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The IVA Initiative: An imperative for College Teachers of Algebra, of Developmental Mathematics, and of School/College Teachers Based on a mathematical monograph from The American Institute for the Improvement of MAthematics LEarning and Instruction a.k.a The MALEI Mathematics Institute © January, 2008 The MALEI Mathematics Institute

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WHAT ALL TEACHERS OF MATHEMATICS SHOULD KNOW IVAF Implementing the Vector-Algebraic Foundations of Curricular Mathematics

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What K-teachers should know “A” is for The term, “alligator”, is the NOMEN for all things of a specific kind.

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What K-teachers should know “A” is for The term, “alligator”, is the NOMEN for all things of a specific kind.

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What K-teachers should know “A” is for 3A = {}

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What K-teachers should know “A” is for 3A = {} A QUANTITY has a NUMERAT0R of a DENOMINATION

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What K-teachers should know “A” is for 3A = {} A QUANTITY has a NUMERAT0R and a DENOMINATION

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What K-teachers should know 3 + 5 + 2 = 3A +5E + 2D Even the very, very young have no trouble with INVENTORIES

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What K-teachers should know 3 + 5 + 2 = 3A +5E + 2D Even the very, very young have no trouble with INVENTORIES

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What K-teachers should know 3 + 5 + 2 = 3A +5E + 2D Even the very, very young have no trouble with INVENTORIES

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What K-teachers should know 3 + 5 + 2 = 3A +5E + 2D Even the very, very young have no trouble with INVENTORIES

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What K-teachers should know 3 + 5 + 2 = 3A +5E + 2D Even the very, very young have no trouble with INVENTORIES

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What K-teachers should know The vector algebra of INVENTORIES.

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What K-teachers should know The vector algebra of INVENTORIES Is about how many of each kind

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What G-1 teachers should know The BASIS for the VECTOR SPACE = [ 9A+7E+6D ] & SUBTRACTION (when it works)

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What G-1 teachers should know The SPACE of such inventories (?) = [ 9A+7E+6D ] & SUBTRACTION (when it works)

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What G-1 teachers should know VECTOR ADDITION [ 3A+5E+2D ] + [ 6A+2E+4D ] = [ 9A+7E+6D ] & SUBTRACTION (when it works)

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What G-1 teachers should know VECTOR ADDITION [ 3A+5E+2D ] + [ 6A+2E+4D ] = [ 9A+7E+6D ] & SUBTRACTION (when it works)

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What G-1 teachers should know VECTOR ADDITION [ 3A+5E+2D ] + [ 6A+2E+4D ] = [ 9A+7E+6D ] & SUBTRACTION (when it works)

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What G-2 teachers should know For REPEATED ADDITION The “ ” is read “of” 7 [3A+5E+2D] = 21A+35E+14D

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What G-2 teachers should know For REPEATED ADDITION The “ ” is read “of” 7 [3A+5E+2D] = 21A+35E+14D

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What G-2 teachers should know For REPEATED ADDITION The “ ” is read “of” 7 [3A+5E+2D] = 21A+35E+14D and its SCALER MULTIPLICATION

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What G-2 teachers should know SCALER DIVISION of inventories. 5A + 8E + 3D. 4 21A + 35E + 14D. 20A + 32E + 12D.. 1A + 3E + 2D. Is REMAINDER division

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What G-2 teachers should know SCALER DIVISION of inventories. 5A + 8E + 3D. 4 21A + 35E + 14D. 20A + 32E + 12D.. 1A + 3E + 2D. Is REMAINDER division

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What G-2 teachers should know SCALER DIVISION of inventories. 5A + 8E + 3D. 4 21A + 35E + 14D. 20A + 32E + 12D.. 1A + 3E + 2D. Is REMAINDER division

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What G-2 teachers should know SCALER DIVISION of inventories. 5A + 8E + 3D. 4 21A + 35E + 14D. 20A + 32E + 12D.. 1A + 3E + 2D. Is REMAINDER division

