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Imagine your class wants to run a food truck for a weekend festival to raise money for a class trip. Before you can write a single equation to predict your earnings, you face a huge question: where do you even start?

To turn a messy real-world scenario into math, we have to isolate the specific things we can count or measure.

What separates an everyday object from a mathematical quantity?

A **quantity** is any measurable or countable attribute of an object or situation, consisting of a numerical value attached to a specific **unit of measurement**.

<!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;
      padding: 12px;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      min-height: 100vh;
      user-select: none;
    }

    .main {
      width: 100%;
      max-width: 480px;
      display: flex;
      flex-direction: column;
      gap: 12px;
    }

    /* Object Selector Tabs */
    .tabs {
      display: flex;
      gap: 6px;
      background: #e9ecef;
      padding: 4px;
      border-radius: 10px;
    }
    .tab {
      flex: 1;
      border: none;
      background: transparent;
      padding: 8px 4px;
      border-radius: 8px;
      font-size: 14px;
      font-weight: 600;
      color: #6b7280;
      cursor: pointer;
      display: flex;
      align-items: center;
      justify-content: center;
      gap: 4px;
      transition: background 0.2s, color 0.2s, box-shadow 0.2s;
    }
    .tab.active {
      background: #ffffff;
      color: #1e2945;
      box-shadow: 0 2px 6px rgba(0,0,0,0.06);
    }

    /* Attribute selection row */
    .attr-row {
      display: flex;
      gap: 6px;
      justify-content: center;
    }
    .attr-btn {
      flex: 1;
      border: 1.5px solid #e2e8f0;
      background: #ffffff;
      padding: 7px 6px;
      border-radius: 8px;
      font-size: 13px;
      font-weight: 600;
      color: #475569;
      cursor: pointer;
      text-align: center;
      transition: border-color 0.2s, background 0.2s, color 0.2s;
    }
    .attr-btn.active {
      border-color: #22b7ff;
      background: #eef8ff;
      color: #0077b6;
    }

    /* Anatomy Equation Card */
    .equation-card {
      background: #ffffff;
      border-radius: 12px;
      padding: 16px 12px;
      box-shadow: 0 2px 10px rgba(0,0,0,0.04);
      display: flex;
      flex-direction: column;
      align-items: center;
      gap: 12px;
    }

    .equation-header {
      font-size: 12px;
      font-weight: 700;
      text-transform: uppercase;
      letter-spacing: 0.8px;
      color: #94a3b8;
    }

    .equation-boxes {
      display: flex;
      align-items: center;
      justify-content: center;
      gap: 6px;
      width: 100%;
    }

    .box {
      border-radius: 10px;
      padding: 10px 8px;
      text-align: center;
      display: flex;
      flex-direction: column;
      justify-content: center;
      align-items: center;
    }

    .box-val {
      flex: 1;
      background: #eff6ff;
      border: 1.5px solid #93c5fd;
    }
    .box-unit {
      flex: 1.3;
      background: #ecfdf5;
      border: 1.5px solid #6ee7b7;
    }
    .box-total {
      flex: 1.5;
      background: #f1f5f9;
      border: 1.5px solid #cbd5e1;
    }

    .box-label {
      font-size: 11px;
      font-weight: 700;
      text-transform: uppercase;
      letter-spacing: 0.5px;
      margin-bottom: 4px;
    }
    .val-label { color: #2563eb; }
    .unit-label { color: #059669; }
    .total-label { color: #475569; }

    .box-content {
      font-size: 17px;
      font-weight: 800;
      line-height: 1.2;
      min-height: 22px;
      display: flex;
      align-items: center;
      justify-content: center;
    }
    .val-content { color: #1d4ed8; }
    .unit-content { color: #047857; }
    .total-content { color: #0f172a; }

