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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

Imagine you want to draw a tower for 10 toy cars, but your grid paper only has 5 boxes going up. How can all 10 cars fit on the page?

A **scale** tells us how much each square on our grid paper counts for. When we pick the right scale, all our data fits neatly.

<!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>Choosing a Suitable Scale</title>
<style>
  * { box-sizing: border-box; margin: 0; padding: 0; font-family: system-ui, -apple-system, sans-serif; }
  body {
    background: transparent;
    display: flex;
    flex-direction: column;
    align-items: center;
    justify-content: center;
    padding: 10px;
    color: #1e2945;
  }
  .controls {
    display: flex;
    gap: 8px;
    margin-bottom: 12px;
    flex-wrap: wrap;
    justify-content: center;
  }
  .btn {
    background: #ffffff;
    border: 2px solid #e2e8f0;
    border-radius: 20px;
    padding: 6px 14px;
    font-size: 14px;
    font-weight: 600;
    color: #475569;
    cursor: pointer;
    transition: all 0.25s ease;
  }
  .btn:hover {
    border-color: #22b7ff;
    color: #0284c7;
  }
  .btn.active {
    background: #22b7ff;
    border-color: #22b7ff;
    color: #ffffff;
    box-shadow: 0 4px 10px rgba(34, 183, 255, 0.3);
  }
  .diagram-area {
    width: 100%;
    max-width: 380px;
    height: 300px;
    position: relative;
    background: #ffffff;
    border-radius: 12px;
    box-shadow: 0 2px 10px rgba(0,0,0,0.04);
    overflow: hidden;
  }
  svg {
    width: 100%;
    height: 100%;
    display: block;
  }
  .grid-line {
    stroke: #e2e8f0;
    stroke-width: 1;
  }
  .axis {
    stroke: #64748b;
    stroke-width: 2;
  }
  .tick-text {
    font-size: 11px;
    fill: #64748b;
    font-weight: 500;
    text-anchor: end;
    dominant-baseline: central;
    transition: opacity 0.2s ease;
  }
  .bar {
    fill: #22b7ff;
    transition: y 0.4s cubic-bezier(0.34, 1.56, 0.64, 1), height 0.4s cubic-bezier(0.34, 1.56, 0.64, 1), fill 0.3s ease;
    rx: 4;
  }
  .target-line {
    stroke: #ef4444;
    stroke-dasharray: 4 4;
    stroke-width: 1.5;
  }
  .status-badge {
    position: absolute;
    top: 14px;
    right: 14px;
    padding: 4px 10px;
    border-radius: 12px;
    font-size: 13px;
    font-weight: 700;
    display: flex;
    align-items: center;
    gap: 4px;
    transition: all 0.3s ease;
  }
  .badge-overflow {
    background: #fee2e2;
    color: #dc2626;
  }
  .badge-perfect {
    background: #dcfce7;
    color: #15803d;
  }
  .badge-squished {
    background: #fef3c7;
    color: #b45309;
  }
  .header-data {
    position: absolute;
    top: 14px;
    left: 14px;
    font-size: 14px;
    font-weight: 700;
    color: #1e2945;
    background: #f1f5f9;
    padding: 4px 10px;
    border-radius: 12px;
  }
</style>
</head>
<body>

<div class="controls">
  <button class="btn" onclick="setScale(1)">1s</button>
  <button class="btn active" onclick="setScale(2)">2s</button>
  <button class="btn" onclick="setScale(5)">5s</button>
</div>

<div class="diagram-area">
  <div class="header-data">Data: 18</div>
  <div id="badge" class="status-badge badge-perfect">Fits neatly ✓</div>
  
  <svg viewBox="0 0 360 280">
    
    <g id="gridLines"></g>

    
    <line x1="60" y1="240" x2="330" y2="240" class="axis" />
    <line x1="60" y1="240" x2="60" y2="35" class="axis" />

    
    <g id="tickLabels"></g>

    
    <rect id="dataBar" class="bar" x="140" y="60" width="80" height="180" />

    
    <g id="overflowMarker" style="opacity: 0; transition: opacity 0.3s ease;">
      <line x1="130" y1="40" x2="230" y2="40" class="target-line" />
      <text x="180" y="28" fill="#ef4444" font-size="12" font-weight="700" text-anchor="middle">Needs 18 squares!</text>
    </g>
  </svg>
</div>

<script>
  const gridLinesG = document.getElementById('gridLines');
  const tickLabelsG = document.getElementById('tickLabels');
  const bar = document.getElementById('dataBar');
  const badge = document.getElementById('badge');
  const overflowMarker = document.getElementById('overflowMarker');
  const buttons = document.querySelectorAll('.btn');

  // Build grid lines
  for (let i = 0; i <= 10; i++) {
    const y = 240 - (i * 20);
    const line = document.createElementNS('http://www.w3.org/2000/svg', 'line');
    line.setAttribute('x1', '60');
    line.setAttribute('y1', y);
    line.setAttribute('x2', '330');
    line.setAttribute('y2', y);
    line.setAttribute('class', 'grid-line');
    gridLinesG.appendChild(line);
  }

  const scaleConfigs = {
    1: {
      ticks: [0, 2, 4, 6, 8, 10], // step 2 labels
      step: 1,
      barHeight: 200, // clamped visually at max top
      barY: 40,
      barColor: '#ef4444',
      badgeText: 'Off grid ✕',
      badgeClass: 'badge-overflow',
      overflow: true
    },
    2: {
      ticks: [0, 4, 8, 12, 16, 20],
      step: 2,
      barHeight: 180, // 18 / 2 * 20px = 180px
      barY: 60,       // 240 - 180 = 60
      barColor: '#22b7ff',
      badgeText: 'Fits neatly ✓',
      badgeClass: 'badge-perfect',
      overflow: false
    },
    5: {
      ticks: [0, 10, 20, 30, 40, 50],
      step: 5,
      barHeight: 72,  // 18 / 5 * 20px = 72px
      barY: 168,      // 240 - 72 = 168
      barColor: '#f59e0b',
      badgeText: 'Too squished',
      badgeClass: 'badge-squished',
      overflow: false
    }
  };

  function updateTicks(scale) {
    tickLabelsG.innerHTML = '';
    for (let i = 0; i <= 10; i++) {
      const val = i * scale;
      // Show alternate ticks for clean spacing
      if (i % 2 === 0 || scale === 1 && i % 2 === 0) {
        const y = 240 - (i * 20);
        const text = document.createElementNS('http://www.w3.org/2000/svg', 'text');
        text.setAttribute('x', '52');
        text.setAttribute('y', y);
        text.setAttribute('class', 'tick-text');
        text.textContent = val;
        tickLabelsG.appendChild(text);
      }
    }
  }

  function setScale(scale) {
    buttons.forEach(b => {
      b.classList.toggle('active', b.textContent.startsWith(scale + 's'));
    });

    const cfg = scaleConfigs[scale];
    updateTicks(scale);

    bar.setAttribute('y', cfg.barY);
    bar.setAttribute('height', cfg.barHeight);
    bar.style.fill = cfg.barColor;

    badge.textContent = cfg.badgeText;
    badge.className = 'status-badge ' + cfg.badgeClass;

    overflowMarker.style.opacity = cfg.overflow ? '1' : '0';
  }

  // Initial load
  setScale(2);
</script>

</body>
</html>

What happens when our numbers are small enough to count one by one?

If your biggest number is 4 and you have 6 boxes, each box can stand for **1**. Counting by ones makes your graph easy and clear to read.

