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Tutoring abilities

Imagine a frog jumping across stepping stones numbered 0, 1, 2, and 3. What happens if the frog lands right in the middle between two stones?

A **number line** is a straight path where every number has its own spot in order. We can split the space between whole numbers into equal smaller steps.

<!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">
  <link rel="stylesheet" href="{{VENDOR_CDN}}/katex.min.css">
  <script src="{{VENDOR_CDN}}/katex.min.js"></script>
  <style>
    * {
      box-sizing: border-box;
      margin: 0;
      padding: 0;
      -webkit-tap-highlight-color: transparent;
    }
    body {
      font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, Helvetica, Arial, sans-serif;
      background: #f8f9fa;
      color: #1e2945;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      min-height: 100vh;
      padding: 16px;
      user-select: none;
    }
    .wrapper {
      width: 100%;
      max-width: 580px;
      display: flex;
      flex-direction: column;
      align-items: center;
      gap: 20px;
    }
    /* Fraction Preset Buttons */
    .btn-group {
      display: flex;
      flex-wrap: wrap;
      justify-content: center;
      gap: 8px;
      width: 100%;
    }
    .frac-btn {
      background: #ffffff;
      border: 2px solid #e6e6e6;
      border-radius: 20px;
      padding: 6px 16px;
      font-size: 16px;
      font-weight: 600;
      color: #1e2945;
      cursor: pointer;
      box-shadow: 0 1px 3px rgba(0,0,0,0.05);
      transition: background 0.2s, border-color 0.2s, box-shadow 0.2s, color 0.2s;
    }
    .frac-btn.active {
      border-color: #22b7ff;
      background: #e8f7ff;
      color: #0088cc;
      box-shadow: 0 0 0 2px rgba(34, 183, 255, 0.25);
    }

    /* Diagram Stage */
    .stage {
      width: 100%;
      background: #ffffff;
      border-radius: 16px;
      box-shadow: 0 2px 10px rgba(0,0,0,0.04);
      border: 1px solid #e8ecf1;
      padding: 24px 20px 20px;
      position: relative;
    }

    /* Math Target Display */
    .target-display {
      display: flex;
      justify-content: center;
      align-items: center;
      min-height: 48px;
      margin-bottom: 8px;
    }
    .math-pill {
      display: inline-flex;
      align-items: center;
      gap: 8px;
      background: #f0fdf4;
      border: 1.5px solid #10b981;
      color: #065f46;
      padding: 4px 16px;
      border-radius: 24px;
      font-size: 20px;
      font-weight: 700;
    }

    /* SVG Area */
    .svg-container {
      width: 100%;
      height: 180px;
      position: relative;
    }
    svg {
      width: 100%;
      height: 100%;
      overflow: visible;
    }

    /* Ticks and click targets */
    .tick-hit {
      cursor: pointer;
    }
    .tick-hit:hover circle {
      fill: #22b7ff;
      opacity: 0.3;
    }

    /* Step Controller bar */
    .nav-bar {
      display: flex;
      align-items: center;
      justify-content: center;
      gap: 12px;
    }
    .nav-btn {
      background: #ffffff;
      border: 1px solid #d1d5db;
      border-radius: 50%;
      width: 36px;
      height: 36px;
      display: flex;
      align-items: center;
      justify-content: center;
      cursor: pointer;
      font-size: 16px;
      color: #4b5563;
      box-shadow: 0 1px 2px rgba(0,0,0,0.05);
      transition: background 0.15s, border-color 0.15s;
    }
    .nav-btn:hover {
      background: #f3f4f6;
      border-color: #9ca3af;
    }
    .indicator-dots {
      display: flex;
      gap: 6px;
    }
    .dot {
      width: 8px;
      height: 8px;
      border-radius: 50%;
      background: #d1d5db;
      cursor: pointer;
      transition: background 0.2s;
    }
    .dot.active {
      background: #22b7ff;
      width: 18px;
      border-radius: 4px;
    }
  </style>
</head>
<body>

<div class="wrapper">
  
  <div class="btn-group" id="btnGroup">
    <button class="frac-btn active" data-index="0">½</button>
    <button class="frac-btn" data-index="1">¾</button>
    <button class="frac-btn" data-index="2">1</button>
    <button class="frac-btn" data-index="3">1 ½</button>
    <button class="frac-btn" data-index="4">1 ¾</button>
    <button class="frac-btn" data-index="5">2</button>
  </div>

  
  <div class="stage">
    <div class="target-display">
      <div class="math-pill" id="mathDisplay"></div>
    </div>

    <div class="svg-container">
      <svg viewBox="0 0 500 160" id="numberLineSvg">
        
        <defs>
          <marker id="arrow" viewBox="0 0 10 10" refX="6" refY="5" markerWidth="6" markerHeight="6" orient="auto-start-reverse">
            <path d="M 0 1 L 8 5 L 0 9 z" fill="#1e2945" />
          </marker>
          <marker id="jump-arrow" viewBox="0 0 10 10" refX="6" refY="5" markerWidth="5" markerHeight="5" orient="auto-start-reverse">
            <path d="M 0 1 L 8 5 L 0 9 z" fill="#22b7ff" />
          </marker>
        </defs>

