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System default

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Tutoring abilities

Imagine you are in an elevator deep underground on floor **-3**.

If you press the up button to go up 2 floors, where will you land?

A **number line** is a straight path of numbers where negative numbers are on the left and positive numbers are on the right.

Whenever we **add a positive number**, we always move **to the right** or **up** toward the bigger numbers.

<!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="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, Helvetica, Arial, sans-serif;
      background: transparent;
      color: #1e2945;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      padding: 16px;
      min-height: 100%;
    }

    .equation-box {
      font-size: 28px;
      margin-bottom: 20px;
      padding: 8px 20px;
      background: #ffffff;
      border: 1px solid #e2e8f0;
      border-radius: 12px;
      box-shadow: 0 2px 8px rgba(0,0,0,0.04);
      display: flex;
      align-items: center;
      gap: 6px;
    }

    .highlight-start { color: #f59e0b; font-weight: bold; }
    .highlight-add { color: #22b7ff; font-weight: bold; }
    .highlight-result { color: #10b981; font-weight: bold; }

    .diagram-area {
      width: 100%;
      max-width: 540px;
      background: #ffffff;
      border: 1px solid #e2e8f0;
      border-radius: 16px;
      padding: 24px 16px 16px 16px;
      box-shadow: 0 4px 12px rgba(0,0,0,0.03);
      display: flex;
      flex-direction: column;
      align-items: center;
    }

    svg {
      width: 100%;
      height: 150px;
      overflow: visible;
    }

    .tick-label {
      font-size: 14px;
      fill: #64748b;
      font-weight: 500;
      text-anchor: middle;
      user-select: none;
    }

    .tick-label.active {
      fill: #1e2945;
      font-weight: 700;
      font-size: 15px;
    }

    .hop-arc {
      fill: none;
      stroke: #22b7ff;
      stroke-width: 3;
      stroke-linecap: round;
      stroke-dasharray: 120;
      stroke-dashoffset: 120;
      transition: stroke-dashoffset 0.5s ease-out;
    }

    .hop-arc.visible {
      stroke-dashoffset: 0;
    }

    .hop-arrow {
      fill: #22b7ff;
      opacity: 0;
      transition: opacity 0.3s ease-out 0.4s;
    }

    .hop-arrow.visible {
      opacity: 1;
    }

    .hop-label {
      font-size: 13px;
      font-weight: 700;
      fill: #0284c7;
      text-anchor: middle;
      opacity: 0;
      transition: opacity 0.3s ease-out 0.2s;
    }

    .hop-label.visible {
      opacity: 1;
    }

    .marker {
      transition: transform 0.4s cubic-bezier(0.34, 1.56, 0.64, 1);
    }

    .controls {
      display: flex;
      gap: 8px;
      margin-top: 16px;
      flex-wrap: wrap;
      justify-content: center;
    }

    .btn {
      background: #f1f5f9;
      border: 1px solid #cbd5e1;
      color: #475569;
      padding: 8px 16px;
      border-radius: 20px;
      font-size: 14px;
      font-weight: 600;
      cursor: pointer;
      display: flex;
      align-items: center;
      gap: 6px;
      transition: background 0.2s, color 0.2s, box-shadow 0.2s;
    }

    .btn:hover {
      background: #e2e8f0;
      color: #1e2945;
    }

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

  <div class="equation-box" id="mathExp"></div>

  <div class="diagram-area">
    <svg viewBox="0 0 460 120" id="numLineSvg">
      
      <line x1="20" y1="80" x2="440" y2="80" stroke="#cbd5e1" stroke-width="3" stroke-linecap="round" />
      
      
      <polygon points="16,80 24,75 24,85" fill="#94a3b8" />
      <polygon points="444,80 436,75 436,85" fill="#94a3b8" />

      
      
      <g id="ticksGroup"></g>

      
      <path id="hop1" class="hop-arc" d="M 110 70 Q 140 28 170 70" />
      <polygon id="arr1" class="hop-arrow" points="172,70 162,63 166,73" />
      <text id="lbl1" class="hop-label" x="140" y="34">+1</text>

      
      <path id="hop2" class="hop-arc" d="M 170 70 Q 200 28 230 70" />
      <polygon id="arr2" class="hop-arrow" points="232,70 222,63 226,73" />
      <text id="lbl2" class="hop-label" x="200" y="34">+1</text>

