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System default

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Tutoring abilities

If you have 9 square toy blocks, can you arrange them into one giant square?

Yes! You place 3 blocks in each row and make 3 rows, because $3 \times 3 = 9$.

When you multiply a number by itself, we call it **squaring** a number.

<!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>
<style>
  * { box-sizing: border-box; margin: 0; padding: 0; }
  body {
    font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
    background: #f8f9fa;
    color: #1e2945;
    padding: 12px;
    display: flex;
    flex-direction: column;
    align-items: center;
    justify-content: space-between;
    min-height: 100%;
    user-select: none;
    -webkit-user-select: none;
  }
  .controls {
    display: flex;
    gap: 8px;
    justify-content: center;
    flex-wrap: wrap;
    width: 100%;
  }
  .btn-pill {
    padding: 6px 14px;
    border-radius: 20px;
    border: 1px solid #d1d5db;
    background: #ffffff;
    font-size: 14px;
    font-weight: 600;
    color: #4b5563;
    cursor: pointer;
    transition: all 0.2s ease;
  }
  .btn-pill.active {
    background: #22b7ff;
    color: #ffffff;
    border-color: #22b7ff;
    box-shadow: 0 2px 6px rgba(34, 183, 255, 0.35);
  }
  .main-stage {
    display: flex;
    align-items: center;
    justify-content: center;
    gap: 20px;
    width: 100%;
    flex: 1;
    margin: 8px 0;
    flex-wrap: wrap;
  }
  .grid-wrapper {
    position: relative;
    padding: 24px 8px 8px 24px;
    display: flex;
    align-items: center;
    justify-content: center;
  }
  .bracket-top {
    position: absolute;
    top: 2px;
    left: 24px;
    right: 8px;
    height: 18px;
    border-top: 2px solid #22b7ff;
    border-left: 2px solid #22b7ff;
    border-right: 2px solid #22b7ff;
    display: flex;
    align-items: center;
    justify-content: center;
  }
  .bracket-top span {
    background: #f8f9fa;
    padding: 0 4px;
    font-size: 13px;
    font-weight: 700;
    color: #0284c7;
    transform: translateY(-10px);
  }
  .bracket-left {
    position: absolute;
    left: 2px;
    top: 24px;
    bottom: 8px;
    width: 18px;
    border-left: 2px solid #22b7ff;
    border-top: 2px solid #22b7ff;
    border-bottom: 2px solid #22b7ff;
    display: flex;
    align-items: center;
    justify-content: center;
  }
  .bracket-left span {
    background: #f8f9fa;
    padding: 2px 0;
    font-size: 13px;
    font-weight: 700;
    color: #0284c7;
    transform: translateX(-10px);
  }
  .grid-board {
    display: grid;
    gap: 4px;
    background: #ffffff;
    padding: 6px;
    border-radius: 10px;
    box-shadow: 0 1px 4px rgba(0,0,0,0.08);
    width: 160px;
    height: 160px;
  }
  .tile {
    background: #e0f2fe;
    border-radius: 4px;
    border: 1px solid #bae6fd;
    display: flex;
    align-items: center;
    justify-content: center;
    font-size: 12px;
    font-weight: 600;
    color: #0369a1;
    transition: background-color 0.25s ease, transform 0.25s ease;
  }
  .tile.highlight {
    background: #22b7ff;
    color: #ffffff;
    border-color: #0284c7;
  }
  .info-panel {
    display: flex;
    flex-direction: column;
    gap: 10px;
    align-items: center;
    min-width: 140px;
  }
  .equation-box {
    background: #ffffff;
    border: 1px solid #e5e7eb;
    border-radius: 10px;
    padding: 10px 16px;
    font-size: 20px;
    box-shadow: 0 1px 3px rgba(0,0,0,0.05);
    min-height: 48px;
    display: flex;
    align-items: center;
    justify-content: center;
  }
  .toggle-group {
    display: flex;
    background: #e5e7eb;
    border-radius: 8px;
    padding: 3px;
    gap: 2px;
  }
  .toggle-btn {
    border: none;
    background: transparent;
    padding: 5px 12px;
    border-radius: 6px;
    font-size: 13px;
    font-weight: 600;
    color: #4b5563;
    cursor: pointer;
    transition: all 0.2s ease;
  }
  .toggle-btn.active {
    background: #ffffff;
    color: #0284c7;
    box-shadow: 0 1px 2px rgba(0,0,0,0.1);
  }
</style>
</head>
<body>