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What G-2 teachers should know SCALER DIVISION of inventories. 5A + 8E + 3D. 4 21A + 35E + 14D. 20A + 32E + 12D.. 1A + 3E + 2D. Is REMAINDER division

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What G-2 teachers should know SCALER DIVISION of inventories. 5A + 8E + 3D. 4 21A + 35E + 14D. 20A + 32E + 12D.. 1A + 3E + 2D. Is REMAINDER division

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What G-2 teachers should know SCALER DIVISION of inventories. 5A + 8E + 3D. 4 21A + 35E + 14D. 20A + 32E + 12D.. 1A + 3E + 2D. Is REMAINDER division

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What G-2 teachers should know SCALER DIVISION of inventories. 5A + 8E + 3D. 4 21A + 35E + 14D. 20A + 32E + 12D.. 1A + 3E + 2D. Is REMAINDER division

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What G-2 teachers should know SCALER DIVISION of inventories. 5A + 8E + 3D. 4 21A + 35E + 14D. 20A + 32E + 12D.. 1A + 3E + 2D. Is REMAINDER division

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What G-2 teachers should know SCALER DIVISION of inventories. 5A + 8E + 3D. 4 21A + 35E + 14D. 20A + 32E + 12D.. 1A + 3E + 2D. Is REMAINDER division

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THE CONTEXT: www.reformingalgebra.org

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An E-book in Progress: THE VECTOR-ALGEBRAIC FOUNDATIONS OF ARITHMETIC Written for mathematicians who teach mathematics to teachers of mathematics, K-Collegebra

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What G-2 teachers should know TODAY’S SPECIALS @ 2 ¢ @ 5 ¢ @ 3 ¢

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What G-2 teachers should know TODAY’S SPECIALS Denomination VALUES. @ 2 ¢ @ 5 ¢ @ 3 ¢

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What G-2 teachers should know TODAY’S SPECIALS Denomination VALUES. @ 2 ¢ @ 5 ¢ @ 3 ¢ are VARIABLE

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What G-2 teachers should know @ 2 ¢ @ 5 ¢ @ 3 ¢ EVALUATION OF INVENTORY VECTORS 3 ∘ (A) + 5 ∘ (E) + 2 ∘ (D) = 3(2) + 5(5) + 2(3) = 37 LINEAR EQUATIONS 3A + 5E + 2D = 37

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What G-2 teachers should know @ 2 ¢ @ 5 ¢ @ 3 ¢ EVALUATION OF INVENTORY VECTORS 3 ∘ (A) + 5 ∘ (E) + 2 ∘ (D) = 3(2) + 5(5) + 2(3) = 37 LINEAR EQUATIONS 3A + 5E + 2D = 37

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What G-2 teachers should know @ 2 ¢ @ 5 ¢ @ 3 ¢ EVALUATION OF INVENTORY VECTORS 3 ∘ (A) + 5 ∘ (E) + 2 ∘ (D) = 3(2) + 5(5) + 2(3) = 37 LINEAR EQUATIONS 3A + 5E + 2D = 37

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What G-2 teachers should know @ 2 ¢ @ 5 ¢ @ 3 ¢ EVALUATION OF INVENTORY VECTORS 3 ∘ (A) + 5 ∘ (E) + 2 ∘ (D) = 3(2) + 5(5) + 2(3) = 37 LINEAR EQUATIONS 3A + 5E + 2D = 37

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What G-2 teachers should know @ 2 ¢ @ 5 ¢ @ 3 ¢ EVALUATION OF INVENTORY VECTORS 3 ∘ (A) + 5 ∘ (E) + 2 ∘ (D) = 3(2) + 5(5) + 2(3) = 37 LINEAR EQUATIONS 3A + 5E + 2D = 37

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More abstractly : If icons have values, those are variable S Is for Spades ♠ Value = __ H is for Hearts ♥ Value = __ D is for Diamonds ♦ Value = __ C Is for Clubs ♣ Value = __ 7S+ 4H+ 6D+ 0C What G-2 teachers should also know

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What G-3 teachers should know The vector algebra of MEASUREMENTS Invokes conversion tables

