Sketch a new set of base pieces for each product, and then show regrouping. a. Multiply 168 by 3.

To solve the problem of multiplying 168 by 3 using base pieces and regrouping, let's break it down into steps.

### Step 1: Break Down 168 into Base Pieces
- **Hundreds**: 168 has 1 hundred, so we represent it as 1 base piece of 100.
- **Tens**: 168 has 6 tens, so we represent it as 6 base pieces of 10.
- **Units (Ones)**: 168 has 8 units, so we represent it as 8 base pieces of 1.

This gives us the following:
- 1 base piece of 100
- 6 base pieces of 10
- 8 base pieces of 1

### Step 2: Multiply Each Group by 3
Now, we multiply each set of base pieces by 3:

- **Hundreds**: \(1 \times 3 = 3\) base pieces of 100
- **Tens**: \(6 \times 3 = 18\) base pieces of 10
- **Units (Ones)**: \(8 \times 3 = 24\) base pieces of 1

This gives us:
- 3 base pieces of 100
- 18 base pieces of 10
- 24 base pieces of 1

### Step 3: Regroup
To make the final result easier to interpret, we regroup the tens and ones:

- **Tens to Hundreds**: 18 tens can be regrouped as \(1\) base piece of 100 and \(8\) tens.
- **Ones to Tens**: 24 ones can be regrouped as \(2\) base pieces of 10 and \(4\) ones.

### Final Result
- **Hundreds**: \(3 + 1 = 4\) base pieces of 100
- **Tens**: \(8 + 2 = 10\) base pieces of 10 (which is regrouped again as \(1\) base piece of 100 and \(0\) tens)
- **Units (Ones)**: 4 base pieces of 1

Final regrouped answer:
- 504, which corresponds to 168 × 3.

Would you like more details or have any other questions?

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Here are some related questions to deepen your understanding:

1. How would the process change if we multiplied 168 by 4 instead of 3?
2. What is the significance of regrouping in multiplication?
3. Can you explain why regrouping is necessary when the digits exceed 9?
4. How would you verify the result of \(168 \times 3\) using a different method?
5. What happens if you multiply a three-digit number by a two-digit number using base pieces?

**Tip:** When regrouping, always double-check each step to ensure the accuracy of your final answer.

Explore how to multiply 168 by 3 using base pieces and regrouping. Understand the concept of breaking numbers into base pieces (hundreds, tens, ones) and regrouping to simplify multiplication. Suitable for Grades 4-6.

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