foil - university of albertaurban/projects/dice/pmvertifoil.pdf · –6a–10 + 12a2+20a...
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FOIL
Frustrate, thwart,baffle, impede
From
Prevent success(c. late 17C)
↑Obscure a scent (by trampling over)
(c. late 13C)
OF fouler: to trample, tread down, full↑
L fullare: to clean cloth (by treading on it)↑
L fullo: fuller, one who cleans cloth
Cover, enhance,counter, contrast
From(One who)
Enhance(s) by contrasting(c. late 16C)
↑Cover, provide backing
(to offset, keep from, cover)↑
Thin sheet of metal(c. late 14C)
L folium: leaf, blade
Metallicwrap(1946)
Sword, fencing
From
Fencing, sword
Unknown origin(c. late 16C)
First, Outside, Inside, Last(F.O.I.L.)
Acrostic mnemonicfor a special case
(double binomials only)
of thedistributive multiplication
of polynomials
From
William Betz – Algebra for Today(c. 1929)
Do you multiply multidigit numbers
such as 12 and 36
horizontally
like
¹36 × 12 = 72 + ¹360 = 432
or vertically
like
¹36× 12 72
+ ¹360 432
?
Why do you multiply that way?
How do you do it,
step by step?
Which way is easier ...
To do?
To understand?
To skim and check?
What if we multiplied 23 and 456?
¹¹4¹¹56 × 23 = 1368 + 9120 = 10488
or
¹¹4¹¹56 × 23 1368
+ 9120 10488
We canhorizontally and vertically multiply
the expanded forms of thesemultidigit numbers.
36 × 12 (30+6)(10+2)
300+60+60+12432
23 × 456(20+3)(400+50+6)
8000+1000+120+1200+150+1810488
(30+6)(10+2)F O I L
30(10)+30(2)+6(10)+6(2)
300+60+60+12432
(20+3)(400+50+6)
20(400)+20(50)+20(6)+3(400)+3(50)+3(6)
8000+1000+120+1200+150+1810488
F FM O I SM L
F irst termsO utside termsI nside termsL ast terms
F irst & M iddle termsS econd & M iddle terms
HorizontalDistributiveMultiplication
30+6× 10+230(2)+6(2)
+ 30(10)+6(10)
60+12+ 300+60 300+120+12
432
Emulation of thevertical multiplicationof multidigit numbers
400+50+6× 20+3
400(3)+50(3)+6(3)+ 400(20)+50(20)+6(20)
1200+150+18+ 8000+1000+120
8000+2200+270+1810488
What if the polynomials have variables in them?
Multiplying algebraicpolynomials
3a+5× 4a–2–6a–10
+ 12a2+20a 12a2+14a–10
Vertical multiplication
(3a+5)(4a–2)12a2+14a–10
Horizontal multiplication(FOIL)
(2r–1)(r2–4r–6)2r3–9r2–8r+6
(distributive multiplication without FOIL)r2–4r–6× 2r–1
–r2 + 4r+6+ 2r3–8r2–12r
2r3–9r2 – 8r+6
Vertical FOILGeneral (all polynomial multiplications) Specific (multiply only pairs of binomials)Necessary (Hewitt 1999) Arbitrary (given wisdom) (Hewitt 1999)Intuit Memorize
Figure out Drill / Practice
Recognize Insular (One-time / Q-type specific)
Familiar (≈ multidigit number mult.) New / Added
Understand Know
Modify / Tailor Use as is / Transfer
Explore, Analyze, Synthesize, Interpret Math mathmath
Mind Calculator
Pattern, Process, History, Connections
Complement of common factoringTo check in a test:
Glance Redo
Written / Work to see Mental / No work to see
Source of error ! Where error ?
Similarity = Familiarity
400+50+6× 20+3
400(3)+50(3)+6(3)+ 400(20)+50(20)+6(20)
1200+150+18+ 8000+1000+120
8000+2200+270+1810488
r2–4r–6× 2r–1
–r2 + 4r+6+ 2r3–8r2–12r
2r3–9r2 – 8r+6
456 × 23 1368
+ 912 10488
Finally, a trick
456 × 23 1368
+ 912 10488
(r2)(–4r)(–6)× (2r)(–1)
(–r2 )(+ 4r)(+6)+ (2r3)(–8r2)(–12r)
(2r3)(–9r2 )( –8r)(+6)
≈
Treat each value plusthe ± operand in frontof it as a signed digit.
Then multiply the values like multidigit numbers and add the partial products like ± terms.
FOIL
Frustrate, thwart,baffle, impede
Fencing, sword
Cover, enhancecounter, contrast
Acrostic mnemonicof a specific situation
erroneously generalizedas a method of
polynomial multiplication
VERTICALPOLYNOMIAL
MULTIPLICATION
Familiar IntuitiveNecessary
(Hewitt 1999)
VisualAssessable
Extendible
Pattern
Process
ProblemSolving
Engaging
Math
(2+x)0
1x0
Point~
Vertex
1
(2+x)1
2x0+1x1
Line segment~
Edge
2+q
(2+x)2
4x0+4x1+1x2
Square~
Face
2+q×2+q
2q+q2
+ 4+2q 4+4q+q2
(2+x)3
8x0+12x1+6x2+1x3
Cube~
Solid
4+4q+q2
×2+q 4q+4q2+q3
+ 8+ 8q+2q2 8+12q+6q2+q3
(2+x)4
16x0+32x1+24x2+8x3+1x4
Tesseract~
4D hyperobject
8+12q+6q2+q3
×2+q 8q+12q2+6q3+q4
+ 16+24q+12q2+2q3 16+32q+24q2+8q3+q4