Area and the distributive property
Use tiling to demonstrate the distributive property: the area of a rectangle with sides a and (b+c) equals a×b + a×c; use area models to represent the distributive property
What a learner can do afterwards
- Tile a 3×(4+2) rectangle and show it decomposes into 3×4 and 3×2
- Use an area model to compute 6×13 as 6×10 + 6×3
- Draw an area model showing 5×(7+3) = 5×7 + 5×3
1 · Read
A rectangle's area is one side times the other side. But a big rectangle, like a school garden that is 6 meters by 13 meters, can be hard to multiply in your head. Split one side into two smaller parts, find the area of each part, then add them together to get the same total.
To work out 6×13, split 13 into 10 and 3. Then 6×13 = 6×10 + 6×3 = 60 + 18 = 78.
You can split either side of the rectangle, not just one way. Splitting only breaks the counting into easier steps. It does not change how much space is inside.
Split one side of a rectangle into parts, multiply each part separately, then add the results to get the same total area.
Rewritten readings were written for this dataset.
More to read
Area: FAQ
Area questions repeat the same moves. Cover with unit squares, count them, then shortcut with length times width. The 3 by (4 + 2) rectangle is the star example: split it into 3 by 4 and 3 by 2, giving 12 + 6 = 18. Splitting never changes the cover. It only cuts the count into friendly chunks. Ask which split helps before you multiply. Friendly chunks finish first.
Multiply by using place value strategies and the distributive property
The distributive property is splitting with permission. 6 x 13 becomes 6 x 10 plus 6 x 3, which is 60 + 18 = 78. Area models draw the split as two blocks side by side. One block shows 6 by 10, the other 6 by 3. Both blocks together are the full 6 by 13 wall. Colour each block and label its area. Colours prove the split covers everything with no gaps and no overlaps.
Split a rectangle to find its area
Measure both sides using the same unit. Rectangle area is length × width. A wall measuring 5 by 10 covers 50 square units. Counting squares and multiplying give the same area. Split 10 into 7 + 3: 5 x (7 + 3) = 5 x 7 + 5 x 3 = 35 + 15 = 50. Multiply 5 by both parts, then add. Leaving out the second part misses 15 squares. This is the distributive property.
Key ideas: divide by using the break apart and distribute strategy
Breaking apart works for division too, which proves splitting is structural, not a one off trick. Friendly chunks divide cleanly where the whole resists. The same discipline guards area models. Check that chunks rebuild the original side: 4 + 2 must equal the 6 you split. Check that every chunk got multiplied. Two checks, zero leaks. Structure plus checking beats raw speed.
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2 · Watch
به انگلیسیTake it off screen
Where it sits
Where this leads
18 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.