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What G-3 teachers should know A fluid measurements space: Gal, Qt, Pt, C, FlOz, Tbs, Tsp - - - - - - Unit-vector conversions: 1G=4Q; 1Q=2P; 1P=2C; 1C=8F; …

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What G-3 teachers should know A fluid measurements space: Gal, Qt, Pt, C, FlOz, Tbs, Tsp - - - - - - Unit-vector conversions: 1G=4Q; 1Q=2P; 1P=2C; 1C=8F; …

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What G-3 teachers should know In fluid measurements space: [G, Q, P, C, F, T, t] Each vector is equivalent to a reduced (or proper) vector 17C+11F+10t ~ 4Q+1P+4F+1T+1t

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What G-3 teachers should know In fluid measurements space: [G, Q, P, C, F, T, t] Each vector is equivalent to a reduced (or proper) vector 17C+11F+10t ~ 4Q+1P+4F+1T+1t

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What G-3 teachers also should know U.S. COINS: Values are legislated PV = Place Value PV=100PV=50PV=25PV=10PV=5PV=1 4$+5H+7Q+3D+7N +8 ¢

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What G-3 teachers also should know U.S. COINS: $(USANS), H(alves), Q(uarters), D(imes), N(ickles), ¢(ents) Vector Basis: $, H, Q, D, N, ¢ 6$ + 9H + 13Q + 4D + 9N + 39¢ THE VECTOR VALUE = SUM OF QUANTITY VALUES

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What G-3 teachers also should know VECTOR CONVERSION EQUIVALENCE CLASSES IMPROPER: 6$ + 9H + 13Q + 4D + 9N + 39¢

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What G-3 teachers also should know VECTOR CONVERSION EQUIVALENCE CLASSES IMPROPER: 6$ + 9H + 13Q + 4D + 9N + 39¢ PROPER: 14$ + 1H + 1Q + 2D + ØN + 4¢ LOWEST SUM OF THE NUMERATORS

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What G-3 teachers also should know VECTOR CONVERSION BY CARRYING: 4D + 9N + 39¢ 1$ + 2D + ØN + 4¢ IT CAN BE DONE & LEARNED SEPARATELY FROM ADDING/MULTIPLYING

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What G-3 teachers also should know “PROPER” ADDITION = VECTOR ADDITION & FULL CARRYING. [1Q+3D+7N+9¢] + [5Q+7D+4N+8¢] =

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What G-3 teachers also should know “PROPER” ADDITION = VECTOR ADDITION & FULL CARRYING. [1Q+3D+7N+9¢] + [5Q+7D+4N+8¢] = 6Q+10D+11N+17¢ =

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What G-3 teachers also should know “PROPER” ADDITION = VECTOR ADDITION & FULL CARRYING. [1Q+3D+7N+9¢] + [5Q+7D+4N+8¢] = 6Q+10D+11N+17¢ = 3$ + ØQ + 2D + ØN + 2¢

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What G-4 teachers should also know In English, pronounce the following number: 4796 (In Old English, “th” was spelled “Y”; The singles, “S”, is s-ilent)

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What G-4 teachers should also know In English, pronounce the following number: 4796 (In Old English, “th” was spelled “Y”; The singles, “S”, is s-ilent) 4796 = 4Y, 7H, 9T,6(S) …

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What G-4 teachers should also know In English, pronounce the following number: 4796 (In Old English, “th” was spelled “Y”; The singles, “S”, is s-ilent) 4796 = 4Y, 7H, 9T,6(S) … short for the vector, 4Y + 7H + 9T + 6(S) = (4,7,9,6)

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What G-4 teachers should also know In English, pronounce the following number: 4796 (In Old English, “th” was spelled “Y”; The singles, “S”, is s-ilent) 4796 = 4Y, 7H, 9T,6(S) … short for the vector, 4Y + 7H + 9T + 6(S) = (4,7,9,6) ARABIC NUMERALS ARE PLACE-BASED VECTORS

