me.
minus.
me.
equ als
me.
SUBTRACTING COUNTING NUMBERS
The third scene in a series of articles
on elementary mathematics.
written by eugene maier
designed and illustrated by Tyson Smith
t‹ˆ@•–ˆ˜„šŒ•”@•‰@™›…š˜„†šŒ•”@†„”@ˆŒš‹ˆ˜@…ˆ@œŒˆžˆ‡@„™@š‹ˆ@–˜•†ˆ™™@•‰@Bš„‘Œ”Š@„ž„ B@
•˜@š‹„š@•‰@B‰Œ”‡Œ”Š@š‹ˆ@‡Œ‰‰ˆ˜ˆ”†ˆBZ@X@–@U@†„”@…ˆ@š‹•›Š‹š@•‰@„™@ž‹„šG™@’ˆ‰š@ž‹ˆ”@U@šŒ’ˆ@
„˜ˆ@˜ˆ“•œˆ‡@‰˜•“@„@–Œ’ˆ@•‰@X@šŒ’ˆL@•˜@X@– U@†„”@…ˆ@š‹•›Š‹š@•‰@„™@š‹ˆ@‡Œ‰‰ˆ˜ˆ”†ˆ@
…ˆšžˆˆ”@„@–Œ’ˆ@•‰@X@šŒ’ˆ@„”‡@„@–Œ’ˆ@•‰@U@šŒ’ˆN@i”@…•š‹@†„™ˆ™L@ŒšG™@S@šŒ’ˆN
Taking Away
8-5=3
Finding The Difference
8-5=3
the difference
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FYI:
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‹„™@WT@“„˜…’ˆ™@„”‡@j•‹”@‹„™@SXL@‹•ž@“„” @“•˜ˆ@“„˜…’ˆ™@‡•ˆ™@m„˜ @
‹„œˆ@š‹„”@j•‹”_B
@
t‹ˆ@‡Œ‰‰ˆ˜ˆ”†ˆ@“•‡ˆ’@•‰@™›…š˜„†šŒ•”@Œ™@„’™•@˜ˆ‰ˆ˜˜ˆ‡@š•@„™@š‹ˆ@†•“–„˜Œ™•”@
“•‡ˆ’@™Œ”†ˆ@šž•@„“•›”š™@„˜ˆ@…ˆŒ”Š@†•“–„˜ˆ‡@žŒš‹@•”ˆ@„”•š‹ˆ˜N
b„™ˆ@šˆ”@–Œˆ†ˆ™@†„”@…ˆ@›™ˆ‡@š•@
Œ’’›™š˜„šˆ@…•š‹@–˜•†ˆ™™ˆ™N
s›––•™ˆL@‰Œ˜™šL@žˆ@†•“–›šˆ@
253
RUS@½@QVX@… @š‹ˆ@š„‘ˆM„ž„ @
“ˆš‹•‡N@a@†•’’ˆ†šŒ•”@‰•˜@RUS@
†•”š„Œ”™@R@“„š™L@U@™š˜Œ–™@„”‡@S@
›”Œš™N@f˜•“@š‹Œ™@žˆ@ž„”š@š•@š„‘ˆ@
„ž„ @š‹ˆ@ˆ—›Œœ„’ˆ”š@•‰@„@†•’’ˆ†šŒ•”@ 253
‰•˜@QVXL@”„“ˆ’ @Q@“„šL@V@™š˜Œ–™@„”‡@
X@›”Œš™N@sŒ”†ˆ@š‹ˆ@†•’’ˆ†šŒ•”@‰•˜@RUS@
‡•ˆ™”Gš@†•”š„Œ”@V@™š˜Œ–™@„”‡@X@›”Œš™L@
