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Imagine you have half of a giant chocolate chip cookie, and you want to give a third of your piece to a friend. How much of the whole cookie does your friend get?

Finding a fraction of another fraction means we multiply them: $\frac{1}{3} \times \frac{1}{2}$.

<!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>Multiplying Proper Fractions</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, sans-serif;
      background: #f8f9fa;
      color: #1e2945;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: space-between;
      min-height: 100%;
      padding: 16px;
    }

    .equation-bar {
      display: flex;
      align-items: center;
      justify-content: center;
      gap: 8px;
      font-size: 1.25rem;
      background: #ffffff;
      padding: 10px 20px;
      border-radius: 12px;
      box-shadow: 0 2px 8px rgba(0,0,0,0.06);
      min-height: 52px;
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    .math-term {
      padding: 2px 6px;
      border-radius: 6px;
      transition: background 0.3s ease, color 0.3s ease;
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    .term-half.active { background: #e0f2fe; color: #0284c7; }
    .term-third.active { background: #fef3c7; color: #d97706; }
    .term-result.active { background: #dcfce7; color: #15803d; font-weight: bold; }

    .stage-container {
      position: relative;
      width: 240px;
      height: 240px;
      margin: 12px auto;
      background: #ffffff;
      border: 2px solid #cbd5e1;
      border-radius: 12px;
      overflow: hidden;
      box-shadow: 0 4px 12px rgba(0,0,0,0.05);
    }

    /* Grid layout: 3 rows, 2 cols = 6 cells */
    .grid-mesh {
      position: absolute;
      inset: 0;
      display: grid;
      grid-template-columns: 1fr 1fr;
      grid-template-rows: 1fr 1fr 1fr;
    }

    .cell {
      border: 1px dashed #cbd5e1;
      transition: background 0.4s ease, transform 0.2s ease;
      position: relative;
    }

    /* Step states */
    /* Step 1: 1/2 vertical highlight */
    .highlight-col {
      position: absolute;
      top: 0;
      bottom: 0;
      left: 0;
      width: 50%;
      background: rgba(34, 183, 255, 0.28);
      border-right: 2px solid #0284c7;
      opacity: 0;
      transition: opacity 0.4s ease;
      pointer-events: none;
    }

    /* Step 2: 1/3 horizontal highlight */
    .highlight-row {
      position: absolute;
      top: 0;
      left: 0;
      right: 0;
      height: 33.333%;
      background: rgba(245, 158, 11, 0.28);
      border-bottom: 2px solid #d97706;
      opacity: 0;
      transition: opacity 0.4s ease;
      pointer-events: none;
    }

    /* Step 3: Overlap tile */
    .overlap-tile {
      position: absolute;
      top: 0;
      left: 0;
      width: 50%;
      height: 33.333%;
      background: rgba(16, 185, 129, 0.55);
      border: 2px solid #10b981;
      opacity: 0;
      transition: opacity 0.4s ease;
      display: flex;
      align-items: center;
      justify-content: center;
      font-weight: 700;
      color: #064e3b;
      font-size: 1.1rem;
      pointer-events: none;
    }

    .dim-label {
      position: absolute;
      font-size: 0.85rem;
      font-weight: 600;
      color: #475569;
      opacity: 0;
      transition: opacity 0.3s ease;
    }
    .dim-bottom {
      bottom: -28px;
      left: 25%;
      transform: translateX(-50%);
    }
    .dim-left {
      left: -32px;
      top: 16.666%;
      transform: translateY(-50%);
    }

    /* Step Controls */
    .controls {
      display: flex;
      flex-direction: column;
      align-items: center;
      gap: 12px;
      width: 100%;
      max-width: 380px;
    }

    .step-buttons {
      display: flex;
      width: 100%;
      gap: 6px;
      justify-content: center;
    }

    .step-btn {
      flex: 1;
      padding: 8px 4px;
      font-size: 0.85rem;
      font-weight: 600;
      border: 1px solid #e2e8f0;
      background: #ffffff;
      color: #64748b;
      border-radius: 8px;
      cursor: pointer;
      text-align: center;
      transition: all 0.2s ease;
      user-select: none;
    }

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

    .nav-row {
      display: flex;
      align-items: center;
      justify-content: space-between;
      width: 100%;
      padding: 0 4px;
    }

    .nav-btn {
      background: #ffffff;
      border: 1px solid #cbd5e1;
      border-radius: 8px;
      padding: 6px 14px;
      font-size: 0.9rem;
      font-weight: 600;
      cursor: pointer;
      color: #334155;
      display: flex;
      align-items: center;
      gap: 4px;
    }
    .nav-btn:hover { background: #f1f5f9; }
    .nav-btn:disabled { opacity: 0.4; cursor: not-allowed; }

