學海：數學 Learn: Mathematics - 第 8 課：加法 - 接住數 | Addition as Counting On

一、不一定要由 1 數起31Please respect copyright.ＰＥＮＡＮＡPTGLJNpFik

上一課我們學了加法：31Please respect copyright.ＰＥＮＡＮＡwYOu3Pioo8
> 2 + 3 = 531Please respect copyright.ＰＥＮＡＮＡ0vIrl2QuCR

可以理解成：31Please respect copyright.ＰＥＮＡＮＡy05AvGn3SA
> 兩堆物件合在一起，變成總數。31Please respect copyright.ＰＥＮＡＮＡVIuJMI9ZYn

但如果數量開始變大，每次都由 1 數起會很慢。

例如：31Please respect copyright.ＰＥＮＡＮＡCc5ZtYKE0D
> 你已經知道盒入面有 8 粒糖果。31Please respect copyright.ＰＥＮＡＮＡxvHuqaomyf
> 朋友再給你 3 粒糖果。31Please respect copyright.ＰＥＮＡＮＡJ1fZRgdJR8

你不需要重新數：31Please respect copyright.ＰＥＮＡＮＡx9axSHFuhL
> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1131Please respect copyright.ＰＥＮＡＮＡC57QkDvAHP

你可以由已知的 8 開始，接住數新加的 3 粒：31Please respect copyright.ＰＥＮＡＮＡjUbqIEWk1y
> 9, 10, 1131Please respect copyright.ＰＥＮＡＮＡhuW2kyTTaL

所以：31Please respect copyright.ＰＥＮＡＮＡ5jSm6PoaA3
> 8 + 3 = 1131Please respect copyright.ＰＥＮＡＮＡgwkg7Anhpp

這個方法叫：31Please respect copyright.ＰＥＮＡＮＡfz8l5hGSVb
> 接住數 | Counting On31Please respect copyright.ＰＥＮＡＮＡGubfWxQynm

二、接住數是甚麼？31Please respect copyright.ＰＥＮＡＮＡtWeKF1M7zZ

接住數的意思是：31Please respect copyright.ＰＥＮＡＮＡbn2l9uSUzq
> 已經知道一部分有幾多，就由那個數開始，向前數另一部分。31Please respect copyright.ＰＥＮＡＮＡWAYm7It5oR

例如：31Please respect copyright.ＰＥＮＡＮＡqCj29iCUrv
> 6 + 231Please respect copyright.ＰＥＮＡＮＡqFctz4Lpax

你可以由 6 開始，向前行 2 步：31Please respect copyright.ＰＥＮＡＮＡqpLgyAxnff
> 第 1 步：6 → 731Please respect copyright.ＰＥＮＡＮＡ8m2oVpfVDL
> 第 2 步：7 → 831Please respect copyright.ＰＥＮＡＮＡoxCj5zvS5G

所以：31Please respect copyright.ＰＥＮＡＮＡLyl9CzDNG5
> 6 + 2 = 831Please respect copyright.ＰＥＮＡＮＡgbz2OdYbZL

重點是：31Please respect copyright.ＰＥＮＡＮＡRFCmFjN7RD
> 6 是起點，不是第一步。31Please respect copyright.ＰＥＮＡＮＡJGhusxkaUU
> 加 2，意思是行 2 步。31Please respect copyright.ＰＥＮＡＮＡZUVHz0gMqU

三、接住數和數線31Please respect copyright.ＰＥＮＡＮＡSSNcxTgmfW

用數線看：31Please respect copyright.ＰＥＮＡＮＡ1yYGtezRVK
> 0 —— 1 —— 2 —— 3 —— 4 —— 5 —— 6 —— 7 —— 831Please respect copyright.ＰＥＮＡＮＡ8ZV99kw5Ip

做：31Please respect copyright.ＰＥＮＡＮＡRn4QeFBWwJ
> 5 + 331Please respect copyright.ＰＥＮＡＮＡxhDGwkuWWm

即是：31Please respect copyright.ＰＥＮＡＮＡH5aLb4Jd3n
> 由 5 開始，向右行 3 步。31Please respect copyright.ＰＥＮＡＮＡnU1bl4bdJt

