>3 × 2 = 6</strong> — “three groups of two.”
  </p>
  <p>
    You can lay the dots in a <strong>rectangle</strong>: 3 rows with 2 dots in each row fills a
    block of 6 dots.
  </p>

  <h2 id="division" class="book-prose__heading">Division</h2>
  <p>
    <strong>Division</strong> is about <strong>sharing equally</strong> or making equal groups. If
    12 biscuits are shared among 4 friends, each friend gets the same number. We write
    <strong>12 ÷ 4 = 3</strong> — “twelve split into four equal piles gives three in each pile.”
  </p>
  <p>
    Division is the partner of multiplication. If <strong>3 × 4 = 12</strong>, then
    <strong>12 ÷ 3 = 4</strong> and <strong>12 ÷ 4 = 3</strong>. One picture of dots can answer both
    kinds of question.
  </p>

  <div class="book-callout" role="note">
    <p class="book-callout__label">Fair shares</p>
    <p>
      When every pile has the same number of dots, the share is <strong>fair</strong>. If dots are
      left over, the total does not split evenly — those extras are the <strong>remainder</strong>.
    </p>
  </div>

  <div class="math-widget-mount" data-widget="division-sharing" data-pagefind-ignore></div>

  <h2 id="past-ten" class="book-prose__heading">Past ten: tens and hundreds</h2>
  <p>
    You can count higher than ten on your fingers — people use <strong>place value</strong>. Ten
    single dots can be thought of as <strong>one group of ten</strong>. Ten groups of ten make
    <strong>one hundred</strong>.
  </p>
  <figure class="counting-place-ladder" aria-label="Place value: ten ones make one ten, ten tens make one hundred">
    <ol class="counting-place-ladder__steps">
      <li class="counting-place-step">
        <span class="counting-place-step__label">1 one</span>
        <span class="counting-dots" role="img" aria-hidden="true"><span class="counting-dots__dot"></span></span>
      </li>
      <li class="counting-place-step counting-place-step--arrow" aria-hidden="true">× 10</li>
      <li class="counting-place-step">
        <span class="counting-place-step__label">1 ten</span>
        <span class="counting-place-blocks counting-place-blocks--ten" role="img" aria-hidden="true">
          <span class="counting-place-blocks__rod"></span>
        </span>
      </li>
      <li class="counting-place-step counting-place-step--arrow" aria-hidden="true">× 10</li>
      <li class="counting-place-step">
        <span class="counting-place-step__label">1 hundred</span>
        <span class="counting-place-blocks counting-place-blocks--hundred" role="img" aria-hidden="true">
          <span class="counting-place-blocks__grid"></span>
        </span>
      </li>
    </ol>
    <figcaption class="counting-pattern-board__caption">
      Each step bundles ten of the smaller unit. That is why our digits work in columns (ones, tens,
      hundreds…).
    </figcaption>
  </figure>
  <p>
    The number <strong>47</strong> means 4 tens and 7 ones — forty-seven dots, without drawing every
    dot one by one.
  </p>

  <h2 id="big-numbers" class="book-prose__heading">Big numbers in real life</h2>
  <p>
    The world is huge. Counting every grain of sand on a beach one by one would take forever, so we
    use <strong>big number words</strong> to talk about amounts we cannot easily picture dot by dot.
    They matter because governments, scientists, and shops all work with large totals.
  </p>

  <div class="book-table-wrap">
    <table class="book-table counting-big-table">
      <thead>
        <tr>
          <th scope="col">Word</th>
          <th scope="col">How we write it</th>
          <th scope="col">Why it matters</th>
        </tr>
      </thead>
      <tbody>
        <tr>
          <td>hundred</td>
          <td>100</td>
          <td>About how many people fit in a big sports field; pages in a long novel.</td>
        </tr>
        <tr>
          <td>thousand</td>
          <td>1,000</td>
          <td>Steps you might walk in a long hike; students in a large school district.</td>
        </tr>
        <tr>
          <td>million</td>
          <td>1,000,000</td>
          <td>People in a large city; dollars in a national budget line item.</td>
        </tr>
        <tr>
          <td>billion</td>
          <td>1,000,000,000</td>
          <td>World population is several <em>billion</em>; stars in our galaxy are hundreds of billions.</td>
        </tr>
      </tbody>
    </table>
  </div>

  <p>
    Each big word is <strong>one thousand times</strong> the word before (ten hundreds make a
    thousand; one thousand thousands make a million). We bundle in tens and thousands so we can
    compare and add without losing track.
  </p>

  <ul>
    <li><strong>Money</strong> — prices and taxes use thousands and millions.</li>
    <li><strong>Distance</strong> — kilometres between cities add up to thousands.</li>
    <li><strong>Time</strong> — minutes in a year are in the hundreds of thousands.</li>
    <li><strong>Nature</strong> — insects in a forest, cells in your body — counts go huge fast.</li>
  </ul>

  <div class="math-widget-mount" data-widget="big-numbers-scale" data-pagefind-ignore></div>

  <h2 id="try-it-yourself" class="book-prose__heading">Try it yourself</h2>
  <p>
    Use the tools below: count dots, add and multiply, <strong>share equally</strong>, and zoom from
    one dot up to a billion to see how big numbers compare.
  </p>
</div>

<div class="math-widget-mount" data-widget="counting-demo" data-pagefind-ignore></div>
<div class="math-widget-mount" data-widget="addition-groups" data-pagefind-ignore></div>
<div class="math-widget-mount" data-widget="multiplication-array" data-pagefind-ignore></div>