Find the blocks that represent each of the following base five numbers. Draw a picture of your block set. Convert each of them to base 10 numbers. a. 234 five b. 1043 five c. 2340 five Picture Base 10 answer
The Cor:
The Correct Answer and Explanation is:
To solve this problem, convert each base five number to base ten, and visualize each using base five blocks. In base five, the place values are powers of 5:
- The rightmost digit is multiplied by 5⁰ (1),
- The next digit is multiplied by 5¹ (5),
- Then 5² (25),
- Then 5³ (125), and so on.
Each digit in a base five number tells how many blocks of that power of five are present.
a. 234 five
Base Ten Conversion:
2 × 25 (5²) = 50
3 × 5 (5¹) = 15
4 × 1 (5⁰) = 4
Total = 50 + 15 + 4 = 69
Blocks:
- 2 blocks of 25
- 3 blocks of 5
- 4 single unit blocks
b. 1043 five
Base Ten Conversion:
1 × 125 (5³) = 125
0 × 25 (5²) = 0
4 × 5 (5¹) = 20
3 × 1 (5⁰) = 3
Total = 125 + 0 + 20 + 3 = 148
Blocks:
- 1 block of 125
- 0 blocks of 25
- 4 blocks of 5
- 3 single unit blocks
c. 2340 five
Base Ten Conversion:
2 × 125 (5³) = 250
3 × 25 (5²) = 75
4 × 5 (5¹) = 20
0 × 1 (5⁰) = 0
Total = 250 + 75 + 20 + 0 = 345
Blocks:
- 2 blocks of 125
- 3 blocks of 25
- 4 blocks of 5
- 0 unit blocks
Explanation
Base five is a positional number system where each digit represents a power of five, starting from the right. Just like the familiar base ten system uses powers of ten (ones, tens, hundreds), base five uses ones, fives, twenty-fives, one hundred twenty-fives, and so on. When converting a base five number to base ten, multiply each digit by the corresponding power of five based on its position and then add all the values.
For example, in the number 234 (base five), the digits represent:
2 × 25 + 3 × 5 + 4 × 1 = 69.
This shows how the base five system breaks numbers into groups of five, twenty-five, and so on, instead of ten, one hundred, and so on as in base ten.
Using base blocks helps visualize this system. Each block size corresponds to a power of five:
- Unit blocks represent ones (5⁰),
- Rods or bars represent fives (5¹),
- Flats represent twenty-fives (5²),
- Cubes or larger blocks represent one hundred twenty-fives (5³).
This concrete model is especially helpful for learners to see how numbers grow in different bases. It emphasizes the grouping nature of base systems and shows how number representation changes depending on the base used.
Therefore, converting and drawing numbers in block form is not just an abstract activity, but one that builds deep conceptual understanding of place value systems
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