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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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Tutoring abilities

Imagine multiplying $200,000$ by $3,000,000$ — counting all those zeros by hand is slow and easy to mess up.

Scientific notation gives us a shortcut: write numbers as a front number times a power of 10, like $2 \times 10^5$.

<!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;
      font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, Helvetica, Arial, sans-serif;
    }

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

    .main-stage {
      width: 100%;
      max-width: 540px;
      display: flex;
      flex-direction: column;
      gap: 16px;
    }

    /* Tabs for Example selection */
    .tab-bar {
      display: flex;
      justify-content: center;
      gap: 8px;
    }

    .tab-btn {
      padding: 6px 14px;
      border: 1px solid #e2e8f0;
      background: #ffffff;
      color: #64748b;
      font-size: 13px;
      font-weight: 600;
      border-radius: 20px;
      cursor: pointer;
      box-shadow: 0 1px 2px rgba(0,0,0,0.04);
      transition: all 0.2s ease;
    }

    .tab-btn.active {
      background: #1e2945;
      color: #ffffff;
      border-color: #1e2945;
    }

    /* Equation visualization box */
    .calc-card {
      background: #ffffff;
      border: 1px solid #e2e8f0;
      border-radius: 12px;
      padding: 20px 16px;
      min-height: 220px;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      box-shadow: 0 4px 12px rgba(0,0,0,0.03);
      position: relative;
    }

    .step-pill {
      position: absolute;
      top: 12px;
      left: 14px;
      background: #eff6ff;
      color: #0284c7;
      font-size: 12px;
      font-weight: 700;
      padding: 3px 10px;
      border-radius: 12px;
      text-transform: uppercase;
      letter-spacing: 0.5px;
    }

    .display-math {
      font-size: 24px;
      margin: 12px 0;
      text-align: center;
      min-height: 48px;
      display: flex;
      align-items: center;
      justify-content: center;
    }

    /* Visual streams breakdown */
    .streams-grid {
      display: grid;
      grid-template-columns: 1fr 1fr;
      gap: 12px;
      width: 100%;
      margin-top: 8px;
    }

    .stream-box {
      border-radius: 8px;
      padding: 10px;
      text-align: center;
      transition: opacity 0.3s ease, transform 0.3s ease;
    }

    .stream-front {
      background: #f0f9ff;
      border: 1px solid #bae6fd;
      color: #0369a1;
    }

    .stream-power {
      background: #faf5ff;
      border: 1px solid #e9d5ff;
      color: #7e22ce;
    }

    .stream-label {
      font-size: 11px;
      font-weight: 700;
      text-transform: uppercase;
      letter-spacing: 0.5px;
      margin-bottom: 4px;
    }

    .stream-math {
      font-size: 18px;
      font-weight: 600;
    }

    /* Controls */
    .controls {
      display: flex;
      align-items: center;
      justify-content: center;
      gap: 12px;
      margin-top: 8px;
    }

    .btn {
      display: inline-flex;
      align-items: center;
      justify-content: center;
      padding: 8px 16px;
      border-radius: 8px;
      font-size: 14px;
      font-weight: 600;
      cursor: pointer;
      border: 1px solid #cbd5e1;
      background: #ffffff;
      color: #334155;
      user-select: none;
      transition: background 0.15s ease, border-color 0.15s ease;
    }

    .btn:hover {
      background: #f1f5f9;
    }

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

    .btn-primary:hover {
      background: #0369a1;
    }

    .stepper-dots {
      display: flex;
      gap: 8px;
    }

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

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

  <div class="main-stage">
    
    <div class="tab-bar">
      <button class="tab-btn active" id="tab0" onclick="setProblem(0)">(2 × 10³) × (4 × 10⁵)</button>
      <button class="tab-btn" id="tab1" onclick="setProblem(1)">(3 × 10⁴) × (3 × 10²)</button>
    </div>

    
    <div class="calc-card">
      <span class="step-pill" id="step-tag">Step 1</span>
      
      <div class="display-math" id="main-math"></div>

      <div class="streams-grid" id="streams-container">
        <div class="stream-box stream-front" id="box-front">
          <div class="stream-label">Front Numbers</div>
          <div class="stream-math" id="math-front"></div>
        </div>
        <div class="stream-box stream-power" id="box-power">
          <div class="stream-label">Powers of 10</div>
          <div class="stream-math" id="math-power"></div>
        </div>
      </div>
    </div>

