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Tutoring abilities

Imagine you have a giant bag of 12 cookies to share with your friends. If more friends show up to eat, do you get to eat **more** cookies or **fewer** cookies?

When numbers change together in the real world, they follow special patterns called **relationships**.

A **direct proportion** happens when two amounts grow or shrink at the exact same pace. If you double one amount, the other amount doubles too!

<!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, sans-serif;
    color: #1e2945;
    background: transparent;
    padding: 12px;
    display: flex;
    flex-direction: column;
    align-items: center;
    justify-content: space-between;
    min-height: 100%;
    user-select: none;
  }

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

  .step-btn {
    flex: 1;
    padding: 8px 4px;
    border-radius: 8px;
    border: 1px solid #e2e8f0;
    background: #ffffff;
    font-size: 14px;
    font-weight: 600;
    color: #64748b;
    cursor: pointer;
    text-align: center;
    box-shadow: 0 1px 2px rgba(0,0,0,0.04);
    transition: all 0.2s ease;
  }

  .step-btn.active {
    background: #22b7ff;
    color: #ffffff;
    border-color: #22b7ff;
    box-shadow: 0 3px 8px rgba(34,183,255,0.3);
  }

  .stage {
    width: 100%;
    max-width: 440px;
    height: 170px;
    display: flex;
    flex-wrap: wrap;
    align-items: center;
    justify-content: center;
    gap: 12px;
    padding: 10px;
  }

  .pack {
    background: #ffffff;
    border: 2px solid #bae6fd;
    border-radius: 12px;
    padding: 10px 14px;
    display: flex;
    flex-direction: column;
    align-items: center;
    gap: 4px;
    box-shadow: 0 4px 12px rgba(0,0,0,0.05);
    animation: popIn 0.3s ease-out;
  }

  .pack-label {
    font-size: 13px;
    font-weight: 700;
    color: #0284c7;
  }

  .stars-row {
    font-size: 20px;
    letter-spacing: 2px;
    line-height: 1;
  }

  @keyframes popIn {
    from { transform: scale(0.85); opacity: 0; }
    to { transform: scale(1); opacity: 1; }
  }

  .formula-card {
    width: 100%;
    max-width: 440px;
    background: #ffffff;
    border: 1px solid #e2e8f0;
    border-radius: 12px;
    padding: 12px 16px;
    display: flex;
    align-items: center;
    justify-content: space-around;
    box-shadow: 0 2px 6px rgba(0,0,0,0.04);
  }

  .metric {
    display: flex;
    flex-direction: column;
    align-items: center;
    gap: 2px;
  }

  .metric-label {
    font-size: 13px;
    font-weight: 600;
    color: #64748b;
  }

  .metric-val {
    font-size: 24px;
    font-weight: 800;
    color: #1e2945;
  }

  .mult-badge {
    background: #f0f9ff;
    border: 1px solid #bae6fd;
    color: #0284c7;
    font-weight: 800;
    font-size: 15px;
    padding: 6px 12px;
    border-radius: 20px;
  }
</style>
</head>
<body>

  <div class="nav-bar">
    <button class="step-btn active" onclick="setCount(1)">1 Pack</button>
    <button class="step-btn" onclick="setCount(2)">2 Packs</button>
    <button class="step-btn" onclick="setCount(3)">3 Packs</button>
    <button class="step-btn" onclick="setCount(4)">4 Packs</button>
  </div>

  <div class="stage" id="stage"></div>

  <div class="formula-card">
    <div class="metric">
      <span class="metric-label">Packs</span>
      <span class="metric-val" id="numPacks">1</span>
    </div>

    <div class="mult-badge">× 3</div>

    <div class="metric">
      <span class="metric-label">Stars</span>
      <span class="metric-val" id="numStars" style="color:#0284c7;">3</span>
    </div>
  </div>

<script>
  let count = 1;

  function setCount(n) {
    count = n;
    
    // Update buttons
    const btns = document.querySelectorAll('.step-btn');
    btns.forEach((btn, idx) => {
      btn.classList.toggle('active', idx + 1 === n);
    });

    // Update numbers
    document.getElementById('numPacks').textContent = n;
    document.getElementById('numStars').textContent = n * 3;

    // Render packs
    const stage = document.getElementById('stage');
    stage.innerHTML = '';
    for (let i = 1; i <= n; i++) {
      const pack = document.createElement('div');
      pack.className = 'pack';
      pack.innerHTML = `
        <div class="pack-label">Pack ${i}</div>
        <div class="stars-row">⭐⭐⭐</div>
      `;
      stage.appendChild(pack);
    }
  }

  setCount(1);
</script>
</body>
</html>

If 1 pack of stickers has 3 stars, then 2 packs have 6 stars, and 3 packs have 9 stars. Both numbers go **up together** by multiplying!