    .op {
      font-size: 18px;
      font-weight: 800;
      color: #94a3b8;
    }

    /* Comparison Bar */
    .compare-card {
      display: flex;
      gap: 8px;
      width: 100%;
    }
    .compare-item {
      flex: 1;
      background: #ffffff;
      border-radius: 10px;
      padding: 10px 8px;
      box-shadow: 0 2px 6px rgba(0,0,0,0.03);
      display: flex;
      flex-direction: column;
      align-items: center;
      text-align: center;
      gap: 4px;
    }
    .compare-tag {
      font-size: 11px;
      font-weight: 700;
      padding: 2px 6px;
      border-radius: 4px;
      text-transform: uppercase;
    }
    .tag-object {
      background: #fee2e2;
      color: #b91c1c;
    }
    .tag-quantity {
      background: #dbeafe;
      color: #1d4ed8;
    }
    .compare-text {
      font-size: 14px;
      font-weight: 700;
      color: #334155;
    }
  </style>
</head>
<body>

  <div class="main">
    
    <div class="tabs">
      <button class="tab active" data-obj="0">🍕 Pizza</button>
      <button class="tab" data-obj="1">🏃 Run</button>
      <button class="tab" data-obj="2">⛽ Gas</button>
    </div>

    
    <div class="attr-row" id="attrRow"></div>

    
    <div class="equation-card">
      <div class="equation-header">Quantity Breakdown</div>
      <div class="equation-boxes">
        <div class="box box-val">
          <div class="box-label val-label">Value</div>
          <div class="box-content val-content" id="valEl">8</div>
        </div>
        <div class="op">+</div>
        <div class="box box-unit">
          <div class="box-label unit-label">Unit</div>
          <div class="box-content unit-content" id="unitEl">slices</div>
        </div>
        <div class="op">=</div>
        <div class="box box-total">
          <div class="box-label total-label">Quantity</div>
          <div class="box-content total-content" id="totalEl">8 slices</div>
        </div>
      </div>
    </div>

    
    <div class="compare-card">
      <div class="compare-item">
        <span class="compare-tag tag-object">Thing / Object</span>
        <span class="compare-text" id="objName">Pizza</span>
      </div>
      <div class="compare-item">
        <span class="compare-tag tag-quantity">Model Quantity</span>
        <span class="compare-text" id="attrSummary">8 slices</span>
      </div>
    </div>
  </div>

  <script>
    const data = [
      {
        name: "Pizza",
        attrs: [
          { label: "Count", val: "8", unit: "slices", full: "8 slices" },
          { label: "Cost", val: "$16", unit: "dollars", full: "$16.00" },
          { label: "Weight", val: "450", unit: "grams", full: "450 g" }
        ]
      },
      {
        name: "Run",
        attrs: [
          { label: "Distance", val: "5", unit: "km", full: "5 km" },
          { label: "Time", val: "24", unit: "min", full: "24 min" },
          { label: "Speed", val: "12.5", unit: "km/h", full: "12.5 km/h" }
        ]
      },
      {
        name: "Gas Station",
        attrs: [
          { label: "Volume", val: "12", unit: "gal", full: "12 gal" },
          { label: "Price", val: "$3.50", unit: "/ gal", full: "$3.50/gal" },
          { label: "Total", val: "$42", unit: "dollars", full: "$42.00" }
        ]
      }
    ];

    let currentObj = 0;
    let currentAttr = 0;

    const tabs = document.querySelectorAll('.tab');
    const attrRow = document.getElementById('attrRow');
    const valEl = document.getElementById('valEl');
    const unitEl = document.getElementById('unitEl');
    const totalEl = document.getElementById('totalEl');
    const objName = document.getElementById('objName');
    const attrSummary = document.getElementById('attrSummary');

    function renderAttrs() {
      attrRow.innerHTML = '';
      data[currentObj].attrs.forEach((attr, idx) => {
        const btn = document.createElement('button');
        btn.className = `attr-btn ${idx === currentAttr ? 'active' : ''}`;
        btn.textContent = attr.label;
        btn.onclick = () => {
          currentAttr = idx;
          updateView();
        };
        attrRow.appendChild(btn);
      });
    }

    function updateView() {
      const obj = data[currentObj];
      const attr = obj.attrs[currentAttr];

