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Override for the tutor's lesson content generation calls. Auto uses the system default (Claude Sonnet 5 for lesson text, Opus 5 for diagrams).

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System default

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Tutoring abilities

If a speedy puppy and a slow turtle run a race, how can a picture show who is winning?

We can draw a **distance-time graph**, which is a picture that shows how far something travels as time ticks by.

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

    body {
      background: #f8f9fa;
      color: #1e2945;
      padding: 10px;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: space-between;
      min-height: 100%;
    }

    .main-container {
      width: 100%;
      max-width: 500px;
      display: flex;
      flex-direction: column;
      gap: 8px;
    }

    /* Race Tracks */
    .tracks-panel {
      background: #ffffff;
      border-radius: 10px;
      padding: 8px 12px;
      box-shadow: 0 1px 3px rgba(0,0,0,0.06);
      display: flex;
      flex-direction: column;
      gap: 6px;
    }

    .track-row {
      position: relative;
      height: 28px;
      background: #f1f5f9;
      border-radius: 14px;
      display: flex;
      align-items: center;
      padding: 0 4px;
      overflow: hidden;
    }

    .track-runner {
      position: absolute;
      top: 2px;
      left: 4px;
      font-size: 18px;
      line-height: 24px;
      transition: left 0.05s linear;
      z-index: 2;
    }

    .track-flag {
      position: absolute;
      right: 8px;
      font-size: 14px;
      opacity: 0.4;
    }

    /* Graph Panel */
    .graph-panel {
      background: #ffffff;
      border-radius: 10px;
      padding: 8px;
      box-shadow: 0 1px 3px rgba(0,0,0,0.06);
      position: relative;
    }

    svg {
      width: 100%;
      height: auto;
      display: block;
    }

    .axis-line {
      stroke: #94a3b8;
      stroke-width: 2;
      stroke-linecap: round;
    }

    .grid-line {
      stroke: #f1f5f9;
      stroke-width: 1;
      stroke-dasharray: 2 2;
    }

    .graph-text {
      font-size: 11px;
      fill: #64748b;
      font-weight: 600;
    }

    .cheetah-path {
      stroke: #2563eb;
      stroke-width: 3.5;
      stroke-linecap: round;
    }

    .tortoise-path {
      stroke: #16a34a;
      stroke-width: 3.5;
      stroke-linecap: round;
    }

    .glow-dot {
      transition: cx 0.05s linear, cy 0.05s linear;
    }

    /* Controls */
    .controls-row {
      display: flex;
      align-items: center;
      gap: 10px;
      background: #ffffff;
      padding: 6px 12px;
      border-radius: 10px;
      box-shadow: 0 1px 3px rgba(0,0,0,0.06);
    }

    .play-btn {
      background: #2563eb;
      color: #ffffff;
      border: none;
      width: 34px;
      height: 34px;
      border-radius: 50%;
      display: flex;
      align-items: center;
      justify-content: center;
      cursor: pointer;
      font-size: 14px;
      flex-shrink: 0;
    }

    .play-btn:active {
      background: #1d4ed8;
    }

    .slider {
      flex: 1;
      accent-color: #2563eb;
      cursor: pointer;
      height: 6px;
    }

    .badge {
      font-size: 11px;
      font-weight: 700;
      padding: 3px 8px;
      border-radius: 6px;
      white-space: nowrap;
    }

    .badge-fast {
      background: #dbeafe;
      color: #1e40af;
    }

    .badge-slow {
      background: #dcfce7;
      color: #166534;
    }
  </style>
</head>
<body>
  <div class="main-container">
    
    <div class="tracks-panel">
      <div class="track-row">
        <span class="track-runner" id="runnerCheetah">🐆</span>
        <span class="track-flag">🏁 100m</span>
      </div>
      <div class="track-row">
        <span class="track-runner" id="runnerTortoise">🐢</span>
        <span class="track-flag">🏁 100m</span>
      </div>
    </div>

