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, to 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.

### Problem

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.

### Teaching sequence

- Start by writing the first three lines of the pattern on the board.
*What is the next line in this pattern? And the next?* - Pose 1(a) of the problem.

Share answers and approaches used. - Pose 1 (b) and brainstorm:
*What approaches could you use to find 61?* - Pose the problem for the students to work on.
- Focus questions that can help the students get started include:
*How can we set this up?*

What information do we know?

What mathematical knowledge could we apply to this problem? - As the students work ask questions that focus on the approach or method they are using and on the patterns that they are observing in the problem.
*What approach are you using? Why did you select that one? Is it effective?*

Have you seen a pattern of numbers like this before? - Ask the students to justify their reasoning by writing a concluding statement to explain their answer.
- Share and discuss answers.

#### Extension to the problem

- Write down a method or a rule for finding the location of any given number. (That is for giving its row and position in that row.)
- Invent your own array of numbers and find general patterns.

### Solution

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: 9^{th} 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 postiion 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 13^{th} row.

The 13^{th} row has 13 numbers, so working backwards locates 86 at (13, 8).

To locate 1,387, try

so, 1,387 is in the 53^{rd} row at (53, 9).

Method 4: Solve an equation.

which means 1,387 is in the 53^{rd} row, and hence it can be located at (53, 9).

Extension: (a) 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.