Avvio alla moltiplicazione (in 2a) e sviluppi progressivi (in 3a e 4a) Como, 7 febbraio 2012

La progressione della moltiplicazione (Classe 3a )   Questa è un'altra opzione per creare diapositive introduttive che utilizzano transizioni.

La moltiplicazione e l'addizione sono due operazioni essenzialmente diverse, anche se, operativamente, posso sostituire la moltiplicazione con un'addizione iterata (che l’allievo può scoprire per conto proprio, contrariamente al concetto di prodotto). L'addizione corrisponde alla riunione di due insiemi disgiunti. La moltiplicazione è la messa in relazione di ogni elemento di un insieme con ogni elemento di un altro insieme, essa è legata al prodotto cartesiano. 7 12 ______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

OBIETTIVI (Apprendimento della moltiplicazione dalla 2a alla 4a elementare) - Capire il concetto di moltiplicazione come prodotto di una combinatoria in cui ogni elemento di un insieme è messo in relazione con i singoli elementi di un altro insieme. - Introdurre l'operatore x simultaneamente alla duplice scrittura: 3 x 4 = 3 . x 4 - Istituzionalizzare il termine prodotto. - Scoprire uguaglianze e diversità tra addizione e moltiplicazione.1 Se aggiungo un elemento ad un insieme, risultato (nell'addizione) e prodotto (nella moltiplicazione) come cambiano? Se inverto i termini dell'operazione, cosa succede? (commutatività) [1] Osservazione: L'addizione reiterata (somma di addendi uguali) interviene in un secondo tempo. Essa non è in nessun caso suggerita dal docente, ma viene accettata al momento in cui è proposta dagli allievi (ciò avviene sempre, in genere).

OBIETTIVI (Apprendimento della moltiplicazione dalla 2a alla 4a elementare) Costruire e capire cosa sia una tavola a doppia entrata attraverso l'uso di varie rappresentazioni grafiche. Costruire la tavola pitagorica e analizzare le sue particolarità (coppie di prodotti, simmetria).1 Apprendimento delle caselline (automatismo). Costruzione di algoritmi spontanei per la risoluzione di moltiplicazioni scritte. Primi accenni al concetto di area (Si tratta di aspetti impliciti nelle azioni precedenti che si ricollegano con quanto affrontato in geometria). Quando ho trovato la forma della "coperta rettangolare" e ho determinato lunghezza e larghezza, non posso calcolare la somma, ma devo trovare il prodotto. (Basandoci sulle "griglie" costruite possiamo inoltre associare i concetti di verticale e orizzontale.) [1] E' consigliabile costruire prima la tavola soltanto con i primi 5 numeri in modo che si ha un numero inferiori di dati sui quali fare le prime scoperte (non viene quindi insegnata prima la casellina del 2, poi del 3 ecc…. . Si inizierà con le caselline “facili” per passare poi a quelle “difficili”)

IL GIOCO DELLE COMBINAZIONI ______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ Accidenti! Che disordine! Prova a metterli in ordine per tipo

IL GIOCO DELLE COMBINAZIONI ______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ Adesso per colore Ora per colore e per tipo

IL GIOCO DELLE COMBINAZIONI Adesso proviamo con 4 colori e 6 oggetti ______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ Per tipo Per colore

… E adesso per tipo e colore ______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

ISTITUZIONALIZZAZIONE Questa è la rappresentazione che noi usiamo ______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

IL GIOCO DELL’ANTICIPAZIONE Prova tu adesso ad anticipare la soluzione 5 MEZZI DI TRASPORTO, 8 COLORI SOLUZIONE: .................................... 5 OGGETTI, 3 COLORI SOLUZIONE: .................................... ______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ ..... ADESSO CON 7 OGGETTI SOLUZIONE: ....................................