    
    <div class="graph-panel">
      <svg viewBox="0 0 340 190">
        
        <line x1="45" y1="30" x2="310" y2="30" class="grid-line" />
        <line x1="45" y1="70" x2="310" y2="70" class="grid-line" />
        <line x1="45" y1="110" x2="310" y2="110" class="grid-line" />
        <line x1="111" y1="20" x2="111" y2="150" class="grid-line" />
        <line x1="177" y1="20" x2="177" y2="150" class="grid-line" />
        <line x1="244" y1="20" x2="244" y2="150" class="grid-line" />

        
        <line x1="45" y1="150" x2="315" y2="150" class="axis-line" />
        <line x1="45" y1="150" x2="45" y2="15" class="axis-line" />

        
        <text x="315" y="172" text-anchor="end" class="graph-text">Time (s) →</text>
        <text x="12" y="18" class="graph-text">Distance (m) ↑</text>

        
        <text x="45" y="165" text-anchor="middle" class="graph-text">0</text>
        <text x="177" y="165" text-anchor="middle" class="graph-text">2.5</text>
        <text x="310" y="165" text-anchor="middle" class="graph-text">5</text>

        <text x="38" y="153" text-anchor="end" class="graph-text">0</text>
        <text x="38" y="93" text-anchor="end" class="graph-text">50</text>
        <text x="38" y="33" text-anchor="end" class="graph-text">100</text>

        
        <line x1="45" y1="150" x2="310" y2="30" class="cheetah-path" opacity="0.3" stroke-dasharray="4 3" />
        <line id="cheetahLine" x1="45" y1="150" x2="45" y2="150" class="cheetah-path" />

        
        <line x1="45" y1="150" x2="310" y2="120" class="tortoise-path" opacity="0.3" stroke-dasharray="4 3" />
        <line id="tortoiseLine" x1="45" y1="150" x2="45" y2="150" class="tortoise-path" />

        
        <circle id="cheetahDot" cx="45" cy="150" r="5" fill="#2563eb" class="glow-dot" />
        <circle id="tortoiseDot" cx="45" cy="150" r="5" fill="#16a34a" class="glow-dot" />

        
        <text x="210" y="55" fill="#2563eb" font-size="11" font-weight="bold">🐆 Steeper (Fast)</text>
        <text x="235" y="138" fill="#16a34a" font-size="11" font-weight="bold">🐢 Gentle (Slow)</text>
      </svg>
    </div>

    
    <div class="controls-row">
      <button class="play-btn" id="playBtn" aria-label="Play/Pause">▶</button>
      <input type="range" id="timeSlider" class="slider" min="0" max="5" step="0.02" value="0">
      <span class="badge badge-fast" id="speedBadge">Steeper = Faster</span>
    </div>
  </div>

  <script>
    const playBtn = document.getElementById('playBtn');
    const timeSlider = document.getElementById('timeSlider');
    const runnerCheetah = document.getElementById('runnerCheetah');
    const runnerTortoise = document.getElementById('runnerTortoise');
    const cheetahLine = document.getElementById('cheetahLine');
    const tortoiseLine = document.getElementById('tortoiseLine');
    const cheetahDot = document.getElementById('cheetahDot');
    const tortoiseDot = document.getElementById('tortoiseDot');

    const ORIGIN_X = 45;
    const ORIGIN_Y = 150;
    const GRAPH_W = 265; // from x=45 to x=310
    const GRAPH_H = 120; // from y=150 to y=30
    const MAX_TIME = 5.0;

    let time = 0;
    let isPlaying = false;
    let animFrame = null;
    let lastTimestamp = 0;

    function updateState(t) {
      time = Math.max(0, Math.min(MAX_TIME, t));
      timeSlider.value = time;

      // Track positions (Cheetah: 0 to 100m, Tortoise: 0 to 25m)
      const maxTrackTravel = 86; // percentage across track
      const cPercent = (time / MAX_TIME) * maxTrackTravel;
      const tPercent = ((time * 5) / 100) * maxTrackTravel;

      runnerCheetah.style.left = `calc(${cPercent}% + 4px)`;
      runnerTortoise.style.left = `calc(${tPercent}% + 4px)`;

