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Override for the tutor's lesson content generation calls. Auto uses the system default (Claude Sonnet 5 for lesson text, Opus 5 for diagrams).

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System default

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Tutoring abilities

Imagine measuring your desk: it is 3 feet long, but it is also 36 inches long. The desk didn't magically stretch — you just swapped to smaller measuring sticks.

When you measure something in smaller units, you need **more** of them to cover the same length. Because the actual length never changes, the ratio between the two units stays permanently locked.

<!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: system-ui, -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, sans-serif;
      background: #ffffff;
      color: #1e2945;
      padding: 12px 8px;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      min-height: 100%;
      user-select: none;
    }

    .diagram-container {
      width: 100%;
      max-width: 600px;
      display: flex;
      flex-direction: column;
      gap: 12px;
    }

    svg {
      width: 100%;
      height: auto;
      display: block;
      overflow: visible;
    }

    /* Interactive Buttons */
    .controls {
      display: flex;
      justify-content: center;
      gap: 8px;
      flex-wrap: wrap;
    }

    .btn-step {
      background: #f1f5f9;
      border: 1px solid #cbd5e1;
      border-radius: 20px;
      padding: 6px 14px;
      font-size: 13px;
      font-weight: 600;
      color: #475569;
      cursor: pointer;
      transition: background 0.2s ease, color 0.2s ease, box-shadow 0.2s ease;
    }

    .btn-step:hover {
      background: #e2e8f0;
    }

    .btn-step.active {
      background: #2563eb;
      border-color: #2563eb;
      color: #ffffff;
      box-shadow: 0 2px 6px rgba(37, 99, 235, 0.25);
    }

    .callout-badge {
      text-anchor: middle;
      font-weight: 700;
      font-size: 13px;
    }

    .segment-clickable {
      cursor: pointer;
      transition: opacity 0.25s ease;
    }
  </style>
</head>
<body>

<div class="diagram-container">
  <svg viewBox="0 0 580 300" id="rulerSvg">
    <defs>
      
      <linearGradient id="woodGrad" x1="0%" y1="0%" x2="0%" y2="100%">
        <stop offset="0%" stop-color="#e3be8e"/>
        <stop offset="50%" stop-color="#d4a366"/>
        <stop offset="100%" stop-color="#c59253"/>
      </linearGradient>

      
      <linearGradient id="blueGrad" x1="0%" y1="0%" x2="0%" y2="100%">
        <stop offset="0%" stop-color="#3b82f6"/>
        <stop offset="100%" stop-color="#1d4ed8"/>
      </linearGradient>

      
      <linearGradient id="orangeGrad" x1="0%" y1="0%" x2="0%" y2="100%">
        <stop offset="0%" stop-color="#f97316"/>
        <stop offset="100%" stop-color="#c2410c"/>
      </linearGradient>

      
      <filter id="glow" x="-10%" y="-10%" width="120%" height="120%">
        <feDropShadow dx="0" dy="2" stdDeviation="3" flood-color="#2563eb" flood-opacity="0.2"/>
      </filter>
    </defs>

    
    <g stroke="#94a3b8" stroke-dasharray="3 3" stroke-width="1.5">
      <line x1="50" y1="35" x2="50" y2="225" />
      <line x1="210" y1="75" x2="210" y2="225" />
      <line x1="370" y1="75" x2="370" y2="225" />
      <line x1="530" y1="35" x2="530" y2="225" />
    </g>

    
    <g id="yardTier">
      <rect x="50" y="35" width="480" height="38" rx="5" fill="url(#woodGrad)" stroke="#8a5a2b" stroke-width="1.5"/>
      <text x="290" y="59" fill="#452203" font-size="15" font-weight="700" text-anchor="middle" letter-spacing="0.5">1 YARD (yd)</text>
    </g>

    
    <g id="feetTier">
      
      <g class="segment-clickable" id="segFt1" onclick="setHighlight(0)">
        <rect id="ftBox1" x="50" y="90" width="160" height="42" fill="#eff6ff" stroke="#2563eb" stroke-width="2" rx="4"/>
        <text x="130" y="116" fill="#1d4ed8" font-size="15" font-weight="700" text-anchor="middle">1 ft</text>
      </g>
      