      
      <circle cx="110" cy="80" r="7" fill="#f59e0b" />
      
      
      <g id="activeMarker" class="marker" transform="translate(110, 80)">
        <circle r="12" fill="#22b7ff" fill-opacity="0.25" />
        <circle r="6" fill="#0284c7" />
      </g>
    </svg>

    <div class="controls">
      <button class="btn active" id="btn0" onclick="setStep(0)">Start: −3</button>
      <button class="btn" id="btn1" onclick="setStep(1)">+1 Step →</button>
      <button class="btn" id="btn2" onclick="setStep(2)">+2 Steps → −1</button>
    </div>
  </div>

  <script>
    const values = [-4, -3, -2, -1, 0, 1, 2];
    const startX = 50;
    const stepX = 60;
    const ticksGroup = document.getElementById('ticksGroup');

    values.forEach((val, i) => {
      const x = startX + i * stepX;
      
      // Tick line
      const line = document.createElementNS('http://www.w3.org/2000/svg', 'line');
      line.setAttribute('x1', x);
      line.setAttribute('y1', 73);
      line.setAttribute('x2', x);
      line.setAttribute('y2', 87);
      line.setAttribute('stroke', '#94a3b8');
      line.setAttribute('stroke-width', '2');
      ticksGroup.appendChild(line);

      // Tick label
      const text = document.createElementNS('http://www.w3.org/2000/svg', 'text');
      text.setAttribute('x', x);
      text.setAttribute('y', 106);
      text.setAttribute('class', 'tick-label' + (val === -3 ? ' active' : ''));
      text.setAttribute('id', `tick-${val}`);
      text.textContent = val > 0 ? `+${val}` : `${val}`;
      ticksGroup.appendChild(text);
    });

    const markerPositions = [110, 170, 230];
    const marker = document.getElementById('activeMarker');

    function updateEquation(step) {
      const exp = document.getElementById('mathExp');
      if (step === 0) {
        katex.render('\\textcolor{#f59e0b}{-3}', exp);
      } else if (step === 1) {
        katex.render('\\textcolor{#f59e0b}{-3} + \\textcolor{#22b7ff}{1} = \\mathbf{-2}', exp);
      } else {
        katex.render('\\textcolor{#f59e0b}{-3} + \\textcolor{#22b7ff}{2} = \\textcolor{#10b981}{\\mathbf{-1}}', exp);
      }
    }

    function setStep(step) {
      // Update buttons
      for (let i = 0; i <= 2; i++) {
        document.getElementById(`btn${i}`).classList.toggle('active', i === step);
      }

      // Move marker
      marker.setAttribute('transform', `translate(${markerPositions[step]}, 80)`);

      // Hop 1 visibility
      const h1 = document.getElementById('hop1');
      const a1 = document.getElementById('arr1');
      const l1 = document.getElementById('lbl1');
      if (step >= 1) {
        h1.classList.add('visible');
        a1.classList.add('visible');
        l1.classList.add('visible');
      } else {
        h1.classList.remove('visible');
        a1.classList.remove('visible');
        l1.classList.remove('visible');
      }

      // Hop 2 visibility
      const h2 = document.getElementById('hop2');
      const a2 = document.getElementById('arr2');
      const l2 = document.getElementById('lbl2');
      if (step >= 2) {
        h2.classList.add('visible');
        a2.classList.add('visible');
        l2.classList.add('visible');
      } else {
        h2.classList.remove('visible');
        a2.classList.remove('visible');
        l2.classList.remove('visible');
      }

      // Highlight target ticks
      values.forEach(v => {
        const el = document.getElementById(`tick-${v}`);
        if (el) {
          const isCurrent = (step === 0 && v === -3) || (step === 1 && v === -2) || (step === 2 && v === -1);
          el.classList.toggle('active', isCurrent);
          if (step === 2 && v === -1) {
            el.setAttribute('fill', '#10b981');
          } else {
            el.removeAttribute('fill');
          }
        }
      });

      updateEquation(step);
    }

    // Initialize
    updateEquation(0);

    // Auto cycle cleanly every 3.5s
    let currentStep = 0;
    let autoTimer = setInterval(() => {
      currentStep = (currentStep + 1) % 3;
      setStep(currentStep);
    }, 3500);

    // Pause auto timer on direct student click
    document.querySelectorAll('.btn').forEach(btn => {
      btn.addEventListener('click', () => {
        clearInterval(autoTimer);
      });
    });
  </script>
</body>
</html>

Starting at $-3$ and hopping $2$ steps to the right lands you on $-1$, so $-3 + 2 = -1$.

