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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).

Lesson generation model

System default

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Tutoring abilities

If you type $2 + 3 \times 4^2$ into two different calculators, one might give you $50$ while another gives you $400$.

To make sure every mathematician, computer, and scientist gets the exact same answer, we follow a strict agreed-upon sequence called the **order of operations**.

Before we solve that opening puzzle, let's make sure we know exactly how exponents work on their own.

An **exponent** is a small number written above and to the right of a **base** number that tells you how many times to multiply the base by itself.

In the expression $4^3$, the base is $4$ and the exponent is $3$, which means $4 \times 4 \times 4 = 64$.

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <meta name="aspect-ratio" content="4:3">
  <style>
    * {
      box-sizing: border-box;
      margin: 0;
      padding: 0;
    }
    body {
      font-family: system-ui, -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
      color: #1e2945;
      background: #ffffff;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      min-height: 100vh;
      padding: 16px;
    }

    .diagram-grid {
      width: 100%;
      max-width: 500px;
      display: flex;
      flex-direction: column;
      gap: 16px;
      align-items: center;
    }

    /* Main expression card */
    .expression-box {
      position: relative;
      width: 100%;
      background: #f8fafc;
      border: 1px solid #e6e6e6;
      border-radius: 12px;
      padding: 24px 16px 20px;
      display: flex;
      flex-direction: column;
      align-items: center;
    }

    .math-display {
      display: flex;
      align-items: baseline;
      justify-content: center;
      font-weight: 800;
      line-height: 1;
      margin-bottom: 8px;
    }

    .base-num {
      font-size: 68px;
      color: #2563eb;
      cursor: pointer;
      padding: 4px 8px;
      border-radius: 8px;
      transition: background-color 0.2s ease, box-shadow 0.2s ease;
    }
    .base-num:hover, .base-num.active {
      background: #dbeafe;
      box-shadow: 0 0 0 2px #2563eb;
    }

    .exponent-num {
      font-size: 40px;
      color: #d97706;
      cursor: pointer;
      vertical-align: super;
      margin-left: 2px;
      padding: 2px 6px;
      border-radius: 6px;
      transition: background-color 0.2s ease, box-shadow 0.2s ease;
    }
    .exponent-num:hover, .exponent-num.active {
      background: #fef3c7;
      box-shadow: 0 0 0 2px #d97706;
    }

    /* Callout labels */
    .callouts {
      display: grid;
      grid-template-columns: 1fr 1fr;
      gap: 12px;
      width: 100%;
      margin-top: 12px;
    }

    .callout {
      padding: 10px 12px;
      border-radius: 8px;
      font-size: 13px;
      line-height: 1.35;
      text-align: center;
      transition: opacity 0.2s ease, transform 0.2s ease;
    }

    .callout-base {
      background: #eff6ff;
      border: 1px solid #bfdbfe;
      color: #1d4ed8;
    }
    .callout-base strong {
      display: block;
      font-size: 14px;
      color: #1e40af;
      margin-bottom: 2px;
    }

    .callout-exp {
      background: #fffbeb;
      border: 1px solid #fde68a;
      color: #b45309;
    }
    .callout-exp strong {
      display: block;
      font-size: 14px;
      color: #92400e;
      margin-bottom: 2px;
    }

    /* Expanded form */
    .expanded-card {
      width: 100%;
      background: #eff6ff;
      border: 1px solid #bfdbfe;
      border-radius: 12px;
      padding: 14px 16px;
      text-align: center;
    }

    .expanded-label {
      font-size: 12px;
      font-weight: 700;
      text-transform: uppercase;
      letter-spacing: 0.5px;
      color: #3b82f6;
      margin-bottom: 6px;
    }

    .expanded-math {
      font-size: 22px;
      font-weight: 700;
      color: #1e2945;
      display: flex;
      align-items: center;
      justify-content: center;
      gap: 6px;
      flex-wrap: wrap;
    }

    .factor {
      color: #2563eb;
      background: #ffffff;
      padding: 2px 8px;
      border-radius: 6px;
      border: 1px solid #93c5fd;
    }