<!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: #f8f9fa;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      min-height: 100%;
      padding: 12px;
      user-select: none;
    }

    .main-wrap {
      width: 100%;
      max-width: 440px;
      display: flex;
      flex-direction: column;
      align-items: center;
      gap: 12px;
    }

    /* Controls */
    .btn-group {
      display: flex;
      gap: 6px;
      background: #e5e7eb;
      padding: 3px;
      border-radius: 10px;
      width: 100%;
      justify-content: center;
    }
    .btn {
      flex: 1;
      padding: 6px 8px;
      font-size: 13px;
      font-weight: 600;
      border: none;
      background: transparent;
      color: #4b5563;
      border-radius: 7px;
      cursor: pointer;
      transition: background 0.2s, color 0.2s;
      text-align: center;
    }
    .btn.active {
      background: #ffffff;
      color: #1e2945;
      box-shadow: 0 1px 3px rgba(0,0,0,0.1);
    }

    /* Graph Area */
    .graph-stage {
      width: 100%;
      display: flex;
      justify-content: center;
      align-items: center;
      gap: 16px;
      background: #ffffff;
      padding: 14px 20px;
      border-radius: 12px;
      border: 1px solid #e5e7eb;
    }

    .axis-container {
      display: flex;
      align-items: flex-end;
      height: 216px; /* 6 boxes * 36px */
      position: relative;
    }

    /* Y-Axis labels */
    .y-axis {
      display: flex;
      flex-direction: column-reverse;
      justify-content: space-between;
      height: 216px;
      padding-right: 8px;
      text-align: right;
      font-size: 13px;
      font-weight: 600;
      color: #4b5563;
    }
    .y-tick {
      height: 0;
      display: flex;
      align-items: center;
      justify-content: flex-end;
      position: relative;
    }
    .y-tick span {
      position: absolute;
      right: 0;
      transform: translateY(-50%);
      font-variant-numeric: tabular-nums;
    }

    /* Grid Paper */
    .grid-paper {
      width: 140px;
      height: 216px;
      display: flex;
      flex-direction: column-reverse;
      border-left: 2px solid #1e2945;
      border-bottom: 2px solid #1e2945;
      position: relative;
      background: 
        linear-gradient(to top, rgba(34, 183, 255, 0.08) 1px, transparent 1px),
        linear-gradient(to right, rgba(34, 183, 255, 0.08) 1px, transparent 1px);
      background-size: 35px 36px;
      overflow: visible;
    }

    .grid-box-marker {
      position: absolute;
      left: 0;
      right: 0;
      border-top: 1px solid #cbd5e1;
    }

    /* Data Bar */
    .bar-wrap {
      position: absolute;
      left: 35px;
      bottom: 0;
      width: 70px;
      height: 100%;
      display: flex;
      align-items: flex-end;
    }
    .bar {
      width: 100%;
      height: 0;
      border-radius: 6px 6px 0 0;
      transition: height 0.35s cubic-bezier(0.4, 0, 0.2, 1), background 0.35s ease;
      display: flex;
      align-items: flex-start;
      justify-content: center;
      padding-top: 4px;
      color: #ffffff;
      font-size: 13px;
      font-weight: 700;
      box-shadow: inset 0 -2px 0 rgba(0,0,0,0.1);
    }

    /* Overflow Indicator */
    .overflow-warning {
      position: absolute;
      top: -24px;
      left: 50%;
      transform: translateX(-50%);
      background: #ef4444;
      color: #ffffff;
      font-size: 11px;
      font-weight: 700;
      padding: 2px 8px;
      border-radius: 10px;
      opacity: 0;
      transition: opacity 0.2s;
      white-space: nowrap;
      pointer-events: none;
    }
    .overflow-warning.show {
      opacity: 1;
    }

    /* Feedback Banner */
    .status-badge {
      display: flex;
      align-items: center;
      justify-content: center;
      gap: 6px;
      font-size: 14px;
      font-weight: 600;
      padding: 6px 14px;
      border-radius: 20px;
      transition: all 0.25s;
    }
    .status-fit { background: #dcfce7; color: #15803d; }
    .status-squish { background: #fef3c7; color: #b45309; }
    .status-overflow { background: #fee2e2; color: #b91c1c; }
  </style>
</head>
<body>

<div class="main-wrap">
  
  <div class="btn-group" id="dataGroup">
    <button class="btn active" onclick="setData(4)">Data: 4</button>
    <button class="btn" onclick="setData(8)">Data: 8</button>
  </div>

  
  <div class="graph-stage">
    <div class="axis-container">
      <div class="y-axis" id="yAxis"></div>
      <div class="grid-paper">
        <div class="grid-box-marker" style="top: 0;"></div>
        <div class="grid-box-marker" style="top: 36px;"></div>
        <div class="grid-box-marker" style="top: 72px;"></div>
        <div class="grid-box-marker" style="top: 108px;"></div>
        <div class="grid-box-marker" style="top: 144px;"></div>
        <div class="grid-box-marker" style="top: 180px;"></div>
        
        <div class="bar-wrap">
          <div class="overflow-warning" id="overflowTag">⚠️ Off Grid!</div>
          <div class="bar" id="dataBar">4</div>
        </div>
      </div>
    </div>
  </div>

  
  <div class="btn-group" id="scaleGroup">
    <button class="btn active" onclick="setScale(1)">Scale: 1s</button>
    <button class="btn" onclick="setScale(2)">Scale: 2s</button>
    <button class="btn" onclick="setScale(5)">Scale: 5s</button>
  </div>

  
  <div class="status-badge status-fit" id="statusBadge">
    <span>✓ Fits clearly (4 boxes)</span>
  </div>
</div>

<script>
  let currentData = 4;
  let currentScale = 1;

  const BOX_HEIGHT = 36; // px
  const TOTAL_BOXES = 6;

  function renderAxis() {
    const yAxis = document.getElementById('yAxis');
    yAxis.innerHTML = '';
    for (let i = 0; i <= TOTAL_BOXES; i++) {
      const tick = document.createElement('div');
      tick.className = 'y-tick';
      tick.innerHTML = `<span>${i * currentScale}</span>`;
      yAxis.appendChild(tick);
    }
  }

  function update() {
    renderAxis();

    const boxesNeeded = currentData / currentScale;
    const bar = document.getElementById('dataBar');
    const badge = document.getElementById('statusBadge');
    const overflow = document.getElementById('overflowTag');

    bar.textContent = currentData;

    if (boxesNeeded > TOTAL_BOXES) {
      // Off grid
      bar.style.height = (TOTAL_BOXES * BOX_HEIGHT) + 'px';
      bar.style.background = '#ef4444';
      overflow.classList.add('show');
      badge.className = 'status-badge status-overflow';
      badge.innerHTML = `<span>✗ Scale too small (Needs ${boxesNeeded} boxes)</span>`;
    } else {
      overflow.classList.remove('show');
      bar.style.height = (boxesNeeded * BOX_HEIGHT) + 'px';

      if (boxesNeeded >= 3) {
        // Ideal fit (uses 50% - 100% of grid)
        bar.style.background = '#22b7ff';
        badge.className = 'status-badge status-fit';
        badge.innerHTML = `<span>✓ Clear fit (${boxesNeeded} boxes)</span>`;
      } else {
        // Squished (uses < 50% of grid)
        bar.style.background = '#f59e0b';
        badge.className = 'status-badge status-squish';
        badge.innerHTML = `<span>⚠️ Too squished (${boxesNeeded} boxes)</span>`;
      }
    }
  }

  function setData(val) {
    currentData = val;
    document.querySelectorAll('#dataGroup .btn').forEach(btn => {
      btn.classList.toggle('active', btn.textContent.includes(val));
    });
    update();
  }

  function setScale(scale) {
    currentScale = scale;
    document.querySelectorAll('#scaleGroup .btn').forEach(btn => {
      btn.classList.toggle('active', btn.textContent.includes(scale + 's'));
    });
    update();
  }

  // Init
  update();
</script>

</body>
</html>

What if your numbers get a little bigger, like 8 or 10?

