算数に強くなる!水道方式【水道方式とは何か】

算数を理解するための最善の方法

×3位数のかけ算(型分け)

×3位数のかけ算は、百位の乗数を、部分積を百位から記述する手続きが理解できれば、これまでと同様のアルゴリズムで計算することができます。
部分積の和では、3段のたて書きの加法を行います。くり上がった場合は4段の加法となり、ここで誤答が生じることも多くなります。
生徒が誤った答えを出したときには、計算のどの段階で間違えたのかを発見させ、まちがい傾向の分析を促すような指導も必要です。

3位数×3位数の型分け(★は積が4位数の問題を含む)

[1]★
 \setstretch{0.7} \begin{array}{rr} & \square\square\square  \\ \times &  \square\square\square \\ \hline & \square\square\square  \\ & \square\square\square\hspace*{0.8em}  \\ & \square\square\square\hspace*{1.6em}  \\ \hline & \square\square\square\square\square   \\ \end{array} \begin{array}{rr} & 231  \\ \times &  223 \\ \hline \end{array} \begin{array}{rr} & 112   \\ \times &  431 \\ \hline \end{array} \begin{array}{rr} & 324   \\ \times &  212 \\ \hline \end{array}
[2]★
 \setstretch{0.7} \begin{array}{rr} & \square\square\square  \\ \times &  \square\square\square \\ \hline & \square\square\square\square  \\ & \square\square\square\hspace*{0.8em}  \\ & \square\square\square\hspace*{1.6em}  \\ \hline & \square\square\square\square\square   \\ \end{array} \begin{array}{rr} & 321  \\ \times &  114 \\ \hline \end{array} \begin{array}{rr} & 429   \\ \times &  215 \\ \hline \end{array} \begin{array}{rr} & 714   \\ \times &  118 \\ \hline \end{array}
[3]★
 \setstretch{0.7} \begin{array}{rr} & \square\square\square  \\ \times &  \square\square\square \\ \hline & \square\square\square  \\ & \square\square\square\square\hspace*{0.8em}  \\ & \square\square\square\hspace*{1.6em}  \\ \hline & \square\square\square\square\square   \\ \end{array} \begin{array}{rr} & 321  \\ \times &  243 \\ \hline \end{array} \begin{array}{rr} & 602   \\ \times &  171 \\ \hline \end{array} \begin{array}{rr} & 342   \\ \times &  182 \\ \hline \end{array}
[4]
 \setstretch{0.7} \begin{array}{rr} & \square\square\square  \\ \times &  \square\square\square \\ \hline & \square\square\square  \\ & \square\square\square\hspace*{0.8em}  \\ & \square\square\square\square\hspace*{1.6em}  \\ \hline & \square\square\square\square\square\square   \\ \end{array} \begin{array}{rr} & 213  \\ \times &  634 \\ \hline \end{array} \begin{array}{rr} & 422   \\ \times &  721 \\ \hline \end{array} \begin{array}{rr} & 423   \\ \times &  922 \\ \hline \end{array}
[5]★
 \setstretch{0.7} \begin{array}{rr} & \square\square\square  \\ \times &  \square\square\square \\ \hline & \square\square\square\square \\ & \square\square\square\square\hspace*{0.8em}  \\ & \square\square\square\hspace*{1.6em}  \\ \hline & \square\square\square\square\square   \\ \end{array} \begin{array}{rr} & 932  \\ \times &  123 \\ \hline \end{array} \begin{array}{rr} & 486   \\ \times &  254 \\ \hline \end{array} \begin{array}{rr} & 276   \\ \times &  385 \\ \hline \end{array}
[6]
 \setstretch{0.7} \begin{array}{rr} & \square\square\square  \\ \times &  \square\square\square \\ \hline & \square\square\square \\ & \square\square\square\square\hspace*{0.8em}  \\ & \square\square\square\square\hspace*{1.6em}  \\ \hline & \square\square\square\square\square\square   \\ \end{array} \begin{array}{rr} & 411  \\ \times &  352 \\ \hline \end{array} \begin{array}{rr} & 986   \\ \times &  231 \\ \hline \end{array} \begin{array}{rr} & 763   \\ \times &  821 \\ \hline \end{array}
[7]
 \setstretch{0.7} \begin{array}{rr} & \square\square\square  \\ \times &  \square\square\square \\ \hline & \square\square\square\square \\ & \square\square\square\hspace*{0.8em}  \\ & \square\square\square\square\hspace*{1.6em}  \\ \hline & \square\square\square\square\square\square   \\ \end{array} \begin{array}{rr} & 463  \\ \times &  324 \\ \hline \end{array} \begin{array}{rr} & 564   \\ \times &  213 \\ \hline \end{array} \begin{array}{rr} & 513   \\ \times &  314 \\ \hline \end{array}
[8]
 \setstretch{0.7} \begin{array}{rr} & \square\square\square  \\ \times &  \square\square\square \\ \hline & \square\square\square\square \\ & \square\square\square\square\hspace*{0.8em}  \\ & \square\square\square\square\hspace*{1.6em}  \\ \hline & \square\square\square\square\square\square   \\ \end{array} \begin{array}{rr} & 911  \\ \times &  423 \\ \hline \end{array} \begin{array}{rr} & 624  \\ \times &  536 \\ \hline \end{array} \begin{array}{rr} & 367   \\ \times &  498 \\ \hline \end{array}
[9]★
 \setstretch{0.7} \begin{array}{rr} & \square\square\square  \\ \times &  \square\square\hspace*{0.13em}0\hspace*{0.13em} \\ \hline & \hspace*{0.14em}0\hspace*{0.15em}\hspace*{0.14em}0\hspace*{0.15em}\hspace*{0.14em}0\hspace*{0.15em}  \\ & \square\square\square\hspace*{0.8em}  \\ & \square\square\square\hspace*{1.6em}  \\ \hline & \square\square\square\square\hspace*{0.14em}0\hspace*{0.15em}    \\ \end{array}