    .play-btn {
      background: #f1f5f9;
      border: none;
      border-radius: 20px;
      padding: 6px 16px;
      font-size: 0.85rem;
      font-weight: 600;
      cursor: pointer;
      color: #475569;
    }

    .cell-count {
      position: absolute;
      font-size: 0.75rem;
      color: #94a3b8;
      font-weight: 600;
      bottom: 4px;
      right: 6px;
      opacity: 0;
      transition: opacity 0.3s ease;
    }
  </style>
</head>
<body>

  
  <div class="equation-bar">
    <span class="math-term term-third" id="eq-third"></span>
    <span>&times;</span>
    <span class="math-term term-half" id="eq-half"></span>
    <span>=</span>
    <span class="math-term term-result" id="eq-result">?</span>
  </div>

  
  <div style="position: relative;">
    <div class="stage-container" id="grid-stage">
      <div class="grid-mesh">
        <div class="cell"><span class="cell-count">1</span></div>
        <div class="cell"><span class="cell-count">2</span></div>
        <div class="cell"><span class="cell-count">3</span></div>
        <div class="cell"><span class="cell-count">4</span></div>
        <div class="cell"><span class="cell-count">5</span></div>
        <div class="cell"><span class="cell-count">6</span></div>
      </div>

      <div class="highlight-col" id="col-half"></div>
      <div class="highlight-row" id="row-third"></div>
      <div class="overlap-tile" id="tile-overlap">
        <span id="overlap-label"></span>
      </div>
    </div>
  </div>

  
  <div class="controls">
    <div class="step-buttons">
      <button class="step-btn active" onclick="setStep(0)">1. Whole</button>
      <button class="step-btn" onclick="setStep(1)">2. Half</button>
      <button class="step-btn" onclick="setStep(2)">3. Third</button>
      <button class="step-btn" onclick="setStep(3)">4. Product</button>
    </div>

    <div class="nav-row">
      <button class="nav-btn" id="prev-btn" onclick="prevStep()">←</button>
      <button class="play-btn" id="play-btn" onclick="toggleAutoPlay()">▶ Auto</button>
      <button class="nav-btn" id="next-btn" onclick="nextStep()">→</button>
    </div>
  </div>

  <script>
    let currentStep = 0;
    const totalSteps = 4;
    let autoInterval = null;

    function renderMath() {
      if (window.katex) {
        katex.render("\\frac{1}{3}", document.getElementById('eq-third'));
        katex.render("\\frac{1}{2}", document.getElementById('eq-half'));
        katex.render("\\frac{1}{6}", document.getElementById('overlap-label'));
      } else {
        setTimeout(renderMath, 100);
      }
    }

    function updateView() {
      // Step buttons
      const buttons = document.querySelectorAll('.step-btn');
      buttons.forEach((btn, idx) => {
        btn.classList.toggle('active', idx === currentStep);
      });

      // Elements
      const colHalf = document.getElementById('col-half');
      const rowThird = document.getElementById('row-third');
      const overlap = document.getElementById('tile-overlap');
      const eqThird = document.getElementById('eq-third');
      const eqHalf = document.getElementById('eq-half');
      const eqResult = document.getElementById('eq-result');
      const cellCounts = document.querySelectorAll('.cell-count');

      // Reset
      colHalf.style.opacity = '0';
      rowThird.style.opacity = '0';
      overlap.style.opacity = '0';
      eqThird.classList.remove('active');
      eqHalf.classList.remove('active');
      eqResult.classList.remove('active');
      cellCounts.forEach(c => c.style.opacity = '0');

      if (window.katex) {
        if (currentStep === 3) {
          katex.render("\\frac{1}{6}", eqResult);
        } else {
          eqResult.textContent = '?';
        }
      }

      if (currentStep === 0) {
        // Step 1: 1 Whole
      } else if (currentStep === 1) {
        // Step 2: Show 1/2 vertical
        colHalf.style.opacity = '1';
        eqHalf.classList.add('active');
      } else if (currentStep === 2) {
        // Step 3: Show 1/3 horizontal on top of 1/2
        colHalf.style.opacity = '1';
        rowThird.style.opacity = '1';
        eqHalf.classList.add('active');
        eqThird.classList.add('active');
      } else if (currentStep === 3) {
        // Step 4: Overlap product 1/6
        colHalf.style.opacity = '0.35';
        rowThird.style.opacity = '0.35';
        overlap.style.opacity = '1';
        eqThird.classList.add('active');
        eqHalf.classList.add('active');
        eqResult.classList.add('active');
        cellCounts.forEach(c => c.style.opacity = '1');
      }

