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Tutoring abilities

Imagine you have a big pile of puppy weights, and you want to turn a chunky bar graph into a sleek mountain profile.

A **frequency polygon** is a special line graph that shows how continuous data is spread out across different groups.

Here, **frequency** means how many items are in a group, while **polygon** means a closed shape made of straight lines.

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <meta name="aspect-ratio" content="4:3">
  <title>Frequency Polygon</title>
  <style>
    * {
      box-sizing: border-box;
      margin: 0;
      padding: 0;
    }
    body {
      font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
      color: #1e2945;
      background: transparent;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      width: 100%;
      min-height: 100%;
      padding: 10px;
    }
    .diagram-wrap {
      width: 100%;
      max-width: 480px;
      display: flex;
      flex-direction: column;
      align-items: center;
      gap: 12px;
    }
    .svg-container {
      width: 100%;
      background: #ffffff;
      border-radius: 12px;
      box-shadow: 0 4px 16px rgba(30, 41, 69, 0.06);
      padding: 8px;
    }
    svg {
      width: 100%;
      height: auto;
      display: block;
      overflow: visible;
    }
    .axis-line {
      stroke: #1e2945;
      stroke-width: 2;
      stroke-linecap: round;
    }
    .grid-line {
      stroke: #e6e6e6;
      stroke-width: 1;
      stroke-dasharray: 3,3;
    }
    .axis-text {
      font-size: 11px;
      fill: #6b7280;
      text-anchor: middle;
      font-weight: 500;
    }
    .axis-title {
      font-size: 12px;
      fill: #1e2945;
      font-weight: 600;
    }
    .bar {
      fill: #22b7ff;
      opacity: 0.25;
      transition: opacity 0.4s ease, fill 0.4s ease;
    }
    .bar-border {
      stroke: #22b7ff;
      stroke-width: 1.5;
      fill: none;
      opacity: 0.6;
      transition: opacity 0.4s ease;
    }
    .poly-fill {
      fill: rgba(255, 122, 0, 0.15);
      transition: opacity 0.4s ease;
      opacity: 0;
    }
    .poly-line {
      fill: none;
      stroke: #ff7a00;
      stroke-width: 3;
      stroke-linecap: round;
      stroke-linejoin: round;
      transition: stroke-dashoffset 0.6s ease, opacity 0.4s ease;
    }
    .midpoint-dot {
      fill: #ff7a00;
      stroke: #ffffff;
      stroke-width: 2;
      transition: transform 0.3s ease, opacity 0.4s ease;
      transform-origin: center;
    }
    .midpoint-guide {
      stroke: #ff7a00;
      stroke-width: 1.5;
      stroke-dasharray: 2,2;
      transition: opacity 0.3s ease;
    }
    .btn-group {
      display: flex;
      flex-wrap: wrap;
      gap: 6px;
      justify-content: center;
      width: 100%;
    }
    .step-btn {
      background: #f8f9fa;
      border: 1px solid #e6e6e6;
      color: #1e2945;
      padding: 6px 12px;
      font-size: 13px;
      font-weight: 600;
      border-radius: 8px;
      cursor: pointer;
      transition: background 0.2s, color 0.2s, box-shadow 0.2s;
    }
    .step-btn.active {
      background: #1e2945;
      color: #ffffff;
      box-shadow: 0 2px 8px rgba(30, 41, 69, 0.2);
    }
  </style>
</head>
<body>

<div class="diagram-wrap">
  <div class="svg-container">
    <svg viewBox="0 0 400 270">
      
      <line class="grid-line" x1="45" y1="184" x2="375" y2="184" />
      <line class="grid-line" x1="45" y1="127" x2="375" y2="127" />
      <line class="grid-line" x1="45" y1="70" x2="375" y2="70" />

      
      <g id="barsGroup">
        <rect class="bar" x="50" y="184" width="64" height="57" />
        <rect class="bar-border" x="50" y="184" width="64" height="57" />

        <rect class="bar" x="114" y="127" width="64" height="114" />
        <rect class="bar-border" x="114" y="127" width="64" height="114" />

        <rect class="bar" x="178" y="70" width="64" height="171" />
        <rect class="bar-border" x="178" y="70" width="64" height="171" />

        <rect class="bar" x="242" y="155.5" width="64" height="85.5" />
        <rect class="bar-border" x="242" y="155.5" width="64" height="85.5" />

        <rect class="bar" x="306" y="212.5" width="64" height="28.5" />
        <rect class="bar-border" x="306" y="212.5" width="64" height="28.5" />
      </g>

      
      <polygon id="polyFill" class="poly-fill" points="82,241 82,184 146,127 210,70 274,155.5 338,212.5 338,241" />

      
      <polyline id="polyLine" class="poly-line" points="82,184 146,127 210,70 274,155.5 338,212.5" />