    
    <div class="controls">
      <button class="btn" onclick="prevStep()">←</button>
      <div class="stepper-dots">
        <div class="dot active" onclick="goToStep(0)"></div>
        <div class="dot" onclick="goToStep(1)"></div>
        <div class="dot" onclick="goToStep(2)"></div>
        <div class="dot" onclick="goToStep(3)"></div>
      </div>
      <button class="btn btn-primary" id="next-btn" onclick="nextStep()">Next →</button>
    </div>
  </div>

  <script>
    const problems = [
      {
        a: 2, p: 3,
        b: 4, q: 5,
        prod: 8, sum: 8
      },
      {
        a: 3, p: 4,
        b: 3, q: 2,
        prod: 9, sum: 6
      }
    ];

    let currentProbIdx = 0;
    let currentStep = 0;
    let autoPlayTimer = null;

    function renderKatex(tex, elId) {
      const el = document.getElementById(elId);
      if (!el) return;
      katex.render(tex, el, { throwOnError: false, displayMode: false });
    }

    function updateView() {
      const prob = problems[currentProbIdx];
      const tag = document.getElementById('step-tag');
      const boxFront = document.getElementById('box-front');
      const boxPower = document.getElementById('box-power');
      const dots = document.querySelectorAll('.dot');
      const nextBtn = document.getElementById('next-btn');

      dots.forEach((dot, idx) => {
        dot.classList.toggle('active', idx === currentStep);
      });

      nextBtn.textContent = currentStep === 3 ? 'Restart ↺' : 'Next →';

      if (currentStep === 0) {
        tag.textContent = 'Problem';
        renderKatex(`({\\color{#0369a1}${prob.a}} \\times {\\color{#7e22ce}10^{${prob.p}}}) \\times ({\\color{#0369a1}${prob.b}} \\times {\\color{#7e22ce}10^{${prob.q}}})`, 'main-math');
        
        boxFront.style.opacity = '0.3';
        boxPower.style.opacity = '0.3';
        renderKatex(`{\\color{#0369a1}${prob.a}} \\text{ and } {\\color{#0369a1}${prob.b}}`, 'math-front');
        renderKatex(`{\\color{#7e22ce}10^{${prob.p}}} \\text{ and } {\\color{#7e22ce}10^{${prob.q}}}`, 'math-power');
      } 
      else if (currentStep === 1) {
        tag.textContent = 'Step 1: Regroup';
        renderKatex(`({\\color{#0369a1}${prob.a} \\times ${prob.b}}) \\times ({\\color{#7e22ce}10^{${prob.p}} \\times 10^{${prob.q}}})`, 'main-math');
        
        boxFront.style.opacity = '1';
        boxPower.style.opacity = '1';
        renderKatex(`{\\color{#0369a1}${prob.a} \\times ${prob.b}}`, 'math-front');
        renderKatex(`{\\color{#7e22ce}10^{${prob.p}} \\times 10^{${prob.q}}}`, 'math-power');
      } 
      else if (currentStep === 2) {
        tag.textContent = 'Step 2: Multiply & Add';
        renderKatex(`{\\color{#0369a1}${prob.prod}} \\times {\\color{#7e22ce}10^{${prob.p}+${prob.q}}}`, 'main-math');
        
        boxFront.style.opacity = '1';
        boxPower.style.opacity = '1';
        renderKatex(`{\\color{#0369a1}${prob.prod}}`, 'math-front');
        renderKatex(`{\\color{#7e22ce}10^{${prob.sum}}}`, 'math-power');
      } 
      else if (currentStep === 3) {
        tag.textContent = 'Result';
        renderKatex(`{\\mathbf{${prob.prod} \\times 10^{${prob.sum}}}}`, 'main-math');
        
        boxFront.style.opacity = '1';
        boxPower.style.opacity = '1';
        renderKatex(`{\\color{#0369a1}${prob.prod}}`, 'math-front');
        renderKatex(`{\\color{#7e22ce}10^{${prob.sum}}}`, 'math-power');
      }
    }

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

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

    function prevStep() {
      currentStep = (currentStep - 1 + 4) % 4;
      updateView();
    }

    function setProblem(idx) {
      currentProbIdx = idx;
      currentStep = 0;
      document.getElementById('tab0').classList.toggle('active', idx === 0);
      document.getElementById('tab1').classList.toggle('active', idx === 1);
      updateView();
    }

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

How do we multiply two of these numbers together in two quick steps?