      <g class="segment-clickable" id="segFt2" onclick="setHighlight(1)">
        <rect id="ftBox2" x="210" y="90" width="160" height="42" fill="#eff6ff" stroke="#2563eb" stroke-width="2" rx="4"/>
        <text x="290" y="116" fill="#1d4ed8" font-size="15" font-weight="700" text-anchor="middle">2 ft</text>
      </g>
      
      <g class="segment-clickable" id="segFt3" onclick="setHighlight(2)">
        <rect id="ftBox3" x="370" y="90" width="160" height="42" fill="#eff6ff" stroke="#2563eb" stroke-width="2" rx="4"/>
        <text x="450" y="116" fill="#1d4ed8" font-size="15" font-weight="700" text-anchor="middle">3 ft</text>
      </g>
    </g>

    
    <g id="inchesTier">
      
      <rect x="50" y="150" width="480" height="48" rx="4" fill="#fff7ed" stroke="#ea580c" stroke-width="2"/>

      
      <g id="inchTicks" stroke="#ea580c" stroke-linecap="round"></g>

      
      <text x="210" y="186" fill="#c2410c" font-size="13" font-weight="800" text-anchor="middle">12 in</text>
      <text x="370" y="186" fill="#c2410c" font-size="13" font-weight="800" text-anchor="middle">24 in</text>
      <text x="530" y="186" fill="#c2410c" font-size="13" font-weight="800" text-anchor="end">36 in</text>
    </g>

    
    <g id="highlightOverlay" pointer-events="none">
      
      <rect id="hlBox" x="50" y="88" width="160" height="112" rx="6" fill="rgba(37,99,235,0.08)" stroke="#2563eb" stroke-width="2.5" stroke-dasharray="6 4" />
      
      
      <g id="hlCallout" transform="translate(130, 245)">
        <rect x="-56" y="-18" width="112" height="28" rx="14" fill="#1e2945" />
        <text x="0" y="1" fill="#ffffff" class="callout-badge" text-anchor="middle" dominant-baseline="middle">1 ft = 12 in</text>
        <path d="M -6 -18 L 0 -24 L 6 -18 Z" fill="#1e2945" />
      </g>
    </g>
  </svg>

  
  <div class="controls">
    <button class="btn-step active" onclick="setHighlight(0)">1st Foot (12 in)</button>
    <button class="btn-step" onclick="setHighlight(1)">2nd Foot (24 in)</button>
    <button class="btn-step" onclick="setHighlight(2)">3rd Foot (36 in)</button>
    <button class="btn-step" onclick="setHighlight('all')">Full Yard (3 ft = 36 in)</button>
  </div>
</div>

<script>
  // Generate 36 precise inch tick marks
  const tickGroup = document.getElementById('inchTicks');
  const startX = 50;
  const totalWidth = 480;
  const inchWidth = totalWidth / 36; // 13.333px per inch

  for (let i = 0; i <= 36; i++) {
    const x = startX + i * inchWidth;
    const isMajor = (i % 12 === 0);
    const isMid = (i % 6 === 0 && !isMajor);
    
    const tickHeight = isMajor ? 20 : (isMid ? 13 : 8);
    const strokeW = isMajor ? 2 : 1;
    const opacity = isMajor ? 1 : 0.65;

    const line = document.createElementNS('http://www.w3.org/2000/svg', 'line');
    line.setAttribute('x1', x);
    line.setAttribute('y1', 150);
    line.setAttribute('x2', x);
    line.setAttribute('y2', 150 + tickHeight);
    line.setAttribute('stroke-width', strokeW);
    line.setAttribute('opacity', opacity);
    tickGroup.appendChild(line);
  }

  // Interactive Highlighting
  const hlBox = document.getElementById('hlBox');
  const hlCallout = document.getElementById('hlCallout');
  const calloutText = hlCallout.querySelector('text');
  const calloutPath = hlCallout.querySelector('path');
  const buttons = document.querySelectorAll('.btn-step');

  const positions = [
    { x: 50, w: 160, centerX: 130, text: '1 ft = 12 in' },
    { x: 210, w: 160, centerX: 290, text: '1 ft = 12 in' },
    { x: 370, w: 160, centerX: 450, text: '1 ft = 12 in' },
    { x: 50, w: 480, centerX: 290, text: '1 yd = 3 ft = 36 in' }
  ];

  function setHighlight(index) {
    const idx = (index === 'all') ? 3 : index;
    const pos = positions[idx];