When the page is too short, we can make each grid box count for **2**. We skip-count: 0, 2, 4, 6, 8, 10.

<!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;
      font-family: system-ui, -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
      user-select: none;
    }

    body {
      padding: 12px;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      gap: 12px;
      color: #1e2945;
      background: transparent;
    }

    .toolbar {
      display: flex;
      gap: 8px;
      align-items: center;
      justify-content: center;
      flex-wrap: wrap;
    }

    .btn {
      padding: 6px 14px;
      font-size: 14px;
      font-weight: 600;
      border: 1px solid #d1d5db;
      border-radius: 20px;
      background: #ffffff;
      color: #1e2945;
      cursor: pointer;
      transition: background 0.2s, color 0.2s, border-color 0.2s, box-shadow 0.2s;
    }

    .btn.active {
      background: #22b7ff;
      color: #ffffff;
      border-color: #22b7ff;
      box-shadow: 0 2px 6px rgba(34, 183, 255, 0.35);
    }

    .comparison {
      display: flex;
      gap: 16px;
      justify-content: center;
      align-items: flex-start;
      width: 100%;
      max-width: 440px;
    }

    .grid-card {
      flex: 1;
      background: #ffffff;
      border: 1px solid #e5e7eb;
      border-radius: 12px;
      padding: 10px 8px 12px;
      display: flex;
      flex-direction: column;
      align-items: center;
      gap: 8px;
      box-shadow: 0 1px 3px rgba(0,0,0,0.05);
    }

    .grid-title {
      font-size: 14px;
      font-weight: 700;
    }

    .grid-wrap {
      display: flex;
      align-items: flex-end;
      gap: 4px;
      height: 200px;
      position: relative;
    }

    .axis-y {
      display: flex;
      flex-direction: column-reverse;
      justify-content: space-between;
      height: 180px;
      padding-bottom: 0;
      font-size: 12px;
      font-weight: 600;
      color: #6b7280;
      text-align: right;
      width: 22px;
    }

    .grid-body {
      width: 80px;
      height: 180px;
      border-left: 2px solid #1e2945;
      border-bottom: 2px solid #1e2945;
      position: relative;
      background-size: 100% 36px;
      background-image: linear-gradient(to top, #e5e7eb 1px, transparent 1px);
    }

    .bar {
      position: absolute;
      bottom: 0;
      left: 12px;
      right: 12px;
      border-radius: 4px 4px 0 0;
      transition: height 0.4s cubic-bezier(0.4, 0, 0.2, 1), background-color 0.4s;
    }

    .bar-fit {
      background: #22b7ff;
    }

    .bar-overflow {
      background: #ef4444;
      border-top: 3px dashed #991b1b;
    }

    .status-badge {
      display: flex;
      align-items: center;
      gap: 4px;
      font-size: 13px;
      font-weight: 700;
      height: 24px;
    }

    .status-ok {
      color: #059669;
    }

    .status-bad {
      color: #dc2626;
    }
  </style>
</head>
<body>

  <div class="toolbar">
    <button class="btn" onclick="setItems(6)" id="btn-6">6 Items</button>
    <button class="btn active" onclick="setItems(10)" id="btn-10">10 Items</button>
    <button class="btn" onclick="setItems(20)" id="btn-20">20 Items</button>
  </div>

  <div class="comparison">
    
    <div class="grid-card">
      <div class="grid-title">Scale: 1</div>
      <div class="grid-wrap">
        <div class="axis-y" id="axis-1">
          <span>5</span>
          <span>4</span>
          <span>3</span>
          <span>2</span>
          <span>1</span>
          <span>0</span>
        </div>
        <div class="grid-body">
          <div class="bar" id="bar-1"></div>
        </div>
      </div>
      <div class="status-badge" id="badge-1"></div>
    </div>

    
    <div class="grid-card">
      <div class="grid-title" id="title-right">Scale: 2</div>
      <div class="grid-wrap">
        <div class="axis-y" id="axis-2">
          <span>10</span>
          <span>8</span>
          <span>6</span>
          <span>4</span>
          <span>2</span>
          <span>0</span>
        </div>
        <div class="grid-body">
          <div class="bar" id="bar-2"></div>
        </div>
      </div>
      <div class="status-badge" id="badge-2"></div>
    </div>
  </div>

  <script>
    let currentVal = 10;

    function setItems(val) {
      currentVal = val;
      
      document.querySelectorAll('.toolbar .btn').forEach(b => b.classList.remove('active'));
      document.getElementById('btn-' + val).classList.add('active');

      const maxGridBoxes = 5;
      const boxHeightPx = 36;
      const maxGridPx = 180;

      // Scale 1 (1 box = 1 item, max capacity = 5)
      const bar1 = document.getElementById('bar-1');
      const badge1 = document.getElementById('badge-1');
      const height1 = (val / 1) * boxHeightPx;

      if (val <= 5) {
        bar1.style.height = height1 + 'px';
        bar1.className = 'bar bar-fit';
        badge1.className = 'status-badge status-ok';
        badge1.innerHTML = '✓ Fits';
      } else {
        bar1.style.height = maxGridPx + 'px';
        bar1.className = 'bar bar-overflow';
        badge1.className = 'status-badge status-bad';
        badge1.innerHTML = '✕ Too tall';
      }

      // Scale 2 or 5 depending on target
      const titleRight = document.getElementById('title-right');
      const axis2 = document.getElementById('axis-2');
      const bar2 = document.getElementById('bar-2');
      const badge2 = document.getElementById('badge-2');

      let scale2 = (val === 20) ? 5 : 2;
      titleRight.textContent = 'Scale: ' + scale2;

      axis2.innerHTML = '';
      for (let i = 5; i >= 0; i--) {
        const span = document.createElement('span');
        span.textContent = i * scale2;
        axis2.appendChild(span);
      }

      const height2 = (val / scale2) * boxHeightPx;
      if (height2 <= maxGridPx) {
        bar2.style.height = height2 + 'px';
        bar2.className = 'bar bar-fit';
        badge2.className = 'status-badge status-ok';
        badge2.innerHTML = '✓ Fits';
      } else {
        bar2.style.height = maxGridPx + 'px';
        bar2.className = 'bar bar-overflow';
        badge2.className = 'status-badge status-bad';
        badge2.innerHTML = '✕ Too tall';
      }
    }

    setItems(10);
  </script>
</body>
</html>

What if you have even more items, like 20 stickers?

For big groups of items, each square can stand for **5**. Skip-counting by fives lets you fit 25 items in just 5 squares!

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

    body {
      background: #ffffff;
      color: #1e2945;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: space-between;
      min-height: 100%;
      padding: 12px;
      user-select: none;
    }

    .controls {
      display: flex;
      gap: 8px;
      width: 100%;
      max-width: 360px;
      justify-content: center;
    }

    .btn-scale {
      flex: 1;
      padding: 8px 10px;
      font-size: 14px;
      font-weight: 700;
      border-radius: 10px;
      border: 2px solid #e2e8f0;
      background: #f8f9fa;
      color: #64748b;
      cursor: pointer;
      transition: all 0.25s ease;
      text-align: center;
    }

    .btn-scale.active {
      background: #22b7ff;
      border-color: #22b7ff;
      color: #ffffff;
      box-shadow: 0 4px 10px rgba(34, 183, 255, 0.3);
    }

    .graph-stage {
      position: relative;
      width: 100%;
      max-width: 360px;
      height: 270px;
      display: flex;
      align-items: flex-end;
      justify-content: center;
      padding-bottom: 20px;
      padding-left: 45px;
    }