What happens when one number goes UP, but the other number goes DOWN?

## Inverse Proportion: The Sharing Pattern

An **inverse proportion** happens when one number goes up while the other number goes down. If you double the helpers, the time needed gets cut in half!

<!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>Inverse Proportion Diagram</title>
<style>
  * { box-sizing: border-box; margin: 0; padding: 0; }
  body {
    font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, Helvetica, Arial, sans-serif;
    background: #f8f9fa;
    color: #1e2945;
    display: flex;
    flex-direction: column;
    align-items: center;
    justify-content: space-between;
    min-height: 100vh;
    padding: 14px 10px;
    user-select: none;
    -webkit-user-select: none;
  }

  .nav-tabs {
    display: flex;
    gap: 6px;
    flex-wrap: wrap;
    justify-content: center;
    width: 100%;
    max-width: 480px;
  }

  .tab-btn {
    flex: 1;
    min-width: 60px;
    background: #ffffff;
    border: 1.5px solid #e2e8f0;
    border-radius: 8px;
    padding: 8px 4px;
    font-size: 14px;
    font-weight: 600;
    color: #64748b;
    cursor: pointer;
    text-align: center;
    transition: all 0.2s ease;
  }

  .tab-btn.active {
    background: #f97316;
    border-color: #f97316;
    color: #ffffff;
    box-shadow: 0 2px 8px rgba(249, 115, 22, 0.35);
  }

  .stage {
    flex: 1;
    width: 100%;
    max-width: 520px;
    display: flex;
    align-items: center;
    justify-content: center;
    min-height: 220px;
    padding: 8px 0;
  }

  .kids-grid {
    display: grid;
    gap: 8px;
    width: 100%;
    transition: all 0.3s ease;
  }

  .kid-card {
    background: #ffffff;
    border-radius: 10px;
    padding: 8px 10px;
    box-shadow: 0 1px 3px rgba(0,0,0,0.06);
    border: 1px solid #edf2f7;
    display: flex;
    align-items: center;
    gap: 8px;
  }

  .kid-avatar {
    font-size: 22px;
    line-height: 1;
    flex-shrink: 0;
  }

  .cookies-box {
    display: flex;
    flex-wrap: wrap;
    gap: 3px;
    align-items: center;
    flex: 1;
  }

  .cookie-item {
    font-size: 17px;
    line-height: 1;
    animation: popIn 0.25s ease-out;
  }

  .per-kid-count {
    font-size: 13px;
    font-weight: 700;
    color: #f97316;
    background: #fff7ed;
    padding: 2px 6px;
    border-radius: 6px;
    margin-left: auto;
    flex-shrink: 0;
  }

  @keyframes popIn {
    0% { transform: scale(0.5); opacity: 0; }
    100% { transform: scale(1); opacity: 1; }
  }

  .math-bar {
    width: 100%;
    max-width: 480px;
    background: #ffffff;
    border: 1.5px solid #fed7aa;
    border-radius: 10px;
    padding: 10px 14px;
    display: flex;
    align-items: center;
    justify-content: space-around;
    box-shadow: 0 2px 6px rgba(249, 115, 22, 0.08);
  }

  .calc-item {
    display: flex;
    flex-direction: column;
    align-items: center;
  }

  .calc-val {
    font-size: 18px;
    font-weight: 800;
    color: #1e2945;
  }

  .calc-val.highlight {
    color: #f97316;
  }

  .calc-lbl {
    font-size: 11px;
    font-weight: 600;
    color: #94a3b8;
    text-transform: uppercase;
    letter-spacing: 0.5px;
  }

  .calc-op {
    font-size: 18px;
    font-weight: 700;
    color: #cbd5e1;
  }
</style>
</head>
<body>