      
      <g id="guidesGroup" opacity="0">
        <line class="midpoint-guide" x1="82" y1="241" x2="82" y2="184" />
        <line class="midpoint-guide" x1="146" y1="241" x2="146" y2="127" />
        <line class="midpoint-guide" x1="210" y1="241" x2="210" y2="70" />
        <line class="midpoint-guide" x1="274" y1="241" x2="274" y2="155.5" />
        <line class="midpoint-guide" x1="338" y1="241" x2="338" y2="212.5" />
      </g>

      
      <g id="dotsGroup" opacity="0">
        <circle class="midpoint-dot" cx="82" cy="184" r="5" />
        <circle class="midpoint-dot" cx="146" cy="127" r="5" />
        <circle class="midpoint-dot" cx="210" cy="70" r="5" />
        <circle class="midpoint-dot" cx="274" cy="155.5" r="5" />
        <circle class="midpoint-dot" cx="338" cy="212.5" r="5" />
      </g>

      
      <line class="axis-line" x1="48" y1="241" x2="375" y2="241" />
      <line class="axis-line" x1="50" y1="243" x2="50" y2="35" />

      
      <text class="axis-text" x="38" y="245">0</text>
      <text class="axis-text" x="38" y="188">4</text>
      <text class="axis-text" x="38" y="131">8</text>
      <text class="axis-text" x="38" y="74">12</text>
      <text class="axis-title" x="20" y="24" text-anchor="start">Frequency</text>

      
      <text class="axis-text" x="50" y="258">0</text>
      <text class="axis-text" x="114" y="258">10</text>
      <text class="axis-text" x="178" y="258">20</text>
      <text class="axis-text" x="242" y="258">30</text>
      <text class="axis-text" x="306" y="258">40</text>
      <text class="axis-text" x="370" y="258">50</text>
    </svg>
  </div>

  
  <div class="btn-group">
    <button class="step-btn active" onclick="setStep(0)">1. Bars</button>
    <button class="step-btn" onclick="setStep(1)">2. Midpoints</button>
    <button class="step-btn" onclick="setStep(2)">3. Lines</button>
    <button class="step-btn" onclick="setStep(3)">4. Polygon</button>
  </div>
</div>

<script>
  const barsGroup = document.getElementById('barsGroup');
  const guidesGroup = document.getElementById('guidesGroup');
  const dotsGroup = document.getElementById('dotsGroup');
  const polyLine = document.getElementById('polyLine');
  const polyFill = document.getElementById('polyFill');
  const buttons = document.querySelectorAll('.step-btn');

  // Total length of polyline for smooth draw animation
  const polyLength = polyLine.getTotalLength();
  polyLine.style.strokeDasharray = polyLength;
  polyLine.style.strokeDashoffset = polyLength;

  function setStep(step) {
    buttons.forEach((btn, idx) => {
      btn.classList.toggle('active', idx === step);
    });

    if (step === 0) {
      // Just Bars
      barsGroup.style.opacity = '1';
      guidesGroup.style.opacity = '0';
      dotsGroup.style.opacity = '0';
      polyLine.style.strokeDashoffset = polyLength;
      polyFill.style.opacity = '0';
    } else if (step === 1) {
      // Midpoints plotted
      barsGroup.style.opacity = '0.6';
      guidesGroup.style.opacity = '1';
      dotsGroup.style.opacity = '1';
      polyLine.style.strokeDashoffset = polyLength;
      polyFill.style.opacity = '0';
    } else if (step === 2) {
      // Lines connecting
      barsGroup.style.opacity = '0.3';
      guidesGroup.style.opacity = '0.3';
      dotsGroup.style.opacity = '1';
      polyLine.style.strokeDashoffset = '0';
      polyFill.style.opacity = '0';
    } else if (step === 3) {
      // Full Polygon highlight
      barsGroup.style.opacity = '0.1';
      guidesGroup.style.opacity = '0';
      dotsGroup.style.opacity = '1';
      polyLine.style.strokeDashoffset = '0';
      polyFill.style.opacity = '1';
    }
  }

  // Auto-cycle gently every 3.5s
  let current = 0;
  let timer = setInterval(() => {
    current = (current + 1) % 4;
    setStep(current);
  }, 3500);

  // Pause auto-cycle on user click
  buttons.forEach((btn, idx) => {
    btn.addEventListener('click', () => {
      clearInterval(timer);
      current = idx;
      setStep(idx);
    });
  });
</script>

</body>
</html>

If a group covers a range like 10 to 20, where exactly along the bottom axis do we put our dot?

Because a group covers a whole interval of numbers, we place our dot directly in the center, called the **class midpoint**.

To find the middle, add the bottom and top limits of the group together, then divide by $2$.