      document.getElementById('prev-btn').disabled = (currentStep === 0);
      document.getElementById('next-btn').disabled = (currentStep === totalSteps - 1);
    }

    function setStep(step) {
      currentStep = step;
      updateView();
    }

    function nextStep() {
      if (currentStep < totalSteps - 1) {
        currentStep++;
        updateView();
      }
    }

    function prevStep() {
      if (currentStep > 0) {
        currentStep--;
        updateView();
      }
    }

    function toggleAutoPlay() {
      const playBtn = document.getElementById('play-btn');
      if (autoInterval) {
        clearInterval(autoInterval);
        autoInterval = null;
        playBtn.textContent = '▶ Auto';
      } else {
        playBtn.textContent = '⏸ Pause';
        autoInterval = setInterval(() => {
          currentStep = (currentStep + 1) % totalSteps;
          updateView();
        }, 3200);
      }
    }

    window.addEventListener('DOMContentLoaded', () => {
      renderMath();
      updateView();
    });
  </script>
</body>
</html>

What happens when our fractions have bigger numbers on top and bottom?

To multiply fractions, multiply the top numbers (the **numerators**) together, and multiply the bottom numbers (the **denominators**) together.

For example, if you plant flowers in $\frac{2}{3}$ of a garden bed and $\frac{3}{4}$ of those flowers are roses, you multiply $\frac{2}{3} \times \frac{3}{4} = \frac{2 \times 3}{3 \times 4} = \frac{6}{12}$, which simplifies to $\frac{1}{2}$.

Multiplying big numbers and then simplifying takes extra work. Can we make the numbers smaller before we even multiply?

**Cross-canceling** means dividing a top number and a bottom number by their shared factor before multiplying.

In $\frac{2}{3} \times \frac{3}{4}$, the top $3$ and bottom $3$ divide by $3$ to become $1$. The top $2$ and bottom $4$ divide by $2$ to become $1$ and $2$.

<!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>Cross-Cancelling Fractions</title>
  <style>
    * { box-sizing: border-box; margin: 0; padding: 0; }
    body {
      font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, Helvetica, Arial, sans-serif;
      background: transparent;
      color: #1e2945;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      min-height: 100%;
      padding: 12px;
      user-select: none;
    }

    .diagram-wrap {
      width: 100%;
      max-width: 480px;
      display: flex;
      flex-direction: column;
      align-items: center;
      gap: 16px;
    }

    /* Controls Bar */
    .controls {
      display: flex;
      align-items: center;
      justify-content: center;
      gap: 8px;
      width: 100%;
    }
    .btn-step {
      border: 1px solid #e2e8f0;
      background: #ffffff;
      color: #64748b;
      font-size: 13px;
      font-weight: 600;
      padding: 6px 14px;
      border-radius: 20px;
      cursor: pointer;
      box-shadow: 0 1px 2px rgba(0,0,0,0.04);
      transition: all 0.25s ease;
    }
    .btn-step.active {
      background: #0284c7;
      color: #ffffff;
      border-color: #0284c7;
      box-shadow: 0 2px 6px rgba(2, 132, 199, 0.3);
    }
    .play-btn {
      width: 32px;
      height: 32px;
      border-radius: 50%;
      border: 1px solid #e2e8f0;
      background: #ffffff;
      color: #1e2945;
      display: flex;
      align-items: center;
      justify-content: center;
      cursor: pointer;
      font-size: 12px;
    }

    /* Equation Board */
    .board {
      position: relative;
      background: #ffffff;
      border-radius: 16px;
      padding: 24px 20px;
      width: 100%;
      display: flex;
      align-items: center;
      justify-content: center;
      gap: 14px;
      box-shadow: 0 4px 16px rgba(0, 0, 0, 0.05);
      border: 1px solid #f1f5f9;
    }

    /* SVG Cross Overlay */
    .connector-svg {
      position: absolute;
      top: 0;
      left: 0;
      width: 100%;
      height: 100%;
      pointer-events: none;
      z-index: 1;
    }
    .cross-line {
      stroke-dasharray: 60;
      stroke-dashoffset: 60;
      transition: stroke-dashoffset 0.4s ease, opacity 0.3s ease;
      opacity: 0;
    }
    .cross-line.active {
      stroke-dashoffset: 0;
      opacity: 0.35;
    }

