Square and Cube Numbers - Worksheet | Maths Year 6 - Free Printable

Square and Cube Numbers Worksheet Solution

#### Bronze Level

1. Luke has 25 small cubes. He says the biggest solid cube he could make would use all 25 cubes because \(5 \times 5 = 25\). Is he correct? Explain your answer, with diagrams.

- Explanation:
- Luke is incorrect. The expression \(5 \times 5 = 25\) represents a square number, not a cube number.
- A cube number is formed by multiplying a number by itself three times (\(n \times n \times n\)). For example, the smallest cube number greater than 25 is \(3^3 = 27\).
- To form a solid cube using small cubes, the total number of cubes must be a perfect cube (e.g., \(1^3 = 1\), \(2^3 = 8\), \(3^3 = 27\), etc.).
- Since 25 is not a perfect cube, Luke cannot form a solid cube using exactly 25 cubes.

- Diagram:
- A \(5 \times 5\) square can be visualized as a flat, two-dimensional shape with 25 smaller squares.
- A \(3 \times 3 \times 3\) cube requires 27 smaller cubes to form a three-dimensional shape.

- Answer: No, Luke is incorrect. The biggest solid cube he could make would require 27 cubes (\(3^3\)).

2. Lucy is trying to work out if \(8^2\) is bigger than \(5^3\). She says that \(8^2\) is bigger because \(8^2 = 8 \times 2 = 16\) and \(5^3 = 5 \times 3 = 15\). Is she correct? Explain your answer.

- Explanation:
- Lucy's calculations are incorrect.
- \(8^2\) means \(8 \times 8 = 64\), not \(8 \times 2 = 16\).
- \(5^3\) means \(5 \times 5 \times 5 = 125\), not \(5 \times 3 = 15\).
- Comparing the correct values:
- \(8^2 = 64\)
- \(5^3 = 125\)
- Clearly, \(5^3 = 125\) is greater than \(8^2 = 64\).

- Answer: No, Lucy is incorrect. \(5^3 = 125\) is bigger than \(8^2 = 64\).

3. Fill in the missing square numbers to make the calculations correct.
- (a) \( ? + 46 = 110 \)
- (b) \( 356 - ? = 307 \)

- Solution:
- (a) Let the missing number be \(x\):
\[
x + 46 = 110
\]
\[
x = 110 - 46 = 64
\]
- \(64\) is a square number (\(8^2 = 64\)).

- (b) Let the missing number be \(y\):
\[
356 - y = 307
\]
\[
y = 356 - 307 = 49
\]
- \(49\) is a square number (\(7^2 = 49\)).

- Answer:
- (a) \(64\)
- (b) \(49\)

#### Silver Level

1. Place 5 odd and 5 even numbers in the Carroll diagram.

- Carroll Diagram:
\[
\begin{array}{|c|c|c|}
\hline
& \text{Squared} & \text{Not squared} \\
\hline
\text{100-200} & & \\
\hline
\text{100 or less} & & \\
\hline
\end{array}
\]

- Odd and Even Numbers:
- Odd numbers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, ...
- Even numbers: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, ...

- Filling the Diagram:
- 100-200:
- Squared: 121 (\(11^2\)), 144 (\(12^2\)), 169 (\(13^2\)), 196 (\(14^2\))
- Not squared: 101, 103, 105, 107, 109 (odd), 102, 104, 106, 108, 110 (even)
- 100 or less:
- Squared: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100
- Not squared: 2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 17, 18, 19, ...

- Final Diagram:
\[
\begin{array}{|c|c|c|}
\hline
& \text{Squared} & \text{Not squared} \\
\hline
\text{100-200} & 121, 144, 169, 196 & 101, 103, 105, 107, 109 \\
\hline
\text{100 or less} & 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 & 2, 3, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 17, 18, 19, ... \\
\hline
\end{array}
\]

2. On a hundred square, shade in all the square numbers. Then shade in all the multiples of 4, in a different colour. What do you notice?