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <meta name="aspect-ratio" content="4:3">
  <title>Finding Midpoints</title>
  <style>
    * {
      box-sizing: border-box;
      margin: 0;
      padding: 0;
      font-family: system-ui, -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
    }
    body {
      background: transparent;
      color: #1e2945;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      min-height: 100%;
      padding: 12px;
      user-select: none;
    }
    .diagram-root {
      width: 100%;
      max-width: 560px;
      display: flex;
      flex-direction: column;
      gap: 16px;
      align-items: center;
    }
    .btn-group {
      display: flex;
      gap: 8px;
      width: 100%;
      justify-content: center;
      flex-wrap: wrap;
    }
    .interval-btn {
      background: #f1f5f9;
      border: 1px solid #cbd5e1;
      color: #475569;
      padding: 8px 16px;
      border-radius: 20px;
      font-size: 14px;
      font-weight: 600;
      cursor: pointer;
      transition: background 0.2s, color 0.2s, border-color 0.2s, box-shadow 0.2s;
    }
    .interval-btn:hover {
      background: #e2e8f0;
    }
    .interval-btn.active {
      background: #22b7ff;
      border-color: #0284c7;
      color: #ffffff;
      box-shadow: 0 2px 8px rgba(34, 183, 255, 0.35);
    }
    /* Formula Card */
    .calc-card {
      display: flex;
      align-items: center;
      justify-content: center;
      gap: 12px;
      background: #ffffff;
      border: 1px solid #e2e8f0;
      border-radius: 12px;
      padding: 12px 20px;
      font-size: 18px;
      font-weight: 700;
      color: #1e2945;
      width: 100%;
      min-height: 64px;
    }
    .fraction {
      display: inline-flex;
      flex-direction: column;
      vertical-align: middle;
      text-align: center;
      font-size: 16px;
    }
    .fraction .top {
      border-bottom: 2px solid #1e2945;
      padding: 0 4px 2px;
    }
    .fraction .bottom {
      padding: 2px 4px 0;
    }
    .val-low { color: #0284c7; }
    .val-high { color: #6366f1; }
    .val-mid { color: #059669; }

    /* Number Line Canvas Container */
    .scale-box {
      width: 100%;
      background: #ffffff;
      border: 1px solid #e2e8f0;
      border-radius: 12px;
      padding: 24px 16px 16px;
      position: relative;
    }
    .track-wrapper {
      position: relative;
      height: 70px;
      margin: 0 16px;
    }
    .axis-line {
      position: absolute;
      top: 36px;
      left: 0;
      right: 0;
      height: 4px;
      background: #cbd5e1;
      border-radius: 2px;
    }
    .highlight-span {
      position: absolute;
      top: 34px;
      height: 8px;
      background: rgba(34, 183, 255, 0.4);
      border-radius: 4px;
      transition: left 0.4s ease, width 0.4s ease;
    }
    .pin {
      position: absolute;
      top: 8px;
      transform: translateX(-50%);
      display: flex;
      flex-direction: column;
      align-items: center;
      transition: left 0.4s ease;
      z-index: 2;
    }
    .pin-badge {
      background: #059669;
      color: #ffffff;
      font-size: 12px;
      font-weight: 700;
      padding: 3px 8px;
      border-radius: 10px;
      white-space: nowrap;
      box-shadow: 0 2px 6px rgba(5, 150, 105, 0.3);
    }
    .pin-line {
      width: 2px;
      height: 16px;
      background: #059669;
    }
    .pin-dot {
      width: 10px;
      height: 10px;
      border-radius: 50%;
      background: #059669;
      border: 2px solid #ffffff;
      box-shadow: 0 0 0 1px #059669;
      margin-top: -1px;
    }
    .ticks {
      position: relative;
      width: 100%;
      height: 20px;
      margin-top: 12px;
    }
    .tick {
      position: absolute;
      top: 0;
      transform: translateX(-50%);
      display: flex;
      flex-direction: column;
      align-items: center;
    }
    .tick-line {
      width: 1px;
      height: 6px;
      background: #94a3b8;
    }
    .tick-label {
      font-size: 12px;
      color: #64748b;
      margin-top: 4px;
      font-weight: 500;
    }
  </style>
</head>
<body>
  <div class="diagram-root">
    
    <div class="btn-group" id="btnGroup">
      <button class="interval-btn active" data-low="10" data-high="20">10 ≤ x &lt; 20</button>
      <button class="interval-btn" data-low="20" data-high="30">20 ≤ x &lt; 30</button>
      <button class="interval-btn" data-low="30" data-high="50">30 ≤ x &lt; 50</button>
    </div>

    
    <div class="calc-card" id="calcDisplay">
      <span>Midpoint =</span>
      <div class="fraction">
        <span class="top"><span class="val-low" id="lblLow">10</span> + <span class="val-high" id="lblHigh">20</span></span>
        <span class="bottom">2</span>
      </div>
      <span>=</span>
      <div class="fraction">
        <span class="top" id="lblSum">30</span>
        <span class="bottom">2</span>
      </div>
      <span>=</span>
      <span class="val-mid" id="lblMid">15</span>
    </div>