步驟：31Please respect copyright.ＰＥＮＡＮＡ5PHD7JaU7k
> 5 → 6：第 1 步31Please respect copyright.ＰＥＮＡＮＡDCPWMMc3iH
> 6 → 7：第 2 步31Please respect copyright.ＰＥＮＡＮＡkRQBW4Wrur
> 7 → 8：第 3 步31Please respect copyright.ＰＥＮＡＮＡDUaByfiaeR

所以：31Please respect copyright.ＰＥＮＡＮＡNh5G5K0lJ3
> 5 + 3 = 831Please respect copyright.ＰＥＮＡＮＡJESD11YkWw

四、簡單例子31Please respect copyright.ＰＥＮＡＮＡBywDpRmy5u

例子 1：4 + 2

由 4 開始，向前行 2 步：31Please respect copyright.ＰＥＮＡＮＡQVUo4lrIKi
> 4 → 5：第 1 步31Please respect copyright.ＰＥＮＡＮＡtcihCKQ1K1
> 5 → 6：第 2 步31Please respect copyright.ＰＥＮＡＮＡjm7v33eUb9

所以：31Please respect copyright.ＰＥＮＡＮＡTaTF6vRd7s
> 4 + 2 = 631Please respect copyright.ＰＥＮＡＮＡ9AnaCM32ct

例子 2：7 + 1

由 7 開始：向前行 1 步：31Please respect copyright.ＰＥＮＡＮＡkrtwbprdPM
> 7 → 831Please respect copyright.ＰＥＮＡＮＡ7Ns1vLtdYQ

所以：31Please respect copyright.ＰＥＮＡＮＡCHTNsBWE6I
> 7 + 1 = 831Please respect copyright.ＰＥＮＡＮＡUUUgICwErs

其實這個就是：31Please respect copyright.ＰＥＮＡＮＡLpyjhJ1ke6
> 7 多一是 831Please respect copyright.ＰＥＮＡＮＡ56hQOadqML

例子 3：3 + 4

由 3 開始，向前行 4 步：31Please respect copyright.ＰＥＮＡＮＡpyQxlNGCuU
> 3 → 4：第 1 步31Please respect copyright.ＰＥＮＡＮＡ5MASDrGlY0
> 4 → 5：第 2 步31Please respect copyright.ＰＥＮＡＮＡDYhIQzV3D4
> 5 → 6：第 3 步31Please respect copyright.ＰＥＮＡＮＡSxxLbLDriR
> 6 → 7：第 4 步31Please respect copyright.ＰＥＮＡＮＡaIi8izqo5I

所以：31Please respect copyright.ＰＥＮＡＮＡZCEnIUHjRl
> 3 + 4 = 731Please respect copyright.ＰＥＮＡＮＡ0MhRRDPAyo

五、接住數可能會有的錯誤31Please respect copyright.ＰＥＮＡＮＡN6kXAWeCUe

錯誤：將起點計成第一步

例如：31Please respect copyright.ＰＥＮＡＮＡkufwpULgNX
> 6 + 331Please respect copyright.ＰＥＮＡＮＡSy7x8ezH1T

錯誤數法：31Please respect copyright.ＰＥＮＡＮＡOIpTdfs8s4
> 6, 7, 831Please respect copyright.ＰＥＮＡＮＡyGfzlilI2y

然後答：31Please respect copyright.ＰＥＮＡＮＡ0qo3RMcBUt
> 831Please respect copyright.ＰＥＮＡＮＡkVlAcKI0I5

但這個其實只是數了 3 個「出現的數」，不是行了 3 步。

正確做法：31Please respect copyright.ＰＥＮＡＮＡKyzqZzODV6
> 6 → 7：第 1 步31Please respect copyright.ＰＥＮＡＮＡZcbNKiUua2
> 7 → 8：第 2 步31Please respect copyright.ＰＥＮＡＮＡvGKq2b57lf
> 8 → 9：第 3 步31Please respect copyright.ＰＥＮＡＮＡICnuQlWUCR

所以：31Please respect copyright.ＰＥＮＡＮＡpyOyVTjdBR
> 6 + 3 = 931Please respect copyright.ＰＥＮＡＮＡCIAg5Lw46D