žˆ@Œ”†˜ˆ„™ˆ@š‹ˆ@”›“…ˆ˜@•‰@™š˜Œ–™@
„”‡@›”Œš™@Œ”@š‹ˆ@†•’’ˆ†šŒ•”@‰•˜@RUS@
… @š˜„‡Œ”Š@Œ”@„@“„š@‰•˜@QP@™š˜Œ–™@
„”‡@š‹ˆ”@š˜„‡Œ”Š@Œ”@•”ˆ@•‰@š‹ˆ@
˜ˆ™›’šŒ”Š@QU@™š˜Œ–™@‰•˜@›”Œš™L@
˜ˆ™›’šŒ”Š@Œ”@„@†•’’ˆ†šŒ•”@•‰@RUS@†•”™Œ™šŒ”Š@•‰@Q@
“„šL@QT@™š˜Œ–™@„”‡@QS@›”Œš™N@rˆ“•œŒ”Š@Q@“„šL@
V@™š˜Œ–™@„”‡@X@›”Œš™@‰˜•“@š‹Œ™@†•’’ˆ†šŒ•”@’ˆ„œˆ™@
X@™š˜Œ–™@„”‡@U@›”Œš™N@hˆ”†ˆ@RUS@½@QVX@]@XUN
Mats
Strips
253 - 168 = 85
Units
a@œ„˜Œ„šŒ•”@•‰@š‹ˆ@–˜ˆœŒ•›™@“ˆš‹•‡L@ž‹ˆ˜ˆ@—›„”šŒšŒˆ™@
„˜ˆ@˜ˆ“•œˆ‡@–Œˆ†ˆ@… @–Œˆ†ˆ@˜„š‹ˆ˜@š‹„”@„’’@„š@•”†ˆL@
’ˆ„‡™@š•@„@™š„”‡„˜‡@–„–ˆ˜M„”‡M–ˆ”†Œ’@–˜•†ˆ‡›˜ˆZ@wˆ@
’Œ™š@„@†•’’ˆ†šŒ•”@‰•˜@RUS@„”‡@…ˆ”ˆ„š‹@Œš@žˆ@’Œ™š@š‹ˆ@
†•’’ˆ†šŒ•”@žˆ@ž„”š@š•@š„‘ˆ@„ž„ L@”„“ˆ’ @QVXL@„”‡@
‡˜„ž@„@’Œ”ˆ@›”‡ˆ˜@ŒšN@wˆ@…ˆŠŒ”@… @‰Œ˜™š@˜ˆ“•œŒ”Š@X@
›”Œš™@‰˜•“@RUSN@i”@•˜‡ˆ˜@š•@‡•@š‹Œ™L@žˆ@ˆŸ†‹„”Šˆ@„@
™š˜Œ–@‰•˜@QP@›”Œš™L@†‹„”ŠŒ”Š@š‹ˆ@”›“…ˆ˜@•‰@™š˜Œ–™@Œ”@
š‹ˆ@†•’’ˆ†šŒ•”@‰•˜@RUS@‰˜•“@U@š•@T@„”‡@š‹ˆ@”›“…ˆ˜@•‰@
›”Œš™@š•@QSN@wˆ@˜ˆ“•œˆ@š‹ˆ@X@›”Œš™@„”‡@˜ˆ†•˜‡@ž‹„šG™@
’ˆ‰š@…ˆ’•ž@š‹ˆ@’Œ”ˆN@wˆ@”ˆŸš@ž„”š@š•@˜ˆ“•œˆ@V@™š˜Œ–™N@
t•@‡•@š‹Œ™@žˆ@ˆŸ†‹„”Šˆ@„@“„š@Œ”@•›˜@•˜ŒŠŒ”„’@
†•’’ˆ†šŒ•”@‰•˜@QP@™š˜Œ–™L@†‹„”ŠŒ”Š@š‹ˆ@”›“…ˆ˜@•‰@
“„š™@š•@Q@„”‡@š‹ˆ@”›“…ˆ˜@•‰@™š˜Œ–™@š•@QTN@wˆ@”•ž@
˜ˆ“•œˆ@V@™š˜Œ–™L@’ˆ„œŒ”Š@X@ž‹Œ†‹@žˆ@˜ˆ†•˜‡@…ˆ”ˆ„š‹@
š‹ˆ@’Œ”ˆN@fŒ”„’’ @žˆ@˜ˆ“•œˆ@š‹ˆ@“„šN
Mats Strips Units
1
2
1
0