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

    // Update highlight rectangle
    hlBox.setAttribute('x', pos.x);
    hlBox.setAttribute('width', pos.w);

    // Update Callout
    hlCallout.setAttribute('transform', `translate(${pos.centerX}, 245)`);
    calloutText.textContent = pos.text;

    // Adjust callout bubble width for longer "1 yd = 3 ft = 36 in" text
    const rect = hlCallout.querySelector('rect');
    if (idx === 3) {
      rect.setAttribute('x', '-85');
      rect.setAttribute('width', '170');
    } else {
      rect.setAttribute('x', '-56');
      rect.setAttribute('width', '112');
    }

    // Foot segment tinting
    for (let f = 1; f <= 3; f++) {
      const box = document.getElementById(`ftBox${f}`);
      if (idx === 3 || idx === (f - 1)) {
        box.setAttribute('fill', '#dbeafe');
        box.setAttribute('stroke', '#2563eb');
      } else {
        box.setAttribute('fill', '#f8fafc');
        box.setAttribute('stroke', '#94a3b8');
      }
    }
  }

  // Initialize
  setHighlight(0);
</script>

</body>
</html>

What happens if you measure two desks, three desks, or ten desks? Does the relationship change?

A **proportional relationship** is a connection between two quantities where their ratio is always constant. If you double the number of feet, you automatically double the number of inches.

The number that you multiply by to switch between units is called a **conversion factor**. Since $1\text{ foot} = 12\text{ inches}$, the conversion factor is $\frac{12\text{ in}}{1\text{ ft}}$ (or 12 inches per foot).

<!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: system-ui, -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
      background: #ffffff;
      color: #1e2945;
      padding: 12px;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      min-height: 100%;
      user-select: none;
    }
    .layout {
      width: 100%;
      max-width: 620px;
      display: flex;
      flex-direction: column;
      gap: 12px;
    }
    .top-controls {
      display: flex;
      justify-content: space-between;
      align-items: center;
      padding: 0 4px;
      flex-wrap: wrap;
      gap: 8px;
    }
    .badge {
      font-size: 13px;
      font-weight: 700;
      color: #0284c7;
      background: #e0f2fe;
      padding: 4px 10px;
      border-radius: 999px;
      display: inline-flex;
      align-items: center;
      gap: 4px;
    }
    .btn-group {
      display: flex;
      gap: 6px;
    }
    .step-btn {
      background: #f1f5f9;
      border: 1px solid #cbd5e1;
      border-radius: 6px;
      font-size: 12px;
      font-weight: 600;
      padding: 4px 8px;
      cursor: pointer;
      color: #475569;
      transition: background 0.2s ease, color 0.2s ease, border-color 0.2s ease;
    }
    .step-btn:hover {
      background: #e2e8f0;
    }
    .step-btn.active {
      background: #0284c7;
      color: #ffffff;
      border-color: #0284c7;
    }
    .main-grid {
      display: grid;
      grid-template-columns: 1.2fr 0.8fr;
      gap: 12px;
      align-items: center;
    }
    @media (max-width: 520px) {
      .main-grid {
        grid-template-columns: 1fr;
      }
    }
    .line-container {
      background: #f8fafc;
      border: 1px solid #e2e8f0;
      border-radius: 10px;
      padding: 10px 6px;
      position: relative;
    }
    svg {
      width: 100%;
      height: auto;
      display: block;
      overflow: visible;
    }
    .table-container {
      background: #f8fafc;
      border: 1px solid #e2e8f0;
      border-radius: 10px;
      padding: 6px;
    }
    table {
      width: 100%;
      border-collapse: collapse;
      font-size: 13px;
      text-align: center;
    }
    th {
      padding: 6px 4px;
      color: #64748b;
      font-weight: 700;
      font-size: 12px;
      border-bottom: 2px solid #cbd5e1;
    }
    td {
      padding: 5px 4px;
      font-weight: 600;
      color: #334155;
      border-bottom: 1px solid #e2e8f0;
      transition: background 0.2s ease, color 0.2s ease;
    }
    tr:last-child td {
      border-bottom: none;
    }
    tr.active td {
      background: #e0f2fe;
      color: #0369a1;
      font-weight: 700;
    }
    tr.active td:first-child {
      border-top-left-radius: 6px;
      border-bottom-left-radius: 6px;
    }
    tr.active td:last-child {
      border-top-right-radius: 6px;
      border-bottom-right-radius: 6px;
    }
    .unit-ft { color: #2563eb; }
    .unit-in { color: #d97706; }
    .rate { color: #059669; font-weight: 700; }
  </style>
</head>
<body>

