一:數學不只問「有多少」53Please respect copyright.PENANAQKehpzw6SF
上一課我們問過:53Please respect copyright.PENANAoSQMAvfz9k
> 「這裏有多少個?」53Please respect copyright.PENANAjuRuFYUImJ
今課開始問另一種問題:53Please respect copyright.PENANAzi0ahHGwko
> 「哪個較多?」「哪個較少?」「是否一樣多?」53Please respect copyright.PENANAQFNw7dA3zK
例如:53Please respect copyright.PENANARnKhCOYfkx
> 🍎🍎🍎53Please respect copyright.PENANAyTqCgqwNnA
> 🧃🧃53Please respect copyright.PENANAV9K22SP4uE
你不只可以說:53Please respect copyright.PENANA17wv4bSxjb
> 有 3 個蘋果、有 2 盒果汁53Please respect copyright.PENANAPchEXaB707
你也可以比較:53Please respect copyright.PENANAdYv1ugwFnP
> 蘋果比果汁多。果汁比蘋果少。
這就是今天的主題:53Please respect copyright.PENANAguVNHArUU1
> 比較 | Comparison53Please respect copyright.PENANAgSeCurDrM0
二:多、少、相等53Please respect copyright.PENANA3UnfTYm1uH
當我們比較兩組物件時,可以有三種結果。
1. 多於 | More53Please respect copyright.PENANADLHri93Qp0
如果 A 組比 B 組多,我們會說:53Please respect copyright.PENANARjs0ah1tji
> A 多於 B | A is more than B
例如:53Please respect copyright.PENANAb7NUlpqz85
> 🐱🐱🐱53Please respect copyright.PENANA5qKwJuNFXl
> 🐟🐟53Please respect copyright.PENANAdSX57BQA4B
貓有 3 隻,魚有 2 條。
所以:53Please respect copyright.PENANAZQieTVYCCv
> 貓多於魚。53Please respect copyright.PENANA694719aMG3
//
2. 少於 | Less than53Please respect copyright.PENANAayLOURCpPc
如果 A 組比 B 組少,我們會說:53Please respect copyright.PENANAZRUW9bFlyb
> A 少於 B | A is less than B53Please respect copyright.PENANAXdTuxJxN0f
例如:53Please respect copyright.PENANAS5HL0n88Ra
> 🐟🐟53Please respect copyright.PENANA0vP3UjBVqD
> 🐱🐱🐱53Please respect copyright.PENANAtWVbDe5IRI
魚有 2 條,貓有 3 隻。
所以:53Please respect copyright.PENANAFET1gKNRVp
> 魚少於貓。53Please respect copyright.PENANABovzoaCIFB
留意:53Please respect copyright.PENANAjkmRGSjSDO
> 「貓多於魚」和「魚少於貓」其實是同一回事,由兩個不同的方向理解。53Please respect copyright.PENANA2DoRXzPKeX
//
3. 相等 | Equal53Please respect copyright.PENANAbaAMlUM4l1
如果兩組一樣多,我們會說:53Please respect copyright.PENANAWMaXzbb6H7
> A 等於 B | A is equal to B53Please respect copyright.PENANAM3uEqfAdLD
例如:53Please respect copyright.PENANAPVQK60WcFy
> 🍎🍎🍎53Please respect copyright.PENANAZ9QlnomBNJ
> 🧃🧃🧃53Please respect copyright.PENANAJr2hptInzd
蘋果有 3 個,果汁有 3 盒。
所以:53Please respect copyright.PENANA9aJX8cGSfM
> 蘋果和果汁一樣多。53Please respect copyright.PENANACAxggjjYMw
在數學上,它叫做:53Please respect copyright.PENANA6eGgBohJz2
> 相等數量 | Equal quantity53Please respect copyright.PENANAkXNoqKu9nh
三:比較不一定要先數出數字53Please respect copyright.PENANAQ1Vzi6JYbD
上一課講過一個好重要的方法:53Please respect copyright.PENANAUliQRrLtpy
> 一一對應 | One-to-one correspondence53Please respect copyright.PENANAJOBhcD3Fif
即是每一個物件,配另一邊一個物件。
例如:53Please respect copyright.PENANAwtNr1K16ey
> 🐱 ↔ 🐟53Please respect copyright.PENANA3GUvI8gQdd
> 🐱 ↔ 🐟53Please respect copyright.PENANAVPT6iFXBsZ
> 🐱 ↔ 🐟53Please respect copyright.PENANAQwFus3onQw
> 🐱 ↔ ?53Please respect copyright.PENANAMY2gR4SbqI
