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System default

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Tutoring abilities

Imagine you have 3 red apples in a basket. A friend drops in a mystery bag, and suddenly you count 5 apples in all!

How can we write this story using math clues to find what is inside the bag?

An **equation** is a math sentence that shows two sides are equal using an equal sign ($=$).

When we do not know a number yet, we can use a letter like $n$ or a box to stand for the **unknown** amount.

<!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;
    background: #f8f9fa;
    color: #1e2945;
    display: flex;
    flex-direction: column;
    align-items: center;
    justify-content: center;
    min-height: 100vh;
    padding: 12px;
  }

  .workspace {
    width: 100%;
    max-width: 540px;
    display: flex;
    flex-direction: column;
    gap: 16px;
    align-items: center;
  }

  /* Visual equation shelf */
  .shelf {
    width: 100%;
    display: flex;
    align-items: center;
    justify-content: space-evenly;
    background: #ffffff;
    border-radius: 16px;
    padding: 18px 8px;
    box-shadow: 0 2px 8px rgba(0,0,0,0.04);
  }

  .group {
    display: flex;
    flex-direction: column;
    align-items: center;
    gap: 8px;
    min-width: 64px;
  }

  .items-box {
    display: flex;
    align-items: center;
    justify-content: center;
    gap: 6px;
    min-height: 48px;
    padding: 6px 10px;
    background: #f1f5f9;
    border-radius: 12px;
    transition: all 0.3s ease;
  }

  .active-target {
    box-shadow: 0 0 0 2px #22b7ff;
    background: #e0f4ff;
  }

  /* Counters */
  .dot {
    width: 14px;
    height: 14px;
    border-radius: 50%;
    background: #22b7ff;
    transition: transform 0.3s ease;
  }

  .dot.green {
    background: #10b981;
  }

  .dot.purple {
    background: #8b5cf6;
  }

  /* Mystery Bag */
  .bag {
    width: 44px;
    height: 44px;
    background: #8b5cf6;
    color: #ffffff;
    border-radius: 10px;
    display: flex;
    align-items: center;
    justify-content: center;
    font-size: 20px;
    font-weight: 700;
    cursor: pointer;
    user-select: none;
    transition: background 0.3s ease, transform 0.3s ease;
  }

  .bag.revealed {
    background: #ede9fe;
    color: #8b5cf6;
    border: 2px dashed #8b5cf6;
  }

  .op {
    font-size: 22px;
    font-weight: 700;
    color: #94a3b8;
  }

  /* Equation display */
  .equation-bar {
    width: 100%;
    display: flex;
    align-items: center;
    justify-content: space-evenly;
    background: #ffffff;
    border-radius: 14px;
    padding: 12px 8px;
    font-size: 26px;
    font-weight: 700;
    box-shadow: 0 2px 8px rgba(0,0,0,0.04);
  }

  .eq-num {
    display: inline-flex;
    align-items: center;
    justify-content: center;
    min-width: 44px;
    height: 44px;
    border-radius: 10px;
    transition: all 0.3s ease;
  }

  .eq-3 { color: #0284c7; background: #e0f2fe; }
  .eq-n { color: #7c3aed; background: #f3e8ff; }
  .eq-5 { color: #059669; background: #d1fae5; }
  .eq-op { color: #64748b; }

  /* Interactive Stepper Tabs */
  .nav {
    display: flex;
    gap: 8px;
    width: 100%;
    justify-content: center;
    flex-wrap: wrap;
  }

  .step-btn {
    border: none;
    outline: none;
    padding: 8px 14px;
    border-radius: 20px;
    font-size: 13px;
    font-weight: 600;
    background: #e2e8f0;
    color: #475569;
    cursor: pointer;
    transition: all 0.2s ease;
  }

  .step-btn.active {
    background: #22b7ff;
    color: #ffffff;
  }
</style>
</head>
<body>

<div class="workspace">
  
  <div class="shelf">
    <div class="group">
      <div class="items-box" id="box-start">
        <span class="dot"></span>
        <span class="dot"></span>
        <span class="dot"></span>
      </div>
    </div>

    <div class="op">+</div>

    <div class="group">
      <div class="bag" id="bag" onclick="toggleReveal()">n</div>
    </div>