      // Graph Coordinates
      const curX = ORIGIN_X + (time / MAX_TIME) * GRAPH_W;
      const cY = ORIGIN_Y - (time / MAX_TIME) * GRAPH_H;
      const tY = ORIGIN_Y - ((time * 5) / 100) * GRAPH_H;

      cheetahLine.setAttribute('x2', curX);
      cheetahLine.setAttribute('y2', cY);
      cheetahDot.setAttribute('cx', curX);
      cheetahDot.setAttribute('cy', cY);

      tortoiseLine.setAttribute('x2', curX);
      tortoiseLine.setAttribute('y2', tY);
      tortoiseDot.setAttribute('cx', curX);
      tortoiseDot.setAttribute('cy', tY);
    }

    function tick(timestamp) {
      if (!isPlaying) return;
      if (!lastTimestamp) lastTimestamp = timestamp;
      const dt = (timestamp - lastTimestamp) / 1000;
      lastTimestamp = timestamp;

      let nextTime = time + dt * 1.2; // ~4s total run
      if (nextTime >= MAX_TIME) {
        updateState(MAX_TIME);
        setTimeout(() => {
          if (isPlaying) {
            updateState(0);
            lastTimestamp = performance.now();
            animFrame = requestAnimationFrame(tick);
          }
        }, 1200);
        return;
      }

      updateState(nextTime);
      animFrame = requestAnimationFrame(tick);
    }

    function togglePlay() {
      isPlaying = !isPlaying;
      playBtn.textContent = isPlaying ? '⏸' : '▶';
      if (isPlaying) {
        if (time >= MAX_TIME) updateState(0);
        lastTimestamp = performance.now();
        animFrame = requestAnimationFrame(tick);
      } else {
        cancelAnimationFrame(animFrame);
      }
    }

    playBtn.addEventListener('click', togglePlay);

    timeSlider.addEventListener('input', (e) => {
      if (isPlaying) {
        isPlaying = false;
        playBtn.textContent = '▶';
        cancelAnimationFrame(animFrame);
      }
      updateState(parseFloat(e.target.value));
    });

    // Auto-start demonstration calmly
    setTimeout(() => {
      togglePlay();
    }, 600);
  </script>
</body>
</html>

How do we know which animal is faster just by looking at the line?

The **gradient** is how steep a line is, just like climbing a steep hill.

A steeper line means you cover more distance in less time, which means a **faster speed**.

<!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>Distance-Time Graphs & Speed</title>
<style>
  * { box-sizing: border-box; margin: 0; padding: 0; }
  body {
    font-family: system-ui, -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, sans-serif;
    background: #f8f9fa;
    color: #1e2945;
    display: flex;
    justify-content: center;
    align-items: center;
    min-height: 100%;
    padding: 12px;
  }
  .app-container {
    width: 100%;
    max-width: 540px;
    display: flex;
    flex-direction: column;
    gap: 12px;
  }
  
  /* Simulation Track */
  .tracks {
    background: #ffffff;
    border-radius: 10px;
    padding: 8px 12px;
    box-shadow: 0 1px 3px rgba(0,0,0,0.06);
    display: flex;
    flex-direction: column;
    gap: 6px;
  }
  .track-lane {
    display: flex;
    align-items: center;
    position: relative;
    height: 28px;
  }
  .lane-label {
    width: 68px;
    font-size: 13px;
    font-weight: 600;
    flex-shrink: 0;
  }
  .lane-bar {
    flex: 1;
    height: 6px;
    background: #e9ecef;
    border-radius: 3px;
    position: relative;
    margin: 0 16px 0 6px;
  }
  .runner {
    position: absolute;
    top: 50%;
    transform: translate(-50%, -50%);
    font-size: 18px;
    line-height: 1;
    will-change: left;
    transition: left 0.05s linear;
  }