 \setstretch{0.7} \begin{array}{rr} & \square\square\square  \\ \times &  \square\square\hspace*{0.13em}0\hspace*{0.13em} \\ \hline & \square\square\square\hspace*{0.14em}0\hspace*{0.15em}  \\ & \square\square\square\hspace*{0.8em}  \\ \hline & \square\square\square\square\hspace*{0.14em}0\hspace*{0.15em}    \\ \end{array}
\begin{array}{rr} & 236  \\ \times &  410 \\ \hline \end{array} \begin{array}{rr} & 183  \\ \times &  240 \\ \hline \end{array} \begin{array}{rr} & 223   \\ \times &  420 \\ \hline \end{array}
[10]★
 \setstretch{0.7} \begin{array}{rr} & \square\square\square  \\ \times &  \square\hspace*{0.13em}0\hspace*{0.13em}\square \\ \hline & \square\square\square  \\ & \hspace*{0.14em}0\hspace*{0.15em}\hspace*{0.14em}0\hspace*{0.15em}\hspace*{0.14em}0\hspace*{0.15em}\hspace*{0.8em}  \\ & \square\square\square\hspace*{1.6em}  \\ \hline & \square\square\square\square\square  \\ \end{array}

 \setstretch{0.7} \begin{array}{rr} & \square\square\square  \\ \times &  \square\hspace*{0.13em}0\hspace*{0.13em}\square \\ \hline & \square\square\square  \\ & \square\square\square\hspace*{1.6em}  \\ \hline & \square\square\square\square\square  \\ \end{array}
\begin{array}{rr} & 234  \\ \times &  301 \\ \hline \end{array} \begin{array}{rr} & 316  \\ \times &  402 \\ \hline \end{array} \begin{array}{rr} & 648   \\ \times &  302 \\ \hline \end{array}
[11]★
 \setstretch{0.7} \begin{array}{rr} & \square\square\square  \\ \times &  \square\hspace*{0.15em}0\hspace*{0.15em}\hspace*{0.15em}0\hspace*{0.15em} \\ \hline & \hspace*{0.14em}0\hspace*{0.15em}\hspace*{0.14em}0\hspace*{0.15em}\hspace*{0.14em}0\hspace*{0.15em}  \\ & \hspace*{0.14em}0\hspace*{0.15em}\hspace*{0.14em}0\hspace*{0.15em}\hspace*{0.14em}0\hspace*{0.15em}\hspace*{0.8em}  \\ & \square\square\square\hspace*{1.6em}  \\ \hline & \square\square\square\hspace*{0.14em}0\hspace*{0.15em}\hspace*{0.14em}0\hspace*{0.15em}  \\ \end{array}

 \setstretch{0.7} \begin{array}{rr} & \square\square\square  \\ \times &  \square\hspace*{0.15em}0\hspace*{0.15em}\hspace*{0.15em}0\hspace*{0.15em} \\ \hline & \square\square\square\hspace*{0.14em}0\hspace*{0.15em}\hspace*{0.14em}0\hspace*{0.15em}  \\ \end{array}
\begin{array}{rr} & 241  \\ \times &  300 \\ \hline \end{array} \begin{array}{rr} & 612  \\ \times &  400 \\ \hline \end{array} \begin{array}{rr} & 977   \\ \times &  600 \\ \hline \end{array}


2位数×3位数の型分け

[12]
 \setstretch{0.7} \begin{array}{rr} & \square\square  \\ \times &  \square\square\square \\ \hline & \square\square  \\ & \square\square\hspace*{0.8em}  \\ & \square\square\hspace*{1.6em}  \\ \hline & \square\square\square\square   \\ \end{array} \begin{array}{rr} & 47  \\ \times &  222 \\ \hline \end{array} \begin{array}{rr} & 41   \\ \times &  232 \\ \hline \end{array} \begin{array}{rr} & 99   \\ \times &  111 \\ \hline \end{array}
