Multiplication: An Adventure in Number Sense

More Patterns to Discover!

Mentor: There are many other interesting ideas and tricks connected with multiplication. For example, ancient Greeks studied so-called "square numbers," that is, the numbers represented by 1, 2, 3, 4 and so on. Here is why they are called square:

Ancient Greeks also studied triangular numbers: 1, 3, 6, 10... Here is how you get these numbers: The triangular numbers are connected to a topic we talked about before. What is this topic? If you do not feel like searching for it, try to find a formula for triangular numbers. Every square number is a sum of two triangular numbers (for example, 9 = 3 + 6). Can you show why it is so, using marbles or graph paper?

Mentor: There are also some silly ways to remember multiplication facts, like this one:

5, 6, 7, 8 so 56=7*8 which reads: "Five, six, seven, eight so fifty six is seven times eight"

Ask people to share their favorite silly ways to remember multiplication facts. You can make a fine collection of little verses, songs, and so on. What kinds of people would you expect to know the most ways to memorize facts?

Mentor: I hope you play around with the numbers enough to find a lot of personal connection with various patterns and facts about them.

Select any four numbers in the table above that are next to each other inside a square, for example: 14 | 21 16 | 24 or 1 | 2 2 | 4 or 3 | 6 4 | 8 Now multiply them in pairs that are in the opposite corners, and subtract the second product from the first, for example 14*24-16*21=?  or  1*4-2*2=?  or 3*8-4*6=?   What do you get? Why?

Four numbers arranged in a square pattern like that: a b c d are an example of a matrix, and the quantity a*d - c*b is called the matrix's determinant: A matrix, in two dimensions, is a rectangular table filled with numbers or other symbols.

Select any rectangle of numbers in the table, for example, as we did: 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 Now take only the corner numbers from the rectangle: 4 12 8 24 Find the determinant of the matrix you get this way, in our example, 4*24-8*12. What do you get? Why? Try it with a different rectangle.

Select any square array of numbers in the table such that the diagonal of that square is on the table's diagonal, for example, as we did: 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 Now add all the numbers you selected, in our example, 4+6+8+6+9+12+8+12+16=81. 81 is one of the square numbers, because 9*9=81. Try to select several different squares of numbers on the table in the same manner (on the diagonal). Do you always get a square number if you add all the numbers in your selected square? Why?

Remember the Off Diagonal pattern? Now look for the numbers that are one, two or even more steps off the diagonal, as highlighted here: 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 Now fill in this table, if you feel like it: How many steps off the diagonal did you go to get your number? What is the difference between your number and the number on the diagonal from which you started? 1 1 2 ... 3 ... ... ... You may want to construct a multiplication table that goes up to a number higher than 10 in order to be able to go more steps off the diagonal. Do you see the pattern in the table? Can you explain it?

Student: But what about memorization?

Mentor: If you really want to do it, you can memorize the 13 facts that we have left uncolored in the table. I would advise you to solve a lot of interesting problems instead. If you use multiplication facts, after a while it will be easier for you to recall them than to look them up in a table. Your brain will automatically memorize them by that time. By the way, did you notice that we can use several different colors for some of the facts, because several patterns work for them?

Student: Oh yes, for example, we could do 9*5 by the five pattern or by the nine pattern.

Find as many multiplication facts for which there are several patterns as you can.