  /* Graph Container */
  .graph-card {
    background: #ffffff;
    border-radius: 10px;
    padding: 8px 12px 4px 8px;
    box-shadow: 0 1px 3px rgba(0,0,0,0.06);
    position: relative;
  }
  svg {
    width: 100%;
    height: auto;
    display: block;
    overflow: visible;
  }
  .grid-line {
    stroke: #edf2f7;
    stroke-width: 1;
  }
  .axis-line {
    stroke: #94a3b8;
    stroke-width: 1.5;
  }
  .axis-label {
    font-size: 12px;
    fill: #64748b;
    font-weight: 500;
  }
  .line-fast {
    stroke: #0284c7;
    stroke-width: 3.5;
    stroke-linecap: round;
  }
  .line-slow {
    stroke: #f59e0b;
    stroke-width: 3.5;
    stroke-linecap: round;
  }
  .line-stop {
    stroke: #94a3b8;
    stroke-width: 3.5;
    stroke-linecap: round;
  }
  .scrubber-line {
    stroke: #6366f1;
    stroke-width: 1.5;
    stroke-dasharray: 4 3;
  }

  /* Controls */
  .controls {
    display: flex;
    align-items: center;
    justify-content: space-between;
    gap: 12px;
    background: #ffffff;
    border-radius: 10px;
    padding: 8px 14px;
    box-shadow: 0 1px 3px rgba(0,0,0,0.06);
  }
  .btn-play {
    background: #0284c7;
    color: #ffffff;
    border: none;
    border-radius: 6px;
    padding: 6px 14px;
    font-size: 14px;
    font-weight: 600;
    cursor: pointer;
    display: flex;
    align-items: center;
    gap: 6px;
    min-width: 84px;
    justify-content: center;
  }
  .btn-play:active {
    background: #0369a1;
  }
  .slider-box {
    flex: 1;
    display: flex;
    align-items: center;
    gap: 10px;
  }
  .slider {
    flex: 1;
    accent-color: #0284c7;
    cursor: pointer;
  }
  .time-readout {
    font-size: 14px;
    font-weight: 600;
    color: #475569;
    min-width: 48px;
    text-align: right;
    font-variant-numeric: tabular-nums;
  }
</style>
</head>
<body>

<div class="app-container">
  
  <div class="tracks">
    <div class="track-lane">
      <span class="lane-label" style="color:#0284c7;">⚡ Fast</span>
      <div class="lane-bar">
        <div class="runner" id="runnerFast" style="left: 0%;">🏃</div>
      </div>
    </div>
    <div class="track-lane">
      <span class="lane-label" style="color:#d97706;">🐢 Slow</span>
      <div class="lane-bar">
        <div class="runner" id="runnerSlow" style="left: 0%;">🚶</div>
      </div>
    </div>
    <div class="track-lane">
      <span class="lane-label" style="color:#64748b;">🛑 Stopped</span>
      <div class="lane-bar">
        <div class="runner" id="runnerStop" style="left: 50%;">🧍</div>
      </div>
    </div>
  </div>

  
  <div class="graph-card">
    <svg viewBox="0 0 360 210">
      
      <line x1="45" y1="25" x2="335" y2="25" class="grid-line" />
      <line x1="45" y1="65" x2="335" y2="65" class="grid-line" />
      <line x1="45" y1="105" x2="335" y2="105" class="grid-line" />
      <line x1="45" y1="145" x2="335" y2="145" class="grid-line" />
      
      <line x1="117.5" y1="25" x2="117.5" y2="185" class="grid-line" />
      <line x1="190" y1="25" x2="190" y2="185" class="grid-line" />
      <line x1="262.5" y1="25" x2="262.5" y2="185" class="grid-line" />
      <line x1="335" y1="25" x2="335" y2="185" class="grid-line" />

      
      <line x1="45" y1="185" x2="335" y2="185" class="axis-line" />
      <line x1="45" y1="25" x2="45" y2="185" class="axis-line" />

      
      <text x="40" y="188" class="axis-label" text-anchor="end">0</text>
      <text x="40" y="109" class="axis-label" text-anchor="end">30</text>
      <text x="40" y="29" class="axis-label" text-anchor="end">60</text>

      <text x="45" y="202" class="axis-label" text-anchor="middle">0</text>
      <text x="190" y="202" class="axis-label" text-anchor="middle">3</text>
      <text x="335" y="202" class="axis-label" text-anchor="middle">6</text>

