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Imagine a toy rocket starts with some speed, gets a boost from its engine, and finishes flying super fast.

In physics, we write this as $v = u + at$, where $v$ is the **final speed**, $u$ is the **starting speed**, and $at$ is the speed boost.

<!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>Rearranging Formulas: One-Step</title>
  <link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.9/katex.min.css">
  <script src="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.16.9/katex.min.js"></script>
  <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;
      min-height: 100%;
      padding: 12px;
      user-select: none;
    }

    .main-stage {
      width: 100%;
      max-width: 480px;
      display: flex;
      flex-direction: column;
      align-items: center;
      gap: 16px;
    }

    /* Equation Display Box */
    .equation-card {
      width: 100%;
      background: #ffffff;
      border-radius: 14px;
      box-shadow: 0 4px 16px rgba(0,0,0,0.06);
      border: 1px solid #e2e8f0;
      padding: 24px 16px;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      min-height: 150px;
      position: relative;
      overflow: hidden;
    }

    .step-content {
      position: absolute;
      width: 100%;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      gap: 12px;
      opacity: 0;
      pointer-events: none;
      transition: opacity 0.35s ease, transform 0.35s ease;
      transform: translateY(8px);
    }

    .step-content.active {
      opacity: 1;
      pointer-events: auto;
      transform: translateY(0);
    }

    .math-row {
      font-size: 26px;
      display: flex;
      align-items: center;
      justify-content: center;
      gap: 6px;
    }

    .target {
      color: #2563eb;
      font-weight: bold;
      background: #eff6ff;
      padding: 2px 8px;
      border-radius: 8px;
      border: 1px solid #bfdbfe;
    }

    .op-badge {
      font-size: 13px;
      font-weight: 600;
      color: #f97316;
      background: #fff7ed;
      border: 1px solid #fed7aa;
      padding: 3px 10px;
      border-radius: 12px;
      letter-spacing: 0.5px;
    }

    .strike {
      position: relative;
      color: #94a3b8;
    }
    .strike::after {
      content: '';
      position: absolute;
      left: -2px;
      right: -2px;
      top: 50%;
      height: 2px;
      background: #ef4444;
      transform: rotate(-12deg);
    }

    /* Step Navigation Bar */
    .nav-bar {
      display: flex;
      align-items: center;
      justify-content: center;
      gap: 8px;
      width: 100%;
    }

    .step-btn {
      background: #ffffff;
      border: 1px solid #e2e8f0;
      color: #64748b;
      font-size: 14px;
      font-weight: 600;
      padding: 8px 16px;
      border-radius: 10px;
      cursor: pointer;
      box-shadow: 0 1px 3px rgba(0,0,0,0.04);
      transition: all 0.2s ease;
    }

    .step-btn.active {
      background: #2563eb;
      color: #ffffff;
      border-color: #2563eb;
      box-shadow: 0 2px 8px rgba(37,99,235,0.25);
    }

    .play-btn {
      background: #f1f5f9;
      border: 1px solid #cbd5e1;
      color: #334155;
      font-size: 14px;
      width: 36px;
      height: 36px;
      border-radius: 10px;
      display: flex;
      align-items: center;
      justify-content: center;
      cursor: pointer;
      margin-left: 4px;
    }
  </style>
</head>
<body>

  <div class="main-stage">
    <div class="equation-card">
      
      
      <div class="step-content active" id="step-0">
        <span class="op-badge" style="color:#2563eb; background:#eff6ff; border-color:#bfdbfe;">Target: u</span>
        <div class="math-row">
          <span>v</span>
          <span>=</span>
          <span class="target">u</span>
          <span>+</span>
          <span>at</span>
        </div>
      </div>

      
      <div class="step-content" id="step-1">
        <span class="op-badge">− at (both sides)</span>
        <div class="math-row">
          <span>v − at</span>
          <span>=</span>
          <span class="target">u</span>
          <span class="strike">+ at − at</span>
        </div>
      </div>