  
  <div class="toggle-group">
    <button class="toggle-btn active" id="modeSquare" onclick="setMode('square')">Squaring</button>
    <button class="toggle-btn" id="modeRoot" onclick="setMode('root')">Square Root</button>
  </div>

  
  <div class="main-stage">
    <div class="grid-wrapper">
      <div class="bracket-top"><span id="topLabel">3</span></div>
      <div class="bracket-left"><span id="leftLabel">3</span></div>
      <div class="grid-board" id="gridBoard"></div>
    </div>

    <div class="info-panel">
      <div class="equation-box" id="eqBox"></div>
    </div>
  </div>

  
  <div class="controls">
    <button class="btn-pill" onclick="setSize(2)" id="btn2">2</button>
    <button class="btn-pill active" onclick="setSize(3)" id="btn3">3</button>
    <button class="btn-pill" onclick="setSize(4)" id="btn4">4</button>
    <button class="btn-pill" onclick="setSize(5)" id="btn5">5</button>
  </div>

<script>
  let currentSize = 3;
  let currentMode = 'square';

  function renderKaTeX(tex, element) {
    if (window.katex) {
      katex.render(tex, element, { throwOnError: false });
    } else {
      element.textContent = tex;
    }
  }

  function update() {
    // Update active size button
    [2, 3, 4, 5].forEach(n => {
      const btn = document.getElementById('btn' + n);
      if (btn) btn.classList.toggle('active', n === currentSize);
    });

    // Update mode buttons
    document.getElementById('modeSquare').classList.toggle('active', currentMode === 'square');
    document.getElementById('modeRoot').classList.toggle('active', currentMode === 'root');

    // Labels
    document.getElementById('topLabel').textContent = currentSize;
    document.getElementById('leftLabel').textContent = currentSize;

    // Grid build
    const grid = document.getElementById('gridBoard');
    grid.innerHTML = '';
    grid.style.gridTemplateColumns = `repeat(${currentSize}, 1fr)`;
    grid.style.gridTemplateRows = `repeat(${currentSize}, 1fr)`;

    const total = currentSize * currentSize;
    for (let i = 1; i <= total; i++) {
      const tile = document.createElement('div');
      tile.className = 'tile';
      tile.textContent = i;
      grid.appendChild(tile);
    }

    // Equation
    const eqBox = document.getElementById('eqBox');
    if (currentMode === 'square') {
      renderKaTeX(`${currentSize}^2 = ${total}`, eqBox);
    } else {
      renderKaTeX(`\\sqrt{${total}} = ${currentSize}`, eqBox);
    }
  }

  function setSize(n) {
    currentSize = n;
    update();
  }

  function setMode(mode) {
    currentMode = mode;
    update();
  }

  window.addEventListener('DOMContentLoaded', update);
</script>
</body>
</html>

What if someone tells you the total number of blocks, but hides the side length?

A **square root** is the opposite of squaring: it asks, "What number multiplied by itself gives this total?"

We write this using the **radical symbol** $\sqrt{}$, so $\sqrt{9} = 3$ because $3 \times 3 = 9$.