<div class="nav-tabs">
  <button class="tab-btn active" onclick="setCount(1)">1 Kid</button>
  <button class="tab-btn" onclick="setCount(2)">2 Kids</button>
  <button class="tab-btn" onclick="setCount(3)">3 Kids</button>
  <button class="tab-btn" onclick="setCount(4)">4 Kids</button>
  <button class="tab-btn" onclick="setCount(6)">6 Kids</button>
</div>

<div class="stage">
  <div class="kids-grid" id="kidsGrid"></div>
</div>

<div class="math-bar">
  <div class="calc-item">
    <div class="calc-val" id="numKids">1</div>
    <div class="calc-lbl">Kids</div>
  </div>
  <div class="calc-op">×</div>
  <div class="calc-item">
    <div class="calc-val" id="perKid">12</div>
    <div class="calc-lbl">Each</div>
  </div>
  <div class="calc-op">=</div>
  <div class="calc-item">
    <div class="calc-val highlight">12</div>
    <div class="calc-lbl">Total</div>
  </div>
</div>

<script>
  const avatars = ['🧑', '👧', '👦', '🧒', '👱', '👩'];
  const TOTAL = 12;
  let currentKids = 1;

  function render() {
    const cookiesEach = TOTAL / currentKids;
    const grid = document.getElementById('kidsGrid');
    
    // Adjust layout columns based on kid count
    if (currentKids <= 2) {
      grid.style.gridTemplateColumns = '1fr';
    } else {
      grid.style.gridTemplateColumns = 'repeat(2, 1fr)';
    }

    grid.innerHTML = '';
    for (let i = 0; i < currentKids; i++) {
      const card = document.createElement('div');
      card.className = 'kid-card';

      const avatar = document.createElement('div');
      avatar.className = 'kid-avatar';
      avatar.textContent = avatars[i % avatars.length];

      const cookiesBox = document.createElement('div');
      cookiesBox.className = 'cookies-box';
      for (let c = 0; c < cookiesEach; c++) {
        const cookie = document.createElement('span');
        cookie.className = 'cookie-item';
        cookie.textContent = '🍪';
        cookiesBox.appendChild(cookie);
      }

      const countBadge = document.createElement('div');
      countBadge.className = 'per-kid-count';
      countBadge.textContent = cookiesEach;

      card.appendChild(avatar);
      card.appendChild(cookiesBox);
      card.appendChild(countBadge);
      grid.appendChild(card);
    }

    document.getElementById('numKids').textContent = currentKids;
    document.getElementById('perKid').textContent = cookiesEach;

    document.querySelectorAll('.tab-btn').forEach((btn, idx) => {
      const counts = [1, 2, 3, 4, 6];
      btn.classList.toggle('active', counts[idx] === currentKids);
    });
  }

  function setCount(n) {
    currentKids = n;
    render();
  }

  render();
</script>

</body>
</html>

With our 12 cookies, 1 kid gets 12 cookies, but 2 kids get 6 each, and 3 kids get 4 each. In inverse proportion, multiplying the two numbers always gives the **same total**.

Do all numbers follow these neat multiplying rules, or do some behave differently?

A **non-proportional** relationship does not follow a strict multiplication pattern. For example, when you turn 6 years old, you are not twice as tall as when you were 3!

<!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: transparent;
      padding: 12px;
      display: flex;
      flex-direction: column;
      align-items: center;
      gap: 12px;
      user-select: none;
    }

    .nav-toggle {
      display: flex;
      gap: 8px;
      width: 100%;
      max-width: 420px;
    }

    .toggle-btn {
      flex: 1;
      padding: 8px 12px;
      font-size: 14px;
      font-weight: 700;
      border: 2px solid transparent;
      border-radius: 8px;
      cursor: pointer;
      background: #f1f5f9;
      color: #64748b;
      transition: all 0.2s ease;
      display: flex;
      align-items: center;
      justify-content: center;
      gap: 6px;
    }

    .toggle-btn.active-prop {
      background: #ecfdf5;
      color: #059669;
      border-color: #10b981;
    }

    .toggle-btn.active-non {
      background: #fff7ed;
      color: #ea580c;
      border-color: #f97316;
    }

    .main-grid {
      display: grid;
      grid-template-columns: 1fr 1fr;
      gap: 12px;
      width: 100%;
      max-width: 460px;
      align-items: center;
    }

    @media (max-width: 380px) {
      .main-grid {
        grid-template-columns: 1fr;
      }
    }