    /* Grid Area */
    .grid-container {
      position: relative;
      width: 220px;
      height: 240px;
      border-left: 3px solid #1e2945;
      border-bottom: 3px solid #1e2945;
      background: 
        repeating-linear-gradient(to top, transparent, transparent 39px, #e2e8f0 39px, #e2e8f0 40px),
        repeating-linear-gradient(to right, transparent, transparent 39px, #e2e8f0 39px, #e2e8f0 40px);
      background-size: 40px 40px;
    }

    /* Axis Ticks */
    .axis-y {
      position: absolute;
      left: -48px;
      top: 0;
      bottom: 0;
      width: 42px;
      display: flex;
      flex-direction: column-reverse;
      justify-content: space-between;
      align-items: flex-end;
      font-size: 13px;
      font-weight: 700;
      color: #64748b;
      padding-bottom: 2px;
    }

    .tick-label {
      height: 0;
      display: flex;
      align-items: center;
      justify-content: flex-end;
      transition: color 0.2s ease;
    }

    .tick-label.target {
      color: #22b7ff;
      font-weight: 800;
    }

    /* Bars */
    .bar-wrapper {
      position: absolute;
      bottom: 0;
      left: 70px;
      width: 80px;
      height: 100%;
      display: flex;
      flex-direction: column-reverse;
      align-items: center;
    }

    .bar {
      width: 100%;
      background: linear-gradient(180deg, #22b7ff 0%, #009be5 100%);
      border-radius: 6px 6px 0 0;
      transition: height 0.4s cubic-bezier(0.34, 1.2, 0.64, 1);
      position: relative;
    }

    .bar.overflow {
      background: linear-gradient(180deg, #ef4444 0%, #dc2626 100%);
    }

    .bar-label {
      position: absolute;
      top: -24px;
      left: 50%;
      transform: translateX(-50%);
      font-size: 14px;
      font-weight: 800;
      color: #1e2945;
      white-space: nowrap;
      transition: all 0.3s ease;
    }

    .bar.overflow .bar-label {
      color: #dc2626;
      top: 6px;
      background: rgba(255, 255, 255, 0.9);
      padding: 2px 6px;
      border-radius: 4px;
    }

    /* Result pill */
    .status-pill {
      display: flex;
      align-items: center;
      gap: 6px;
      padding: 6px 14px;
      border-radius: 20px;
      font-size: 14px;
      font-weight: 700;
      background: #f1f5f9;
      color: #475569;
      transition: all 0.25s ease;
    }

    .status-pill.good {
      background: #ecfdf5;
      color: #059669;
    }

    .status-pill.bad {
      background: #fef2f2;
      color: #dc2626;
    }
  </style>
</head>
<body>

  
  <div class="controls">
    <button class="btn-scale" data-scale="1">Scale: 1</button>
    <button class="btn-scale active" data-scale="5">Scale: 5</button>
    <button class="btn-scale" data-scale="10">Scale: 10</button>
  </div>

  
  <div class="graph-stage">
    <div class="grid-container">
      <div class="axis-y" id="axisY">
        
      </div>
      <div class="bar-wrapper">
        <div class="bar" id="dataBar" style="height: 200px;">
          <div class="bar-label" id="barLabel">25 items</div>
        </div>
      </div>
    </div>
  </div>

  
  <div class="status-pill good" id="statusPill">
    <span>✓ Fits in 5 squares</span>
  </div>

  <script src="https://cdn.jsdelivr.net/npm/classace-diagram.js@1.0.0/dist/classace-diagram.js"></script>
  <script>
    const scales = {
      '1': {
        ticks: [0, 1, 2, 3, 4, 5, 6],
        heightPx: 240, // 25 would be 1000px (overflows ceiling)
        overflow: true,
        pillText: '✕ Needs 25 squares (Too tall)',
        pillClass: 'bad',
        targetVal: '25'
      },
      '5': {
        ticks: [0, 5, 10, 15, 20, 25, 30],
        heightPx: 200, // 5 * 40px = 200px
        overflow: false,
        pillText: '✓ Fits in 5 squares',
        pillClass: 'good',
        targetVal: '25'
      },
      '10': {
        ticks: [0, 10, 20, 30, 40, 50, 60],
        heightPx: 100, // 2.5 * 40px = 100px
        overflow: false,
        pillText: '✓ Fits in 2.5 squares',
        pillClass: 'good',
        targetVal: '25'
      }
    };

    const axisY = document.getElementById('axisY');
    const dataBar = document.getElementById('dataBar');
    const barLabel = document.getElementById('barLabel');
    const statusPill = document.getElementById('statusPill');
    const buttons = document.querySelectorAll('.btn-scale');

    function applyScale(scaleKey) {
      const cfg = scales[scaleKey];

      // Update buttons
      buttons.forEach(btn => {
        btn.classList.toggle('active', btn.dataset.scale === scaleKey);
      });

      // Update Axis Ticks
      axisY.innerHTML = '';
      cfg.ticks.forEach(val => {
        const span = document.createElement('span');
        span.className = 'tick-label' + (val === 25 ? ' target' : '');
        span.textContent = val;
        axisY.appendChild(span);
      });

      // Update Bar Height & Overflow State
      dataBar.style.height = cfg.heightPx + 'px';
      dataBar.classList.toggle('overflow', cfg.overflow);

      if (cfg.overflow) {
        barLabel.textContent = '25 (Off grid!)';
      } else {
        barLabel.textContent = '25 items';
      }

      // Update Status Pill
      statusPill.className = 'status-pill ' + cfg.pillClass;
      statusPill.textContent = cfg.pillText;
    }

    buttons.forEach(btn => {
      btn.addEventListener('click', () => {
        applyScale(btn.dataset.scale);
      });
    });

    // Initialize with Scale: 5
    applyScale('5');
  </script>
</body>
</html>

Can we change how much we count by in the middle of our grid?

To **Maintain Equal Spacing**, every physical grid square along an axis must advance by identical **Uniform Increments**. Changing step sizes midway distorts your graph and makes data misleading.

<!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: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
    background: #f8f9fa;
    color: #1e2945;
    display: flex;
    flex-direction: column;
    align-items: center;
    justify-content: center;
    min-height: 100%;
    padding: 12px;
    user-select: none;
  }
  .controls {
    display: flex;
    gap: 8px;
    margin-bottom: 12px;
    flex-wrap: wrap;
    justify-content: center;
  }
  .scale-btn {
    background: #ffffff;
    border: 2px solid #e2e8f0;
    border-radius: 8px;
    padding: 6px 14px;
    font-size: 15px;
    font-weight: 700;
    color: #475569;
    cursor: pointer;
    transition: all 0.2s ease;
  }
  .scale-btn:hover {
    border-color: #22b7ff;
  }
  .scale-btn.active {
    background: #22b7ff;
    border-color: #22b7ff;
    color: #ffffff;
  }
  .scale-btn.bad-btn.active {
    background: #ef4444;
    border-color: #ef4444;
    color: #ffffff;
  }
  .diagram-stage {
    position: relative;
    width: 100%;
    max-width: 380px;
    height: 290px;
    background: #ffffff;
    border: 1px solid #e2e8f0;
    border-radius: 12px;
    box-shadow: 0 4px 12px rgba(0,0,0,0.04);
    display: flex;
    align-items: center;
    justify-content: center;
  }
  svg {
    width: 100%;
    height: 100%;
  }
  .step-label {
    font-size: 12px;
    font-weight: 700;
    transition: all 0.3s ease;
  }
  .axis-num {
    font-size: 14px;
    font-weight: 700;
    fill: #1e2945;
    text-anchor: end;
    dominant-baseline: central;
    transition: all 0.25s ease;
  }
  .badge {
    position: absolute;
    bottom: 12px;
    right: 14px;
    font-size: 14px;
    font-weight: 700;
    padding: 4px 10px;
    border-radius: 6px;
    transition: all 0.2s ease;
  }
  .badge.good {
    background: #dcfce7;
    color: #15803d;
  }
  .badge.bad {
    background: #fee2e2;
    color: #b91c1c;
  }
</style>
</head>
<body>

