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