<!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, Helvetica, Arial, sans-serif;
    background: #f8f9fa;
    color: #1e2945;
    display: flex;
    flex-direction: column;
    align-items: center;
    justify-content: space-between;
    min-height: 100%;
    padding: 12px;
    user-select: none;
  }

  /* Number Selector */
  .selector-row {
    display: flex;
    gap: 8px;
    justify-content: center;
    width: 100%;
    max-width: 420px;
  }
  .num-btn {
    flex: 1;
    padding: 6px 0;
    font-size: 15px;
    font-weight: 600;
    background: #ffffff;
    border: 2px solid #e6e6e6;
    border-radius: 8px;
    color: #1e2945;
    cursor: pointer;
    transition: all 0.2s ease;
  }
  .num-btn.active {
    background: #22b7ff;
    border-color: #22b7ff;
    color: #ffffff;
    box-shadow: 0 2px 8px rgba(34, 183, 255, 0.35);
  }

  /* Main Interactive Stage */
  .stage {
    display: flex;
    align-items: center;
    justify-content: center;
    gap: 12px;
    width: 100%;
    max-width: 460px;
    margin: 8px 0;
  }

  /* Side & Area Cards */
  .panel {
    background: #ffffff;
    border-radius: 12px;
    border: 1px solid #e6e6e6;
    padding: 10px;
    display: flex;
    flex-direction: column;
    align-items: center;
    justify-content: center;
    box-shadow: 0 2px 6px rgba(0,0,0,0.04);
    flex: 1;
    min-height: 190px;
  }
  .panel-title {
    font-size: 13px;
    font-weight: 700;
    color: #6b7280;
    text-transform: uppercase;
    letter-spacing: 0.5px;
    margin-bottom: 8px;
  }

  /* Side visualization */
  .side-view {
    display: flex;
    flex-direction: column;
    align-items: center;
    justify-content: center;
    gap: 4px;
    height: 120px;
  }
  .side-blocks {
    display: flex;
    gap: 3px;
  }
  .side-unit {
    width: 22px;
    height: 22px;
    background: #22b7ff;
    border-radius: 4px;
    transition: all 0.3s ease;
  }
  .side-val {
    font-size: 20px;
    font-weight: 700;
    color: #1e2945;
    margin-top: 6px;
  }

  /* Area Grid */
  .grid-view {
    display: flex;
    flex-direction: column;
    align-items: center;
    justify-content: center;
    height: 120px;
  }
  .grid-container {
    display: grid;
    gap: 3px;
    transition: all 0.3s ease;
  }
  .grid-cell {
    width: 20px;
    height: 20px;
    background: #e0f2fe;
    border: 1px solid #7dd3fc;
    border-radius: 3px;
    transition: background 0.3s ease, transform 0.3s ease;
  }
  .grid-cell.highlight {
    background: #22b7ff;
  }
  .area-val {
    font-size: 20px;
    font-weight: 700;
    color: #1e2945;
    margin-top: 6px;
  }

  /* Middle Arrow Actions */
  .ops-column {
    display: flex;
    flex-direction: column;
    gap: 12px;
    justify-content: center;
    align-items: center;
  }
  .op-badge {
    display: flex;
    align-items: center;
    gap: 4px;
    padding: 6px 10px;
    background: #ffffff;
    border: 1px solid #e6e6e6;
    border-radius: 20px;
    font-size: 13px;
    font-weight: 700;
    color: #1e2945;
    box-shadow: 0 2px 4px rgba(0,0,0,0.04);
  }
  .op-badge.square {
    color: #0284c7;
    border-color: #bae6fd;
    background: #f0f9ff;
  }
  .op-badge.root {
    color: #059669;
    border-color: #a7f3d0;
    background: #ecfdf5;
  }