        
        <line x1="30" y1="110" x2="470" y2="110" stroke="#1e2945" stroke-width="3" marker-start="url(#arrow)" marker-end="url(#arrow)" stroke-linecap="round" />

        
        <g id="jumpsLayer"></g>

        
        <g id="ticksLayer"></g>

        
        <g id="markerLayer" style="transition: transform 0.4s cubic-bezier(0.34, 1.56, 0.64, 1);">
          <circle cx="0" cy="110" r="7" fill="#10b981" stroke="#ffffff" stroke-width="2.5" />
          <path d="M 0 100 L 0 70" stroke="#10b981" stroke-width="2.5" stroke-dasharray="3,3" />
        </g>
      </svg>
    </div>
  </div>

  
  <div class="nav-bar">
    <button class="nav-btn" id="prevBtn" aria-label="Previous">◀</button>
    <div class="indicator-dots" id="dotsContainer"></div>
    <button class="nav-btn" id="nextBtn" aria-label="Next">▶</button>
  </div>
</div>

<script src="{{VENDOR_CDN}}/classace-diagram.js"></script>
<script>
  const PRESETS = [
    { value: 0.5,  tex: "\\frac{1}{2}",         denom: 2, whole: 0, steps: 1 },
    { value: 0.75, tex: "\\frac{3}{4}",         denom: 4, whole: 0, steps: 3 },
    { value: 1.0,  tex: "1",                    denom: 2, whole: 1, steps: 0 },
    { value: 1.5,  tex: "1\\frac{1}{2}",        denom: 2, whole: 1, steps: 1 },
    { value: 1.75, tex: "1\\frac{3}{4}",        denom: 4, whole: 1, steps: 3 },
    { value: 2.0,  tex: "2",                    denom: 2, whole: 2, steps: 0 }
  ];

  let currentIndex = 0;

  // Geometry: Line range is 0 to 2
  // 0 is at X = 70, 1 is at X = 250, 2 is at X = 430
  const X_START = 70;
  const X_END = 430;
  const Y_LINE = 110;

  function valToX(val) {
    return X_START + (val / 2.0) * (X_END - X_START);
  }

  function renderTicks(denom) {
    const layer = document.getElementById('ticksLayer');
    layer.innerHTML = '';

    // Draw sub-ticks based on denominator (2 for halves, 4 for quarters)
    const totalParts = 2 * denom;
    for (let i = 0; i <= totalParts; i++) {
      const val = i / denom;
      const x = valToX(val);
      const isWhole = (i % denom === 0);
      const tickHeight = isWhole ? 24 : 12;
      const strokeWidth = isWhole ? 2.5 : 1.5;
      const strokeColor = isWhole ? '#1e2945' : '#9ca3af';

      // Tick Line
      const line = document.createElementNS('http://www.w3.org/2000/svg', 'line');
      line.setAttribute('x1', x);
      line.setAttribute('y1', Y_LINE - tickHeight / 2);
      line.setAttribute('x2', x);
      line.setAttribute('y2', Y_LINE + tickHeight / 2);
      line.setAttribute('stroke', strokeColor);
      line.setAttribute('stroke-width', strokeWidth);
      line.setAttribute('stroke-linecap', 'round');
      layer.appendChild(line);

      // Whole Number Label
      if (isWhole) {
        const text = document.createElementNS('http://www.w3.org/2000/svg', 'text');
        text.setAttribute('x', x);
        text.setAttribute('y', Y_LINE + 32);
        text.setAttribute('text-anchor', 'middle');
        text.setAttribute('font-size', '18');
        text.setAttribute('font-weight', '700');
        text.setAttribute('fill', '#1e2945');
        text.textContent = Math.round(val);
        layer.appendChild(text);
      }

      // Invisible hit area for interactive tapping
      const hitG = document.createElementNS('http://www.w3.org/2000/svg', 'g');
      hitG.setAttribute('class', 'tick-hit');
      hitG.innerHTML = `<circle cx="${x}" cy="${Y_LINE}" r="16" fill="transparent"/>`;
      hitG.addEventListener('click', () => {
        // Find closest preset or match
        const foundIdx = PRESETS.findIndex(p => Math.abs(p.value - val) < 0.01);
        if (foundIdx !== -1) setPreset(foundIdx);
      });
      layer.appendChild(hitG);
    }
  }

  function renderJumps(preset) {
    const layer = document.getElementById('jumpsLayer');
    layer.innerHTML = '';

    const targetVal = preset.value;
    const denom = preset.denom;
    const stepSize = 1 / denom;
    const numSteps = Math.round(targetVal / stepSize);

    for (let i = 0; i < numSteps; i++) {
      const x1 = valToX(i * stepSize);
      const x2 = valToX((i + 1) * stepSize);
      const midX = (x1 + x2) / 2;
      const radius = (x2 - x1) / 2;
      const arcHeight = Math.min(radius * 0.75, 28);
      const apexY = Y_LINE - arcHeight;

