Grade 1 · Geometry · MAT-01-GEO-002
Compose and Decompose 2D Shapes
Compose larger 2D figures from triangles, squares, rectangles, and other simple pieces.
Learning objectives
What you should be able to do
- Compose larger 2D figures from triangles, squares, rectangles, and other simple pieces.
- Decompose composite figures into recognizable component shapes.
- Explain how rotation and rearrangement can change the composite whole without changing the pieces.
Prerequisite check
Make sure the foundation is ready.
Checks shape recognition and property reasoning.
Learn
Build the idea from meaning, not memorization.
Composition joins shape parts into a new whole
Two triangles can be arranged to make a square or rectangle in suitable cases. Several small shapes can form one larger figure whose outer boundary may have a different name from its parts.
Decomposition reverses the process
A rectangle can be divided into smaller rectangles, squares, or triangles depending on how it is partitioned. Recognizing the hidden component shapes strengthens spatial reasoning.
Rotation and translation help pieces fit
A piece can slide or turn without changing its defining attributes. The goal is to align edges so the target is covered without unintended gaps or overlaps.
The same pieces can create different wholes
Rearranging a fixed set of pieces may produce multiple composite figures. This prepares students for area, fractions, transformations, and later geometric reasoning.
Interactive math lab
Change it. See what stays true.
Geometry Explorer
- drag pieces
- rotate
- trace outer boundary
- split whole
Worked examples
Twenty different ways to see the concept work.
Two congruent right triangles arranged along a diagonal.
SquareTheir outer boundary can form four equal sides.
Two squares side by side.
RectangleThe shared edge becomes internal.
Four small squares in a 2×2 array.
Larger squareA composite square.
Two rectangles end to end.
Longer rectangleAligned edges create one larger whole.
Split a square along a diagonal.
Two trianglesDecomposition creates two triangular parts.
Split rectangle vertically in half.
Two rectanglesEach part is rectangular.
Rotate a triangle before joining.
Triangle identity stays sameOnly orientation changes.
Slide a square to another location.
Still squareTranslation preserves attributes.
Can pieces overlap in an exact target composition?
NoOverlaps double-cover part of the target.
Can target composition contain gaps?
NoThe target should be fully covered.
Three small squares in a row.
RectangleOuter boundary is longer than tall.
One square plus two triangles around it.
Composite figureThe whole may not share a simple basic name.
A rectangle divided into 4 equal smaller rectangles.
4 partsThe whole is decomposed systematically.
Can same two triangles form different wholes?
YesDifferent edge alignments create different outer boundaries.
Why inspect outer boundary?
It identifies the composite wholeInternal edges show parts.
What happens to a shared edge after joining two shapes?
It becomes internalIt may no longer be part of the outer boundary.
Which transformation helps fit a sideways piece?
RotationTurn it without changing shape.
Which action changes location without turning?
Translation/slideMove it straight to a new place.
How can decomposition help count parts?
It makes component shapes explicitThe whole is understood through smaller regions.
Explain composition/decomposition relationship.
One combines parts; the other separates a whole into partsThey are complementary spatial processes.
Practice
Guided practice with hints
01Join two squares along full edges.
Hint: What outer shape forms?
Answer: Rectangle
02Split a square diagonally.
Hint: What are the two parts?
Answer: Triangles
03Piece does not fit orientation.
Hint: What can you do without changing its shape?
Answer: Rotate it.
04What should exact target avoid?
Hint: Inspect all boundaries.
Answer: Gaps and overlaps.
05What is an internal edge?
Hint: An edge where two pieces meet.
Answer: Shared boundary inside the whole.
06Can same pieces make another figure?
Hint: Rearrange them.
Answer: Yes
Practice
Independent practice
012 squares side-by-side
Answer: Rectangle
02Square diagonal split
Answer: 2 triangles
034 squares 2×2
Answer: Larger square
04Does rotation change shape type?
Answer: No
05Do gaps belong in exact composition?
Answer: No
06Do overlaps belong?
Answer: No
07What does decompose mean?
Answer: Separate a whole into component shapes.
08What does compose mean?
Answer: Combine shapes into a larger figure.
Common mistakes
Learn to catch the error, not just the answer.
Trace only the outside of the completed figure when naming the whole.
Recheck defining attributes after turning.
Use edge alignment and inspect the entire target area.
Applications & challenge
Use the idea beyond a single exercise.
Applications
- Use pattern blocks to build and decompose pictures.
- Partition paper rectangles into smaller geometric pieces and name them.
Challenge problems
Why can two triangles produce a square in one arrangement and a different shape in another?
The same pieces can create different outer boundaries.
How can an internal edge help reveal decomposition?
It shows where the whole can be separated into component shapes.
Can a composite figure lack a familiar basic shape name?
Yes; it can still be described by its component shapes and boundary.
Why does this matter for later area?
Area can be found by decomposing complex figures into simpler regions.
Flashcards
Retrieve it from memory.
Flashcards
Card 1 of 4
Game mode
Shape Workshop
Build target 2D figures, identify hidden parts, and create alternate arrangements.
Practice
Mastery check
01Two squares side by side make?
Answer: Rectangle
02Diagonal split of square makes?
Answer: Two triangles
03Does rotation change identity?
Answer: No
04What is compose?
Answer: Join shape parts.
05What is decompose?
Answer: Split a shape whole.
06Why trace outer boundary?
Answer: To identify completed figure.
07Should exact composition have gaps?
Answer: No
08Can same pieces make different wholes?
Answer: Yes
Terminology
Words to know
- compose
- Combine shapes into a larger figure.
- decompose
- Separate a figure into component shapes.
- internal edge
- An edge shared by pieces inside a composite figure.
- outer boundary
- The complete outside edge of a figure.
Continue learning
Connected concepts
Curriculum references
Standards and instructional references
California Common Core State Standards for Mathematics — Grade 1 Geometry
Grade 1 defining attributes, composition, partitioning, and spatial reasoning.