      
      <div class="step-content" id="step-2">
        <span class="op-badge" style="color:#16a34a; background:#f0fdf4; border-color:#bbf7d0;">✓ u Isolated</span>
        <div class="math-row">
          <span class="target">u</span>
          <span>=</span>
          <span>v − at</span>
        </div>
      </div>

    </div>

    
    <div class="nav-bar">
      <button class="step-btn active" data-step="0">1. Original</button>
      <button class="step-btn" data-step="1">2. Balance</button>
      <button class="step-btn" data-step="2">3. Solution</button>
      <button class="play-btn" id="play-btn" aria-label="Play/Pause">⏸</button>
    </div>
  </div>

  <script>
    const steps = document.querySelectorAll('.step-content');
    const buttons = document.querySelectorAll('.step-btn');
    const playBtn = document.getElementById('play-btn');

    let currentStep = 0;
    let isPlaying = true;
    let timer = null;

    function showStep(index) {
      currentStep = index;
      steps.forEach((s, i) => s.classList.toggle('active', i === currentStep));
      buttons.forEach((b, i) => b.classList.toggle('active', i === currentStep));
    }

    function nextStep() {
      showStep((currentStep + 1) % steps.length);
    }

    function startTimer() {
      stopTimer();
      timer = setInterval(nextStep, 3500);
    }

    function stopTimer() {
      if (timer) {
        clearInterval(timer);
        timer = null;
      }
    }

    buttons.forEach((btn) => {
      btn.addEventListener('click', () => {
        const step = parseInt(btn.getAttribute('data-step'), 10);
        showStep(step);
        if (isPlaying) {
          startTimer();
        }
      });
    });

    playBtn.addEventListener('click', () => {
      isPlaying = !isPlaying;
      playBtn.textContent = isPlaying ? '⏸' : '▶';
      if (isPlaying) {
        startTimer();
      } else {
        stopTimer();
      }
    });

    // Auto-cycle start
    startTimer();
  </script>
</body>
</html>

What if you already know the final speed, but you need to find the mystery starting speed $u$?

The letter sitting alone on one side of an equals sign is called the **subject** of the formula.

To make $u$ the subject, we use an **inverse operation** — a math action that undoes another action, just like subtraction undoes addition.

In the 1600s, Galileo Galilei studied rolling balls on ramps and showed that moving objects follow these exact balance rules.

