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Enhance basic math notation content by adding a historical overview of mathematical symbols, including the evolution of the equals sign and the introduction of zero. Update comparison section with clearer explanations and examples, and improve styling for visual elements in the math chapter. Adjust chapter structure to allow for flexible content ordering based on importance.

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content/math/basic-math-notation/_index.md
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@ -17,6 +17,45 @@ strand = "Number & Algebra"
arithmetic yet; you only need to <strong>read</strong> the symbols like words on a sign.
</p>
<h2 id="history" class="book-prose__heading">History</h2>
<p>
People counted with fingers and words long before they agreed on one way to write math. Ancient
Egyptians and Babylonians used pictures and marks for amounts. Over centuries, different
countries used different symbols for the same ideas.
</p>
<figure class="book-figure">
<img
class="book-figure__img"
src="/chapters/math/basic-math-notation/basic-math-notation-history.png"
alt="Cartoon timeline: counting long ago, Egyptian and Babylonian marks, different symbols around the world, then equals, plus and minus, times and divide, and brackets for grouping"
width="960"
height="640"
loading="lazy"
decoding="async"
/>
<figcaption class="book-figure__caption">
From finger counting and ancient marks to short symbols on one line: how +, −, ×, ÷, =, and ( )
became the signs you read today.
</figcaption>
</figure>
<p>
The <strong>equals sign</strong> (=) was popularised in the 1500s by Welsh mathematician
<strong>Robert Recorde</strong>. He chose two lines because “no two things can be more equal” than
parallel lines — a way to show both sides match.
</p>
<p>
The <strong>plus</strong> and <strong>minus</strong> signs (+ and −) became common in Europe
around the same era, replacing longer Latin words. The <strong>times</strong> sign (×) and
<strong>division</strong> sign (÷) appeared later, when printing books needed short, clear marks
that fit on one line.
</p>
<p>
Brackets grew important when formulas got longer — mathematicians needed a visible rule for
“which part first,” just like you do in <strong>(2 + 3) + 4</strong>.
</p>
<h2 id="digits-and-words" class="book-prose__heading">Digits, words, and spaces</h2>
<p>
The symbols <strong>0</strong> through <strong>9</strong> are called <strong>digits</strong>. Each
@ -218,28 +257,135 @@ strand = "Number & Algebra"
<h2 id="compare" class="book-prose__heading">Less than and greater than (&lt; &gt;)</h2>
<p>
Sometimes we compare two amounts without adding or subtracting. The open mouth of the symbol
points at the <strong>bigger</strong> number.
Sometimes we compare two amounts without adding or subtracting. Think of
<strong>&lt;</strong> and <strong>&gt;</strong> as a hungry mouth: the open side points at the
<strong>bigger</strong> number — it “wants” the larger pile.
</p>
<ul>
<li><strong>&lt;</strong> means <strong>less than</strong>. Example: <strong>3 &lt; 7</strong> — three is less than seven.</li>
<li><strong>&gt;</strong> means <strong>greater than</strong>. Example: <strong>9 &gt; 5</strong> — nine is greater than five.</li>
<li><strong>&lt;</strong> means <strong>less than</strong> — the mouth opens toward the bigger number on the right.</li>
<li><strong>&gt;</strong> means <strong>greater than</strong> — the mouth opens toward the bigger number on the left.</li>
</ul>
<h3 id="compare-examples" class="book-prose__heading">More examples</h3>
<p>Read each line like a short sentence. The symbol tells you which side is bigger:</p>
<ul>
<li><strong>3 &lt; 7</strong> — “three is less than seven.”</li>
<li><strong>9 &gt; 5</strong> — “nine is greater than five.”</li>
<li><strong>2 &lt; 6</strong> — “two is less than six.”</li>
<li><strong>8 &gt; 3</strong> — “eight is greater than three.”</li>
<li><strong>5 &lt; 5</strong> is <strong>not</strong> true — five is the same as five, not less.</li>
<li><strong>4 &lt; 4</strong> is also false — use <strong>=</strong> when both sides match.</li>
</ul>
<p>
You may also see <strong>≤</strong> (less than or equal) and <strong>≥</strong> (greater than or
equal). The extra line means “or the same amount.”
The same idea works either way: <strong>7 &gt; 4</strong> and <strong>4 &lt; 7</strong> both say
that seven is the bigger amount.
</p>
<h3 id="compare-chain" class="book-prose__heading">A longer compare line</h3>
<p>
You can chain comparisons when several numbers line up from <strong>biggest to smallest</strong>
(or the other way around). Read it in order, one piece at a time:
</p>
<p>
<strong>9 &gt; 7 &gt; 4 &gt; 1</strong> — “nine is greater than seven, greater than four, greater
than one.” Each number is bigger than the one to its right.
</p>
<p>
In words: nine is greater than seven; seven is greater than four; four is greater than one. The
whole chain means <strong>nine wins</strong> and <strong>one is smallest</strong>.
