Compose & Decompose Shapes | 1st Grade Math

Compose and Decompose Shapes - Grade 1 Math
Grade 1 Math  ›  Geometry  ›  Lesson 27
Grade 1 Math - Geometry

Compose & Decompose Shapes

Learn how smaller two-dimensional and three-dimensional shapes can be joined to make larger composite shapes, and how composite shapes can be taken apart or analyzed to find their component shapes.

Students explore more than one way to build a target shape, identify the pieces inside composite figures, and connect composing with the reverse idea of decomposing.

Compose shapes Decompose shapes Composite shapes 2D & 3D geometry 1st Grade Math

๐ŸŽฏ What You Will Learn

Compose 2D shapes
Join smaller shapes to make a target shape.
Find component shapes
Identify the smaller shapes inside a composite figure.
Build in more than one way
Explore different pieces or arrangements that can make the same target shape.
Explore composite solids
Join cubes and other solids to make larger 3D figures.
For parents & teachers — standards alignment: CCSS 1.G.A.2 directly addresses composing two-dimensional shapes (including rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) and three-dimensional shapes (including cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create composite shapes and then compose new shapes from those composites. Texas Grade 1 TEKS 1.6(F) directly addresses composing two-dimensional shapes by joining two, three, or four figures to produce a target shape in more than one way when possible. In this lesson, decompose is used as the intuitive reverse action—taking apart a composition or identifying its component shapes. Equal-share partitioning belongs to Lesson 28 (CCSS 1.G.A.3 and Texas TEKS 1.6(G)–1.6(H)), while Texas explicitly formalizes general two-dimensional decomposition in Grade 2 TEKS 2.8(E).

๐Ÿง  Key Vocabulary

compose — join smaller shapes to make a larger shape.
decompose — take a composite shape apart or identify the smaller shapes inside it.
composite shape — a shape made from two or more smaller shapes.
component shape — one of the smaller shapes used inside a composite shape.
target shape — the larger shape you are trying to build.
arrangement — the way pieces are positioned to make a shape.

๐Ÿงฉ Compose: Put Shapes Together

Two equal squares
→
One rectangle

When pieces meet with no gaps or overlaps, their outside boundary can create a new target shape. Here, two equal squares placed side by side compose one rectangle.

๐Ÿ”Ž Decompose: Find the Parts

Compose Start with the pieces → join them → make the whole.
Decompose Start with the whole → take it apart or identify → find the pieces.
Composite house
→
Triangle + Rectangle

The target figure contains two component shapes.

๐Ÿ”„ One Target Shape, More Than One Way

Four equal squares
→
One larger square
Important idea: a target shape can sometimes be made in more than one way. The pieces may be different, or the same kinds of pieces may be arranged differently, while the outer boundary of the target shape stays the same.

⭕ Curved Pieces Can Compose Shapes Too

Four quarter-circles can make one circle

๐ŸงŠ Compose 3D Solids

Under Common Core, Grade 1 students also compose solid figures. For example, two equal cubes joined face to face can make a rectangular prism, and several cubes can be joined to make a larger composite solid.

Common mistake: composing does not mean simply placing shapes near each other. The pieces must be joined in an arrangement that creates the intended target shape.

✅ Four Questions to Ask

What are the pieces?Name the component shapes.
How do they fit?Turn or move pieces without changing their shape names.
What is the target?Look at the whole outside boundary.
Another way?Ask whether the same target can be built differently.

๐ŸŒŸ 25 Worked Examples

Try to picture or build each arrangement first. Then open “Show solution” to check your reasoning.
1 Compose two squares
Place two equal squares side by side so they share a full side. What larger shape do they make?
Show solution Hide solution

Two equal squares placed side by side make a rectangle.

2 Compose four squares
Arrange four equal squares in a 2-by-2 array. What larger shape do they make?
Show solution Hide solution

A 2-by-2 array of equal squares makes a larger square.

3 Compose two matching triangles
A square is cut along one diagonal into two matching triangles. If the two triangles are joined back along the cut edge, what shape do they make?
Show solution Hide solution

They reconstruct the original square.

4 Compose two half-circles
Join two matching half-circles along their straight edges. What shape do they make?
Show solution Hide solution

They make one circle.

5 Compose four quarter-circles
Arrange four matching quarter-circles so their curved edges form one complete round boundary. What shape do they make?
Show solution Hide solution

They make one circle.

6 Compose a house shape
Place a triangle on top of a rectangle so they share one full edge. What kind of figure is created?
Show solution Hide solution

They create a composite 2D shape that looks like a simple house.

7 Compose a larger rectangle
Place two matching rectangles end to end along equal sides. What can they make?
Show solution Hide solution

They can make a larger rectangle.

8 Compose an L-shaped figure
Join three equal squares so two are in a row and the third is directly below the left square. What kind of figure is made?
Show solution Hide solution

The three squares make an L-shaped composite figure.

9 Extension: compose a hexagon from triangles
Six matching equilateral triangles meet at one central point with no gaps or overlaps. What outer shape do they form?
Show solution Hide solution

They form a hexagon.

