Carson uses cubes to make a pattern of shapes. The first shape has a height of 2 cubes and a length of 2 cubes, as shown below. For each step in the pattern, Carson adds 1 cube to the height and 1 cube to the length. Which sentence about the shapes in Carson’s pattern is true? Step 1 height length The total number of cubes in each shape will be even. The total number of cubes in each shape will alternate between even and odd. The length of each shape will alternate between even numbers and odd numbers. The length of each shape will be an even number
The Correct Answer and Explanation is:
Correct Answer:
The length of each shape will alternate between even numbers and odd numbers.
Explanation
The problem describes a pattern of shapes made from cubes. In Step 1, the shape has a height of 2 cubes and a length of 2 cubes. With each step in the pattern, Carson adds 1 cube to the height and 1 cube to the length. This means:
- Step 1: Height = 2, Length = 2
- Step 2: Height = 3, Length = 3
- Step 3: Height = 4, Length = 4
- Step 4: Height = 5, Length = 5
- … and so on.
Let’s examine the lengths over multiple steps:
- Step 1: Length = 2 → Even
- Step 2: Length = 3 → Odd
- Step 3: Length = 4 → Even
- Step 4: Length = 5 → Odd
As seen here, the length alternates between even and odd numbers with each step. This is because Carson adds 1 cube to the length each time, and increasing any number by 1 always switches its parity (even to odd or odd to even). Therefore, the correct statement is:
“The length of each shape will alternate between even numbers and odd numbers.”
Let’s also briefly examine the other options:
- “The total number of cubes in each shape will be even.”
❌ Not true — the total number of cubes increases by 2 each step (1 in height and 1 in length), so it alternates between even and even if it starts from an even number (2+2 = 4). But the arrangement doesn’t guarantee the total stays even consistently. - “The total number of cubes will alternate between even and odd.”
❌ Also incorrect, because increasing by 2 keeps parity the same. - “The length will always be even.”
❌ False — it alternates between even and odd.
✅ Correct choice: “The length of each shape will alternate between even numbers and odd numbers.”
