Grade 3 · Measurement & Data · MAT-03-MD-011
Find Composite Area with the Distributive Property
Represent find composite area with the distributive property with words, symbols, diagrams, or models appropriate to Grade 3.
Learning objectives
What you should be able to do
- Represent find composite area with the distributive property with words, symbols, diagrams, or models appropriate to Grade 3.
- Solve problems involving find composite area with the distributive property accurately and explain why the method works.
- Check results for reasonableness using a second representation, inverse relationship, benchmark, or property when appropriate.
Prerequisite check
Make sure the foundation is ready.
Checks conceptual readiness before new Grade 3 reasoning.
Checks whether prior skills are available for use, not only recalled by name.
Learn
Build the idea from meaning, not memorization.
Meaning of Find Composite Area with the Distributive Property
A composite rectilinear region can be split into non-overlapping rectangles whose areas add to the total.
A reliable reasoning strategy
Choose a useful split, find each rectangle's area, and add the partial areas.
Representations that should agree
This geometric decomposition is the distributive property made visible. A correct model, equation, verbal explanation, or diagram should preserve the same quantities and relationships.
How to check the result
Check whether the answer fits the quantities, units, and structure of the problem. Use a second representation or inverse relationship when available, and explain what would make an answer unreasonable.
Interactive math lab
Change it. See what stays true.
Geometry Explorer
- build a model
- change one quantity
- predict before calculating
- explain why the result changes
Worked examples
Twenty different ways to see the concept work.
A shape is split into rectangles 3 by 4 and 3 by 4. What is the total area?
3 × 4 = 12; 3 × 4 = 12; 12 + 12 = 24Add the areas of non-overlapping rectangles.
A shape is split into rectangles 4 by 6 and 4 by 3. What is the total area?
4 × 6 = 24; 4 × 3 = 12; 24 + 12 = 36Add the areas of non-overlapping rectangles.
A shape is split into rectangles 5 by 3 and 5 by 2. What is the total area?
5 × 3 = 15; 5 × 2 = 10; 15 + 10 = 25Add the areas of non-overlapping rectangles.
A shape is split into rectangles 6 by 5 and 6 by 1. What is the total area?
6 × 5 = 30; 6 × 1 = 6; 30 + 6 = 36Add the areas of non-overlapping rectangles.
A shape is split into rectangles 2 by 2 and 2 by 4. What is the total area?
2 × 2 = 4; 2 × 4 = 8; 4 + 8 = 12Add the areas of non-overlapping rectangles.
A shape is split into rectangles 3 by 4 and 3 by 3. What is the total area?
3 × 4 = 12; 3 × 3 = 9; 12 + 9 = 21Add the areas of non-overlapping rectangles.
A shape is split into rectangles 4 by 6 and 4 by 2. What is the total area?
4 × 6 = 24; 4 × 2 = 8; 24 + 8 = 32Add the areas of non-overlapping rectangles.
A shape is split into rectangles 5 by 3 and 5 by 1. What is the total area?
5 × 3 = 15; 5 × 1 = 5; 15 + 5 = 20Add the areas of non-overlapping rectangles.
A shape is split into rectangles 6 by 5 and 6 by 4. What is the total area?
6 × 5 = 30; 6 × 4 = 24; 30 + 24 = 54Add the areas of non-overlapping rectangles.
A shape is split into rectangles 2 by 2 and 2 by 3. What is the total area?
2 × 2 = 4; 2 × 3 = 6; 4 + 6 = 10Add the areas of non-overlapping rectangles.
A shape is split into rectangles 3 by 4 and 3 by 2. What is the total area?
3 × 4 = 12; 3 × 2 = 6; 12 + 6 = 18Add the areas of non-overlapping rectangles.
A shape is split into rectangles 4 by 6 and 4 by 1. What is the total area?
4 × 6 = 24; 4 × 1 = 4; 24 + 4 = 28Add the areas of non-overlapping rectangles.
A shape is split into rectangles 5 by 3 and 5 by 4. What is the total area?
5 × 3 = 15; 5 × 4 = 20; 15 + 20 = 35Add the areas of non-overlapping rectangles.
A shape is split into rectangles 6 by 5 and 6 by 3. What is the total area?
6 × 5 = 30; 6 × 3 = 18; 30 + 18 = 48Add the areas of non-overlapping rectangles.
A shape is split into rectangles 2 by 2 and 2 by 2. What is the total area?
2 × 2 = 4; 2 × 2 = 4; 4 + 4 = 8Add the areas of non-overlapping rectangles.
A shape is split into rectangles 3 by 4 and 3 by 1. What is the total area?
3 × 4 = 12; 3 × 1 = 3; 12 + 3 = 15Add the areas of non-overlapping rectangles.
A shape is split into rectangles 4 by 6 and 4 by 4. What is the total area?
4 × 6 = 24; 4 × 4 = 16; 24 + 16 = 40Add the areas of non-overlapping rectangles.
A shape is split into rectangles 5 by 3 and 5 by 3. What is the total area?