<div class="controls">
  <button class="scale-btn active" data-scale="2">+2</button>
  <button class="scale-btn" data-scale="5">+5</button>
  <button class="scale-btn" data-scale="10">+10</button>
  <button class="scale-btn bad-btn" data-scale="bad">Mixed ✗</button>
</div>

<div class="diagram-stage">
  <svg viewBox="0 0 340 260">
    <defs>
      
      <pattern id="grid" width="36" height="36" patternUnits="userSpaceOnUse">
        <path d="M 36 0 L 0 0 0 36" fill="none" stroke="#e2e8f0" stroke-width="1.2"/>
      </pattern>
    </defs>

    
    <rect x="76" y="20" width="180" height="216" fill="url(#grid)" />
    <rect x="76" y="20" width="180" height="216" fill="none" stroke="#cbd5e1" stroke-width="1.5"/>

    
    <line x1="76" y1="20" x2="76" y2="236" stroke="#1e2945" stroke-width="2.5" stroke-linecap="round"/>
    
    <line x1="76" y1="236" x2="256" y2="236" stroke="#1e2945" stroke-width="2.5" stroke-linecap="round"/>

    
    <line x1="70" y1="236" x2="76" y2="236" stroke="#1e2945" stroke-width="2"/>
    <line x1="70" y1="200" x2="76" y2="200" stroke="#1e2945" stroke-width="2"/>
    <line x1="70" y1="164" x2="76" y2="164" stroke="#1e2945" stroke-width="2"/>
    <line x1="70" y1="128" x2="76" y2="128" stroke="#1e2945" stroke-width="2"/>
    <line x1="70" y1="92"  x2="76" y2="92"  stroke="#1e2945" stroke-width="2"/>
    <line x1="70" y1="56"  x2="76" y2="56"  stroke="#1e2945" stroke-width="2"/>
    <line x1="70" y1="20"  x2="76" y2="20"  stroke="#1e2945" stroke-width="2"/>

    
    <text class="axis-num" x="62" y="236">0</text>
    <text id="n1" class="axis-num" x="62" y="200">2</text>
    <text id="n2" class="axis-num" x="62" y="164">4</text>
    <text id="n3" class="axis-num" x="62" y="128">6</text>
    <text id="n4" class="axis-num" x="62" y="92">8</text>
    <text id="n5" class="axis-num" x="62" y="56">10</text>
    <text id="n6" class="axis-num" x="62" y="20">12</text>

    
    <g id="arcs">
      
      <path id="arc1" d="M 82 230 C 102 230, 102 206, 82 206" fill="none" stroke="#22b7ff" stroke-width="2.5" stroke-linecap="round"/>
      <text id="lbl1" class="step-label" x="110" y="218" fill="#0284c7" dominant-baseline="central">+2</text>

      
      <path id="arc2" d="M 82 194 C 102 194, 102 170, 82 170" fill="none" stroke="#22b7ff" stroke-width="2.5" stroke-linecap="round"/>
      <text id="lbl2" class="step-label" x="110" y="182" fill="#0284c7" dominant-baseline="central">+2</text>

      
      <path id="arc3" d="M 82 158 C 102 158, 102 134, 82 134" fill="none" stroke="#22b7ff" stroke-width="2.5" stroke-linecap="round"/>
      <text id="lbl3" class="step-label" x="110" y="146" fill="#0284c7" dominant-baseline="central">+2</text>

      
      <path id="arc4" d="M 82 122 C 102 122, 102 98, 82 98" fill="none" stroke="#22b7ff" stroke-width="2.5" stroke-linecap="round"/>
      <text id="lbl4" class="step-label" x="110" y="110" fill="#0284c7" dominant-baseline="central">+2</text>

      
      <path id="arc5" d="M 82 86 C 102 86, 102 62, 82 62" fill="none" stroke="#22b7ff" stroke-width="2.5" stroke-linecap="round"/>
      <text id="lbl5" class="step-label" x="110" y="74" fill="#0284c7" dominant-baseline="central">+2</text>

      
      <path id="arc6" d="M 82 50 C 102 50, 102 26, 82 26" fill="none" stroke="#22b7ff" stroke-width="2.5" stroke-linecap="round"/>
      <text id="lbl6" class="step-label" x="110" y="38" fill="#0284c7" dominant-baseline="central">+2</text>
    </g>
  </svg>

  <div id="badge" class="badge good">Equal Step ✓</div>
</div>

<script>
  const scales = {
    '2': {
      nums: [2, 4, 6, 8, 10, 12],
      steps: ['+2', '+2', '+2', '+2', '+2', '+2'],
      valid: true,
      badge: 'Equal Step ✓'
    },
    '5': {
      nums: [5, 10, 15, 20, 25, 30],
      steps: ['+5', '+5', '+5', '+5', '+5', '+5'],
      valid: true,
      badge: 'Equal Step ✓'
    },
    '10': {
      nums: [10, 20, 30, 40, 50, 60],
      steps: ['+10', '+10', '+10', '+10', '+10', '+10'],
      valid: true,
      badge: 'Equal Step ✓'
    },
    'bad': {
      nums: [2, 5, 6, 10, 20, 25],
      steps: ['+2', '+3', '+1', '+4', '+10', '+5'],
      valid: false,
      badge: 'Unequal Step ✗'
    }
  };

  const btns = document.querySelectorAll('.scale-btn');
  const badge = document.getElementById('badge');

  function updateScale(key) {
    const data = scales[key];
    
    // Update numbers
    data.nums.forEach((val, i) => {
      document.getElementById(`n${i+1}`).textContent = val;
    });

    // Update steps & colors
    const strokeColor = data.valid ? '#22b7ff' : '#ef4444';
    const textColor = data.valid ? '#0284c7' : '#dc2626';

    data.steps.forEach((step, i) => {
      const lbl = document.getElementById(`lbl${i+1}`);
      const arc = document.getElementById(`arc${i+1}`);
      lbl.textContent = step;
      lbl.setAttribute('fill', textColor);
      arc.setAttribute('stroke', strokeColor);
    });

    // Update badge
    badge.textContent = data.badge;
    badge.className = `badge ${data.valid ? 'good' : 'bad'}`;
  }

  btns.forEach(btn => {
    btn.addEventListener('click', () => {
      btns.forEach(b => b.classList.remove('active'));
      btn.classList.add('active');
      updateScale(btn.dataset.scale);
    });
  });
</script>

</body>
</html>

How do you calculate the exact scale step when working with larger datasets?

How do we calculate the exact scale step systematically for any dataset?

Identify the **Maximum Value** and **Minimum Value** in your data. The **Data Range** is $\text{Maximum Value} - \text{Minimum Value}$ (or from $0$ to the maximum when starting at zero).