    .op {
      color: #64748b;
      font-size: 18px;
    }

    .result {
      color: #0f172a;
      background: #dbeafe;
      padding: 2px 10px;
      border-radius: 6px;
      border: 1px solid #60a5fa;
    }

    /* Warning trap */
    .warning-card {
      width: 100%;
      background: #fef2f2;
      border: 1px solid #fecaca;
      border-radius: 12px;
      padding: 12px 16px;
      display: flex;
      align-items: center;
      justify-content: center;
      gap: 8px;
      font-size: 13.5px;
      font-weight: 600;
      color: #991b1b;
      text-align: center;
    }

    .trap-strike {
      text-decoration: line-through 2px #dc2626;
      color: #dc2626;
      font-weight: 700;
      margin: 0 4px;
    }

    .warn-icon {
      font-size: 16px;
      flex-shrink: 0;
    }
  </style>
</head>
<body>

  <div class="diagram-grid">
    
    <div class="expression-box">
      <div class="math-display">
        <span class="base-num" id="baseEl" title="Base">4</span>
        <span class="exponent-num" id="expEl" title="Exponent">3</span>
      </div>

      <div class="callouts">
        <div class="callout callout-base" id="calloutBase">
          <strong>Base (4)</strong>
          The factor being multiplied
        </div>
        <div class="callout callout-exp" id="calloutExp">
          <strong>Exponent (3)</strong>
          Repeated 3 times
        </div>
      </div>
    </div>

    
    <div class="expanded-card">
      <div class="expanded-label">Expanded Form</div>
      <div class="expanded-math">
        <span class="factor">4</span>
        <span class="op">×</span>
        <span class="factor">4</span>
        <span class="op">×</span>
        <span class="factor">4</span>
        <span class="op">=</span>
        <span class="result">64</span>
      </div>
    </div>

    
    <div class="warning-card">
      <span class="warn-icon">⚠️</span>
      <span>Common Trap: 4³ is <strong>NOT</strong> <span class="trap-strike">4 × 3 = 12</span></span>
    </div>
  </div>

  <script>
    const baseEl = document.getElementById('baseEl');
    const expEl = document.getElementById('expEl');
    const calloutBase = document.getElementById('calloutBase');
    const calloutExp = document.getElementById('calloutExp');

    function highlight(target) {
      if (target === 'base') {
        baseEl.classList.add('active');
        expEl.classList.remove('active');
        calloutBase.style.transform = 'scale(1.02)';
        calloutExp.style.opacity = '0.5';
        calloutBase.style.opacity = '1';
      } else if (target === 'exp') {
        expEl.classList.add('active');
        baseEl.classList.remove('active');
        calloutExp.style.transform = 'scale(1.02)';
        calloutBase.style.opacity = '0.5';
        calloutExp.style.opacity = '1';
      } else {
        baseEl.classList.remove('active');
        expEl.classList.remove('active');
        calloutBase.style.transform = 'none';
        calloutExp.style.transform = 'none';
        calloutBase.style.opacity = '1';
        calloutExp.style.opacity = '1';
      }
    }

    baseEl.addEventListener('mouseenter', () => highlight('base'));
    baseEl.addEventListener('mouseleave', () => highlight(null));
    baseEl.addEventListener('click', () => highlight('base'));

    expEl.addEventListener('mouseenter', () => highlight('exp'));
    expEl.addEventListener('mouseleave', () => highlight(null));
    expEl.addEventListener('click', () => highlight('exp'));
  </script>
</body>
</html>

So where do exponents fit in when an expression has a whole mix of plus signs, times signs, and parentheses?

In the early 1900s, textbook authors standardized the **order of operations** (often remembered as **PEMDAS**) so mathematical statements would have only one universal meaning.

We always simplify operations with the highest binding strength first: grouping symbols come first, followed by repeated multiplication (exponents), then multiplication and division, and finally addition and subtraction.