[13]
 \setstretch{0.7} \begin{array}{rr} & \square\square  \\ \times &  \square\square\square \\ \hline & \square\square\square  \\ & \square\square\square\hspace*{0.8em}  \\ & \square\square\square\hspace*{1.6em}  \\ \hline & \square\square\square\square\square   \\ \end{array} \begin{array}{rr} & 86  \\ \times &  347 \\ \hline \end{array} \begin{array}{rr} & 63   \\ \times &  867 \\ \hline \end{array} \begin{array}{rr} & 73   \\ \times &  234 \\ \hline \end{array}
[14]
 \setstretch{0.7} \begin{array}{rr} & \square\square  \\ \times & \square\square\hspace*{0.15em}0\hspace*{0.15em}  \\ \hline & \hspace*{1.5em}0\hspace*{0.15em}\hspace*{0.15em}0\hspace*{0.15em}  \\ & \square\square\hspace*{0.8em}  \\ & \square\square\hspace*{1.6em}  \\ \hline & \square\square\square\hspace*{0.17em}0\hspace*{0.15em}  \\ \end{array} \begin{array}{rr} & 24  \\ \times &  340 \\ \hline \end{array} \begin{array}{rr} & 92   \\ \times &  570 \\ \hline \end{array} \begin{array}{rr} & 36   \\ \times &  220 \\ \hline \end{array}
[15]
 \setstretch{0.7} \begin{array}{rr} & \square\square  \\ \times & \square\hspace*{0.15em}0\hspace*{0.15em}\square  \\ \hline & \square\square  \\ & \hspace*{1.5em}0\hspace*{0.15em}\hspace*{0.15em}0\hspace*{0.15em}\hspace*{0.8em}  \\ & \square\square\hspace*{1.6em}  \\ \hline & \square\square\square\square \\ \end{array} \begin{array}{rr} & 24  \\ \times &  302 \\ \hline \end{array} \begin{array}{rr} & 75   \\ \times &  801 \\ \hline \end{array} \begin{array}{rr} & 36   \\ \times &  409 \\ \hline \end{array}
[16]
 \setstretch{0.7} \begin{array}{rr} & \square\square  \\ \times & \square\hspace*{0.15em}0\hspace*{0.15em}\hspace*{0.15em}0\hspace*{0.15em}  \\ \hline & \hspace*{0.15em}0\hspace*{0.15em}\hspace*{0.15em}0\hspace*{0.15em}  \\ & \hspace*{0.15em}0\hspace*{0.15em}\hspace*{0.15em}0\hspace*{0.15em}\hspace*{0.8em}  \\ & \square\square\hspace*{1.6em}  \\ \hline & \square\square\hspace*{0.15em}0\hspace*{0.15em}\hspace*{0.15em}0\hspace*{0.15em} \\ \end{array} \begin{array}{rr} & 76  \\ \times &  800 \\ \hline \end{array} \begin{array}{rr} & 94   \\ \times &  500 \\ \hline \end{array} \begin{array}{rr} & 84   \\ \times &  100 \\ \hline \end{array}
[17]
 \setstretch{0.7} \begin{array}{rr} & \square  \\ \times & \square\square\square  \\ \hline & \square\square  \\ & \square\square\hspace*{0.8em}  \\ & \square\square\hspace*{1.6em}  \\ \hline & \square\square\square\square \\ \end{array}