      // Update active tabs
      tabs.forEach((t, i) => t.classList.toggle('active', i === currentObj));

      // Update active attribute buttons
      const btns = attrRow.querySelectorAll('.attr-btn');
      btns.forEach((b, i) => b.classList.toggle('active', i === currentAttr));

      // Update contents
      valEl.textContent = attr.val;
      unitEl.textContent = attr.unit;
      totalEl.textContent = attr.full;

      objName.textContent = obj.name;
      attrSummary.textContent = attr.full;
    }

    tabs.forEach((tab, idx) => {
      tab.onclick = () => {
        currentObj = idx;
        currentAttr = 0;
        renderAttrs();
        updateView();
      };
    });

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

Notice that "pizza" is just a thing, but the **number of slices** or the **cost per box in dollars** is a precise quantity we can calculate with.

Once you know what counts as a quantity, how do you sort them out when reading a word problem?

## Sorting Constants, Variables, and Goals

When building a model, every quantity falls into one of three roles: a fixed value (**constant**), an amount that can change (**independent variable**), or the final outcome you want to measure (**dependent variable**).

<!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;
    }

    body {
      padding: 12px;
      background: transparent;
      color: #1e2945;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      min-height: 100%;
    }

    .flow-container {
      width: 100%;
      max-width: 480px;
      display: flex;
      flex-direction: column;
      gap: 16px;
    }

    /* 3 Role Cards Grid */
    .roles-grid {
      display: grid;
      grid-template-columns: repeat(3, 1fr);
      gap: 8px;
    }

    .role-card {
      background: #ffffff;
      border: 1.5px solid #e6e6e6;
      border-radius: 10px;
      padding: 10px 6px;
      text-align: center;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: space-between;
      gap: 6px;
      box-shadow: 0 1px 3px rgba(0,0,0,0.04);
      min-height: 120px;
    }

    .role-card.constant {
      border-color: #94a3b8;
      background: #f8fafc;
    }

    .role-card.independent {
      border-color: #22b7ff;
      background: #f0f9ff;
    }

    .role-card.dependent {
      border-color: #10b981;
      background: #f0fdf4;
    }

    .badge {
      font-size: 11px;
      font-weight: 700;
      letter-spacing: 0.5px;
      text-transform: uppercase;
      padding: 2px 6px;
      border-radius: 4px;
    }

    .badge-constant {
      background: #e2e8f0;
      color: #475569;
    }

    .badge-independent {
      background: #bae6fd;
      color: #0369a1;
    }

    .badge-dependent {
      background: #bbf7d0;
      color: #15803d;
    }

    .role-icon {
      font-size: 22px;
      line-height: 1;
    }

    .role-val {
      font-size: 18px;
      font-weight: 700;
      color: #1e2945;
    }

    .role-desc {
      font-size: 12px;
      color: #64748b;
      font-weight: 500;
    }

    /* Equation / Model Display */
    .model-box {
      background: #ffffff;
      border: 1.5px solid #e2e8f0;
      border-radius: 10px;
      padding: 12px;
      display: flex;
      align-items: center;
      justify-content: center;
      gap: 6px;
      font-size: 18px;
      font-weight: 700;
      box-shadow: 0 1px 3px rgba(0,0,0,0.04);
    }

    .eq-chip {
      padding: 4px 8px;
      border-radius: 6px;
      display: inline-flex;
      align-items: center;
      justify-content: center;
      transition: background-color 0.2s ease;
    }

    .eq-const {
      background: #f1f5f9;
      color: #475569;
      border: 1px solid #cbd5e1;
    }

    .eq-indep {
      background: #e0f2fe;
      color: #0284c7;
      border: 1px solid #7dd3fc;
    }

    .eq-dep {
      background: #dcfce7;
      color: #16a34a;
      border: 1px solid #86efac;
    }