魚已經配完,但還剩餘一隻貓。
所以:53Please respect copyright.PENANAT4ropZtCl4
> 貓比較多。53Please respect copyright.PENANA8xh9Vn2cWO
> 魚比較少。53Please respect copyright.PENANA5xvw4pgkhs
這個方法有時比直接數更清楚。
因為比較數量的核心不是「背記數字」,而是:53Please respect copyright.PENANAodbY34Hqac
> 「兩邊可不可以完整地配對?」53Please respect copyright.PENANADlmxrfFifG
四:簡單例子53Please respect copyright.PENANACOZ72c5eT6
例子 1:哪一邊多?53Please respect copyright.PENANAbZkaJEezzm
A 組:53Please respect copyright.PENANAqpajpmSAOt
> ⭐⭐⭐⭐53Please respect copyright.PENANAOfacfJju9a
B 組:53Please respect copyright.PENANAiKSdfYKRb8
> ⭐⭐53Please respect copyright.PENANAcvB7HVJpY8
步驟:
A 組數到:53Please respect copyright.PENANAWRwvwpfU0y
> 1、2、3、453Please respect copyright.PENANA2t5Bo6rDhZ
B 組數到:53Please respect copyright.PENANAdGrIk65hsH
> 1、253Please respect copyright.PENANAEvzMrieuu0
因為 4 比 2 多,所以:53Please respect copyright.PENANAVPNXl8NT0m
> A 組多;B 組少。53Please respect copyright.PENANAdeKDXmDjuE
//
例子 2:是否一樣多?
A 組:53Please respect copyright.PENANAWxDTjPt30T
> ▲▲▲53Please respect copyright.PENANAUlbjRW0EB1
B 組:53Please respect copyright.PENANACoTQXsHPyq
> ●●●53Please respect copyright.PENANAFpIxmEnlmH
用配對:53Please respect copyright.PENANABDeEDWHKkc
> ▲ ↔ ●53Please respect copyright.PENANADhyE41Otev
> ▲ ↔ ●53Please respect copyright.PENANAw6v0Sj0Xsi
> ▲ ↔ ●53Please respect copyright.PENANAAHzqmGJzQY
沒有剩餘的三角形,也沒有剩餘的圓形。
所以:53Please respect copyright.PENANAeLKczBdtHO
> A 組和 B 組一樣多。53Please respect copyright.PENANAadiXSl4sYc
即是:53Please respect copyright.PENANACizhpA45jo
> 3 等於 3。53Please respect copyright.PENANAt4YJ3aJ9Ac
五:引入符號:=、>、<53Please respect copyright.PENANAdTn4WRZV5F
數學會利用符號簡短表示比較。53Please respect copyright.PENANAljJzBnE81c
1. 等於符號:=53Please respect copyright.PENANAXutZvbvLC0
> 3 = 3
讀作:53Please respect copyright.PENANAkjz1pq52eW
> 「3 等於 3」53Please respect copyright.PENANAJiojgFOzcC
意思是:53Please respect copyright.PENANAYiIR6faiOu
> 左邊和右邊一樣大。53Please respect copyright.PENANAywu0O4i4Zo
//
2. 大於符號:>53Please respect copyright.PENANADvNKCIajp2
> 4 > 2
讀作:53Please respect copyright.PENANAj1dwpa7i1U
> 「4 大於 2」53Please respect copyright.PENANAWRgzbqqbf3
意思是:53Please respect copyright.PENANAqVUbS8D2IY
> 4 比 2 多。53Please respect copyright.PENANAZifzbHnZdZ
//
3. 小於符號:<53Please respect copyright.PENANAngsfjuWAZr
> 2 < 453Please respect copyright.PENANA5fGxAh2mWS
讀作:53Please respect copyright.PENANAOp90d5RI9V
> 「2 小於 4」53Please respect copyright.PENANAEVdD8OtFfd
意思是:53Please respect copyright.PENANAuujX8FFRIp
> 2 比 4 少。53Please respect copyright.PENANAt3bHTPLUdW
六:如何記住 > 和 <?53Please respect copyright.PENANAGXQhDjIjuN
可以將符號想像成一張開口。53Please respect copyright.PENANAAhLqC3gPEN
開口那邊,通常向往比較大的那個數。53Please respect copyright.PENANAE3oAxJa23K
例如:53Please respect copyright.PENANA16ZSQDLziJ
> 5 > 253Please respect copyright.PENANAOLTswNVnqT