 \setstretch{0.7} \begin{array}{rr} & \square  \\ \times & \square\square\square  \\ \hline & \square\square\square\square  \\ \end{array}
\begin{array}{rr} & 8  \\ \times &  324 \\ \hline \end{array} \begin{array}{rr} & 6   \\ \times &  278 \\ \hline \end{array} \begin{array}{rr} & 84   \\ \times &  100 \\ \hline \end{array}
[18]
 \setstretch{0.7} \begin{array}{rr} & \square  \\ \times & \square\square\square  \\ \hline & \square  \\ & \square\hspace*{0.8em}  \\ & \square\square\hspace*{1.6em}  \\ \hline & \square\square\square\square \\ \end{array}

 \setstretch{0.7} \begin{array}{rr} & \square  \\ \times & \square\square\square  \\ \hline & \square\square\square\square  \\ \end{array}
\begin{array}{rr} & 7  \\ \times &  311 \\ \hline \end{array} \begin{array}{rr} & 6   \\ \times &  811 \\ \hline \end{array} \begin{array}{rr} & 4   \\ \times &  422 \\ \hline \end{array}
[19]
 \setstretch{0.7} \begin{array}{rr} & \square  \\ \times & \square\square\square  \\ \hline & \square  \\ & \square\hspace*{0.8em}  \\ & \square\hspace*{1.6em}  \\ \hline & \square\square\square \\ \end{array} \begin{array}{rr} & 2  \\ \times &  213 \\ \hline \end{array} \begin{array}{rr} & 4   \\ \times &  211 \\ \hline \end{array} \begin{array}{rr} & 3   \\ \times &  321 \\ \hline \end{array}
[20]
 \setstretch{0.7} \begin{array}{rr} & \square  \\ \times & \square\square\hspace*{0.15em}0\hspace*{0.15em}  \\ \hline & \hspace*{0.15em}0\hspace*{0.15em}  \\ & \square\square\hspace*{0.8em}  \\ & \square\square\hspace*{1.6em}  \\ \hline & \square\square\square\hspace*{0.15em}0\hspace*{0.15em} \\ \end{array}