First, multiply the **coefficients** (the front numbers). Then, add the **exponents** (the small power numbers on the tens).

Why do we add exponents? Because $10^2 \times 10^3 = (10 \times 10) \times (10 \times 10 \times 10) = 10^5$, which is just $2 + 3 = 5$ tens.

<!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>Adding Exponents Visualization</title>
  <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;
      font-family: system-ui, -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
      -webkit-tap-highlight-color: transparent;
    }

    body {
      background-color: #f8f9fa;
      color: #1e2945;
      padding: 12px;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      min-height: 100%;
    }

    .main-layout {
      width: 100%;
      max-width: 520px;
      display: flex;
      flex-direction: column;
      gap: 12px;
      align-items: center;
    }

    /* Preset Tabs */
    .tabs {
      display: flex;
      gap: 8px;
      justify-content: center;
      width: 100%;
      flex-wrap: wrap;
    }

    .tab-btn {
      background: #ffffff;
      border: 1.5px solid #e2e8f0;
      border-radius: 8px;
      padding: 6px 12px;
      font-size: 14px;
      font-weight: 600;
      color: #64748b;
      cursor: pointer;
      transition: all 0.2s ease;
      box-shadow: 0 1px 2px rgba(0,0,0,0.04);
    }

    .tab-btn.active {
      border-color: #22b7ff;
      background: #f0f9ff;
      color: #0284c7;
      box-shadow: 0 0 0 2px rgba(34, 183, 255, 0.2);
    }

    /* Display Area */
    .stage {
      width: 100%;
      background: #ffffff;
      border: 1px solid #e6e6e6;
      border-radius: 12px;
      padding: 16px 12px;
      display: flex;
      flex-direction: column;
      align-items: center;
      gap: 14px;
      box-shadow: 0 2px 6px rgba(0,0,0,0.03);
    }

    .math-row {
      font-size: 20px;
      display: flex;
      align-items: center;
      justify-content: center;
      min-height: 32px;
    }

    /* Chips visualizer */
    .expansion-area {
      display: flex;
      align-items: center;
      justify-content: center;
      gap: 8px;
      flex-wrap: wrap;
      width: 100%;
    }

    .group-box {
      display: flex;
      flex-direction: column;
      align-items: center;
      gap: 6px;
      padding: 8px 10px;
      border-radius: 10px;
      transition: all 0.3s ease;
    }

    .group-a {
      background: #f0f9ff;
      border: 1.5px dashed #0284c7;
    }

    .group-b {
      background: #f0fdf4;
      border: 1.5px dashed #16a34a;
    }

    .chips-row {
      display: flex;
      align-items: center;
      gap: 4px;
      flex-wrap: wrap;
      justify-content: center;
    }

    .chip {
      width: 32px;
      height: 32px;
      border-radius: 6px;
      display: flex;
      align-items: center;
      justify-content: center;
      font-size: 13px;
      font-weight: 700;
      transition: transform 0.3s ease;
    }

    .chip-a {
      background: #0284c7;
      color: #ffffff;
    }

    .chip-b {
      background: #16a34a;
      color: #ffffff;
    }

    .op-sign {
      font-size: 18px;
      font-weight: 700;
      color: #94a3b8;
    }

    .group-label {
      font-size: 12px;
      font-weight: 600;
    }

    .label-a { color: #0284c7; }
    .label-b { color: #16a34a; }

    /* Total result container */
    .combined-box {
      width: 100%;
      background: #faf5ff;
      border: 1.5px solid #c084fc;
      border-radius: 10px;
      padding: 10px;
      display: flex;
      flex-direction: column;
      align-items: center;
      gap: 8px;
    }

    .combined-chips {
      display: flex;
      align-items: center;
      justify-content: center;
      gap: 4px;
      flex-wrap: wrap;
    }

    .count-tag {
      font-size: 13px;
      font-weight: 700;
      color: #7e22ce;
      display: flex;
      align-items: center;
      gap: 6px;
    }