<!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="1:1">
  <style>
    * {
      box-sizing: border-box;
      margin: 0;
      padding: 0;
    }

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

    .ladder-container {
      width: 100%;
      max-width: 440px;
      display: flex;
      flex-direction: column;
      gap: 10px;
      position: relative;
    }

    .tier {
      display: flex;
      align-items: center;
      justify-content: space-between;
      padding: 10px 14px;
      border-radius: 12px;
      background: #ffffff;
      border: 2px solid transparent;
      box-shadow: 0 2px 6px rgba(0, 0, 0, 0.05);
      cursor: pointer;
      transition: transform 0.2s ease, box-shadow 0.2s ease, border-color 0.2s ease;
    }

    .tier:hover {
      box-shadow: 0 4px 12px rgba(0, 0, 0, 0.1);
    }

    .tier.active {
      transform: scale(1.02);
      box-shadow: 0 6px 16px rgba(0, 0, 0, 0.12);
    }

    /* Tier Specific Colors */
    .tier-1 {
      border-color: #8b5cf6;
      background: #faf5ff;
    }
    .tier-1 .badge {
      background: #7c3aed;
      color: #ffffff;
    }
    .tier-1 .tier-title {
      color: #5b21b6;
    }

    .tier-2 {
      border-color: #3b82f6;
      background: #eff6ff;
    }
    .tier-2 .badge {
      background: #2563eb;
      color: #ffffff;
    }
    .tier-2 .tier-title {
      color: #1e40af;
    }

    .tier-3 {
      border-color: #f59e0b;
      background: #fffbeb;
    }
    .tier-3 .badge {
      background: #d97706;
      color: #ffffff;
    }
    .tier-3 .tier-title {
      color: #92400e;
    }

    .tier-4 {
      border-color: #10b981;
      background: #ecfdf5;
    }
    .tier-4 .badge {
      background: #059669;
      color: #ffffff;
    }
    .tier-4 .tier-title {
      color: #065f46;
    }

    .tier-left {
      display: flex;
      align-items: center;
      gap: 10px;
    }

    .badge {
      width: 28px;
      height: 28px;
      border-radius: 8px;
      display: flex;
      align-items: center;
      justify-content: center;
      font-weight: 700;
      font-size: 15px;
      flex-shrink: 0;
    }

    .tier-content {
      display: flex;
      flex-direction: column;
    }

    .tier-title {
      font-size: 15px;
      font-weight: 700;
      line-height: 1.2;
    }

    .tier-subtitle {
      font-size: 12px;
      font-weight: 600;
      color: #6b7280;
      margin-top: 2px;
      display: flex;
      align-items: center;
      gap: 4px;
    }

    .tier-right {
      display: flex;
      align-items: center;
      justify-content: flex-end;
    }

    .symbol-pill {
      background: #ffffff;
      border: 1px solid rgba(0, 0, 0, 0.08);
      padding: 4px 10px;
      border-radius: 8px;
      font-size: 15px;
      font-weight: 700;
      letter-spacing: 0.5px;
      color: #1e2945;
      box-shadow: 0 1px 2px rgba(0, 0, 0, 0.04);
    }

    /* Priority Arrow Bar */
    .flow-bar {
      display: flex;
      align-items: center;
      justify-content: space-between;
      padding: 0 8px;
      font-size: 12px;
      font-weight: 700;
      color: #64748b;
      text-transform: uppercase;
      letter-spacing: 0.8px;
    }

    .flow-line {
      flex: 1;
      height: 2px;
      background: #cbd5e1;
      margin: 0 10px;
      position: relative;
    }

    .flow-line::after {
      content: '▼';
      position: absolute;
      right: -4px;
      top: -6px;
      font-size: 9px;
      color: #94a3b8;
    }

    /* Interactive Example Box */
    .example-card {
      margin-top: 8px;
      padding: 10px 14px;
      background: #ffffff;
      border: 1px solid #e2e8f0;
      border-radius: 12px;
      display: flex;
      flex-direction: column;
      gap: 8px;
      box-shadow: 0 2px 5px rgba(0, 0, 0, 0.04);
    }

    .example-header {
      display: flex;
      justify-content: space-between;
      align-items: center;
      font-size: 12px;
      font-weight: 700;
      color: #64748b;
    }