<!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;
      font-family: system-ui, -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
      user-select: none;
      -webkit-user-select: none;
    }

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

    .container {
      width: 100%;
      max-width: 520px;
      display: flex;
      flex-direction: column;
      gap: 16px;
    }

    /* Mode Selector Buttons */
    .tabs {
      display: flex;
      background: #e9ecef;
      padding: 4px;
      border-radius: 10px;
      gap: 4px;
    }

    .tab-btn {
      flex: 1;
      border: none;
      background: transparent;
      padding: 8px 12px;
      font-size: 14px;
      font-weight: 600;
      color: #6b7280;
      border-radius: 8px;
      cursor: pointer;
      transition: all 0.25s ease;
      text-align: center;
    }

    .tab-btn.active {
      background: #ffffff;
      color: #0284c7;
      box-shadow: 0 2px 6px rgba(0, 0, 0, 0.06);
    }

    /* Dataset Cards */
    .data-row {
      display: flex;
      justify-content: space-between;
      gap: 6px;
    }

    .data-item {
      flex: 1;
      background: #ffffff;
      border: 1.5px solid #e5e7eb;
      border-radius: 10px;
      padding: 8px 4px;
      text-align: center;
      position: relative;
      transition: all 0.3s ease;
    }

    .data-val {
      font-size: 16px;
      font-weight: 700;
      color: #1e2945;
    }

    .data-item.is-min {
      border-color: #f97316;
      background: #fff7ed;
    }

    .data-item.is-min .data-val {
      color: #ea580c;
    }

    .data-item.is-max {
      border-color: #0284c7;
      background: #f0f9ff;
    }

    .data-item.is-max .data-val {
      color: #0284c7;
    }

    .badge {
      font-size: 10px;
      font-weight: 700;
      text-transform: uppercase;
      margin-top: 2px;
      display: block;
    }

    .badge-min { color: #ea580c; }
    .badge-max { color: #0284c7; }

    /* Visual Number Line */
    .viz-card {
      background: #ffffff;
      border-radius: 12px;
      border: 1px solid #e5e7eb;
      padding: 20px 16px 16px;
      display: flex;
      flex-direction: column;
      gap: 18px;
    }

    .axis-area {
      position: relative;
      height: 64px;
      margin: 0 10px;
    }

    .axis-line {
      position: absolute;
      bottom: 12px;
      left: 0;
      right: 0;
      height: 3px;
      background: #cbd5e1;
      border-radius: 2px;
    }

    .axis-ticks {
      position: absolute;
      bottom: 0;
      left: 0;
      right: 0;
      display: flex;
      justify-content: space-between;
    }

    .tick {
      display: flex;
      flex-direction: column;
      align-items: center;
      font-size: 12px;
      font-weight: 600;
      color: #64748b;
    }

    .tick::before {
      content: "";
      width: 2px;
      height: 6px;
      background: #cbd5e1;
      margin-bottom: 2px;
    }

    /* Range Span Bracket / Bar */
    .range-bar {
      position: absolute;
      top: 6px;
      height: 28px;
      background: rgba(34, 183, 255, 0.18);
      border: 2px solid #0284c7;
      border-radius: 6px;
      transition: all 0.4s cubic-bezier(0.4, 0, 0.2, 1);
      display: flex;
      align-items: center;
      justify-content: center;
    }

    .range-label {
      font-size: 12px;
      font-weight: 700;
      color: #0369a1;
      white-space: nowrap;
      padding: 0 4px;
    }

    .pin {
      position: absolute;
      bottom: 7px;
      width: 10px;
      height: 10px;
      border-radius: 50%;
      transform: translateX(-50%);
      transition: all 0.4s ease;
      z-index: 2;
    }

    .pin-start {
      background: #ea580c;
      box-shadow: 0 0 0 3px rgba(234, 88, 12, 0.2);
    }

    .pin-end {
      background: #0284c7;
      box-shadow: 0 0 0 3px rgba(2, 132, 199, 0.2);
    }

    /* Formula Banner */
    .formula-banner {
      background: #f8fafc;
      border: 1.5px solid #e2e8f0;
      border-radius: 10px;
      padding: 12px 14px;
      display: flex;
      align-items: center;
      justify-content: center;
      gap: 10px;
      font-size: 17px;
      font-weight: 700;
    }

    .calc-tag {
      padding: 4px 8px;
      border-radius: 6px;
      line-height: 1;
    }

    .tag-max {
      background: #e0f2fe;
      color: #0284c7;
    }

    .tag-min {
      background: #ffedd5;
      color: #ea580c;
    }

    .tag-result {
      background: #0284c7;
      color: #ffffff;
      padding: 5px 10px;
    }

    .math-op {
      color: #64748b;
      font-weight: 800;
    }
  </style>
</head>
<body>

  <div class="container">
    
    <div class="tabs">
      <button class="tab-btn active" id="tabMin" onclick="setMode('min')">From Minimum (4)</button>
      <button class="tab-btn" id="tabZero" onclick="setMode('zero')">From Zero (0)</button>
    </div>

    
    <div class="data-row">
      <div class="data-item">
        <div class="data-val">11</div>
      </div>
      <div class="data-item is-min" id="item4">
        <div class="data-val">4</div>
        <span class="badge badge-min">Min</span>
      </div>
      <div class="data-item">
        <div class="data-val">17</div>
      </div>
      <div class="data-item is-max">
        <div class="data-val">24</div>
        <span class="badge badge-max">Max</span>
      </div>
      <div class="data-item">
        <div class="data-val">9</div>
      </div>
      <div class="data-item">
        <div class="data-val">15</div>
      </div>
    </div>

    
    <div class="viz-card">
      <div class="axis-area">
        
        <div class="range-bar" id="rangeBar">
          <span class="range-label" id="rangeLabel">Range = 20</span>
        </div>

        
        <div class="pin pin-start" id="pinStart"></div>
        <div class="pin pin-end" id="pinEnd"></div>

        
        <div class="axis-line"></div>
        <div class="axis-ticks">
          <div class="tick">0</div>
          <div class="tick">5</div>
          <div class="tick">10</div>
          <div class="tick">15</div>
          <div class="tick">20</div>
          <div class="tick">25</div>
        </div>
      </div>

      
      <div class="formula-banner">
        <span>Range</span>
        <span class="math-op">=</span>
        <span class="calc-tag tag-max" id="dispMax">24</span>
        <span class="math-op">−</span>
        <span class="calc-tag tag-min" id="dispMin">4</span>
        <span class="math-op">=</span>
        <span class="calc-tag tag-result" id="dispRes">20</span>
      </div>
    </div>
  </div>

  <script>
    function setMode(mode) {
      const tabMin = document.getElementById('tabMin');
      const tabZero = document.getElementById('tabZero');
      const bar = document.getElementById('rangeBar');
      const label = document.getElementById('rangeLabel');
      const pinStart = document.getElementById('pinStart');
      const pinEnd = document.getElementById('pinEnd');
      const dispMin = document.getElementById('dispMin');
      const dispRes = document.getElementById('dispRes');

      // Max value is always 24 on a 0-25 axis (24/25 = 96%)
      const maxPct = (24 / 25) * 100;
      pinEnd.style.left = maxPct + '%';

      if (mode === 'min') {
        tabMin.classList.add('active');
        tabZero.classList.remove('active');

        const minPct = (4 / 25) * 100; // 16%
        bar.style.left = minPct + '%';
        bar.style.width = (maxPct - minPct) + '%';
        pinStart.style.left = minPct + '%';
        
        label.textContent = 'Range = 20';
        dispMin.textContent = '4';
        dispRes.textContent = '20';
      } else {
        tabZero.classList.add('active');
        tabMin.classList.remove('active');

        bar.style.left = '0%';
        bar.style.width = maxPct + '%';
        pinStart.style.left = '0%';

        label.textContent = 'Range = 24';
        dispMin.textContent = '0';
        dispRes.textContent = '24';
      }
    }

    // Initialize
    setMode('min');
  </script>
</body>
</html>

First, **Count Available Grid Divisions** ($N$) along the axis. Then find the **Minimum Step Size** needed using $\text{Step}_{\text{min}} = \frac{\text{Data Range}}{N}$.

Always **Round Up** your minimum step to convenient **Friendly Intervals** in the $1, 2, 5 \times 10^k$ family (such as $1, 2, 5, 10, 20, 50$). Never round down, or points will spill off the grid.