      
      <text x="190" y="209" class="axis-label" text-anchor="middle" font-weight="600" fill="#334155">Time (s)</text>
      <text x="12" y="105" class="axis-label" text-anchor="middle" font-weight="600" fill="#334155" transform="rotate(-90 12 105)">Distance (m)</text>

      
      <line x1="45" y1="185" x2="335" y2="25" stroke="#e0f2fe" stroke-width="3.5" stroke-linecap="round" />
      <line x1="45" y1="185" x2="335" y2="131.7" stroke="#fef3c7" stroke-width="3.5" stroke-linecap="round" />
      <line x1="45" y1="105" x2="335" y2="105" stroke="#f1f5f9" stroke-width="3.5" stroke-linecap="round" />

      
      <line id="graphFast" x1="45" y1="185" x2="45" y2="185" class="line-fast" />
      <line id="graphSlow" x1="45" y1="185" x2="45" y2="185" class="line-slow" />
      <line id="graphStop" x1="45" y1="105" x2="45" y2="105" class="line-stop" />

      
      <text x="325" y="20" fill="#0284c7" font-size="12" font-weight="700" text-anchor="end">Steep = Fast</text>
      <text x="338" y="128" fill="#d97706" font-size="12" font-weight="700" text-anchor="end">Gentle = Slow</text>
      <text x="338" y="99" fill="#64748b" font-size="12" font-weight="700" text-anchor="end">Flat = Stopped</text>

      
      <line id="scrubLine" x1="45" y1="25" x2="45" y2="185" class="scrubber-line" />
      
      
      <circle id="dotFast" cx="45" cy="185" r="5" fill="#0284c7" />
      <circle id="dotSlow" cx="45" cy="185" r="5" fill="#f59e0b" />
      <circle id="dotStop" cx="45" cy="105" r="5" fill="#94a3b8" />
    </svg>
  </div>

  
  <div class="controls">
    <button class="btn-play" id="playBtn">▶ Play</button>
    <div class="slider-box">
      <input type="range" id="timeSlider" class="slider" min="0" max="6" step="0.05" value="0">
      <span class="time-readout" id="timeDisplay">0.0 s</span>
    </div>
  </div>
</div>

<script>
  const DURATION = 6;
  const X_START = 45;
  const X_END = 335;
  const Y_ZERO = 185;
  const Y_MAX = 25; // 60m
  const Y_MID = 105; // 30m
  const Y_SLOW_END = 131.67; // 20m

  let currentTime = 0;
  let isPlaying = false;
  let lastTimestamp = null;
  let animFrameId = null;

  const playBtn = document.getElementById('playBtn');
  const slider = document.getElementById('timeSlider');
  const timeDisplay = document.getElementById('timeDisplay');

  const runnerFast = document.getElementById('runnerFast');
  const runnerSlow = document.getElementById('runnerSlow');
  const runnerStop = document.getElementById('runnerStop');

  const graphFast = document.getElementById('graphFast');
  const graphSlow = document.getElementById('graphSlow');
  const graphStop = document.getElementById('graphStop');

  const dotFast = document.getElementById('dotFast');
  const dotSlow = document.getElementById('dotSlow');
  const dotStop = document.getElementById('dotStop');
  const scrubLine = document.getElementById('scrubLine');

  function updateVisuals(t) {
    const fraction = Math.min(Math.max(t / DURATION, 0), 1);
    const curX = X_START + fraction * (X_END - X_START);

    // Positions (0 to 60 meters scale)
    const distFast = fraction * 60; // 0 to 60m
    const distSlow = fraction * 20; // 0 to 20m
    const distStop = 30;            // 30m constant

    // Top Runners (0% to 100% of lane)
    runnerFast.style.left = `${(distFast / 60) * 100}%`;
    runnerSlow.style.left = `${(distSlow / 60) * 100}%`;
    runnerStop.style.left = `${(distStop / 60) * 100}%`;