<div class="layout">
  <div class="top-controls">
    <div class="badge">× 12 Conversion Factor</div>
    <div class="btn-group" id="btnGroup"></div>
  </div>

  <div class="main-grid">
    
    <div class="line-container">
      <svg viewBox="0 0 360 200" id="numLineSvg">
        
        <defs>
          <marker id="arrow-blue" viewBox="0 0 6 6" refX="5" refY="3" markerWidth="4" markerHeight="4" orient="auto">
            <path d="M 0 0 L 6 3 L 0 6 z" fill="#2563eb"/>
          </marker>
          <marker id="arrow-amber" viewBox="0 0 6 6" refX="5" refY="3" markerWidth="4" markerHeight="4" orient="auto">
            <path d="M 0 0 L 6 3 L 0 6 z" fill="#d97706"/>
          </marker>
          <marker id="arrow-center" viewBox="0 0 6 6" refX="5" refY="3" markerWidth="5" markerHeight="5" orient="auto">
            <path d="M 0 0 L 6 3 L 0 6 z" fill="#0284c7"/>
          </marker>
        </defs>

        
        <path id="topCurve" d="M 125 40 Q 175 15 225 40" fill="none" stroke="#2563eb" stroke-width="1.5" stroke-dasharray="3,3" marker-end="url(#arrow-blue)" />
        <text id="topCurveText" x="175" y="24" font-size="11" font-weight="700" fill="#2563eb" text-anchor="middle">× 3</text>

        
        <text x="65" y="60" font-size="11" font-weight="700" fill="#2563eb" text-anchor="end" dominant-baseline="middle">Feet (x)</text>
        <line x1="75" y1="60" x2="335" y2="60" stroke="#94a3b8" stroke-width="2.5" stroke-linecap="round" />

        
        <g id="centerArrowGroup" transform="translate(225, 65)">
          <line x1="0" y1="3" x2="0" y2="67" stroke="#0284c7" stroke-width="2" marker-end="url(#arrow-center)"/>
          <rect x="-38" y="22" width="76" height="22" rx="4" fill="#ffffff" stroke="#0284c7" stroke-width="1.5"/>
          <text x="0" y="37" font-size="11" font-weight="700" fill="#0284c7" text-anchor="middle">× 12</text>
        </g>

        
        <text x="65" y="140" font-size="11" font-weight="700" fill="#d97706" text-anchor="end" dominant-baseline="middle">Inches (y)</text>
        <line x1="75" y1="140" x2="335" y2="140" stroke="#94a3b8" stroke-width="2.5" stroke-linecap="round" />

        
        <path id="botCurve" d="M 125 160 Q 175 185 225 160" fill="none" stroke="#d97706" stroke-width="1.5" stroke-dasharray="3,3" marker-end="url(#arrow-amber)" />
        <text id="botCurveText" x="175" y="184" font-size="11" font-weight="700" fill="#d97706" text-anchor="middle">× 3</text>

        
        
        <g class="tick-group" data-idx="0" transform="translate(75, 0)">
          <line x1="0" y1="53" x2="0" y2="67" stroke="#64748b" stroke-width="2" />
          <text x="0" y="48" font-size="12" font-weight="700" fill="#1e2945" text-anchor="middle">0</text>
          <line x1="0" y1="133" x2="0" y2="147" stroke="#64748b" stroke-width="2" />
          <text x="0" y="159" font-size="12" font-weight="700" fill="#1e2945" text-anchor="middle">0</text>
        </g>