    <div class="op">=</div>

    <div class="group">
      <div class="items-box" id="box-total">
        <span class="dot green"></span>
        <span class="dot green"></span>
        <span class="dot green"></span>
        <span class="dot green"></span>
        <span class="dot green"></span>
      </div>
    </div>
  </div>

  
  <div class="equation-bar">
    <span class="eq-num eq-3" id="eq-part-1">3</span>
    <span class="eq-op">+</span>
    <span class="eq-num eq-n" id="eq-part-2">n</span>
    <span class="eq-op">=</span>
    <span class="eq-num eq-5" id="eq-part-3">5</span>
  </div>

  
  <div class="nav">
    <button class="step-btn active" onclick="setStep(0)">1. Start (3)</button>
    <button class="step-btn" onclick="setStep(1)">2. Add Unknown (+ n)</button>
    <button class="step-btn" onclick="setStep(2)">3. Total (= 5)</button>
    <button class="step-btn" onclick="setStep(3)">✦ Reveal n</button>
  </div>
</div>

<script>
  let currentStep = 0;
  const bag = document.getElementById('bag');
  const boxStart = document.getElementById('box-start');
  const boxTotal = document.getElementById('box-total');
  const eq1 = document.getElementById('eq-part-1');
  const eq2 = document.getElementById('eq-part-2');
  const eq3 = document.getElementById('eq-part-3');
  const btns = document.querySelectorAll('.step-btn');

  function clearHighlights() {
    boxStart.classList.remove('active-target');
    boxTotal.classList.remove('active-target');
    bag.style.transform = 'scale(1)';
    eq1.style.transform = 'scale(1)';
    eq2.style.transform = 'scale(1)';
    eq3.style.transform = 'scale(1)';
  }

  function setStep(index) {
    currentStep = index;
    btns.forEach((btn, i) => {
      btn.classList.toggle('active', i === index);
    });

    clearHighlights();

    if (index === 0) {
      boxStart.classList.add('active-target');
      eq1.style.transform = 'scale(1.15)';
      resetBag();
    } else if (index === 1) {
      bag.style.transform = 'scale(1.15)';
      eq2.style.transform = 'scale(1.15)';
      resetBag();
    } else if (index === 2) {
      boxTotal.classList.add('active-target');
      eq3.style.transform = 'scale(1.15)';
      resetBag();
    } else if (index === 3) {
      revealBag();
    }
  }

  function resetBag() {
    bag.classList.remove('revealed');
    bag.innerHTML = 'n';
    eq2.innerHTML = 'n';
  }

  function revealBag() {
    bag.classList.add('revealed');
    bag.innerHTML = '<div style="display:flex;gap:4px;"><span class="dot purple"></span><span class="dot purple"></span></div>';
    eq2.innerHTML = '2';
    eq2.style.transform = 'scale(1.15)';
  }

  function toggleReveal() {
    if (bag.classList.contains('revealed')) {
      resetBag();
      setStep(1);
    } else {
      setStep(3);
    }
  }

  // Initial render
  setStep(0);
</script>

</body>
</html>

We started with 3, added the mystery bag $n$, and ended with 5, giving us $3 + n = 5$.

What if a story talks about giving things away instead of getting more?

Leo bakes 8 tasty cookies. He gives some cookies to his sister, and now he has 6 cookies left.

<!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>Constructing Equations</title>
  <link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.16.8/dist/katex.min.css">
  <script defer src="https://cdn.jsdelivr.net/npm/katex@0.16.8/dist/katex.min.js"></script>
  <style>
    * {
      box-sizing: border-box;
      margin: 0;
      padding: 0;
      font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
    }

    body {
      background: #f8f9fa;
      color: #1e2945;
      padding: 16px;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: space-between;
      min-height: 100%;
      user-select: none;
    }

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

    /* Nav Bar */
    .controls {
      display: flex;
      justify-content: center;
      gap: 8px;
    }

    .step-btn {
      padding: 6px 14px;
      font-size: 14px;
      font-weight: 600;
      border-radius: 20px;
      border: 1px solid #d1d5db;
      background: #ffffff;
      color: #4b5563;
      cursor: pointer;
      box-shadow: 0 1px 2px rgba(0,0,0,0.05);
      transition: all 0.25s ease;
    }

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

    /* Stage Area */
    .stage {
      background: #ffffff;
      border: 1px solid #e5e7eb;
      border-radius: 16px;
      padding: 20px 16px;
      box-shadow: 0 4px 12px rgba(0,0,0,0.03);
      display: flex;
      flex-direction: column;
      align-items: center;
      gap: 16px;
    }

