Geometry Part 3: Composing and Decomposing

by C. Elkins, OK Math and Reading Lady Composing and decomposing geometric shapes (2D and 3D) should be centered around concrete and pictorial methods. In this and upcoming posts, I will illustrate some methods using various manipulatives and line drawings which help students take a shape apart or put shapes together. If you refer back to  Geometry Part 1: The Basics, all grade levels KG-5th have standards dealing with this issue. Some of the experiences I plan to share will also help students relate to multiplication, division, fractions, area, and other geometry concepts (such as rotations, reflections, slides).

Refer to Geometry Part 2: van Hiele levels to determine if the activities you are choosing are appropriate for Level 0, 1, or 2 students.

One Inch Color Tiles: 1.  Can you make a larger square out of several individual squares?

• Level 0 students will be using the visual aspect of making it look like a square.
• Level 1 students will be checking properties to see if their squares are indeed squares (with the same number of tiles on each side).
• Level 2 students will be noticing they are creating an array (ex: 3 x 3 = 9) and perhaps learning about squared numbers. 3 squared = 9. They might be able to predict the total number of tiles needed when given just the length of one side.

2.  How many rectangles can you make using 2 or more squares? (Level 0-1) • Level 1:  Are the green and blue rectangles the same size (using properties to determine)?

3. How many different ways can you make a rectangle using 12 tiles?  24 tiles?  Record on graph paper. (Level 1 or 2)

• Connect to perimeter and area lessons by noticing the area might be the same, but the perimeter changes. • Connect to factoring lessons.  Example:  12 can be factored as 1 x 12, 3 x 4, and 2 x 6; 24 can be factored as 1 x 24, 2 x 12, 3 x 8, 4 x 6
• Connect to multiplication commutative property (3 x 4 = 4 x 3).

4. Make a rectangle using 24 tiles. If you decompose it, how many different smaller rectangles or squares can you make?  Explore with tiles first, then record results on graph paper.

5.  How can you use color tiles to decompose a polygon into smaller rectangular shapes?  For students to master this skill, they must have a good understanding that opposite sides of rectangles are equal length. This is also crucial for determining area. Here are some resources using one inch tiles:

Pattern Blocks: Put pattern blocks together to create another shape:

• Students at a level 0 would be choosing shapes to combine in any manner. They may or may not match equal edges together. It may or may not make a recognizable design / shape.
• Students moving into a level 1 are paying attention to the properties of the shapes they are combining. What is a side? What are corners? Equal size edges must match together.
• Students combine shapes that fit exactly into an outline. Easy pattern block task cards (link below)

This encompasses composing (because they are using smaller pieces to build a larger shape), but also decomposing if the student shows multiple ways to fill in a shape — 2 trapezoids = 1 hexagon; 3 rhombus = 1 hexagon; 6 triangles = 1 hexagon. OR, 1 hexagon = 1 trapezoid and 3 triangles; 1 hexagon = 1 trapezoid, 1 rhombus, and 1 triangle, etc.  This exploration is also beneficial to understanding equivalent fractions.

• Students at Level 1-2 may create shapes which have more than one property and/or fit into more than one category: symmetrical, a shape using a particular number and type of pattern blocks, those in which shapes must be manipulated (rotate, reflect, slide) to achieve the results, or shapes which substitute blocks (example: 1 triangle and 2 rhombus can make a trapezoid). Composing 2D shapes with pattern blocks (link below)

Here are some resources using pattern blocks:

Do you have any ideas to share about composing / decomposing using tiles or pattern blocks? More on composing / decomposing next time (Tangrams, etc.)