Learning Session 3 – Tiling Patterns
1. This pattern uses square numbers. The number of black squares is the square of the pattern
number. The number of white squares is the square of one less than the pattern number.
Number of black squares = n2
Number of white squares = (n – 1)2
2. Bee hexagons
There are different ways of describing the relationship. Two are shown here for ring 3, but there are
more.
If there are 6 corners of a hexagon ring, then the number on each of 6
between the corners is one less than the ring number.
Number of cells = 6 × (r – 1) + 6
sides
If the ring is thought of as being made up of 6 rows of cells, then the
number in each row is the same as the cell number.
Number of cells = 6 × r
3. Draw a tiling pattern that can be represented mathematically and
describe the structure of the pattern in a way that shows the
mathematical relationship in the pattern.
4. Take a look at the picture to the right. This is the eighth stage of the
pattern. Draw the first three steps of the pattern.
The pattern shows that when you add consecutive odd numbers starting with one, you always get a
square number as the total
1=1
1+3=4
1+3+5=9
The relationship can be explained many ways. If the previous odd numbers add to a square, then
the next odd number is 2 times the side length of the square plus one, making the square one unit
longer on each side.
learning session three – tiling patterns - teacher.docx | Page 1
If each odd number is thought of as twice a counting number plus one (2n + 1), then the square is
made up of a diagonal row of single squares plus twice a counting number each time you add a new
layer.
Hyperlink 3. 4
Hint: Try starting at the bottom left corner and add up the numbers in each stripe as you move to
the top right.
What sort of numbers do the stripes add to?
1=1
1+3=4
1 + 3 + 5 = 9…
3. Students could debate this, and give evidence supporting their reasoning. In fact,
mathematically, 0.999…. does equal 1, because the fractions extend into infinity, so you cannot
define a point where 0.999… is not equal to one.
4. If you look at the L shape of 3 quarters of the square, 1 of the 3 quarters is unshaded, or one
third of the area. If we look at the remaining quarter, and look at the L shape of 3 sixteenths, 1
of the 3 sixteenths is unshaded, or one third of the area, and so on and so on. No matter how
small the final L shape is, one third of it will be unshaded, so the relationship is:
1/4 + 1/16
+ 1/64 + ... = 1/3
Hyperlink 3.6
Hint: Look at the square divided into quarters. Ignore the top right quarter. What fraction
of the L shape is un-shaded? Now look at the top right quarter, divide it into quarters and
just look at the L shape. What fraction is un-shaded?
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5.
1/2 + 1/4 + 1/8 + 1/16 + ... = 1
Hyperlink 3.7
Hint: Divide the square into half and colour one half, then divide the uncoloured half again and
colour half of it, repeat as many times as you can.
6. 1/3 + 1/9 + 1/27 + 1/81 + ... = 1/2
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