This problem solving activity has an algebra focus.
This array of numbers is built using the following pattern.
Where would you find the following numbers? That is, in which row and which position/column in that row?
a. 37 b. 61 c. 86
Use your method to locate the number 1,387.
A range of approaches will achieve a solution to this problem. These include writing out all the numbers until the given number is reached and a more sophisticated approach using triangular numbers.
As students work through this problem, different patterns will become evident. As this happens, encourage and support the students to express these patterns algebraically.
This array of numbers is built using the following pattern.
Where would you find the following numbers? That is, in which row and which position/column in that row?
a. 37 b. 61 c. 86
Use your method to locate the number 1,387.
There are a number of ways of doing this problem and so it should be useful to use with a class with a range of abilities.
Method 1: Build the table to the required number.
This equates to testing all possible combinations which will generate the answers to 37, 61 and 89 but clearly would be extremely tedious for a number such as 1,387. Of course, it will be impossible to find a general rule to locate any number using this approach.
1 | |||||||||||
2 | 3 | ||||||||||
4 | 5 | 6 | |||||||||
7 | 8 | 9 | 10 | ||||||||
11 | 12 | 13 | 14 | 15 | |||||||
16 | 17 | 18 | 19 | 20 | 21 | ||||||
22 | 23 | 24 | 25 | 26 | 27 | 28 | |||||
29 | 30 | 31 | 12 | 33 | 34 | 35 | 36 | ||||
37 | 38 | 39 | 40 | 41 | 42 | 43 | 44 | 45 | |||
46 | 47 | 48 | 49 | 50 | 51 | 52 | 53 | 54 | 55 | ||
56 | 57 | 58 | 59 | 60 | 61 | 62 | 63 | 64 | 65 | 66 | |
67 | 68 | 69 | 70 | 71 | 72 | 73 | 74 | 75 | 76 | 77 | 78 |
79 | 80 | 81 | 82 | 83 | 84 | 85 | 86 | … |
From this method, the solutions are
37: 9th row, position 1 (9, 1)
61: (11, 6)
86: (13, 8)
Note: A computer could be set ‘to work’ to build the table and locate any number. The position of 1 387 might be found using a spreadsheet.
Method 2: Look for a pattern.
The first number of each row increases steadily.
Row number | First number in row | Increase from previous row |
1 | 1 | 0 |
2 | 2 | 1 |
3 | 4 | 2 |
4 | 7 | 3 |
5 | 11 | 4 |
6 | 16 | 5 |
7 | 22 | 6 |
Extending this pattern will indicate the row for any given number. But this is still tedious for finding 1,387. And it still won’t tell you where any given number is.
Method 3: Adopt the strategy ‘have I seen a similar problem like this before?’, combined with ‘Guess and Check’.
Notice that at the end of each row the numbers are 1, 3, 6, 10, 15, 21, These are the well known Triangular Numbers. The formula for the nth one of these is
So to locate 86, say,
So 86 is located in the 13th row.
The 13th row has 13 numbers, so working backwards locates 86 at (13, 8).
To locate 1,387, try
so, 1,387 is in the 53rd row at (53, 9).
Method 4: Solve an equation.
This means 1,387 is in the 53rd row. Hence, it can be located at (53, 9).
Method 3 gives a general approach using the triangular numbers. Is it possible to find a formula though, which will give the position of the number n? Let us know if you or one of your students finds such a rule.
Printed from https://meaningfulmaths.nt.edu.au/mmws/nz/resource/rows-numbers at 8:58pm on the 26th February 2024