Question

10. Dan explained that the middle tower is always the same number as the step number. He
also pointed out that the two arms on each side of the tower contain one less block than
the step number
Create an equation that fits Dan's way of seeing the relationship.
11. Sarah counted the number of tiles at each step and made a table. She explained that the
number of tiles in each figure was always 3 times the step number minus 2.
step
number
1
2
3
4
5
6
number
of tiles
1
4
10
13
16
Create an equation that fits Sarah's way of seeing the relationship.

Answer 1

The given pattern is an illustration of a linear equation. The students' equations are:

• Dan's: $n = s + 2(s - 1)$.
• Sally's: $n = 3s - 2$

Given that:

$n \to$ number of tiles

$s\to$ step number

See attachment for the pattern, that is being analyzed by the students.

Dan

For each step, the number of tower is the same as the step number.

This means that:

$n \to s$

2 arms contain one less block means: 2 x (s - 1) blocks for both arms.

So, Dan's equation is::

$n = s + 2 \times (s - 1)$

$n = s + 2(s - 1)$

Sally

First, we calculate the slope (m) of the table

$m = \frac{n_2 - n_1}{s_2 - s_1}$

So, we have:

$m = \frac{4-1}{2-1}$

$m = \frac{3}{1}$

$m =3$

The equation is then calculated using:

$n = m(s - s_1) + n_1$

This gives:

$n = 3(s - 1) + 1$

Open bracket

$n = 3s - 3 +1$

$n = 3s - 2$

Hence;

Dan's equation is: $n = s + 2(s - 1)$

Sally's equation is: $n = 3s - 2$

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