    // Graph Lines & Dots
    const curYFast = Y_ZERO - (distFast / 60) * (Y_ZERO - Y_MAX);
    const curYSlow = Y_ZERO - (distSlow / 60) * (Y_ZERO - Y_MAX);
    const curYStop = Y_MID;

    graphFast.setAttribute('x2', curX);
    graphFast.setAttribute('y2', curYFast);
    dotFast.setAttribute('cx', curX);
    dotFast.setAttribute('cy', curYFast);

    graphSlow.setAttribute('x2', curX);
    graphSlow.setAttribute('y2', curYSlow);
    dotSlow.setAttribute('cx', curX);
    dotSlow.setAttribute('cy', curYSlow);

    graphStop.setAttribute('x2', curX);
    graphStop.setAttribute('y2', curYStop);
    dotStop.setAttribute('cx', curX);
    dotStop.setAttribute('cy', curYStop);

    scrubLine.setAttribute('x1', curX);
    scrubLine.setAttribute('x2', curX);

    slider.value = t.toFixed(2);
    timeDisplay.textContent = `${t.toFixed(1)} s`;
  }

  function loop(timestamp) {
    if (!lastTimestamp) lastTimestamp = timestamp;
    const dt = (timestamp - lastTimestamp) / 1000;
    lastTimestamp = timestamp;

    currentTime += dt;
    if (currentTime >= DURATION) {
      currentTime = DURATION;
      updateVisuals(currentTime);
      pause();
      return;
    }

    updateVisuals(currentTime);
    if (isPlaying) {
      animFrameId = requestAnimationFrame(loop);
    }
  }

  function play() {
    if (currentTime >= DURATION) {
      currentTime = 0;
    }
    isPlaying = true;
    playBtn.textContent = '⏸ Pause';
    lastTimestamp = performance.now();
    animFrameId = requestAnimationFrame(loop);
  }

  function pause() {
    isPlaying = false;
    playBtn.textContent = '▶ Play';
    if (animFrameId) {
      cancelAnimationFrame(animFrameId);
      animFrameId = null;
    }
  }

  playBtn.addEventListener('click', () => {
    if (isPlaying) pause();
    else play();
  });

  slider.addEventListener('input', (e) => {
    if (isPlaying) pause();
    currentTime = parseFloat(e.target.value);
    updateVisuals(currentTime);
  });

  // Initial draw
  updateVisuals(0);
</script>
</body>
</html>

Can we find the exact speed number from the graph?

**Speed** tells us how many meters something moves in $1$ second.

To find the speed, divide the distance traveled by the time it took: $$\text{Speed} = \frac{\text{Distance}}{\text{Time}}$$

<!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;
    color: #1e2945;
    background: transparent;
    padding: 12px;
    display: flex;
    flex-direction: column;
    align-items: center;
    gap: 12px;
    user-select: none;
  }
  .visual-container {
    width: 100%;
    max-width: 440px;
    display: flex;
    flex-direction: column;
    gap: 10px;
  }
  .track-card {
    background: #ffffff;
    border-radius: 12px;
    padding: 10px 14px;
    box-shadow: 0 2px 8px rgba(0,0,0,0.06);
    display: flex;
    flex-direction: column;
    gap: 4px;
  }
  .track-lane {
    position: relative;
    height: 32px;
    background: #f1f5f9;
    border-radius: 8px;
    border: 1px dashed #cbd5e1;
    overflow: hidden;
  }
  .track-runner {
    position: absolute;
    top: 50%;
    transform: translate(-50%, -50%);
    font-size: 20px;
    transition: left 0.3s ease;
    will-change: left;
  }
  .track-markers {
    display: flex;
    justify-content: space-between;
    font-size: 11px;
    color: #64748b;
    padding: 0 4px;
  }
  .graph-card {
    background: #ffffff;
    border-radius: 12px;
    padding: 12px;
    box-shadow: 0 2px 8px rgba(0,0,0,0.06);
    display: flex;
    flex-direction: column;
    align-items: center;
  }
  svg {
    width: 100%;
    height: auto;
    max-height: 210px;
    overflow: visible;
  }
  .math-bar {
    display: flex;
    align-items: center;
    justify-content: center;
    gap: 8px;
    font-size: 15px;
    font-weight: 600;
    padding: 6px 12px;
    background: #f8fafc;
    border-radius: 8px;
    border: 1px solid #e2e8f0;
  }
  .val-d { color: #0284c7; }
  .val-t { color: #8b5cf6; }
  .val-s { color: #10b981; }
  .controls {
    display: flex;
    gap: 8px;
    justify-content: center;
  }
  .step-btn {
    border: 1px solid #e2e8f0;
    background: #ffffff;
    color: #1e2945;
    font-size: 13px;
    font-weight: 600;
    padding: 6px 14px;
    border-radius: 8px;
    cursor: pointer;
    box-shadow: 0 1px 2px rgba(0,0,0,0.05);
    transition: background 0.2s, box-shadow 0.2s;
  }
  .step-btn.active {
    background: #22b7ff;
    color: #ffffff;
    border-color: #22b7ff;
    box-shadow: 0 2px 6px rgba(34, 183, 255, 0.4);
  }
</style>
</head>
<body>