@@@@@@@@@@@@@@@@@@@i”@™†‹••’L@š‹ˆ@–˜•†ˆ™™@•‰@ˆŸ†‹„”ŠŒ”Š@„@–Œˆ†ˆ@‰•˜@QP@
Gene
says:
14
™“„’’ˆ˜@–Œˆ†ˆ™@Œ™@•‰šˆ”@˜ˆ‰ˆ˜˜ˆ‡@š•@„™@B…•˜˜•žŒ”ŠLB@ž‹Œ†‹@™š˜Œ‘ˆ™@“ˆ@„™@
„@“Œ™”•“ˆ˜N@u™›„’’ @Œ‰@•”ˆ@…•˜˜•ž™@™•“ˆš‹Œ”ŠL@š‹ˆ@Œ“–’Œ†„šŒ•”@Œ™@š‹„š@
Œš@žŒ’’@…ˆ@˜ˆš›˜”ˆ‡N
5
6
8
1
3
8
5
t•@†•“–›šˆ@RUS@½@QVX@… @š‹ˆ@
‡Œ‰‰ˆ˜ˆ”†ˆ@“ˆš‹•‡L@…„™ˆ@šˆ”@
†•’’ˆ†šŒ•”™@‰•˜@RUS@„”‡@QVX@
„˜ˆ@‡Œ™–’„ ˆ‡@„”‡@š‹ˆ”@š‹ˆŒ˜@
‡Œ‰‰ˆ˜ˆ”†ˆ@Œ™@‡ˆšˆ˜“Œ”ˆ‡N@i”@
š‹ˆ@Œ’’›™š˜„šŒ•”@™‹•ž”L@„@
†•’’ˆ†šŒ•”@‰•˜@QVX@Hš‹ˆ@’ŒŠ‹š@
Š˜„ @˜ˆŠŒ•”I@‹„™@…ˆˆ”@
™›–ˆ˜Œ“–•™ˆ‡@•”@„@†•’’ˆ†šŒ•”@
‰•˜@RUSN@@t‹ˆ@‡Œ‰‰ˆ˜ˆ”†ˆL@X@
™š˜Œ–™@„”‡@U@›”Œš™@Œ™@™‹•ž”@Œ”@…’„†‘N@n•šŒ†ˆ@
š‹„š@š‹ˆ@‡Œ‰‰ˆ˜ˆ”†ˆ@XU@Œ™@ž‹„š@“›™š@…ˆ@
„‡‡ˆ‡@š•@QVX@š•@Šˆš@RUSN
168
170
200
253
2
30
53
85
253 compared to 168
the difference is 85
t‹Œ™@’„™š@˜ˆ“„˜‘@’ˆ„‡™@š•@„@‰„Œ˜’ @™Œ“–’ˆ@–„–ˆ˜M„”‡M–ˆ”†Œ’@
–˜•†ˆ‡›˜ˆ@‰•˜@‰Œ”‡Œ”Š@„@‡Œ‰‰ˆ˜ˆ”†ˆ@š‹„š@‡•ˆ™”Gš@˜ˆ—›Œ˜ˆ@“„‘Œ”Š@
„” @ˆŸ†‹„”Šˆ™N@wˆ@…ˆŠŒ”@… @˜ˆ†•˜‡Œ”Š@QVX@„”‡@š‹ˆ”@žˆ@„‡‡@
—›„”šŒšŒˆ™@š•@š‹„š@„“•›”š@›”šŒ’@žˆ@Šˆš@š•@RUSN@i”@š‹ˆ@ˆŸ„“–’ˆ@
™‹•ž”L@@žˆ@‰Œ˜™š@„‡‡ˆ‡@R@š•@Šˆš@š•@QWPL@š‹ˆ”@žˆ@„‡‡ˆ‡@SP@š•@Šˆš@
š•@RPP@„”‡@‰Œ”„’’ @US@š•@Šˆš@š•@•›˜@‡ˆ™Œ˜ˆ‡@Š•„’@•‰@RUSN@t‹ˆ@
‡Œ‰‰ˆ˜ˆ”†ˆ@Œ™@š‹ˆ@™›“@•‰@š‹ˆ@„“•›”š™@„‡‡ˆ‡N
@@@@@@@@@@a™@Œ”@š‹ˆ@–˜ˆœŒ•›™@ˆŸ„“–’ˆL@‰Œ”‡Œ”Š@š‹ˆ@
FYI...@
‡Œ‰‰ˆ˜ˆ”†ˆ@…ˆšžˆˆ”@šž•@”›“…ˆ˜™@„“•›”š™@š•@
‰Œ”‡Œ”Š@ž‹„š@“›™š@…ˆ@„‡‡ˆ‡@š•@š‹ˆ@™“„’’ˆ˜@”›“…ˆ˜@