        
        <g class="tick-group" data-idx="1" transform="translate(125, 0)">
          <line x1="0" y1="53" x2="0" y2="67" stroke="#64748b" stroke-width="2" />
          <circle id="c-top-1" cx="0" cy="60" r="4" fill="#2563eb" opacity="0"/>
          <text x="0" y="48" font-size="12" font-weight="700" fill="#2563eb" text-anchor="middle">1</text>
          <line x1="0" y1="133" x2="0" y2="147" stroke="#64748b" stroke-width="2" />
          <circle id="c-bot-1" cx="0" cy="140" r="4" fill="#d97706" opacity="0"/>
          <text x="0" y="159" font-size="12" font-weight="700" fill="#d97706" text-anchor="middle">12</text>
        </g>

        
        <g class="tick-group" data-idx="2" transform="translate(175, 0)">
          <line x1="0" y1="53" x2="0" y2="67" stroke="#64748b" stroke-width="2" />
          <circle id="c-top-2" cx="0" cy="60" r="4" fill="#2563eb" opacity="0"/>
          <text x="0" y="48" font-size="12" font-weight="700" fill="#2563eb" text-anchor="middle">2</text>
          <line x1="0" y1="133" x2="0" y2="147" stroke="#64748b" stroke-width="2" />
          <circle id="c-bot-2" cx="0" cy="140" r="4" fill="#d97706" opacity="0"/>
          <text x="0" y="159" font-size="12" font-weight="700" fill="#d97706" text-anchor="middle">24</text>
        </g>

        
        <g class="tick-group" data-idx="3" transform="translate(225, 0)">
          <line x1="0" y1="53" x2="0" y2="67" stroke="#64748b" stroke-width="2" />
          <circle id="c-top-3" cx="0" cy="60" r="5" fill="#2563eb"/>
          <text x="0" y="48" font-size="12" font-weight="700" fill="#2563eb" text-anchor="middle">3</text>
          <line x1="0" y1="133" x2="0" y2="147" stroke="#64748b" stroke-width="2" />
          <circle id="c-bot-3" cx="0" cy="140" r="5" fill="#d97706"/>
          <text x="0" y="159" font-size="12" font-weight="700" fill="#d97706" text-anchor="middle">36</text>
        </g>

        
        <g class="tick-group" data-idx="4" transform="translate(275, 0)">
          <line x1="0" y1="53" x2="0" y2="67" stroke="#64748b" stroke-width="2" />
          <circle id="c-top-4" cx="0" cy="60" r="4" fill="#2563eb" opacity="0"/>
          <text x="0" y="48" font-size="12" font-weight="700" fill="#2563eb" text-anchor="middle">4</text>
          <line x1="0" y1="133" x2="0" y2="147" stroke="#64748b" stroke-width="2" />
          <circle id="c-bot-4" cx="0" cy="140" r="4" fill="#d97706" opacity="0"/>
          <text x="0" y="159" font-size="12" font-weight="700" fill="#d97706" text-anchor="middle">48</text>
        </g>

        
        <g class="tick-group" data-idx="5" transform="translate(325, 0)">
          <line x1="0" y1="53" x2="0" y2="67" stroke="#64748b" stroke-width="2" />
          <circle id="c-top-5" cx="0" cy="60" r="4" fill="#2563eb" opacity="0"/>
          <text x="0" y="48" font-size="12" font-weight="700" fill="#2563eb" text-anchor="middle">5</text>
          <line x1="0" y1="133" x2="0" y2="147" stroke="#64748b" stroke-width="2" />
          <circle id="c-bot-5" cx="0" cy="140" r="4" fill="#d97706" opacity="0"/>
          <text x="0" y="159" font-size="12" font-weight="700" fill="#d97706" text-anchor="middle">60</text>
        </g>
      </svg>
    </div>

    
    <div class="table-container">
      <table>
        <thead>
          <tr>
            <th class="unit-ft">Feet (x)</th>
            <th class="unit-in">Inches (y)</th>
            <th class="rate">y / x</th>
          </tr>
        </thead>
        <tbody id="tableBody">
          
        </tbody>
      </table>
    </div>
  </div>
</div>

<script>
  const data = [
    { ft: 1, in: 12, ratio: "12 / 1 = 12", xPos: 125 },
    { ft: 2, in: 24, ratio: "24 / 2 = 12", xPos: 175 },
    { ft: 3, in: 36, ratio: "36 / 3 = 12", xPos: 225 },
    { ft: 4, in: 48, ratio: "48 / 4 = 12", xPos: 275 },
    { ft: 5, in: 60, ratio: "60 / 5 = 12", xPos: 325 }
  ];

  let activeIdx = 2; // Default to 3 ft
  const tbody = document.getElementById('tableBody');
  const btnGroup = document.getElementById('btnGroup');

