Jim's Table

Problem

Jim has some squared paper handy. He put the numbers 1 to 10 along the top and the even numbers down the side. He then started adding the numbers together. As he filled in the numbers he began to see patterns.

Find some patterns for yourself. Use them to complete Jim’s table.

Teaching sequence

1. Introduce the problem to the class. Ask them
   What number do you think goes here?
   And what about here?
   Put these numbers onto the table.
2. Let the students work on the problem with a partner. Check that they understand what they are supposed to be doing.  
3. As a whole class share the students’ ideas and solutions.
4. Encourage them to go on to try the Extension problem.
5. Discuss the answers that the students get in a class situation. Possibly ask them to make up questions of their own.

Extension to the problem

How many times does 4 appear by itself in the body of the table? 

How many times does the digit 4 occur in the table?

Which number (as opposed to digit) occurs most in your table?

Solution

Jim’s completed table is shown below.

There are a large number of patterns that can be found here. We have highlighted a few. These are consecutive numbers (in green) both horizontally and diagonally, odd numbers (in blue), even numbers (in red) and increasing in threes (in pink). Encourage them especially to look for patterns that occur in the diagonals.

Solution to the extension:

By searching you can see that there is only 4 in the body of the table. But you can justify this by noting that there can be no 4 to the right of the 4s column or below the 4s row. So the only 4s occur above and to the left of the purple lines. There is only one 4 here.

To see how many times the digit 4 occurs, it is probably best to go systematically through each column and count them. There are none in the first column (nor any of the other ‘odd’ columns), two in the second, one in the fourth, one in the sixth, one in the eighth, and two in the tenth. This gives a total of seven. (You could follow this up by asking if we put some more columns into Jim’s table, what would be the next column to have only one 4 in it.)

There are several numbers that occur five times. These are 12, 13, 14, 15 and 16.

There are, of course, a lot more questions that you could ask here. For instance, are there more 3s than 4? Which numbers only appear twice? Which digit occurs most?