Teaching One Variable’s Algebraic Course Sketch

( 8th Grade )

Central idea is to show permanent link between 1) geometry : area computing andmagnitude – i.e. named numbers - computing 2) arithmetic operations with « pure numbers »and algebraic computing : this last link is based upon place value and polynomial algebra ( infact monomial degree)

This to show that it’s possible to make pupils think about link between “structures” (without “ à priori” beginning from an abstract definition) and to give them powerful tools

On the other side , it should be important to know how to incorporate this course inpresent curriculum, since it is not based on an algebraic view but a functional view ofpolynomials. A good solution should be not to begin with multiplication but with addition , li keit was done in curriculum before the sixties, but this implies a diff iculty to integrate it in presentprogression which should be drastically changed if we want to use the entire powerful li nkbetween 10n et Xn . Those old methods’ great interest was to make easier algebraic computingwhich could be leaned on arithmetic operations. As I begin directly with polynomialmultiplication , I reduce my number’s choice so there’s no carrying and I introduce laternegative numbers.

At last , some more points

- there’s a possibili ty to extend this kind of computing to X negative powers , using decimals’multiplication

- we can use directly polynomials’ multiplication to justify distributivity ( for development )

- we can use, too, correspondence between integers’ Euclidean division and polynomials’division to justify distributivity for factorization ( factorization is an Euclidean division withremainder = 0).

Main diff iculty is to appreciate what is important for that such an introduction should not be agadget ( “mathematic curiosities” ) and remains compatible with current national curriculum.

Base Ten’s place value and muliplication

We show a geometric picture for multiplication of 1311 by 212. This picture is corresponding tostandard multiplication algorithm as in Arab multiplication as in standard one ( sometimesnamed “ Italian”)

Fig 1( Arab Multiplication, pure numbers)

Fig 2: ( “classic” multiplication : "à l'italienne", pure numbers )

We get a double equivalence between ten’s powers diagonal alignment in Arab multiplicationand the same one in Italian multiplication .

Form taken by this equivalence when multiplication is understood as area computation usingnamed numbers’ multiplication is :

Fig 3 ( Arab multiplication, named numbers):

or Fig 4 ( Italian multiplication, named numbers )

We should remember that :

1 km × 1 hm = 10 hm × 1 hm = 10 hm² ;1 km × 1 dam = 10 hm × 0,1 hm = 1 hm²;1 km × 1 m = 100 dam × 0,1 dam = 10 dam²;1 hm × 1 m = 10 dam × 0,1 dam = 1 dam²;1dam × 1 m = 10 m × 1 m = 10 m²;

One unknown X polynomials’ Multiplication

So , you can see that all formulas in Figure 1 and 2 . remain true if you replace 10n by 2n ,5n , 7n , 15n ….. ( apart those in green , i.e. those meaning base ten writings of 1311, 212 andtheir product 277 932 )

For example, if we replace 10 by 2 in the two members of the following equali ty ( which onlymeans that 1311× 212 = 277 932 ) :

( 1 × 103 + 3 × 102 + 1× 101 + 1 × 100 ) × ( 2 × 102 + 1 × 101 + 2× 100 ) = 2 × 105 + 7 × 104 + 7× 103 + 9 × 102 + 3 × 101 + 2 × 100

Left side becomes:

( 1 × 23 + 3 × 22 + 1× 21 + 1 × 20 ) × ( 2 × 22 + 1 × 21 + 2× 20 ) =

( 8 + 3 × 4 + 2 + 1) × ( 2 × 4 + 2 + 2 = 23 × 12 = 276

Right side becomes :

2 × 25 + 7 × 24 + 7 × 23 + 9 × 22 + 3 × 21 + 2 × 20 =

2 × 32 + 7 × 16 + 7 × 8 + 9× 4 + 3 × 2 + 2 =

64 + 112 + 56 + 36 + 6 + 2 = 276

So , we have :

( 1 × 23 + 3 × 22 + 1× 21 + 1 × 20 ) × ( 2 × 22 + 1 × 21 + 2× 20 ) = 2 × 25 + 7 × 24 + 7 × 23 + 9 × 22

+ 3 × 21 + 2 × 20

We can consider that all formulas remain true if we replace 10 by X . So , we have :

Fig 5 :

Fig 6

We need two steps to reach polynomial’s multiplication:

- explaining carrying : by a simple verification, we can show that one can carry only if 10 × Xn =Xn+1 , i.e. X = 10 . In that case, it’s quite interesting to come back on the example – Fig 4 -usinglength and area named numbers to show different manners for regrouping terms ( i.e. to showthat one must understand a calculus, and don’ t compute anyhow).

- introducing negative numbers as monomials’ coefficients.

So , progression’s end could be, for a first step :

Fig 7

and :

Fig 8

20 Janvier 200

Michel Delord