  // Build UI
  data.forEach((d, idx) => {
    // Buttons
    const btn = document.createElement('button');
    btn.className = `step-btn ${idx === activeIdx ? 'active' : ''}`;
    btn.textContent = `${d.ft} ft`;
    btn.onclick = () => selectStep(idx);
    btnGroup.appendChild(btn);

    // Table rows
    const tr = document.createElement('tr');
    tr.id = `row-${idx}`;
    tr.className = idx === activeIdx ? 'active' : '';
    tr.style.cursor = 'pointer';
    tr.onclick = () => selectStep(idx);
    tr.innerHTML = `
      <td class="unit-ft">${d.ft}</td>
      <td class="unit-in">${d.in}</td>
      <td class="rate">${d.ratio}</td>
    `;
    tbody.appendChild(tr);
  });

  function selectStep(idx) {
    activeIdx = idx;
    const d = data[idx];

    // Update buttons
    document.querySelectorAll('.step-btn').forEach((b, i) => {
      b.classList.toggle('active', i === idx);
    });

    // Update table rows
    data.forEach((_, i) => {
      const r = document.getElementById(`row-${i}`);
      if (r) r.classList.toggle('active', i === idx);
    });

    // Update SVG elements
    const centerArrow = document.getElementById('centerArrowGroup');
    centerArrow.setAttribute('transform', `translate(${d.xPos}, 65)`);

    // Update Top & Bottom Curves
    const topCurve = document.getElementById('topCurve');
    const botCurve = document.getElementById('botCurve');
    const topText = document.getElementById('topCurveText');
    const botText = document.getElementById('botCurveText');

    if (idx === 0) {
      // 1 ft: hide multiplier curve
      topCurve.style.display = 'none';
      botCurve.style.display = 'none';
      topText.style.display = 'none';
      botText.style.display = 'none';
    } else {
      topCurve.style.display = 'block';
      botCurve.style.display = 'block';
      topText.style.display = 'block';
      botText.style.display = 'block';

      const midX = (125 + d.xPos) / 2;
      topCurve.setAttribute('d', `M 125 40 Q ${midX} 15 ${d.xPos} 40`);
      botCurve.setAttribute('d', `M 125 160 Q ${midX} 185 ${d.xPos} 160`);

      topText.setAttribute('x', midX);
      botText.setAttribute('x', midX);
      topText.textContent = `× ${d.ft}`;
      botText.textContent = `× ${d.ft}`;
    }

    // Dots on ticks
    for (let i = 1; i <= 5; i++) {
      const topC = document.getElementById(`c-top-${i}`);
      const botC = document.getElementById(`c-bot-${i}`);
      if (i === d.ft) {
        topC.setAttribute('opacity', '1');
        botC.setAttribute('opacity', '1');
      } else if (i === 1 && idx > 0) {
        topC.setAttribute('opacity', '0.4');
        botC.setAttribute('opacity', '0.4');
      } else {
        topC.setAttribute('opacity', '0');
        botC.setAttribute('opacity', '0');
      }
    }
  }

  // Initial draw
  selectStep(2);
</script>

</body>
</html>

What if you start with inches and want to find feet instead? How do you pick which way to multiply or divide?

Because $12\text{ inches}$ and $1\text{ foot}$ name the exact same amount, the fraction $\frac{12\text{ in}}{1\text{ ft}}$ equals 1 whole. Multiplying by this fraction changes the label without changing the actual measurement.

To convert $48\text{ inches}$ to feet, set up your conversion factor so the old unit cancels out: $48\text{ in} \times \frac{1\text{ ft}}{12\text{ in}} = \frac{48}{12}\text{ ft} = 4\text{ ft}$.