    /* Custom Stepper Controls */
    .controls {
      display: flex;
      gap: 16px;
      align-items: center;
      justify-content: center;
      width: 100%;
    }

    .exp-control {
      display: flex;
      align-items: center;
      gap: 6px;
      background: #ffffff;
      border: 1px solid #e2e8f0;
      border-radius: 8px;
      padding: 4px 8px;
    }

    .stepper-btn {
      width: 24px;
      height: 24px;
      border-radius: 4px;
      border: 1px solid #cbd5e1;
      background: #f8fafc;
      color: #334155;
      font-size: 14px;
      font-weight: 700;
      display: flex;
      align-items: center;
      justify-content: center;
      cursor: pointer;
      user-select: none;
    }

    .stepper-btn:active {
      background: #e2e8f0;
    }

    .exp-indicator {
      font-size: 14px;
      font-weight: 700;
      min-width: 36px;
      text-align: center;
    }
  </style>
</head>
<body>

<div class="main-layout">
  
  <div class="tabs" id="tabGroup">
    <button class="tab-btn active" onclick="setExponents(2, 3, this)">10² × 10³</button>
    <button class="tab-btn" onclick="setExponents(3, 4, this)">10³ × 10⁴</button>
    <button class="tab-btn" onclick="setExponents(1, 4, this)">10¹ × 10⁴</button>
    <button class="tab-btn" onclick="setExponents(2, 2, this)">10² × 10²</button>
  </div>

  
  <div class="stage">
    
    <div class="math-row" id="topMath"></div>

    
    <div class="expansion-area">
      
      <div class="group-box group-a">
        <div class="chips-row" id="chipsA"></div>
        <span class="group-label label-a" id="labelA"></span>
      </div>

      <div class="op-sign">×</div>

      
      <div class="group-box group-b">
        <div class="chips-row" id="chipsB"></div>
        <span class="group-label label-b" id="labelB"></span>
      </div>
    </div>

    
    <div class="combined-box">
      <div class="count-tag" id="countLabel"></div>
      <div class="combined-chips" id="allChips"></div>
      <div class="math-row" id="bottomMath" style="font-size: 18px;"></div>
    </div>
  </div>

  
  <div class="controls">
    <div class="exp-control">
      <span style="font-size:13px; font-weight:600; color:#0284c7;">a:</span>
      <button class="stepper-btn" onclick="adjustExp('a', -1)">−</button>
      <span class="exp-indicator" id="indA" style="color:#0284c7;">2</span>
      <button class="stepper-btn" onclick="adjustExp('a', 1)">+</button>
    </div>
    <div class="exp-control">
      <span style="font-size:13px; font-weight:600; color:#16a34a;">b:</span>
      <button class="stepper-btn" onclick="adjustExp('b', -1)">−</button>
      <span class="exp-indicator" id="indB" style="color:#16a34a;">3</span>
      <button class="stepper-btn" onclick="adjustExp('b', 1)">+</button>
    </div>
  </div>
</div>

<script>
  let expA = 2;
  let expB = 3;

  function render() {
    // 1. Math Header
    const topStr = `10^{\\mathbf{\\color{#0284c7}{${expA}}}} \\times 10^{\\mathbf{\\color{#16a34a}{${expB}}}}`;
    katex.render(topStr, document.getElementById('topMath'), { throwOnError: false });

    // 2. Indicators & Labels
    document.getElementById('indA').textContent = expA;
    document.getElementById('indB').textContent = expB;
    document.getElementById('labelA').textContent = `${expA} tens`;
    document.getElementById('labelB').textContent = `${expB} tens`;

    // 3. Chips Group A
    const boxA = document.getElementById('chipsA');
    boxA.innerHTML = '';
    for (let i = 0; i < expA; i++) {
      if (i > 0) {
        const dot = document.createElement('span');
        dot.className = 'op-sign';
        dot.style.fontSize = '12px';
        dot.textContent = '×';
        boxA.appendChild(dot);
      }
      const chip = document.createElement('div');
      chip.className = 'chip chip-a';
      chip.textContent = '10';
      boxA.appendChild(chip);
    }