    .eq-op {
      color: #94a3b8;
      font-weight: 600;
    }

    /* Interactive Slider Control */
    .control-panel {
      background: #ffffff;
      border: 1px solid #e6e6e6;
      border-radius: 10px;
      padding: 12px 16px;
      display: flex;
      flex-direction: column;
      gap: 8px;
    }

    .control-header {
      display: flex;
      justify-content: space-between;
      align-items: center;
      font-size: 13px;
      font-weight: 600;
      color: #334155;
    }

    .slider {
      -webkit-appearance: none;
      appearance: none;
      width: 100%;
      height: 8px;
      border-radius: 4px;
      background: #e2e8f0;
      outline: none;
      cursor: pointer;
    }

    .slider::-webkit-slider-thumb {
      -webkit-appearance: none;
      appearance: none;
      width: 22px;
      height: 22px;
      border-radius: 50%;
      background: #22b7ff;
      border: 2px solid #ffffff;
      box-shadow: 0 2px 4px rgba(0,0,0,0.15);
      cursor: pointer;
    }

    .slider::-moz-range-thumb {
      width: 22px;
      height: 22px;
      border-radius: 50%;
      background: #22b7ff;
      border: 2px solid #ffffff;
      box-shadow: 0 2px 4px rgba(0,0,0,0.15);
      cursor: pointer;
    }

    /* Preset Quick Buttons */
    .btn-row {
      display: flex;
      gap: 6px;
      justify-content: center;
    }

    .btn-step {
      flex: 1;
      padding: 6px;
      border: 1px solid #cbd5e1;
      background: #f8fafc;
      border-radius: 6px;
      font-size: 12px;
      font-weight: 600;
      color: #475569;
      cursor: pointer;
      text-align: center;
    }

    .btn-step.active {
      background: #22b7ff;
      color: #ffffff;
      border-color: #0284c7;
    }
  </style>
</head>
<body>

<div class="flow-container">
  
  <div class="roles-grid">
    
    <div class="role-card constant">
      <span class="badge badge-constant">Constant</span>
      <div class="role-icon">🔒</div>
      <div class="role-val">$5</div>
      <div class="role-desc">Fixed Value</div>
    </div>

    
    <div class="role-card independent">
      <span class="badge badge-independent">Independent</span>
      <div class="role-icon">🎛️</div>
      <div class="role-val" id="indepVal">3 hrs</div>
      <div class="role-desc">Input (Changes)</div>
    </div>

    
    <div class="role-card dependent">
      <span class="badge badge-dependent">Dependent</span>
      <div class="role-icon">🎯</div>
      <div class="role-val" id="depVal">$15</div>
      <div class="role-desc">Output (Outcome)</div>
    </div>
  </div>

  
  <div class="model-box">
    <span class="eq-chip eq-const">$5/hr</span>
    <span class="eq-op">×</span>
    <span class="eq-chip eq-indep" id="eqX">3 hrs</span>
    <span class="eq-op">=</span>
    <span class="eq-chip eq-dep" id="eqY">$15</span>
  </div>

  
  <div class="control-panel">
    <div class="control-header">
      <span>Adjust Input:</span>
      <span id="sliderLabel" style="color: #0284c7; font-weight: 700;">3 hrs</span>
    </div>
    <input type="range" id="inputSlider" class="slider" min="1" max="8" value="3" step="1">
    
    <div class="btn-row">
      <button class="btn-step" data-val="1">1 hr</button>
      <button class="btn-step" data-val="2">2 hrs</button>
      <button class="btn-step active" data-val="3">3 hrs</button>
      <button class="btn-step" data-val="5">5 hrs</button>
      <button class="btn-step" data-val="8">8 hrs</button>
    </div>
  </div>
</div>