</p>
<figure class="notation-compare-chain" role="group" aria-label="Nine dots, then seven, then four, then one; each group has fewer dots than the group before">
<ol class="notation-compare-chain__ladder">
<li class="notation-compare-chain__step">
<span class="notation-compare-chain__digit">9</span>
<span class="notation-dots notation-dots--chain" role="img" aria-hidden="true">
<span class="notation-dots__dot"></span><span class="notation-dots__dot"></span><span class="notation-dots__dot"></span><span class="notation-dots__dot"></span><span class="notation-dots__dot"></span><span class="notation-dots__dot"></span><span class="notation-dots__dot"></span><span class="notation-dots__dot"></span><span class="notation-dots__dot"></span>
</span>
</li>
<li class="notation-compare-chain__sign" aria-hidden="true">&gt;</li>
<li class="notation-compare-chain__step">
<span class="notation-compare-chain__digit">7</span>
<span class="notation-dots notation-dots--chain" role="img" aria-hidden="true">
<span class="notation-dots__dot"></span><span class="notation-dots__dot"></span><span class="notation-dots__dot"></span><span class="notation-dots__dot"></span><span class="notation-dots__dot"></span><span class="notation-dots__dot"></span><span class="notation-dots__dot"></span>
</span>
</li>
<li class="notation-compare-chain__sign" aria-hidden="true">&gt;</li>
<li class="notation-compare-chain__step">
<span class="notation-compare-chain__digit">4</span>
<span class="notation-dots notation-dots--chain" role="img" aria-hidden="true">
<span class="notation-dots__dot"></span><span class="notation-dots__dot"></span><span class="notation-dots__dot"></span><span class="notation-dots__dot"></span>
</span>
</li>
<li class="notation-compare-chain__sign" aria-hidden="true">&gt;</li>
<li class="notation-compare-chain__step">
<span class="notation-compare-chain__digit">1</span>
<span class="notation-dots notation-dots--chain" role="img" aria-hidden="true">
<span class="notation-dots__dot"></span>
</span>
</li>
</ol>
<figcaption class="notation-compare-chain__caption">
Nine dots &gt; seven dots &gt; four dots &gt; one dot — the pictures shrink the same way the
numbers do.
</figcaption>
</figure>
<p>
You can flip the story for “smallest first”: <strong>1 &lt; 4 &lt; 7 &lt; 9</strong> — “one is
less than four, less than seven, less than nine.” Same order, different mouths.
</p>
<div class="book-callout" role="note">
<p class="book-callout__label">Or the same amount</p>
<p>
You may also see <strong>≤</strong> (less than or equal) and <strong>≥</strong> (greater than or
equal). The extra line means “or the same amount,” like <strong>5 ≤ 5</strong>.
</p>
</div>
<h2 id="comma-decimal" class="book-prose__heading">Comma and decimal point</h2>
<p>
In big numbers like <strong>1,000</strong>, a <strong>comma</strong> groups digits in threes so
the number is easier to read — one thousand, not “one zero zero zero” as separate ideas.
</p>
<p>
A <strong>decimal point</strong> <strong>.</strong> sits between a whole part and a smaller part.
<strong>3.50</strong> might mean three dollars and fifty cents: the part after the point is not
whole dollars. You will learn more about decimals later; for now, remember the dot splits the
number into two pieces.
A <strong>decimal point</strong> <strong>.</strong> is a tiny dot that acts like a
<strong>fence</strong> in a number. Everything on the <strong>left</strong> counts whole things
you can line up (1, 2, 3, …). Everything on the <strong>right</strong> tells a
<strong>piece</strong> of one more thing — not another full thing.
</p>
<p>
Take <strong>3.5</strong>. Read it as <strong>“three and a half”</strong>: three full ones, plus
half of the next one. The digit <strong>5</strong> after the point does <em>not</em> mean “five
more marbles” — it names how much of the <em>next</em> whole you have filled in.
</p>
<figure class="notation-decimal-demo" role="group" aria-label="Three point five shown as three whole dots and half of a fourth">
<p class="notation-decimal-demo__label">
<span class="notation-decimal-demo__whole">3</span><span class="notation-decimal-demo__point" aria-hidden="true">.</span><span class="notation-decimal-demo__fraction">5</span>
<span class="notation-decimal-demo__words">three and a half</span>
</p>
<div class="notation-decimal-demo__picture">
<div class="notation-decimal-demo__wholes" aria-hidden="true">
<span class="notation-dots notation-dots--decimal" role="presentation">
<span class="notation-dots__dot"></span><span class="notation-dots__dot"></span><span class="notation-dots__dot"></span>
</span>
<span class="notation-decimal-demo__fence" aria-hidden="true">·</span>
<span class="notation-decimal-demo__half" role="presentation">
<span class="notation-decimal-demo__half-bar">
<span class="notation-decimal-demo__half-on"></span>
<span class="notation-decimal-demo__half-off"></span>
</span>
</span>
</div>
</div>
<figcaption class="notation-decimal-demo__caption">
Left of the fence: three whole dots. Right of the fence: half of the next dot filled in — that
is what <strong>3.5</strong> means.
</figcaption>
</figure>
<p>
In a shop, <strong>3.50</strong> often means three dollars and fifty cents: dollars are whole
steps on the left, cents are small pieces on the right. You will learn more about decimals later;
for now, remember the point splits “how many wholes” from “how much of the next one.”
