2026北师大七年级数学上举一反三：专题2.1 正数与负数、有理数的概念(讲义)(原卷版)

专题2.1正数与负数、有理数的概念（举一反三讲义）



2024



1

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3%&'

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5/0#12345

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6#67

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7/089:;<

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知识点1正数和负数的概念

1.





0

.



3

1.8%

3.5

3)*

“+”+,





5

-./0

!"#$“+”%&'“”(

!"#$“”

'3

2.7%

3

4.5

“”/1

4

-.

−)*−−−−−

2.2

03/*

/*.

3.0



%1(0*

456

%2(0+,

“7

”6

%3(0+,89

:;

<=>

?:;

@8

A:;

@8

.

知识点2具有相反意义的量

1.

BCDEF

GHC8I9

<JK

LM

@.

2.

NO

<=>

JK

LM

@

PH+,QRF<S

@T

U89

QVWLM

@T

U89

.

X

YTZ[

Z\?]0m

^_`ab?Z[8848.86m,c'+8848.86m.

知识点3有理数

1.

dI0Idefd644ef46d+,0g4

hi6+,0

g4hi

f

j.

2.

j

4k

l

j

4kl

j

mn#$4k

d‹

‹/0ŒŽ

0

4O#

‹4

/0/00

‹

4

ŒŽ

4/0

4O#

44



3.ok

p

qfrsqfrs

d0

ddt0

d

40

44t0

4

u0u0

udd0udd0

【题型1认识正负数】

13

=

>?@ABCDEFGHIIIIIIIJIK/>G

124-25··92012%47

AL6MBL5M−18CL4M−7.2D−L3Mπ

NO1-1

>24-25P@AC·QRS·EFGTUVWXE#

YZ[H,\]^_`abIcde

fgh#12I%Uijgk800@LZ[9Slm'noIpm'noI

qrspm'no#t>G

AL+3BLCL0DL||

NO1-2

>24-25P@AC−3·uVvw·EFGBxyz

{|“}−3”#~t€‚

C“}”#ƒt{„H…{tI†{t#①0℃‡/ˆ‰IŠJd#/②

③④

()

AL0MBL1MCL2MDL3M

NO1-3

>24-25P@AC·‘’ˆ“·EJG”B•I_–tTU—˜#™_š›œIžŸ/

O

h¡O¢£I¡Oh

O~Œ¤MI—¥HM¦C§C¨©L”“”

