Introduction
Area as Additive is an important Grade 3 math skill because students are moving from simple answers toward explaining how the math works.
In this lesson, students use models, real questions, worked examples, practice problems, and two online quizzes to build confidence with area as additive.
What Is Area as Additive?
Area as Additive means measuring how much flat space a figure covers by using equal-sized square units.
The goal is not only to get the answer. Students should be able to show the idea, explain the strategy, and check whether the answer makes sense.
Understanding Area as Additive
Before solving, students should slow down and decide what each number, shape, unit, or label represents.
- Use square units that cover the figure without gaps or overlaps.
- Count rows and columns when the unit squares are arranged in an array.
- Connect repeated addition to multiplication when finding area.
- Break complex figures into smaller rectangles when that makes the work clearer.
Visual Models
Visual Model 1
Question: An L-shaped garden is divided into two rectangles. The top rectangle has length \(5\) m and width \(3\) m. The bottom rectangle has length \(4\) m and width \(2\) m. What is the total area of the garden?
- A. \(14\) sq m
- B. \(23\) sq m
- C. \(38\) sq m
- D. \(120\) sq m
Why it works: Find each rectangle: top is \(5 \times 3 = 15\) sq m; bottom is \(4 \times 2 = 8\) sq m. Total: \(15+8=23\) sq m.
Answer: \(23\) sq m
Visual Model 2
Question: A composite shape is made by joining two rectangles. What is the total area?
- A. \(6\) sq units
- B. \(9\) sq units
- C. \(12\) sq units
- D. \(15\) sq units
Why it works: Left rectangle: \(3 \times 2 = 6\) sq units. Right rectangle: \(3 \times 1 = 3\) sq units. Total: \(6+3=9\) sq units.
Answer: \(9\) sq units
Worked Examples
Example 1
Question: A floor plan shows an L-shaped room. One part is \(4\) units long and \(3\) units wide. The other part is \(6\) units long and \(2\) units wide. What is the total floor area?
- A. \(24\) sq units
- B. \(18\) sq units
- C. \(12\) sq units
- D. \(36\) sq units
- Part 1: \(4 \times 3 = 12\) sq units.
- Part 2: \(6 \times 2 = 12\) sq units.
- Total: \(12+12=24\) sq units.
Answer: \(24\) sq units
Example 2
Question: A composite shape has areas of \(18\) sq units and \(11\) sq units. Find the total area.
- A. \(29\) sq units
- B. \(18\) sq units
- C. \(7\) sq units
- D. \(198\) sq units
- Add the areas: \(18+11=29\) square units.
- Choice B (only one part) and C (subtraction) are common errors.
Answer: \(29\) sq units
Example 3
Question: An L-shaped patio is shown with dashed lines dividing it into two rectangles. The first rectangle is \(5\) ft by \(4\) ft. The second is \(3\) ft by \(2\) ft. What is the total area of the patio?
- A. \(26\) sq ft
- B. \(20\) sq ft
- C. \(14\) sq ft
- D. \(40\) sq ft
- Top rectangle: \(5 \times 4 = 20\) sq ft.
- Bottom rectangle: \(3 \times 2 = 6\) sq ft.
- Total: \(20+6=26\) sq ft.
Answer: \(26\) sq ft
Real-World Word Problems
Problem 1
Question: A classroom floor is T-shaped. The top part is \(6\) units by \(2\) units. The bottom part is \(4\) units by \(3\) units. What is the total floor area?
- A. \(24\) sq units
- B. \(12\) sq units
- C. \(36\) sq units
- D. \(30\) sq units
Why it works: Top rectangle: \(6 \times 2 = 12\) sq units. Bottom rectangle: \(4 \times 3 = 12\) sq units. Total: \(12+12=24\) sq units.
Answer: \(24\) sq units
Problem 2
Question: An L-shaped garden with dashed decomposition lines shows: The bottom part is \(5\) units by \(2\) units. The top part is \(2\) units by \(2\) units. What is the total area?
