friend or freak?a brief curriculum on

multiplication

Willem Uittenbogaard

theorem 1• stop learning the algorithms

• no standard procedures

• no columnwise arithmetic

• agree or disagree

stop learning the algorithms

we can do it by:

• calculating by heart

• clever calculating

• sensible use of a calculator

also suitable for +, x and :

• multiplication as an example

• for each of the other operations it’s the same story

all multiplications are freaks!

a list:• 37 × 249• 53 × 187• 13 × 619

approach?

with a calculator or a algorithm?

children, often…….

• are trying

• use a not understood algorithm

• type something on a calculator

• are even calculating table products with a calculator

• figure out between products with a calculator

friend or freak?another list:

• 17 × 237

• 10 × 237

• 83 × 346

• 100 × 346

• 1000 × 129

are there any friends?yes, 10 ×, 100 × and 1000 ×

how do you do that?

by moving the zeroes

the money context can help

does it always work?

→ friends of 10

friend or freak?another list:

• 17 × 239………… freak

• 10 × 169………… friend of 10

• 27 × 153………… freak

• 20 × 60…………...?

• 40 × 70…………...?

are there any new friends?yes, 20 x 60 and 40 x 70how do you do that?20 x 60 = 2 x 600 = 1200use the money context.you can do 2 x 6 plus two zeroes.does it always work?

→ tables with zeroes

friend or freakagain a list:

• 27 x 473…………..freak

• 100 x 73…………..friend of 10

• 30 x 80……………tables with zeroes

• 9 x 34………………?

• 11 x 27……………..?

• 101 x 27……………?

can we make new friends?may be 9 × 34 and 11 × 27

how can we do it?

9 × and 11 × are both close to 10 ×

9 × is one time less and 11 × one time more:

340 – 34 and 270 + 27

and you can do them with your head

→ almost friends of 10

first we have to practice!do I recognize them all?• 100 × 69• 11 × 54• 37 × 83• 50 × 90

have a good look at the numbers and then choose

your strategy

improve the knowledge of tables and maintain it!

friend or freak?again a list:

• 24 × 25……………?

• 12 × 35……………?

• 14 × 55……………?

no friend, or …?

you can make 12 rows of 50 and then again another 6 rowsof 100, then it becomes a friend of 10.does it always work? because of the 24, that is even andthe five.

→ halve and double

25

242412

25 25

a nice couple of friends

• 10 x 37

• 9 x 47

• 11 x 67

• 200 x 60

• 16 x 35

• 1001 x 123

→ a nice couple

theorem 2

• I think all children can learn this!

• not as a trick, but with insight and applicable!

• agree or disagree

are there any more friends? what do you think of:

• 10 x 10

• 11 x 11

• 13 x 13

• ……….

• 20 x 20worth a research; try to memorize

→ squared numbers are friends too!

10

10

1 1 1

1

1

1

11 x 11 = 10 x 10 + 2 x 10 + 1

12 x 12 = 11 x 11 + 2 x 11 + 1

and what about these ones?

• 5 x 5 • 15 x 15• 25 x 25• 35 x 35• ……• 75 x 75• …

not as an algorithm, but as a subject of researchlook for the patterns, find the rules, understand themand use them.

→”best” friends

30 5

30

5

30 x 5

30 x 5

5 x 5

35 x 35 = 30 x 30 + 2 x 5 x 30 + 5 x 5 =

40 x 30 + 5 x 5 = 1225

one has more friends than the other

again a list:• 24 x 12,5 = 12 x 25 = 6 x 50 = 300• 125 x 840 = 250 x 420 = 500 x 210 = 1000 x 105

= 105.000• 3 x 210 = 9 x 70 = 630

• 54 x 56 = 50 x 60 + 4 x 6; is it correct? always?

→ far friends?

what do we do with this freak?

can we do something about:7 x 234yes we can:7 x 200……..tables with zeroes7 x 30 ……...tables with zeroes7 x 4 ……….tableand then add: 1400 + 210 + 28not columnwise, but in your head:1610 + 28 = 1638 (no algorithm for +)

→ freak becomes friend

and the rest of the freaks?

23 x 347……..?

with a calculator!

and take enough time to do it!

so always give lists with problems!

how to do it?

→calculator becomes friend!

theorem 3• the more friends you have, the easier it is!

• we do not need standard procedures or columnwise arithmetic

• agree or disagree

finally• a lot of attention for knowledge of tables!

• strategies with their own names

• practice with lists of problems

• not bear, but attached to a suitable context

• let the children do their reasoning!

• no escape in algorithms

• sensible use of the calculator

why like this?• because of my experience in working with

grade 5 pupils of River East Elementary School in Manhattan, NY

• thanks to the children and Peter Markovitz, their teacher

• with warmth I remember their eagerness and enthousiasm to find out something new, week after week

for who?• áll children in the primary schools

• everyone in his own way: with more or less friends

• at the end you only have friends: the calculator can easily become a good friend

• each cell phone has a calculator

et voilàa brief curriculum multiplication in 15 minutes