    /* Fraction Styles */
    .fraction {
      display: inline-flex;
      flex-direction: column;
      align-items: center;
      vertical-align: middle;
      font-size: 26px;
      font-weight: 700;
      line-height: 1;
      position: relative;
      z-index: 2;
    }
    .numerator, .denominator {
      position: relative;
      width: 38px;
      height: 38px;
      display: flex;
      align-items: center;
      justify-content: center;
    }
    .fraction-bar {
      width: 100%;
      height: 3px;
      background: #1e2945;
      border-radius: 2px;
      margin: 2px 0;
    }
    .operator {
      font-size: 22px;
      font-weight: 700;
      color: #94a3b8;
      z-index: 2;
    }

    /* Number & Strike Elements */
    .num-val {
      position: absolute;
      transition: opacity 0.3s ease, transform 0.3s ease, color 0.3s ease;
    }
    .strike-line {
      position: absolute;
      width: 0%;
      height: 3px;
      border-radius: 2px;
      transform: rotate(-35deg);
      transition: width 0.3s cubic-bezier(0.4, 0, 0.2, 1);
    }

    /* Simplified replacement tag */
    .sub-tag {
      position: absolute;
      font-size: 15px;
      font-weight: 800;
      border-radius: 6px;
      padding: 1px 6px;
      opacity: 0;
      transform: scale(0.5);
      transition: all 0.3s cubic-bezier(0.34, 1.56, 0.64, 1);
    }
    .tag-top { top: -14px; }
    .tag-bottom { bottom: -14px; }

    /* Colors */
    .color-pair-a { color: #8b5cf6; } /* 3 & 3 */
    .color-pair-b { color: #0284c7; } /* 2 & 4 */

    .strike-a { background: #8b5cf6; }
    .strike-b { background: #0284c7; }

    .tag-a { background: #ede9fe; color: #7c3aed; }
    .tag-b { background: #e0f2fe; color: #0284c7; }

    /* Active Strike / Cancel States */
    .cancelled .strike-line { width: 34px; }
    .cancelled .num-val { opacity: 0.35; }
    .cancelled .sub-tag { opacity: 1; transform: scale(1); }

    /* Result Fraction */
    .result-group {
      display: flex;
      align-items: center;
      gap: 12px;
      opacity: 0;
      transform: translateX(6px);
      transition: all 0.4s ease;
    }
    .result-group.visible {
      opacity: 1;
      transform: translateX(0);
    }
    .result-frac .numerator { color: #059669; }
    .result-frac .denominator { color: #059669; }
    .result-frac .fraction-bar { background: #059669; }

    /* Visual Grid Diagram */
    .visual-area {
      background: #ffffff;
      border-radius: 12px;
      padding: 12px 18px;
      display: flex;
      align-items: center;
      justify-content: center;
      gap: 16px;
      width: 100%;
      border: 1px solid #f1f5f9;
      box-shadow: 0 2px 8px rgba(0,0,0,0.03);
    }
    .grid-box {
      width: 112px;
      height: 64px;
      display: grid;
      grid-template-columns: repeat(4, 1fr);
      grid-template-rows: repeat(3, 1fr);
      border: 2px solid #cbd5e1;
      border-radius: 6px;
      overflow: hidden;
      background: #f8fafc;
    }
    .cell {
      border: 0.5px solid #e2e8f0;
      transition: background 0.3s ease;
    }
    .cell.in-col { background: #f1f5f9; }
    .cell.overlap { background: #10b981; }

    .grid-label {
      font-size: 15px;
      font-weight: 700;
      color: #334155;
      display: flex;
      align-items: center;
      gap: 6px;
    }
    .badge-pill {
      font-size: 12px;
      padding: 2px 8px;
      border-radius: 12px;
      background: #ecfdf5;
      color: #059669;
      font-weight: 700;
    }
  </style>
</head>
<body>

  <div class="diagram-wrap">
    
    <div class="controls">
      <button class="btn-step active" id="btn0">1. Original</button>
      <button class="btn-step" id="btn1">2. ÷ 3</button>
      <button class="btn-step" id="btn2">3. ÷ 2</button>
      <button class="btn-step" id="btn3">4. Result</button>
      <button class="play-btn" id="playBtn" title="Play / Pause">⏸</button>
    </div>

    
    <div class="board" id="board">
      <svg class="connector-svg" id="svgLayer">
        
        <line id="lineA" class="cross-line" stroke="#8b5cf6" stroke-width="2.5" stroke-linecap="round"/>
        
        <line id="lineB" class="cross-line" stroke="#0284c7" stroke-width="2.5" stroke-linecap="round"/>
      </svg>

