學海：數學 Learn: Mathematics - 第 6 課：向前數與向後數 | Counting Forward and Backward

一、數數不一定由 1 開始36Please respect copyright.ＰＥＮＡＮＡMo6FlC4QGo

最初我們數數，通常是：36Please respect copyright.ＰＥＮＡＮＡpuF6FID7fk
> 1, 2, 3, 4, 5,...36Please respect copyright.ＰＥＮＡＮＡLghg8fvxrr

但其實數數可以由任何一個數開始。

例如由 4 開始向前數：36Please respect copyright.ＰＥＮＡＮＡqw5RIJl5pG
> 4, 5, 6, 7, 8,...36Please respect copyright.ＰＥＮＡＮＡyOscIPRH3I

又可以由 8 開始倒數：36Please respect copyright.ＰＥＮＡＮＡKSYD4KYkuD
> 8, 7, 6, 5, 4,...36Please respect copyright.ＰＥＮＡＮＡ0KsgbQJDs8

上一課我們學過：36Please respect copyright.ＰＥＮＡＮＡPgfAXLUjD0
> 多一 = 向右行一格36Please respect copyright.ＰＥＮＡＮＡDRfn6swUoi
> 少一 = 向左行一格36Please respect copyright.ＰＥＮＡＮＡeVO2btitS0

今課就是將這件事連續做。

二、向前數36Please respect copyright.ＰＥＮＡＮＡVz2Jfc3IH4

向前數即是每次多一。

例如：36Please respect copyright.ＰＥＮＡＮＡAIw4kkWV9K
> 2, 3, 4, 5, 6,...36Please respect copyright.ＰＥＮＡＮＡ9pEE2Ou36D

每一步都是多一：36Please respect copyright.ＰＥＮＡＮＡklYq7WUvfg
> 2 → 336Please respect copyright.ＰＥＮＡＮＡJ1GM8qohSp
> 3 → 436Please respect copyright.ＰＥＮＡＮＡNBEhDH2VVq
> 4 → 536Please respect copyright.ＰＥＮＡＮＡOiR6CVe2KP
> 5 → 636Please respect copyright.ＰＥＮＡＮＡLdVgfdKbsq

在數線上：36Please respect copyright.ＰＥＮＡＮＡSN1hF2BZHW
> 0 —— 1 —— 2 —— 3 —— 4 —— 5 —— 636Please respect copyright.ＰＥＮＡＮＡ5VWKLgsz8c
> 2 → 3 → 4 → 5 → 636Please respect copyright.ＰＥＮＡＮＡl1F8vF01ny

所以：36Please respect copyright.ＰＥＮＡＮＡls3nbVQWEW
> 向前數 = 沿住數線向右行36Please respect copyright.ＰＥＮＡＮＡrjmxVzcn4w

三、倒數36Please respect copyright.ＰＥＮＡＮＡWV2NNXE1fG

倒數即是每次少一。36Please respect copyright.ＰＥＮＡＮＡjpLqseGcIs

例如：36Please respect copyright.ＰＥＮＡＮＡkUbsHhhyaD
> 6, 5, 4, 3, 2,...36Please respect copyright.ＰＥＮＡＮＡGknry51mNb

每一步都是少一：36Please respect copyright.ＰＥＮＡＮＡ2wtsGswtmy
> 6 → 536Please respect copyright.ＰＥＮＡＮＡcdTaUdfplw
> 5 → 436Please respect copyright.ＰＥＮＡＮＡeFHFhjh6Ky
> 4 → 336Please respect copyright.ＰＥＮＡＮＡ1V9YovCtKd
> 3 → 236Please respect copyright.ＰＥＮＡＮＡYX8NtvYj54

在數線上：36Please respect copyright.ＰＥＮＡＮＡPAajQJ1axz
> 0 —— 1 —— 2 —— 3 —— 4 —— 5 —— 636Please respect copyright.ＰＥＮＡＮＡDNMuBn0j7g
> 2 ← 3 ← 4 ← 5 ← 636Please respect copyright.ＰＥＮＡＮＡAx7iDm6C5Y

所以：36Please respect copyright.ＰＥＮＡＮＡxkk3irCORB
> 倒數 = 沿住數線向左行

四、簡單例子：由指定數開始向前數36Please respect copyright.ＰＥＮＡＮＡkesikO18Y1

例子 1：36Please respect copyright.ＰＥＮＡＮＡULcmUdfBYX

由 3 開始，向前數 5 個數：36Please respect copyright.ＰＥＮＡＮＡFJm7LE8A0E
> 3, 4, 5, 6, 736Please respect copyright.ＰＥＮＡＮＡ112iSTsD6Y

