Refactor math notation and integer foundations content: Update basic math notation to replace half-bar representation with a dot-based notation for clarity. Enhance integer foundations by revising definitions, adding historical context, and improving explanations of operations, absolute value, and percentages. Include new examples and applications to better illustrate concepts and their relevance in real-world scenarios.

content/math/integer-foundations/core.html

@@ -1,35 +1,102 @@
 <div class="book-prose">
   <p>
-    Integers extend whole numbers to include values below zero. They are useful for temperature,
-    elevation, and money balances.
+    Integers extend whole numbers so we can model values above, below, and exactly at a reference
+    point. They appear in temperature, sea level, game scores, electricity, and money balances.
+    Without integers, many real situations would be awkward to calculate.
   </p>
 
-  <h2 id="number-line" class="book-prose__heading">Directed numbers on a number line</h2>
+  <h2 id="history" class="book-prose__heading">A short history of the number line idea</h2>
   <p>
-    Positive numbers are to the right of zero, negative numbers are to the left. Moving right
-    increases value; moving left decreases value.
+    The modern number line grew over time. Early printed works arranged numbers in order, and later
+    mathematicians described arithmetic as movement along a line. By the late 1600s, this “walk
+    forward / backward” idea was used to explain addition and subtraction in a geometric way.
+  </p>
+  <p>
+    The key shift was not just drawing numbers in order, but treating position on a line as a model
+    for operations. That model is now foundational in school mathematics.
+  </p>
+
+  <h2 id="number-line" class="book-prose__heading">Drawing and reading a number line</h2>
+  <p>
+    A number line is a straight line with equally spaced marks. Numbers increase to the right and
+    decrease to the left. Zero is the center reference point for positive and negative values.
+  </p>
+  <ul>
+    <li><strong>Right of 0:</strong> positive integers (\(+1, +2, +3, \dots\)).</li>
+    <li><strong>Left of 0:</strong> negative integers (\(-1, -2, -3, \dots\)).</li>
+    <li><strong>At 0:</strong> neither positive nor negative.</li>
+  </ul>
+  <p>
+    In coordinate graphs, both axes are number lines: the horizontal \(x\)-axis and vertical
+    \(y\)-axis.
   </p>
   <div class="book-callout book-callout--formula" role="note">
     <p class="book-callout__label">Compare quickly</p>
     <p class="book-formula">\(a &gt; b\) means \(a\) is to the right of \(b\) on the number line.</p>
   </div>
 
-  <h2 id="absolute-value" class="book-prose__heading">Absolute value and opposites</h2>
+  <h2 id="compare" class="book-prose__heading">Comparing integers with position</h2>
   <p>
-    Absolute value is distance from zero: \(|-7|=7\). Opposites have equal distance from zero and
-    opposite signs, such as \(+5\) and \(-5\).
-  </p>
-
-  <h2 id="add-subtract" class="book-prose__heading">Adding and subtracting integers</h2>
-  <p>
-    You can think of addition and subtraction as movement on the number line.
+    Position gives order immediately:
   </p>
   <ul>
-    <li>\(a+b\): start at \(a\), move by \(b\).</li>
-    <li>\(a-b = a+(-b)\): subtracting is adding the opposite.</li>
+    <li>If \(a\) is to the right of \(b\), then \(a &gt; b\).</li>
+    <li>If \(a\) is to the left of \(b\), then \(a &lt; b\).</li>
   </ul>
   <p>
-    Example: \(6-(-4)=6+4=10\).
+    Example: \(-2 &gt; -5\) because \(-2\) is to the right of \(-5\), even though both are negative.
+  </p>
+  <div class="math-widget-mount" data-widget="integer-compare-demo" data-pagefind-ignore></div>
+
+  <h2 id="absolute-value" class="book-prose__heading">Absolute value and opposites</h2>
+  <p>
+    Absolute value means distance from zero, so it is never negative: \(|-7|=7\), \(|7|=7\).
+    Opposite numbers are equally far from zero in opposite directions, such as \(+5\) and \(-5\).
+  </p>
+  <p>
+    Distance between two integers can be read from the line:
+    \[
+      \text{distance between } a \text{ and } b = |a-b|.
+    \]
+    For example, the distance from \(-4\) to \(3\) is \(|-4-3|=7\).
+  </p>
+
+  <h2 id="operations" class="book-prose__heading">Operations as movement and grouping</h2>
+  <h3 class="book-prose__heading">Addition and subtraction</h3>
+  <p>
+    Think of arithmetic as moves on the line:
+  </p>
+  <ul>
+    <li>\(a+b\): start at \(a\), move \(b\) units (right if \(b&gt;0\), left if \(b&lt;0\)).</li>
+    <li>\(a-b = a+(-b)\): subtracting means adding the opposite.</li>
+  </ul>
+  <p>
+    Examples:
+    \[
+      -3+8=5,\qquad 4-9=-5,\qquad 6-(-4)=10.
+    \]
+  </p>
+  <h3 class="book-prose__heading">Multiplication</h3>
+  <p>
+    Multiplication can be seen as repeated equal jumps. For instance, \(5\times3\) means three jumps
+    of length 5 to the right, ending at 15.
+  </p>
+  <p>
+    Sign rules come from direction:
+  </p>
+  <ul>
+    <li>\((+)\times(+)=+\)</li>
+    <li>\((+)\times(-)=-\)</li>
+    <li>\((-)\times(+)=-\)</li>
+    <li>\((-)\times(-)=+\)</li>
+  </ul>
+  <h3 class="book-prose__heading">Division</h3>
+  <p>
+    Division asks how many equal lengths fit into another length. For example, \(6\div2=3\) because
+    three jumps of 2 reach 6. Integer sign rules mirror multiplication:
+    \[
+      (+)\div(+)=+,\quad (+)\div(-)=-,\quad (-)\div(+)=-,\quad (-)\div(-)=+.
+    \]
   </p>
 
   <h2 id="zero-principle" class="book-prose__heading">Zero principle with tiles</h2>
@@ -60,4 +127,23 @@
       </p>
     </div>
   </details>
+
+  <h2 id="why-zero-negative" class="book-prose__heading">Why zero and negatives matter</h2>
+  <p>
+    Zero and negative numbers make arithmetic coherent. Before negatives were accepted, people often
+    restricted results to non-negative values, which forced case-by-case rules and extra wording.
+  </p>
+  <p>
+    With integers, one set of laws works smoothly:
+  </p>
+  <ul>
+    <li>Subtraction is always possible: \(a-b=a+(-b)\).</li>
+    <li>Opposites are explicit: every \(a\) has \(-a\), and \(a+(-a)=0\).</li>
+    <li>Distributive structure stays consistent across signs:
+      \(k(a+b)=ka+kb\), even when values are negative.</li>
+  </ul>
+  <p>
+    This consistency is why integer arithmetic is not just “extra symbols”; it is the foundation for
+    algebra, equations, and later topics like vectors and coordinate geometry.
+  </p>
 </div>