Multiplication: An Adventure in Number Sense

   

Off Diagonal

Mentor: The next facts I am going to color have a really neat pattern. Here is what we have so far:

1

2

3

4

5

6

7

8

9

10

1

1*1=1

1*2=2

1*3=3

1*4=4

1*5=5

1*6=6

1*7=7

1*8=8

1*9=9

1*10=10

2

2*1=2

2*2=4

2*3=6

2*4=8

2*5=10

2*6=12

2*7=14

2*8=16

2*9=18

2*10=20

3

3*1=3

3*2=6

3*3=9

3*4=12

3*5=15

3*6=18

3*7=21

3*8=24

3*9=27

3*10=30

4

4*1=4

4*2=8

4*3=12

4*4=16

4*5=20

4*6=24

4*7=28

4*8=32

4*9=36

4*10=40

5

5*1=5

5*2=10

5*3=15

 5*4=20

5*5=25

5*6=30

5*7=35

5*8=40

5*9=45

5*10=50

6

6*1=6

6*2=12

6*3=18

6*4=24

6*5=30

6*6=36

6*7=42

6*8=48

6*9=54

6*10=60

7

7*1=7

7*2=14

7*3=21

7*4=28

7*5=35

7*6=42

7*7=49

7*8=56

7*9=63

7*10=70

8

8*1=8

8*2=16

8*3=24

8*4=32

8*5=40

8*6=48

8*7=56

8*8=64

8*9=72

8*10=80

9

9*1=9

9*2=18

9*3=27

9*4=36

9*5=45

9*6=54

9*7=63

9*8=72

9*9=81

9*10=90

10

10*1=10

10*2=20

10*3=30

10*4=40

10*5=50

10*6=60

10*7=70

10*8=80

10*9=90

10*10=100

Mentor: The next pattern is easier to see if we write the table in the other form. Select any number on the diagonal of the table, for example, 25 or 49. Look at the numbers that are to the North-East and to the South-West of it:

1

2

3

4

5

6

7

8

9

10

1

1

2

3

4

5

6

7

8

9

10

2

2

4

6

8

10

12

14

16

18

20

3

3

6

9

12

15

18

21

24

27

30

4

4

8

12

16

20

 24

28

32

36

40

5

5

10

15

20

25

30

35

40

45

50

6

6

12

18

24

30

36

42

48

54

60

7

7

14

21

28

35

42

49

56

63

70

8

8

16

24

32

40

48

56

64

72

80

9

9

18

27

36

45

54

63

72

81

90

10

10

20

30

40

50

60

70

80

90

100

Student: These numbers are one less than the diagonal numbers! Here is this whole pattern in the table:

1

2

3

4

5

6

7

8

9

10

1

1

2

3

4

5

6

7

8

9

10

2

2

4

6

8

10

12

14

16

18

20

3

3

6

9

12

15

18

21

24

27

30

4

4

8

12

16

20

 24

28

32

36

40

5

5

10

15

20

25

30

35

40

45

50

6

6

12

18

24

30

36

42

48

54

60

7

7

14

21

28

35

42

49

56

63

70

8

8

16

24

32

40

48

56

64

72

80

9

9

18

27

36

45

54

63

72

81

90

10

10

20

30

40

50

60

70

80

90

100

Will this always work? Why does it work? Hint: Note that any number on the diagonal is always a square number, that is, it is of the form a*a for some number a. How would you express the numbers to the North-East and to the South-West in terms of a?  

Student: I see how we can use it! If I remember that 5*5=25, then I get 4*6 and of course 6*4 for free!

How many facts are described by this pattern in the 10 by 10 table (ignore the colors for a while)? How did you count? How many facts would be described by this pattern in the 5 by 5 multiplication table? In the 7 by 8 table? In the 20 by 30 table? Do you see a pattern here?  

Mentor: Of course it also works with larger numbers, so you can use it to easily calculate 19*21 or 51*49 and so on. Anyway, I color two more facts in the big table (using dark gray) and now we are down to no more than 13 to memorize!

1

2

3

4

5

6

7

8

9

10

1

1*1=1

1*2=2

1*3=3

1*4=4

1*5=5

1*6=6

1*7=7

1*8=8

1*9=9

1*10=10

2

2*1=2

2*2=4

2*3=6

2*4=8

2*5=10

2*6=12

2*7=14

2*8=16

2*9=18

2*10=20

3

3*1=3

3*2=6

3*3=9

3*4=12

3*5=15

3*6=18

3*7=21

3*8=24

3*9=27

3*10=30

4

4*1=4

4*2=8

4*3=12

4*4=16

4*5=20

4*6=24

4*7=28

4*8=32

4*9=36

4*10=40

5

5*1=5

5*2=10

5*3=15

 5*4=20

5*5=25

5*6=30

5*7=35

5*8=40

5*9=45

5*10=50

6

6*1=6

6*2=12

6*3=18

6*4=24

6*5=30

6*6=36

6*7=42

6*8=48

6*9=54

6*10=60

7

7*1=7

7*2=14

7*3=21

7*4=28

7*5=35

7*6=42

7*7=49

7*8=56

7*9=63

7*10=70

8

8*1=8

8*2=16

8*3=24

8*4=32

8*5=40

8*6=48

8*7=56

8*8=64

8*9=72

8*10=80

9

9*1=9

9*2=18

9*3=27

9*4=36

9*5=45

9*6=54

9*7=63

9*8=72

9*9=81

9*10=90

10

10*1=10

10*2=20

10*3=30

10*4=40

10*5=50

10*6=60

10*7=70

10*8=80

10*9=90

10*10=100

   

     

© Copyright 1998 by Maria Droujkova and Dmitri Droujkov. All rights reserved. No part of these materials should ever be used in any situation that involves compulsory teaching. See also copyright notes and student rights.