<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="UTF-8">
<meta name="viewport" content="width=device-width, initial-scale=1.0">
<meta name="aspect-ratio" content="4:3">
<link rel="stylesheet" href="{{VENDOR_CDN}}/katex.min.css">
<script src="{{VENDOR_CDN}}/katex.min.js"></script>
<script src="{{VENDOR_CDN}}/classace-diagram.js"></script>
<style>
  * { box-sizing: border-box; margin: 0; padding: 0; }
  body {
    font-family: system-ui, -apple-system, sans-serif;
    color: #1e2945;
    background: #ffffff;
    display: flex;
    flex-direction: column;
    align-items: center;
    justify-content: center;
    min-height: 100%;
    padding: 12px;
    user-select: none;
  }
  .app-container {
    width: 100%;
    max-width: 520px;
    display: flex;
    flex-direction: column;
    gap: 12px;
  }
  .op-tabs {
    display: flex;
    gap: 6px;
    background: #f1f5f9;
    padding: 4px;
    border-radius: 10px;
  }
  .tab-btn {
    flex: 1;
    border: none;
    background: transparent;
    padding: 8px 4px;
    border-radius: 8px;
    font-size: 15px;
    font-weight: 600;
    color: #64748b;
    cursor: pointer;
    text-align: center;
  }
  .tab-btn.active {
    background: #ffffff;
    color: #2563eb;
    box-shadow: 0 1px 3px rgba(0,0,0,0.1);
  }
  .stage {
    background: #f8fafc;
    border: 1px solid #e2e8f0;
    border-radius: 12px;
    padding: 16px;
    display: flex;
    flex-direction: column;
    align-items: center;
    gap: 12px;
    position: relative;
    overflow: hidden;
  }
  .scale-svg {
    width: 100%;
    max-width: 320px;
    height: 90px;
  }
  .beam {
    stroke: #64748b;
    stroke-width: 4;
    stroke-linecap: round;
  }
  .pivot {
    fill: #334155;
  }
  .pan-rope {
    stroke: #94a3b8;
    stroke-width: 2;
  }
  .pan-plate {
    stroke: #2563eb;
    stroke-width: 3;
    fill: #dbeafe;
    rx: 4;
  }
  .equation-box {
    display: flex;
    align-items: center;
    justify-content: center;
    gap: 12px;
    min-height: 48px;
    font-size: 20px;
  }
  .side-card {
    background: #ffffff;
    border: 2px solid #e2e8f0;
    border-radius: 8px;
    padding: 6px 14px;
    min-width: 80px;
    text-align: center;
    box-shadow: 0 2px 4px rgba(0,0,0,0.03);
    transition: all 0.3s ease;
  }
  .side-card.highlight {
    border-color: #2563eb;
    background: #eff6ff;
  }
  .side-card.target {
    border-color: #10b981;
    background: #ecfdf5;
  }
  .eq-sign {
    font-weight: 700;
    color: #64748b;
    font-size: 22px;
  }
  .op-badge {
    display: inline-flex;
    align-items: center;
    justify-content: center;
    background: #fee2e2;
    color: #dc2626;
    border-radius: 6px;
    padding: 3px 10px;
    font-size: 13px;
    font-weight: 600;
    min-height: 24px;
  }
  .op-badge.inverse {
    background: #dbeafe;
    color: #1d4ed8;
  }
  .stepper-nav {
    display: flex;
    align-items: center;
    justify-content: space-between;
    width: 100%;
    padding: 0 4px;
  }
  .step-dots {
    display: flex;
    gap: 6px;
  }
  .dot {
    width: 24px;
    height: 6px;
    border-radius: 3px;
    background: #cbd5e1;
    cursor: pointer;
    transition: background 0.3s ease;
  }
  .dot.active {
    background: #2563eb;
  }
  .ctrl-btn {
    border: 1px solid #cbd5e1;
    background: #ffffff;
    padding: 6px 14px;
    border-radius: 6px;
    font-size: 13px;
    font-weight: 600;
    color: #1e2945;
    cursor: pointer;
    display: flex;
    align-items: center;
    gap: 4px;
  }
  .ctrl-btn:hover {
    background: #f8fafc;
  }
</style>
</head>
<body>

<div class="app-container">
  <div class="op-tabs">
    <button class="tab-btn active" id="tab0">+ a</button>
    <button class="tab-btn" id="tab1">− a</button>
    <button class="tab-btn" id="tab2">× a</button>
    <button class="tab-btn" id="tab3">÷ a</button>
  </div>

  <div class="stage">
    <svg class="scale-svg" viewBox="0 0 320 90">
      <polygon points="160,35 150,85 170,85" class="pivot" />
      <line x1="40" y1="35" x2="280" y2="35" class="beam" />
      <line x1="60" y1="35" x2="40" y2="65" class="pan-rope" />
      <line x1="60" y1="35" x2="80" y2="65" class="pan-rope" />
      <rect x="30" y="65" width="60" height="6" class="pan-plate" />
      <line x1="260" y1="35" x2="240" y2="65" class="pan-rope" />
      <line x1="260" y1="35" x2="280" y2="65" class="pan-rope" />
      <rect x="230" y="65" width="60" height="6" class="pan-plate" />
      <circle cx="160" cy="35" r="4" fill="#ffffff" stroke="#334155" stroke-width="2" />
    </svg>