      // Arc path
      const path = document.createElementNS('http://www.w3.org/2000/svg', 'path');
      const d = `M ${x1} ${Y_LINE} Q ${midX} ${apexY - 14} ${x2} ${Y_LINE}`;
      path.setAttribute('d', d);
      path.setAttribute('fill', 'none');
      path.setAttribute('stroke', '#22b7ff');
      path.setAttribute('stroke-width', '2.5');
      path.setAttribute('marker-end', 'url(#jump-arrow)');
      path.style.opacity = '0.9';
      layer.appendChild(path);

      // Fraction increment label inside arc
      const stepText = document.createElementNS('http://www.w3.org/2000/svg', 'text');
      stepText.setAttribute('x', midX);
      stepText.setAttribute('y', apexY - 6);
      stepText.setAttribute('text-anchor', 'middle');
      stepText.setAttribute('font-size', '11');
      stepText.setAttribute('font-weight', '600');
      stepText.setAttribute('fill', '#0088cc');
      stepText.textContent = `+1/${denom}`;
      layer.appendChild(stepText);
    }
  }

  function updateMarker(val) {
    const x = valToX(val);
    const marker = document.getElementById('markerLayer');
    marker.setAttribute('transform', `translate(${x}, 0)`);
  }

  function setPreset(index) {
    currentIndex = (index + PRESETS.length) % PRESETS.length;
    const p = PRESETS[currentIndex];

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

    // Update dots
    document.querySelectorAll('.dot').forEach((dot, i) => {
      dot.classList.toggle('active', i === currentIndex);
    });

    // Render KaTeX formula
    const mathEl = document.getElementById('mathDisplay');
    if (window.katex) {
      katex.render(p.tex, mathEl, { throwOnError: false });
    } else {
      mathEl.textContent = p.tex;
    }

    renderTicks(p.denom);
    renderJumps(p);
    updateMarker(p.value);
  }

  // Build dots
  const dotsContainer = document.getElementById('dotsContainer');
  PRESETS.forEach((_, i) => {
    const dot = document.createElement('div');
    dot.className = 'dot' + (i === 0 ? ' active' : '');
    dot.addEventListener('click', () => setPreset(i));
    dotsContainer.appendChild(dot);
  });

  // Attach button events
  document.querySelectorAll('.frac-btn').forEach(btn => {
    btn.addEventListener('click', (e) => {
      const idx = parseInt(e.currentTarget.getAttribute('data-index'), 10);
      setPreset(idx);
    });
  });

  document.getElementById('prevBtn').addEventListener('click', () => setPreset(currentIndex - 1));
  document.getElementById('nextBtn').addEventListener('click', () => setPreset(currentIndex + 1));

  // Initial draw
  setPreset(0);
</script>

</body>
</html>

How do we find a spot like $1\frac{1}{2}$ on the line?

A **mixed number** has a whole number and a fraction part, like $1\frac{1}{2}$.

To place $1\frac{1}{2}$, first take a big jump to the whole number **1**, and then take half a step forward.

<!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">
  <link rel="stylesheet" href="{{VENDOR_CDN}}/katex.min.css">
  <script src="{{VENDOR_CDN}}/katex.min.js"></script>
  <style>
    * { box-sizing: border-box; margin: 0; padding: 0; }
    body {
      font-family: system-ui, -apple-system, sans-serif;
      background: transparent;
      color: #1e2945;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      min-height: 100%;
      padding: 12px;
    }
    .diagram-wrap {
      width: 100%;
      max-width: 480px;
      display: flex;
      flex-direction: column;
      align-items: center;
      gap: 16px;
    }
    .display-box {
      background: #ffffff;
      border-radius: 12px;
      padding: 8px 16px;
      box-shadow: 0 2px 8px rgba(0, 0, 0, 0.05);
      border: 1px solid #e5e7eb;
      font-size: 20px;
      font-weight: 600;
      min-height: 48px;
      display: flex;
      align-items: center;
      justify-content: center;
      gap: 8px;
    }
    .svg-container {
      width: 100%;
      background: #ffffff;
      border-radius: 16px;
      padding: 16px 8px;
      box-shadow: 0 4px 12px rgba(0, 0, 0, 0.04);
      border: 1px solid #e5e7eb;
    }
    svg {
      width: 100%;
      height: auto;
      display: block;
      overflow: visible;
    }
    .tick-label {
      font-size: 13px;
      fill: #4b5563;
      text-anchor: middle;
      font-family: system-ui, sans-serif;
    }
    .whole-label {
      font-size: 16px;
      font-weight: 700;
      fill: #1e2945;
    }
    .jump-path {
      fill: none;
      stroke-width: 3.5;
      stroke-linecap: round;
      stroke-dasharray: 400;
      stroke-dashoffset: 400;
      transition: stroke-dashoffset 0.6s cubic-bezier(0.4, 0, 0.2, 1);
    }
    .jump-path.active {
      stroke-dashoffset: 0;
    }
    .jump-1 { stroke: #22b7ff; }
    .jump-2 { stroke: #10b981; }
    .jump-tag {
      font-size: 14px;
      font-weight: 700;
      opacity: 0;
      transform: translateY(4px);
      transition: opacity 0.4s ease, transform 0.4s ease;
      text-anchor: middle;
    }
    .jump-tag.active {
      opacity: 1;
      transform: translateY(0);
    }
    .tag-1 { fill: #0284c7; }
    .tag-2 { fill: #059669; }
    .marker-dot {
      fill: #1e2945;
      transition: transform 0.5s cubic-bezier(0.34, 1.56, 0.64, 1);
      transform-origin: center;
    }
    .target-pin {
      opacity: 0;
      transform: scale(0.6);
      transition: opacity 0.4s ease, transform 0.4s cubic-bezier(0.34, 1.56, 0.64, 1);
      transform-origin: 280px 145px;
    }
    .target-pin.active {
      opacity: 1;
      transform: scale(1);
    }
    .btn-group {
      display: flex;
      gap: 8px;
      width: 100%;
      justify-content: center;
    }
    .step-btn {
      flex: 1;
      max-width: 120px;
      padding: 8px 12px;
      font-size: 13px;
      font-weight: 600;
      border: 1px solid #e5e7eb;
      background: #f8f9fa;
      color: #4b5563;
      border-radius: 8px;
      cursor: pointer;
      display: flex;
      align-items: center;
      justify-content: center;
      gap: 4px;
      box-shadow: 0 1px 2px rgba(0, 0, 0, 0.04);
      transition: background 0.2s, color 0.2s, box-shadow 0.2s;
    }
    .step-btn.active {
      background: #22b7ff;
      color: #ffffff;
      border-color: #22b7ff;
      box-shadow: 0 2px 6px rgba(34, 183, 255, 0.35);
    }
  </style>
</head>
<body>