  /* Bottom Equation Bar */
  .equations-bar {
    display: flex;
    gap: 10px;
    width: 100%;
    max-width: 420px;
    justify-content: center;
  }
  .eq-card {
    flex: 1;
    background: #ffffff;
    border: 1px solid #e6e6e6;
    border-radius: 10px;
    padding: 8px 12px;
    text-align: center;
    font-size: 16px;
    font-weight: 700;
  }
  .eq-card.sq {
    border-left: 4px solid #0284c7;
  }
  .eq-card.rt {
    border-left: 4px solid #059669;
  }
  .accent-sq { color: #0284c7; }
  .accent-rt { color: #059669; }
</style>
</head>
<body>

  
  <div class="selector-row">
    <button class="num-btn" data-n="2">2</button>
    <button class="num-btn active" data-n="3">3</button>
    <button class="num-btn" data-n="4">4</button>
    <button class="num-btn" data-n="5">5</button>
  </div>

  
  <div class="stage">
    
    <div class="panel">
      <div class="panel-title">Side Length</div>
      <div class="side-view">
        <div class="side-blocks" id="sideBlocks"></div>
        <div class="side-val" id="sideVal">3</div>
      </div>
    </div>

    
    <div class="ops-column">
      <div class="op-badge square">
        <span>x²</span>
        <span>→</span>
      </div>
      <div class="op-badge root">
        <span>←</span>
        <span>√</span>
      </div>
    </div>

    
    <div class="panel">
      <div class="panel-title">Square Area</div>
      <div class="grid-view">
        <div class="grid-container" id="gridContainer"></div>
        <div class="area-val" id="areaVal">9</div>
      </div>
    </div>
  </div>

  
  <div class="equations-bar">
    <div class="eq-card sq">
      <span class="accent-sq" id="eqSquare">3² = 9</span>
    </div>
    <div class="eq-card rt">
      <span class="accent-rt" id="eqRoot">√9 = 3</span>
    </div>
  </div>

<script>
  const buttons = document.querySelectorAll('.num-btn');
  const sideBlocks = document.getElementById('sideBlocks');
  const gridContainer = document.getElementById('gridContainer');
  const sideVal = document.getElementById('sideVal');
  const areaVal = document.getElementById('areaVal');
  const eqSquare = document.getElementById('eqSquare');
  const eqRoot = document.getElementById('eqRoot');

  function update(n) {
    // Update active button
    buttons.forEach(b => {
      b.classList.toggle('active', parseInt(b.dataset.n) === n);
    });

    const area = n * n;

    // Render 1D side blocks
    sideBlocks.innerHTML = '';
    for (let i = 0; i < n; i++) {
      const unit = document.createElement('div');
      unit.className = 'side-unit';
      sideBlocks.appendChild(unit);
    }
    sideVal.textContent = n;

    // Render 2D grid
    gridContainer.innerHTML = '';
    gridContainer.style.gridTemplateColumns = `repeat(${n}, 1fr)`;
    for (let i = 0; i < area; i++) {
      const cell = document.createElement('div');
      cell.className = 'grid-cell highlight';
      gridContainer.appendChild(cell);
    }
    areaVal.textContent = area;

    // Update Equations
    eqSquare.textContent = `${n}² = ${area}`;
    eqRoot.textContent = `√${area} = ${n}`;
  }

  buttons.forEach(btn => {
    btn.addEventListener('click', () => {
      update(parseInt(btn.dataset.n));
    });
  });

  // Initial load
  update(3);
</script>
</body>
</html>

Can we use this reverse trick for even bigger numbers like 36 or 100?

Numbers like 1, 4, 9, 16, and 25 are called **perfect squares** because they form neat, solid squares with whole numbers.

To find $\sqrt{49}$, look for the number that makes 49 when multiplied by itself: $7 \times 7 = 49$, so $\sqrt{49} = 7$.