    <div class="op-badge" id="actionBadge">...</div>

    <div class="equation-box">
      <div class="side-card" id="leftSide"></div>
      <div class="eq-sign">=</div>
      <div class="side-card" id="rightSide"></div>
    </div>
  </div>

  <div class="stepper-nav">
    <button class="ctrl-btn" id="prevBtn">← Back</button>
    <div class="step-dots">
      <div class="dot active" id="dot0"></div>
      <div class="dot" id="dot1"></div>
      <div class="dot" id="dot2"></div>
    </div>
    <button class="ctrl-btn" id="nextBtn">Next →</button>
  </div>
</div>

<script>
const data = [
  {
    name: "+ a",
    steps: [
      { left: "y", right: "x + a", badge: "Target: x", isInv: false, lClass: "", rClass: "highlight" },
      { left: "y - a", right: "x + a - a", badge: "Both sides − a", isInv: true, lClass: "highlight", rClass: "highlight" },
      { left: "y - a", right: "x", badge: "x is the subject ✦", isInv: true, lClass: "", rClass: "target" }
    ]
  },
  {
    name: "− a",
    steps: [
      { left: "y", right: "x - a", badge: "Target: x", isInv: false, lClass: "", rClass: "highlight" },
      { left: "y + a", right: "x - a + a", badge: "Both sides + a", isInv: true, lClass: "highlight", rClass: "highlight" },
      { left: "y + a", right: "x", badge: "x is the subject ✦", isInv: true, lClass: "", rClass: "target" }
    ]
  },
  {
    name: "× a",
    steps: [
      { left: "y", right: "ax", badge: "Target: x", isInv: false, lClass: "", rClass: "highlight" },
      { left: "\\frac{y}{a}", right: "\\frac{ax}{a}", badge: "Both sides ÷ a", isInv: true, lClass: "highlight", rClass: "highlight" },
      { left: "\\frac{y}{a}", right: "x", badge: "x is the subject ✦", isInv: true, lClass: "", rClass: "target" }
    ]
  },
  {
    name: "÷ a",
    steps: [
      { left: "y", right: "\\frac{x}{a}", badge: "Target: x", isInv: false, lClass: "", rClass: "highlight" },
      { left: "y \\cdot a", right: "\\frac{x}{a} \\cdot a", badge: "Both sides × a", isInv: true, lClass: "highlight", rClass: "highlight" },
      { left: "ay", right: "x", badge: "x is the subject ✦", isInv: true, lClass: "", rClass: "target" }
    ]
  }
];

let curOp = 0;
let curStep = 0;

function render() {
  const op = data[curOp];
  const step = op.steps[curStep];

  const leftEl = document.getElementById('leftSide');
  const rightEl = document.getElementById('rightSide');
  const badgeEl = document.getElementById('actionBadge');

  leftEl.className = 'side-card ' + step.lClass;
  rightEl.className = 'side-card ' + step.rClass;

  katex.render(step.left, leftEl, { throwOnError: false });
  katex.render(step.right, rightEl, { throwOnError: false });

  badgeEl.textContent = step.badge;
  badgeEl.className = 'op-badge' + (step.isInv ? ' inverse' : '');

  for (let i = 0; i < 4; i++) {
    document.getElementById('tab' + i).className = 'tab-btn' + (i === curOp ? ' active' : '');
  }
  for (let i = 0; i < 3; i++) {
    document.getElementById('dot' + i).className = 'dot' + (i === curStep ? ' active' : '');
  }
}

for (let i = 0; i < 4; i++) {
  document.getElementById('tab' + i).onclick = () => {
    curOp = i;
    curStep = 0;
    render();
  };
}

for (let i = 0; i < 3; i++) {
  document.getElementById('dot' + i).onclick = () => {
    curStep = i;
    render();
  };
}

document.getElementById('prevBtn').onclick = () => {
  curStep = (curStep + 2) % 3;
  render();
};

document.getElementById('nextBtn').onclick = () => {
  curStep = (curStep + 1) % 3;
  render();
};

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

How do we write this down with algebra steps without breaking the balance?