<script>
  const RATE = 5;
  const slider = document.getElementById('inputSlider');
  const indepVal = document.getElementById('indepVal');
  const depVal = document.getElementById('depVal');
  const eqX = document.getElementById('eqX');
  const eqY = document.getElementById('eqY');
  const sliderLabel = document.getElementById('sliderLabel');
  const buttons = document.querySelectorAll('.btn-step');

  function updateModel(hours) {
    const total = RATE * hours;
    
    indepVal.textContent = hours + (hours === 1 ? ' hr' : ' hrs');
    depVal.textContent = '$' + total;
    
    eqX.textContent = hours + (hours === 1 ? ' hr' : ' hrs');
    eqY.textContent = '$' + total;
    
    sliderLabel.textContent = hours + (hours === 1 ? ' hr' : ' hrs');
    slider.value = hours;

    buttons.forEach(btn => {
      if (parseInt(btn.getAttribute('data-val')) === hours) {
        btn.classList.add('active');
      } else {
        btn.classList.remove('active');
      }
    });
  }

  slider.addEventListener('input', (e) => {
    updateModel(parseInt(e.target.value));
  });

  buttons.forEach(btn => {
    btn.addEventListener('click', () => {
      const val = parseInt(btn.getAttribute('data-val'));
      updateModel(val);
    });
  });
</script>

</body>
</html>

Let's test this classification on a concrete example to see how each quantity gets defined.

## Worked Example: The T-Shirt Fundraiser

Suppose a club pays a $50 flat design fee for screen printing plus $6 per blank shirt, and sells finished shirts for $15 each. Let's define the key quantities needed to model the club's profit.

<!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>Defining Quantities for Descriptive Modeling</title>
  <link rel="stylesheet" href="{{VENDOR_CDN}}/katex.min.css">
  <style>
    * {
      box-sizing: border-box;
      margin: 0;
      padding: 0;
      font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, Helvetica, Arial, sans-serif;
      user-select: none;
    }
    body {
      background: #f8f9fa;
      color: #1e2945;
      padding: 12px;
      display: flex;
      flex-direction: column;
      gap: 12px;
      min-height: 100%;
    }

    /* Slider / Input Control */
    .control-panel {
      background: #ffffff;
      border: 1px solid #e2e8f0;
      border-radius: 10px;
      padding: 10px 14px;
      display: flex;
      align-items: center;
      justify-content: space-between;
      gap: 12px;
    }
    .slider-group {
      display: flex;
      align-items: center;
      gap: 10px;
      flex: 1;
    }
    input[type="range"] {
      flex: 1;
      accent-color: #22b7ff;
      cursor: pointer;
      height: 6px;
    }
    .val-badge {
      background: #e0f2fe;
      color: #0369a1;
      font-weight: 700;
      font-size: 15px;
      padding: 4px 10px;
      border-radius: 6px;
      min-width: 68px;
      text-align: center;
    }

    /* Quantities Breakdown Grid */
    .quantities-grid {
      display: grid;
      grid-template-columns: repeat(3, 1fr);
      gap: 8px;
    }
    .q-card {
      background: #ffffff;
      border: 1px solid #e2e8f0;
      border-radius: 10px;
      padding: 10px 8px;
      display: flex;
      flex-direction: column;
      align-items: center;
      text-align: center;
      gap: 4px;
    }
    .q-title {
      font-size: 13px;
      font-weight: 600;
      color: #64748b;
      text-transform: uppercase;
      letter-spacing: 0.5px;
    }
    .q-formula {
      font-size: 14px;
      color: #1e2945;
    }
    .q-val {
      font-size: 18px;
      font-weight: 700;
    }

    .c-rev { color: #0284c7; }
    .c-cost { color: #d97706; }
    .c-prof-pos { color: #16a34a; }
    .c-prof-neg { color: #dc2626; }