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <meta name="aspect-ratio" content="4:3">
  <style>
    * { box-sizing: border-box; margin: 0; padding: 0; }
    body {
      font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
      background: #f8f9fa;
      color: #1e2945;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      min-height: 100%;
      padding: 12px;
    }

    .app-wrap {
      width: 100%;
      max-width: 480px;
      display: flex;
      flex-direction: column;
      align-items: center;
      gap: 14px;
    }

    /* Number Selector */
    .selector-row {
      display: flex;
      align-items: center;
      gap: 8px;
    }
    .num-btn {
      width: 40px;
      height: 36px;
      border: 1px solid #d1d5db;
      background: #ffffff;
      color: #1e2945;
      font-weight: 600;
      font-size: 15px;
      border-radius: 8px;
      cursor: pointer;
      display: flex;
      align-items: center;
      justify-content: center;
      transition: all 0.2s ease;
    }
    .num-btn.active {
      background: #22b7ff;
      color: #ffffff;
      border-color: #22b7ff;
      box-shadow: 0 2px 6px rgba(34, 183, 255, 0.35);
    }

    /* Interactive Display Arena */
    .stage {
      width: 100%;
      background: #ffffff;
      border-radius: 12px;
      border: 1px solid #e5e7eb;
      box-shadow: 0 2px 8px rgba(0,0,0,0.04);
      padding: 16px;
      display: flex;
      flex-direction: column;
      align-items: center;
      gap: 14px;
    }

    /* Grid Area */
    .grid-container {
      position: relative;
      display: flex;
      align-items: center;
      justify-content: center;
      padding: 24px 12px 12px 28px;
    }

    .side-label {
      position: absolute;
      font-weight: 700;
      color: #0284c7;
      font-size: 16px;
      transition: all 0.3s ease;
    }
    .side-label.top {
      top: 0;
      left: 28px;
      right: 12px;
      text-align: center;
    }
    .side-label.left {
      left: 4px;
      top: 24px;
      bottom: 12px;
      display: flex;
      align-items: center;
    }

    .grid {
      display: grid;
      gap: 3px;
      background: #f1f5f9;
      padding: 4px;
      border-radius: 8px;
      border: 2px solid #0284c7;
      transition: all 0.3s ease;
    }
    .cell {
      background: #bae6fd;
      border-radius: 3px;
      transition: background-color 0.25s ease, transform 0.25s ease;
    }
    .cell.highlight-edge {
      background: #0284c7;
    }

    /* Inverse Flow Relations */
    .flow-card {
      width: 100%;
      display: flex;
      flex-direction: column;
      gap: 8px;
    }

    .flow-row {
      display: flex;
      align-items: center;
      justify-content: space-between;
      padding: 8px 14px;
      border-radius: 8px;
      background: #f8fafc;
      border: 1px solid #e2e8f0;
      font-size: 15px;
      transition: all 0.2s ease;
    }
    .flow-row.active {
      background: #e0f2fe;
      border-color: #38bdf8;
    }

    .flow-label {
      font-size: 13px;
      color: #64748b;
      font-weight: 600;
      display: flex;
      align-items: center;
      gap: 4px;
    }
    .flow-math {
      font-weight: 700;
      font-size: 16px;
      color: #0f172a;
      letter-spacing: 0.5px;
    }
    .accent {
      color: #0284c7;
    }
  </style>
</head>
<body>

<div class="app-wrap">
  
  <div class="selector-row">
    <button class="num-btn" data-val="3">3</button>
    <button class="num-btn" data-val="4">4</button>
    <button class="num-btn" data-val="5">5</button>
    <button class="num-btn active" data-val="7">7</button>
  </div>

  
  <div class="stage">
    <div class="grid-container">
      <div class="side-label top" id="labelTop">7</div>
      <div class="side-label left" id="labelLeft">7</div>
      <div class="grid" id="gridBox"></div>
    </div>

    
    <div class="flow-card">
      <div class="flow-row" id="rowSquare">
        <span class="flow-label"><span>Square</span> →</span>
        <span class="flow-math"><span class="accent" id="sqBase">7</span>² = <span id="sqArea">49</span></span>
      </div>
      <div class="flow-row active" id="rowRoot">
        <span class="flow-label">← <span>Square Root</span></span>
        <span class="flow-math">√<span id="rootArea">49</span> = <span class="accent" id="rootBase">7</span></span>
      </div>
    </div>
  </div>
</div>