      
      <div class="fraction">
        <div class="numerator" id="node2">
          <span class="num-val">2</span>
          <span class="strike-line strike-b"></span>
          <span class="sub-tag tag-b tag-top">1</span>
        </div>
        <div class="fraction-bar"></div>
        <div class="denominator" id="node3b">
          <span class="num-val">3</span>
          <span class="strike-line strike-a"></span>
          <span class="sub-tag tag-a tag-bottom">1</span>
        </div>
      </div>

      <div class="operator">×</div>

      
      <div class="fraction">
        <div class="numerator" id="node3t">
          <span class="num-val">3</span>
          <span class="strike-line strike-a"></span>
          <span class="sub-tag tag-a tag-top">1</span>
        </div>
        <div class="fraction-bar"></div>
        <div class="denominator" id="node4">
          <span class="num-val">4</span>
          <span class="strike-line strike-b"></span>
          <span class="sub-tag tag-b tag-bottom">2</span>
        </div>
      </div>

      
      <div class="result-group" id="resultGroup">
        <div class="operator">=</div>
        <div class="fraction result-frac">
          <div class="numerator">1</div>
          <div class="fraction-bar"></div>
          <div class="denominator">2</div>
        </div>
      </div>
    </div>

    
    <div class="visual-area">
      <div class="grid-box" id="gridBox">
        
        <div class="cell" id="c0"></div><div class="cell" id="c1"></div><div class="cell" id="c2"></div><div class="cell" id="c3"></div>
        <div class="cell" id="c4"></div><div class="cell" id="c5"></div><div class="cell" id="c6"></div><div class="cell" id="c7"></div>
        <div class="cell" id="c8"></div><div class="cell" id="c9"></div><div class="cell" id="c10"></div><div class="cell" id="c11"></div>
      </div>
      <div class="grid-label">
        <span id="gridText">2/3 of 3/4</span>
        <span class="badge-pill" id="gridBadge">6/12 = 1/2</span>
      </div>
    </div>
  </div>

  <script src="{{VENDOR_CDN}}/classace-diagram.js"></script>
  <script>
    const node2 = $('#node2');
    const node3b = $('#node3b');
    const node3t = $('#node3t');
    const node4 = $('#node4');
    const lineA = $('#lineA');
    const lineB = $('#lineB');
    const resultGroup = $('#resultGroup');
    const board = $('#board');

    // Update connector line coordinates relative to board
    function updateConnectors() {
      const bRect = board.getBoundingClientRect();
      const r2 = node2.getBoundingClientRect();
      const r3b = node3b.getBoundingClientRect();
      const r3t = node3t.getBoundingClientRect();
      const r4 = node4.getBoundingClientRect();

      // Pair A: 3 (bottom-left) to 3 (top-right)
      lineA.setAttribute('x1', (r3b.left + r3b.width/2) - bRect.left);
      lineA.setAttribute('y1', (r3b.top + r3b.height/2) - bRect.top);
      lineA.setAttribute('x2', (r3t.left + r3t.width/2) - bRect.left);
      lineA.setAttribute('y2', (r3t.top + r3t.height/2) - bRect.top);

      // Pair B: 2 (top-left) to 4 (bottom-right)
      lineB.setAttribute('x1', (r2.left + r2.width/2) - bRect.left);
      lineB.setAttribute('y1', (r2.top + r2.height/2) - bRect.top);
      lineB.setAttribute('x2', (r4.left + r4.width/2) - bRect.left);
      lineB.setAttribute('y2', (r4.top + r4.height/2) - bRect.top);
    }

    // Grid cells overlap indices: columns 0, 1, 2 (first 3 cols) and rows 0, 1 (top 2 rows)
    // Row 0: c0, c1, c2, c3
    // Row 1: c4, c5, c6, c7
    // Row 2: c8, c9, c10, c11
    const overlapCells = [0, 1, 2, 4, 5, 6];
    const col3Cells = [0, 1, 2, 4, 5, 6, 8, 9, 10];

    function setGridState(step) {
      for (let i = 0; i < 12; i++) {
        const el = $('#c' + i);
        el.className = 'cell';
        if (step === 0) {
          // clean
        } else if (step === 1 || step === 2) {
          if (col3Cells.includes(i)) el.classList.add('in-col');
          if (overlapCells.includes(i)) el.classList.add('overlap');
        } else if (step === 3) {
          if (overlapCells.includes(i)) el.classList.add('overlap');
        }
      }
    }

    function applyStep(index) {
      $$('.btn-step').forEach((b, i) => b.classList.toggle('active', i === index));