</p>
<h2 id="fractions" class="book-prose__heading">Fractions (½ and 1/2)</h2>
@ -296,28 +442,6 @@ strand = "Number & Algebra"
</div>
<div class="book-prose">
<h2 id="history" class="book-prose__heading">History</h2>
<p>
People counted with fingers and words long before they agreed on one way to write math. Ancient
Egyptians and Babylonians used pictures and marks for amounts. Over centuries, different
countries used different symbols for the same ideas.
</p>
<p>
The <strong>equals sign</strong> (=) was popularised in the 1500s by Welsh mathematician
<strong>Robert Recorde</strong>. He chose two lines because “no two things can be more equal” than
parallel lines — a way to show both sides match.
</p>
<p>
The <strong>plus</strong> and <strong>minus</strong> signs (+ and −) became common in Europe
around the same era, replacing longer Latin words. The <strong>times</strong> sign (×) and
<strong>division</strong> sign (÷) appeared later, when printing books needed short, clear marks
that fit on one line.
</p>
<p>
Brackets grew important when formulas got longer — mathematicians needed a visible rule for
“which part first,” just like you do in <strong>(2 + 3) + 4</strong>.
</p>
<h2 id="derivation" class="book-prose__heading">Derivation</h2>
<h3 id="why-symbols" class="book-prose__heading">Why symbols instead of only words?</h3>

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content/math/basic-math-notation/chapter.scss
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@ -36,3 +36,162 @@
font-size: 0.9rem;
color: var(--muted);
}
// Dot pictures for compare chains (same idea as counting chapter)
.notation-dots {
display: inline-flex;
flex-wrap: wrap;
align-items: center;
gap: 4px;
vertical-align: middle;
}
.notation-dots__dot {
display: block;
width: 10px;
height: 10px;
border-radius: 50%;
background: #0f766e;
box-shadow: 0 1px 0 rgba(0, 0, 0, 0.12);
flex-shrink: 0;
}
.notation-dots--chain {
max-width: 88px;
justify-content: center;
}
.notation-dots--decimal .notation-dots__dot {
width: 14px;
height: 14px;
}
.notation-compare-chain {
margin: 1.25rem 0;
padding: 16px 14px;
text-align: center;
border: 1px solid var(--line);
border-radius: 12px;
background: var(--card-bg);
}
.notation-compare-chain__ladder {
display: flex;
flex-wrap: wrap;
align-items: flex-end;
justify-content: center;
gap: 10px 12px;
margin: 0;
padding: 0;
list-style: none;
}
.notation-compare-chain__step {
display: flex;
flex-direction: column;
align-items: center;
gap: 6px;
min-width: 52px;
}
.notation-compare-chain__digit {
font-family: var(--font-mono, ui-monospace, monospace);
font-size: 1.15rem;
font-weight: 700;
}
.notation-compare-chain__sign {
align-self: center;
font-size: 1.35rem;
font-weight: 700;
color: var(--soft-ink);
line-height: 1;
padding-bottom: 18px;
}
.notation-compare-chain__caption {
margin-top: 0.75rem;
font-size: 0.9rem;
color: var(--muted);
}
.notation-decimal-demo {
margin: 1.25rem 0;
padding: 16px 14px;
text-align: center;
border: 1px solid var(--line);
border-radius: 12px;
background: var(--card-bg);
}
.notation-decimal-demo__label {
margin: 0 0 12px;
font-family: var(--font-mono, ui-monospace, monospace);
font-size: 1.5rem;
font-weight: 700;
line-height: 1.2;
}
.notation-decimal-demo__point {
color: #b45309;
padding: 0 2px;
}
.notation-decimal-demo__fraction {
color: var(--soft-ink);
}
.notation-decimal-demo__words {
display: block;
margin-top: 4px;
font-family: var(--font-sans, system-ui, sans-serif);
font-size: 0.95rem;
font-weight: 500;
color: var(--muted);
}
.notation-decimal-demo__picture {
display: flex;
justify-content: center;
}
.notation-decimal-demo__wholes {
display: inline-flex;
flex-wrap: wrap;
align-items: center;
justify-content: center;
gap: 10px;
}
.notation-decimal-demo__fence {
font-size: 1.75rem;
font-weight: 800;
line-height: 1;
color: #b45309;
}
.notation-decimal-demo__half-bar {
display: inline-flex;
width: 56px;
height: 18px;
border-radius: 6px;
overflow: hidden;
border: 2px solid rgba(15, 118, 110, 0.4);
vertical-align: middle;
}
.notation-decimal-demo__half-on {
flex: 1;
background: #0f766e;
}
.notation-decimal-demo__half-off {
flex: 1;
background: rgba(20, 184, 166, 0.1);
}
.notation-decimal-demo__caption {
margin-top: 0.75rem;
font-size: 0.9rem;
color: var(--muted);
}

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@ -6,6 +6,45 @@
arithmetic yet; you only need to <strong>read</strong> the symbols like words on a sign.
</p>
<h2 id="history" class="book-prose__heading">History</h2>
<p>
People counted with fingers and words long before they agreed on one way to write math. Ancient
Egyptians and Babylonians used pictures and marks for amounts. Over centuries, different
countries used different symbols for the same ideas.