開口向住 5,因為 5 比較大。
又例如:53Please respect copyright.PENANAO1o6c1BVYN
> 2 < 553Please respect copyright.PENANAmVWqMtpAJG
開口都是向住 5,因為 5 比較大。53Please respect copyright.PENANA2RWSbyYEdh
所以你可以這樣記:53Please respect copyright.PENANAszwZxVJdn3
> 「開口吃大數。」53Please respect copyright.PENANAbiQ886SvFA
但不好只靠口訣,最重要是要理解:53Please respect copyright.PENANAj75KXkyRhU
> 「左邊是否比右邊多?」53Please respect copyright.PENANAL1m3mbCLd3
> 「左邊是否比右邊少?」53Please respect copyright.PENANAVM2iZG4apU
> 「兩邊是否一樣?」53Please respect copyright.PENANADjWxYB3HZ5
七:例子,填入正確符號53Please respect copyright.PENANArNDJqLVOW0
例子 3:53Please respect copyright.PENANAMRFNm9bRwJ
> 3 _ 553Please respect copyright.PENANAkuty4CmUud
思考:53Please respect copyright.PENANAEf0Uol4DdF
3 是三件,5 是五件。53Please respect copyright.PENANAyEUM7DlpDB
> 3 比 5 少。53Please respect copyright.PENANAhi6rhqmYkk
所以:53Please respect copyright.PENANAEAb0RE0N99
> 3 < 553Please respect copyright.PENANARVKGUVFXvB
//
例子 4:53Please respect copyright.PENANAXQkICF1x9F
> 6 _ 253Please respect copyright.PENANAWteOJkfLcB
思考:53Please respect copyright.PENANAKuruqBS3VU
6 比 2 多。53Please respect copyright.PENANApYavZmQqOm
所以:53Please respect copyright.PENANAw9PKTbObSS
> 6 > 253Please respect copyright.PENANA0KjsO6nW3t
//
例子 5:53Please respect copyright.PENANAXQUpXyW4iE
> 4 _ 453Please respect copyright.PENANA1iNs4RXpFq
思考:53Please respect copyright.PENANA9A5V6TgYeT
兩邊都是 4。
所以:53Please respect copyright.PENANAPxHIcn2wUr
> 4 = 453Please respect copyright.PENANAiZyekLomjc
八:需要思考的例子,看似比較長,是不是一定較多?53Please respect copyright.PENANAG85TaXFKy1
A 組:53Please respect copyright.PENANA70lSVBryKB
> ●●●●53Please respect copyright.PENANAXk1QoB3PTd
B 組:53Please respect copyright.PENANAvExsWYunxo
> ● ● ●53Please respect copyright.PENANAv4fiAblISy
B 組排得很疏,好像佔用了更多空間。
問題:53Please respect copyright.PENANA5La37TkHce
> 「B 組是不是比 A 組多?」53Please respect copyright.PENANAvgXt5V1eLV
步驟 1:不好只靠視覺長度53Please respect copyright.PENANA3YRdgDutwg
B 組雖然排得長,但「長」不等於「多」。53Please respect copyright.PENANAywvZUdRLyi
步驟 2:數 A 組53Please respect copyright.PENANAFB05bvWrtc
> 1、2、3、453Please respect copyright.PENANAclbEWcco48
A 組有 4 個。53Please respect copyright.PENANAyLaXsuDJoK
步驟 3:數 B 組53Please respect copyright.PENANAafPQJMl8t3
> 1、2、353Please respect copyright.PENANAgU38zjllh1
B 組有 2 個。53Please respect copyright.PENANAzyWgvigyAT
結論:53Please respect copyright.PENANAfqR4UeZQ8m
> A 組多於 B 組。53Please respect copyright.PENANAiOXpedygbt
用符號寫:53Please respect copyright.PENANAVNXAnmTvr6
> 4 > 353Please respect copyright.PENANAp9Tpy7cjr3
這個例子提醒你:53Please respect copyright.PENANAzCLWWGU4AA
> 「數量和排列方式不是同一回事。」53Please respect copyright.PENANAnoutp2Zfgq
九:更深入,多了多少?53Please respect copyright.PENANAipRhlcULMa
比較不只可以講「哪一邊多」,也可以問:53Please respect copyright.PENANA3abmn3YS8P
> 「多了多少?」53Please respect copyright.PENANAxCG2blzhu5
