## Key Concepts

- Estimation, Standard algorithm, partial products, and word problems

**Estimation, standard algorithm, partial products, and word problems**

### Estimation:

Estimation means finding a number that is close enough to the right answer.

### Example:

Estimate the product of 7 × 489 by rounding to the nearest hundred.

7 × 489

489 –> 500

Multiply 7 × 500 = 3500

Multiply 7 × 489 = 3,423

3,423 is close to 3,500. It is reasonable.

**What is meant by a standard algorithm?**

A standard algorithm or method is a specific method of computation that is conventionally taught for solving particular mathematical problems.

Standard algorithm is a way of doing multiplication by using partial products or multiplying in parts.

**What is meant by partial products?**

A model that breaks numbers down into their factors or place values to make multiplication easier.

**Example 1**:

Multiply 215 × 9

Step1: Multiply 200 × 9 = 1800

Step2: Multiply 10 × 9 = 90

Step3: Multiply 5 × 9 = 45

Step4: Add 1800 + 90 + 45 =1935

**Example2:**

Each T-Shirts’ cost is $485. Find the cost of 6 T-Shirts?

Each frock’s cost is $4,480. Find the cost of 3 frocks?

Also, find the total cost for T-Shirts and Frocks.

**Solution:**

6 × 485 = c

Estimate: 6 × 485 is about 6 × 480= 2,880

Break apart 485 using place value and the distributive property.

6 × 485 = 6 × (400 + 85)

= 6 × 400 + 6 × 85

= 2400 + 510

= 2,910

The total cost of T-Shirts is $ 2,910.

Find the cost of the dress 3 × 4,480 = y

Estimate 3 × 4,480 is about 3 × 4,500 =13,440

Use an area model partial products.

**Example 2:**

Find the product of 3 × (7+4) using the distributive property

Using the distributive law, we:

- Multiply, or distribute, the outer term to the inner terms.
- Combine like terms.
- Solve the equation.

3(7+4) = 3(7) + 3(4)

= 21 + 12

= 33

**Example 3:**

Let us understand this concept with distributive property examples

For example, 3(2 + 5) = 3 (7) = 21

Or

By distributive law

3(2 + 5) = 3 × 2 + 3 × 5

= 6 + 15

= 21

Here we are distributing the process of multiplying 3 evenly between 2 and 5. We observe that whether we follow the order of the operation or distributive law or not, the result is the same.

**Example 4:**

** **Multiply 9 × 4860 using the standard algorithm model.

**Solution:**

**Steps:**

- Multiply the ones

- Multiply the tens

- Multiply the hundreds, add any extra hundreds that we were carried over.

- Multiply the thousands, add any extra thousands.

**Example 5:**

Use partial products to multiply 7 × $332.

**Solution:**

**Step 1:** Estimate the product.

332 rounds to 300; 7 × $300 = 2100

**Step 2:** Multiply the 3 hundreds, or 300, by 7.

7 × 300=$2100

**Step 3:** Multiply the 3 tens, or 30, by 7.

7 × 30 =$ 210

**Step 4:** Multiply the 2 ones, or 2, by 7.

7 × 2=$14

**Step 5:** Add the partial products.

So, 7 × $332 = $2,324. Since $2,324 is close to the estimate of $2,100

**Example 6: **

Use partial products to multiply 4 × $534.

**Step 1:** Estimate the product.

4 × $534 is about estimate 4 × $ 500 = $2000

**Step 2:** Multiply the 5 hundreds, or 500, by 4.

4 × 500 = 2000

**Step 3:** Multiply the 3 tens, or 30, by 4

4 × 30 =120

**Step 4:** Multiply the 4 ones, or 4, by 4.

4 × 4 = 16

**Step 5:** Add the partial products.

So, 4 × $ 534 = $2,136. Since $2,136 is close to the estimate of $2,000.

### Exercise:

1. Solve the following using partial products:

(a) 78 × 8 (b) 6 × 765 (c) 4 × 456

2. Solve the following using distributive property:

(a) 4 × 564 (b) 2 × 465 (c) 3 × 590

3. Find each product by choosing an appropriate strategy.

(a) 3 × 45 (b) 6 × 876 (c) 5 × 25

4. Estimate the sum by rounding the number to the nearest thousand and then multiply.

(a) 3 x 3,953 (b) 5 × 5,458

5. Solve the following using standard algorithm:

(a) 5 × 34 (b) 6 × 345 (c) 7 × 543

6. A theatre can seat 460 people at one time. If a movie in the theatre fills every seat, every day for 3 weeks, how many tickets were sold?

7. A banana contains 105 calories. Last week, Nate and Haley ate a total of 14 bananas. How many calories did they eat?

An office needs to buy 20 new printers and 100 packages of paper. Each printer costs $300. Each package of paper costs $25. What is the cost of the printers and paper combined?

9. A company sells jerseys for $50 each. In October, they see 32 jerseys. In April, they sold 55 jerseys. What is the total amount of money they earned by selling jerseys in October and April combined?

10. A toy shop sells 4 toys that cost $200 each. Find the cost of 4 toys?

**Concept Map**:

### What have we learned:

- Understand estimation
- Understand standard algorithm
- Understand partial products
- Understand word problems