</p>
<figure class="book-figure">
<img
class="book-figure__img"
src="/chapters/math/basic-math-notation/basic-math-notation-history.png"
alt="Cartoon timeline: counting long ago, Egyptian and Babylonian marks, different symbols around the world, then equals, plus and minus, times and divide, and brackets for grouping"
width="960"
height="640"
loading="lazy"
decoding="async"
/>
<figcaption class="book-figure__caption">
From finger counting and ancient marks to short symbols on one line: how +, −, ×, ÷, =, and ( )
became the signs you read today.
</figcaption>
</figure>
<p>
The <strong>equals sign</strong> (=) was popularised in the 1500s by Welsh mathematician
<strong>Robert Recorde</strong>. He chose two lines because “no two things can be more equal” than
parallel lines — a way to show both sides match.
</p>
<p>
The <strong>plus</strong> and <strong>minus</strong> signs (+ and −) became common in Europe
around the same era, replacing longer Latin words. The <strong>times</strong> sign (×) and
<strong>division</strong> sign (÷) appeared later, when printing books needed short, clear marks
that fit on one line.
</p>
<p>
Brackets grew important when formulas got longer — mathematicians needed a visible rule for
“which part first,” just like you do in <strong>(2 + 3) + 4</strong>.
</p>
<h2 id="digits-and-words" class="book-prose__heading">Digits, words, and spaces</h2>
<p>
The symbols <strong>0</strong> through <strong>9</strong> are called <strong>digits</strong>. Each
@ -207,28 +246,135 @@
<h2 id="compare" class="book-prose__heading">Less than and greater than (&lt; &gt;)</h2>
<p>
Sometimes we compare two amounts without adding or subtracting. The open mouth of the symbol
points at the <strong>bigger</strong> number.
Sometimes we compare two amounts without adding or subtracting. Think of
<strong>&lt;</strong> and <strong>&gt;</strong> as a hungry mouth: the open side points at the
<strong>bigger</strong> number — it “wants” the larger pile.
</p>
<ul>
<li><strong>&lt;</strong> means <strong>less than</strong>. Example: <strong>3 &lt; 7</strong> — three is less than seven.</li>
<li><strong>&gt;</strong> means <strong>greater than</strong>. Example: <strong>9 &gt; 5</strong> — nine is greater than five.</li>
<li><strong>&lt;</strong> means <strong>less than</strong> — the mouth opens toward the bigger number on the right.</li>
<li><strong>&gt;</strong> means <strong>greater than</strong> — the mouth opens toward the bigger number on the left.</li>
</ul>
<h3 id="compare-examples" class="book-prose__heading">More examples</h3>
<p>Read each line like a short sentence. The symbol tells you which side is bigger:</p>
<ul>
<li><strong>3 &lt; 7</strong> — “three is less than seven.”</li>
<li><strong>9 &gt; 5</strong> — “nine is greater than five.”</li>
<li><strong>2 &lt; 6</strong> — “two is less than six.”</li>
<li><strong>8 &gt; 3</strong> — “eight is greater than three.”</li>
<li><strong>5 &lt; 5</strong> is <strong>not</strong> true — five is the same as five, not less.</li>
<li><strong>4 &lt; 4</strong> is also false — use <strong>=</strong> when both sides match.</li>
</ul>
<p>
You may also see <strong>≤</strong> (less than or equal) and <strong>≥</strong> (greater than or
equal). The extra line means “or the same amount.”
The same idea works either way: <strong>7 &gt; 4</strong> and <strong>4 &lt; 7</strong> both say
that seven is the bigger amount.
</p>
<h3 id="compare-chain" class="book-prose__heading">A longer compare line</h3>
<p>
You can chain comparisons when several numbers line up from <strong>biggest to smallest</strong>
(or the other way around). Read it in order, one piece at a time:
</p>
<p>
<strong>9 &gt; 7 &gt; 4 &gt; 1</strong> — “nine is greater than seven, greater than four, greater
than one.” Each number is bigger than the one to its right.
</p>
<p>
In words: nine is greater than seven; seven is greater than four; four is greater than one. The
whole chain means <strong>nine wins</strong> and <strong>one is smallest</strong>.
</p>
<figure class="notation-compare-chain" role="group" aria-label="Nine dots, then seven, then four, then one; each group has fewer dots than the group before">
<ol class="notation-compare-chain__ladder">
<li class="notation-compare-chain__step">
<span class="notation-compare-chain__digit">9</span>
<span class="notation-dots notation-dots--chain" role="img" aria-hidden="true">
<span class="notation-dots__dot"></span><span class="notation-dots__dot"></span><span class="notation-dots__dot"></span><span class="notation-dots__dot"></span><span class="notation-dots__dot"></span><span class="notation-dots__dot"></span><span class="notation-dots__dot"></span><span class="notation-dots__dot"></span><span class="notation-dots__dot"></span>
</span>
</li>
<li class="notation-compare-chain__sign" aria-hidden="true">&gt;</li>
<li class="notation-compare-chain__step">
<span class="notation-compare-chain__digit">7</span>
<span class="notation-dots notation-dots--chain" role="img" aria-hidden="true">
<span class="notation-dots__dot"></span><span class="notation-dots__dot"></span><span class="notation-dots__dot"></span><span class="notation-dots__dot"></span><span class="notation-dots__dot"></span><span class="notation-dots__dot"></span><span class="notation-dots__dot"></span>
</span>
</li>
<li class="notation-compare-chain__sign" aria-hidden="true">&gt;</li>
<li class="notation-compare-chain__step">
<span class="notation-compare-chain__digit">4</span>
<span class="notation-dots notation-dots--chain" role="img" aria-hidden="true">
<span class="notation-dots__dot"></span><span class="notation-dots__dot"></span><span class="notation-dots__dot"></span><span class="notation-dots__dot"></span>
</span>
</li>
<li class="notation-compare-chain__sign" aria-hidden="true">&gt;</li>
<li class="notation-compare-chain__step">
<span class="notation-compare-chain__digit">1</span>
<span class="notation-dots notation-dots--chain" role="img" aria-hidden="true">
<span class="notation-dots__dot"></span>
</span>
</li>
</ol>
<figcaption class="notation-compare-chain__caption">
Nine dots &gt; seven dots &gt; four dots &gt; one dot — the pictures shrink the same way the
numbers do.