    .math-steps {
      display: flex;
      align-items: center;
      justify-content: center;
      gap: 8px;
      font-size: 17px;
      font-weight: 700;
      min-height: 32px;
    }

    .highlight-exp {
      color: #2563eb;
      background: #dbeafe;
      padding: 1px 6px;
      border-radius: 6px;
    }

    .highlight-mult {
      color: #d97706;
      background: #fef3c7;
      padding: 1px 6px;
      border-radius: 6px;
    }

    .highlight-add {
      color: #059669;
      background: #d1fae5;
      padding: 1px 6px;
      border-radius: 6px;
    }

    .step-btn {
      background: #f1f5f9;
      border: 1px solid #cbd5e1;
      color: #334155;
      padding: 4px 10px;
      border-radius: 6px;
      font-size: 12px;
      font-weight: 600;
      cursor: pointer;
      display: flex;
      align-items: center;
      gap: 4px;
    }

    .step-btn:hover {
      background: #e2e8f0;
    }
  </style>
</head>
<body>

  <div class="ladder-container">
    <div class="flow-bar">
      <span>First Priority</span>
      <div class="flow-line"></div>
      <span>Order of Operations</span>
    </div>

    
    <div class="tier tier-1" id="t1" onclick="selectTier(1)">
      <div class="tier-left">
        <div class="badge">1</div>
        <div class="tier-content">
          <div class="tier-title">Parentheses</div>
          <div class="tier-subtitle">Grouping & brackets</div>
        </div>
      </div>
      <div class="tier-right">
        <span class="symbol-pill">( ) [ ]</span>
      </div>
    </div>

    
    <div class="tier tier-2" id="t2" onclick="selectTier(2)">
      <div class="tier-left">
        <div class="badge">2</div>
        <div class="tier-content">
          <div class="tier-title">Exponents</div>
          <div class="tier-subtitle">Powers & roots</div>
        </div>
      </div>
      <div class="tier-right">
        <span class="symbol-pill">x² &radic;</span>
      </div>
    </div>

    
    <div class="tier tier-3" id="t3" onclick="selectTier(3)">
      <div class="tier-left">
        <div class="badge">3</div>
        <div class="tier-content">
          <div class="tier-title">Multiply &amp; Divide</div>
          <div class="tier-subtitle">Left to Right &rarr;</div>
        </div>
      </div>
      <div class="tier-right">
        <span class="symbol-pill">&times; &divide;</span>
      </div>
    </div>

    
    <div class="tier tier-4" id="t4" onclick="selectTier(4)">
      <div class="tier-left">
        <div class="badge">4</div>
        <div class="tier-content">
          <div class="tier-title">Add &amp; Subtract</div>
          <div class="tier-subtitle">Left to Right &rarr;</div>
        </div>
      </div>
      <div class="tier-right">
        <span class="symbol-pill">+ &minus;</span>
      </div>
    </div>

    
    <div class="example-card">
      <div class="example-header">
        <span id="stepLabel">STEP 1: EXPONENTS</span>
        <button class="step-btn" onclick="nextExampleStep()">Next Step &rarr;</button>
      </div>
      <div class="math-steps" id="mathDisplay">
        2 + 3 &times; <span class="highlight-exp">4²</span>
      </div>
    </div>
  </div>

  <script>
    const steps = [
      {
        tier: 2,
        label: "STEP 1: EXPONENTS (4² = 16)",
        html: "2 + 3 &times; <span class='highlight-exp'>4²</span> &rarr; 2 + 3 &times; 16"
      },
      {
        tier: 3,
        label: "STEP 2: MULTIPLY (3 &times; 16 = 48)",
        html: "2 + <span class='highlight-mult'>3 &times; 16</span> &rarr; 2 + 48"
      },
      {
        tier: 4,
        label: "STEP 3: ADD (2 + 48 = 50)",
        html: "<span class='highlight-add'>2 + 48</span> &rarr; <strong>50</strong>"
      }
    ];

    let currentStep = 0;

    function updateView() {
      const s = steps[currentStep];
      document.getElementById('stepLabel').innerHTML = s.label;
      document.getElementById('mathDisplay').innerHTML = s.html;