<div class="diagram-wrap">
  <div class="display-box" id="mathDisplay"></div>

  <div class="svg-container">
    <svg viewBox="0 0 400 180">
      <defs>
        <marker id="arrow" viewBox="0 0 10 10" refX="6" refY="5" markerWidth="6" markerHeight="6" orient="auto-start-reverse">
          <path d="M 0 1 L 8 5 L 0 9 z" fill="#9ca3af"/>
        </marker>
        <marker id="arrow-blue" viewBox="0 0 10 10" refX="5" refY="5" markerWidth="5" markerHeight="5" orient="auto-start-reverse">
          <path d="M 0 1 L 7 5 L 0 9 z" fill="#22b7ff"/>
        </marker>
        <marker id="arrow-green" viewBox="0 0 10 10" refX="5" refY="5" markerWidth="5" markerHeight="5" orient="auto-start-reverse">
          <path d="M 0 1 L 7 5 L 0 9 z" fill="#10b981"/>
        </marker>
      </defs>

      
      <path id="pathJump1" class="jump-path jump-1" d="M 40 145 Q 120 40 200 145" marker-end="url(#arrow-blue)"/>
      <path id="pathJump2" class="jump-path jump-2" d="M 200 145 Q 240 85 280 145" marker-end="url(#arrow-green)"/>

      
      <text id="tagJump1" class="jump-tag tag-1" x="120" y="70">+1</text>
      <text id="tagJump2" class="jump-tag tag-2" x="240" y="98">+½</text>

      
      <line x1="20" y1="145" x2="380" y2="145" stroke="#9ca3af" stroke-width="3" stroke-linecap="round" marker-end="url(#arrow)"/>

      
      
      <line x1="40" y1="135" x2="40" y2="155" stroke="#1e2945" stroke-width="2.5" stroke-linecap="round"/>
      <text x="40" y="172" class="tick-label whole-label">0</text>

      
      <line x1="120" y1="139" x2="120" y2="151" stroke="#9ca3af" stroke-width="2" stroke-linecap="round"/>
      <text x="120" y="168" class="tick-label">½</text>

      
      <line x1="200" y1="135" x2="200" y2="155" stroke="#1e2945" stroke-width="2.5" stroke-linecap="round"/>
      <text x="200" y="172" class="tick-label whole-label">1</text>

      
      <line x1="280" y1="139" x2="280" y2="151" stroke="#10b981" stroke-width="2" stroke-linecap="round"/>
      <text x="280" y="168" class="tick-label" style="fill:#059669; font-weight:700;">1½</text>

      
      <line x1="360" y1="135" x2="360" y2="155" stroke="#1e2945" stroke-width="2.5" stroke-linecap="round"/>
      <text x="360" y="172" class="tick-label whole-label">2</text>

      
      <g id="targetPin" class="target-pin">
        <circle cx="280" cy="145" r="7" fill="#10b981"/>
        <circle cx="280" cy="145" r="3" fill="#ffffff"/>
      </g>

      
      <circle id="posDot" class="marker-dot" cx="40" cy="145" r="5"/>
    </svg>
  </div>