</figcaption>
</figure>
<p>
You can flip the story for “smallest first”: <strong>1 &lt; 4 &lt; 7 &lt; 9</strong> — “one is
less than four, less than seven, less than nine.” Same order, different mouths.
</p>
<div class="book-callout" role="note">
<p class="book-callout__label">Or the same amount</p>
<p>
You may also see <strong>≤</strong> (less than or equal) and <strong>≥</strong> (greater than or
equal). The extra line means “or the same amount,” like <strong>5 ≤ 5</strong>.
</p>
</div>
<h2 id="comma-decimal" class="book-prose__heading">Comma and decimal point</h2>
<p>
In big numbers like <strong>1,000</strong>, a <strong>comma</strong> groups digits in threes so
the number is easier to read — one thousand, not “one zero zero zero” as separate ideas.
</p>
<p>
A <strong>decimal point</strong> <strong>.</strong> sits between a whole part and a smaller part.
<strong>3.50</strong> might mean three dollars and fifty cents: the part after the point is not
whole dollars. You will learn more about decimals later; for now, remember the dot splits the
number into two pieces.
A <strong>decimal point</strong> <strong>.</strong> is a tiny dot that acts like a
<strong>fence</strong> in a number. Everything on the <strong>left</strong> counts whole things
you can line up (1, 2, 3, …). Everything on the <strong>right</strong> tells a
<strong>piece</strong> of one more thing — not another full thing.
</p>
<p>
Take <strong>3.5</strong>. Read it as <strong>“three and a half”</strong>: three full ones, plus
half of the next one. The digit <strong>5</strong> after the point does <em>not</em> mean “five
more marbles” — it names how much of the <em>next</em> whole you have filled in.
</p>
<figure class="notation-decimal-demo" role="group" aria-label="Three point five shown as three whole dots and half of a fourth">
<p class="notation-decimal-demo__label">
<span class="notation-decimal-demo__whole">3</span><span class="notation-decimal-demo__point" aria-hidden="true">.</span><span class="notation-decimal-demo__fraction">5</span>
<span class="notation-decimal-demo__words">three and a half</span>
</p>
<div class="notation-decimal-demo__picture">
<div class="notation-decimal-demo__wholes" aria-hidden="true">
<span class="notation-dots notation-dots--decimal" role="presentation">
<span class="notation-dots__dot"></span><span class="notation-dots__dot"></span><span class="notation-dots__dot"></span>
</span>
<span class="notation-decimal-demo__fence" aria-hidden="true">·</span>
<span class="notation-decimal-demo__half" role="presentation">
<span class="notation-decimal-demo__half-bar">
<span class="notation-decimal-demo__half-on"></span>
<span class="notation-decimal-demo__half-off"></span>
</span>
</span>
</div>
</div>
<figcaption class="notation-decimal-demo__caption">
Left of the fence: three whole dots. Right of the fence: half of the next dot filled in — that
is what <strong>3.5</strong> means.
</figcaption>
</figure>
<p>
In a shop, <strong>3.50</strong> often means three dollars and fifty cents: dollars are whole
steps on the left, cents are small pieces on the right. You will learn more about decimals later;
for now, remember the point splits “how many wholes” from “how much of the next one.”
</p>
<h2 id="fractions" class="book-prose__heading">Fractions (½ and 1/2)</h2>
@ -283,197 +429,3 @@
<li><strong>(1 + 2) × 3 = 9</strong> — brackets first: one plus two is three; three times three is nine</li>
</ul>
</div>
<div class="book-prose">
<h2 id="history" class="book-prose__heading">History</h2>
<p>
People counted with fingers and words long before they agreed on one way to write math. Ancient
Egyptians and Babylonians used pictures and marks for amounts. Over centuries, different
countries used different symbols for the same ideas.
</p>
<p>
The <strong>equals sign</strong> (=) was popularised in the 1500s by Welsh mathematician
<strong>Robert Recorde</strong>. He chose two lines because “no two things can be more equal” than
parallel lines — a way to show both sides match.
</p>
<p>
The <strong>plus</strong> and <strong>minus</strong> signs (+ and −) became common in Europe
around the same era, replacing longer Latin words. The <strong>times</strong> sign (×) and
<strong>division</strong> sign (÷) appeared later, when printing books needed short, clear marks
that fit on one line.
</p>
<p>
Brackets grew important when formulas got longer — mathematicians needed a visible rule for
“which part first,” just like you do in <strong>(2 + 3) + 4</strong>.