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What G-4 teachers should also know LINEAR ALGEBRAIC PROPER ARABIC ARITHMETIC 2487 + 3942 = (2,4,8,7) + (3,9,4,2) = (5,13,12,9) ~ (5,14,2,9) ~ (6,4,2,9) = 6429

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What G-4 teachers should also know LINEAR ALGEBRAIC PROPER ARABIC ARITHMETIC 2487 + 3942 = (2,4,8,7) + (3,9,4,2) = (5,13,12,9) ~ (5,14,2,9) ~ (6,4,2,9) = 6429

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What G-4 teachers should also know LINEAR ALGEBRAIC PROPER ARABIC ARITHMETIC 2487 + 3942 = (2,4,8,7) + (3,9,4,2) = (5,13,12,9) ~ (5,14,2,9) ~ (6,4,2,9) = 6429

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What G-4 teachers should also know LINEAR ALGEBRAIC PROPER ARABIC ARITHMETIC 2487 + 3942 = (2,4,8,7) + (3,9,4,2) = (5,13,12,9) ~ (5,14,2,9) ~ (6,4,2,9) = 6429

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What G-4 teachers should also know LINEAR ALGEBRAIC PROPER ARABIC ARITHMETIC 2487 + 3942 = (2,4,8,7) + (3,9,4,2) = (5,13,12,9) ~ (5,14,2,9) ~ (6,4,2,9) = 6429

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What G-4 teachers should also know LINEAR ALGEBRAIC PROPER ARABIC ARITHMETIC 2487 + 3942 = (2,4,8,7) + (3,9,4,2) = (5,13,12,9) ~ (5,14,2,9) ~ (6,4,2,9) = 6429

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What G-5 teachers should know The vector algebra of FRACTIONS: the numbers got by dividing whole-scaler spaces by natural numbers

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What G-5 teachers should know Fraction DENOMINATIONS are: H(alves) or T(hirds) or F(ourths) or f(ifths) or S(ixths) or s(evenths) or E(ighths) or N(inths) or t(enths) or e(levenths) or … Vector basis: H,T,F,f,S,s,E,N,t,e,… (We’ll soon run out of letters!)

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What G-5 teachers should know Fraction DENOMINATIONS are: H(alves) or T(hirds) or F(ourths) or f(ifths) or S(ixths) or s(evenths) or E(ighths) or N(inths) or t(enths) or e(levenths) or … Vector basis: H,T,F,f,S,s,E,N,t,e,… (We’ll soon run out of letters!)

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What G-5 teachers should know Fraction DENOMINATIONS are: H(alves) or T(hirds) or F(ourths) or f(ifths) or S(ixths) or s(evenths) or E(ighths) or N(inths) or t(enths) or e(levenths) or … Vector basis: H,T,F,f,S,s,E,N,t,e,… (We’ll soon run out of letters!)

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What G-5 teachers should know Fraction DENOMINATIONS are: H(alves) or T(hirds) or F(ourths) or f(ifths) or S(ixths) or s(evenths) or E(ighths) or N(inths) or t(enths) or e(levenths) or … Vector basis: H,T,F,f,S,s,E,N,t,e,… (We’ll soon run out of letters!)

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What G-5 teachers should know Fraction DENOMINATIONS are: H(alves) or T(hirds) or F(ourths) or f(ifths) or S(ixths) or s(evenths) or E(ighths) or N(inths) or t(enths) or e(levenths) or … Vector basis: H,T,F,f,S,s,E,N,t,e,… (We’ll soon run out of letters!)