10 Find the parts of a composite shape
A house-like figure is made from one rectangle and one triangle. What two component shapes can you identify?
Show solution Hide solution

The parts are one rectangle and one triangle.

11 Decompose a rectangle into squares
A rectangle is exactly two equal squares wide and one square tall. Into what shapes can it be taken apart?
Show solution Hide solution

It can be decomposed into two equal squares.

12 Decompose a square into triangles
A square has a diagonal drawn from one vertex to the opposite vertex. What two shapes are formed?
Show solution Hide solution

The diagonal decomposes the square into two triangles.

13 Recognize half-circle components
A composite circle was built from two matching half-circle pieces joined along their straight edges. If the pieces are separated, what two component shapes do you get?
Show solution Hide solution

You get two half-circles.

14 Recognize quarter-circle components
A composite circle was built from four matching quarter-circle pieces. If the pieces are separated, what component shapes do you get?
Show solution Hide solution

You get four quarter-circles.

15 Same target, different composition
A rectangle can be made from two equal squares. Can it also be made from two matching triangles that came from cutting that rectangle along a diagonal?
Show solution Hide solution

Yes. The same target rectangle can be composed in more than one way.

16 Choose pieces for a target shape
You want to make a larger square using equal small squares. Which works: 2 squares in a row or 4 squares in a 2-by-2 array?
Show solution Hide solution

The 4 squares in a 2-by-2 array make a larger square.

17 Recognize an impossible composition
Can two whole circles, without cutting or overlapping, be combined to make exactly one square?
Show solution Hide solution

No. Their curved boundaries do not create the four straight sides of a square.

18 Compose with three shapes
A rectangle has the same height as each of two equal squares. One square is attached to each short side, and all three pieces are aligned in one row. What kind of figure is formed?
Show solution Hide solution

The three pieces form a larger rectangle.

19 Compose with four triangles
Four matching right triangles that come from dividing a square into four triangular regions are put back together. What target shape is restored?
Show solution Hide solution

They restore the original square.

20 Decompose and recompose
A 2-by-2 square made from four equal small squares is taken apart and then rebuilt. Does the target shape change?
Show solution Hide solution

No. If the pieces are returned to the same arrangement, they compose the same larger square.

21 Compose cubes
Join two equal cubes face to face. What solid can the pair form?
Show solution Hide solution

The two cubes can form a rectangular prism.

22 Compose four cubes
Arrange four equal cubes in one straight row, face to face. What kind of solid is formed?
Show solution Hide solution

They form a longer rectangular prism.

23 Find cubes inside a composite solid
A rectangular prism is built from three equal cubes in one row. If it is taken apart, what smaller solids do you get?
Show solution Hide solution

You get three cubes.

24 Compose a composite solid
Place a cone on top of a cylinder so the cone's circular base matches the cylinder's circular face. What have you made?
Show solution Hide solution

You have made a composite 3D solid from a cone and a cylinder.

25 Explain compose and decompose
What is the difference between composing and decomposing shapes?
Show solution Hide solution

To compose means to join smaller shapes to make a larger shape. To decompose means to take a composite shape apart or identify the smaller shapes inside it.

๐ŸŽฎ Independent Practice — 30 Problems

  1. Two equal squares are placed side by side. What larger shape do they make?
  2. Four equal squares are arranged in a 2-by-2 array. What larger shape do they make?
  3. Two matching half-circles are joined along their straight edges. What shape do they make?
  4. Four matching quarter-circles are arranged to complete a round boundary. What shape do they make?
  5. A triangle is placed on top of a rectangle. Name the two component shapes.
  6. Three equal squares are joined in an L arrangement. How many component shapes are used?
  7. A rectangle is exactly two equal squares wide. Into what two shapes can it be decomposed?
  8. A square is divided by one diagonal. What two shapes are formed?
  9. A composite circle is made from two matching half-circle pieces. What component shapes are inside it?
  10. A composite circle is made from four matching quarter-circle pieces. What component shapes are inside it?
  11. Can the same target rectangle sometimes be composed in more than one way? Answer yes or no.
  12. Which makes a larger square: 2 equal squares in one row or 4 equal squares in a 2-by-2 array?
  13. A simple house shape is made from one triangle and one rectangle. If decomposed, how many pieces are there?
  14. Extension: Six matching equilateral triangles meet around one point with no gaps or overlaps. What outer shape can they form?
  15. Three equal squares are arranged in one straight row. What larger shape do they form?
  16. Two matching rectangles are joined along equal sides. What larger shape can they form?
  17. A composite figure contains one triangle and two squares. How many component shapes does it contain?
  18. A figure is made from four equal squares. If one square is removed, how many squares remain?
  19. If only the geometric target shape matters, does changing a piece's color change whether it can fit the same way?
  20. Does rotating a piece change its shape name when its defining attributes stay the same?
  21. Two equal cubes are joined face to face. What solid can they form?
  22. Four equal cubes are joined in one straight row. What solid can they form?
  23. A rectangular prism is built from three equal cubes. If decomposed, what smaller solids appear?
  24. A cone sits on a cylinder with matching circular faces. Name the two component solids.
  25. Which word means 'join smaller shapes to make a larger shape': compose or decompose?
  26. Which word means 'take a composite shape apart into component shapes': compose or decompose?
  27. A larger square is made from four equal small squares. How many small squares are inside it?
  28. A rectangle was cut along a diagonal into two matching triangles. If the pieces are joined back along the cut, what shape returns?
  29. Name one target shape that can be made from two equal squares.
  30. Extension: Explain one way the same target shape can sometimes be composed from different sets or arrangements of pieces.