</p>
<h2 id="derivation" class="book-prose__heading">Derivation</h2>
<h3 id="why-symbols" class="book-prose__heading">Why symbols instead of only words?</h3>
<p>
Writing “three plus four equals seven” every time is long. Symbols are a <strong>shorthand</strong>:
the same meaning in less space, so you can compare many sentences quickly.
</p>
<h3 id="equals-meaning" class="book-prose__heading">What equals really says</h3>
<p>
If the left side of \(=\) counts to \(a\) and the right side counts to \(b\), then \(a = b\)
means both sides name the <strong>same number</strong>. If they were different, the sentence would
be false — like claiming \(3 + 4 = 8\) when seven dots are the true total.
</p>
<h3 id="brackets-order" class="book-prose__heading">Why brackets change the order</h3>
<p>
Without a rule, \(2 + 3 \times 4\) could be read in two ways. Brackets fix one group:
\((2 + 3) \times 4 = 5 \times 4 = 20\). Later you will learn more order rules; for now,
brackets always mean “work inside first.”
</p>
<h2 id="checkpoints" class="book-prose__heading">Checkpoints</h2>
<p>Pause and answer in your head or with a partner.</p>
<div class="book-callout" role="note">
<p class="book-callout__label">Checkpoint</p>
<p>What is the name of the sign in \(7 - 3 = 4\)? What does it tell you to do?</p>
</div>
<div class="book-callout" role="note">
<p class="book-callout__label">Checkpoint</p>
<p>Say out loud: \(4 \times 2 = 8\). Which word goes with \(\times\)?</p>
</div>
<div class="book-callout" role="note">
<p class="book-callout__label">Checkpoint</p>
<p>In \((5 + 1) + 2\), which part do you work out first? What is the final total?</p>
</div>
<div class="book-callout" role="note">
<p class="book-callout__label">Checkpoint</p>
<p>Is \(6 &gt; 2\) true? What does the open side of \(>\) point toward?</p>
</div>
<div class="book-callout" role="note">
<p class="book-callout__label">Checkpoint</p>
<p>What does \(50\%\) mean in words (“out of …”)?</p>
</div>
<h2 id="applications" class="book-prose__heading">Real-life applications</h2>
<ul>
<li><strong>Recipes</strong> — “add 2 cups flour + 1 cup sugar” is a plus sentence in the kitchen.</li>
<li><strong>Scores</strong> — “Team A: 12, Team B: 8” uses comparison; 12 &gt; 8 means A scored more.</li>
<li><strong>Sharing</strong> — “12 sweets ÷ 4 children” is fair division at a party.</li>
<li><strong>Shopping</strong> — prices like $3.50 use a decimal point between dollars and cents.</li>
<li><strong>Sales</strong> — “25% off” uses percent out of one hundred.</li>
<li><strong>Patterns</strong> — “2, 4, 6, …” on a worksheet uses ellipsis for “keep going.”</li>
</ul>
<h2 id="question-bank" class="book-prose__heading">Question bank</h2>
<h3 id="questions-easy" class="book-prose__heading">Easy</h3>
<details class="book-question">
<summary class="book-question__prompt">What is the plus sign?</summary>
<div class="book-question__solution">
<p><strong>Answer:</strong> A cross-shaped mark <strong>+</strong> that means add or put together.</p>
</div>
</details>
<details class="book-question">
<summary class="book-question__prompt">How do you say \(5 = 5\) out loud?</summary>
<div class="book-question__solution">
<p><strong>Answer:</strong> “Five equals five.”</p>
</div>
</details>
<details class="book-question">
<summary class="book-question__prompt">Which symbol means “take away”?</summary>
<div class="book-question__solution">
<p><strong>Answer:</strong> Minus: <strong>−</strong> (or <strong>-</strong> on a keyboard).</p>
</div>
</details>
<h3 id="questions-intermediate" class="book-prose__heading">Intermediate</h3>
<details class="book-question">
<summary class="book-question__prompt">Say \(9 \div 3 = 3\) in words.</summary>
<div class="book-question__solution">
<p><strong>Answer:</strong> “Nine divided by three equals three.”</p>
</div>
</details>
<details class="book-question">
<summary class="book-question__prompt">What does \(\times\) mean in \(4 \times 2\)?</summary>
<div class="book-question__solution">
<p><strong>Answer:</strong> Multiply — four groups of two (or two groups of four), total eight.</p>
</div>
</details>
<details class="book-question">
<summary class="book-question__prompt">Compute \((3 + 2) + 4\).</summary>
<div class="book-question__solution">
<p><strong>Answer:</strong> \(9\).</p>
<p><strong>Steps:</strong> Inside brackets: \(3 + 2 = 5\). Then \(5 + 4 = 9\).</p>
</div>
</details>
<details class="book-question">
<summary class="book-question__prompt">Is \(4 &lt; 9\) true or false?</summary>
<div class="book-question__solution">
<p><strong>Answer:</strong> True — four is less than nine.</p>
</div>
</details>
<h3 id="questions-difficult" class="book-prose__heading">Difficult</h3>
<details class="book-question">
<summary class="book-question__prompt">Which is bigger: \(7\) or \(5\)? Write a true comparison using \(>\) or \(&lt;\).</summary>
<div class="book-question__solution">
<p><strong>Answer:</strong> Seven is bigger. Examples: \(7 &gt; 5\) or \(5 &lt; 7\).</p>
</div>
</details>
<details class="book-question">
<summary class="book-question__prompt">What does \(1/2\) mean in words?</summary>
<div class="book-question__solution">
<p><strong>Answer:</strong> One half — one part out of two equal parts.</p>
</div>
</details>
<details class="book-question">
<summary class="book-question__prompt">A shirt costs \(20\) dollars. The tag says \(25\%\) off. What does \(25\%\) mean?</summary>
<div class="book-question__solution">
<p><strong>Answer:</strong> Twenty-five out of every hundred — a quarter of the price is taken off (you will learn to compute the exact dollars later).</p>
</div>
</details>
<h3 id="questions-hardcore" class="book-prose__heading">Hardcore</h3>
<details class="book-question">
<summary class="book-question__prompt">Write \(2 \times 3 + 1\) in words two ways: (a) if you multiply first, (b) if you add inside brackets as \((2 \times 3) + 1\) only — are the totals the same as \(2 \times (3 + 1)\)?</summary>
<div class="book-question__solution">
<p><strong>Answer:</strong> \((2 \times 3) + 1 = 7\). \(2 \times (3 + 1) = 2 \times 4 = 8\). They are <strong>not</strong> the same — brackets change the result.</p>
<p><strong>Explanation:</strong> Brackets show which group to do first; this is why notation matters.</p>
</div>
</details>
<details class="book-question">
<summary class="book-question__prompt">Name two other symbols that can mean “multiply” besides \(\times\).</summary>
<div class="book-question__solution">
<p><strong>Answer:</strong> Dot \(\cdot\) and asterisk \(*\) (on keyboards/calculators).</p>
</div>
</details>
<details class="book-question">
<summary class="book-question__prompt">Read \(1{,}000 + 500 = 1{,}500\) aloud. What job does the comma do in \(1{,}000\)?</summary>
<div class="book-question__solution">
<p><strong>Answer:</strong> “One thousand plus five hundred equals one thousand five hundred.” The comma groups digits in threes to show thousands.</p>
</div>
</details>
<h2 id="references" class="book-prose__heading">References</h2>
<ul>
<li>Recorde, Robert — early use of the equals sign in <em>The Whetstone of Witte</em> (1557).</li>
<li>Cajori, Florian — <em>A History of Mathematical Notations</em> (origins of +, −, ×, ÷).</li>
<li>National Council of Teachers of Mathematics — developing operation sense and notation in elementary grades.</li>
</ul>
</div>

171
content/math/basic-math-notation/supplement.html Normal file
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@ -0,0 +1,171 @@
<div class="book-prose">
<h2 id="derivation" class="book-prose__heading">Derivation</h2>
<h3 id="why-symbols" class="book-prose__heading">Why symbols instead of only words?</h3>
<p>
Writing “three plus four equals seven” every time is long. Symbols are a <strong>shorthand</strong>:
the same meaning in less space, so you can compare many sentences quickly.
</p>
<h3 id="equals-meaning" class="book-prose__heading">What equals really says</h3>
<p>
If the left side of \(=\) counts to \(a\) and the right side counts to \(b\), then \(a = b\)
means both sides name the <strong>same number</strong>. If they were different, the sentence would
be false — like claiming \(3 + 4 = 8\) when seven dots are the true total.
</p>
<h3 id="brackets-order" class="book-prose__heading">Why brackets change the order</h3>
<p>
Without a rule, \(2 + 3 \times 4\) could be read in two ways. Brackets fix one group:
\((2 + 3) \times 4 = 5 \times 4 = 20\). Later you will learn more order rules; for now,
brackets always mean “work inside first.”
</p>
<h2 id="checkpoints" class="book-prose__heading">Checkpoints</h2>
<p>Pause and answer in your head or with a partner.</p>
<div class="book-callout" role="note">
<p class="book-callout__label">Checkpoint</p>
<p>What is the name of the sign in \(7 - 3 = 4\)? What does it tell you to do?</p>
</div>
<div class="book-callout" role="note">
<p class="book-callout__label">Checkpoint</p>
<p>Say out loud: \(4 \times 2 = 8\). Which word goes with \(\times\)?</p>
</div>
<div class="book-callout" role="note">
<p class="book-callout__label">Checkpoint</p>
<p>In \((5 + 1) + 2\), which part do you work out first? What is the final total?</p>
</div>
<div class="book-callout" role="note">
<p class="book-callout__label">Checkpoint</p>
<p>Is \(6 &gt; 2\) true? What does the open side of \(>\) point toward?</p>
</div>
<div class="book-callout" role="note">
<p class="book-callout__label">Checkpoint</p>
<p>What does \(50\%\) mean in words (“out of …”)?</p>
</div>
<h2 id="applications" class="book-prose__heading">Real-life applications</h2>
<ul>
<li><strong>Recipes</strong> — “add 2 cups flour + 1 cup sugar” is a plus sentence in the kitchen.</li>
<li><strong>Scores</strong> — “Team A: 12, Team B: 8” uses comparison; 12 &gt; 8 means A scored more.</li>
<li><strong>Sharing</strong> — “12 sweets ÷ 4 children” is fair division at a party.</li>
<li><strong>Shopping</strong> — prices like $3.50 use a decimal point between dollars and cents.</li>
<li><strong>Sales</strong> — “25% off” uses percent out of one hundred.</li>
<li><strong>Patterns</strong> — “2, 4, 6, …” on a worksheet uses ellipsis for “keep going.”</li>
</ul>
<h2 id="question-bank" class="book-prose__heading">Question bank</h2>
<h3 id="questions-easy" class="book-prose__heading">Easy</h3>
<details class="book-question">
<summary class="book-question__prompt">What is the plus sign?</summary>
<div class="book-question__solution">
<p><strong>Answer:</strong> A cross-shaped mark <strong>+</strong> that means add or put together.</p>
</div>
</details>
<details class="book-question">
<summary class="book-question__prompt">How do you say \(5 = 5\) out loud?</summary>
<div class="book-question__solution">
<p><strong>Answer:</strong> “Five equals five.”</p>
</div>
</details>
<details class="book-question">
<summary class="book-question__prompt">Which symbol means “take away”?</summary>
<div class="book-question__solution">
<p><strong>Answer:</strong> Minus: <strong>−</strong> (or <strong>-</strong> on a keyboard).</p>
</div>
</details>
<h3 id="questions-intermediate" class="book-prose__heading">Intermediate</h3>
<details class="book-question">
<summary class="book-question__prompt">Say \(9 \div 3 = 3\) in words.</summary>
<div class="book-question__solution">
<p><strong>Answer:</strong> “Nine divided by three equals three.”</p>
</div>
</details>
<details class="book-question">
<summary class="book-question__prompt">What does \(\times\) mean in \(4 \times 2\)?</summary>
<div class="book-question__solution">
<p><strong>Answer:</strong> Multiply — four groups of two (or two groups of four), total eight.</p>
</div>
</details>
<details class="book-question">
<summary class="book-question__prompt">Compute \((3 + 2) + 4\).</summary>
<div class="book-question__solution">
<p><strong>Answer:</strong> \(9\).</p>
<p><strong>Steps:</strong> Inside brackets: \(3 + 2 = 5\). Then \(5 + 4 = 9\).</p>
</div>
</details>
<details class="book-question">
<summary class="book-question__prompt">Is \(4 &lt; 9\) true or false?</summary>
<div class="book-question__solution">
<p><strong>Answer:</strong> True — four is less than nine.</p>
</div>
</details>
<h3 id="questions-difficult" class="book-prose__heading">Difficult</h3>
<details class="book-question">
<summary class="book-question__prompt">Which is bigger: \(7\) or \(5\)? Write a true comparison using \(>\) or \(&lt;\).</summary>
<div class="book-question__solution">
<p><strong>Answer:</strong> Seven is bigger. Examples: \(7 &gt; 5\) or \(5 &lt; 7\).</p>
</div>
</details>
<details class="book-question">
<summary class="book-question__prompt">What does \(1/2\) mean in words?</summary>
<div class="book-question__solution">
<p><strong>Answer:</strong> One half — one part out of two equal parts.</p>
</div>
</details>
<details class="book-question">
<summary class="book-question__prompt">A shirt costs \(20\) dollars. The tag says \(25\%\) off. What does \(25\%\) mean?</summary>
<div class="book-question__solution">
<p><strong>Answer:</strong> Twenty-five out of every hundred — a quarter of the price is taken off (you will learn to compute the exact dollars later).</p>
</div>
</details>
<h3 id="questions-hardcore" class="book-prose__heading">Hardcore</h3>
<details class="book-question">
<summary class="book-question__prompt">Write \(2 \times 3 + 1\) in words two ways: (a) if you multiply first, (b) if you add inside brackets as \((2 \times 3) + 1\) only — are the totals the same as \(2 \times (3 + 1)\)?</summary>
<div class="book-question__solution">
<p><strong>Answer:</strong> \((2 \times 3) + 1 = 7\). \(2 \times (3 + 1) = 2 \times 4 = 8\). They are <strong>not</strong> the same — brackets change the result.</p>
<p><strong>Explanation:</strong> Brackets show which group to do first; this is why notation matters.</p>
</div>
</details>
<details class="book-question">
<summary class="book-question__prompt">Name two other symbols that can mean “multiply” besides \(\times\).</summary>
<div class="book-question__solution">
<p><strong>Answer:</strong> Dot \(\cdot\) and asterisk \(*\) (on keyboards/calculators).</p>
</div>
</details>
<details class="book-question">
<summary class="book-question__prompt">Read \(1{,}000 + 500 = 1{,}500\) aloud. What job does the comma do in \(1{,}000\)?</summary>
<div class="book-question__solution">
<p><strong>Answer:</strong> “One thousand plus five hundred equals one thousand five hundred.” The comma groups digits in threes to show thousands.</p>
</div>
</details>
<h2 id="references" class="book-prose__heading">References</h2>
<ul>
<li>Recorde, Robert — early use of the equals sign in <em>The Whetstone of Witte</em> (1557).</li>
<li>Cajori, Florian — <em>A History of Mathematical Notations</em> (origins of +, −, ×, ÷).</li>
<li>National Council of Teachers of Mathematics — developing operation sense and notation in elementary grades.</li>
</ul>
</div>