Œ”@•˜‡ˆ˜@š•@Šˆš@š‹ˆ@’„˜Šˆ˜@”›“…ˆ˜N@t‹›™L@™›…š˜„†šŒ•”@
†„”@…ˆ@œŒˆžˆ‡@„™@‰Œ”‡Œ”Š@š‹ˆ@“Œ™™Œ”Š@„‡‡ˆ”‡@Œ”@
the
„”@„‡‡ŒšŒ•”@–˜•…’ˆ“N@f•˜@ˆŸ„“–’ˆ@QR@½@W@Œ™@š‹ˆ@
”›“…ˆ˜@š‹„š@“›™š@…ˆ@–’„†ˆ‡@Œ”@š‹ˆ@…•Ÿ@š•@“„‘ˆ
W@K@@@@@@@]@QR@„@†•˜˜ˆ†š@™š„šˆ“ˆ”šN
@
the difference
stays the same
7+
= 12
missing addend is 5
12 - 7 = 5
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‰•’’•žŒ”Š@†•“–›š„šŒ•”@˜ˆ™›’š™@Œ”@„@™Œ“–’ˆ@
™›…š˜„†šŒ•”Z@QXU@½@VW@]@QXX@½@WP@]@QQXN
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Œ’’›™š˜„šˆ‡@•”@š‹ˆ@˜ŒŠ‹šN@t•@†•“–›šˆ@XSU@½@VTYL@˜„š‹ˆ˜@
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‰•’’•žŒ”ŠZ@sŒ”†ˆ@U@Œ™@’ˆ™™@š‹„”@YL@Œ”†˜ˆ„™ˆ@š‹ˆ@›”Œš™@
‡ŒŠŒš@U@Œ”@š‹ˆ@š•–@”›“…ˆ˜@š•@QU@„”‡@š‹ˆ@šˆ”™@‡ŒŠŒš@Œ”@
š‹ˆ@…•šš•“@”›“…ˆ˜@š•@U[@š„‘ˆ@Y@‰˜•“@QU@„”‡@˜ˆ†•˜‡@
š‹ˆ@˜ˆ™›’š@HVIN@t‹ˆ”L@™Œ”†ˆ@S@Œ™@’ˆ™™@š‹„”@UL@Œ”†˜ˆ„™ˆ@š‹ˆ@
šˆ”™@‡ŒŠŒš@S@Œ”@š‹ˆ@š•–@”›“…ˆ˜@š•@QS@„”‡@Œ”†˜ˆ„™ˆ@š‹ˆ@
‹›”‡˜ˆ‡™@‡ŒŠŒš@V@Œ”@š‹ˆ@…•šš•“@”›“…ˆ˜@š•@W[@š„‘ˆ@U@
‰˜•“@QS@„”‡@˜ˆ†•˜‡@š‹ˆ@˜ˆ™›’š@HXIN@fŒ”„’’ @š„‘ˆ@W@‰˜•“@
X@„”‡@˜ˆ†•˜‡@š‹„š@˜ˆ™›’š@HQIN
Mats Strips Units
1
8
76
3
54
1
8
1
5
9
6
t‹ˆ@˜ˆ„™•”@š‹Œ™@„’Š•˜Œš‹“@ž•˜‘™@Œ™@š‹„š@š‹ˆ@™„“ˆ@„“•›”š@
Œ™@…ˆŒ”Š@„‡‡ˆ‡@š•@…•š‹@”›“…ˆ˜™L@ž‹Œ†‹@‡•ˆ™”Gš@†‹„”Šˆ@š‹ˆ@
‡Œ‰‰ˆ˜ˆ”†ˆ@…›š@‡•ˆ™@“„‘ˆ™@Œš@ˆ„™Œˆ˜@š•@†•“–›šˆN@i”@š‹ˆ@