一個簡單的檢查方法是：31Please respect copyright.ＰＥＮＡＮＡjdgolCiOUX
> 加 3，應該比原本大 3。31Please respect copyright.ＰＥＮＡＮＡHEjiUXmk19
> 6 變 9，剛好大 3。31Please respect copyright.ＰＥＮＡＮＡ3xMzrMY6Yn

六、如何選擇起點更快？31Please respect copyright.ＰＥＮＡＮＡy8lekzHH6S

因為加法可以交換次序：31Please respect copyright.ＰＥＮＡＮＡr5yZcPJan4
> 2 + 8 = 8 + 231Please respect copyright.ＰＥＮＡＮＡkgV0BDTPog

所以接住數時，通常可以由較大的數開始，數較小的數。

例如：31Please respect copyright.ＰＥＮＡＮＡMvY7fQI6Zi
> 2 + 831Please respect copyright.ＰＥＮＡＮＡZqZ8hYJuJ5

方法 A：由 2 開始，行 8 步。31Please respect copyright.ＰＥＮＡＮＡKIAgo2MhRM
> 2 → 3 → 4 → 5 → 6 → 7 → 8 → 9 → 1031Please respect copyright.ＰＥＮＡＮＡdAmMeNCGeP

答案：31Please respect copyright.ＰＥＮＡＮＡQDz90aNzOI
> 1031Please respect copyright.ＰＥＮＡＮＡJJ1MW5G2fY

方法 B：換一換次序，想成：31Please respect copyright.ＰＥＮＡＮＡWA7Xqv5dOA
> 8 + 231Please respect copyright.ＰＥＮＡＮＡ9EBLrrx0la

由 8 開始，行 2 步：31Please respect copyright.ＰＥＮＡＮＡj5grCgUc9s
> 8 → 9 → 1031Please respect copyright.ＰＥＮＡＮＡVsrRTqJSzo

答案一樣；31Please respect copyright.ＰＥＮＡＮＡIZq92bAjlq
> 10

但方法 B 快很多。

所以做加法時可以用這個策略：31Please respect copyright.ＰＥＮＡＮＡ71fhN4mhuW
> 由大數開始，接住數小的數。31Please respect copyright.ＰＥＮＡＮＡVQjNkm6BAM

七、1 + 931Please respect copyright.ＰＥＮＡＮＡ2ovWynmdLB

問題：31Please respect copyright.ＰＥＮＡＮＡ3sDeet3Wde
> 1 + 9 = ?31Please respect copyright.ＰＥＮＡＮＡzLxPPOSJJ0

你可以由 1 開始行 9 步，但很慢。

因為加法可以交換次序：31Please respect copyright.ＰＥＮＡＮＡ312oodH91v
> 1 + 9 = 9 + 131Please respect copyright.ＰＥＮＡＮＡRm4IIWzmoa

由 9 開始，行 1 步：31Please respect copyright.ＰＥＮＡＮＡ8sqHgqDvJ9
> 9 → 1031Please respect copyright.ＰＥＮＡＮＡfWPJsNgNAQ

所以：31Please respect copyright.ＰＥＮＡＮＡEUsEUWovvd
> 1 + 9 = 1031Please respect copyright.ＰＥＮＡＮＡNg2QUh4krP

不是盲目跟左邊開始，而是選擇更容易的方向理解。

八、接住數和「合併」是同一回事31Please respect copyright.ＰＥＮＡＮＡgO20wCoY9z

有時你可能會問：31Please respect copyright.ＰＥＮＡＮＡcnP48uWSZI
> 加法到底是合併，還是接住數？31Please respect copyright.ＰＥＮＡＮＡDZ3im2r73v

答案是：31Please respect copyright.ＰＥＮＡＮＡckWtqMJQ2Q
> 兩個都對，只是角度不同。31Please respect copyright.ＰＥＮＡＮＡsQOYguggvK

例如：31Please respect copyright.ＰＥＮＡＮＡqSx2QXATHY
> 4 + 331Please respect copyright.ＰＥＮＡＮＡ8hcHCrGSms