๐Ÿ† Lesson Test — 15 Questions

  1. A larger rectangle is built from three equal squares in one straight row. If the rectangle is taken apart, what component shapes are found?
  2. A larger square is built from four equal small squares. One small square is removed. Is the target square still complete?
  3. Two matching triangles that came from cutting one square along a diagonal are joined back along the matching cut. What target shape returns?
  4. A composite circle is made from four quarter-circle pieces. If one piece is removed, how many pieces remain, and is the circle complete?
  5. A house-like figure is made from one triangle and one rectangle. If the triangle is removed, which component shape remains?
  6. Two equal cubes are joined face to face, and then a third equal cube is added in the same row. What solid can the three cubes form?
  7. A rectangular prism is built from five equal cubes in one row. If it is taken apart completely, how many cubes are obtained?
  8. Two composite rectangles, each made from two equal squares, are joined end to end to make one longer rectangle. How many small squares are used altogether?
  9. Which action is composing: separating a house into a triangle and rectangle, or joining a triangle and rectangle to make a house?
  10. Which action is decomposing: joining four squares to make a larger square, or identifying the four small squares inside the larger square?
  11. Four equal square pieces make a geometric target shape. If only their colors change, can the same geometric arrangement still make that target?
  12. A triangle piece is rotated before it is fitted into a composite figure. Is it still a triangle?
  13. Can two equal squares placed in one straight row make a larger square? Explain.
  14. On Monday a rectangle is built from two equal squares. On Tuesday the same rectangle is rebuilt from two matching triangles. What important composition idea does this show?
  15. Explain how composing and decomposing are related.

๐Ÿ” Complete Answer Key

Show practice answer keyHide practice answer key

Practice Answers

1. Rectangle.
2. Larger square.
3. Circle.
4. Circle.
5. Triangle and rectangle.
6. 3.
7. Two equal squares.
8. Two triangles.
9. Two half-circles.
10. Four quarter-circles.
11. Yes.
12. 4 equal squares in a 2-by-2 array.
13. 2 pieces.
14. Hexagon.
15. Rectangle.
16. Larger rectangle.
17. 3 component shapes.
18. 3 squares.
19. No. Color is non-defining.
20. No. Orientation does not change the shape name.
21. Rectangular prism.
22. Rectangular prism.
23. Three cubes.
24. Cone and cylinder.
25. Compose.
26. Decompose.
27. 4.
28. Rectangle.
29. Rectangle.
30. Possible answer: a rectangle can be made from two equal squares or from two matching triangles that came from cutting the rectangle along a diagonal.
Show test answer keyHide test answer key

Test Answers

1. Three equal squares.
2. No. One of the four component squares is missing.
3. The original square.
4. 3 pieces remain, and the circle is not complete.
5. The rectangle.
6. A rectangular prism.
7. 5 cubes.
8. 4 small squares.
9. Joining the triangle and rectangle to make a house.
10. Identifying the four small squares inside the larger square.
11. Yes. Color is non-defining; the pieces still have the same sizes and shapes.
12. Yes. Rotation does not change the piece's shape name.
13. No. Two equal squares in one row make a rectangle that is twice as wide as it is tall.
14. The same target shape can sometimes be composed in more than one way, using different component shapes or arrangements.
15. Composing joins component shapes to make a whole; decomposing reverses that idea by taking the whole apart or identifying its components.

๐Ÿ’ก Common Mistakes to Watch For

Leaving gaps between pieces
Pieces must fit together to make the intended target shape.
Thinking rotation changes the shape
Turning a triangle or square does not change its shape name.
Counting only the outside shape
When decomposing, look for the component shapes inside the whole.
Assuming there is only one solution
Some target shapes can be composed in more than one way.
What comes next: Lesson 28 focuses on Halves, Fourths & Quarters, where circles and rectangles are partitioned into equal shares.

๐Ÿ”— Related Lessons

Lesson 26 — 3D Shapes & Solid Figures Review solid figures before composing larger 3D shapes. Lesson 25 — 2D Shapes & Defining Attributes Review sides, vertices, orientation, and common 2D shape names.

๐Ÿ–จ️ Printable Companion

A future Compose & Decompose Shapes Practice Pack can include cut-and-build shape cards, tangram-style compositions, target-shape challenges, multiple-solution puzzles, cube-building tasks, decomposition diagrams, mixed review, and a complete answer key.

← Lesson 26: 3D Shapes & Solid Figures Grade 1 Math Hub Next: Halves, Fourths & Quarters

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