要小心，題目稱「數 5 個數」，第一個數 3 都要計進去。

數法是：36Please respect copyright.ＰＥＮＡＮＡgAajCkv1pF
> 第 1 個：336Please respect copyright.ＰＥＮＡＮＡD7VZB9FCvl
> 第 2 個：436Please respect copyright.ＰＥＮＡＮＡNBHsALeVh3
> 第 3 個：536Please respect copyright.ＰＥＮＡＮＡ3RWRQHCgvb
> 第 4 個：636Please respect copyright.ＰＥＮＡＮＡACe0P8g9kd
> 第 5 個：736Please respect copyright.ＰＥＮＡＮＡQwrrtsjNMj

所以答案是：36Please respect copyright.ＰＥＮＡＮＡEVwIyOcaYJ
> 3, 4, 5, 6, 736Please respect copyright.ＰＥＮＡＮＡVTUthu1gGM

例子 2：

由 6 開始，向前數 4 個數：36Please respect copyright.ＰＥＮＡＮＡj6oCXeSjQw
> 第 1 個：636Please respect copyright.ＰＥＮＡＮＡ7hQnPl4pWb
> 第 2 個：736Please respect copyright.ＰＥＮＡＮＡWvYoCotmVg
> 第 3 個：836Please respect copyright.ＰＥＮＡＮＡ2Tqd5sQehO
> 第 4 個：936Please respect copyright.ＰＥＮＡＮＡN5cikUDWC4

答案：36Please respect copyright.ＰＥＮＡＮＡCoPt5Nizc2
> 6, 7, 8, 936Please respect copyright.ＰＥＮＡＮＡfzIF1lHN5h

五、簡單例子：由指定數開始倒數36Please respect copyright.ＰＥＮＡＮＡlfse0Rbm65

例子 3：36Please respect copyright.ＰＥＮＡＮＡBCf5GQKLDy

由 8 開始，倒數 5 個數：36Please respect copyright.ＰＥＮＡＮＡ73tDBPTIyH
> 第 1 個：836Please respect copyright.ＰＥＮＡＮＡheQED9PS9q
> 第 2 個：736Please respect copyright.ＰＥＮＡＮＡTXxRtshXy3
> 第 3 個：636Please respect copyright.ＰＥＮＡＮＡRLwXBSueZi
> 第 4 個：536Please respect copyright.ＰＥＮＡＮＡMgtVs8Ts9C
> 第 5 個：436Please respect copyright.ＰＥＮＡＮＡWTulM2jPWe

答案：36Please respect copyright.ＰＥＮＡＮＡRSyQiuMxc1
> 8, 7, 6, 5, 436Please respect copyright.ＰＥＮＡＮＡEkWF2AbYgy

例子 4：

由 4 開始，倒數到 0：36Please respect copyright.ＰＥＮＡＮＡAK5hqJo6yM
> 4, 3, 2, 1, 036Please respect copyright.ＰＥＮＡＮＡnUoNYM9OJz

這裏 0 是可以數到的，因為 0 是一個正式數字，代表「沒有」。36Please respect copyright.ＰＥＮＡＮＡO38FCHI2zT

六、兩種常見題目要分清楚36Please respect copyright.ＰＥＮＡＮＡKaP9CVrALO

有兩種題目好像相似，但其實不同。

題型 A：數出幾多個數36Please respect copyright.ＰＥＮＡＮＡGqplMgye9e

例如：36Please respect copyright.ＰＥＮＡＮＡYWNjtcRHdR
> 由 2 開始，向前數 4 個數。36Please respect copyright.ＰＥＮＡＮＡHX4NN0DSxx

答案：36Please respect copyright.ＰＥＮＡＮＡpbjSb7jHqu
> 2, 3, 4, 536Please respect copyright.ＰＥＮＡＮＡDjRzvHcIT9

因為 2 本身是第 1 個。

題型 B：向前行幾步36Please respect copyright.ＰＥＮＡＮＡQupmlhtThA

例如：36Please respect copyright.ＰＥＮＡＮＡJ0PYxmDXni
> 由 2 開始，向前行 4 步。36Please respect copyright.ＰＥＮＡＮＡD3oOWkJEM8

步驟：36Please respect copyright.ＰＥＮＡＮＡS4cAE7ycGq
> 第 1 步：2 → 336Please respect copyright.ＰＥＮＡＮＡMS6HkdRjsb
> 第 2 步：3 → 436Please respect copyright.ＰＥＮＡＮＡAO18jrJUyC
> 第 3 步：4 → 536Please respect copyright.ＰＥＮＡＮＡwCYuFVMpZL
> 第 4 步：5 → 636Please respect copyright.ＰＥＮＡＮＡKyifIaucHk