<script>
  let currentN = 7;
  const maxBoxSize = 160;

  function renderGrid(n) {
    const gridBox = document.getElementById('gridBox');
    gridBox.innerHTML = '';
    gridBox.style.gridTemplateColumns = `repeat(${n}, 1fr)`;
    gridBox.style.gridTemplateRows = `repeat(${n}, 1fr)`;
    
    // Scale container smoothly
    const cellSize = Math.floor(maxBoxSize / Math.max(n, 4));
    const totalSize = cellSize * n + (n - 1) * 3 + 8;
    gridBox.style.width = totalSize + 'px';
    gridBox.style.height = totalSize + 'px';

    const total = n * n;
    for (let i = 0; i < total; i++) {
      const cell = document.createElement('div');
      cell.className = 'cell';
      gridBox.appendChild(cell);
    }

    // Update labels and math
    document.getElementById('labelTop').textContent = n;
    document.getElementById('labelLeft').textContent = n;
    document.getElementById('sqBase').textContent = n;
    document.getElementById('sqArea').textContent = total;
    document.getElementById('rootArea').textContent = total;
    document.getElementById('rootBase').textContent = n;
  }

  // Toggle & select
  const buttons = document.querySelectorAll('.num-btn');
  buttons.forEach(btn => {
    btn.addEventListener('click', () => {
      buttons.forEach(b => b.classList.remove('active'));
      btn.classList.add('active');
      currentN = parseInt(btn.dataset.val, 10);
      renderGrid(currentN);
    });
  });

  // Highlight side vs area on row hover/tap
  const rowSquare = document.getElementById('rowSquare');
  const rowRoot = document.getElementById('rowRoot');

  function setMode(mode) {
    const cells = document.querySelectorAll('.cell');
    if (mode === 'root') {
      rowRoot.classList.add('active');
      rowSquare.classList.remove('active');
      cells.forEach((c, idx) => {
        // highlight first row and first column to show side length extraction
        if (idx < currentN || idx % currentN === 0) {
          c.classList.add('highlight-edge');
        } else {
          c.classList.remove('highlight-edge');
        }
      });
    } else {
      rowSquare.classList.add('active');
      rowRoot.classList.remove('active');
      cells.forEach(c => c.classList.remove('highlight-edge'));
    }
  }

  rowSquare.addEventListener('click', () => setMode('square'));
  rowRoot.addEventListener('click', () => setMode('root'));

  // Initial render
  renderGrid(currentN);
  setMode('root');
</script>

</body>
</html>

Be careful: $\sqrt{16}$ is not $16 \div 2 = 8$.

Square root asks what number times **itself** equals 16, which is $4$, because $4 \times 4 = 16$.

Next time you see $\sqrt{144}$, just ask: "What twin numbers multiply to 144?" — and you'll find 12 right away.

math

Understanding Square Roots as the Inverse of Squaring. Define the square root symbol and find exact square roots of perfect squares up to 144.

Understanding Square Roots and Finding Exact Square Roots of Perfect Squares

CfE Third Level Mathematics: Powers, Roots, and Scientific Notation

Square Roots of Perfect Squares (Levels and Progressive Practice)

Pedagogical sequence utilizes the Concrete-Pictorial-Abstract (CPA) framework: start with physical/virtual square arrays to build spatial intuition connecting area to side length, transition to function machines highlighting the inverse relationship to multiplication facts, and solidify mental fluency of perfect square roots up to $144$ ($1^2$ to $12^2$) aligned with CfE MTH 3-06a and MNU 3-03b.

Understanding Square Roots as the Inverse of Squaring

Undoing the Square

Perfect Squares Up to 144