- Square Numbers (1-100):
- \(1^2 = 1\), \(2^2 = 4\), \(3^2 = 9\), \(4^2 = 16\), \(5^2 = 25\), \(6^2 = 36\), \(7^2 = 49\), \(8^2 = 64\), \(9^2 = 81\), \(10^2 = 100\)
- These are: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100.

- Multiples of 4 (1-100):
- These are: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96, 100.

- Observation:
- All square numbers that are multiples of 4 (e.g., 4, 16, 36, 64, 100) are also multiples of 4.
- However, not all multiples of 4 are square numbers (e.g., 8, 12, 20, 24, etc.).

- Answer: Square numbers that are multiples of 4 are also multiples of 4, but not all multiples of 4 are square numbers.

#### Gold Level

1. Complete the table below. What relationship do you see between square and cube numbers?

- Table:
\[
\begin{array}{|c|c|c|c|c|}
\hline
& 6 \times 6 & 6^3 & & \\
\hline
& & 64 & 8^3 & \\
\hline
5^2 & & & 5 \times 5 \times 5 & \\
\hline
& 2 \times 2 & & 2^3 & \\
\hline
7^2 & & & & 64 \\
\hline
\end{array}
\]

- Completed Table:
\[
\begin{array}{|c|c|c|c|c|}
\hline
& 6 \times 6 & 6^3 & & \\
\hline
& & 64 & 8^3 & \\
\hline
5^2 & & & 5 \times 5 \times 5 & \\
\hline
& 2 \times 2 & & 2^3 & \\
\hline
7^2 & & & & 64 \\
\hline
\end{array}
\]
- Row 1: \(6 \times 6 = 36\), \(6^3 = 216\)
- Row 2: \(64\) is \(8^2\), \(8^3 = 512\)
- Row 3: \(5^2 = 25\), \(5 \times 5 \times 5 = 125\)
- Row 4: \(2 \times 2 = 4\), \(2^3 = 8\)
- Row 5: \(7^2 = 49\), \(64\) is \(4^3\)

- Relationship:
- Square numbers are the result of multiplying a number by itself (\(n \times n\)).
- Cube numbers are the result of multiplying a number by itself three times (\(n \times n \times n\)).
- Some numbers can be both square and cube (e.g., \(64 = 8^2 = 4^3\)).

2. Look at the hundred square from Silver question 2, is there a pattern for cube numbers on the grid?

- Pattern:
- Cube numbers up to 100 are: \(1^3 = 1\), \(2^3 = 8\), \(3^3 = 27\), \(4^3 = 64\).
- These numbers do not follow a regular pattern on the hundred square grid like square numbers do. They are more spread out and less frequent.

- Answer: There is no clear repeating pattern for cube numbers on the hundred square grid.

3. How many rows and columns would you need so you could find \(5^3\)?

- Explanation:
- \(5^3 = 5 \times 5 \times 5 = 125\).
- To represent \(125\) in a grid, you would need a grid large enough to contain 125 cells.
- A \(125 \times 1\) grid or a \(5 \times 25\) grid or any other factor pair of 125 would work.

- Answer: You would need a grid with at least 5 rows and 25 columns (or any configuration that totals 125 cells).

---

Final Answers:

1. Bronze:
- (1) No, Luke is incorrect. The biggest solid cube he could make would require 27 cubes.
- (2) No, Lucy is incorrect. \(5^3 = 125\) is bigger than \(8^2 = 64\).
- (3) (a) \(64\), (b) \(49\).

2. Silver:
- (1) Completed Carroll diagram as explained.
- (2) Square numbers that are multiples of 4 are also multiples of 4, but not all multiples of 4 are square numbers.

3. Gold:
- (1) Relationship: Square numbers are \(n \times n\), cube numbers are \(n \times n \times n\).
- (2) No clear pattern for cube numbers on the hundred square grid.
- (3) At least 5 rows and 25 columns.

Boxed Final Answer:
\[
\boxed{64, 49, 5 \text{ rows and } 25 \text{ columns}}
\]