  <div class="btn-group">
    <button class="step-btn active" id="btn0" onclick="setStep(0)">Start (0)</button>
    <button class="step-btn" id="btn1" onclick="setStep(1)">+ 1 Jump</button>
    <button class="step-btn" id="btn2" onclick="setStep(2)">+ ½ Step</button>
  </div>
</div>

<script>
  const equations = [
    "0",
    "0 + 1 = 1",
    "1 + \\frac{1}{2} = 1\\frac{1}{2}"
  ];

  const dotPositions = [40, 200, 280];
  let currentStep = 0;
  let autoTimer = null;

  function renderMath(step) {
    const el = document.getElementById('mathDisplay');
    katex.render(equations[step], el, { throwOnError: false });
  }

  function setStep(step, userAction = true) {
    currentStep = step;
    renderMath(step);

    // Update buttons
    [0, 1, 2].forEach(i => {
      document.getElementById('btn' + i).classList.toggle('active', i === step);
    });

    // Update Jump Visuals
    const j1 = document.getElementById('pathJump1');
    const tag1 = document.getElementById('tagJump1');
    const j2 = document.getElementById('pathJump2');
    const tag2 = document.getElementById('tagJump2');
    const pin = document.getElementById('targetPin');
    const dot = document.getElementById('posDot');

    if (step === 0) {
      j1.classList.remove('active');
      tag1.classList.remove('active');
      j2.classList.remove('active');
      tag2.classList.remove('active');
      pin.classList.remove('active');
      dot.setAttribute('cx', dotPositions[0]);
      dot.style.display = 'block';
    } else if (step === 1) {
      j1.classList.add('active');
      tag1.classList.add('active');
      j2.classList.remove('active');
      tag2.classList.remove('active');
      pin.classList.remove('active');
      dot.setAttribute('cx', dotPositions[1]);
      dot.style.display = 'block';
    } else if (step === 2) {
      j1.classList.add('active');
      tag1.classList.add('active');
      j2.classList.add('active');
      tag2.classList.add('active');
      pin.classList.add('active');
      dot.setAttribute('cx', dotPositions[2]);
    }

    if (userAction) {
      clearInterval(autoTimer);
    }
  }

  // Initial render
  renderMath(0);

  // Auto-play cycling with calm pacing
  autoTimer = setInterval(() => {
    const next = (currentStep + 1) % 3;
    setStep(next, false);
  }, 3200);
</script>

</body>
</html>

What if we count every single half-step starting from zero instead of taking one big jump?

An **improper fraction** has a top number that is equal to or bigger than the bottom number, like $\frac{3}{2}$.

To place $\frac{3}{2}$, count three half-steps from zero: one-half ($\frac{1}{2}$), two-halves ($\frac{2}{2} = 1$), and three-halves ($\frac{3}{2}$).

<!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>Placing 3/2 on a Number Line</title>
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      background: transparent;
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      justify-content: center;
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      border-radius: 20px;
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      gap: 6px;
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    }
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      background: #eef9ff;
      border-color: #22b7ff;
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      box-shadow: 0 2px 6px rgba(34,183,255,0.25);
    }
    .svg-container {
      width: 100%;
      background: #ffffff;
      border-radius: 12px;
      box-shadow: 0 2px 8px rgba(0,0,0,0.04);
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      width: 100%;
      height: auto;
      display: block;
      overflow: visible;
    }
    .hop-path {
      fill: none;
      stroke-width: 3;
      stroke-linecap: round;
      stroke-dasharray: 6,6;
      transition: stroke 0.3s ease, opacity 0.3s ease, stroke-dasharray 0.3s ease;
    }
    .hop-label {
      font-size: 13px;
      font-weight: 700;
      text-anchor: middle;
      transition: opacity 0.3s ease;
    }
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      transition: transform 0.4s cubic-bezier(0.34, 1.56, 0.64, 1);
    }
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      align-items: center;
      gap: 12px;
    }
    .play-btn {
      background: #ffffff;
      border: 1px solid #e2e8f0;
      border-radius: 20px;
      padding: 5px 12px;
      font-size: 13px;
      font-weight: 600;
      color: #1e2945;
      cursor: pointer;
      display: flex;
      align-items: center;
      gap: 4px;
    }
    .play-btn:hover {
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    }
  </style>
</head>
<body>

<div class="diagram-wrap">
  <div class="step-buttons">
    <button class="step-btn active" id="btn-0" onclick="setStep(0)">
      <span>1st half</span>
      <span>(+½)</span>
    </button>
    <button class="step-btn" id="btn-1" onclick="setStep(1)">
      <span>2nd half</span>
      <span>(²⁄₂ = 1)</span>
    </button>
    <button class="step-btn" id="btn-2" onclick="setStep(2)">
      <span>3rd half</span>
      <span>(³⁄₂ = 1½)</span>
    </button>
  </div>

  <div class="svg-container">
    <svg viewBox="0 0 560 210" aria-label="Number line showing halves to 3/2">
      <defs>
        
