Studio Aletheia · The Center for Quantitative Studies
6.MGSR.1.1SC 6th Grade Math
Finding Area, Piece by Piece
Lesson 28 · New Content, Area of Triangles & Quadrilaterals
Every triangle, parallelogram, and trapezoid you will ever measure can be broken into rectangles and triangles you already know how to handle. Today you'll learn the strategy mathematicians actually use: decompose the shape, find each piece's area, and add it back together.
Learning Targets and Success Criteria
Any shape's area can be found by breaking it into pieces you already know how to measure.
What I will learn
- I can find the area of a square and rectangle by multiplying length times width.
- I can find the area of a triangle by relating it to half of a rectangle.
- I can find the area of a parallelogram by decomposing it into a rectangle.
- I can find the area of a trapezoid by decomposing it into a rectangle and two triangles.
What success looks like
- I can find the area of a rectangle given its side lengths.
- I can explain why a triangle's area is half of a related rectangle's area.
- I can decompose a parallelogram into pieces with a known area.
- I can decompose a trapezoid and add the pieces to find total area.
Find the area of triangles, squares, rectangles, parallelograms, and trapezoids using appropriate strategies.
The words we'll use in today's lesson.
These terms will carry through today's mini-lesson, Data Lab, hands-on activity, journal, and exit challenge, and they'll keep coming back all year.
Trapezoid
A four-sided figure (quadrilateral) with exactly one pair of parallel sides.
Area
The amount of flat space a two-dimensional figure covers, measured in square units.
Where this shows up in the real world.
Thinking like a mathematician is not just a school skill. It's what people get paid to do every day, in jobs you may not have heard of yet.
Every Floor Plan Starts With Area
Before an architect can tell a client how much a room will cost to build, they need its area, in square feet. The same decomposing strategy you'll learn today, breaking an odd-shaped room into rectangles and triangles, is exactly what architects do on every single floor plan.
Decompose, Then Add
Any shape's area can be found by breaking it into pieces you already know how to measure.
You already know how to find the area of a rectangle: multiply its length by its width. That one fact is the key to finding the areaThe amount of flat space a two-dimensional figure covers, measured in square units. of almost any flat shape, because almost every shape can be decomposed, or broken apart, into rectangles and triangles.
A triangle is exactly half of a rectangle that shares its base and height, so you can find a triangle's area by finding that rectangle's area and cutting it in half. A parallelogram looks slanted, but if you slide a triangular sliver from one side to the other, it becomes a rectangle with the exact same area. A trapezoidA four-sided figure with exactly one pair of parallel sides. is trickier since its two parallel sides are different lengths, but you can decompose it into one rectangle in the middle and a triangle on each end, find the area of all three pieces, and add them together.
Notice the pattern: every strategy here uses the same big idea, decompose into known shapes, find each piece, then add. That's Structure & Patterns at work: once you can decompose a trapezoid, you can decompose almost any figure a mathematician throws at you.
Adapted from Studio Aletheia's The Center for Quantitative Studies curriculum library, drawing on mathematical resources and the SC CCR Mathematics Standards.
Materials for the Data Lab.
- A. Grid paper
- B. Ruler
- C. Scissors (to physically decompose a shape, optional)
- D. Your Data Journal
- E. A calculator (optional)
For every area problem this week: sketch the shape, decompose it into known pieces, label each piece's dimensions, find each piece's area, then add.
Decompose a Trapezoid
Today's Data Lab walks through decomposing one trapezoid step by step before you try it on your own.
Decompose and Solve
Now practice the full strategy on a new set of figures using your Data Journal.
Data Journal Entry
Accountability Checklist
How Do You Find the Area of a Trapezoid?
Explain, in your own words, how to find the area of a trapezoid by decomposing it into simpler shapes.
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