What happens if we take a giant jump that goes past zero?

Let's solve **$-2 + 5$** by starting at $-2$ and jumping $5$ steps to the right.

<!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>Adding a Positive Integer</title>
  <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;
      background: transparent;
      color: #1e2945;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: space-between;
      min-height: 100%;
      padding: 12px 16px;
      user-select: none;
    }

    .header-box {
      display: flex;
      align-items: center;
      justify-content: center;
      gap: 12px;
      margin-top: 4px;
      margin-bottom: 8px;
    }

    .math-pill {
      background: #ffffff;
      border: 1px solid #e2e8f0;
      border-radius: 999px;
      padding: 6px 20px;
      font-size: 22px;
      box-shadow: 0 2px 6px rgba(0, 0, 0, 0.05);
      display: inline-flex;
      align-items: center;
      justify-content: center;
    }

    .svg-container {
      width: 100%;
      max-width: 580px;
      flex-grow: 1;
      display: flex;
      align-items: center;
      justify-content: center;
    }

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

    .axis-line {
      stroke: #64748b;
      stroke-width: 2.5;
      stroke-linecap: round;
    }

    .tick {
      stroke: #94a3b8;
      stroke-width: 2;
    }

    .tick-zero {
      stroke: #1e2945;
      stroke-width: 3;
    }

    .tick-label {
      font-size: 15px;
      font-weight: 500;
      fill: #64748b;
      text-anchor: middle;
      dominant-baseline: middle;
      font-family: inherit;
    }

    .tick-label.zero {
      font-weight: 700;
      fill: #1e2945;
      font-size: 17px;
    }

    .tick-label.active-pos {
      fill: #0284c7;
      font-weight: 700;
    }

    .tick-label.active-neg {
      fill: #d97706;
      font-weight: 700;
    }

    .arc-path {
      fill: none;
      stroke-width: 3;
      stroke-linecap: round;
      stroke-dasharray: 6 6;
      opacity: 0;
      transition: opacity 0.4s ease, stroke-dashoffset 0.6s ease;
    }

    .arc-solid {
      fill: none;
      stroke-width: 3.5;
      stroke-linecap: round;
      opacity: 0;
      transition: opacity 0.4s ease;
    }

    .arc-1 {
      stroke: #d97706;
    }

    .arc-2 {
      stroke: #0284c7;
    }

    .arc-label {
      font-size: 14px;
      font-weight: 700;
      opacity: 0;
      transition: opacity 0.4s ease;
      text-anchor: middle;
    }

    .label-1 {
      fill: #b45309;
    }

    .label-2 {
      fill: #0369a1;
    }

    .marker-dot {
      fill: #22b7ff;
      stroke: #ffffff;
      stroke-width: 3;
      filter: drop-shadow(0 2px 4px rgba(34, 183, 255, 0.4));
      transition: cx 0.5s cubic-bezier(0.34, 1.56, 0.64, 1), cy 0.5s ease;
    }

    .marker-pulse {
      fill: #22b7ff;
      opacity: 0.25;
      transition: cx 0.5s cubic-bezier(0.34, 1.56, 0.64, 1), cy 0.5s ease;
    }

    .stepper-bar {
      display: flex;
      flex-wrap: wrap;
      align-items: center;
      justify-content: center;
      gap: 8px;
      margin-top: 8px;
      margin-bottom: 4px;
      width: 100%;
    }

    .step-btn {
      background: #ffffff;
      border: 1.5px solid #e2e8f0;
      border-radius: 8px;
      padding: 6px 12px;
      font-size: 13px;
      font-weight: 600;
      color: #475569;
      cursor: pointer;
      display: inline-flex;
      align-items: center;
      gap: 6px;
      transition: all 0.2s ease;
    }

