How to Use Area Models and the Distributive Property to Multiply?

How to Use Area Models and the Distributive Property to Multiply? Is a multiple-choice math challenge from Effortless Math. Read the problem, choose from the answer options below, then open the solution to see the full step-by-step working and check your reasoning against it.

Step 1: Understand the Area Model:

Step 2: Break down the Numbers:

Step 3: Use the Distributive Property:

Step 4: Draw the Area Model:

Step 5: Multiply and Add:

Using Area Models and the Distributive Property to Multiply, Example 1:

Solution:

(20 × 4 = 80)

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(4 × 4 = 16)

(80 + 16 = 96)

Therefore, (24 × 4 = 96).

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Worked examples with area models

Example 1: 24 × 4. Split 24 by place value into 20 + 4. Draw a rectangle 4 units tall and split its length into a 20 part and a 4 part.

Add the two areas: 80 + 16 = 96. In distributive property form: 4 × (20 + 4) = 4 × 20 + 4 × 4 = 96.

Example 2: 23 × 14. Split both numbers: 23 = 20 + 3 and 14 = 10 + 4. Now the rectangle has four parts.

Add all four partial products: 200 + 30 + 80 + 12 = 322. So 23 × 14 = 322.

Splitting by place value (tens and ones) keeps every small product easy to do in your head.

Practice: use an area model

1. 7 × 38
2. 5 × 126
3. 15 × 12
4. 3(x + 5)

Answers:

1. 7 × 30 + 7 × 8 = 210 + 56 = 266
2. 5 × 100 + 5 × 20 + 5 × 6 = 500 + 100 + 30 = 630
3. (10 + 5)(10 + 2) = 100 + 20 + 50 + 10 = 180
4. 3 × x + 3 × 5 = 3x + 15

Frequently Asked Questions

How do you use the distributive property with an area model?

Draw a rectangle, label one side with one factor, and split the other side into friendly parts. Find the area of each smaller rectangle, then add. For 8 × 36, split 36 into 30 + 6: the parts are 8 × 30 = 240 and 8 × 6 = 48, so 8 × 36 = 288. The picture shows why 8 × (30 + 6) = 8 × 30 + 8 × 6.

How do you use the distributive property to find the area?

Split a rectangle into two smaller rectangles, find each area, and add them. A garden 9 feet wide and 14 feet long can be split into 9 × 10 and 9 × 4. That gives 90 + 36 = 126 square feet, the same as 9 × 14. This works well when one side is awkward to multiply but easy to break apart.

How do you use the area model to distribute 7 × 9?

Split the 9 into two easier numbers, such as 5 + 4, and draw a 7-by-9 rectangle cut into a 7-by-5 part and a 7-by-4 part. The areas are 35 and 28, and 35 + 28 = 63. So 7 × 9 = 7 × (5 + 4) = 63. You could also use 10 − 1: 70 − 7 = 63.

How do you use the area model to distribute 5(3x + 2)?

Draw a rectangle with height 5 and split its length into 3x and 2. The left box has area 5 × 3x = 15x, and the right box has area 5 × 2 = 10. Add the boxes to get 5(3x + 2) = 15x + 10. The two terms cannot combine, because 15x and 10 are not like terms.

How do you use area models to multiply polynomials?

Write the terms of one polynomial along the top and the other down the side, then fill in a box for each pair. For (x + 4)(2x + 1), the boxes are 2x², x, 8x and 4. Combine the like terms x and 8x to get 2x² + 9x + 4. Each box is one product from distributing every term.

Can you use area models and the distributive property to divide?

Yes. Here you know the area and one side, and you find the missing side in parts. For 84 ÷ 4, split 84 into 80 + 4. Since 80 ÷ 4 = 20 and 4 ÷ 4 = 1, the missing side is 20 + 1 = 21. Check by multiplying: 4 × 21 = 84.