    .panel-box {
      background: #ffffff;
      border-radius: 10px;
      padding: 10px;
      box-shadow: 0 1px 4px rgba(0,0,0,0.06);
      border: 1px solid #e2e8f0;
      display: flex;
      flex-direction: column;
      align-items: center;
      min-height: 190px;
    }

    .table-view {
      width: 100%;
      border-collapse: collapse;
      text-align: center;
      font-size: 13px;
    }

    .table-view th {
      color: #64748b;
      font-size: 12px;
      font-weight: 600;
      padding: 4px 2px 6px;
      border-bottom: 2px solid #e2e8f0;
    }

    .table-view td {
      padding: 5px 2px;
      border-bottom: 1px solid #f1f5f9;
      font-variant-numeric: tabular-nums;
      font-weight: 600;
    }

    .ratio-col {
      font-weight: 700;
      border-radius: 4px;
      padding: 2px 4px;
    }

    .ratio-match {
      color: #059669;
      background: #d1fae5;
    }

    .ratio-diff {
      color: #dc2626;
      background: #fee2e2;
    }

    .graph-container {
      position: relative;
      width: 100%;
      aspect-ratio: 1 / 1;
      max-width: 190px;
    }

    svg {
      width: 100%;
      height: 100%;
      overflow: visible;
    }

    .status-badge {
      width: 100%;
      max-width: 420px;
      padding: 8px 12px;
      border-radius: 8px;
      font-size: 13px;
      font-weight: 700;
      text-align: center;
      display: flex;
      align-items: center;
      justify-content: center;
      gap: 8px;
      transition: all 0.2s ease;
    }

    .status-prop {
      background: #ecfdf5;
      color: #047857;
      border: 1px solid #a7f3d0;
    }

    .status-non {
      background: #fff7ed;
      color: #c2410c;
      border: 1px solid #fed7aa;
    }
  </style>
</head>
<body>

  <div class="nav-toggle">
    <button id="btnProp" class="toggle-btn active-prop" onclick="setMode('prop')">
      <span>✓</span> Proportional
    </button>
    <button id="btnNon" class="toggle-btn" onclick="setMode('non')">
      <span>✕</span> Non-Proportional
    </button>
  </div>

  <div class="main-grid">
    <div class="panel-box">
      <table class="table-view">
        <thead>
          <tr>
            <th>x</th>
            <th>y</th>
            <th>y ÷ x</th>
          </tr>
        </thead>
        <tbody id="tableBody">
          
        </tbody>
      </table>
    </div>

    <div class="panel-box">
      <div class="graph-container">
        <svg viewBox="-5 -5 110 110" id="graphSvg">
          
          <line x1="0" y1="0" x2="100" y2="0" stroke="#f1f5f9" stroke-width="1.5" />
          <line x1="0" y1="25" x2="100" y2="25" stroke="#f1f5f9" stroke-width="1.5" />
          <line x1="0" y1="50" x2="100" y2="50" stroke="#f1f5f9" stroke-width="1.5" />
          <line x1="0" y1="75" x2="100" y2="75" stroke="#f1f5f9" stroke-width="1.5" />
          <line x1="25" y1="0" x2="25" y2="100" stroke="#f1f5f9" stroke-width="1.5" />
          <line x1="50" y1="0" x2="50" y2="100" stroke="#f1f5f9" stroke-width="1.5" />
          <line x1="75" y1="0" x2="75" y2="100" stroke="#f1f5f9" stroke-width="1.5" />
          <line x1="100" y1="0" x2="100" y2="100" stroke="#f1f5f9" stroke-width="1.5" />

          
          <line x1="0" y1="100" x2="105" y2="100" stroke="#64748b" stroke-width="2" marker-end="url(#arrow)" />
          <line x1="0" y1="100" x2="0" y2="-5" stroke="#64748b" stroke-width="2" marker-end="url(#arrow)" />
          
          
          <text x="-4" y="106" font-size="8" fill="#94a3b8" font-weight="bold">0</text>
          
          
          <line id="plotLine" x1="0" y1="100" x2="100" y2="0" stroke="#10b981" stroke-width="3" stroke-linecap="round" />
          <g id="plotPoints"></g>
        </svg>
      </div>
    </div>
  </div>

  <div id="statusBadge" class="status-badge status-prop">
    <span>Constant Multiplier: ×3 &bull; Starts at (0,0)</span>
  </div>