- A. \(4\) sq units
- B. \(14\) sq units
- C. \(10\) sq units
- D. \(20\) sq units
Why it works: Bottom: \(5 \times 2 = 10\) sq units. Top: \(2 \times 2 = 4\) sq units. Total: \(10+4=14\) sq units.
Answer: \(14\) sq units
Common Mistakes
- Counting only the outside squares instead of all squares inside the figure.
- Leaving gaps or overlaps when using unit squares.
- Multiplying side lengths before checking whether the figure is a rectangle.
- Forgetting to write square units with an area answer.
Strategy Tips
- Trace the rectangle or figure before counting.
- Use rows and columns to organize unit squares.
- Write an equation after the model makes sense.
- Check whether the answer needs square units.
Practice Questions
Question 1
A shape is made of two rectangles. Rectangle 1 has area \(12\) sq units, and Rectangle 2 has area \(15\) sq units. What is the total area?
- A. \(15\) sq units
- B. \(24\) sq units
- C. \(27\) sq units
- D. \(180\) sq units
Question 2
Which shows a correct way to find the total area of an L-shaped figure?
- A. Multiply the length times the width once
- B. Break it into rectangles and add their areas
- C. Multiply the perimeter by 2
- D. Count only the longest parts
Question 3
Sam's bedroom is shaped like an L. One part is \(3\) m long and \(4\) m wide. The other part is \(5\) m long and \(2\) m wide. What is the total area of his bedroom?
- A. \(12\) sq m
- B. \(10\) sq m
- C. \(22\) sq m
- D. \(120\) sq m
Question 4
A composite shape is divided by dashed lines. Part A has area \(7\) sq units. Part B has area \(9\) sq units. What is the total?
- A. \(9\) sq units
- B. \(16\) sq units
- C. \(63\) sq units
- D. \(2\) sq units
Question 5
An L-shaped pool is made of two rectangles. The first is \(8\) ft long and \(3\) ft wide. The second is \(4\) ft long and \(2\) ft wide. What is the total area?
- A. \(24\) sq ft
- B. \(8\) sq ft
- C. \(48\) sq ft
- D. \(32\) sq ft
Question 6
A composite figure shows: What is the total area?
- A. \(12\) sq units
- B. \(4\) sq units
- C. \(16\) sq units
- D. \(18\) sq units
Full Answer Explanations Click to show all answers and explanations
Question 1
Answer: \(27\) sq units
When a composite shape is divided into non-overlapping rectangles, the total area equals the sum of the individual rectangle areas. Here: \(12+15=27\) square units.
Question 2
Answer: Break it into rectangles and add their areas
Composite shapes are found by dividing them into non-overlapping rectangles, finding each area, and adding them together. Choice A fails because an L-shape is not a single rectangle. Choices C and D are incorrect methods.
Question 3
Answer: \(22\) sq m
Part 1: \(3 \times 4 = 12\) sq m. Part 2: \(5 \times 2 = 10\) sq m. Total: \(12+10=22\) sq m.
Question 4
Answer: \(16\) sq units
Add the parts: \(7+9=16\) square units. Choice A (forgot Part A) and C (multiply instead of add) are common errors.
Question 5
Answer: \(32\) sq ft
Rectangle 1: \(8 \times 3 = 24\) sq ft. Rectangle 2: \(4 \times 2 = 8\) sq ft. Total: \(24+8=32\) sq ft.
Question 6
Answer: \(16\) sq units
Bottom rectangle: \(4 \times 3 = 12\) sq units. Top rectangle: \(2 \times 2 = 4\) sq units. Total: \(12+4=16\) sq units.
Connection to Standards
This lesson supports Grade 3 math expectations for reasoning, modeling, problem solving, and explaining answers clearly. It connects classroom skills to the kind of questions students see on state math assessments.
Summary
Area as Additive becomes easier when students connect the question to a model, use clear steps, and explain why the answer fits.
GOLDEN RULE
Area means every square unit inside the figure.
