15. Patterns

9, 16, 64, 81

9 = 3 × 3, 16 = 4 × 4, 64 = 8 × 8, 81 = 9 × 9. The others are not square numbers. For example, 50 lies between 49 (7 × 7) and 64 (8 × 8), and 72 and 80 lie between 64 and 81.

3, 6, 15, 21

The triangular numbers are 1 + 2 = 3, + 3 = 6, + 4 = 10, + 5 = 15, + 6 = 21, + 7 = 28, + 8 = 36, + 9 = 45 … Of these, only 3, 6, 15 and 21 are in the list. 42 is not triangular (it lies between 36 and 45).

36 is both a square number (6 × 6 = 36) and a triangular number (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = 36).

The square numbers are 2 × 2 = 4 (1st), 3 × 3 = 9 (2nd), 4 × 4 = 16 (3rd), … The side is always one more than the position. So the 10th square number is 11 × 11 = 121.

The list: 4, 9, 16, 25, 36, 49, 64, 81, 100, 121.

The triangular numbers are 1 + 2 = 3 (1st), 1 + 2 + 3 = 6 (2nd), … The 10th one adds up to 11: 1 + 2 + 3 + … + 11 = 66.

The list: 3, 6, 10, 15, 21, 28, 36, 45, 55, 66.

1. How will you decide if a given number is a square number? Multiply numbers by themselves (1 × 1, 2 × 2, 3 × 3, …). If one of these products equals the given number, it is a square number. If the number falls between two such products, it is not. For example, 40 lies between 36 and 49, so it is not a square number.
2. How will you decide if a given number is a triangular number? Keep adding 1 + 2 + 3 + 4 + … If the total reaches the given number exactly, it is triangular; if the total jumps past it, it is not. Another way: double the number. If the result is the product of two consecutive numbers, the number is triangular (for example, 2 × 21 = 42 = 6 × 7, so 21 is triangular).
3. How many square numbers are there? There are endless (infinitely many) square numbers, because every counting number, however large, can be multiplied by itself.
4. How many triangular numbers are there? There are also infinitely many triangular numbers, because we can always add the next counting number to get the next one.

• Square numbers: a chessboard (8 × 8 = 64 squares), tiles on a square floor, a square rangoli of dots, a window with square panes, egg or chocolate trays arranged in a square.
• Triangular numbers: pipes or logs stacked in a pile, bowls or glasses stacked in a pyramid, ten-pin bowling pins (10), snooker balls in the triangle (15), fruit piled on a cart.

These are hands-on activities. Some tips:

• Join the 6 cm × 4 cm notched shape so that the tab of one piece fits into the notch of the next. Alternate the coloured and plain pieces, as in the textbook example.
• For the four shapes in the box, turn or flip each piece so that its edges match the next one. Keep adding pieces until the whole area is covered with no gaps and no overlaps.
• A pattern like floor tiles is formed when the same arrangement repeats in every direction.

• Border 1: a four-petal flower followed by a curling stem that joins the next flower. Keep repeating flower – stem – flower – stem to the end of the box.
• Border 2: two shapes take turns. One is a leaf opening downward with a small circle above it; the next is a leaf opening upward with a small circle below it. Continue down – up – down – up.
• Border 3: a small flower sits on top of each pair of arches, and the arches join side by side. Keep adding arch pairs with a flower above each.