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What G-5 teachers should know 2F+3F = ? 7E-4E = ? 1H+2T is an inventory vector. But 1H=3S & 2T = 4S; So 1H+2T = 7S

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What G-5 teachers should know 2F+3F = ? 7E-4E = ? 1H+2T is an inventory vector. But 1H=3S & 2T = 4S; So 1H+2T = 7S

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What G-5 teachers should know 2F+3F = ? 7E-4E = ? 1H+2T is an inventory vector. But 1H=3S & 2T = 4S; So 1H+2T = 7S

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What G-5 teachers should know 2F+3F = ? 7E-4E = ? 1H+2T is an inventory vector. But 1H=3S & 2T = 4S; So 1H+2T = 7S

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What G-5 teachers should know 2F+3F = ? 7E-4E = ? 1H+2T is an inventory vector. But 1H=3S & 2T = 4S; So 1H+2T = 7S

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What G-5 teachers should know PHONIC DENOMINATIONS: 9ths, 8ths, 7ths, 6ths, … 5ths, 4ths, 3ths, 2ths, 1ths 2(4ths) +3(4ths) = 5(4ths)

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What G-5 teachers should know PHONIC DENOMINATIONS: 9ths, 8ths, 7ths, 6ths, … 5ths, 4ths, 3ths, 2ths, 1ths 2(4ths) +3(4ths) = 5(4ths)

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What G-5 teachers should know PHONIC DENOMINATIONS: 9ths, 8ths, 7ths, 6ths, … 5ths, 4ths, 3ths, 2ths, 1ths 2(4ths) +3(4ths) = 5(4ths)

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What G-5 teachers should know PHONIC DENOMINATIONS: 9ths, 8ths, 7ths, 6ths, … 5ths, 4ths, 3ths, 2ths, 1ths 2(4ths) + 3(4ths) = 5(4ths)

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What G-5 teachers should know PHONIC DENOMINATIONS: 9ths, 8ths, 7ths, 6ths, … 5ths, 4ths, 3ths, 2ths, 1ths 2(4ths) + 3(4ths) = 5(4ths)

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What G-5 teachers should know PHONIC DENOMINATIONS: 9ths, 8ths, 7ths, 6ths, … 5ths, 4ths, 3ths, 2ths, 1ths 2(4ths) + 3(4ths) = 5(4ths) 2(4) + 3(4) =5(4)

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What G-5 teachers should know Fraction denominations can be assigned to PLACES 0. __ __ __ __ __ __ __ __ __ __ __ __ __

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What G-5 teachers should know A DECIMAL POINT is a pair of place-based vectors: Each place value is an integer- power of ten. Each of the placed numerators is a whole number 0. 7 32 5 642 0 0 0 0 …

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What G-5 teachers should know In a DIGITAL decimal point all numerators are less that ten. Those are the PROPER decimal points. 0. 7 3 2 5 6 4 0 0 0 0 …

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What G-5 teachers should know In the English phonics, most decimal points are pronounced as IMPROPER vectors. 0. 7 3 2 is pronounced as 0. 0 0 732

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What G-5 teachers should know 0. __ __ __ __ __ __ __ __ __ __ __ __ __

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What G-5 teachers should know A FRACTIONAL point uses all fraction denominations The place-numbers are the fraction’s DENOMINATORS. __. __ __ __ __ __ __ __ __ __ __ __ … 1 2 3 4 5 6 7 8 9 10 11 12

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What G-5 teachers should know The scalers are the fraction’s NUMERATORS 3(4) = 0. 0 0 3 0 0 0 0 0 0 0 0 0 0 0 …

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What G-5 teachers should know All fraction conversions Are VECTOR CONVERSIONS (just as with measurements) 0. __ __ __ __ __ __ __ __ __ __ __ __ __

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What G-5 teachers should know Fraction addition and subtraction are VECTOR addition and subtraction 0. __ __ __ __ __ __ __ __ __ __ __ __ __

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What G-5 teachers should know Multiplication of fractions by whole numbers is SCALER MULTIPLICATION of vectors, by scalers. 0. __ __ __ __ __ __ __ __ __ __ __ __ __

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WHERE WOULD YOU LIKE TO RUN WITH IT? FOUNDATIONS OF MATHEAMATICS? LINEAR ALGEBRA? TEACHER EDUCATION? COLLEGE ALGEBRA? LIBERAL ARTS? DEVELOPMENTAL MATHEMATICS? PUBLIC LITERACY?

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For details, visit www.reformingalgebra.org

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THANKS FOR GETTING THIS GLIMPSE OF THE IVAF INITIATIVE

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