    
    <div class="scale-box">
      <div class="track-wrapper">
        <div class="axis-line"></div>
        <div class="highlight-span" id="spanBar"></div>
        <div class="pin" id="midPin">
          <div class="pin-badge" id="pinText">Midpoint: 15</div>
          <div class="pin-line"></div>
          <div class="pin-dot"></div>
        </div>
      </div>
      <div class="ticks" id="ticksContainer"></div>
    </div>
  </div>

  <script>
    const MAX_VAL = 60;
    const ticksContainer = document.getElementById('ticksContainer');
    const buttons = document.querySelectorAll('.interval-btn');
    const lblLow = document.getElementById('lblLow');
    const lblHigh = document.getElementById('lblHigh');
    const lblSum = document.getElementById('lblSum');
    const lblMid = document.getElementById('lblMid');
    const spanBar = document.getElementById('spanBar');
    const midPin = document.getElementById('midPin');
    const pinText = document.getElementById('pinText');

    // Generate axis ticks 0, 10, 20, 30, 40, 50, 60
    for (let v = 0; v <= MAX_VAL; v += 10) {
      const pct = (v / MAX_VAL) * 100;
      const tick = document.createElement('div');
      tick.className = 'tick';
      tick.style.left = pct + '%';
      tick.innerHTML = '<div class="tick-line"></div><div class="tick-label">' + v + '</div>';
      ticksContainer.appendChild(tick);
    }

    function setRange(low, high) {
      const sum = low + high;
      const mid = sum / 2;

      lblLow.textContent = low;
      lblHigh.textContent = high;
      lblSum.textContent = sum;
      lblMid.textContent = mid;
      pinText.textContent = 'Midpoint: ' + mid;

      const leftPct = (low / MAX_VAL) * 100;
      const widthPct = ((high - low) / MAX_VAL) * 100;
      const midPct = (mid / MAX_VAL) * 100;

      spanBar.style.left = leftPct + '%';
      spanBar.style.width = widthPct + '%';
      midPin.style.left = midPct + '%';
    }

    buttons.forEach(btn => {
      btn.addEventListener('click', () => {
        buttons.forEach(b => b.classList.remove('active'));
        btn.classList.add('active');
        const low = parseFloat(btn.dataset.low);
        const high = parseFloat(btn.dataset.high);
        setRange(low, high);
      });
    });

    // Initial setup
    setRange(10, 20);
  </script>
</body>
</html>

How do we organize all our midpoints and counts before plotting them on a grid?

Suppose we measure the heights of $20$ seedlings: $0\text{ to }10\text{ cm}$ (count $4$), $10\text{ to }20\text{ cm}$ (count $10$), and $20\text{ to }30\text{ cm}$ (count $6$).

Each group turns into an $(x, y)$ coordinate, where $x$ is the **midpoint** and $y$ is the **frequency**.

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <meta name="aspect-ratio" content="4:3">
  <style>
    * { box-sizing: border-box; margin: 0; padding: 0; }
    body {
      font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
      color: #1e2945;
      background: transparent;
      padding: 8px;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      min-height: 100%;
    }

    .diagram-wrap {
      width: 100%;
      max-width: 560px;
      display: flex;
      flex-direction: column;
      gap: 10px;
    }

    /* Table styling */
    .data-table {
      width: 100%;
      border-collapse: collapse;
      background: #ffffff;
      border-radius: 10px;
      overflow: hidden;
      font-size: 13px;
      box-shadow: 0 1px 4px rgba(0,0,0,0.06);
    }
    .data-table th, .data-table td {
      padding: 6px 8px;
      text-align: center;
      border-bottom: 1px solid #f0f0f0;
    }
    .data-table th {
      background: #f8f9fa;
      color: #6b7280;
      font-weight: 600;
      font-size: 11px;
      text-transform: uppercase;
      letter-spacing: 0.5px;
    }
    .data-table tr.active {
      background: #fff8f0;
    }
    .data-table tr.active td {
      color: #ff7a00;
      font-weight: 700;
    }
    .coord-badge {
      background: #f1f5f9;
      padding: 2px 6px;
      border-radius: 6px;
      font-weight: 600;
      display: inline-block;
      transition: all 0.25s ease;
    }
    tr.active .coord-badge {
      background: #ff7a00;
      color: #ffffff;
    }