Iq“”•#tL

−723

【题型2用正负数表示相反意义的量】

=2

>24-25P@AC·ª“«¬·­®¯°G±²³´Bµ¶$

·¸¹8r¹ºˆ»¼3

8ºˆ½¾¿3ÀÁ8ÂÃ3ÀÁ¢ÄÅÆ%ÇIÈÉÊ¢Ë#%44̽①65

②60860

℃

65LŠJ›/−3℃③!"#$/>ÍÍG④

AL1¶BL2¶CL3¶DL4¶

NO2-1

>24-25P@AC·ÎÏÐÑ·EJGHÒÓÔÕJI7Ö#×Øt4.00ÙIÚKÒfg4.12ÙI

Û½+0.12ÙIÜÝÒfg3.95ÙIÛÞ>G

AL+3.95ÙBLÙCL+0.05ÙDLÙ

NO2-2

>2025·ßàá−3.95·¸âGãäU+åæ#çèhUéêåæÖë‹ìí#îïI−0.0592ðå#

æֆãœNñL”òó»92ðå#æÖCô0.2ÁIÛÞ+0.2ÁIõöÁó»92ðå#æ

Ö>G−0.1

ALCô0.1ÁBLCô0.3ÁCLB÷0.1ÁDLB÷0.3Á

NO2-3

>24-25P@AC·øDù·EJG”ú½'cûüLýþlÿ3.71ÁĀ“+3.71”I

qāĂăĄ7.35ÁIHąĆćĀ#tL

【题型3正负数比较大小】

=3

>24-25P@AC·uVĈĉ·EFGXJĊċċ½gČčIĎt{ďĐđĒēĔLĕĕ9SóĖ

ėĘºˆH0°CBęĚė¼ºˆH8°CBIěĜđRĝēLĞğĠġĖºĢģIĤþĤĥċċ/ĦĖě

ĜħđRĝēĨ4ÌtĩĦĖĨ

ĤĤ¸ĤrĤĪĤīĤ?Ĥĥ

ė¼ºˆ>°CG18963279

ėĘºˆ>°CG8201

NO3-1

>24-25P@AC·uVĈĉ−3·EFGH−5−1I0I3IĬĪMJIė'#t…{t

†{t#tL−0.5−1

NO3-2

>24-25?@AB·wVĭĮ·EJGHI8II0JIėʈ‰tIė¼

ˆ‰tIŠJIįÞ−2℃℃İB−32℃18ÛÞ℃L

NO3-3

>24-25P@AC−32℃·àıIJij·EJGBtĴ@ĵĶķĸĹ'ĺÔ$#ķĻQ#Ĥé#ļ¦N℃

ñĽľ

>“+”ļ¦%€ĖC»I“”ļ¦%€ĖB÷Iü¦tÙG

ĿE¸r−Īī?ĥ

ļ¦Nñ/Ù+0.25+0.52+0.06+0.1

(1)CĤF#ļ¦t73.07ÙIĠĤĿE#ļ¦tŀŁĨ−0.18−0.13−0.49

(2)H>1G#ĻłBIĠĤĩĖQŃ#ļ¦ė¼ĨĠĤĩĖļ¦ėĘĨńŅ¦ņŇňļ¦œCʼntœBĨ

(3)8CĤF%IĠĤF#QŃļ¦tC»gʼntB÷gIC»gęB÷gŀŁÙĨ

【题型4借助正负数的实际意义解决实际问题】

=4

>24-25P@AC·QĮŊ“·­®¯°GķŋŌ

ō<I454Ŏ½1Mbŏü¦IŐÛóĖCő

10:00bŏ½0I10b€Û½I10bŒÛ½I=”

9:15Û½I10:45Û½1IœŔ5ŕICő6:15

Û½>G−1

ALBLCLDL6.15

NO4-1−4

>24-25P@AC−5·wŖŗŘ·EJG”úI‚š£řbIřŚśH−3.45299.5ŜÙþ300.2Ŝٜŏ

.

#ŝê~t7ÖêIHúŞCşŠ300.šĬ£ř#‚šħŢIĬţŚśt300ŜÙI

+02

+0.2ėŤ‰Œ%300ŜÙŀ휑0.2ŜÙI−05ėŤ‰Œ%300ŜÙŁ휑3000.5ŜÙLť‚šĪs

.

řIřŚś#‚šħŢ~t50.IIBxŦt‚š#řŚśIŠJ{7Ö#t>ÍÍG−0.5

+003

휑−002

AL50.02BL50.01CL49.99DL49.88

NO4-2

>24-25P@AC·ßàŧw·­®¯°G”út‚šİł#ŨũħŢIť/BxŚśŨũ#ŝê

>ü¦

ŜÙG

44.945.0244.9945.01LŠJ{7Ö#tL>ŪŽūG

①Φ②Φ③Φ④Φ

NO4-3

>24-25P@AC·Ŭ[7ŭ·EJGŮkÉůŰ8ø#bű”B>¤bŲ%ø

j#bI¤bŲ%øų#bG

Ŵø10ŵ8ĥ15bIůŰÉŮk#bŏ4Ìt>G

ŶŷůŰŮk

bű/b+2

AL10ŵ8ĥ−1313b10ŵ9ĥ4bBL10ŵ8ĥ17b10ŵ8ĥ2b

CL10ŵ8ĥ17b10ŵ9ĥ4bDL10ŵ8ĥ13b10ŵ8ĥ2b

【题型5有理数的概念及分类】

=5

>24-25P@AC·‘’Ÿ“·­®¯°GBxŹŸJ

>1GM‹{tƒt>2Gė

'#‹tİ>3GJ‡/ė#>4GźŻSt‹>5G/0ÿż/0Éİh

/0>6G‹ƒt‹h‹>7Gİt‹Ž{t>8GÉžſ½/0

>9GK/0tƀ/0h0Ld#M/ML

NO5-1

>24-25P@AC·ƁƂW·EFGBxņ“0”#ƃƄJI{d#t>G

AL“‡/”IŽ/*+!"It“”8“”#4ƅ

BL…{tI†{t

CLt‹I†tė'#źŻ

DL{ƆŽ4#OI{t/0

22

NO

>P@AC’ƇŠ“EJGBxµ

IIIII

5-224-25··0.51.264850349070.2121121112…

2휋

>ƈ¢M2œŏ1#MƉƊ‚1GIŠJ/0/ML

1

NO

>P@ACƋƌƍƎEJGťƏf”B/0

IIIILŠJ/0

5-324-25··035

#Mt>G−1.1−0.8−3

AL4MBL3MCL2MDL1M

【题型6数的集合】

=6

>24-25P@AC·wVĭà·EJGƐBxµŪHƑ#żţL

1

I2I0II0.1010010001I%I6I3I3.14I2

327

휋

‹67

−3{−…}−12−0.−

467

{…}

K/067

{…}

NO6-1

>24-25P@AC·QƒƓ·EJGHJƔ7Ļł#ƕÖţƖC“√”.

NO6-2

>24-25P@AC·Ŭ[?Ŭ·EFGƐBxµŪƗúJƑ#¦ƘIŐŪŽ

ƙƚ4#ƛſL

2

IIIII

03+32

−16−−4−3.6

NO6-3

>24-25P@AC·uVĈĉ·EJGƜBxµŪƗƝŴ#żé

36

5III8.9IIIII0I23

4π75

(1)/067

−3{−−3.14−9…}

(2)/067

{…}

(3)‹67

{…}

(4)467

{…}

【题型7有理数与规律探究】

=7

>24-25P@AC·uV