A focused Grade 3 learning session for Tiling and the Distributive Property using visual models, worked examples, strategy guidance, and a short mastery check.
Student goal: I can use an area model and the distributive property.
Tiling and the Distributive Property means measuring how much flat space a figure covers by using equal-sized square units.
Question: Look at this rectangle split by a vertical line: Which distributive equation matches?
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: \(4\times(3+3)=4\times6=24\)
Question: Look at this tiling diagram: Which sum of the two smaller areas gives the total area?
Answer: \((3\times3)+(4\times3)=9+12=21\)
Question: A rectangle with a vertical split looks like this: What is the left rectangle's width?
Answer: \(5\) units
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?
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: \(3\times(10+6)=3\times16=48\)
Tiling and the Distributive Property becomes easier when students connect the question to a model, use clear steps, and explain why the answer fits.
Area means every square unit inside the figure.
Use scratch paper. The explanation matters more than simply seeing whether the choice is correct.
A rectangle is \(6\) units long and \(8\) units wide. Which shows its area broken into two smaller rectangles?
Answer A: \(6\times8=6\times(3+5)=(6\times3)+(6\times5)\)
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.
A rectangle has length \(5\) and width \(7\). If we split the width into \(4+3\), which equation shows the area?
Answer A: \(5\times7=5\times(4+3)=(5\times4)+(5\times3)\)
\(5\times(4+3)=(5\times4)+(5\times3)=20+15=35\). Splitting the width lets us show the distributive property with area.
A rectangle is tiled by two smaller rectangles. One is \(4\times6\) and the other is \(4\times2\). What is the total area?
Answer B: \(32\)
\((4\times6)+(4\times2)=24+8=32\). Both rectangles share width \(4\), so the total width is \(6+2=8\).
Which of these shows tiling that proves \(3\times9=3\times(5+4)\)?
Answer A: 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 \(3\).
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?
Answer B: \(20\)
One part is \(4\times10=40\). Total is \(60\). So the other part has area \(60-40=20\).
Which equation matches the tiling shown by a rectangle with one line cutting it vertically into two parts?
Answer A: \(7\times(3+2)=(7\times3)+(7\times2)\)
A vertical line in a rectangle splits its width or length. Here the length is \(7\) and the other dimension splits into \(3+2\).
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?
Answer A: \(16\)
Total is \(8\times5=40\). One part is \(8\times3=24\). Other part is \(40-24=16\), which is \(8\times2\).
Which shows the distributive property for \(4\times(6+1)\)?
Answer A: \((4\times6)+(4\times1)=24+4=28\)
The distributive property multiplies \(4\) by each part separately, then adds: \(4\times(6+1)=(4\times6)+(4\times1)\).
A rectangle's area can be written as \(3\times(5+4)\). What is the total area?
Answer C: \(27\)
\(3\times(5+4)=3\times9=27\) or \((3\times5)+(3\times4)=15+12=27\).
Which pair of rectangles can be tiled to show \(5\times(7+2)=(5\times7)+(5\times2)\)?
Answer A: One \(5\times7\) and one \(5\times2\)
Both rectangles must have the same dimension (width \(5\)) so they share a side when tiled together.
A rectangle has area \(6\times(3+2)\). Which calculation uses the distributive property correctly?
Answer B: \((6\times3)+(6\times2)=18+12=30\)
Use the distributive property: multiply \(6\) by each part, then add.