      for (let i = 1; i <= 4; i++) {
        const el = document.getElementById('t' + i);
        if (i === s.tier) {
          el.classList.add('active');
        } else {
          el.classList.remove('active');
        }
      }
    }

    function nextExampleStep() {
      currentStep = (currentStep + 1) % steps.length;
      updateView();
    }

    function selectTier(tierNum) {
      for (let i = 1; i <= 4; i++) {
        document.getElementById('t' + i).classList.toggle('active', i === tierNum);
      }
      const foundIdx = steps.findIndex(s => s.tier === tierNum);
      if (foundIdx !== -1) {
        currentStep = foundIdx;
        document.getElementById('stepLabel').innerHTML = steps[currentStep].label;
        document.getElementById('mathDisplay').innerHTML = steps[currentStep].html;
      }
    }

    updateView();
  </script>
</body>
</html>

Let's put this hierarchy to work on a multi-step expression that combines several operations.

Let's evaluate the expression $5 + (8 - 2)^2 \div 4$ step by 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="1:1">
  <title>Order of Operations</title>
  <style>
    * {
      box-sizing: border-box;
      margin: 0;
      padding: 0;
    }
    body {
      font-family: system-ui, -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
      color: #1e2945;
      background: transparent;
      padding: 8px;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: space-between;
      min-height: 100%;
      box-sizing: border-box;
    }
    .stepper-nav {
      display: flex;
      align-items: center;
      justify-content: center;
      gap: 6px;
      width: 100%;
      max-width: 380px;
      margin-bottom: 6px;
    }
    .tab-btn {
      flex: 1;
      padding: 5px 3px;
      font-size: 12px;
      font-weight: 700;
      border: 1px solid #e2e8f0;
      border-radius: 8px;
      background: #f8fafc;
      color: #475569;
      cursor: pointer;
      display: flex;
      flex-direction: column;
      align-items: center;
      gap: 2px;
      transition: background-color 0.2s ease, border-color 0.2s ease, color 0.2s ease;
    }
    .tab-btn .tab-num {
      font-size: 10px;
      font-weight: 600;
      text-transform: uppercase;
      opacity: 0.9;
    }
    .tab-btn.active {
      background: #2563eb;
      color: #ffffff;
      border-color: #2563eb;
      box-shadow: 0 2px 6px rgba(37, 99, 235, 0.25);
    }
    .steps-container {
      width: 100%;
      max-width: 380px;
      display: flex;
      flex-direction: column;
      gap: 6px;
      flex: 1;
    }
    .step-row {
      display: flex;
      align-items: center;
      justify-content: space-between;
      padding: 6px 10px;
      border-radius: 8px;
      border: 1px solid #e2e8f0;
      background: #ffffff;
      color: #334155;
      opacity: 0.88;
      transition: all 0.2s ease;
      cursor: pointer;
      gap: 6px;
    }
    .step-row.active {
      opacity: 1;
      border-color: #2563eb;
      background: #eff6ff;
      color: #1e2945;
      box-shadow: 0 2px 6px rgba(37, 99, 235, 0.12);
    }
    .step-row.final.active {
      border-color: #16a34a;
      background: #f0fdf4;
      box-shadow: 0 2px 6px rgba(22, 163, 74, 0.15);
    }
    .op-tag {
      font-size: 11px;
      font-weight: 700;
      padding: 3px 6px;
      border-radius: 5px;
      background: #f1f5f9;
      color: #334155;
      border: 1px solid #cbd5e1;
      white-space: nowrap;
      flex-shrink: 0;
    }
    .step-row.active .op-tag {
      background: #2563eb;
      color: #ffffff;
      border-color: #2563eb;
    }
    .step-row.final.active .op-tag {
      background: #16a34a;
      color: #ffffff;
      border-color: #16a34a;
    }
    .math-expr {
      font-size: 14px;
      font-family: "Cambria Math", "Latin Modern Math", "Times New Roman", serif;
      font-weight: 600;
      letter-spacing: 0.3px;
      display: flex;
      align-items: center;
      justify-content: flex-end;
      gap: 3px;
      white-space: nowrap;
      flex: 1;
    }
    .arr {
      color: #64748b;
      font-size: 12px;
      margin: 0 2px;
    }
    .hl {
      padding: 1px 4px;
      border-radius: 4px;
      background: #dbeafe;
      color: #1d4ed8;
      font-weight: 700;
      white-space: nowrap;
    }
    .res {
      padding: 1px 4px;
      border-radius: 4px;
      background: #fef08a;
      color: #854d0e;
      font-weight: 700;
      white-space: nowrap;
    }
    .step-row:not(.active) .hl {
      background: #e2e8f0;
      color: #334155;
    }
    .step-row:not(.active) .res {
      background: #e2e8f0;
      color: #334155;
    }
    .final-val {
      font-size: 17px;
      font-weight: 800;
      color: #16a34a;
      padding: 1px 8px;
      background: #dcfce7;
      border-radius: 6px;
      border: 1px solid #86efac;
    }
    .controls {
      display: flex;
      align-items: center;
      justify-content: space-between;
      width: 100%;
      max-width: 380px;
      margin-top: 6px;
      gap: 8px;
    }
    .ctrl-btn {
      padding: 6px 12px;
      font-size: 12px;
      font-weight: 600;
      border: 1px solid #cbd5e1;
      border-radius: 6px;
      background: #ffffff;
      color: #1e2945;
      cursor: pointer;
      display: flex;
      align-items: center;
      justify-content: center;
      gap: 4px;
      transition: background 0.15s ease;
    }
    .ctrl-btn:active {
      background: #f1f5f9;
    }
    .play-btn {
      padding: 6px 10px;
      min-width: 78px;
    }

