# Fill the Number Grid

**Fill the Number Grid **

**Objectives**

The following number game is an excellent opening exercise for a whole

class to tackle. It is a discovery exercise in which all students can
participate.

As the grid is filled in, some patterns will emerge which can be seen even by

students who consider themselves 'weak' in math . Thus, everyone in the

class gets a chance to find a correct answer. But the actual rule for filling in

the grid also yields some results which may prove challenging to even the

math 'whizzes' in the class. [We will not reveal the rule for filling in the
grid

till later in the **Procedure** section, in case you would like to figure out
the

rule yourself first.]

**Procedure**

Present students with the following grid written on the
chalkboard. Their job

is to fill in the grid.

12 | ||||||||||||

11 | ||||||||||||

10 | 70 | |||||||||||

9 | ||||||||||||

8 | ||||||||||||

7 | ||||||||||||

6 | ||||||||||||

5 | 20 | |||||||||||

4 | ||||||||||||

3 | 24 | |||||||||||

2 | ||||||||||||

1 | ||||||||||||

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |

The ground rules for this game are similar to those for **
What's the Rule?**

Students make suggestions for numbers they would like to place in the grid.

The grid is set up like a set of coordinate axes , so students name the box they

would like to fill by saying the 'x' number first, followed by the 'y' number.

For example, box (4,5) has already been filled with the number 20. This

gives students practice for naming coordinates in a standard x, y graph . If a

student guesses a number correctly, that number is placed in the grid. We

make the rule that only students who raise their hands get to put numbers in

the grid. Students who just call out numbers are ignored. In this way, the

teacher can 'reward' specific students by allowing them to fill in several

numbers in a row once they find a specific pattern. It also makes it possible

to get more students involved, rather than letting a few aggressive students

dominate the action.

The three numbers that have already been placed in the
grid are a bit

misleading. They have been chosen to make it look like this is just a table of

multiplication facts . However, if you look at the completed grid below, you

will quickly see that this is not the case.

12 | 12 | 12 | 12 | 12 | 60 | 12 | 84 | 24 | 36 | 60 | 132 | 12 |

11 | 11 | 22 | 33 | 44 | 55 | 66 | 77 | 88 | 99 | 110 | 11 | 132 |

10 | 10 | 10 | 30 | 20 | 10 | 30 | 70 | 40 | 90 | 10 | 110 | 60 |

9 | 9 | 18 | 9 | 36 | 45 | 18 | 63 | 72 | 9 | 90 | 99 | 36 |

8 | 8 | 8 | 24 | 8 | 40 | 24 | 56 | 8 | 72 | 40 | 88 | 24 |

7 | 7 | 14 | 21 | 28 | 35 | 42 | 7 | 56 | 63 | 70 | 77 | 84 |

6 | 6 | 6 | 6 | 12 | 30 | 6 | 42 | 24 | 18 | 30 | 66 | 12 |

5 | 5 | 10 | 15 | 20 | 5 | 30 | 35 | 40 | 45 | 10 | 55 | 60 |

4 | 4 | 4 | 12 | 4 | 20 | 12 | 28 | 8 | 36 | 20 | 44 | 12 |

3 | 3 | 6 | 3 | 12 | 15 | 6 | 21 | 24 | 9 | 30 | 33 | 12 |

2 | 2 | 2 | 6 | 4 | 10 | 6 | 14 | 8 | 18 | 10 | 11 | 12 |

1 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |

The numbers in the grid are the Least Common Multiples
(LCM) of the bold

face numbers along the bottom and left axes. That is, 18 is the smallest

number that both 6 and 9 will go into evenly. 18 is also the smallest number

that 2 and 9 will go into evenly.

*Example 1:* You can get these least common multiples
by using a technique

that is sort of a repeated upside-down division. Example 1 shows this

technique for finding the least common multiple for the numbers 6 and 12:

First, divide both numbers by the smallest prime number (2) that will go into

both evenly. Write down the quotients 3 and 6 underneath 6 and 12,

respectively. Then divide both of those numbers by the smallest prime

number (3) that will go into them evenly. Again write down 1 and 2 under

them. Now multiply the divisors and the last two quotients together to get the

least common multiple : (2)(3)(1)(2) = 12. That is, 12 is the smallest number

that both 6 and 12 will go into evenly.

*Example 2:* Here is another example, finding the LCM
for 6 and 9:

Only one division is possible. Therefore, the LCM is : (3)(2)(3) = 18.

*Example 3:* And here is an example for finding the
LCM for 2 and 9:

Since there is no number that goes into both 2 and 9 evenly, simply multiply

them together to get the LCM: (2)(9) = 18.

*Examples 4 and 5: *Here are two that require more
steps to show that you just

keep going till you cannot go any more:

So the LCM for 8 and 12 is (2)(2)(2)(3) = 24. And the LCM for 24 and 36

is (2)(2)(3)(2)(3) = 72.

As you can see in the grid, there are some interesting
patterns. For example,

the main diagonal from lower left to upper right is the numbers themselves.

The left-hand column and the bottom row are all multiples of 1; therefore,

they are also the numbers themselves. Rows and columns involving prime

numbers (e.g., 7 and 11) do look like the simple multiplication facts, except

for the special cases of (7,7) and (11,11). These special cases, of course, can

be important clues to understanding the whole grid. Rows and columns

involving non-primes (e.g., 4s, 6s, and 12s) are the ones that are surprising

and/or confusing and, therefore, the most interesting.

**Extensions**

One teacher who used this exercise (Mary Hebrank, Duke
School for

Children) presented it a bit differently . She began the exercise as outlined

above, but then, since her students did not finish it right away, she devised a

poster with the grid on it. Students could come to her with their suggested

numbers, and if they were correct, she would add them to the grid with

post-it notes. The grid was finally filled after several days. Students were

also encouraged to write out and submit their explanations of how the grid

worked.

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