合併角度：31Please respect copyright.ＰＥＮＡＮＡ7t1sMQOGhP
> 4 個物件 + 3 個物件 = 7 個物件31Please respect copyright.ＰＥＮＡＮＡb9XR9RqupS

接住數角度：31Please respect copyright.ＰＥＮＡＮＡ8P8P5fCVik
> 由 4 開始，向前行 3 步，到 731Please respect copyright.ＰＥＮＡＮＡ5p49ht2lvP

兩種方法都描述著同一件事：31Please respect copyright.ＰＥＮＡＮＡVbApMNI4EP
> 4 + 3 = 731Please respect copyright.ＰＥＮＡＮＡ6akwViZVaj

所以數學入面，常常會有多個模型解釋同一個概念。

九、用接住數解故事題31Please respect copyright.ＰＥＮＡＮＡV8RKsWQJDX

例子 4：

你有 6 張卡，31Please respect copyright.ＰＥＮＡＮＡG5B5J7N5h3
朋友再給你 4 張。31Please respect copyright.ＰＥＮＡＮＡzFJyJVdu0Q
你現在有多少張？31Please respect copyright.ＰＥＮＡＮＡKmYyxodh7j

步驟 1：找已經起點31Please respect copyright.ＰＥＮＡＮＡzlb8A4wnZ8
原本有：31Please respect copyright.ＰＥＮＡＮＡYNudVORlPL
> 6 張31Please respect copyright.ＰＥＮＡＮＡ2GISW3Z79C

步驟 2：找新增數量31Please respect copyright.ＰＥＮＡＮＡ6n1lBTNAbb
朋友給你：31Please respect copyright.ＰＥＮＡＮＡIpScSCoRVT
> 4 張31Please respect copyright.ＰＥＮＡＮＡRnfzr58gha

所以是：31Please respect copyright.ＰＥＮＡＮＡXUWCJCR53t
> 6 + 431Please respect copyright.ＰＥＮＡＮＡ1drfp5MRBo

步驟 3：由 6 接住數 4 步31Please respect copyright.ＰＥＮＡＮＡBPQNAov3Ai
> 6 → 7：第 1 步31Please respect copyright.ＰＥＮＡＮＡYPlw2TiZ5M
> 7 → 8：第 2 步31Please respect copyright.ＰＥＮＡＮＡFJL7fkRSud
> 8 → 9：第 3 步31Please respect copyright.ＰＥＮＡＮＡCd44xwLlEA
> 9 → 10：第 4 步31Please respect copyright.ＰＥＮＡＮＡlsvGKtQTnz

答案：31Please respect copyright.ＰＥＮＡＮＡrphr4vGgBD
> 你現在有 10 張卡。31Please respect copyright.ＰＥＮＡＮＡPzj03Y1pGT

十一、那個方法較簡單？31Please respect copyright.ＰＥＮＡＮＡB6URkKbPum

問題：31Please respect copyright.ＰＥＮＡＮＡdRbpuRpIro
> 4 + 931Please respect copyright.ＰＥＮＡＮＡdGrgeTgtOG

方法 A：由 4 開始，行 9 步。31Please respect copyright.ＰＥＮＡＮＡNhG6skjQo8
> 4 → 5 → 6 → 7 → 8 → 9 → 10 → 11 → 12 → 1331Please respect copyright.ＰＥＮＡＮＡVkhaSdMkvI

答案是：31Please respect copyright.ＰＥＮＡＮＡd7F8EK6T57
> 1331Please respect copyright.ＰＥＮＡＮＡYZ0YbP1XgD

方法 B：交換次序：31Please respect copyright.ＰＥＮＡＮＡGLmmdqufiD
> 4 + 9 = 9 + 431Please respect copyright.ＰＥＮＡＮＡfxgEvYrNWe

由 9 開始，行 4 步：31Please respect copyright.ＰＥＮＡＮＡBISvPK0Spq
> 9 → 10：第 1 步31Please respect copyright.ＰＥＮＡＮＡqcmDBpSi3r
> 10 → 11：第 2 步31Please respect copyright.ＰＥＮＡＮＡA7RfozA2HL
> 11 → 12：第 3 步31Please respect copyright.ＰＥＮＡＮＡ9Aew8bBBro
> 12 → 13：第 4 步31Please respect copyright.ＰＥＮＡＮＡBRzW1voKXR