    .step-btn:hover {
      border-color: #cbd5e1;
      background: #f8fafc;
    }

    .step-btn.active {
      border-color: #22b7ff;
      color: #0284c7;
      background: #f0f9ff;
      box-shadow: 0 0 0 2px rgba(34, 183, 255, 0.2);
    }

    .ctrl-btn {
      background: #ffffff;
      border: 1.5px solid #e2e8f0;
      border-radius: 8px;
      width: 32px;
      height: 32px;
      display: inline-flex;
      align-items: center;
      justify-content: center;
      font-size: 14px;
      color: #475569;
      cursor: pointer;
      transition: all 0.2s ease;
    }

    .ctrl-btn:hover {
      background: #f8fafc;
      border-color: #cbd5e1;
    }

    .visible {
      opacity: 1 !important;
    }
  </style>
</head>
<body>

  <div class="header-box">
    <div class="math-pill" id="mathDisplay"></div>
  </div>

  <div class="svg-container">
    <svg viewBox="0 0 580 200">
      <defs>
        <marker id="arrow-right" viewBox="0 0 10 10" refX="6" refY="5" markerWidth="6" markerHeight="6" orient="auto-start-reverse">
          <path d="M 1 2 L 7 5 L 1 8 z" fill="#64748b" />
        </marker>
        <marker id="arrow-left" viewBox="0 0 10 10" refX="4" refY="5" markerWidth="6" markerHeight="6" orient="auto-start-reverse">
          <path d="M 9 2 L 3 5 L 9 8 z" fill="#64748b" />
        </marker>
        <marker id="jump-arrow-1" viewBox="0 0 10 10" refX="6" refY="5" markerWidth="5" markerHeight="5" orient="auto">
          <path d="M 1 2 L 7 5 L 1 8 z" fill="#d97706" />
        </marker>
        <marker id="jump-arrow-2" viewBox="0 0 10 10" refX="6" refY="5" markerWidth="5" markerHeight="5" orient="auto">
          <path d="M 1 2 L 7 5 L 1 8 z" fill="#0284c7" />
        </marker>
      </defs>

      
      <line x1="30" y1="150" x2="550" y2="150" class="axis-line" marker-start="url(#arrow-left)" marker-end="url(#arrow-right)" />

      
      
      <line x1="50" y1="142" x2="50" y2="158" class="tick" />
      <text x="50" y="176" class="tick-label">-3</text>

      <line x1="110" y1="138" x2="110" y2="162" class="tick" />
      <text x="110" y="176" class="tick-label active-neg" id="tickNeg2">-2</text>

      <line x1="170" y1="142" x2="170" y2="158" class="tick" />
      <text x="170" y="176" class="tick-label">-1</text>

      <line x1="230" y1="136" x2="230" y2="164" class="tick-zero" />
      <text x="230" y="176" class="tick-label zero">0</text>

      <line x1="290" y1="142" x2="290" y2="158" class="tick" />
      <text x="290" y="176" class="tick-label">+1</text>

      <line x1="350" y1="142" x2="350" y2="158" class="tick" />
      <text x="350" y="176" class="tick-label">+2</text>

      <line x1="410" y1="138" x2="410" y2="162" class="tick" />
      <text x="410" y="176" class="tick-label active-pos" id="tickPos3">+3</text>

      <line x1="470" y1="142" x2="470" y2="158" class="tick" />
      <text x="470" y="176" class="tick-label">+4</text>

      <line x1="530" y1="142" x2="530" y2="158" class="tick" />
      <text x="530" y="176" class="tick-label">+5</text>

      
      <path id="arc1" d="M 110 148 C 130 90, 210 90, 226 144" class="arc-solid arc-1" marker-end="url(#jump-arrow-1)" />
      <text id="lblArc1" x="170" y="85" class="arc-label label-1">+2 steps</text>

      
      <path id="arc2" d="M 230 148 C 260 65, 380 65, 406 144" class="arc-solid arc-2" marker-end="url(#jump-arrow-2)" />
      <text id="lblArc2" x="320" y="60" class="arc-label label-2">+3 steps</text>

      
      <circle id="markerPulse" cx="110" cy="150" r="16" class="marker-pulse" />
      <circle id="markerDot" cx="110" cy="150" r="8" class="marker-dot" />
    </svg>
  </div>

