Use Area Models for Distribution - 3rd Grade Math

1. A rectangle is divided into two sections along its length. One section is 5 units long and the other is 3 units long, and the rectangle is 4 units wide. What is the total area in square units?
   Answer: 32 square units Adding the two lengths, 5 plus 3, gives 8 units total, and 4 times 8 equals 32 square units. Choice B (20) only counts the area of one section instead of both. Choice C (28) comes from an incorrect total length before multiplying. Choice D (60) comes from multiplying the two section lengths together instead of adding them.
2. Keisha's garden has a width of 3 units. Its length is split into two parts: 4 units and 2 units. What is the total area of the garden, in square units?
   Answer: 18 square units Keisha's garden has a width of 3 units, split into length parts of 4 and 2 units, for a total length of 6 units. The area is 3 times 6, which equals 18 square units, so A is correct. Choice B (24) does not match multiplying 3 by 6. Choice C (14) is close to but not equal to the correct product. Choice D (12) only accounts for one of the two length parts.
3. A rectangle has a width of 6 units. Its length is split into two parts: 2 units and 4 units. What is the total area of the rectangle, found by adding the area of each part?
   Answer: 36 square units The rectangle's width is 6, split into two length parts of 2 and 4. The area of the first part is 6 times 2, or 12, and the area of the second part is 6 times 4, or 24. Adding them, 12 plus 24 equals 36 square units, so A is correct. Choice B (12) is only the area of the first part. Choice C (24) is only the area of the second part. Choice D (30) does not match adding the two partial areas correctly.
4. Which equation correctly uses the distributive property to find the area for $2\times(6+3)$?
   Answer: $2\times(6+3)=(2\times6)+(2\times3)=12+6=18$ square units The distributive property splits 6+3 into two parts, so 2×(6+3) equals (2×6)+(2×3), which is 12+6=18 square units, matching choice B. Choice A subtracts instead of adding the two parts. Choice C forgets to multiply the 3 by 2 before adding. Choice D turns the addition inside the parentheses into another multiplication instead of distributing.
5. A rectangle is 4 units wide. Its length is split into two parts: 3 units and 2 units. What is the total area?
   Answer: 20 square units The total length is 3 plus 2, or 5 units, and area equals length times width, so 4 times 5 equals 20 square units. Choice A only uses one of the two length parts. Choice C comes from a multiplication error. Choice D comes from adding instead of multiplying the dimensions.
6. Maya's rectangular floor has a length of 5 units and a width of $(4+4)$ units. What is the total area of the floor, in square units?
   Answer: 40 square units The floor's width is 4+4=8 units, so the area is 5 x 8 = 40 square units, making Choice A correct. Choice B (45) adds an extra unit that isn't part of either dimension. Choice C (20) uses only one of the 4's instead of adding them together first. Choice D (25) comes from treating the width as 5 instead of 8.
7. A rectangular playground has an area that can be calculated as $8 \times (5 + 2)$. If the playground is divided into a basketball court and a sandbox, and the basketball court has area $8 \times 5$, what is the area of the sandbox?
   Answer: $8 \times 2 = 16 \text{ square units}$ When you see a problem about breaking apart areas, think about how the distributive property works with multiplication. The total playground area is $8 \times (5 + 2)$, which means the playground is 8 units wide and has a total length of $(5 + 2) = 7$ units. Since the basketball court takes up $8 \times 5$ square units of the total area, you can find the sandbox area by using the distributive property. When you distribute the 8, you get: $8 \times (5 + 2) = (8 \times 5) + (8 \times 2)$. The basketball court is $8 \times 5$, so the remaining piece—the sandbox—must be $8 \times 2 = 16$ square units. Looking at the wrong answers: Choice A calculates $8 \times 7 = 56$, which is actually the total playground area, not just the sandbox. Choice B gives $8 + 2 = 10$, which incorrectly adds instead of multiplying—remember, area always involves multiplication of length and width. Choice C calculates $5 \times 2 = 10$, which uses the wrong dimensions and ignores that the sandbox shares the 8-unit width with the basketball court. The correct answer is D: $8 \times 2 = 16$ square units. Remember this pattern: when you see an area problem with parts, use the distributive property. If the total area is length times $(a + b)$, then the two parts have areas of length times $a$ and length times $b$.
8. A hallway is 6 feet wide. It is divided into two sections: one section is 4 feet long and the other is 3 feet long. What is the total area of the hallway?
   Answer: 42 square feet The total length is 4 feet plus 3 feet, or 7 feet, and area equals length times width, so 7 times 6 equals 42 square feet. Choice A comes from a miscalculation of the total length. Choice B only uses one section's length instead of the full hallway. Choice C comes from multiplying the width by only part of the length.
9. A rectangle is 3 units wide. Its length is split into two parts: 4 units and 2 units. What is the total area?
   Answer: 18 square units The total length is 4 plus 2, or 6 units, and area equals length times width, so 3 times 6 equals 18 square units. Choice B comes from a multiplication error. Choice C comes from adding instead of multiplying the dimensions. Choice D only uses one of the two length parts.
10. A rectangle is 3 units wide and 6 units long. The 6-unit length is split into two parts: 2 units and 4 units. Which equation uses the distributive property to find the total area?