答案一樣是：31Please respect copyright.ＰＥＮＡＮＡgU9cn06KLx
> 1331Please respect copyright.ＰＥＮＡＮＡG1wmXMwfRw

方法 B 更短。

結論：31Please respect copyright.ＰＥＮＡＮＡzaFqmucFZp
> 接住數不只是計算方法，亦是一種選擇更有效路線的方法。31Please respect copyright.ＰＥＮＡＮＡcCD2MJtwPH

十二、接住數可以跨過 1031Please respect copyright.ＰＥＮＡＮＡ7jrvMdzX1D

由 8 加 5：31Please respect copyright.ＰＥＮＡＮＡcUoP89FbZ4
> 8 + 531Please respect copyright.ＰＥＮＡＮＡPoCXMhrFhR

有時過了 9 去到 10，會覺得特別。31Please respect copyright.ＰＥＮＡＮＡ3dmqMVwzXH
但其實只是繼續向前行。31Please respect copyright.ＰＥＮＡＮＡig4x4KKoZV

步驟：31Please respect copyright.ＰＥＮＡＮＡGpwutKQtCY
> 8 → 9：第 1 步31Please respect copyright.ＰＥＮＡＮＡYNIea4Dtkd
> 9 → 10：第 2 步31Please respect copyright.ＰＥＮＡＮＡF6itP57TXj
> 10 → 11：第 3 步31Please respect copyright.ＰＥＮＡＮＡvclGcIyjlT
> 11 → 12：第 4 步31Please respect copyright.ＰＥＮＡＮＡzj8J2c16oX
> 12 → 13：第 5 步31Please respect copyright.ＰＥＮＡＮＡeZl6T7d0QG

所以：31Please respect copyright.ＰＥＮＡＮＡwUsbGJt4Be
> 8 + 5 = 1331Please respect copyright.ＰＥＮＡＮＡLsUV2WfqGl

10 不是牆壁，只是數線上其中一個位置，與其他數一樣。

十三、接住數和「差多少到某個數」

有時接住數可以幫你猜：31Please respect copyright.ＰＥＮＡＮＡGfsIbJdP4s
> 由一個數去到另一個數，要加幾多？31Please respect copyright.ＰＥＮＡＮＡrV3QJm4xL3

例如：31Please respect copyright.ＰＥＮＡＮＡUfSblBnHuR
> 6 + ? = 1031Please respect copyright.ＰＥＮＡＮＡ7d5ZroSjqg

意思是：31Please respect copyright.ＰＥＮＡＮＡo1z0dGI5Qf
> 由 6 開始，要行幾步才到 10？31Please respect copyright.ＰＥＮＡＮＡEoupJF4N63

數線步驟：31Please respect copyright.ＰＥＮＡＮＡED9fvTDgIB
> 6 → 7：第 1 步31Please respect copyright.ＰＥＮＡＮＡrv3lP5vav4
> 7 → 8：第 2 步31Please respect copyright.ＰＥＮＡＮＡaQorX0iV4S
> 8 → 9：第 3 步31Please respect copyright.ＰＥＮＡＮＡPuPq6mNXUm
> 9 → 10：第 4 步31Please respect copyright.ＰＥＮＡＮＡqqxMdBXG2H

所以：31Please respect copyright.ＰＥＮＡＮＡghmrg7wt0V
> 6 + 4 = 1031Please respect copyright.ＰＥＮＡＮＡJ6K6KofHkW

即是：31Please respect copyright.ＰＥＮＡＮＡ9dbUEd4dZO
> ? = 431Please respect copyright.ＰＥＮＡＮＡusW2xrK6De

十四、三個數相加

問題：31Please respect copyright.ＰＥＮＡＮＡ3fKPZZMEdg
> 3 + 2 + 4 = ?31Please respect copyright.ＰＥＮＡＮＡuugJ0GVjpY
可以逐段接住數。31Please respect copyright.ＰＥＮＡＮＡDQx2cv594F