    @media (min-width: 400px) {
      .step-row {
        padding: 8px 12px;
        gap: 8px;
      }
      .op-tag {
        font-size: 12px;
        padding: 3px 8px;
      }
      .math-expr {
        font-size: 16px;
        gap: 4px;
      }
      .arr {
        font-size: 14px;
        margin: 0 4px;
      }
      .tab-btn {
        font-size: 13px;
        padding: 6px 4px;
      }
      .final-val {
        font-size: 19px;
      }
      .ctrl-btn {
        font-size: 13px;
        padding: 7px 14px;
      }
    }
  </style>
</head>
<body>

  <div class="stepper-nav" role="tablist">
    <button class="tab-btn active" id="tab0" onclick="goToStep(0)">
      <span class="tab-num">Step 1</span>
      <span>( )</span>
    </button>
    <button class="tab-btn" id="tab1" onclick="goToStep(1)">
      <span class="tab-num">Step 2</span>
      <span>x²</span>
    </button>
    <button class="tab-btn" id="tab2" onclick="goToStep(2)">
      <span class="tab-num">Step 3</span>
      <span>÷</span>
    </button>
    <button class="tab-btn" id="tab3" onclick="goToStep(3)">
      <span class="tab-num">Step 4</span>
      <span>+</span>
    </button>
  </div>

  <div class="steps-container">
    <div class="step-row active" id="row0" onclick="goToStep(0)">
      <div class="op-tag">Parentheses</div>
      <div class="math-expr">
        5 + <span class="hl">(8 − 2)</span>² ÷ 4
        <span class="arr">→</span>
        5 + <span class="res">6</span>² ÷ 4
      </div>
    </div>

    <div class="step-row" id="row1" onclick="goToStep(1)">
      <div class="op-tag">Exponents</div>
      <div class="math-expr">
        5 + <span class="hl">6²</span> ÷ 4
        <span class="arr">→</span>
        5 + <span class="res">36</span> ÷ 4
      </div>
    </div>

    <div class="step-row" id="row2" onclick="goToStep(2)">
      <div class="op-tag">Divide</div>
      <div class="math-expr">
        5 + <span class="hl">36 ÷ 4</span>
        <span class="arr">→</span>
        5 + <span class="res">9</span>
      </div>
    </div>

    <div class="step-row final" id="row3" onclick="goToStep(3)">
      <div class="op-tag">Add</div>
      <div class="math-expr">
        <span class="hl">5 + 9</span>
        <span class="arr">=</span>
        <span class="final-val">14</span>
      </div>
    </div>
  </div>