例如:53Please respect copyright.PENANAQqNmcVaI7J
A 組:53Please respect copyright.PENANAzZCytSFM8J
> 🍎🍎🍎🍎🍎53Please respect copyright.PENANA48ohO5LvTq
B 組:53Please respect copyright.PENANAUnkBSVderB
> 🧃🧃🧃53Please respect copyright.PENANAobzUkadoiv
用配對:53Please respect copyright.PENANAaACeLHgM21
> 🍎 ↔ 🧃53Please respect copyright.PENANAhQCg6C2hHu
> 🍎 ↔ 🧃53Please respect copyright.PENANApSLWuYRppD
> 🍎 ↔ 🧃53Please respect copyright.PENANAu0MlnaVGBH
> 🍎 ↔ ?53Please respect copyright.PENANA9jl6G0EdTB
> 🍎 ↔ >53Please respect copyright.PENANA5QbvAKPb7B
果汁配完之後,還有 2 個蘋果未配對。
所以:53Please respect copyright.PENANAKqlRSyhddV
> 蘋果比果汁多 2 個。53Please respect copyright.PENANAlDWV7zDDpU
而果汁就可以說:53Please respect copyright.PENANAbBAudy4fq7
> 果汁比蘋果少 2 個。53Please respect copyright.PENANAiswt0yJcHG
留意:我們現在未正式學習「減法」 ,但你已經可以用「配對後剩餘多少」理解差距。
這個差距將來會變成:53Please respect copyright.PENANAcdX6v0kJEP
> 差 | Difference53Please respect copyright.PENANABVuN2gra85
十:思考例子:三組之間如何比較?53Please respect copyright.PENANAzL8hOBpl90
有三組物件:
A 組:53Please respect copyright.PENANArqzaohwSGR
> ⭐⭐⭐⭐⭐53Please respect copyright.PENANAzoEZzGT7nh
B 組:53Please respect copyright.PENANAkUxXts5XcO
> ▲▲▲
C 組:53Please respect copyright.PENANA1DtEBCmXRm
> ●●●●53Please respect copyright.PENANATTjS9nWJzX
問題:53Please respect copyright.PENANAPlXdGdHOU8
> 「哪組最多?」「哪組最少?」「C 組在中間嗎?」
步驟 1:數每組數量53Please respect copyright.PENANAuF6E46E0TA
A 組:53Please respect copyright.PENANApsk2oSfG1i
> 5 個53Please respect copyright.PENANAhtLd4GXYYn
B 組:53Please respect copyright.PENANA1ZjmWCYRvr
> 3 個53Please respect copyright.PENANAH9A3UZ7hzR
C 組:53Please respect copyright.PENANAwML7tdjYhW
> 4 個53Please respect copyright.PENANAMkz2gB4eS4
步驟 2:比較大小53Please respect copyright.PENANAW0g3ElvtgD
我們知道:53Please respect copyright.PENANAKu3Cpp5hyI
> 5 > 453Please respect copyright.PENANAfiPrM7RzjA
> 4 > 353Please respect copyright.PENANAIfbx1CPXY2
所以排列為:53Please respect copyright.PENANA7aw1beITUN
> 5 > 4 > 353Please respect copyright.PENANAuSHDHyckDO
對應物件:53Please respect copyright.PENANAYrqgFV6bmT
> A 組 > C 組 > B 組53Please respect copyright.PENANAzTIKTXeazd
結論:53Please respect copyright.PENANAD1vIBTe4Kp
最多:53Please respect copyright.PENANA82w39boI4s
> A 組53Please respect copyright.PENANAKGoa8qCCIa
最少:53Please respect copyright.PENANASF8jZ25vl7
> B 組53Please respect copyright.PENANAbdchc4U5ZT
C 組:53Please respect copyright.PENANA2dcKy0aTKz
> 在 A 組和 B 組中間。53Please respect copyright.PENANA0I3DA9DFgi
在這裏,你已經開始用一種新思想:53Please respect copyright.PENANAwoEbUNJWsg
> 排序 | Ordering53Please respect copyright.PENANADvppO32AqM
即是將數量由小至大,或者由大至小排好。
十一:最深一點:如果 A 多於 B、B 多於 C,哪 A 和 C 呢?53Please respect copyright.PENANAAA4g85u3cE
假設:53Please respect copyright.PENANAYPuoCcC75q
> A 多於 B53Please respect copyright.PENANAIGzjhKwCOT
> B 多於 C
哪 A 和 C 比較,哪個多?