    /* Graph Section */
    .graph-card {
      background: #ffffff;
      border-radius: 12px;
      padding: 10px 10px 4px 6px;
      box-shadow: 0 2px 8px rgba(0,0,0,0.04);
      border: 1px solid #f0f0f0;
      position: relative;
    }
    svg {
      width: 100%;
      height: auto;
      display: block;
      overflow: visible;
    }
    .grid-line {
      stroke: #f1f3f5;
      stroke-width: 1;
    }
    .axis {
      stroke: #6b7280;
      stroke-width: 1.5;
      stroke-linecap: round;
    }
    .tick-label {
      font-size: 11px;
      fill: #6b7280;
      font-family: inherit;
    }
    .axis-title {
      font-size: 11px;
      font-weight: 600;
      fill: #1e2945;
      font-family: inherit;
    }
    .poly-line {
      fill: none;
      stroke: #ff7a00;
      stroke-width: 2.5;
      stroke-linejoin: round;
      stroke-linecap: round;
      transition: stroke-dashoffset 0.4s ease;
    }
    .poly-area {
      fill: rgba(255, 122, 0, 0.08);
      transition: opacity 0.3s ease;
    }
    .mid-guide {
      stroke: #ff7a00;
      stroke-width: 1;
      stroke-dasharray: 3,3;
      opacity: 0;
      transition: opacity 0.3s ease;
    }
    .point-dot {
      fill: #ffffff;
      stroke: #ff7a00;
      stroke-width: 2.5;
      r: 4.5;
      transition: r 0.25s ease, fill 0.25s ease;
    }
    .point-dot.current {
      r: 6.5;
      fill: #ff7a00;
    }
    .point-label {
      font-size: 11px;
      font-weight: 700;
      fill: #ff7a00;
      text-anchor: middle;
      opacity: 0;
      transition: opacity 0.25s ease;
    }
    .point-label.visible {
      opacity: 1;
    }

    /* Stepper Controls */
    .controls {
      display: flex;
      align-items: center;
      justify-content: center;
      gap: 8px;
      margin-top: 2px;
    }
    .step-btn {
      background: #ffffff;
      border: 1px solid #e6e6e6;
      border-radius: 8px;
      padding: 6px 12px;
      font-size: 13px;
      font-weight: 600;
      color: #1e2945;
      cursor: pointer;
      display: flex;
      align-items: center;
      gap: 4px;
      transition: background 0.2s, border-color 0.2s;
    }
    .step-btn.active {
      background: #ff7a00;
      border-color: #ff7a00;
      color: #ffffff;
    }
    .play-btn {
      background: #f8f9fa;
      border: 1px solid #e6e6e6;
      border-radius: 8px;
      padding: 6px 12px;
      font-size: 13px;
      cursor: pointer;
      color: #1e2945;
      transition: background 0.2s;
    }
    .play-btn:hover {
      background: #ebeef2;
    }
  </style>
</head>
<body>
  <div class="diagram-wrap">
    
    <table class="data-table">
      <thead>
        <tr>
          <th>Group</th>
          <th>Midpoint ($x$)</th>
          <th>Freq ($y$)</th>
          <th>Point $(x,y)$</th>
        </tr>
      </thead>
      <tbody>
        <tr id="row-0">
          <td>0 – 10</td>
          <td>5</td>
          <td>3</td>
          <td><span class="coord-badge">(5, 3)</span></td>
        </tr>
        <tr id="row-1">
          <td>10 – 20</td>
          <td>15</td>
          <td>7</td>
          <td><span class="coord-badge">(15, 7)</span></td>
        </tr>
        <tr id="row-2">
          <td>20 – 30</td>
          <td>25</td>
          <td>6</td>
          <td><span class="coord-badge">(25, 6)</span></td>
        </tr>
        <tr id="row-3">
          <td>30 – 40</td>
          <td>35</td>
          <td>2</td>
          <td><span class="coord-badge">(35, 2)</span></td>
        </tr>
      </tbody>
    </table>

    
    <div class="graph-card">
      <svg viewBox="0 0 360 200">
        
        <line x1="40" y1="160" x2="330" y2="160" class="grid-line" />
        <line x1="40" y1="125" x2="330" y2="125" class="grid-line" />
        <line x1="40" y1="90"  x2="330" y2="90"  class="grid-line" />
        <line x1="40" y1="55"  x2="330" y2="55"  class="grid-line" />
        <line x1="40" y1="20"  x2="330" y2="20"  class="grid-line" />

        
        <line x1="40"  y1="20" x2="40"  y2="160" class="grid-line" />
        <line x1="110" y1="20" x2="110" y2="160" class="grid-line" />
        <line x1="180" y1="20" x2="180" y2="160" class="grid-line" />
        <line x1="250" y1="20" x2="250" y2="160" class="grid-line" />
        <line x1="320" y1="20" x2="320" y2="160" class="grid-line" />

        
        <line x1="40" y1="160" x2="335" y2="160" class="axis" />
        <line x1="40" y1="160" x2="40" y2="15" class="axis" />

        
        <text x="32" y="163" text-anchor="end" class="tick-label">0</text>
        <text x="32" y="128" text-anchor="end" class="tick-label">2</text>
        <text x="32" y="93"  text-anchor="end" class="tick-label">4</text>
        <text x="32" y="58"  text-anchor="end" class="tick-label">6</text>
        <text x="32" y="23"  text-anchor="end" class="tick-label">8</text>
        <text x="38" y="12"  text-anchor="start" class="axis-title">Frequency (y)</text>