  <div class="controls">
    <button class="ctrl-btn" onclick="prevStep()">◀ Back</button>
    <button class="ctrl-btn play-btn" id="autoBtn" onclick="toggleAuto()">⏸ Pause</button>
    <button class="ctrl-btn" onclick="nextStep()">Next ▶</button>
  </div>

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

    function render() {
      for (let i = 0; i < 4; i++) {
        const tab = document.getElementById('tab' + i);
        const row = document.getElementById('row' + i);
        if (i === currentStep) {
          tab.classList.add('active');
          row.classList.add('active');
        } else {
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        }
      }
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    function goToStep(step) {
      currentStep = step;
      render();
    }

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

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

    function toggleAuto() {
      autoPlaying = !autoPlaying;
      const btn = document.getElementById('autoBtn');
      if (autoPlaying) {
        btn.textContent = '⏸ Pause';
        startTimer();
      } else {
        btn.textContent = '▶ Play';
        clearInterval(timer);
      }
    }

    function startTimer() {
      clearInterval(timer);
      timer = setInterval(() => {
        if (autoPlaying) {
          nextStep();
        }
      }, 3500);
    }

    startTimer();
  </script>
</body>
</html>

We first simplified inside parentheses $(8 - 2 = 6)$, evaluated the exponent $6^2 = 36$, carried out division $36 \div 4 = 9$, and finally added $5 + 9 = 14$.

What happens when an expression contains two operations that share the exact same rank?

Multiplication does not outrank division, and addition does not outrank subtraction — they are equal partners.

When you encounter operations on the same level, you must always evaluate them from **left to right**, exactly like reading a sentence.

<!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>Order of Operations</title>
  <style>
    * {
      box-sizing: border-box;
      margin: 0;
      padding: 0;
    }

    body {
      font-family: system-ui, -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
      color: #1e2945;
      background: transparent;
      padding: 12px;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      min-height: 100%;
    }

    .main-expr {
      font-size: 22px;
      font-weight: 700;
      letter-spacing: 1px;
      padding: 6px 16px;
      background: #f1f5f9;
      border-radius: 20px;
      margin-bottom: 12px;
      display: inline-flex;
      align-items: center;
      gap: 6px;
    }

    .columns {
      display: flex;
      gap: 12px;
      width: 100%;
      max-width: 580px;
    }

    .col-card {
      flex: 1;
      border-radius: 12px;
      padding: 12px;
      display: flex;
      flex-direction: column;
      gap: 8px;
    }

    .col-correct {
      background: #f0fdf4;
      border: 1.5px solid #86efac;
    }

    .col-wrong {
      background: #fef2f2;
      border: 1.5px solid #fca5a5;
    }

    .col-header {
      display: flex;
      align-items: center;
      justify-content: space-between;
      font-size: 13px;
      font-weight: 700;
      padding-bottom: 4px;
      border-bottom: 1px solid rgba(0, 0, 0, 0.06);
    }

    .col-correct .col-header {
      color: #15803d;
    }

    .col-wrong .col-header {
      color: #b91c1c;
    }

    .step-box {
      background: #ffffff;
      border-radius: 8px;
      padding: 6px 10px;
      font-size: 15px;
      font-weight: 600;
      display: flex;
      align-items: center;
      justify-content: space-between;
      box-shadow: 0 1px 3px rgba(0, 0, 0, 0.04);
      min-height: 38px;
    }

    .op-highlight-green {
      background: #dcfce7;
      color: #15803d;
      padding: 1px 5px;
      border-radius: 4px;
    }

    .op-highlight-red {
      background: #fee2e2;
      color: #b91c1c;
      padding: 1px 5px;
      border-radius: 4px;
    }

    .arrow {
      text-align: center;
      color: #94a3b8;
      font-size: 12px;
      line-height: 1;
      margin: -4px 0;
    }

    .result-box {
      margin-top: 2px;
      padding: 8px 10px;
      border-radius: 8px;
      font-size: 18px;
      font-weight: 800;
      display: flex;
      align-items: center;
      justify-content: center;
      gap: 8px;
    }