最後去到：36Please respect copyright.ＰＥＮＡＮＡ5SpPYmwyCz
> 636Please respect copyright.ＰＥＮＡＮＡV7TK82Xr9R

留意分別：36Please respect copyright.ＰＥＮＡＮＡzvjmTQcW8o
> 數 4 個數：2, 3, 4, 536Please respect copyright.ＰＥＮＡＮＡ59Xb0DVuVo
> 行 4 步：2 → 636Please respect copyright.ＰＥＮＡＮＡmDN50V0Jvh

這個很重要。36Please respect copyright.ＰＥＮＡＮＡ18YP8PnuJC
因為「數幾個數」和「行幾多步」差一個起點。36Please respect copyright.ＰＥＮＡＮＡtEABqFf5Zw

七、為何會差一？36Please respect copyright.ＰＥＮＡＮＡtMepCB8arg

問題：36Please respect copyright.ＰＥＮＡＮＡqsUQVw5zWo
> 由 5 開始，向前數 3 個數，最後是幾多？36Please respect copyright.ＰＥＮＡＮＡxZAXTigfUr
> 由 5 開始，向前行 3 步，最後是幾多？36Please respect copyright.ＰＥＮＡＮＡXoIlviLeh5

看來都是「3」，但答案不同。

第一題：向前數 3 個數36Please respect copyright.ＰＥＮＡＮＡ6n6T7vk9mw

> 第 1 個：536Please respect copyright.ＰＥＮＡＮＡ3oC98teQrc
> 第 2 個：636Please respect copyright.ＰＥＮＡＮＡ56xwunqtbO
> 第 3 個：736Please respect copyright.ＰＥＮＡＮＡBTIy1Lhu9n

所以最後是：36Please respect copyright.ＰＥＮＡＮＡ4dswhlgd3M
> 736Please respect copyright.ＰＥＮＡＮＡ2dSwU5h97e

第二題：向前行 3 步36Please respect copyright.ＰＥＮＡＮＡaKVCHjhI5W

> 第 1 步：5 → 636Please respect copyright.ＰＥＮＡＮＡ2mZYSmQM5D
> 第 2 步：6 → 736Please respect copyright.ＰＥＮＡＮＡB6oTqto0Lx
> 第 3 步：7 → 836Please respect copyright.ＰＥＮＡＮＡz5DndsitG8

所以最後是：36Please respect copyright.ＰＥＮＡＮＡs24zxbSuNi
> 836Please respect copyright.ＰＥＮＡＮＡPCh8wrviC4

為何不同？36Please respect copyright.ＰＥＮＡＮＡErboZljNiq

因為：36Please respect copyright.ＰＥＮＡＮＡZFKY6ia3k6
> 數 3 個數，起點本身已經算第 1 個。36Please respect copyright.ＰＥＮＡＮＡ3lokNdPn5M
> 行 3 步，起點只是開始位置，未算 1 步。36Please respect copyright.ＰＥＮＡＮＡbqgIUEdH8C

八、倒數和 036Please respect copyright.ＰＥＮＡＮＡr7yrx8jHwr

由 5 倒數到 0：36Please respect copyright.ＰＥＮＡＮＡK7aSGuF6we
> 5, 4, 3, 2, 1, 036Please respect copyright.ＰＥＮＡＮＡBZbbV4U2Vi

每一步都是少一。

但現在我們暫時不會在 0 之後繼續倒數：36Please respect copyright.ＰＥＮＡＮＡQsAckTSOsd
> 0, ?

因為在「數物件數量」的世界入面，0 已經是最少。

將來學習「負數」時，數線會變成：36Please respect copyright.ＰＥＮＡＮＡlfEupY0mTV
> ... -3 —— -2 —— -1 —— 0 —— 1 —— 2 —— 3 ...36Please respect copyright.ＰＥＮＡＮＡTy1hZVrbl3

到那時候就可以繼續倒數：36Please respect copyright.ＰＥＮＡＮＡIOFecivohw
> 0, -1, -2, -3,...36Please respect copyright.ＰＥＮＡＮＡP8Xw1KLAxD

但現時我們先停在 0。

九、向前數可以幫你「接住數」36Please respect copyright.ＰＥＮＡＮＡWHyJ6GhvJ3

假設你已經知道盒子入面有 5 粒糖果。

朋友再給你 3 粒：36Please respect copyright.ＰＥＮＡＮＡkLELsQ1Lxr
> 🍬🍬🍬36Please respect copyright.ＰＥＮＡＮＡcNJQnVoO6s