<div class="visual-container">
  
  <div class="track-card">
    <div class="track-lane">
      <div id="runner" class="track-runner" style="left: 8%;">🐶</div>
    </div>
    <div class="track-markers">
      <span>0 m</span>
      <span>3 m</span>
      <span>6 m</span>
      <span>9 m</span>
      <span>12 m</span>
    </div>
  </div>

  
  <div class="graph-card">
    <svg viewBox="0 0 320 180" aria-label="Distance-Time Graph">
      
      <g stroke="#f1f5f9" stroke-width="1">
        <line x1="40" y1="20" x2="300" y2="20" />
        <line x1="40" y1="55" x2="300" y2="55" />
        <line x1="40" y1="90" x2="300" y2="90" />
        <line x1="40" y1="125" x2="300" y2="125" />
        <line x1="105" y1="20" x2="105" y2="160" />
        <line x1="170" y1="20" x2="170" y2="160" />
        <line x1="235" y1="20" x2="235" y2="160" />
        <line x1="300" y1="20" x2="300" y2="160" />
      </g>

      
      <line x1="40" y1="160" x2="305" y2="160" stroke="#94a3b8" stroke-width="2" />
      <line x1="40" y1="15" x2="40" y2="160" stroke="#94a3b8" stroke-width="2" />

      
      <text x="305" y="174" fill="#64748b" font-size="11" text-anchor="end" font-weight="600">Time (s)</text>
      <text x="10" y="16" fill="#64748b" font-size="11" font-weight="600">Distance (m)</text>

      
      <text x="40" y="174" fill="#94a3b8" font-size="10" text-anchor="middle">0</text>
      <text x="105" y="174" fill="#94a3b8" font-size="10" text-anchor="middle">1</text>
      <text x="170" y="174" fill="#94a3b8" font-size="10" text-anchor="middle">2</text>
      <text x="235" y="174" fill="#94a3b8" font-size="10" text-anchor="middle">3</text>
      <text x="300" y="174" fill="#94a3b8" font-size="10" text-anchor="middle">4</text>

      <text x="32" y="163" fill="#94a3b8" font-size="10" text-anchor="end">0</text>
      <text x="32" y="128" fill="#94a3b8" font-size="10" text-anchor="end">3</text>
      <text x="32" y="93" fill="#94a3b8" font-size="10" text-anchor="end">6</text>
      <text x="32" y="58" fill="#94a3b8" font-size="10" text-anchor="end">9</text>
      <text x="32" y="23" fill="#94a3b8" font-size="10" text-anchor="end">12</text>

      
      <polygon id="slope-poly" points="40,160 170,160 170,90" fill="#22b7ff" fill-opacity="0.1" stroke="#22b7ff" stroke-width="1.5" stroke-dasharray="3,3" />
      <text id="time-delta" x="105" y="152" fill="#8b5cf6" font-size="11" font-weight="700" text-anchor="middle">2 s</text>
      <text id="dist-delta" x="176" y="128" fill="#0284c7" font-size="11" font-weight="700">6 m</text>