    .cookie-grid {
      display: grid;
      grid-template-columns: repeat(8, 1fr);
      gap: 6px;
      width: 100%;
      max-width: 440px;
      justify-items: center;
    }

    .cookie-slot {
      display: flex;
      flex-direction: column;
      align-items: center;
      gap: 4px;
    }

    .cookie {
      font-size: 30px;
      line-height: 1;
      transition: all 0.4s ease;
      filter: drop-shadow(0 2px 3px rgba(0,0,0,0.1));
    }

    .cookie-num {
      font-size: 12px;
      font-weight: 600;
      color: #9ca3af;
      transition: color 0.3s;
    }

    /* State modifications */
    .cookie.taken {
      opacity: 0.25;
      transform: translateY(-8px) scale(0.85);
      filter: grayscale(0.8);
    }

    .cookie.remaining {
      transform: scale(1.05);
      filter: drop-shadow(0 3px 5px rgba(34, 197, 94, 0.3));
    }

    /* Visual Groups */
    .group-labels {
      display: flex;
      width: 100%;
      max-width: 440px;
      justify-content: space-between;
      gap: 8px;
    }

    .badge {
      flex: 1;
      text-align: center;
      padding: 6px 8px;
      border-radius: 8px;
      font-size: 13px;
      font-weight: 600;
      transition: all 0.3s ease;
    }

    .badge-start {
      background: #eff6ff;
      border: 1px solid #bfdbfe;
      color: #1d4ed8;
    }

    .badge-taken {
      background: #fff7ed;
      border: 1px solid #fed7aa;
      color: #c2410c;
    }

    .badge-left {
      background: #f0fdf4;
      border: 1px solid #bbf7d0;
      color: #15803d;
    }

    /* Equation Section */
    .equation-card {
      background: #ffffff;
      border: 1px solid #e5e7eb;
      border-radius: 14px;
      padding: 14px 20px;
      display: flex;
      align-items: center;
      justify-content: center;
      gap: 12px;
      box-shadow: 0 2px 6px rgba(0,0,0,0.02);
    }

    .eq-part {
      font-size: 26px;
      font-weight: 700;
      padding: 4px 10px;
      border-radius: 8px;
      display: inline-flex;
      align-items: center;
      justify-content: center;
      transition: all 0.3s ease;
    }

    .eq-num-start {
      background: #eff6ff;
      color: #1d4ed8;
      border: 1px solid #93c5fd;
    }

    .eq-op {
      color: #6b7280;
      font-size: 22px;
    }

    .eq-var {
      background: #fff7ed;
      color: #ea580c;
      border: 1px solid #fdba74;
    }

    .eq-res {
      background: #f0fdf4;
      color: #16a34a;
      border: 1px solid #86efac;
    }

    .dimmed {
      opacity: 0.3;
      filter: grayscale(0.5);
    }
  </style>
</head>
<body>

  <div class="main-wrap">
    
    <div class="controls">
      <button class="step-btn active" onclick="setStep(0)">1. Start (8)</button>
      <button class="step-btn" onclick="setStep(1)">2. Gives (c)</button>
      <button class="step-btn" onclick="setStep(2)">3. Left (6)</button>
    </div>

    
    <div class="stage">
      <div class="cookie-grid">
        <div class="cookie-slot"><span class="cookie" id="c1">🍪</span><span class="cookie-num">1</span></div>
        <div class="cookie-slot"><span class="cookie" id="c2">🍪</span><span class="cookie-num">2</span></div>
        <div class="cookie-slot"><span class="cookie" id="c3">🍪</span><span class="cookie-num">3</span></div>
        <div class="cookie-slot"><span class="cookie" id="c4">🍪</span><span class="cookie-num">4</span></div>
        <div class="cookie-slot"><span class="cookie" id="c5">🍪</span><span class="cookie-num">5</span></div>
        <div class="cookie-slot"><span class="cookie" id="c6">🍪</span><span class="cookie-num">6</span></div>
        <div class="cookie-slot"><span class="cookie" id="c7">🍪</span><span class="cookie-num">7</span></div>
        <div class="cookie-slot"><span class="cookie" id="c8">🍪</span><span class="cookie-num">8</span></div>
      </div>