<!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;
      font-family: system-ui, -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, sans-serif;
    }

    body {
      background: #ffffff;
      color: #1e2945;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: space-between;
      min-height: 100%;
      padding: 16px 12px;
      user-select: none;
    }

    .stepper-nav {
      display: flex;
      align-items: center;
      justify-content: center;
      gap: 8px;
      width: 100%;
      max-width: 440px;
    }

    .step-btn {
      flex: 1;
      padding: 8px 10px;
      font-size: 14px;
      font-weight: 600;
      border: 1px solid #e2e8f0;
      background: #f8fafc;
      color: #64748b;
      border-radius: 8px;
      cursor: pointer;
      display: flex;
      align-items: center;
      justify-content: center;
      gap: 6px;
      transition: all 0.2s ease;
    }

    .step-btn.active {
      background: #22b7ff;
      border-color: #22b7ff;
      color: #ffffff;
      box-shadow: 0 2px 6px rgba(34, 183, 255, 0.3);
    }

    .play-btn {
      padding: 8px 12px;
      font-size: 14px;
      border: 1px solid #e2e8f0;
      background: #f8fafc;
      color: #1e2945;
      border-radius: 8px;
      cursor: pointer;
      font-weight: 600;
    }

    .stage-container {
      position: relative;
      width: 100%;
      max-width: 480px;
      height: 220px;
      display: flex;
      align-items: center;
      justify-content: center;
      margin: auto 0;
    }

    .step-panel {
      position: absolute;
      inset: 0;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      opacity: 0;
      pointer-events: none;
      transition: opacity 0.3s ease, transform 0.3s ease;
      transform: translateY(6px);
    }

    .step-panel.active {
      opacity: 1;
      pointer-events: auto;
      transform: translateY(0);
    }

    /* Math typography */
    .math-row {
      display: inline-flex;
      align-items: center;
      font-size: 26px;
      font-weight: 600;
      color: #1e2945;
      line-height: 1;
    }

    .fraction {
      display: inline-flex;
      flex-direction: column;
      align-items: center;
      vertical-align: middle;
      padding: 0 4px;
      margin: 0 4px;
    }

    .numerator {
      padding-bottom: 5px;
      border-bottom: 2.5px solid #1e2945;
      text-align: center;
      width: 100%;
    }

    .denominator {
      padding-top: 5px;
      text-align: center;
      width: 100%;
    }

    .op {
      margin: 0 8px;
      color: #64748b;
      font-weight: 500;
    }

    .unit {
      color: #0284c7;
      margin-left: 2px;
      position: relative;
      display: inline-block;
    }

    /* Cancellation Slash */
    .cancel-unit {
      position: relative;
      display: inline-block;
      color: #0284c7;
    }

    .slash {
      position: absolute;
      top: 50%;
      left: -15%;
      width: 130%;
      height: 3px;
      background-color: #ef4444;
      transform: translateY(-50%) rotate(-35deg);
      border-radius: 2px;
      opacity: 0;
      transform-origin: center;
      transition: opacity 0.3s ease, transform 0.3s ease;
    }

    .step-panel.active .slash {
      opacity: 1;
    }

    /* Step 3 Highlight Box */
    .result-box {
      border: 2.5px solid #16a34a;
      background: #f0fdf4;
      border-radius: 12px;
      padding: 16px 24px;
      box-shadow: 0 4px 12px rgba(22, 163, 74, 0.12);
      display: flex;
      flex-direction: column;
      align-items: center;
      gap: 10px;
    }

    .check-badge {
      display: inline-flex;
      align-items: center;
      justify-content: center;
      background: #16a34a;
      color: #ffffff;
      width: 28px;
      height: 28px;
      border-radius: 50%;
      font-size: 16px;
      font-weight: bold;
      margin-left: 8px;
    }

    .sub-label {
      font-size: 13px;
      font-weight: 600;
      color: #64748b;
      letter-spacing: 0.5px;
      text-transform: uppercase;
      margin-top: 14px;
    }

    @media (max-width: 400px) {
      .math-row {
        font-size: 21px;
      }
      .result-box {
        padding: 12px 16px;
      }
      .step-btn {
        font-size: 13px;
        padding: 6px 8px;
      }
    }
  </style>
</head>
<body>

  
  <div class="stepper-nav">
    <button class="step-btn active" id="btn0" onclick="goToStep(0)">1. Set Up</button>
    <button class="step-btn" id="btn1" onclick="goToStep(1)">2. Cancel</button>
    <button class="step-btn" id="btn2" onclick="goToStep(2)">3. Simplify</button>
    <button class="play-btn" id="playBtn" onclick="togglePlay()">⏸</button>
  </div>

