Solved Problems: Representing Hundredths with Base Ten Blocks

📚 Understanding Hundredths with Base Ten Blocks

Base ten blocks are a fantastic tool for visualizing place value, including hundredths. Let's break down how they work:

• 🧱 Definition: Hundredths represent parts of a whole that has been divided into one hundred equal pieces. Think of it as a fraction where the denominator is 100.
• 📜 History: The concept of place value and using physical objects to represent numbers dates back to ancient civilizations. Base ten blocks are a modern adaptation of these ideas, making abstract concepts more concrete.
• 📐 Key Principle: In a base ten block system representing hundredths, a large flat square typically represents one whole (1), a long rod represents one-tenth (0.1), and a small cube represents one-hundredth (0.01).
• ➕ Addition: To add hundredths, combine the corresponding blocks. For example, to add 0.23 and 0.14, combine 2 rods and 3 cubes with 1 rod and 4 cubes, resulting in 3 rods and 7 cubes, or 0.37.
• ➖ Subtraction: Subtraction works similarly. If you have 0.45 and want to subtract 0.21, remove 2 rods and 1 cube, leaving 2 rods and 4 cubes, or 0.24.
• 📊 Real-World Example: Imagine you have a chocolate bar divided into 100 squares. If you eat 35 squares, you've eaten 35/100 or 0.35 of the bar. Using base ten blocks, you'd represent this with 3 rods and 5 cubes.
• 💡 Tip: Always start by identifying what each block represents in your specific problem. This will prevent confusion and make calculations easier.

🧮 Representing Hundredths Visually

Here's how to visually represent hundredths using base ten blocks:

• 🟩 The Flat (1 Whole): The large flat square represents one whole unit, or 1. This is equivalent to 100 hundredths ($1 = \frac{100}{100} = 1.00$).
• 📏 The Rod (Tenths): Each rod represents one-tenth, or 0.1. It takes ten rods to make one flat ($0.1 = \frac{10}{100}$).
• ▪️ The Cube (Hundredths): Each small cube represents one-hundredth, or 0.01. It takes one hundred cubes to make one flat ($0.01 = \frac{1}{100}$).

➕ Adding Hundredths with Base Ten Blocks

Let's work through some addition problems using base ten blocks:

• 🤝 Example 1: Represent 0.15 + 0.23. Use 1 rod and 5 cubes for 0.15, and 2 rods and 3 cubes for 0.23. Combine them: 3 rods and 8 cubes, totaling 0.38.
• ➕ Example 2: Represent 0.47 + 0.35. Use 4 rods and 7 cubes for 0.47, and 3 rods and 5 cubes for 0.35. Combining gives 7 rods and 12 cubes. Since 10 cubes make a rod, exchange 10 cubes for 1 rod, resulting in 8 rods and 2 cubes, totaling 0.82.

➖ Subtracting Hundredths with Base Ten Blocks

Now, let's try some subtraction problems:

• 📦 Example 1: Represent 0.56 - 0.21. Start with 5 rods and 6 cubes. Remove 2 rods and 1 cube, leaving 3 rods and 5 cubes, totaling 0.35.
• 💔 Example 2: Represent 0.32 - 0.18. Start with 3 rods and 2 cubes. You need to remove 1 rod and 8 cubes. Since you only have 2 cubes, exchange 1 rod for 10 cubes, giving you 2 rods and 12 cubes. Now remove 1 rod and 8 cubes, leaving 1 rod and 4 cubes, totaling 0.14.

➗ Dividing Hundredths with Base Ten Blocks

Division can also be visualized, though it's slightly more abstract:

• ➗ Example 1: Represent 0.6 ÷ 2. Start with 6 rods. Divide the rods into two equal groups. Each group has 3 rods, so 0.6 ÷ 2 = 0.3.
• ➅ Example 2: Represent 0.84 ÷ 4. Start with 8 rods and 4 cubes. Divide the rods and cubes into four equal groups. Each group has 2 rods and 1 cube, so 0.84 ÷ 4 = 0.21.

📝 Practice Quiz

Test your understanding with these problems:

1. ❓ Represent 0.63 using base ten blocks.
2. ➕ Calculate 0.28 + 0.31 using base ten blocks.
3. ➖ Calculate 0.75 - 0.42 using base ten blocks.
4. ➗ Calculate 0.9 ÷ 3 using base ten blocks.
5. 🤔 Calculate 0.56 + 0.27 using base ten blocks.
6. ➗ Calculate 0.72 ÷ 6 using base ten blocks.
7. ➕ Calculate 0.45 + 0.38 using base ten blocks.

✅ Conclusion

Base ten blocks provide a hands-on way to understand and manipulate hundredths, making them an invaluable tool in mathematics education. By understanding the relationships between the blocks, you can confidently tackle addition, subtraction, multiplication, and division problems involving decimals.