Question

b practising

1. these two models show the same pattern in two different ways.

figure 1 figure 2 figure 3

a) describe what changes and what stays the same in each model.

b) write an algebraic pattern rule for each model.

Explanation:

Part (a)

• What stays the same: The overall shape structure (a "U" or "notch" - like shape with a top row of 3 blocks, a middle empty space, and side columns) and the position of the empty/middle block (1 block in the center of the top row's level) are constant. Also, the number of blocks in the top row (3) remains the same across all figures.
• What changes: The number of blocks in the side columns (vertical columns on the left and right) increases. In Figure 1, each side column has 1 block (below the top row), in Figure 2, each side column has 2 blocks, and in Figure 3, each side column has 3 blocks. So the height (number of vertical blocks) of the side columns increases as we move from Figure 1 to Figure 3.

Step 1: Define variables

Let \( n \) be the figure number (where \( n = 1,2,3,\dots \)) and \( T \) be the total number of blocks.

Step 2: Analyze block count

• Top row: 3 blocks (constant for all \( n \)).
• Side columns: Each side column has \( n \) blocks (since in Figure 1 (\( n = 1 \)) each side column has 1 block, Figure 2 (\( n = 2 \)) each has 2, Figure 3 (\( n = 3 \)) each has 3). There are 2 side columns.

Step 3: Formulate the formula

The total number of blocks \( T=3 + 2\times n\).

For the second (colored: gray - yellow) model:

Step 1: Define variables

Let \( n \) be the figure number (where \( n = 1,2,3,\dots \)) and \( T \) be the total number of blocks.

Step 2: Analyze block count

• Top row: 3 blocks (but 1 is yellow (empty - space - like) and 2 are gray, but the count of non - empty top - row blocks is 2? Wait, no. Wait, the structure: top row has 3 blocks (1 yellow (empty) and 2 gray? No, looking at the gray - yellow model: in Figure 1, each side column has 2 blocks (top row + 1 below), Figure 2 each side column has 3 blocks (top row + 2 below), Figure 3 each side column has 4 blocks (top row + 3 below). Wait, re - evaluating:

Wait, in the gray - yellow model:

• Figure 1: Top row (3 blocks: 2 gray, 1 yellow), and each side column has 1 block below the top row. So total blocks: \( 3+2\times1 = 5\)
• Figure 2: Top row (3 blocks: 2 gray, 1 yellow), and each side column has 2 blocks below the top row. So total blocks: \( 3 + 2\times2=7\)
• Figure 3: Top row (3 blocks: 2 gray, 1 yellow), and each side column has 3 blocks below the top row. So total blocks: \( 3+2\times3 = 9\)

Wait, actually, for the gray - yellow model, the number of blocks in each side column (including the top - row block) is \( n + 1\)? Wait, no. Let's re - do:

For the gray - yellow model:

• In Figure 1 (\( n = 1 \)):
• Top row: 3 blocks (2 gray, 1 yellow).
• Side columns: each side column has 2 blocks (top row block + 1 below). So total blocks: \( 3+2\times1=5\)
• In Figure 2 (\( n = 2 \)):
• Top row: 3 blocks (2 gray, 1 yellow).
• Side columns: each side column has 3 blocks (top row block + 2 below). So total blocks: \( 3+2\times2 = 7\)
• In Figure 3 (\( n = 3 \)):
• Top row: 3 blocks (2 gray, 1 yellow).
• Side columns: each side column has 4 blocks (top row block + 3 below). Wait, this is inconsistent with the first model. Wait, no, maybe the first model (red - purple) has the top row as 3 (all filled) and side columns as \( n \) blocks below, and the second model (gray - yellow) has the top row as 3 (with 1 empty) and side columns as \( n + 1\) blocks (including the top - row block).

Wait, let's start over for the gray - yellow model:

Let \( n \) be the figure number.

• Top row: 3 blocks (1 yellow, 2 gray).
• Side columns: Each side column has \( n + 1\) blocks? No, in Figure 1, side columns (gray) have 2 blocks (top row + 1 below), Figure 2: 3 blocks (top row + 2 below), Figure 3: 4 blocks (top row + 3 below). So the number of blocks in each side column (gray) is \( n + 1\), and there are 2 side columns.

So total blocks \( T=3+2\times(n)\)? No, Figure 1: \( 3 + 2\times1=5\) (matches: 2 gray top, 1 yellow top, 2 gray side (1 each column)), Figure 2: \( 3+2\times2 = 7\) (2 gray top, 1 yellow top, 4 gray side (2 each column)), Figure 3: \( 3+2\times3=9\) (2 gray top, 1 yellow top, 6 gray side (3 each column)). Yes, so for the gray - yellow model, the formula is also \( T = 3+2n\), but the interpretation of the blocks (yellow as a "space" and gray as the filled side and partial top) is slightly different, but the count formula is the same in terms of arithmetic.

Answer:

• Stays the same: The top - row block count (3), the central empty - space position, and the overall "notched" shape structure.
• Changes: The number of blocks in each vertical side column (increases by 1 per figure: 1 in Figure 1, 2 in Figure 2, 3 in Figure 3).

Part (b)

For the first (colored: red - purple) model: