🍕 分数运算：换成同一种小块🍕 Fraction Arithmetic: The Same-Size Pieces
上一片里，小数点对齐才能相加减。分披萨也有同一条规矩：1/3 的大块和 1/4 的小块，块头不一样，没法直接数。这一片就教你把它们换成同一种小块——通分。Last leaf, the decimal points had to line up before we could add. Pizza follows the same rule: a 1/3 slice and a 1/4 slice are different sizes, so you can't count them together. This leaf shows how to re-cut them into the same-size pieces — the common denominator.
第 1 步：分数就是“切几块、取几块”Step 1: A fraction is “cut how many, take how many”
先弄明白：1/3 和 1/4，到底哪个大？First, settle this: which is bigger, 1/3 or 1/4?
一个披萨切成 3 份，每份是 1/3；切成 4 份，每份是 1/4。切的份数越多，每块越小——所以 1/3 反而比 1/4 大，尽管 3 比 4 小。分母是“切成几块”，分子是“取了几块”：这就是分数的全部秘密。Cut a pizza into 3 and each piece is 1/3; cut it into 4 and each is 1/4. The more pieces you cut, the smaller each one — so 1/3 is actually bigger than 1/4, even though 3 is smaller than 4. The denominator says how many pieces you cut, the numerator says how many you take: that's the whole secret of fractions.
块头不一样，就不能直接相加——那怎么办？Different sizes can't be added directly — so what do we do?
第 2 步：加减——都换成同一种小块Step 2: To add — re-cut into the same-size pieces
2/3 + 1/4：把披萨都切成 12 小块——每个 1/3 能切成 4 小块，所以 2/3 就是 8 小块；每个 1/4 切成 3 小块，所以 1/4 就是 3 小块。现在块头一样了，数一数：8/12 + 3/12 = 11/12。通分，就是换个更细的切法，让两种块变成同一种。2/3 + 1/4: cut everything into 12ths — each 1/3 becomes 4 small pieces, so 2/3 is 8 pieces; each 1/4 becomes 3 pieces. Now all pieces match, so just count: 8/12 + 3/12 = 11/12. Finding a common denominator just means re-cutting until both fractions use the same-size piece.
加减讲完了，乘除各有自己的画面。Addition done — multiplying and dividing each have their own picture.
第 3 步：乘法取“一部分的一部分”，除法数“能装几份”Step 3: Multiply as “a part of a part”, divide as “how many fit”
乘法是“取一部分的一部分”：2/3 × 3/4，先取披萨的 2/3，再从中取它的 3/4，合起来是整个披萨的 6/12（也就是 1/2）。除法是“能装几份”：1/2 ÷ 1/4，半张披萨正好装下 2 块 1/4。做双倍食谱也一样：3/4 杯面粉 × 2 = 3/2 杯。Multiplication is “a part of a part”: 2/3 × 3/4 means take 2/3 of the pizza, then 3/4 of that — together 6/12 of the whole (that's 1/2). Division asks “how many helpings fit”: 1/2 ÷ 1/4 — half a pizza fits exactly two 1/4 slices. Doubling a recipe works the same way: 3/4 cup of flour × 2 = 3/2 cups.
🎮 你来切披萨（1 分钟）🎮 Your turn: slice the pizza (1 minute)
道理讲完了。拖动两个滑块，把每题的两个分数都换成同样大的小块——绿色正好盖住红色轮廓就对了。Theory done. Drag the two sliders to re-cut both fractions into the same-size pieces — when the green exactly covers the red outline, you've got it.
一句话记住它：分数运算就是让块头统一——加减先通分，乘法是“一部分的一部分”，除法数“能装几份”。Remember it in one line: fraction arithmetic is about matching the piece size — re-cut before adding, multiply as “a part of a part”, divide as “how many fit”.
分母是切成几块，分子是取几块；份数越多，每块越小（1/3 > 1/4）Denominator = pieces cut, numerator = pieces taken; more slices means smaller pieces (1/3 > 1/4)加减：先通分，都换成同一种小块，再数块数Add/subtract: find a common denominator, re-cut into same-size pieces, then count乘法：分子乘分子、分母乘分母；除法：看 1/2 里能装几个 1/4Multiply top times top, bottom times bottom; divide by asking how many 1/4 fit in 1/2
内容参考 OpenStax 等公开教材，多来源核对 · AI 生成、人工审核 · 发现错误欢迎指正，帮这片叶子长得更好。Based on OpenStax and other open textbooks, cross-checked across sources · AI-generated, human-reviewed · Spotted a mistake? Tell us — help this leaf grow.
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