      // Reset states
      node3t.classList.toggle('cancelled', index >= 1);
      node3b.classList.toggle('cancelled', index >= 1);
      lineA.classList.toggle('active', index === 1);

      node2.classList.toggle('cancelled', index >= 2);
      node4.classList.toggle('cancelled', index >= 2);
      lineB.classList.toggle('active', index === 2);

      resultGroup.classList.toggle('visible', index === 3);

      setGridState(index);
    }

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

    function goTo(idx) {
      currentStep = idx;
      applyStep(currentStep);
    }

    function next() {
      currentStep = (currentStep + 1) % 4;
      applyStep(currentStep);
    }

    function startAuto() {
      isPlaying = true;
      $('#playBtn').textContent = '⏸';
      if (timer) clearInterval(timer);
      timer = setInterval(next, 3200);
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      isPlaying = false;
      $('#playBtn').textContent = '▶';
      if (timer) clearInterval(timer);
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    // Bind Controls
    $$('.btn-step').forEach((btn, idx) => {
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        pauseAuto();
        goTo(idx);
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      else {
        next();
        startAuto();
      }
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    window.addEventListener('resize', updateConnectors);
    window.addEventListener('load', () => {
      updateConnectors();
      goTo(0);
      startAuto();
    });
    setTimeout(updateConnectors, 50);
  </script>
</body>
</html>

How do we use this trick to solve a real recipe problem with larger numbers?

A baker has $\frac{5}{6}$ cup of sugar in a bowl and uses $\frac{3}{10}$ of that sugar to make glaze. How much sugar goes into the glaze?

We write the problem as $\frac{3}{10} \times \frac{5}{6}$. Cross-canceling $3$ and $6$ gives $1$ and $2$, while cross-canceling $5$ and $10$ gives $1$ and $2$, leaving $\frac{1 \times 1}{2 \times 2} = \frac{1}{4}$ cup.

<!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>Cross-Canceling Fractions</title>
  <style>
    * {
      box-sizing: border-box;
      margin: 0;
      padding: 0;
      font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
      user-select: none;
    }

    body {
      background: #f8f9fa;
      color: #1e2945;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: space-between;
      min-height: 100%;
      padding: 16px;
      gap: 16px;
    }

    /* Controls Header */
    .controls {
      display: flex;
      gap: 8px;
      align-items: center;
      justify-content: center;
      flex-wrap: wrap;
      width: 100%;
      max-width: 480px;
    }

    .btn {
      background: #ffffff;
      border: 1px solid #d1d5db;
      padding: 6px 14px;
      border-radius: 20px;
      font-size: 14px;
      font-weight: 600;
      color: #4b5563;
      cursor: pointer;
      display: flex;
      align-items: center;
      gap: 6px;
      box-shadow: 0 1px 2px rgba(0,0,0,0.05);
      transition: background-color 0.2s, border-color 0.2s, color 0.2s;
    }

    .btn.active {
      background: #0284c7;
      border-color: #0284c7;
      color: #ffffff;
    }

    .play-btn {
      width: 34px;
      height: 34px;
      padding: 0;
      justify-content: center;
      border-radius: 50%;
    }

    /* Main Stage */
    .stage {
      width: 100%;
      max-width: 520px;
      background: #ffffff;
      border-radius: 16px;
      padding: 24px 16px;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      gap: 20px;
      box-shadow: 0 4px 12px rgba(0,0,0,0.03);
      border: 1px solid #e5e7eb;
      min-height: 240px;
      position: relative;
    }

    /* Math Expression Area */
    .math-row {
      display: flex;
      align-items: center;
      justify-content: center;
      gap: 14px;
      font-size: 32px;
      font-weight: 700;
      position: relative;
      padding: 10px 0;
    }

    .fraction {
      display: inline-flex;
      flex-direction: column;
      align-items: center;
      vertical-align: middle;
      position: relative;
    }

    .num, .den {
      position: relative;
      display: flex;
      align-items: center;
      justify-content: center;
      width: 44px;
      height: 44px;
      border-radius: 8px;
      transition: background-color 0.3s ease, color 0.3s ease;
    }

    .bar {
      width: 100%;
      height: 3px;
      background: #1e2945;
      border-radius: 2px;
      margin: 2px 0;
    }

    .op {
      color: #64748b;
      font-size: 26px;
    }

    /* Cross-canceling badges & slashes */
    .strike {
      position: absolute;
      width: 100%;
      height: 3px;
      background: currentColor;
      transform: rotate(-45deg);
      opacity: 0;
      transition: opacity 0.3s ease;
    }

    .badge {
      position: absolute;
      font-size: 18px;
      font-weight: 800;
      opacity: 0;
      transform: translateY(0);
      transition: opacity 0.3s ease, transform 0.3s ease;
    }