 \setstretch{0.7} \begin{array}{rr} & \square  \\ \times & \square\square\hspace*{0.15em}0\hspace*{0.15em}  \\ \hline & \square\square\hspace*{0.15em}0\hspace*{0.15em}  \\ & \square\square\hspace*{1.6em}  \\ \hline & \square\square\square\hspace*{0.15em}0\hspace*{0.15em} \\ \end{array}
\begin{array}{rr} & 8  \\ \times &  760 \\ \hline \end{array} \begin{array}{rr} & 9   \\ \times &  240 \\ \hline \end{array} \begin{array}{rr} & 4   \\ \times &  670 \\ \hline \end{array}
[21]
 \setstretch{0.7} \begin{array}{rr} & \square  \\ \times & \square\square\hspace*{0.15em}0\hspace*{0.15em}  \\ \hline & \hspace*{0.15em}0\hspace*{0.15em}  \\ & \square\hspace*{0.8em}  \\ & \square\hspace*{1.6em}  \\ \hline & \square\square\hspace*{0.15em}0\hspace*{0.15em} \\ \end{array}

 \setstretch{0.7} \begin{array}{rr} & \square  \\ \times & \square\square\hspace*{0.15em}0\hspace*{0.15em}  \\ \hline & \square\square\hspace*{0.15em}0\hspace*{0.15em}  \\ \end{array}
\begin{array}{rr} & 3  \\ \times &  320 \\ \hline \end{array} \begin{array}{rr} & 9   \\ \times &  110 \\ \hline \end{array} \begin{array}{rr} & 4   \\ \times &  210 \\ \hline \end{array}
[22]
 \setstretch{0.7} \begin{array}{rr} & \square  \\ \times & \square\hspace*{0.15em}0\hspace*{0.15em}\square  \\ \hline & \square\square  \\ & \hspace*{0.15em}0\hspace*{0.15em}\hspace*{0.8em}  \\ & \square\square\hspace*{1.6em}  \\ \hline & \square\square\square\square \\ \end{array}

 \setstretch{0.7} \begin{array}{rr} & \square  \\ \times & \square\hspace*{0.15em}0\hspace*{0.15em}\square  \\ \hline & \square\square\square\square  \\ \end{array}
\begin{array}{rr} & 7  \\ \times &  806 \\ \hline \end{array} \begin{array}{rr} & 6   \\ \times &  903 \\ \hline \end{array} \begin{array}{rr} & 9   \\ \times &  305 \\ \hline \end{array}
[23]
 \setstretch{0.7} \begin{array}{rr} & \square  \\ \times & \square\hspace*{0.15em}0\hspace*{0.15em}\square  \\ \hline & \square\square  \\ & \hspace*{0.15em}0\hspace*{0.15em}\hspace*{0.8em}  \\ & \square\hspace*{1.6em}  \\ \hline & \square\square\square \\ \end{array}

 \setstretch{0.7} \begin{array}{rr} & \square  \\ \times & \square\hspace*{0.15em}0\hspace*{0.15em}\square  \\ \hline & \square\square\square  \\ \end{array}
\begin{array}{rr} & 2  \\ \times &  306 \\ \hline \end{array} \begin{array}{rr} & 8   \\ \times &  109 \\ \hline \end{array} \begin{array}{rr} & 4   \\ \times &  104 \\ \hline \end{array}
[24]
 \setstretch{0.7} \begin{array}{rr} & \square  \\ \times & \square\hspace*{0.15em}0\hspace*{0.15em}\square  \\ \hline & \square  \\ & \hspace*{0.15em}0\hspace*{0.15em}\hspace*{0.8em}  \\ & \square\hspace*{1.6em}  \\ \hline & \square\hspace*{0.15em}0\hspace*{0.15em}\square \\ \end{array}

 \setstretch{0.7} \begin{array}{rr} & \square  \\ \times & \square\hspace*{0.15em}0\hspace*{0.15em}\square  \\ \hline & \square\hspace*{0.15em}0\hspace*{0.15em}\square  \\ \end{array}
\begin{array}{rr} & 2  \\ \times &  403 \\ \hline \end{array} \begin{array}{rr} & 8   \\ \times &  101 \\ \hline \end{array} \begin{array}{rr} & 3   \\ \times &  201 \\ \hline \end{array}

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