    // 4. Chips Group B
    const boxB = document.getElementById('chipsB');
    boxB.innerHTML = '';
    for (let i = 0; i < expB; i++) {
      if (i > 0) {
        const dot = document.createElement('span');
        dot.className = 'op-sign';
        dot.style.fontSize = '12px';
        dot.textContent = '×';
        boxB.appendChild(dot);
      }
      const chip = document.createElement('div');
      chip.className = 'chip chip-b';
      chip.textContent = '10';
      boxB.appendChild(chip);
    }

    // 5. Combined Chips
    const allBox = document.getElementById('allChips');
    allBox.innerHTML = '';
    const total = expA + expB;

    for (let i = 0; i < total; i++) {
      if (i > 0) {
        const dot = document.createElement('span');
        dot.className = 'op-sign';
        dot.style.fontSize = '12px';
        dot.textContent = '×';
        allBox.appendChild(dot);
      }
      const chip = document.createElement('div');
      chip.className = 'chip ' + (i < expA ? 'chip-a' : 'chip-b');
      chip.textContent = '10';
      allBox.appendChild(chip);
    }

    // 6. Total Label & Bottom Math
    document.getElementById('countLabel').textContent = `${expA} + ${expB} = ${total} tens total`;
    const botStr = `= 10^{\\mathbf{\\color{#0284c7}{${expA}} + \\color{#16a34a}{${expB}}}} = 10^{\\mathbf{${total}}}`;
    katex.render(botStr, document.getElementById('bottomMath'), { throwOnError: false });
  }

  function setExponents(a, b, btnEl) {
    expA = a;
    expB = b;
    if (btnEl) {
      document.querySelectorAll('.tab-btn').forEach(b => b.classList.remove('active'));
      btnEl.classList.add('active');
    }
    render();
  }

  function adjustExp(which, delta) {
    if (which === 'a') {
      expA = Math.max(1, Math.min(4, expA + delta));
    } else {
      expB = Math.max(1, Math.min(4, expB + delta));
    }
    // clear active tabs since custom
    document.querySelectorAll('.tab-btn').forEach(b => {
      const match = b.textContent.includes(`${expA}`) && b.textContent.includes(`${expB}`);
      if (match && ((expA === 2 && expB === 3) || (expA === 3 && expB === 4) || (expA === 1 && expB === 4) || (expA === 2 && expB === 2))) {
        b.classList.add('active');
      } else {
        b.classList.remove('active');
      }
    });
    render();
  }

  // Initial render
  window.addEventListener('DOMContentLoaded', render);
  render();
</script>

</body>
</html>

Let's test this rule on $(2 \times 10^3) \times (3 \times 10^4)$.

$$ (2 \times 3) \times (10^3 \times 10^4) = 6 \times 10^{3+4} = 6 \times 10^7 $$

What happens if multiplying the front numbers gives a number that is 10 or bigger?

In scientific notation, the front number must always stay between **1 and 10** (never 10 or larger).

Let's multiply $(4 \times 10^5) \times (5 \times 10^3)$.

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$$ 4 \times 5 = 20 \quad \text{and} \quad 10^5 \times 10^3 = 10^8 $$

Since $20 = 2.0 \times 10^1$, we combine powers to get $2.0 \times 10^{1+8} = 2.0 \times 10^9$.

Whenever your front number gets 10 times bigger, move the decimal left by one spot and add 1 to the exponent!

math

Multiplying Numbers Written in Scientific Notation. Multiply two numbers in scientific notation by multiplying coefficients and adding powers of 10.

Multiplying and Dividing in Scientific Notation

Multiply Numbers Written in Scientific Notation

Calculations Using Scientific Notation (Curriculum for Excellence - Fourth Level / National 5 Maths)

Note on Curriculum Level: While the prompt metadata references Early Level (P1), Curriculum for Excellence standard MTH 4-06b is a Fourth Level (Secondary S1–S3 / Key Stage 3 / Grade 8) mathematics outcome. Pedagogical best practice for mastering this standard focuses on: (1) explicit anchoring to index laws ($a^m \times a^n = a^{m+n}$), (2) conceptual justification using commutative/associative grouping rather than rote algorithmic rules, (3) two-stage checking where learners evaluate the coefficient range separately from exponent calculations to prevent normalization errors, and (4) grounding calculations in real-life science contexts (e.g., astronomical distances and microscopic quantities).

Multiplying Numbers Written in Scientific Notation

The Two-Step Rule

Fixing the Front Number