我們用具體數字看:
> A = 653Please respect copyright.PENANAi8lZmOq6NW
> B = 453Please respect copyright.PENANAbvT5rna9oc
> C = 2
因為:53Please respect copyright.PENANArv2chmbJZL
> 6 > 453Please respect copyright.PENANAdLzNPzuora
> 4 > 253Please respect copyright.PENANA8ebHOedp7e
所以:53Please respect copyright.PENANAd4JrUtbAzd
> 6 > 253Please respect copyright.PENANA1lvbQdlQDp
即是:53Please respect copyright.PENANARtaSdSsDwq
> A 多於 C。53Please respect copyright.PENANAaBn38bRl3p
這個思想叫做:53Please respect copyright.PENANAxDugZXWudn
> 傳遞性 | Transitivity53Please respect copyright.PENANACpclHF81K3
你現在不用記住這個名字,但要明白意思:53Please respect copyright.PENANAQdpOCGIEH1
> 「如果第一個比第二個大,第二個又比第三個大,那麼,第一個一定比第三個大。」53Please respect copyright.PENANA6yCWVaoOGw
用簡單形式寫:53Please respect copyright.PENANA1syLM1fMzP
> 如果 A > B,而且 B > C,53Please respect copyright.PENANARtULHWOSEI
> 那 A > C。53Please respect copyright.PENANACkf7Bqr5cS
這個將來會在代數、幾何、證明裏面不停出現。
十二:容易混淆的地方53Please respect copyright.PENANAqwuqYRkI73
混淆 1:物件大小不等於數量多少53Please respect copyright.PENANAZ7S60S9THU
例如:
A 組:53Please respect copyright.PENANAdjLC1GIGB0
> 🐘🐘53Please respect copyright.PENANALVRgwB2llX
B 組:53Please respect copyright.PENANAWN2Z9srgcH
> 🐜🐜🐜🐜🐜53Please respect copyright.PENANApxY1K6tVYc
大象很大,螞蟻很小。
但數量上:53Please respect copyright.PENANAwcjbmGVG49
> 大象有 2 隻53Please respect copyright.PENANAyl65L0LQwW
> 螞蟻有 5 隻53Please respect copyright.PENANA46GwEp3Lsj
所以:53Please respect copyright.PENANABn6XAa7A5V
> 螞蟻比較多。53Please respect copyright.PENANA5uICwaqJXf
數量比較問的是:53Please respect copyright.PENANAGW9q7dMW5B
> 「有多少個。」53Please respect copyright.PENANAbn2tFV3mn8
不是:53Please respect copyright.PENANAT70ECfL0VK
> 「每個有多大。」53Please respect copyright.PENANA2Egh0x5GRY
//
混淆 2:排得長不等於多53Please respect copyright.PENANAllCIjG2LT8
例如:53Please respect copyright.PENANAInxWZhvx2o
> ● ● ●53Please respect copyright.PENANAsOMHWA0a1P
> ●●●●53Please respect copyright.PENANAEpT4Ifpdre
上面排得長,但只有 3 個。53Please respect copyright.PENANAafZMl8vBXC
下面排得短,但有 4 個。53Please respect copyright.PENANAJwdo7ojvny
所以:53Please respect copyright.PENANAsbYSuPrYtj
> 下面比較多。53Please respect copyright.PENANASddR25qAtL
//
混淆 3:相等不代表完全一樣53Please respect copyright.PENANAVE09MpQyfG
例如:53Please respect copyright.PENANABdbh6eyVmw
> 🍎🍎🍎53Please respect copyright.PENANAqaem5fNJ5A
> 🐟🐟🐟53Please respect copyright.PENANAmW51jgP9X4
蘋果和魚完全不同。
但數量一樣:53Please respect copyright.PENANA392aSI2KTI
> 3 = 353Please respect copyright.PENANAE7fuemd4hk
所以可以說:53Please respect copyright.PENANAQQo8NU3VhV
> 他們數量相等。53Please respect copyright.PENANAo2KeBaio5A
相等可以只是指某一方面相等。53Please respect copyright.PENANAzGnfnPK6SA
今課中指的相等,是:53Please respect copyright.PENANAuhg5FxerY9
> 數量相等。53Please respect copyright.PENANAB3HpVHF3oF
十三:今日小結53Please respect copyright.PENANA0r8P9EBh37
今課你掌握了:53Please respect copyright.PENANAeJo7pJEGfA
1. 比較數量有三種結果:多於、少於、相等。53Please respect copyright.PENANAUfE3bE9XbT
2. 可以用數數比較,也可以用一一對應比較。53Please respect copyright.PENANAAkCWLqeAoa
3. 「=」比示等於,「>」表示大於,「<」表示少於。53Please respect copyright.PENANAQtaSqPKfnp
4. 排列疏密、物件大小、物件種類,都不一定影響數量。53Please respect copyright.PENANArDoPzhZh12
5. 如果 A > B,而且 B > C,那 A > C。這個是比較的傳遞性。53Please respect copyright.PENANAJSSJBIb8Cx