        
        <text x="40"  y="174" text-anchor="middle" class="tick-label">0</text>
        <text x="110" y="174" text-anchor="middle" class="tick-label">10</text>
        <text x="180" y="174" text-anchor="middle" class="tick-label">20</text>
        <text x="250" y="174" text-anchor="middle" class="tick-label">30</text>
        <text x="320" y="174" text-anchor="middle" class="tick-label">40</text>
        <text x="180" y="193" text-anchor="middle" class="axis-title">Continuous Data / Midpoints (x)</text>

        
        
        
        
        
        
        
        

        <line id="guide-0" x1="75" y1="160" x2="75" y2="107.5" class="mid-guide" />
        <line id="guide-1" x1="145" y1="160" x2="145" y2="37.5" class="mid-guide" />
        <line id="guide-2" x1="215" y1="160" x2="215" y2="55" class="mid-guide" />
        <line id="guide-3" x1="285" y1="160" x2="285" y2="125" class="mid-guide" />

        
        <polyline id="polygon-line" class="poly-line" points="75,107.5" />

        
        <g id="pts-group">
          <circle id="pt-0" cx="75" cy="107.5" class="point-dot" style="display:none;" />
          <text id="lbl-0" x="75" y="100" class="point-label">(5, 3)</text>

          <circle id="pt-1" cx="145" cy="37.5" class="point-dot" style="display:none;" />
          <text id="lbl-1" x="145" y="30" class="point-label">(15, 7)</text>

          <circle id="pt-2" cx="215" cy="55" class="point-dot" style="display:none;" />
          <text id="lbl-2" x="215" y="47.5" class="point-label">(25, 6)</text>

          <circle id="pt-3" cx="285" cy="125" class="point-dot" style="display:none;" />
          <text id="lbl-3" x="285" y="117.5" class="point-label">(35, 2)</text>
        </g>
      </svg>
    </div>

    
    <div class="controls">
      <button class="step-btn" data-step="0">1</button>
      <button class="step-btn" data-step="1">2</button>
      <button class="step-btn" data-step="2">3</button>
      <button class="step-btn" data-step="3">4</button>
      <button id="playBtn" class="play-btn">⏸</button>
    </div>
  </div>

  <script src="https://cdn.classace.io/vendor/classace-diagram.js"></script>
  <script>
    const pointsData = [
      { pt: '75,107.5' },
      { pt: '145,37.5' },
      { pt: '215,55' },
      { pt: '285,125' }
    ];

    let currentStep = 0;
    let isPlaying = true;
    let autoTimer = null;

    const stepBtns = document.querySelectorAll('.step-btn');
    const playBtn = document.getElementById('playBtn');
    const polyline = document.getElementById('polygon-line');

    function renderStep(idx) {
      currentStep = idx;

      // Update buttons
      stepBtns.forEach((b, i) => {
        b.classList.toggle('active', i === idx);
      });

      // Update table rows
      for (let i = 0; i < 4; i++) {
        const row = document.getElementById(`row-${i}`);
        row.classList.toggle('active', i === idx);
      }

      // Update SVG elements
      const activePts = pointsData.slice(0, idx + 1).map(d => d.pt).join(' ');
      polyline.setAttribute('points', activePts);

      for (let i = 0; i < 4; i++) {
        const pt = document.getElementById(`pt-${i}`);
        const lbl = document.getElementById(`lbl-${i}`);
        const guide = document.getElementById(`guide-${i}`);

        if (i <= idx) {
          pt.style.display = 'block';
          pt.classList.toggle('current', i === idx);
          lbl.classList.toggle('visible', i === idx);
          guide.style.opacity = (i === idx) ? '1' : '0.2';
        } else {
          pt.style.display = 'none';
          lbl.classList.remove('visible');
          guide.style.opacity = '0';
        }
      }
    }

    function nextStep() {
      currentStep = (currentStep + 1) % 4;
      renderStep(currentStep);
    }

    function startTimer() {
      if (autoTimer) clearInterval(autoTimer);
      autoTimer = setInterval(nextStep, 3200);
      isPlaying = true;
      playBtn.textContent = '⏸';
    }

    function pauseTimer() {
      if (autoTimer) clearInterval(autoTimer);
      autoTimer = null;
      isPlaying = false;
      playBtn.textContent = '▶';
    }

    stepBtns.forEach(btn => {
      btn.addEventListener('click', () => {
        const s = parseInt(btn.getAttribute('data-step'), 10);
        pauseTimer();
        renderStep(s);
      });
    });

    playBtn.addEventListener('click', () => {
      if (isPlaying) {
        pauseTimer();
      } else {
        nextStep();
        startTimer();
      }
    });

    // Initialize
    renderStep(0);
    startTimer();
  </script>
</body>
</html>

Now that we have our coordinate pairs, how do we draw the completed shape?

Plot each coordinate point on the grid and connect them in order using straight line segments.