  <script>
    const dataProp = {
      points: [
        { x: 1, y: 3, calc: '3 ÷ 1 = 3', px: 25, py: 75 },
        { x: 2, y: 6, calc: '6 ÷ 2 = 3', px: 50, py: 50 },
        { x: 3, y: 9, calc: '9 ÷ 3 = 3', px: 75, py: 25 },
        { x: 4, y: 12, calc: '12 ÷ 4 = 3', px: 100, py: 0 }
      ],
      line: { x1: 0, y1: 100, x2: 100, y2: 0, color: '#10b981' },
      isProp: true,
      label: 'Constant Multiplier: ×3 • Starts at (0,0)'
    };

    const dataNon = {
      points: [
        { x: 1, y: 5, calc: '5 ÷ 1 = 5', px: 25, py: 60 },
        { x: 2, y: 7, calc: '7 ÷ 2 = 3.5', px: 50, py: 45 },
        { x: 3, y: 9, calc: '9 ÷ 3 = 3', px: 75, py: 30 },
        { x: 4, y: 11, calc: '11 ÷ 4 = 2.75', px: 100, py: 15 }
      ],
      line: { x1: 0, y1: 75, x2: 100, y2: 15, color: '#f97316' },
      isProp: false,
      label: 'Multiplier Changes • Misses (0,0)'
    };

    function setMode(mode) {
      const isP = mode === 'prop';
      const d = isP ? dataProp : dataNon;

      // Buttons
      document.getElementById('btnProp').className = 'toggle-btn' + (isP ? ' active-prop' : '');
      document.getElementById('btnNon').className = 'toggle-btn' + (!isP ? ' active-non' : '');

      // Status
      const badge = document.getElementById('statusBadge');
      badge.className = 'status-badge ' + (isP ? 'status-prop' : 'status-non');
      badge.innerHTML = (isP ? '✓ ' : '✕ ') + d.label;

      // Table
      const tbody = document.getElementById('tableBody');
      tbody.innerHTML = d.points.map(p => `
        <tr>
          <td>${p.x}</td>
          <td>${p.y}</td>
          <td><span class="ratio-col ${isP ? 'ratio-match' : 'ratio-diff'}">${p.calc.split('=')[1]}</span></td>
        </tr>
      `).join('');

      // Graph Line
      const line = document.getElementById('plotLine');
      line.setAttribute('x1', d.line.x1);
      line.setAttribute('y1', d.line.y1);
      line.setAttribute('x2', d.line.x2);
      line.setAttribute('y2', d.line.y2);
      line.setAttribute('stroke', d.line.color);

      // Graph Points
      const ptsGroup = document.getElementById('plotPoints');
      ptsGroup.innerHTML = d.points.map(p => `
        <circle cx="${p.px}" cy="${p.py}" r="4" fill="${d.line.color}" stroke="#ffffff" stroke-width="1.5" />
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A common mistake is thinking that any numbers going up together must be direct. Always check if dividing them gives the exact same multiplier every time!

Next time you look at a table of numbers, ask yourself: do they multiply to stay equal, or do they share the exact same multiplier?

math

Identifying Proportional and Non-Proportional Relationships. Distinguish between direct proportion, inverse proportion, and non-proportional relationships in data.

Identifying Proportional and Non-Proportional Functions

Identify Direct Variation and Inverse Variation from Data Tables and Equations

Direct and Inverse Proportion: Formulas, Tables, and Graphs

Pedagogical Note regarding Curriculum Level: While labeled under Primary 1 in some administrative schedules, standard MNU 4-08a is strictly a Scottish Curriculum for Excellence Fourth Level outcome (Secondary S1–S3, ages 12–15) due to its abstract demands in algebraic variation, bivariate tabular testing, and non-linear coordinate graphing. Effective instruction utilizes the 'Rule of Four' (Verbal, Numeric/Tabular, Graphical, and Algebraic representations). Begin with concrete data tables where students perform quotient ($\frac{y}{x}$) and product ($x \cdot y$) tests before introducing formal algebraic manipulation ($y = kx$ vs $y = \frac{k}{x}$). Emphasize non-examples (e.g., $y = mx + c$ where $c \neq 0$) to inoculate against the widespread misconception that all linear graphs represent direct proportion.

Identifying Proportional and Non-Proportional Relationships

Direct Proportion: Growing Together

Inverse Proportion: The Sharing Pattern

Non-Proportional: No Perfect Match