<!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>Rounding Scale Step</title>
  <style>
    * { box-sizing: border-box; margin: 0; padding: 0; }
    body {
      font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
      background: transparent;
      color: #1e2945;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      padding: 10px;
      min-height: 100%;
    }

    .header-bar {
      display: flex;
      align-items: center;
      justify-content: center;
      gap: 8px;
      font-size: 15px;
      font-weight: 700;
      margin-bottom: 12px;
      color: #334155;
    }

    .badge-calc {
      background: #f1f5f9;
      border: 1px solid #cbd5e1;
      padding: 4px 10px;
      border-radius: 20px;
      font-size: 14px;
      font-family: ui-monospace, monospace;
    }

    .comparison {
      display: flex;
      gap: 12px;
      width: 100%;
      max-width: 480px;
    }

    .card-option {
      flex: 1;
      background: #ffffff;
      border: 1.5px solid #e2e8f0;
      border-radius: 12px;
      padding: 10px 8px 12px;
      display: flex;
      flex-direction: column;
      align-items: center;
      position: relative;
    }

    .card-option.bad {
      border-color: #fca5a5;
      background: #fffafa;
    }

    .card-option.good {
      border-color: #86efac;
      background: #fafffc;
    }

    .title-row {
      font-size: 14px;
      font-weight: 700;
      margin-bottom: 6px;
      display: flex;
      align-items: center;
      gap: 4px;
    }

    .bad .title-row { color: #dc2626; }
    .good .title-row { color: #16a34a; }

    .chart-container {
      width: 100%;
      height: 230px;
    }

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

    .status-pill {
      margin-top: 8px;
      font-size: 13px;
      font-weight: 700;
      padding: 3px 10px;
      border-radius: 12px;
    }

    .bad .status-pill {
      background: #fee2e2;
      color: #b91c1c;
    }

    .good .status-pill {
      background: #dcfce7;
      color: #15803d;
    }

    .pulse-pt {
      animation: pulse 2s infinite ease-in-out;
    }

    @keyframes pulse {
      0%, 100% { transform: scale(1); }
      50% { transform: scale(1.25); }
    }
  </style>
</head>
<body>

  <div class="header-bar">
    <span>Need: 36 units ÷ 10 sq</span>
    <span class="badge-calc">Step = 3.6</span>
  </div>

  <div class="comparison">
    
    <div class="card-option bad">
      <div class="title-row">
        <span>↓ Step = 3</span>
      </div>

      <div class="chart-container">
        <svg viewBox="0 0 160 210">
          
          <rect x="36" y="8" width="112" height="34" fill="#fee2e2" opacity="0.6" rx="4" stroke="#f87171" stroke-dasharray="3 3" stroke-width="1.5" />
          <text x="92" y="24" text-anchor="middle" font-size="10" font-weight="700" fill="#dc2626">NO GRID</text>

          
          <rect x="36" y="45" width="112" height="150" fill="#ffffff" stroke="#94a3b8" stroke-width="1.5" />

          
          
          <line x1="36" y1="60" x2="148" y2="60" stroke="#f1f5f9" stroke-width="1" />
          <line x1="36" y1="75" x2="148" y2="75" stroke="#f1f5f9" stroke-width="1" />
          <line x1="36" y1="90" x2="148" y2="90" stroke="#f1f5f9" stroke-width="1" />
          <line x1="36" y1="105" x2="148" y2="105" stroke="#f1f5f9" stroke-width="1" />
          <line x1="36" y1="120" x2="148" y2="120" stroke="#f1f5f9" stroke-width="1" />
          <line x1="36" y1="135" x2="148" y2="135" stroke="#f1f5f9" stroke-width="1" />
          <line x1="36" y1="150" x2="148" y2="150" stroke="#f1f5f9" stroke-width="1" />
          <line x1="36" y1="165" x2="148" y2="165" stroke="#f1f5f9" stroke-width="1" />
          <line x1="36" y1="180" x2="148" y2="180" stroke="#f1f5f9" stroke-width="1" />

          
          <line x1="64" y1="45" x2="64" y2="195" stroke="#f8fafc" stroke-width="1" />
          <line x1="92" y1="45" x2="92" y2="195" stroke="#f8fafc" stroke-width="1" />
          <line x1="120" y1="45" x2="120" y2="195" stroke="#f8fafc" stroke-width="1" />

          
          <text x="28" y="198" text-anchor="end" font-size="11" font-weight="600" fill="#64748b">0</text>
          <text x="28" y="123" text-anchor="end" font-size="11" font-weight="600" fill="#64748b">15</text>
          <text x="28" y="49" text-anchor="end" font-size="11" font-weight="700" fill="#dc2626">30</text>

          
          <line x1="36" y1="45" x2="148" y2="45" stroke="#dc2626" stroke-width="2" />

          
          <g class="pulse-pt" style="transform-origin: 92px 15px;">
            <circle cx="92" cy="15" r="6" fill="#ef4444" stroke="#ffffff" stroke-width="1.5" />
            <text x="104" y="19" font-size="11" font-weight="800" fill="#b91c1c">36</text>
          </g>

          <path d="M 92 40 L 92 25" stroke="#ef4444" stroke-width="1.5" stroke-dasharray="2 2" marker-end="url(#arrow-red)" />
        </svg>
      </div>

      <div class="status-pill">✕ Spills Off</div>
    </div>

    
    <div class="card-option good">
      <div class="title-row">
        <span>↑ Step = 5</span>
      </div>

      <div class="chart-container">
        <svg viewBox="0 0 160 210">
          
          <rect x="36" y="45" width="112" height="150" fill="#ffffff" stroke="#94a3b8" stroke-width="1.5" />

          
          <line x1="36" y1="60" x2="148" y2="60" stroke="#f1f5f9" stroke-width="1" />
          <line x1="36" y1="75" x2="148" y2="75" stroke="#f1f5f9" stroke-width="1" />
          <line x1="36" y1="90" x2="148" y2="90" stroke="#f1f5f9" stroke-width="1" />
          <line x1="36" y1="105" x2="148" y2="105" stroke="#f1f5f9" stroke-width="1" />
          <line x1="36" y1="120" x2="148" y2="120" stroke="#f1f5f9" stroke-width="1" />
          <line x1="36" y1="135" x2="148" y2="135" stroke="#f1f5f9" stroke-width="1" />
          <line x1="36" y1="150" x2="148" y2="150" stroke="#f1f5f9" stroke-width="1" />
          <line x1="36" y1="165" x2="148" y2="165" stroke="#f1f5f9" stroke-width="1" />
          <line x1="36" y1="180" x2="148" y2="180" stroke="#f1f5f9" stroke-width="1" />

          
          <line x1="64" y1="45" x2="64" y2="195" stroke="#f8fafc" stroke-width="1" />
          <line x1="92" y1="45" x2="92" y2="195" stroke="#f8fafc" stroke-width="1" />
          <line x1="120" y1="45" x2="120" y2="195" stroke="#f8fafc" stroke-width="1" />

          
          <text x="28" y="198" text-anchor="end" font-size="11" font-weight="600" fill="#64748b">0</text>
          <text x="28" y="123" text-anchor="end" font-size="11" font-weight="600" fill="#64748b">25</text>
          <text x="28" y="49" text-anchor="end" font-size="11" font-weight="700" fill="#16a34a">50</text>

          
          <line x1="36" y1="45" x2="148" y2="45" stroke="#16a34a" stroke-width="2" />

          
          <g class="pulse-pt" style="transform-origin: 92px 87px;">
            <circle cx="92" cy="87" r="6" fill="#22c55e" stroke="#ffffff" stroke-width="1.5" />
            <text x="104" y="91" font-size="11" font-weight="800" fill="#15803d">36</text>
          </g>

          <line x1="36" y1="87" x2="92" y2="87" stroke="#22c55e" stroke-width="1" stroke-dasharray="2 2" />
        </svg>
      </div>

      <div class="status-pill">✓ Fits Grid</div>
    </div>
  </div>

</body>
</html>

For an **Optimal Spread**, choose a scale where the data fills at least 50% (ideally 60%–75%) of the axis length. This ensures the graph is clear and not cramped in one corner.