To close the polygon, bring both ends down to zero frequency at the imaginary midpoints before the first group and after the last group.

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <meta name="aspect-ratio" content="4:3">
  <title>Frequency Polygon</title>
  <style>
    * { box-sizing: border-box; margin: 0; padding: 0; }
    body {
      font-family: system-ui, -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
      color: #1e2945;
      background: transparent;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: space-between;
      min-height: 100%;
      padding: 12px;
      user-select: none;
    }
    .main-wrap {
      width: 100%;
      max-width: 500px;
      display: flex;
      flex-direction: column;
      align-items: center;
      gap: 12px;
    }
    .svg-container {
      width: 100%;
      aspect-ratio: 16 / 11;
      background: #ffffff;
      border-radius: 12px;
      box-shadow: 0 2px 8px rgba(0,0,0,0.06);
      position: relative;
    }
    svg {
      width: 100%;
      height: 100%;
      overflow: visible;
    }
    .axis-line { stroke: #1e2945; stroke-width: 2; stroke-linecap: round; }
    .grid-line { stroke: #f1f5f9; stroke-width: 1; stroke-dasharray: 3 3; }
    .tick-text { font-size: 11px; fill: #64748b; text-anchor: middle; font-weight: 500; }
    .axis-label { font-size: 12px; fill: #1e2945; font-weight: 600; }

    /* Interactive elements */
    .data-line {
      fill: none;
      stroke: #22b7ff;
      stroke-width: 3;
      stroke-linejoin: round;
      stroke-linecap: round;
      transition: stroke-dashoffset 0.6s ease;
    }
    .closed-line {
      fill: none;
      stroke: #ff7a00;
      stroke-width: 2.5;
      stroke-dasharray: 5 4;
      transition: opacity 0.4s ease;
    }
    .area-fill {
      fill: rgba(34, 183, 255, 0.12);
      transition: opacity 0.5s ease;
    }
    .data-point {
      fill: #22b7ff;
      stroke: #ffffff;
      stroke-width: 2;
      transition: transform 0.3s ease, opacity 0.3s ease;
    }
    .zero-point {
      fill: #ff7a00;
      stroke: #ffffff;
      stroke-width: 2;
      transition: transform 0.3s ease, opacity 0.3s ease;
    }
    .point-label {
      font-size: 11px;
      font-weight: 700;
      fill: #1e2945;
      text-anchor: middle;
      transition: opacity 0.3s ease;
    }

    /* Controls */
    .controls {
      display: flex;
      gap: 8px;
      width: 100%;
      justify-content: center;
    }
    .btn-step {
      flex: 1;
      max-width: 140px;
      padding: 8px 10px;
      background: #f1f5f9;
      border: 1px solid #e2e8f0;
      border-radius: 8px;
      font-size: 13px;
      font-weight: 600;
      color: #64748b;
      cursor: pointer;
      text-align: center;
      transition: background 0.2s, color 0.2s, box-shadow 0.2s;
    }
    .btn-step.active {
      background: #22b7ff;
      color: #ffffff;
      border-color: #22b7ff;
      box-shadow: 0 2px 6px rgba(34, 183, 255, 0.35);
    }
    .btn-step:hover:not(.active) {
      background: #e2e8f0;
    }
  </style>
</head>
<body>
  <div class="main-wrap">
    <div class="svg-container">
      <svg viewBox="0 0 460 300" aria-label="Frequency polygon diagram">
        
        <g id="grid">
          <line x1="60" y1="210" x2="430" y2="210" class="grid-line" />
          <line x1="60" y1="165" x2="430" y2="165" class="grid-line" />
          <line x1="60" y1="120" x2="430" y2="120" class="grid-line" />
          <line x1="60" y1="75"  x2="430" y2="75"  class="grid-line" />
          <line x1="60" y1="30"  x2="430" y2="30"  class="grid-line" />
        </g>

        
        <polygon id="polyArea" class="area-fill" points="80,255 138,120 196,45 254,75 312,165 370,255" style="opacity:0;" />

        
        <line x1="60" y1="255" x2="430" y2="255" class="axis-line" />
        <line x1="60" y1="20" x2="60" y2="255" class="axis-line" />

        
        <text x="50" y="259" class="tick-text" style="text-anchor: end;">0</text>
        <text x="50" y="214" class="tick-text" style="text-anchor: end;">2</text>
        <text x="50" y="169" class="tick-text" style="text-anchor: end;">4</text>
        <text x="50" y="124" class="tick-text" style="text-anchor: end;">6</text>
        <text x="50" y="79"  class="tick-text" style="text-anchor: end;">8</text>
        <text x="50" y="34"  class="tick-text" style="text-anchor: end;">10</text>
        <text x="25" y="140" class="axis-label" transform="rotate(-90 25 140)" style="text-anchor: middle;">Frequency</text>

        
        