      <div class="group-labels" id="labels-box">
        
      </div>
    </div>

    
    <div class="equation-card">
      <span class="eq-part eq-num-start" id="part-start">8</span>
      <span class="eq-op" id="part-op">−</span>
      <span class="eq-part eq-var" id="part-var">c</span>
      <span class="eq-op" id="part-eq">=</span>
      <span class="eq-part eq-res" id="part-res">6</span>
    </div>
  </div>

  <script>
    let currentStep = 0;
    const cookies = [
      document.getElementById('c1'),
      document.getElementById('c2'),
      document.getElementById('c3'),
      document.getElementById('c4'),
      document.getElementById('c5'),
      document.getElementById('c6'),
      document.getElementById('c7'),
      document.getElementById('c8')
    ];

    const partStart = document.getElementById('part-start');
    const partOp = document.getElementById('part-op');
    const partVar = document.getElementById('part-var');
    const partEq = document.getElementById('part-eq');
    const partRes = document.getElementById('part-res');
    const labelsBox = document.getElementById('labels-box');
    const buttons = document.querySelectorAll('.step-btn');

    function updateView() {
      // Buttons
      buttons.forEach((btn, idx) => {
        btn.classList.toggle('active', idx === currentStep);
      });

      // Cookies & Equation tokens
      if (currentStep === 0) {
        cookies.forEach(c => {
          c.className = 'cookie';
        });
        labelsBox.innerHTML = `
          <div class="badge badge-start" style="width: 100%;">Start: 8 cookies</div>
        `;
        partStart.classList.remove('dimmed');
        partOp.classList.add('dimmed');
        partVar.classList.add('dimmed');
        partEq.classList.add('dimmed');
        partRes.classList.add('dimmed');
      } else if (currentStep === 1) {
        cookies.slice(0, 6).forEach(c => c.className = 'cookie');
        cookies.slice(6).forEach(c => c.className = 'cookie taken');
        labelsBox.innerHTML = `
          <div class="badge badge-start" style="flex:3;">6 known</div>
          <div class="badge badge-taken" style="flex:1;">Takes away: c</div>
        `;
        partStart.classList.remove('dimmed');
        partOp.classList.remove('dimmed');
        partVar.classList.remove('dimmed');
        partEq.classList.add('dimmed');
        partRes.classList.add('dimmed');
      } else {
        cookies.slice(0, 6).forEach(c => c.className = 'cookie remaining');
        cookies.slice(6).forEach(c => c.className = 'cookie taken');
        labelsBox.innerHTML = `
          <div class="badge badge-left" style="flex:3;">6 cookies left</div>
          <div class="badge badge-taken" style="flex:1;">Given away: c</div>
        `;
        partStart.classList.remove('dimmed');
        partOp.classList.remove('dimmed');
        partVar.classList.remove('dimmed');
        partEq.classList.remove('dimmed');
        partRes.classList.remove('dimmed');
      }
    }

    function setStep(step) {
      currentStep = step;
      updateView();
    }

    // Auto cycle through steps with pedagogical pacing
    let autoTimer = setInterval(() => {
      currentStep = (currentStep + 1) % 3;
      updateView();
    }, 3500);

    // Pause cycling on user tap
    document.querySelectorAll('.step-btn').forEach(btn => {
      btn.addEventListener('click', () => clearInterval(autoTimer));
    });

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

Because Leo started with 8 and took away an unknown number $c$, our equation is $8 - c = 6$.

A common trap is putting the total at the start of an addition story. Remember, the total always belongs by itself on the other side of the equal sign!

Next time you hear a story with numbers, look for the starting amount, what changed, and the total to build your equation.

math

Constructing Equations from Word Problems. Translate written problem scenarios into solvable linear algebraic equations.

Constructing Linear Equations and Functions from Context

Third Level Maths: Forming and Solving Algebraic Equations

Write an Equation from Words and Scenarios

Aligned with Scottish Curriculum for Excellence (CfE) Third Level standard MTH 3-15a. Instructional best practice utilizes the Concrete-Representational-Abstract (CRA) model: starting with physical balance scales or algebra tiles, moving to bar model/strip representations, and finalizing in standard symbolic linear notation ($ax + b = c$). Scaffolding must focus on semantic understanding of relations rather than superficial keyword matching to avoid common syntactic traps.

Constructing Equations from Word Problems

Mystery Numbers

Taking Away