We start with our equation: $$v = u + at$$

Subtract the term $at$ from **both sides** so that $u$ gets left all by itself: $$v - at = u$$

Because both sides are equal, we can flip the equation around: $$u = v - at$$

<!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>Changing the Subject</title>
  <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: space-between;
      min-height: 100%;
      padding: 16px 12px;
    }

    .header-bar {
      display: flex;
      align-items: center;
      justify-content: center;
      gap: 8px;
      margin-bottom: 8px;
    }

    .badge {
      display: inline-flex;
      align-items: center;
      padding: 4px 10px;
      border-radius: 20px;
      font-size: 13px;
      font-weight: 600;
      background: #e0f2fe;
      color: #0284c7;
    }

    .stage {
      position: relative;
      width: 100%;
      max-width: 440px;
      height: 220px;
      background: #ffffff;
      border-radius: 16px;
      box-shadow: 0 4px 20px rgba(0, 0, 0, 0.05);
      border: 1px solid #e5e7eb;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      overflow: hidden;
    }

    .op-banner {
      position: absolute;
      top: 18px;
      padding: 4px 14px;
      border-radius: 12px;
      font-size: 14px;
      font-weight: 700;
      background: #fef2f2;
      color: #dc2626;
      border: 1px dashed #fca5a5;
      opacity: 0;
      transform: translateY(-8px);
      transition: opacity 0.3s ease, transform 0.3s ease;
    }

    .op-banner.visible {
      opacity: 1;
      transform: translateY(0);
    }

    .equation-row {
      display: flex;
      align-items: center;
      justify-content: center;
      gap: 10px;
      font-size: 26px;
      font-weight: 600;
      height: 60px;
      transition: all 0.4s ease;
    }

    .tile {
      display: inline-flex;
      align-items: center;
      justify-content: center;
      min-width: 44px;
      height: 48px;
      padding: 0 10px;
      border-radius: 10px;
      background: #f3f4f6;
      color: #1f2937;
      transition: all 0.35s ease;
    }

    .tile.target {
      background: #dbeafe;
      color: #1d4ed8;
      box-shadow: 0 0 0 2px #3b82f6;
      font-weight: 700;
    }

    .tile.term-op {
      background: #fee2e2;
      color: #dc2626;
      transition: all 0.4s ease;
    }

    .tile.term-plus {
      background: #fef3c7;
      color: #d97706;
    }

    .tile.sym {
      background: transparent;
      min-width: 24px;
      font-size: 24px;
      color: #6b7280;
    }

    .controls {
      display: flex;
      align-items: center;
      justify-content: center;
      gap: 8px;
      width: 100%;
      max-width: 440px;
      margin-top: 14px;
    }

    .step-btn {
      flex: 1;
      padding: 10px 8px;
      font-size: 13px;
      font-weight: 600;
      border-radius: 10px;
      border: 1px solid #e5e7eb;
      background: #ffffff;
      color: #4b5563;
      cursor: pointer;
      display: flex;
      align-items: center;
      justify-content: center;
      gap: 4px;
      transition: all 0.2s ease;
    }

    .step-btn:hover {
      background: #f9fafb;
    }

    .step-btn.active {
      background: #2563eb;
      color: #ffffff;
      border-color: #2563eb;
      box-shadow: 0 2px 8px rgba(37, 99, 235, 0.25);
    }

    .play-btn {
      width: 42px;
      height: 40px;
      border-radius: 10px;
      border: 1px solid #e5e7eb;
      background: #ffffff;
      color: #1e2945;
      font-size: 14px;
      cursor: pointer;
      display: flex;
      align-items: center;
      justify-content: center;
      transition: background 0.2s ease;
    }

    .play-btn:hover {
      background: #f3f4f6;
    }
  </style>
</head>
<body>

  <div class="header-bar">
    <span class="badge">Target: <i>u</i></span>
  </div>