PROVIAMO CON I NUMERI COSTRUISCI TUTTE LE CASELLINE CHE TI VENGONO IN MENTE ______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

La costruzione delle CASELLINE Combina in tutti i modi possibili 1,2,3,4,5 ... fino a 25 ______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

La costruzione delle CASELLINE ______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

IL GIOCO DELLE CASELLINE Sovrapponi le tabelle scoperte fino al 25 1 2 3 4 5 6 8 10 9 12 15 16 20 25 ______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ Mescolare tre giochi e poi rimettere assieme in modo ordinato

CASELLINE FACILI 1 2 3 4 5 6 7 8 9 10 12 14 16 18 20 15 30 40 25 50 60 70 80 90 100 ______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

CASELLINE DIFFICILI 18 21 24 27 28 32 36 30 35 40 45 42 48 54 49 56 63 64 72 81 ______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

Memorizzazione delle caselline (automatismi) Il gioco delle carte colorate ______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ Preparare tante carte come questa

Prova a vedere, quanti sono? Scrivi il risultato nell’ultima casellina ______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

Scrivi il risultato nella casellina rossa … Quanto sei bravo? Scrivi il risultato nella casellina rossa ______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

… Infine solo con i numeri Due modi di rappresentare 4 X 6 = 4 X 6 = ______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ Abbandono della rappresentazione grafica Si mantiene il quadratino (per marcare il collegamento con le precedenti attività) Inevitabilmente questo percorso sarà compiuto dai bambini in tempi diversi. Va seguito il percorso senza interruzione per permettere al bambino di collegare meglio le diverse esperienze. Il lavoro fatto la settimana prima deve quindi essere rievocato.

La moltiplicazione ______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

La moltiplicazione Obiettivo ______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

La moltiplicazione Esempio 1. Situazione introduttiva Abbiamo tanti pezzi di lana, di forma quadrata e desideriamo formare, cucire, delle coperte (rettangolari). Consegna: Dovremo avere tre tipi di coperte: - larghe 4 (per i letti dei bambini), - larghe 7 (per letti a una piazza), - larghe 9 (per letti “francesi”) , - larghe 12 (per i letti a due piazze). La lunghezza può variare, guardate un po’ voi (pensate alle misure dei letti). Cercate la varie soluzioni possibili e calcolate quanti quadrati sono necessari per ogni modello. 2. Situazioni di sviluppo - Con 94 pezze che possibilità ho di ... avendo minor spreco ... - Sono state ordinate 5 coperte del modello “bambini” e 7 del modello “francese” , .... - ecc... 3. Situazioni “astratte” - Con quali numeri (numero di pezze) posso formare delle coperte a una piazza senza nessun spreco ? ______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

Sviluppi ulteriori (in 4a) (Apprendimento della moltiplicazione) - Addizione e moltiplicazione nella scomposizione di quantità numeriche: L'associazione delle due operazioni è quanto è già stato probabilmente messo in atto dagli allievi durante l'invenzione di alcuni algoritmi spontanei. In ogni caso le due operazioni saranno coinvolte al momento dell'apprendimento dell'algoritmo tradizionale.

Sviluppi ulteriori (in 4a) (Apprendimento della moltiplicazione) - Attività con le grandi collezioni: Inversamente: "Trovate il numero corrispondente a 46 x 25" ecc.....

Sviluppi ulteriori (in 4a) (Apprendimento della moltiplicazione) - Attività di approssimazione (controllo) sulla base dei prodotti conosciuti. - "Dall'algoritmo spontaneo all'algoritmo tradizionale." Apprendimento della tecnica tradizionale sulla base delle conoscenze sviluppato durante tutte le attività precedenti. (Non avere fretta di arrivare alla moltiplicazione tradizionale!) Calcolo delle superfici: - Attività di scoperta su altri algoritmi possibili (in particolare quello mussulmano).

ALGORITMO CONVENZIONALE MUSULMANO (Soltanto in 4a-5a!) 835 x 43=35.905 “Di algoritmi convenzionali ce ne sono e ce ne sono stati diversi, a dipendenza delle culture e dei paesi. Ecco quello chiamato “musulmano”. (Portiamo gli allievi ad approfondire il senso dell’algoritmo da un lato e dall’altro proponiamo una interessante attività di ricerca-scoperta. “Ora, dopo aver prima eseguito la moltiplicazione come la svolgiamo noi, guardate quest’altro modo e cercate di capire se funziona e come funziona. Mettetevi a coppie.” 8 3 5 2 1 4 9

15 12 3 + 3 + 3 + 3 = 12 3 + 3 + 3 + 3 + 3 = 15