This worksheet helps students understand factoring by using area models to visualize the distributive property.

Absolutely! Let’s solve this worksheet step by step. The goal is to factor expressions using the area model, which visually represents multiplication and factoring as rectangles (areas = length × width).

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## 🔹 SECTION 1: Fill in missing information for each rectangle

We are given a rectangle divided into two parts. The total area is the sum of the two smaller areas, and we can also express it as the product of its dimensions (length × width). We’ll find the missing dimensions and write the area as both a sum and a product.

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Problem 1:

Given:
- Top row: `x` and `6`
- Left column: `2`
- So, areas: `2×x = 2x`, `2×6 = 12`

✔ Already filled:
- Product: `2(x + 6)`
- Sum: `2x + 12`

✔️ Correct!

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Problem 2:

Given:
- Areas: `5x` and `20`
- Left side: `5`

Since left side is 5, and areas are 5x and 20:
→ Top dimensions must be: `x` and `4` (because 5×x = 5x, 5×4 = 20)

✔ Missing top boxes: `x` and `4`

Area as product: `5(x + 4)`
Area as sum: `5x + 20`

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Problem 3:

Given:
- Areas: `6x` and `48`
- Top right: `8`

So, if one dimension is 8, and area is 48 → other dimension = 48 ÷ 8 = 6

So left side is `6`. Then, since area is `6x`, top left must be `x`.

✔ Missing top left: `x`
✔ Missing left side: `6`

Product: `6(x + 8)`
Sum: `6x + 48`

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Problem 4:

Given:
- Areas: `10x` and `30`
- Top left: `x`

So, if top left is `x`, and area is `10x` → left side = `10`

Then, since area is `30`, top right = 30 ÷ 10 = 3

✔ Missing top right: `3`
✔ Missing left side: `10`

Product: `10(x + 3)`
Sum: `10x + 30`

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## 🔹 SECTION 2: Fill in missing dimensions from the expression given

Now we’re given an expression like `5x + 35 = 5(____)` and need to fill in the rectangle and the blank.

This means we’re factoring out the GCF (greatest common factor) and representing it with a rectangle.

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Problem 5: `5x + 35 = 5(___)`

Factor out 5:
→ `5x + 35 = 5(x + 7)`

Rectangle:
- Left side: `5`
- Top: `x` and `7`
- Areas: `5x` and `35`

✔ Blank: `x + 7`
✔ Rectangle: left = 5, top = x and 7

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Problem 6: `2x + 12 = 2(___)`

Factor out 2:
→ `2x + 12 = 2(x + 6)`

✔ Blank: `x + 6`
✔ Rectangle: left = 2, top = x and 6

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Problem 7: `3x - 21 = ___`

Factor out 3:
→ `3x - 21 = 3(x - 7)`

✔ Blank: `3(x - 7)`
✔ Rectangle: left = 3, top = x and -7 (or you can write as subtraction)

Note: In area models, negative areas are sometimes shown with dashed lines or minus signs — here, just write `-7` in the top right box.

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Problem 8: `7x - 21 = ___`

Factor out 7:
→ `7x - 21 = 7(x - 3)`

✔ Blank: `7(x - 3)`
✔ Rectangle: left = 7, top = x and -3

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Problem 9: `-3x - 15 = -3(___)`

Factor out -3:
→ `-3x - 15 = -3(x + 5)`

Check: -3 × x = -3x, -3 × 5 = -15 → correct.

✔ Blank: `x + 5`
✔ Rectangle: left = -3, top = x and 5
*(Note: Negative side length? Conceptually, we treat it as factoring out the negative sign.)*

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Problem 10: `-5x + 45 = ___`

Factor out -5:
→ `-5x + 45 = -5(x - 9)`

Check: -5 × x = -5x, -5 × (-9) = +45 → correct.

✔ Blank: `-5(x - 9)`
✔ Rectangle: left = -5, top = x and -9

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## 🔹 SECTION 3: Factor these expressions

Just factor out the GCF.

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11. `4x - 16`

GCF of 4 and 16 is 4
→ `4(x - 4)`

✔ Answer: `4(x - 4)`

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12. `-7x - 35`

GCF is 7, but factor out -7 to keep leading coefficient positive inside parentheses:
→ `-7(x + 5)`

✔ Answer: `-7(x + 5)`

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13. `9x - 81`

GCF of 9 and 81 is 9
→ `9(x - 9)`

✔ Answer: `9(x - 9)`

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14. `4x + 18`

GCF of 4 and 18 is 2
→ `2(2x + 9)`

✔ Answer: `2(2x + 9)`

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## ✔ FINAL ANSWERS SUMMARY:

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Section 1:

2.
Top: `x`, `4`
Product: `5(x + 4)`
Sum: `5x + 20`

3.
Top: `x`
Left: `6`
Product: `6(x + 8)`
Sum: `6x + 48`

4.
Top: `3`
Left: `10`
Product: `10(x + 3)`
Sum: `10x + 30`

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Section 2:

5. `5(x + 7)`
6. `2(x + 6)`
7. `3(x - 7)`
8. `7(x - 3)`
9. `-3(x + 5)`
10. `-5(x - 9)`

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Section 3:

11. `4(x - 4)`
12. `-7(x + 5)`
13. `9(x - 9)`
14. `2(2x + 9)`

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Let me know if you’d like me to draw the rectangles or explain any part visually! 😊