    /* Visual Bars */
    .bars-section {
      background: #ffffff;
      border: 1px solid #e2e8f0;
      border-radius: 10px;
      padding: 14px;
      display: flex;
      flex-direction: column;
      gap: 12px;
      flex: 1;
      justify-content: center;
    }
    .bar-row {
      display: flex;
      flex-direction: column;
      gap: 4px;
    }
    .bar-label-row {
      display: flex;
      justify-content: space-between;
      font-size: 13px;
      font-weight: 600;
      color: #475569;
    }
    .bar-track {
      height: 24px;
      background: #f1f5f9;
      border-radius: 6px;
      overflow: hidden;
      display: flex;
      position: relative;
    }
    .bar-seg {
      height: 100%;
      display: flex;
      align-items: center;
      justify-content: center;
      font-size: 11px;
      font-weight: 700;
      color: #ffffff;
      transition: width 0.15s ease;
      overflow: hidden;
      white-space: nowrap;
    }
    .seg-fixed { background: #f59e0b; }
    .seg-variable { background: #fbbf24; color: #78350f; }
    .seg-revenue { background: #0284c7; }

    /* Model Equation Summary */
    .model-equation {
      background: #ffffff;
      border: 1px solid #e2e8f0;
      border-radius: 10px;
      padding: 10px 12px;
      text-align: center;
      display: flex;
      align-items: center;
      justify-content: center;
      font-size: 16px;
    }
  </style>
</head>
<body>

  
  <div class="control-panel">
    <div class="slider-group">
      <span id="label-n"></span>
      <input type="range" id="shirtsInput" min="0" max="20" value="8" step="1">
    </div>
    <div class="val-badge" id="shirtsVal">n = 8</div>
  </div>

  
  <div class="quantities-grid">
    <div class="q-card">
      <span class="q-title">Revenue</span>
      <span class="q-formula" id="formula-rev"></span>
      <span class="q-val c-rev" id="val-rev">$120</span>
    </div>
    <div class="q-card">
      <span class="q-title">Cost</span>
      <span class="q-formula" id="formula-cost"></span>
      <span class="q-val c-cost" id="val-cost">$98</span>
    </div>
    <div class="q-card">
      <span class="q-title">Profit</span>
      <span class="q-formula" id="formula-prof"></span>
      <span class="q-val" id="val-prof">+$22</span>
    </div>
  </div>

  
  <div class="bars-section">
    <div class="bar-row">
      <div class="bar-label-row">
        <span>Total Revenue</span>
        <span id="bar-rev-txt">$120</span>
      </div>
      <div class="bar-track">
        <div class="bar-seg seg-revenue" id="bar-rev" style="width: 40%;">15n</div>
      </div>
    </div>

    <div class="bar-row">
      <div class="bar-label-row">
        <span>Total Cost</span>
        <span id="bar-cost-txt">$98</span>
      </div>
      <div class="bar-track">
        <div class="bar-seg seg-fixed" id="bar-fixed" style="width: 16.6%;">50</div>
        <div class="bar-seg seg-variable" id="bar-var" style="width: 16%;">6n</div>
      </div>
    </div>
  </div>

  
  <div class="model-equation" id="eq-summary"></div>

  <script src="{{VENDOR_CDN}}/katex.min.js"></script>
  <script>
    const slider = document.getElementById('shirtsInput');
    const shirtsVal = document.getElementById('shirtsVal');
    const valRev = document.getElementById('val-rev');
    const valCost = document.getElementById('val-cost');
    const valProf = document.getElementById('val-prof');
    const barRevTxt = document.getElementById('bar-rev-txt');
    const barCostTxt = document.getElementById('bar-cost-txt');
    const barRev = document.getElementById('bar-rev');
    const barFixed = document.getElementById('bar-fixed');
    const barVar = document.getElementById('bar-var');

    function renderMath() {
      if (window.katex) {
        katex.render("n", document.getElementById('label-n'));
        katex.render("15n", document.getElementById('formula-rev'));
        katex.render("6n + 50", document.getElementById('formula-cost'));
        katex.render("R - C", document.getElementById('formula-prof'));
      }
    }

    function update() {
      const n = parseInt(slider.value, 10);
      shirtsVal.textContent = `n = ${n}`;

      const rev = 15 * n;
      const fixed = 50;
      const variable = 6 * n;
      const cost = fixed + variable;
      const profit = rev - cost;

      valRev.textContent = `$${rev}`;
      valCost.textContent = `$${cost}`;

      if (profit >= 0) {
        valProf.textContent = `+$${profit}`;
        valProf.className = 'q-val c-prof-pos';
      } else {
        valProf.textContent = `-$${Math.abs(profit)}`;
        valProf.className = 'q-val c-prof-neg';
      }

      barRevTxt.textContent = `$${rev}`;
      barCostTxt.textContent = `$${cost}`;