        <marker id="arrow" viewBox="0 0 10 10" refX="6" refY="5" markerWidth="6" markerHeight="6" orient="auto-start-reverse">
          <path d="M 0 1.5 L 8 5 L 0 8.5 z" fill="#94a3b8"/>
        </marker>
        <marker id="hop-arrow-blue" viewBox="0 0 10 10" refX="7" refY="5" markerWidth="6" markerHeight="6" orient="auto">
          <path d="M 0 2 L 7 5 L 0 8 z" fill="#22b7ff"/>
        </marker>
        <marker id="hop-arrow-orange" viewBox="0 0 10 10" refX="7" refY="5" markerWidth="6" markerHeight="6" orient="auto">
          <path d="M 0 2 L 7 5 L 0 8 z" fill="#f97316"/>
        </marker>
      </defs>

      
      <line x1="40" y1="130" x2="520" y2="130" stroke="#94a3b8" stroke-width="3" marker-start="url(#arrow)" marker-end="url(#arrow)"/>

      
      
      <line x1="70" y1="114" x2="70" y2="146" stroke="#1e2945" stroke-width="3" stroke-linecap="round"/>
      <text x="70" y="170" text-anchor="middle" font-size="16" font-weight="700" fill="#1e2945">0</text>

      
      <line x1="175" y1="118" x2="175" y2="142" stroke="#64748b" stroke-width="2" stroke-linecap="round"/>
      <text x="175" y="165" text-anchor="middle" font-size="14" font-weight="600" fill="#64748b">½</text>

      
      <line x1="280" y1="114" x2="280" y2="146" stroke="#1e2945" stroke-width="3" stroke-linecap="round"/>
      <text x="280" y="170" text-anchor="middle" font-size="16" font-weight="700" fill="#1e2945">1</text>
      <text x="280" y="190" text-anchor="middle" font-size="12" font-weight="600" fill="#94a3b8">²⁄₂</text>

      
      <line x1="385" y1="118" x2="385" y2="142" stroke="#f97316" stroke-width="2.5" stroke-linecap="round"/>
      <text x="385" y="166" text-anchor="middle" font-size="15" font-weight="800" fill="#f97316">³⁄₂</text>
      <text x="385" y="190" text-anchor="middle" font-size="13" font-weight="700" fill="#ea580c">1½</text>

      
      <line x1="490" y1="114" x2="490" y2="146" stroke="#1e2945" stroke-width="3" stroke-linecap="round"/>
      <text x="490" y="170" text-anchor="middle" font-size="16" font-weight="700" fill="#1e2945">2</text>
      <text x="490" y="190" text-anchor="middle" font-size="12" font-weight="600" fill="#94a3b8">⁴⁄₂</text>

      
      <path id="hop-0" class="hop-path" d="M 70 120 Q 122.5 50 175 120" stroke="#22b7ff" marker-end="url(#hop-arrow-blue)" />
      <text id="hop-lbl-0" class="hop-label" x="122.5" y="65" fill="#0284c7">+½</text>

      
      <path id="hop-1" class="hop-path" d="M 175 120 Q 227.5 50 280 120" stroke="#22b7ff" marker-end="url(#hop-arrow-blue)" />
      <text id="hop-lbl-1" class="hop-label" x="227.5" y="65" fill="#0284c7">+½</text>

      
      <path id="hop-2" class="hop-path" d="M 280 120 Q 332.5 50 385 120" stroke="#f97316" marker-end="url(#hop-arrow-orange)" />
      <text id="hop-lbl-2" class="hop-label" x="332.5" y="65" fill="#ea580c">+½</text>

      
      <g id="target-tag" opacity="0" style="transition: opacity 0.3s ease;">
        <rect x="340" y="8" width="90" height="26" rx="13" fill="#ffedd5" stroke="#f97316" stroke-width="1.5"/>
        <text x="385" y="25" text-anchor="middle" font-size="12" font-weight="800" fill="#c2410c">³⁄₂ = 1½</text>
      </g>

      
      <g id="marker" class="marker-group" transform="translate(175, 130)">
        <circle r="9" fill="#22b7ff" stroke="#ffffff" stroke-width="2.5" />
        <circle r="14" fill="none" stroke="#22b7ff" stroke-width="1.5" opacity="0.4" />
      </g>
    </svg>
  </div>

  <div class="control-bar">
    <button class="play-btn" id="playBtn" onclick="toggleAuto()">
      <span id="playIcon">⏸</span>
      <span id="playText">Auto</span>
    </button>
  </div>
</div>

<script>
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  let isPlaying = true;
  let timer = null;

  const positions = [175, 280, 385];

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    if (step === 2) {
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      ringCircle.setAttribute('stroke', '#f97316');
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  // Initialize
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  startTimer();
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</body>
</html>

Because they name the same total distance, $1\frac{1}{2}$ and $\frac{3}{2}$ share the exact same place on the number line!

Can we use the same rule when spaces are cut into four equal pieces called fourths?

When each whole space is split into **4 equal parts**, each tick mark is one fourth ($\frac{1}{4}$).

To find $2\frac{1}{4}$, jump straight to **2**, and then hop forward **1 small fourth-step**.