    .result-correct {
      background: #22c55e;
      color: #ffffff;
    }

    .result-wrong {
      background: #ef4444;
      color: #ffffff;
    }

    .tag {
      font-size: 11px;
      font-weight: 600;
      padding: 2px 6px;
      border-radius: 4px;
      background: #f8fafc;
      color: #64748b;
    }

    @media (max-width: 360px) {
      .columns {
        flex-direction: column;
      }
      .main-expr {
        font-size: 18px;
      }
      .step-box {
        font-size: 14px;
      }
    }
  </style>
</head>
<body>

  <div class="main-expr">
    12 ÷ 3 × 2<sup>3</sup>
  </div>

  <div class="columns">
    
    <div class="col-card col-correct">
      <div class="col-header">
        <span>Correct (L → R)</span>
        <span>✓</span>
      </div>

      <div class="step-box">
        <span>12 ÷ 3 × <span class="op-highlight-green">2<sup>3</sup></span></span>
        <span class="tag">2<sup>3</sup> = 8</span>
      </div>

      <div class="arrow">↓</div>

      <div class="step-box">
        <span><span class="op-highlight-green">12 ÷ 3</span> × 8</span>
        <span class="tag">12 ÷ 3 = 4</span>
      </div>

      <div class="arrow">↓</div>

      <div class="step-box">
        <span><span class="op-highlight-green">4 × 8</span></span>
        <span class="tag">4 × 8</span>
      </div>

      <div class="result-box result-correct">
        = 32 ✓
      </div>
    </div>

    
    <div class="col-card col-wrong">
      <div class="col-header">
        <span>Mistake (× first)</span>
        <span>✕</span>
      </div>

      <div class="step-box">
        <span>12 ÷ 3 × <span class="op-highlight-green">2<sup>3</sup></span></span>
        <span class="tag">2<sup>3</sup> = 8</span>
      </div>

      <div class="arrow">↓</div>

      <div class="step-box">
        <span>12 ÷ <span class="op-highlight-red">3 × 8</span></span>
        <span class="tag">3 × 8 = 24</span>
      </div>

      <div class="arrow">↓</div>

      <div class="step-box">
        <span><span class="op-highlight-red">12 ÷ 24</span></span>
        <span class="tag">12 ÷ 24</span>
      </div>

      <div class="result-box result-wrong">
        = 0.5 ✕
      </div>
    </div>
  </div>

</body>
</html>

Evaluating from right to left gave $0.5$, which is incorrect because division and multiplication must be solved in the order they appear from left to right.

Now let's revisit that opening challenge and see which calculator was right.

In the expression $2 + 3 \times 4^2$, exponents come first: $4^2 = 16$. Next comes multiplication: $3 \times 16 = 48$. Finally, addition gives $2 + 48 = 50$.

The calculator that gave $400$ added $2 + 3$ first, ignoring the hierarchy — always check that you follow each level of PEMDAS in order.

math

Order of Operations with Whole Number Exponents. Evaluate numerical expressions using order of operations including whole number exponents

Order of Operations Examples: Exponents

Evaluate Numerical Expressions with Parentheses, Brackets, and Exponents (6th Grade TEKS)

Grade 6, Unit 6, Lesson 14: Expressions with Exponents and Order of Operations

Pedagogical strategies for mastery: (1) Use GEMDAS (Grouping, Exponents, Multiplication/Division, Addition/Subtraction) instead of PEMDAS to emphasize all grouping formats (brackets, fraction bars) and prevent treating multiplication as higher priority than division; (2) Require students to use highlighters to underline the specific operation being executed in each step, rewriting the simplified expression directly underneath to enforce the 'inverted triangle' formatting; (3) Incorporate 'Error Analysis' routines (e.g., 'Find and fix the mistake in this student's work') to directly target the common misconception of multiplying base by exponent ($2^3 \neq 6$) and ignoring left-to-right rules.

Order of Operations with Whole Number Exponents

What Is an Exponent?

The Hierarchy of Operations