      // Scale to max capacity ($300 = 20 shirts * $15)
      const maxVal = 300;
      barRev.style.width = `${(rev / maxVal) * 100}%`;
      barFixed.style.width = `${(fixed / maxVal) * 100}%`;
      barVar.style.width = `${(variable / maxVal) * 100}%`;

      if (window.katex) {
        const sign = profit >= 0 ? '+' : '-';
        const pStr = `P = ${rev} - ${cost} = ${profit >= 0 ? '' : '-'}\\$${Math.abs(profit)}`;
        katex.render(pStr, document.getElementById('eq-summary'));
      }
    }

    slider.addEventListener('input', update);

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

With these precise definitions, the relationship becomes straightforward: profit equals total revenue minus total cost, or $P = 15n - (6n + 50)$.

What is the single most common mistake people make when writing down their variable definitions?

A frequent error is defining a variable as an item rather than a number, such as writing "let $t = \text{t-shirts}$" or "let $t = \text{time}$".

<!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">
<title>Defining Quantities</title>
<style>
  * { box-sizing: border-box; margin: 0; padding: 0; font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif; }
  body {
    background: transparent;
    color: #1e2945;
    display: flex;
    flex-direction: column;
    align-items: center;
    justify-content: center;
    min-height: 100%;
    padding: 12px;
  }
  .app-wrap {
    width: 100%;
    max-width: 520px;
    display: flex;
    flex-direction: column;
    gap: 12px;
  }
  .tabs {
    display: flex;
    gap: 8px;
    justify-content: center;
    width: 100%;
  }
  .tab-btn {
    flex: 1;
    padding: 8px 12px;
    border-radius: 8px;
    border: 1px solid #d1d5db;
    background: #ffffff;
    color: #4b5563;
    font-size: 14px;
    font-weight: 600;
    cursor: pointer;
    transition: all 0.2s ease;
    text-align: center;
  }
  .tab-btn:hover {
    background: #f3f4f6;
  }
  .tab-btn.active {
    background: #22b7ff;
    color: #ffffff;
    border-color: #22b7ff;
    box-shadow: 0 2px 8px rgba(34, 183, 255, 0.3);
  }
  .comparison {
    display: flex;
    flex-direction: column;
    gap: 10px;
    width: 100%;
  }
  .box {
    border-radius: 12px;
    padding: 14px 16px;
    display: flex;
    align-items: center;
    justify-content: space-between;
    gap: 12px;
    transition: all 0.3s ease;
  }
  .box-wrong {
    background: #fff1f2;
    border: 1.5px solid #fecdd3;
  }
  .box-right {
    background: #f0fdf4;
    border: 1.5px solid #bbf7d0;
  }
  .tag {
    font-size: 11px;
    font-weight: 700;
    text-transform: uppercase;
    letter-spacing: 0.5px;
    padding: 3px 8px;
    border-radius: 6px;
    display: inline-block;
  }
  .tag-wrong {
    background: #fee2e2;
    color: #dc2626;
  }
  .tag-right {
    background: #dcfce7;
    color: #16a34a;
  }
  .content-side {
    display: flex;
    flex-direction: column;
    gap: 4px;
    flex: 1;
  }
  .expr {
    font-size: 16px;
    font-weight: 700;
    color: #1e2945;
  }
  .formula-parts {
    display: flex;
    flex-wrap: wrap;
    align-items: center;
    gap: 6px;
    margin-top: 4px;
  }
  .chip {
    padding: 4px 8px;
    border-radius: 6px;
    font-size: 13px;
    font-weight: 600;
  }
  .chip-attr {
    background: #e0f2fe;
    color: #0369a1;
    border: 1px solid #bae6fd;
  }
  .chip-unit {
    background: #fef3c7;
    color: #b45309;
    border: 1px solid #fde68a;
  }
  .icon-badge {
    font-size: 24px;
    width: 36px;
    height: 36px;
    display: flex;
    align-items: center;
    justify-content: center;
    flex-shrink: 0;
  }
  .plus {
    font-weight: 700;
    color: #64748b;
    font-size: 13px;
  }
</style>
</head>
<body>