  <div class="stage">
    <div id="opBanner" class="op-banner">− at</div>
    <div id="eqDisplay" class="equation-row"></div>
  </div>

  <div class="controls">
    <button id="btn0" class="step-btn active" onclick="setStep(0)">1. Start</button>
    <button id="btn1" class="step-btn" onclick="setStep(1)">2. − at</button>
    <button id="btn2" class="step-btn" onclick="setStep(2)">3. Flip</button>
    <button id="playBtn" class="play-btn" onclick="togglePlay()" title="Play/Pause">⏸</button>
  </div>

  <script>
    const steps = [
      {
        banner: false,
        html: `
          <span class="tile"><i>v</i></span>
          <span class="tile sym">=</span>
          <span class="tile target"><i>u</i></span>
          <span class="tile term-plus">+ <i>at</i></span>
        `
      },
      {
        banner: true,
        bannerText: "− at",
        html: `
          <span class="tile"><i>v</i></span>
          <span class="tile term-op">− <i>at</i></span>
          <span class="tile sym">=</span>
          <span class="tile target"><i>u</i></span>
        `
      },
      {
        banner: false,
        html: `
          <span class="tile target"><i>u</i></span>
          <span class="tile sym">=</span>
          <span class="tile"><i>v</i></span>
          <span class="tile term-op">− <i>at</i></span>
        `
      }
    ];

    let currentStep = 0;
    let timer = null;
    let isPlaying = true;

    const eqDisplay = document.getElementById('eqDisplay');
    const opBanner = document.getElementById('opBanner');
    const playBtn = document.getElementById('playBtn');

    function render() {
      const s = steps[currentStep];
      eqDisplay.innerHTML = s.html;

      if (s.banner) {
        opBanner.textContent = s.bannerText;
        opBanner.classList.add('visible');
      } else {
        opBanner.classList.remove('visible');
      }

      [0, 1, 2].forEach(i => {
        const b = document.getElementById('btn' + i);
        if (i === currentStep) {
          b.classList.add('active');
        } else {
          b.classList.remove('active');
        }
      });
    }

    function setStep(idx) {
      currentStep = idx;
      render();
    }

    function nextStep() {
      currentStep = (currentStep + 1) % steps.length;
      render();
    }

    function startTimer() {
      if (timer) clearInterval(timer);
      timer = setInterval(nextStep, 3200);
      isPlaying = true;
      playBtn.textContent = '⏸';
    }

    function pauseTimer() {
      if (timer) clearInterval(timer);
      timer = null;
      isPlaying = false;
      playBtn.textContent = '▶';
    }

    function togglePlay() {
      if (isPlaying) {
        pauseTimer();
      } else {
        nextStep();
        startTimer();
      }
    }

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

A common mistake is copying the plus sign by accident — remember, moving an added piece across the equals sign always requires **subtraction**.

Try looking at a formula like $y = x + 5$. What would $x$ equal if you undo the plus 5?

math

Changing the Subject of a Formula: One-Step Operations. Rearrange formulas like v = u + at to make u the subject using inverse operations.

Changing the Subject of a Formula - Third Level / National 4 & 5 Mathematics

Rearrange Formulas to Highlight a Quantity of Interest (Literal Equations)

Equations and Formulas: Balancing and Inverse Operations

Although labeled with an Early Level header in some modular tracking sheets, standard MTH 3-15b represents Scottish Curriculum for Excellence Third Level mathematics (Early Secondary / S1–S3, ages 11–14). The most effective pedagogical progression begins by connecting literal formulas to familiar one-step one-variable equations (e.g., solving $10 = u + 4$ in parallel with $v = u + at$). Use dual-column 'concrete-to-abstract' bridging and physical or visual balance scales to reinforce that entire algebraic terms (like $at$) act as single units under addition and subtraction.

Changing the Subject of a Formula: One-Step Operations

The Balance Rule

Step-by-Step Rearranging