Introduction
Tiling and the Distributive Property 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 tiling and the distributive property.
What Is Tiling and the Distributive Property?
Tiling and the Distributive Property 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 Tiling and the Distributive Property
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: Look at this rectangle split by a vertical line: Which distributive equation matches?
- A. \(4\times(3+3)=4\times6=24\)
- B. \(3+3+4=10\)
- C. \(4\times3\times3=36\)
- D. \(6+4=10\)
Why it works: The height is \(4\) and width is \(6=3+3\). The distributive form is \(4\times(3+3)=(4\times3)+(4\times3)=12+12=24\).
Answer: \(4\times(3+3)=4\times6=24\)
Visual Model 2
Question: A rectangle is split by a horizontal line into two parts: Which equation uses the distributive property to find the total area?
- A. \(5\times2=10\) only
- B. \(5\times(2+4)=(5\times2)+(5\times4)=10+20=30\)
- C. \(2+4+5=11\)
- D. \(6\times5=30\)
Why it works: Both smaller rectangles share width \(5\). Heights are \(2\) and \(4\), totaling \(6\). The distributive property gives \(5\times(2+4)=30\).
Answer: \(5\times(2+4)=(5\times2)+(5\times4)=10+20=30\)
Worked Examples
Example 1
Question: Look at this tiling diagram: Which sum of the two smaller areas gives the total area?
- A. \((3\times3)+(4\times3)=9+12=21\)
- B. \(3\times(3+4)=21\)
- C. \(7+3=10\)
- D. \(3+4+7=14\)
- Add the two smaller rectangle areas: \(3\times3=9\) and \(4\times3=12\), so total is \(21\) square units.
Answer: \((3\times3)+(4\times3)=9+12=21\)
Example 2
Question: A rectangle with a vertical split looks like this: What is the left rectangle's width?
- A. \(5\) units
- B. \(2\) units
- C. \(8\) units
- D. \(6\) units
- Left rectangle: Area = \(10\), Height = \(2\), so Width = \(10\div2=5\).
- Check: \(2\times(5+3)=2\times8=16=10+6\).
Answer: \(5\) units
Example 3
Question: A rectangle is tiled and labeled like this: Which sum of the two smaller areas gives the total area?
- A. \(6\times(3+2)=30\)
- B. \((6\times3)+(6\times2)=18+12=30\)
- C. \(6+5=11\)
- D. \(3+2=5\)
- Top part: \(6\times3=18\).
- Bottom part: \(6\times2=12\).
- Add the areas: \(18+12=30\) square units.
Answer: \((6\times3)+(6\times2)=18+12=30\)
Real-World Word Problems
Problem 1
Question: A rectangular garden can be tiled into two smaller rectangles, each with width \(3\). One rectangle is \(3\) by \(10\) and the other is \(3\) by \(6\). What is the total area?
- A. \(3\times(10+6)=3\times16=48\)
- B. \(3+10+6=19\)
- C. \((10+6)=16\)
- D. \(3\times10\times3\times6=5400\)
Why it works: Both rectangles have width \(3\). Total length is \(10+6=16\). Combined area is \((3\times10)+(3\times6)=30+18=48\).
Answer: \(3\times(10+6)=3\times16=48\)
Problem 2
Question: A rectangle is \(6\) units long and \(8\) units wide. Which shows its area broken into two smaller rectangles?
- A. \(6\times8=6\times(3+5)=(6\times3)+(6\times5)\)
- B. \(6\times8=48\)
- C. \(6+8=14\)
- D. \(6+8+6+8=28\)
Why it works: The distributive property lets us split \(8\) into \(3+5\) and add two smaller areas: \(18+30=48\). This matches tiling the rectangle in two parts.
Answer: \(6\times8=6\times(3+5)=(6\times3)+(6\times5)\)
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 rectangle has length \(5\) and width \(7\). If we split the width into \(4+3\), which equation shows the area?
- A. \(5\times7=5\times(4+3)=(5\times4)+(5\times3)\)
- B. \(5+4+3=12\)
- C. \((5+4)+(5+3)=17\)
- D. \(5\times7=35\)
Question 2
A rectangle is tiled by two smaller rectangles. One is \(4\times6\) and the other is \(4\times2\). What is the total area?
- A. \(48\)
- B. \(32\)
- C. \(20\)
- D. \(12\)
Question 3
Which of these shows tiling that proves \(3\times9=3\times(5+4)\)?
- A. Two rectangles with width \(3\): one is \(3\times5\), the other is \(3\times4\)
- B. Two rectangles with width \(2\): one is \(2\times5\), the other is \(2\times4\)
- C. A single rectangle \(3\times9\)
- D. Two rectangles with length \(9\): one is \(9\times3\), the other is \(9\times2\)
Question 4
A long rectangle has area \(60\) square units. It is split vertically into two parts. One part is \(4\times10\). What is the area of the other part if they share the same length?
- A. \(40\)
- B. \(20\)
- C. \(30\)
- D. \(60\)
Question 5
Which equation matches the tiling shown by a rectangle with one line cutting it vertically into two parts?
- A. \(7\times(3+2)=(7\times3)+(7\times2)\)
- B. \(3\times2=6\)
- C. \(7+3+2=12\)
- D. \((3\times7)+(2\times7)=35+14\)
Question 6
A \(8\times5\) rectangle is tiled into two smaller rectangles by a horizontal line. If one rectangle is \(8\times3\), what is the area of the other?
- A. \(16\)
- B. \(24\)
- C. \(40\)
- D. \(14\)
Full Answer Explanations Click to show all answers and explanations
Question 1
Answer: \(5\times(4+3)=(5\times4)+(5\times3)\)
\(5\times(4+3)=(5\times4)+(5\times3)=20+15=35\). Splitting the width lets us show the distributive property with area.
Question 2
Answer: \(32\)
\((4\times6)+(4\times2)=24+8=32\). Both rectangles share width \(4\), so the total width is \(6+2=8\).
Question 3
Answer: Two rectangles with width \(3\): one is \(3\times5\), the other is \(3\times4\)
The equation \(3\times(5+4)=(3\times5)+(3\times4)\) needs rectangles with the same width of \(3\).
Question 4
Answer: \(20\)
One part is \(4\times10=40\). Total is \(60\). So the other part has area \(60-40=20\).
Question 5
Answer: \(7\times(3+2)=(7\times3)+(7\times2)\)
A vertical line in a rectangle splits its width or length. Here the length is \(7\) and the other dimension splits into \(3+2\).
Question 6
Answer: \(16\)
Total is \(8\times5=40\). One part is \(8\times3=24\). Other part is \(40-24=16\), which is \(8\times2\).
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
Tiling and the Distributive Property 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.