<div class="app-wrap">
  <div class="tabs">
    <button class="tab-btn active" onclick="setExample(0)">👕 T-Shirts</button>
    <button class="tab-btn" onclick="setExample(1)">⏱️ Time</button>
    <button class="tab-btn" onclick="setExample(2)">💧 Water</button>
  </div>

  <div class="comparison">
    
    <div class="box box-wrong">
      <div class="content-side">
        <div><span class="tag tag-wrong">✕ Item / Vague</span></div>
        <div class="expr" id="wrongExpr">let <i>t</i> = t-shirts</div>
      </div>
      <div class="icon-badge">❌</div>
    </div>

    
    <div class="box box-right">
      <div class="content-side">
        <div><span class="tag tag-right">✓ Measurable Quantity</span></div>
        <div class="expr" id="rightExpr">let <i>n</i> =</div>
        <div class="formula-parts">
          <span class="chip chip-attr" id="attrChip">number of shirts</span>
          <span class="plus">+</span>
          <span class="chip chip-unit" id="unitChip">(count)</span>
        </div>
      </div>
      <div class="icon-badge">✅</div>
    </div>
  </div>
</div>

<script>
  const data = [
    {
      wrong: 'let <i>t</i> = t-shirts',
      varName: 'let <i>n</i> =',
      attr: 'number of shirts',
      unit: '(count)'
    },
    {
      wrong: 'let <i>t</i> = time',
      varName: 'let <i>t</i> =',
      attr: 'elapsed time',
      unit: '(minutes)'
    },
    {
      wrong: 'let <i>w</i> = water',
      varName: 'let <i>V</i> =',
      attr: 'volume of water',
      unit: '(liters)'
    }
  ];

  function setExample(index) {
    const btns = document.querySelectorAll('.tab-btn');
    btns.forEach((btn, i) => {
      btn.classList.toggle('active', i === index);
    });

    const item = data[index];
    document.getElementById('wrongExpr').innerHTML = item.wrong;
    document.getElementById('rightExpr').innerHTML = item.varName;
    document.getElementById('attrChip').textContent = item.attr;
    document.getElementById('unitChip').textContent = item.unit;
  }
</script>

</body>
</html>

Always include both the **specific measurable attribute** and the **unit** whenever you define a variable.

Whenever you encounter a word problem, ask yourself: what is changing, what stays fixed, and what units give those numbers meaning?

math

Defining Quantities for Descriptive Modeling. Identify and define the key quantities needed to model a real-world situation.

Defining Appropriate Quantities for Modeling

Algebra 1, Unit 2: Linear Equations, Inequalities, and Systems - Modeling with Quantities

Identify Independent and Dependent Quantities in Real-World Contexts

To develop student mastery of HSN.Q.A.2, instruction must bridge qualitative English context and algebraic formalism. Use a structured 4-Column Quantity Definition Protocol ('Variable Symbol', 'Complete Descriptive Name', 'Unit of Measurement', 'Realistic Domain/Bounds') before writing equations. Incorporate non-examples and deliberate error analysis (e.g., contrasting 'let $d = \text{distance}$' with 'let $d = \text{distance from school in miles}$') to prevent variable-as-label misconceptions.

Defining Quantities for Descriptive Modeling

What Is a Quantity?

Sorting Constants, Variables, and Goals

Worked Example: The T-Shirt Fundraiser