    .badge-top {
      top: -22px;
    }

    .badge-bottom {
      bottom: -22px;
    }

    /* Diagonal Pair Colors */
    .pair-teal {
      color: #0d9488;
    }
    .pair-teal.active-highlight {
      background: #ccfbf1;
    }

    .pair-amber {
      color: #d97706;
    }
    .pair-amber.active-highlight {
      background: #fef3c7;
    }

    .strike-visible {
      opacity: 1;
    }

    .badge-visible {
      opacity: 1;
    }

    /* Visual Model Bar */
    .visual-area {
      display: flex;
      align-items: center;
      gap: 16px;
      width: 100%;
      max-width: 380px;
      padding-top: 12px;
      border-top: 1px dashed #e2e8f0;
      justify-content: center;
    }

    .grid-container {
      display: grid;
      grid-template-columns: repeat(4, 1fr);
      width: 160px;
      height: 40px;
      border-radius: 8px;
      border: 2px solid #0284c7;
      overflow: hidden;
      background: #ffffff;
    }

    .grid-cell {
      border-right: 1px solid #bae6fd;
      transition: background-color 0.4s ease;
    }
    .grid-cell:last-child {
      border-right: none;
    }

    .grid-cell.filled {
      background: #0284c7;
    }

    .visual-label {
      font-size: 15px;
      font-weight: 700;
      color: #0284c7;
    }

    /* Indicator Dots */
    .dots {
      display: flex;
      gap: 8px;
      align-items: center;
      justify-content: center;
    }

    .dot {
      width: 10px;
      height: 10px;
      border-radius: 50%;
      background: #cbd5e1;
      cursor: pointer;
      transition: background-color 0.2s, transform 0.2s;
    }

    .dot.active {
      background: #0284c7;
      transform: scale(1.2);
    }
  </style>
</head>
<body>

  
  <div class="controls">
    <button class="btn play-btn" id="playBtn" aria-label="Play/Pause">⏸</button>
    <button class="btn active" data-step="0">1. Problem</button>
    <button class="btn" data-step="1">2. Cancel 3 &amp; 6</button>
    <button class="btn" data-step="2">3. Cancel 5 &amp; 10</button>
    <button class="btn" data-step="3">4. Result</button>
  </div>

  
  <div class="stage">
    
    <div class="math-row">
      
      <div class="fraction">
        <div class="num pair-teal" id="num1">
          <span>3</span>
          <div class="strike" id="strike1"></div>
          <span class="badge badge-top" id="badge1">1</span>
        </div>
        <div class="bar"></div>
        <div class="den pair-amber" id="den1">
          <span>10</span>
          <div class="strike" id="strike2"></div>
          <span class="badge badge-bottom" id="badge2">2</span>
        </div>
      </div>

      <div class="op">×</div>

      
      <div class="fraction">
        <div class="num pair-amber" id="num2">
          <span>5</span>
          <div class="strike" id="strike3"></div>
          <span class="badge badge-top" id="badge3">1</span>
        </div>
        <div class="bar"></div>
        <div class="den pair-teal" id="den2">
          <span>6</span>
          <div class="strike" id="strike4"></div>
          <span class="badge badge-bottom" id="badge4">2</span>
        </div>
      </div>

      <div class="op" id="equalsOp" style="opacity: 0.3; transition: opacity 0.3s;">=</div>

      
      <div class="fraction" id="resultFrac" style="opacity: 0.2; transition: opacity 0.3s;">
        <div class="num" style="color:#0284c7;" id="resNum">1</div>
        <div class="bar"></div>
        <div class="den" style="color:#0284c7;" id="resDen">4</div>
      </div>
    </div>

    
    <div class="visual-area">
      <div class="grid-container">
        <div class="grid-cell" id="cell1"></div>
        <div class="grid-cell" id="cell2"></div>
        <div class="grid-cell" id="cell3"></div>
        <div class="grid-cell" id="cell4"></div>
      </div>
      <div class="visual-label" id="visualText">1 of 4 equal parts</div>
    </div>

  </div>

  
  <div class="dots">
    <div class="dot active" data-step="0"></div>
    <div class="dot" data-step="1"></div>
    <div class="dot" data-step="2"></div>
    <div class="dot" data-step="3"></div>
  </div>

  <script>
    let currentStep = 0;
    let isPlaying = true;
    let timer = null;

    const stepButtons = document.querySelectorAll('.controls .btn[data-step]');
    const dots = document.querySelectorAll('.dot');
    const playBtn = document.getElementById('playBtn');