<!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>Placing Mixed Numbers on a Number Line</title>
  <link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.16.8/dist/katex.min.css">
  <script defer src="https://cdn.jsdelivr.net/npm/katex@0.16.8/dist/katex.min.js"></script>
  <style>
    * { box-sizing: border-box; margin: 0; padding: 0; }
    body {
      font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, Helvetica, Arial, sans-serif;
      background: #f8f9fa;
      color: #1e2945;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      min-height: 100%;
      padding: 16px;
      user-select: none;
      -webkit-user-select: none;
    }
    .stage {
      width: 100%;
      max-width: 600px;
      display: flex;
      flex-direction: column;
      align-items: center;
      gap: 16px;
    }
    .display-card {
      background: #ffffff;
      border-radius: 12px;
      padding: 12px 24px;
      box-shadow: 0 2px 8px rgba(0,0,0,0.06);
      display: flex;
      align-items: center;
      gap: 12px;
      font-size: 20px;
      font-weight: 600;
      min-height: 54px;
    }
    .target-badge {
      background: #f3e8ff;
      color: #7c3aed;
      padding: 4px 12px;
      border-radius: 8px;
      display: inline-flex;
      align-items: center;
    }
    .svg-container {
      width: 100%;
      background: #ffffff;
      border-radius: 12px;
      box-shadow: 0 2px 8px rgba(0,0,0,0.06);
      padding: 20px 10px;
      position: relative;
    }
    svg {
      width: 100%;
      height: auto;
      display: block;
      overflow: visible;
    }
    .controls {
      display: flex;
      gap: 8px;
      flex-wrap: wrap;
      justify-content: center;
    }
    .btn {
      background: #ffffff;
      border: 1px solid #e2e8f0;
      padding: 8px 16px;
      border-radius: 8px;
      font-size: 14px;
      font-weight: 600;
      color: #1e2945;
      cursor: pointer;
      display: inline-flex;
      align-items: center;
      gap: 6px;
      transition: background 0.2s, border-color 0.2s;
    }
    .btn:hover {
      background: #f1f5f9;
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    }
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      border-color: #8b5cf6;
      color: #ffffff;
    }
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      stroke-linecap: round;
      transition: stroke-dashoffset 0.6s cubic-bezier(0.4, 0, 0.2, 1), opacity 0.3s;
    }
    .arc-whole {
      stroke: #8b5cf6;
      stroke-width: 3.5;
    }
    .arc-part {
      stroke: #10b981;
      stroke-width: 3.5;
    }
    .label-tag {
      font-size: 13px;
      font-weight: 700;
      transition: opacity 0.3s;
    }
    .marker-dot {
      transition: transform 0.4s cubic-bezier(0.34, 1.56, 0.64, 1), opacity 0.3s;
    }
    .space-rect {
      fill: #10b981;
      opacity: 0;
      transition: opacity 0.3s;
    }
  </style>
</head>
<body>
  <div class="stage">
    <div class="display-card">
      <span>Target:</span>
      <span class="target-badge" id="mathTarget"></span>
    </div>

    <div class="svg-container">
      <svg viewBox="0 0 540 180">
        
        <g id="spacesGroup">
          <rect class="space-rect" id="space1" x="370" y="105" width="40" height="20" rx="4" opacity="0" />
        </g>

        
        <line x1="40" y1="120" x2="510" y2="120" stroke="#cbd5e1" stroke-width="4" stroke-linecap="round" />
        
        <polygon points="515,120 505,115 505,125" fill="#cbd5e1" />
        <polygon points="35,120 45,115 45,125" fill="#cbd5e1" />

        
        
        
        
        
        
        <g stroke="#94a3b8" stroke-width="2">
          
          <line x1="95" y1="112" x2="95" y2="128" />
          <line x1="130" y1="112" x2="130" y2="128" />
          <line x1="165" y1="112" x2="165" y2="128" />
          
          <line x1="235" y1="112" x2="235" y2="128" />
          <line x1="270" y1="112" x2="270" y2="128" />
          <line x1="305" y1="112" x2="305" y2="128" />
          
          <line x1="375" y1="112" x2="375" y2="128" />
          <line x1="410" y1="112" x2="410" y2="128" />
          <line x1="445" y1="112" x2="445" y2="128" />
        </g>

        
        <g stroke="#1e2945" stroke-width="3">
          <line x1="60" y1="105" x2="60" y2="135" />
          <line x1="200" y1="105" x2="200" y2="135" />
          <line x1="340" y1="105" x2="340" y2="135" />
          <line x1="480" y1="105" x2="480" y2="135" />
        </g>

        
        <text x="60" y="155" font-size="16" font-weight="700" text-anchor="middle" fill="#1e2945">0</text>
        <text x="200" y="155" font-size="16" font-weight="700" text-anchor="middle" fill="#1e2945">1</text>
        <text x="340" y="155" font-size="16" font-weight="700" text-anchor="middle" fill="#1e2945">2</text>
        <text x="480" y="155" font-size="16" font-weight="700" text-anchor="middle" fill="#1e2945">3</text>