你不一定要由 1 重新數完全部糖果。

你可以由 5 開始接住數：36Please respect copyright.ＰＥＮＡＮＡCqqCbl9zxD
> 原本有 536Please respect copyright.ＰＥＮＡＮＡvaUHKU7Rcx
> 再數新加的 3 粒：36Please respect copyright.ＰＥＮＡＮＡPlajEyp1Vx

> 6, 7, 8

所以總共有：36Please respect copyright.ＰＥＮＡＮＡpxW3HiIvzI
> 8 粒糖36Please respect copyright.ＰＥＮＡＮＡ4U3NVXZWWm

這個方法叫：36Please respect copyright.ＰＥＮＡＮＡUiCpytmjPC
> 接住數 | Counting On36Please respect copyright.ＰＥＮＡＮＡFQs7gVKQaK

它是加法之前非常重要的方法。

將來我們會寫成：36Please respect copyright.ＰＥＮＡＮＡaDqlQK6c6p
> 5 + 3 = 836Please respect copyright.ＰＥＮＡＮＡgkXPWVpoxF

十一、由 4 接住數 3 個新物件36Please respect copyright.ＰＥＮＡＮＡ9Hy98qoYy2

問題：36Please respect copyright.ＰＥＮＡＮＡgSQ1GzoFi4
> 你已經有 4 本書，之後又得到 3 本書。總共有幾多本？36Please respect copyright.ＰＥＮＡＮＡktgjEAYXSK

步驟 1：不用又 1 開始36Please respect copyright.ＰＥＮＡＮＡcYlWNJB1xt

因為你已經知道原本有：36Please respect copyright.ＰＥＮＡＮＡGxbU2lBl35
> 4 本36Please respect copyright.ＰＥＮＡＮＡ7k1ZM7Ayb6

步驟 2：新得到 3 本，所以向前數 3 步36Please respect copyright.ＰＥＮＡＮＡkBMmbzJFk9

> 第 1 本新書：536Please respect copyright.ＰＥＮＡＮＡ0IyBuJZ9fx
> 第 2 本新書：636Please respect copyright.ＰＥＮＡＮＡaKLro8KTvQ
> 第 3 本新書：736Please respect copyright.ＰＥＮＡＮＡ2WRgCoUvyE

結論36Please respect copyright.ＰＥＮＡＮＡeBqOkckkdB

> 總共有 7 本書。36Please respect copyright.ＰＥＮＡＮＡfysyvwYbrK

這裏的答案不是 6，因為新得到的 3 本是令數量行三步：36Please respect copyright.ＰＥＮＡＮＡ4LfCiJzre6
> 4 → 5 → 6 → 736Please respect copyright.ＰＥＮＡＮＡKk1gLB6U54

十二、由 9 倒數 4 步36Please respect copyright.ＰＥＮＡＮＡ4UG1M6uMMO

問題：36Please respect copyright.ＰＥＮＡＮＡlqhb95kIKv
> 你有 9 張貼紙，用了 4 張，剩幾多張？36Please respect copyright.ＰＥＮＡＮＡheJuTZwbi9

步驟 1：由 9 開始36Please respect copyright.ＰＥＮＡＮＡN5E1ZGFJ1n

> 936Please respect copyright.ＰＥＮＡＮＡk89a8H4DI6

步驟 2：用了 4 張，所以倒後行 4 步36Please respect copyright.ＰＥＮＡＮＡ3K6grPzm2Y

> 第 1 張：9 → 836Please respect copyright.ＰＥＮＡＮＡrDf1oF3e1x
> 第 2 張：8 → 736Please respect copyright.ＰＥＮＡＮＡCHe9wvf9rE
> 第 3 張：7 → 636Please respect copyright.ＰＥＮＡＮＡc1cm4GgEEi
> 第 4 張：6 → 536Please respect copyright.ＰＥＮＡＮＡ5mhFnXTAGd

結論36Please respect copyright.ＰＥＮＡＮＡvPwTU2MTbz

> 剩 5 張貼紙

留意：36Please respect copyright.ＰＥＮＡＮＡkmLjWPC63j
> 9, 8, 7, 6, 5

這裏出現 5 個數，但只行了 4 步。36Please respect copyright.ＰＥＮＡＮＡivsfBKNPQF
因為 9 是起點，不是第一步。36Please respect copyright.ＰＥＮＡＮＡLB8A138I40