„…•œˆ@ˆŸ„“–’ˆL@š‹ˆ@”›“…ˆ˜@•‰@›”Œš™@Œ”@„@†•’’ˆ†šŒ•”@‰•˜@
XSU@Œ™@Œ”†˜ˆ„™ˆ‡@… @QP@„”‡L@Œ”@•˜‡ˆ˜@š•@‘ˆˆ–@š‹ˆ@
‡Œ‰‰ˆ˜ˆ”†ˆ@›”†‹„”Šˆ‡L@š‹ˆ@†•’’ˆ†šŒ•”@‰•˜@VTY@Œ™@Œ”†˜ˆ„™ˆ‡@
… @š‹ˆ@™„“ˆ@„“•›”š@… @„‡‡Œ”Š@Q@™š˜Œ–@š•@ŒšN@t‹ˆ”@š‹ˆ@
”›“…ˆ˜@•‰@™š˜Œ–™@Œ”@š‹ˆ@‰Œ˜™š@†•’’ˆ†šŒ•”@Œ™@Œ”†˜ˆ„™ˆ‡@… @QP@
„”‡@š‹ˆ@™ˆ†•”‡@†•’’ˆ†šŒ•”@Œ™@Œ”†˜ˆ„™ˆ‡@š‹ˆ@™„“ˆ@„“•›”š@
… @„‡‡Œ”Š@Q@“„š@š•@ŒšN@a’š‹•›Š‹@š‹ˆ@œ„’›ˆ@•‰@š‹ˆ@
†•’’ˆ†šŒ•”™@Œ™@†‹„”Šˆ‡L@š‹ˆŒ˜@‡Œ‰‰ˆ˜ˆ”†ˆ@˜ˆ“„Œ”™@š‹ˆ@™„“ˆN
@@@@@@@@@@@@@@@@@@@@t‹ˆ˜ˆ@„˜ˆ@šˆ†‹”Œ†„’@”„“ˆ™@‰•˜@š‹ˆ@”›“…ˆ˜™@Œ”œ•’œˆ‡@Œ”@„@™›…š˜„†šŒ•”N@
Gene
says:
t‹ˆ@”›“…ˆ˜@š•@…ˆ@™›…š˜„†šˆ‡@‰˜•“@Œ™@†„’’ˆ‡@š‹ˆ@“Œ”›ˆ”‡@Hš‹„š@ž‹Œ†‹@Œ™@š•@…ˆ@
’ˆ™™ˆ”ˆ‡L@‰˜•“@š‹ˆ@l„šŒ”@“Œ”›ˆ˜ˆL@š•@’ˆ™™ˆ”I@„”‡@š‹ˆ@”›“…ˆ˜@™›…š˜„†šˆ‡@Œ™@†„’’ˆ‡@š‹ˆ@
™›…š˜„‹ˆ”‡@Hš‹„š@ž‹Œ†‹@Œ™@š•@…ˆ@žŒš‹‡˜„ž”L@‰˜•“@š‹ˆ@l„šŒ”@™›…š˜„‹ˆ˜ˆL@š•@
žŒš‹‡˜„žIN@i@‘ˆˆ–@†•”‰›™Œ”Š@š‹ˆ™ˆ@ž•˜‡™@„”‡@”ˆœˆ˜@›™ˆ@š‹ˆ“@žŒš‹•›š@‰Œ˜™š@†‹ˆ†‘Œ”Š@
š‹ˆŒ˜@“ˆ„”Œ”Š@Œ”@„@‡Œ†šŒ•”„˜ L@„™@i@Ž›™š@‡Œ‡N@pˆ˜‹„–™@Œ‰@“ @l„šŒ”@žˆ˜ˆ@…ˆššˆ˜¬
@@@@@@@@t‹ˆ@•–ˆ˜„šŒ•”@•‰@™›…š˜„†šŒ•”@Œ™@”ˆŒš‹ˆ˜@
FYI...
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4
1
7 4
(7 - 4)
(7 - 4) - 1
3-1
2
1
7
(4 - 1)
7 - (4 - 1)
7 - (3)
4
END of SCENE 3: SUBTRACTING COUNTING NUMBERS
For comments and questions please email
Gene Maier at [email protected]
coming up next...
SCENE 4: MULTIPLYING COUNTING NUMBERS