        <text x="80"  y="273" class="tick-text" id="tick0" style="fill:#ff7a00; font-weight:700; opacity:0.4;">5*</text>
        <text x="138" y="273" class="tick-text">15</text>
        <text x="196" y="273" class="tick-text">25</text>
        <text x="254" y="273" class="tick-text">35</text>
        <text x="312" y="273" class="tick-text">45</text>
        <text x="370" y="273" class="tick-text" id="tickLast" style="fill:#ff7a00; font-weight:700; opacity:0.4;">55*</text>
        <text x="245" y="294" class="axis-label" style="text-anchor: middle;">Midpoints</text>

        
        <line id="closeStart" x1="80" y1="255" x2="138" y2="120" class="closed-line" style="opacity:0;" />
        <line id="closeEnd"   x1="312" y1="165" x2="370" y2="255" class="closed-line" style="opacity:0;" />

        
        <polyline id="dataPolyline" class="data-line" points="138,120 196,45 254,75 312,165" style="opacity:0;" />

        
        <g id="dataPoints">
          
          <circle cx="138" cy="120" r="5" class="data-point" />
          <text x="138" y="110" class="point-label">(15, 6)</text>

          
          <circle cx="196" cy="45" r="5" class="data-point" />
          <text x="196" y="35" class="point-label">(25, 9)</text>

          
          <circle cx="254" cy="75" r="5" class="data-point" />
          <text x="254" y="65" class="point-label">(35, 8)</text>

          
          <circle cx="312" cy="165" r="5" class="data-point" />
          <text x="312" y="155" class="point-label">(45, 4)</text>
        </g>

        
        <g id="zeroPoints" style="opacity: 0;">
          <circle cx="80" cy="255" r="5" class="zero-point" />
          <text x="80" y="245" class="point-label" style="fill:#ff7a00;">(5, 0)</text>

          <circle cx="370" cy="255" r="5" class="zero-point" />
          <text x="370" y="245" class="point-label" style="fill:#ff7a00;">(55, 0)</text>
        </g>
      </svg>
    </div>

    
    <div class="controls">
      <button class="btn-step active" id="step1Btn" onclick="setStep(1)">1. Plot Midpoints</button>
      <button class="btn-step" id="step2Btn" onclick="setStep(2)">2. Join Points</button>
      <button class="btn-step" id="step3Btn" onclick="setStep(3)">3. Close at Zero</button>
    </div>
  </div>

  <script>
    const polyline = document.getElementById('dataPolyline');
    const closeStart = document.getElementById('closeStart');
    const closeEnd = document.getElementById('closeEnd');
    const zeroPoints = document.getElementById('zeroPoints');
    const polyArea = document.getElementById('polyArea');
    const tick0 = document.getElementById('tick0');
    const tickLast = document.getElementById('tickLast');

    function setStep(step) {
      document.querySelectorAll('.btn-step').forEach((btn, idx) => {
        btn.classList.toggle('active', idx + 1 === step);
      });

      if (step === 1) {
        polyline.style.opacity = '0';
        closeStart.style.opacity = '0';
        closeEnd.style.opacity = '0';
        zeroPoints.style.opacity = '0';
        polyArea.style.opacity = '0';
        tick0.style.opacity = '0.3';
        tickLast.style.opacity = '0.3';
      } else if (step === 2) {
        polyline.style.opacity = '1';
        closeStart.style.opacity = '0';
        closeEnd.style.opacity = '0';
        zeroPoints.style.opacity = '0';
        polyArea.style.opacity = '0';
        tick0.style.opacity = '0.3';
        tickLast.style.opacity = '0.3';
      } else if (step === 3) {
        polyline.style.opacity = '1';
        closeStart.style.opacity = '1';
        closeEnd.style.opacity = '1';
        zeroPoints.style.opacity = '1';
        polyArea.style.opacity = '1';
        tick0.style.opacity = '1';
        tickLast.style.opacity = '1';
      }
    }
  </script>
</body>
</html>

What is the single most common mistake people make when drawing these graphs?

A common mistake is plotting points on the group edges, such as at $10$ or $20$, instead of the exact middle.

Always make sure each dot sits right above its class midpoint so your line stays perfectly centered and accurate.

math

Creating Frequency Polygons for Grouped Continuous Data. Plot and connect class midpoints against frequencies to construct frequency polygons.

Frequency Polygons - National 4 / GCSE Mathematics

Frequency Polygons: Steps, Examples & Worksheet

What is a Frequency Polygon? Revision Notes & Worked Examples

Teaching frequency polygons for grouped continuous data under CfE standard MTH 4-21a focuses on transitioning learners from discrete bar models to continuous line models. Begin by illustrating histogram bars and marking midpoints on their upper edges, reinforcing the geometric definition of a class midpoint. Emphasize precision in ruler-drawn line segments and discuss why frequency polygons are preferable to bar charts when overlaying multiple comparative distributions on identical axes.

Creating Frequency Polygons for Grouped Continuous Data

Finding the Midpoint

Organizing the Points