十三、向前數和倒數會互相取消36Please respect copyright.ＰＥＮＡＮＡQnCHgfK7XW

如果由 6 開始：

向前數 3 步：36Please respect copyright.ＰＥＮＡＮＡbnIZBVciXC
> 6 → 7 → 8 → 936Please respect copyright.ＰＥＮＡＮＡGKtEVbhpiq

再倒數 3 步：36Please respect copyright.ＰＥＮＡＮＡQnaM7BwtbF
> 9 → 8 → 7 → 636Please respect copyright.ＰＥＮＡＮＡsxqbtNtUFO

最後返回到：36Please respect copyright.ＰＥＮＡＮＡEpFhQLn9EK
> 636Please respect copyright.ＰＥＮＡＮＡY2LCtjx5pp

所以：36Please respect copyright.ＰＥＮＡＮＡIwC19pTuV7
> 向前幾步，再倒後同樣幾步，會返到原點。36Please respect copyright.ＰＥＮＡＮＡFAhzqoYlQl

相反：36Please respect copyright.ＰＥＮＡＮＡub2ucyx6C1

由 6 倒後 2 步：36Please respect copyright.ＰＥＮＡＮＡuypaaUMSch
> 6 → 5 → 436Please respect copyright.ＰＥＮＡＮＡ37CzStOaKL

再向前 2 步：36Please respect copyright.ＰＥＮＡＮＡPIibEXVU1n
> 4 → 5 → 636Please respect copyright.ＰＥＮＡＮＡyxfGkRKECL

又返回到：36Please respect copyright.ＰＥＮＡＮＡkgqxwPmL26
> 636Please respect copyright.ＰＥＮＡＮＡ1SeS9oJn9N

這個就是將來「加法和減法互相相反」的直覺基礎。

十四、一段小故事36Please respect copyright.ＰＥＮＡＮＡ9sJiHvM9Rd

你有 3 粒波子。

朋友給你 4 粒。36Please respect copyright.ＰＥＮＡＮＡT3RD6dM9NT
你之後又送了 2 粒給妹妹。36Please respect copyright.ＰＥＮＡＮＡlF2fZj5v30

問題：36Please respect copyright.ＰＥＮＡＮＡGhFrRJuS6F
> 最後你有幾多粒波子？36Please respect copyright.ＰＥＮＡＮＡM1lN7nK8cA

步驟 1：由 3 開始36Please respect copyright.ＰＥＮＡＮＡe104e9NxQd

> 3

步驟 2：朋友給你 4 粒，所以向前數 4 步36Please respect copyright.ＰＥＮＡＮＡ44bQMEFS3r

> 3 → 4 → 5 → 6 → 736Please respect copyright.ＰＥＮＡＮＡROSRxkclS5

現在有：36Please respect copyright.ＰＥＮＡＮＡe3uEte1jl4
> 7 粒36Please respect copyright.ＰＥＮＡＮＡjQ8w9av4BV

步驟 3：你送走 2 粒，所以倒後數 2 步36Please respect copyright.ＰＥＮＡＮＡf7Rw7NV7Gp

> 7 → 6 → 536Please respect copyright.ＰＥＮＡＮＡZZjRTDbvRy

最後有：36Please respect copyright.ＰＥＮＡＮＡpl39JZ3wFC
> 5 粒波子36Please respect copyright.ＰＥＮＡＮＡ2aUMX3loln

完整移動是：36Please respect copyright.ＰＥＮＡＮＡwlDDxpEwuM
> 3 → 4 → 5 → 6 → 7 → 6 → 536Please respect copyright.ＰＥＮＡＮＡBXHBs3j67H

這個故事雖然未正式用加減法符號，但其實已經是：36Please respect copyright.ＰＥＮＡＮＡaMZfPlVOSd
> 3 加 4，再減 236Please respect copyright.ＰＥＮＡＮＡO1JhjB1hxn

十五、今日小結

今課你掌握了：

1. 向前數即是每次多一，在數線上向右行。36Please respect copyright.ＰＥＮＡＮＡDE8R1otfug
2. 倒數即是每次少一，在數線上向左行。36Please respect copyright.ＰＥＮＡＮＡFzm7LoHCmy
3. 「數幾個數」和「行幾步」不同：起點會不會計進去，要看題目。36Please respect copyright.ＰＥＮＡＮＡTWt449MAYj
4. 接住數可以幫你理解加法。36Please respect copyright.ＰＥＮＡＮＡ8FEHLDlARC
5. 倒數可以幫你理解減法。36Please respect copyright.ＰＥＮＡＮＡy0VoVCgkDa
6. 向前幾步再倒數相同步數，會返回原本位置。36Please respect copyright.ＰＥＮＡＮＡF6MlZklFXq

向前數是連續多一，倒數是連續少一。

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