    // Visual Elements
    const num1 = document.getElementById('num1');
    const den2 = document.getElementById('den2');
    const strike1 = document.getElementById('strike1');
    const strike4 = document.getElementById('strike4');
    const badge1 = document.getElementById('badge1');
    const badge4 = document.getElementById('badge4');

    const num2 = document.getElementById('num2');
    const den1 = document.getElementById('den1');
    const strike3 = document.getElementById('strike3');
    const strike2 = document.getElementById('strike2');
    const badge3 = document.getElementById('badge3');
    const badge2 = document.getElementById('badge2');

    const equalsOp = document.getElementById('equalsOp');
    const resultFrac = document.getElementById('resultFrac');
    const resNum = document.getElementById('resNum');
    const resDen = document.getElementById('resDen');

    const cell1 = document.getElementById('cell1');
    const visualText = document.getElementById('visualText');

    function renderStep(step) {
      currentStep = step;

      // Update Nav Buttons & Dots
      stepButtons.forEach((btn, idx) => {
        btn.classList.toggle('active', idx === step);
      });
      dots.forEach((dot, idx) => {
        dot.classList.toggle('active', idx === step);
      });

      // Reset / Step 0: Base Problem
      const s1Active = step >= 1;
      const s2Active = step >= 2;
      const s3Active = step >= 3;

      // Pair 1: 3 & 6 (Teal)
      num1.classList.toggle('active-highlight', step === 1);
      den2.classList.toggle('active-highlight', step === 1);
      strike1.classList.toggle('strike-visible', s1Active);
      strike4.classList.toggle('strike-visible', s1Active);
      badge1.classList.toggle('badge-visible', s1Active);
      badge4.classList.toggle('badge-visible', s1Active);

      // Pair 2: 5 & 10 (Amber)
      num2.classList.toggle('active-highlight', step === 2);
      den1.classList.toggle('active-highlight', step === 2);
      strike3.classList.toggle('strike-visible', s2Active);
      strike2.classList.toggle('strike-visible', s2Active);
      badge3.classList.toggle('badge-visible', s2Active);
      badge2.classList.toggle('badge-visible', s2Active);

      // Result State
      if (s3Active) {
        equalsOp.style.opacity = '1';
        resultFrac.style.opacity = '1';
        resNum.innerHTML = '1 × 1';
        resDen.innerHTML = '2 × 2';
        setTimeout(() => {
          if (currentStep === 3) {
            resNum.innerHTML = '1';
            resDen.innerHTML = '4';
          }
        }, 800);
        cell1.classList.add('filled');
        visualText.textContent = '1 / 4 cup';
      } else {
        equalsOp.style.opacity = '0.2';
        resultFrac.style.opacity = '0.2';
        resNum.innerHTML = '1';
        resDen.innerHTML = '4';
        cell1.classList.remove('filled');
        visualText.textContent = '1 of 4 equal parts';
      }
    }

    function setStep(step) {
      renderStep(step);
    }

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

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

    function pauseAutoPlay() {
      isPlaying = false;
      playBtn.textContent = '▶';
      clearInterval(timer);
    }

    // Event Listeners
    playBtn.addEventListener('click', () => {
      if (isPlaying) {
        pauseAutoPlay();
      } else {
        startAutoPlay();
      }
    });

    stepButtons.forEach((btn, idx) => {
      btn.addEventListener('click', () => {
        pauseAutoPlay();
        setStep(idx);
      });
    });

    dots.forEach((dot, idx) => {
      dot.addEventListener('click', () => {
        pauseAutoPlay();
        setStep(idx);
      });
    });

    // Start
    renderStep(0);
    startAutoPlay();
  </script>
</body>
</html>

A common mistake is trying to cancel two numbers that are both on top or both on bottom. You can only cancel between one top number and one bottom number.

Whenever you multiply fractions, look diagonally first — shrinking your numbers early makes the whole problem a breeze!

math

Multiplying Proper Fractions in Context. Multiply two proper fractions and simplify by cross-canceling common factors.

Multiplying Fractions and Cancelling Common Factors

Multiplying and Dividing Fractions (CfE Third & Fourth Level)

Fraction Multiplication in Context: Visual and Concrete Problems

Although the prompt references Early Level (P1), standard MTH 4-07b strictly belongs to Fourth Level in Scotland's Curriculum for Excellence (ages 11–14). The pedagogy emphasizes a Concrete-Pictorial-Abstract (CPA) approach: starting with 2D area overlap models to conceptualize fraction multiplication, transitioning to cross-simplification to reinforce factor structures and algebraic fluency, and anchoring the operations in meaningful real-world contexts like recipe scaling, area calculations, and proportions.

Multiplying Proper Fractions in Context

Multiplying Straight Across

The Cross-Canceling Shortcut