步驟 1：先做 3 + 231Please respect copyright.ＰＥＮＡＮＡPB2J2DfUFS

由 3 行 2 步：31Please respect copyright.ＰＥＮＡＮＡj6mUpZP8xf
> 3 → 4：第 1 步31Please respect copyright.ＰＥＮＡＮＡalA3u1k9FJ
> 4 → 5：第 2 步31Please respect copyright.ＰＥＮＡＮＡXiFK7S1hfs

所以：31Please respect copyright.ＰＥＮＡＮＡxL2Nja9gKI
> 3 + 2 = 531Please respect copyright.ＰＥＮＡＮＡ9ayWB6P4LC

步驟 2：再加 431Please respect copyright.ＰＥＮＡＮＡYP5060c4Ty

由 5 行 4 步：31Please respect copyright.ＰＥＮＡＮＡMojI1NL5pu
> 5 → 6：第 1 步31Please respect copyright.ＰＥＮＡＮＡk4JgSMXAH3
> 6 → 7：第 2 步31Please respect copyright.ＰＥＮＡＮＡo5hCsYaICA
> 7 → 8：第 3 步31Please respect copyright.ＰＥＮＡＮＡsO2aUzE8uk
> 8 → 9：第 4 步31Please respect copyright.ＰＥＮＡＮＡthXX7xIYBH

所以：31Please respect copyright.ＰＥＮＡＮＡEopuJWpHcN
> 3 + 2 + 4 = 931Please respect copyright.ＰＥＮＡＮＡRu4wHrWqor

另一種想法是：31Please respect copyright.ＰＥＮＡＮＡ9tgUn427xs
> 先將 2 + 4 變成 631Please respect copyright.ＰＥＮＡＮＡ1CGsKvG8A9
> 再做 3 + 6 變成 931Please respect copyright.ＰＥＮＡＮＡjAXj5Arduc

這個會引出之後的加法結合律。

十五、接住數背後的數學思想31Please respect copyright.ＰＥＮＡＮＡ1Xy9jWz6sY

接住數表面上很簡單，但背後有三個重要思想：31Please respect copyright.ＰＥＮＡＮＡiXbzTvSBjy

第一，數字有順序：31Please respect copyright.ＰＥＮＡＮＡtyCSGPkeDI
> 6 後面是 7，7 後面是 8。31Please respect copyright.ＰＥＮＡＮＡw0HwpBWM9i

第二，加法可以理解成移動：31Please respect copyright.ＰＥＮＡＮＡcTMgIncJVn
> 加 3 = 向右行 3 步。31Please respect copyright.ＰＥＮＡＮＡVtWKOmLsSb

第三，已知的數量可以成為起點：31Please respect copyright.ＰＥＮＡＮＡSPLVNDv24X
> 不需要每次由 1 重新開始。31Please respect copyright.ＰＥＮＡＮＡHgHp2Z50cR

這個思想很重要，因為數學不是每次重新數一次，而是會利用已知的資料，繼續推理。

十六、今日小結31Please respect copyright.ＰＥＮＡＮＡJt0HtFllIV

今課你已掌握：31Please respect copyright.ＰＥＮＡＮＡRbiHm0mNT6

1. 接住數即是由已知數量開始，向前數新增的數量。31Please respect copyright.ＰＥＮＡＮＡ681swyAYpl
2. a + b 可以理解成由 a 開始，向右行 b 步。31Please respect copyright.ＰＥＮＡＮＡSf8lFBcSe1
3. 起點不是第一步：加幾，就要行幾步。31Please respect copyright.ＰＥＮＡＮＡDbg72lv6Vz
4. 加法可以交換次序，所以通常由較大數開始數較小的數會更快。31Please respect copyright.ＰＥＮＡＮＡ5VqFEenU8R
5. 接住數和合併數量是同一個加法概念的兩種理解。31Please respect copyright.ＰＥＮＡＮＡkmUjUrIRI9
6. 接住數可以幫我們處理缺失部分，例如 6 + ? = 10。31Please respect copyright.ＰＥＮＡＮＡcCTmJfDLdf

接住數是用已佑數量做起點，將「加法」變成沿數線向前走。

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