  
  <div class="stage-container">
    
    
    <div class="step-panel active" id="panel0">
      <div class="math-row">
        <span>48<span class="unit">in</span></span>
        <span class="op">×</span>
        <div class="fraction">
          <div class="numerator">1<span class="unit">ft</span></div>
          <div class="denominator">12<span class="unit">in</span></div>
        </div>
      </div>
      <div class="sub-label">Conversion Ratio</div>
    </div>

    
    <div class="step-panel" id="panel1">
      <div class="math-row">
        <span>48<span class="cancel-unit">in<span class="slash"></span></span></span>
        <span class="op">×</span>
        <div class="fraction">
          <div class="numerator">1<span class="unit">ft</span></div>
          <div class="denominator">12<span class="cancel-unit">in<span class="slash"></span></span></div>
        </div>
      </div>
      <div class="sub-label" style="color: #ef4444;">Matching Units Cancel</div>
    </div>

    
    <div class="step-panel" id="panel2">
      <div class="result-box">
        <div class="math-row">
          <div class="fraction" style="margin-right: 8px;">
            <div class="numerator">48 × 1<span class="unit">ft</span></div>
            <div class="denominator">12</div>
          </div>
          <span class="op">=</span>
          <span style="color: #16a34a; font-weight: 700; margin-left: 4px;">4<span class="unit" style="color: #16a34a;">ft</span></span>
          <span class="check-badge">✓</span>
        </div>
      </div>
      <div class="sub-label" style="color: #16a34a;">Simplified Answer</div>
    </div>

  </div>

  <script src="{{VENDOR_CDN}}/classace-diagram.js"></script>
  <script>
    let currentStep = 0;
    const totalSteps = 3;
    let autoInterval = null;
    let isPlaying = true;

    function updateView() {
      for (let i = 0; i < totalSteps; i++) {
        const btn = document.getElementById('btn' + i);
        const panel = document.getElementById('panel' + i);
        if (i === currentStep) {
          btn.classList.add('active');
          panel.classList.add('active');
        } else {
          btn.classList.remove('active');
          panel.classList.remove('active');
        }
      }
    }

    function goToStep(idx) {
      currentStep = idx;
      updateView();
    }

    function nextStep() {
      currentStep = (currentStep + 1) % totalSteps;
      updateView();
    }

    function startAutoPlay() {
      if (autoInterval) clearInterval(autoInterval);
      autoInterval = setInterval(nextStep, 3500);
      isPlaying = true;
      document.getElementById('playBtn').textContent = '⏸';
    }

    function pauseAutoPlay() {
      if (autoInterval) clearInterval(autoInterval);
      autoInterval = null;
      isPlaying = false;
      document.getElementById('playBtn').textContent = '▶';
    }

    function togglePlay() {
      if (isPlaying) {
        pauseAutoPlay();
      } else {
        nextStep();
        startAutoPlay();
      }
    }

    // Initialize auto stepper
    startAutoPlay();
  </script>
</body>
</html>

A common mistake is multiplying when you should divide, which gives $48 \times 12 = 576\text{ feet}$ for a desk. Always ask: *Is a foot bigger than an inch?* Since feet are larger units, you should have fewer of them.

The next time you see a measurement in cups, pounds, or meters, look for the hidden ratio that keeps both sides balanced.

math

Understanding Unit Conversion Within Measurement Systems. Explain why unit conversions are proportional relationships and identify conversion factors

Ratios and Measurement Units

Grade 6 Unit 3: Unit Rates and Percentages - Lesson 3 & 4: Converting Units

Convert and Compare Customary and Metric Units Using Proportions

To build conceptual mastery for TEKS 6.4(H) and 6.1(D), transition students through a concrete-representational-abstract (CRA) sequence. Begin by comparing physical measuring tools (e.g., 1-foot ruler aligned against a 1-yard stick) to demonstrate why a larger unit requires fewer counts than a smaller unit. Use double number lines and ratio tables to establish that unit conversions preserve proportionality before introducing formal algebraic rules ($y = kx$) and dimensional analysis cancellation.

Understanding Unit Conversion Within Measurement Systems

Changing Units, Keeping the Same Size

Why Conversions Are Proportional

Setting Up the Conversion Factor