      
      <line x1="40" y1="160" x2="300" y2="20" stroke="#22b7ff" stroke-width="3" stroke-linecap="round" />

      
      <circle id="dot" cx="170" cy="90" r="6" fill="#0284c7" stroke="#ffffff" stroke-width="2" />
    </svg>
  </div>

  
  <div class="math-bar">
    <span>Speed =</span>
    <span class="val-d" id="calc-d">6 m</span>
    <span>÷</span>
    <span class="val-t" id="calc-t">2 s</span>
    <span>=</span>
    <span class="val-s">3 m/s</span>
  </div>

  
  <div class="controls">
    <button class="step-btn" data-t="1">1 s (3 m)</button>
    <button class="step-btn active" data-t="2">2 s (6 m)</button>
    <button class="step-btn" data-t="3">3 s (9 m)</button>
    <button class="step-btn" data-t="4">4 s (12 m)</button>
  </div>
</div>

<script>
  const data = {
    1: { cx: 105, cy: 125, d: 3, trackPct: 29 },
    2: { cx: 170, cy: 90,  d: 6, trackPct: 50 },
    3: { cx: 235, cy: 55,  d: 9, trackPct: 71 },
    4: { cx: 300, cy: 20,  d: 12, trackPct: 92 }
  };

  const dot = document.getElementById('dot');
  const runner = document.getElementById('runner');
  const poly = document.getElementById('slope-poly');
  const timeDelta = document.getElementById('time-delta');
  const distDelta = document.getElementById('dist-delta');
  const calcD = document.getElementById('calc-d');
  const calcT = document.getElementById('calc-t');
  const buttons = document.querySelectorAll('.step-btn');

  function setTime(t) {
    const cur = data[t];
    
    // Update active button
    buttons.forEach(b => {
      if (parseInt(b.dataset.t, 10) === t) {
        b.classList.add('active');
      } else {
        b.classList.remove('active');
      }
    });

    // Update Graph Elements
    dot.setAttribute('cx', cur.cx);
    dot.setAttribute('cy', cur.cy);
    poly.setAttribute('points', `40,160 ${cur.cx},160 ${cur.cx},${cur.cy}`);
    
    timeDelta.setAttribute('x', (40 + cur.cx) / 2);
    timeDelta.textContent = `${t} s`;

    distDelta.setAttribute('x', cur.cx + 6);
    distDelta.setAttribute('y', (160 + cur.cy) / 2 + 4);
    distDelta.textContent = `${cur.d} m`;

    // Update Track Runner
    runner.style.left = `${cur.trackPct}%`;

    // Update Math
    calcD.textContent = `${cur.d} m`;
    calcT.textContent = `${t} s`;
  }

  buttons.forEach(b => {
    b.addEventListener('click', () => {
      setTime(parseInt(b.dataset.t, 10));
    });
  });
</script>

</body>
</html>

Let us check the puppy: it ran $6$ meters in $2$ seconds, so its speed is $6 \div 2 = 3$ meters per second.

What is an easy mistake to avoid when looking at these lines?

A line going up does not mean an animal is walking up a mountain.

It only means the animal is moving **farther away** from the start line on flat ground.

Next time you look at a race line, check how steep it climbs to see who is zooming ahead!

math

Interpreting Distance-Time Graphs with Constant Speed. Determine speed by finding the gradient of linear sections on a distance-time graph.

Curriculum for Excellence Third Level Maths: Distance, Speed and Time Graphs

Distance-Time Graphs Video & Practice Worksheets

Interpreting Distance-Time Graphs and Calculating Gradient

Ensure students bridge conceptual kinematics and geometric rate-of-change by explicitly using the gradient formula $\text{Gradient} = \frac{\text{Rise}}{\text{Run}} = \frac{\Delta d}{\Delta t}$. Emphasize physical storytelling (narrating each leg of a journey) before numerical computation to eliminate the visual-pathway misconception.

Interpreting Distance-Time Graphs with Constant Speed

Steep Lines Mean Fast Speed

Finding Speed with Numbers