  <div class="stepper-bar">
    <button class="ctrl-btn" id="prevBtn" aria-label="Previous step">◀</button>
    <button class="step-btn active" id="btn0">1. Start at -2</button>
    <button class="step-btn" id="btn1">2. Jump +2 → 0</button>
    <button class="step-btn" id="btn2">3. Jump +3 → +3</button>
    <button class="ctrl-btn" id="nextBtn" aria-label="Next step">▶</button>
  </div>

  <script>
    const equations = [
      "-2",
      "-2 + 2 = 0",
      "-2 + 5 = +3"
    ];

    const markerPositions = [110, 230, 410];
    let currentStep = 0;
    let autoTimer = null;

    function renderMath(step) {
      const el = document.getElementById('mathDisplay');
      if (window.katex) {
        katex.render(equations[step], el, { throwOnError: false });
      } else {
        el.textContent = equations[step];
      }
    }

    function updateStep(step) {
      currentStep = step;
      renderMath(step);

      // Marker
      const x = markerPositions[step];
      document.getElementById('markerDot').setAttribute('cx', x);
      document.getElementById('markerPulse').setAttribute('cx', x);

      // Arc 1
      const arc1 = document.getElementById('arc1');
      const lbl1 = document.getElementById('lblArc1');
      if (step >= 1) {
        arc1.classList.add('visible');
        lbl1.classList.add('visible');
      } else {
        arc1.classList.remove('visible');
        lbl1.classList.remove('visible');
      }

      // Arc 2
      const arc2 = document.getElementById('arc2');
      const lbl2 = document.getElementById('lblArc2');
      if (step >= 2) {
        arc2.classList.add('visible');
        lbl2.classList.add('visible');
      } else {
        arc2.classList.remove('visible');
        lbl2.classList.remove('visible');
      }

      // Buttons
      for (let i = 0; i < 3; i++) {
        const btn = document.getElementById('btn' + i);
        if (i === step) {
          btn.classList.add('active');
        } else {
          btn.classList.remove('active');
        }
      }
    }

    function nextStep() {
      updateStep((currentStep + 1) % 3);
    }

    function prevStep() {
      updateStep((currentStep - 1 + 3) % 3);
    }

    function startAutoPlay() {
      stopAutoPlay();
      autoTimer = setInterval(nextStep, 3500);
    }

    function stopAutoPlay() {
      if (autoTimer) {
        clearInterval(autoTimer);
        autoTimer = null;
      }
    }

    // Attach click handlers
    document.getElementById('btn0').addEventListener('click', () => { stopAutoPlay(); updateStep(0); });
    document.getElementById('btn1').addEventListener('click', () => { stopAutoPlay(); updateStep(1); });
    document.getElementById('btn2').addEventListener('click', () => { stopAutoPlay(); updateStep(2); });
    document.getElementById('nextBtn').addEventListener('click', () => { stopAutoPlay(); nextStep(); });
    document.getElementById('prevBtn').addEventListener('click', () => { stopAutoPlay(); prevStep(); });

    // Initialize
    updateStep(0);
    startAutoPlay();
  </script>
</body>
</html>

The first $2$ steps bring you to **zero**, and the next $3$ steps carry you into the positive numbers to land on **$+3$**.

It is easy to think that $-4 + 1$ equals $-5$, but adding always makes numbers bigger.

When you add, always hop right toward zero: $-4 + 1 = -3$.

math

Adding a Positive Integer. Add positive numbers to negative numbers by moving right or upward on a number line.

Adding Integers with Different Signs on a Number Line

Negative Numbers and Integer Operations (CfE Third Level / KS3)

Playing Connect Three: Adding and Subtracting Positive and Negative Integers

When teaching integers across developmental frameworks, concrete-to-representational-to-abstract (CRA) progression is critical. Ground initial understanding in familiar vertical and horizontal real-world metaphors (thermometers, elevators, altitude, bank balances). Emphasize that 'adding positive' universally corresponds to an upward or rightward vector translation, reinforcing that movement toward positive values makes negative numbers smaller in absolute magnitude but greater in overall value.

Adding a Positive Integer

Moving Up and to the Right

Crossing Past Zero

Watch Out for This Mistake