    Answer: $3 \times (2 + 4) = 3 \times 2 + 3 \times 4$ The rectangle's area can be split into two smaller rectangles: 3 by 2 and 3 by 4, so 3 times (2 plus 4) equals 3 times 2 plus 3 times 4. Choice B mixes up which numbers get multiplied and added. Choice C incorrectly adds the widths and lengths together before multiplying. Choice D confuses multiplication with simple addition of all the parts.
11. This area model is $3$ by $(5+1)$. What is the total area?
    Answer: 18 square units 18 square units is correct because 3 times 6 equals 18. 16 square units is incorrect because it does not match the correct product. 12 square units is incorrect because it does not match multiplying 3 by 6. 15 square units is incorrect because it uses only the 5 and skips adding the 1 before multiplying.
12. A rectangle is 5 units by 8 units. It is split into two smaller rectangles: one 5 by 2 and the other 5 by 6. What is the total area of the original rectangle?
    Answer: 40 square units The two smaller rectangles have areas of 5 times 2, or 10, and 5 times 6, or 30, which add up to 40 square units total. Choice A is only the area of the larger part. Choice B comes from multiplying the two given lengths, 5 times 6 by 2 instead of adding the two areas. Choice C is only the area of the smaller part.
13. Study the area model diagram. Marcus wants to verify that his area calculation is correct by using the distributive property. Which statement about this rectangle is true?
    Answer: The total area equals $6 \times 3 + 6 \times 4 = 42 \text{ square units}$ The rectangle shows width 6 with length divided into sections of 3 and 4. Using the distributive property: $6 \times (3 + 4) = 6 \times 3 + 6 \times 4 = 18 + 24 = 42$. Choice B finds perimeter, choice C incorrectly subtracts areas, and choice D uses wrong dimensions.
14. Sofia's poster is 3 inches wide and 7 inches tall, divided into a 5-inch-tall section and a 2-inch-tall section. What is the total area of the poster?
    Answer: 21 square inches The poster is 3 inches wide and 7 inches tall in total, so the area is 3 times 7, which is 21 square inches, matching choice D. This also matches adding the two sections separately: 3 times 5, plus 3 times 2, which is 15 plus 6, or 21. Choices A, B, and C come from using only part of the poster's height or miscalculating the sections.
15. A rectangle is 6 units wide. It is divided along its length into two sections: one section is 2 units long and the other is 5 units long. What is the total area?
    Answer: 42 square units The two sections have areas of 6 times 2, or 12, and 6 times 5, or 30, which add up to 42 square units total. Choice A is only the area of the larger section. Choice C comes from adding the given lengths instead of finding and adding the two areas. Choice D is only the area of the smaller section.
16. Lisa is using unit squares to tile two connected rectangular sections of her classroom floor. The first section is 5 squares wide and 3 squares long. The second section is 5 squares wide and 4 squares long. They share the same width. How many unit squares does she need in total?
    Answer: $5 \times (3 + 4) = 35 \text{ squares}$ Since both sections have the same width (5), we can use the distributive property: $5 \times 3 + 5 \times 4 = 5 \times (3 + 4) = 5 \times 7 = 35$. Choice B incorrectly adds dimensions, choice C adds perimeter instead of finding area, and choice D multiplies all dimensions together.
17. Look at the rectangle shown. Maya wants to find the total area by breaking it into two smaller rectangles. Which equation shows how the distributive property applies to this area model?
    Answer: $4 \times (3 + 2) = (4 \times 3) + (4 \times 2)$ The distributive property states that multiplying a number by a sum equals the sum of multiplying that number by each addend separately. Here, $4 \times (3 + 2) = (4 \times 3) + (4 \times 2)$. Choice B incorrectly distributes to both factors, choice C uses multiplication instead of addition, and choice D doesn't properly distribute the 4.
18. If Elena splits the width of 8 into 3 and 5, which equation correctly shows the split of the rectangle's area?
    Answer: $5 \times 8 = 5 \times 3 + 5 \times 5$ Since 3 plus 5 equals 8, splitting the width into 3 and 5 correctly matches the original rectangle. Choice B splits the width into 4 and 5, which adds up to 9, not 8. Choice C splits the width into 3 and 6, which adds up to 9, not 8. Choice D splits the width into 2 and 5, which adds up to 7, not 8.
19. Sophie draws a rectangle to show that $4 \times 9 = 4 \times (6 + 3)$. The rectangle is 4 units wide. She splits the 9-unit length into two parts: one part is 6 units long, and the other part is 3 units long. What is the area of the smaller part?
    Answer: $4 \times 3 = 12 \text{ square units}$ The smaller part is 3 units long, and the rectangle is 4 units wide, so its area is 4 times 3, which is 12 square units. Choice A, 6 times 3, uses the wrong length for this part. Choice B, 4 plus 3, adds the dimensions instead of multiplying them. Choice D, 4 times 6, gives the area of the larger part instead of the smaller one.
20. Jamal's garden is 4 meters wide. Its length is made up of two sections: 3 meters and 2 meters. What is the total area?
    Answer: 20 square meters The total length is 3 plus 2, or 5 meters, and area equals length times width, so 4 times 5 equals 20 square meters. Choice A comes from a multiplication error. Choice C only uses one of the two length sections. Choice D comes from adding instead of multiplying the dimensions.

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