        
        <text x="375" y="148" font-size="11" font-weight="600" text-anchor="middle" fill="#10b981" id="subLabel">2¼</text>

        
        <path id="arcWhole" class="arc arc-whole" d="M 60,115 Q 200,30 340,115" stroke-dasharray="320" stroke-dashoffset="320" />
        <g id="wholeTag" opacity="0">
          <rect x="175" y="32" width="50" height="24" rx="12" fill="#8b5cf6" />
          <text x="200" y="48" font-size="12" font-weight="700" text-anchor="middle" fill="#ffffff">+ 2</text>
        </g>

        
        <path id="arcPart" class="arc arc-part" d="M 340,115 Q 357.5,75 375,115" stroke-dasharray="70" stroke-dashoffset="70" />
        <g id="partTag" opacity="0">
          <rect x="343" y="60" width="30" height="20" rx="6" fill="#10b981" />
          <text x="358" y="74" font-size="11" font-weight="700" text-anchor="middle" fill="#ffffff">+¼</text>
        </g>

        
        <g id="pinMarker" transform="translate(60, 120)" class="marker-dot">
          <circle cx="0" cy="0" r="7" fill="#8b5cf6" stroke="#ffffff" stroke-width="2.5" />
        </g>
      </svg>
    </div>

    
    <div class="controls">
      <button class="btn active" id="btn1" onclick="setStep(1)">1. Start at 0</button>
      <button class="btn" id="btn2" onclick="setStep(2)">2. Jump to 2</button>
      <button class="btn" id="btn3" onclick="setStep(3)">3. Hop +¼</button>
    </div>
  </div>

  <script>
    function renderMath() {
      if (window.katex) {
        katex.render("2\\frac{1}{4}", document.getElementById('mathTarget'));
      } else {
        document.getElementById('mathTarget').textContent = "2 ¼";
      }
    }
    window.addEventListener('load', renderMath);

    const arcWhole = document.getElementById('arcWhole');
    const wholeTag = document.getElementById('wholeTag');
    const arcPart = document.getElementById('arcPart');
    const partTag = document.getElementById('partTag');
    const pin = document.getElementById('pinMarker');
    const space1 = document.getElementById('space1');
    const buttons = [
      document.getElementById('btn1'),
      document.getElementById('btn2'),
      document.getElementById('btn3')
    ];

    let currentStep = 1;
    let autoTimer = null;

    function setStep(step, manual = true) {
      if (manual && autoTimer) {
        clearInterval(autoTimer);
        autoTimer = null;
      }
      currentStep = step;

      buttons.forEach((b, i) => {
        if (i + 1 === step) b.classList.add('active');
        else b.classList.remove('active');
      });

      if (step === 1) {
        arcWhole.style.strokeDashoffset = '320';
        wholeTag.style.opacity = '0';
        arcPart.style.strokeDashoffset = '70';
        partTag.style.opacity = '0';
        space1.style.opacity = '0';
        pin.setAttribute('transform', 'translate(60, 120)');
        pin.querySelector('circle').setAttribute('fill', '#8b5cf6');
      } else if (step === 2) {
        arcWhole.style.strokeDashoffset = '0';
        wholeTag.style.opacity = '1';
        arcPart.style.strokeDashoffset = '70';
        partTag.style.opacity = '0';
        space1.style.opacity = '0';
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        pin.querySelector('circle').setAttribute('fill', '#8b5cf6');
      } else if (step === 3) {
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        partTag.style.opacity = '1';
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        pin.querySelector('circle').setAttribute('fill', '#10b981');
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    }

    // Gentle auto-cycle through steps for initial demonstration
    let cycleStep = 1;
    autoTimer = setInterval(() => {
      cycleStep = (cycleStep % 3) + 1;
      setStep(cycleStep, false);
    }, 3200);
  </script>
</body>
</html>

Remember to count the **spaces** between the marks, not just the little lines, so you always know how many equal parts make one whole.

Next time you see a number line, look at how many spaces split each whole number, make your jumps, and spot your landing mark!

math

Placing Fractions and Mixed Numbers on a Number Line. Accurately position mixed numbers and improper fractions along a continuous number line.

Fractions and Mixed Numbers on the Number Line

Fractions and Mixed Numbers on Number Lines

Improper Fractions and Mixed Numbers on a Number Line Simulation

Pedagogical strategy focuses on concrete-pictorial-abstract (CPA) progression aligned with Curriculum for Excellence MTH 3-07c: Start with partitioned physical strips or Cuisenaire rods placed along a measuring tape to bridge linear measurement and fraction concepts. Explicitly teach students to identify the 'unit interval' (distance between consecutive integers) before counting subdivisions to prevent tick-counting errors. Emphasize dual-labeling (plotting both improper fraction $\frac{p}{q}$ and mixed number $a\frac{b}{q}$ at the identical coordinate) to develop conceptual understanding of equivalence and conversion.

Placing Fractions and Mixed Numbers on a Number Line

Mixed Numbers on the Line

Counting Fraction Steps