5 × 3 = 15; 5 × 3 = 15; 15 + 15 = 30Add the areas of non-overlapping rectangles.
A shape is split into rectangles 6 by 5 and 6 by 2. What is the total area?
6 × 5 = 30; 6 × 2 = 12; 30 + 12 = 42Add the areas of non-overlapping rectangles.
A shape is split into rectangles 2 by 2 and 2 by 1. What is the total area?
2 × 2 = 4; 2 × 1 = 2; 4 + 2 = 6Add the areas of non-overlapping rectangles.
Practice
Guided practice with hints
01A shape is split into rectangles 3 by 4 and 3 by 2. What is the total area?
Hint: Choose a useful split, find each rectangle's area, and add the partial areas.
Answer: 18 square units
02A shape is split into rectangles 4 by 6 and 4 by 1. What is the total area?
Hint: Choose a useful split, find each rectangle's area, and add the partial areas.
Answer: 28 square units
03A shape is split into rectangles 5 by 3 and 5 by 4. What is the total area?
Hint: Choose a useful split, find each rectangle's area, and add the partial areas.
Answer: 35 square units
04A shape is split into rectangles 6 by 5 and 6 by 3. What is the total area?
Hint: Choose a useful split, find each rectangle's area, and add the partial areas.
Answer: 48 square units
05A shape is split into rectangles 2 by 2 and 2 by 2. What is the total area?
Hint: Choose a useful split, find each rectangle's area, and add the partial areas.
Answer: 8 square units
06A shape is split into rectangles 3 by 4 and 3 by 1. What is the total area?
Hint: Choose a useful split, find each rectangle's area, and add the partial areas.
Answer: 15 square units
Practice
Independent practice
01A shape is split into rectangles 3 by 4 and 3 by 2. What is the total area?
Answer: 18 square units
02A shape is split into rectangles 4 by 6 and 4 by 1. What is the total area?
Answer: 28 square units
03A shape is split into rectangles 5 by 3 and 5 by 4. What is the total area?
Answer: 35 square units
04A shape is split into rectangles 6 by 5 and 6 by 3. What is the total area?
Answer: 48 square units
05A shape is split into rectangles 2 by 2 and 2 by 2. What is the total area?
Answer: 8 square units
06A shape is split into rectangles 3 by 4 and 3 by 1. What is the total area?
Answer: 15 square units
07A shape is split into rectangles 4 by 6 and 4 by 4. What is the total area?
Answer: 40 square units
08A shape is split into rectangles 5 by 3 and 5 by 3. What is the total area?
Answer: 30 square units
Common mistakes
Learn to catch the error, not just the answer.
Use non-overlapping rectangles and verify they exactly rebuild the original region.
Reconnect the answer to find composite area with the distributive property and verify that the model, calculation, and label describe the same quantity.
Applications & challenge
Use the idea beyond a single exercise.
Applications
- Estimate flooring, wall space, and irregular rectilinear regions.
- Use find composite area with the distributive property to interpret or solve a Grade 3 situation with quantities, diagrams, measurements, or data.
Challenge problems
A shape is split into rectangles 3 by 4 and 3 by 2. What is the total area?
18 square units
A shape is split into rectangles 4 by 6 and 4 by 1. What is the total area?
28 square units
A shape is split into rectangles 5 by 3 and 5 by 4. What is the total area?
35 square units
A shape is split into rectangles 6 by 5 and 6 by 3. What is the total area?
48 square units
Flashcards
Retrieve it from memory.
Flashcards
Card 1 of 4
Game mode
Find Composite Area with the Distributive Property Challenge
Solve and explain Grade 3 problems involving find composite area with the distributive property without relying on answer guessing.
Practice
Mastery check
01A shape is split into rectangles 3 by 4 and 3 by 2. What is the total area?
Answer: 18 square units
02A shape is split into rectangles 4 by 6 and 4 by 1. What is the total area?
Answer: 28 square units
03A shape is split into rectangles 5 by 3 and 5 by 4. What is the total area?
Answer: 35 square units
04A shape is split into rectangles 6 by 5 and 6 by 3. What is the total area?
Answer: 48 square units
05A shape is split into rectangles 2 by 2 and 2 by 2. What is the total area?
Answer: 8 square units
06A shape is split into rectangles 3 by 4 and 3 by 1. What is the total area?
Answer: 15 square units
07A shape is split into rectangles 4 by 6 and 4 by 4. What is the total area?
Answer: 40 square units
08A shape is split into rectangles 5 by 3 and 5 by 3. What is the total area?
Answer: 30 square units
Terminology
Words to know
- composite figure
- A figure made from two or more simpler figures.
- partial area
- The area of one component region.
Continue learning
Connected concepts
Curriculum references
Standards and instructional references
California Common Core State Standards for Mathematics — Grade 3, 3.MD.7.c-d
Primary standards alignment for Find Composite Area with the Distributive Property.2023 Mathematics Framework for California Public Schools
Instructional emphasis on reasoning, representations, problem solving, and equitable access.