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This investigation is about the number of dots in shapes that are regular polygons.
For any dotty polygon
• $p$ is the number of sides
• $n$ is the number of dots on one side
• there are the same number of dots on each side.
Example
This is a dotty triangle.
In this dotty triangle, $p = 3$ and $n = 4$.
(a) Look at the numbers of dots in each row of the example.
Complete this sum for the total number of dots in the dotty triangle.
1 + 2 + 3 + ............ = ............ [2]
(b) For a dotty triangle where $n = 10$, complete this sum and find the total number of dots.
1 + 2 + 3 + ........ + ........ + ........ + ........ + ........ + ........ + ........ = ........ [2]
(c) Show that $\frac{n^2}{2} + \frac{n}{2}$ gives the correct number of dots when $n = 10$. [2]
587
511
523
The diagram shows the first four dotty triangles.
The number of dots added each time is $d$.
$n = 1 \quad n = 2 \quad n = 3 \quad n = 4$
$d = 1 \quad d = 2 \quad d = 3 \quad d = 4$
So, for dotty triangles, $d = n$.
This diagram shows the first three dotty squares.
$n = 1 \quad n = 2 \quad n = 3 \quad n = 4$
$d = 1 \quad d = 3 \quad d = 5 \quad d = 7$
(a) Draw the dotty square for $n = 4$ in the space above. [1]
(b) (i) Write down the total number of dots in each of the first four dotty squares.
................ , ................ , ................ , ................ [1]
(ii) Write down an expression, in terms of $n$, for the total number of dots in the $n$th dotty square.
........................................ [1]
(c) For dotty squares, find a formula for $d$ in terms of $n$.
................................................... [3]
(d) A formula for $d$, in terms of $p$ (the number of sides) and $n$ is
$$d = (p - 2)n - p + 3.$$
By substituting appropriate values for $p$, show that this formula gives
(i) the formula for dotty triangles, [2]
(ii) your formula for dotty squares. [2]
570
580
582
(a) For dotty pentagons, show that the formula in Question 2(d) becomes $d = 3n - 2$.
[1]
(b) This diagram shows the first three dotty pentagons.
$\begin{align*} &\cdot \\[5pt] &\cdot \\ &\cdot \\[15pt] &\cdot \\[5pt] &\cdot &\cdot &\cdot \\[15pt] &\cdot \\[5pt] &\cdot &\cdot &\cdot &\cdot &\cdot \end{align*}$
$d = 1 \quad d = 4 \quad d = 7$
total = 1 \quad total = 5 \quad total = 12
Dotty pentagons grow along the grey lines.
This diagram shows how to form the first three dotty pentagons.
(i) Use $d = 3n - 2$ to find the number of dots that you add to the 3rd dotty pentagon to make the 4th dotty pentagon.
........................................... [2]
(ii) Complete the diagram to show the 4th and 5th dotty pentagons. [2]
(iii) Complete the final statement.
1st pentagon $+$ 4 dots $=$ 2nd pentagon
2nd pentagon $+$ 7 dots $=$ 3rd pentagon
$\vdots$
.......th pentagon $+$ 52 dots $=$ ........th pentagon
[2]
545
597
582
(a) This table shows the total number of dots in some dotty polygons.
Use Question 2, Question 3 and any patterns you notice to help you complete this table.
[Table_1]
b(i) The number of dots in a dotty pentagon × 3 = The number of dots in a dotty triangle
Give two examples from the table that show this statement is true.
(ii) The number of dots in the 4th dotty pentagon × 3 = The number of dots in the $k^{th}$ dotty triangle
Find the value of $k$.
524
546
576