<!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;
    justify-content: center;
    min-height: 100%;
  }

  .grid-container {
    display: flex;
    gap: 16px;
    width: 100%;
    max-width: 620px;
    justify-content: center;
    flex-wrap: wrap;
  }

  .graph-card {
    flex: 1 1 260px;
    background: #ffffff;
    border-radius: 12px;
    padding: 14px 12px 10px;
    box-shadow: 0 2px 8px rgba(0,0,0,0.06);
    border: 1px solid #e6e6e6;
    display: flex;
    flex-direction: column;
    align-items: center;
    position: relative;
  }

  .badge {
    font-size: 13px;
    font-weight: 700;
    padding: 4px 10px;
    border-radius: 20px;
    margin-bottom: 8px;
    display: flex;
    align-items: center;
    gap: 4px;
  }

  .badge-bad {
    background: #fee2e2;
    color: #dc2626;
  }

  .badge-good {
    background: #dcfce7;
    color: #16a34a;
  }

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

  .axis { stroke: #1e2945; stroke-width: 2; stroke-linecap: round; }
  .grid { stroke: #f1f5f9; stroke-width: 1; }
  .tick { stroke: #94a3b8; stroke-width: 1.5; }
  .tick-text { font-size: 11px; fill: #64748b; font-family: monospace; }
  
  .span-zone {
    transition: opacity 0.3s ease;
  }
  .zone-bad { fill: rgba(239, 68, 68, 0.12); stroke: #ef4444; stroke-width: 1.5; stroke-dasharray: 4; }
  .zone-good { fill: rgba(34, 183, 255, 0.15); stroke: #22b7ff; stroke-width: 1.5; }
  
  .point { fill: #22b7ff; stroke: #ffffff; stroke-width: 2; }
  .line-bad { stroke: #ef4444; stroke-width: 2.5; stroke-linecap: round; }
  .line-good { stroke: #22b7ff; stroke-width: 2.5; stroke-linecap: round; }

  .bar-text {
    font-size: 11px;
    font-weight: 700;
  }
</style>
</head>
<body>

<div class="grid-container">
  
  <div class="graph-card">
    <div class="badge badge-bad">✗ 20% Spread</div>
    <svg viewBox="0 0 220 180">
      
      <line x1="35" y1="20" x2="205" y2="20" class="grid"/>
      <line x1="35" y1="55" x2="205" y2="55" class="grid"/>
      <line x1="35" y1="90" x2="205" y2="90" class="grid"/>
      <line x1="35" y1="125" x2="205" y2="125" class="grid"/>

      
      <rect x="35" y="132" width="34" height="28" class="zone-bad" rx="4" />
      
      
      <line x1="35" y1="20" x2="35" y2="160" class="axis"/>
      <line x1="35" y1="160" x2="205" y2="160" class="axis"/>

      
      <line x1="30" y1="160" x2="35" y2="160" class="tick"/>
      <text x="26" y="163" text-anchor="end" class="tick-text">0</text>
      
      <line x1="30" y1="90" x2="35" y2="90" class="tick"/>
      <text x="26" y="93" text-anchor="end" class="tick-text">50</text>
      
      <line x1="30" y1="20" x2="35" y2="20" class="tick"/>
      <text x="26" y="23" text-anchor="end" class="tick-text">100</text>

      
      <polyline points="40,153 50,143 60,149 68,135" fill="none" class="line-bad" />
      <circle cx="40" cy="153" r="3.5" class="point" style="fill:#ef4444;"/>
      <circle cx="50" cy="143" r="3.5" class="point" style="fill:#ef4444;"/>
      <circle cx="60" cy="149" r="3.5" class="point" style="fill:#ef4444;"/>
      <circle cx="68" cy="135" r="3.5" class="point" style="fill:#ef4444;"/>

      
      <path d="M 75 160 L 80 160 L 80 135 L 75 135" fill="none" stroke="#ef4444" stroke-width="1.5" />
      <text x="85" y="151" class="bar-text" fill="#ef4444">Cramped</text>
    </svg>
  </div>

  
  <div class="graph-card">
    <div class="badge badge-good">✓ 70% Spread</div>
    <svg viewBox="0 0 220 180">
      
      <line x1="35" y1="20" x2="205" y2="20" class="grid"/>
      <line x1="35" y1="55" x2="205" y2="55" class="grid"/>
      <line x1="35" y1="90" x2="205" y2="90" class="grid"/>
      <line x1="35" y1="125" x2="205" y2="125" class="grid"/>

      
      <rect x="35" y="35" width="165" height="125" class="zone-good" rx="4" />

      
      <line x1="35" y1="20" x2="35" y2="160" class="axis"/>
      <line x1="35" y1="160" x2="205" y2="160" class="axis"/>

      
      <line x1="30" y1="160" x2="35" y2="160" class="tick"/>
      <text x="26" y="163" text-anchor="end" class="tick-text">0</text>
      
      <line x1="30" y1="90" x2="35" y2="90" class="tick"/>
      <text x="26" y="93" text-anchor="end" class="tick-text">12</text>
      
      <line x1="30" y1="20" x2="35" y2="20" class="tick"/>
      <text x="26" y="23" text-anchor="end" class="tick-text">24</text>

      
      <polyline points="50,131 95,90 145,113 190,48" fill="none" class="line-good" />
      <circle cx="50" cy="131" r="4" class="point"/>
      <circle cx="95" cy="90" r="4" class="point"/>
      <circle cx="145" cy="113" r="4" class="point"/>
      <circle cx="190" cy="48" r="4" class="point"/>

      
      <path d="M 200 160 L 205 160 L 205 48 L 200 48" fill="none" stroke="#22b7ff" stroke-width="1.5" />
      <text x="140" y="32" class="bar-text" fill="#0284c7">Clear Data</text>
    </svg>
  </div>
</div>

</body>
</html>

First, identify the **Maximum Value** and **Minimum Value** in your data. Calculate the **Data Range** using $\text{Maximum Value} - \text{Minimum Value}$ (or from $0$ to the maximum when a zero-baseline is needed).

Next, **Count Available Grid Divisions** ($N$) along the axis. Divide your range by $N$ to find the **Minimum Step Size** ($\text{Step}_{\text{min}} = \frac{\text{Data Range}}{N}$) needed to fit every data point.

Always **Round Up** your calculated step to the nearest convenient value from **Friendly Intervals** like $1, 2, 5, 10, 20, 25, 50,$ or $100$. Never round down, or your data will run off the grid!

For an **Optimal Spread**, your plotted data should fill more than $50\%$ (ideally $60\%\text{--}75\%$) of the axis length. This ensures the graph is clear and easy to read rather than squeezed into a corner.

Before you draw your next graph, count your grid boxes and check your biggest number to pick your perfect step.

math

Choosing a Suitable Scale for Axes on Grid Paper. Determine optimal step sizes and intervals for numerical axes based on data range and paper dimensions.

Graph Labels and Scales: Choosing Appropriate Intervals

Curriculum for Excellence Third Level Mathematics: Constructing Frequency Tables and Graphs (MTH 3-21a)

Scaling and Graph Representation: Developing Proportional Reasoning on Grids

When teaching scale selection under Curriculum for Excellence MTH 3-21a, guide learners through a structured three-step heuristic: (1) Calculate $\frac{\text{Range}}{\text{Squares}}$; (2) Round up to a friendly step ($1, 2, 5 \times 10^k$); (3) Check that data occupies $\ge 50\%$ of the axis without exceeding grid borders. Scaffold the transition from early concrete counting to formal interval arithmetic through comparative analysis of poorly scaled versus well-scaled graphs.

Choosing a Suitable Scale for Axes on Grid Paper

Counting by Ones

Counting by Twos

Counting by Fives