11.5 Classifying Two-Dimensional Figures & Partitioning Shapes into Equal Parts

Classifying Two-Dimensional Figures & Partitioning Shapes into Equal Parts

Section 11.1 introduced polygons and the quadrilateral hierarchy. This section covers what Levels E and M actually assess about them: identifying shapes by attributes, reasoning through the hierarchy of categories, and partitioning a shape into equal-area parts named by unit fractions.

Classifying by Attributes

An attribute is a defining property. TABE items describe a shape by its attributes and ask you to name it, or name a shape and ask which attributes must be true.

Triangles by side and by angle

Every triangle has both a side name and an angle name: a triangle can be an isosceles right triangle, a scalene obtuse triangle, and so on. An equilateral triangle is always acute, with all three angles 60°.

Orientation does not change classification. A square rotated 45° so it stands on a corner is still a square. TABE draws shapes in unusual orientations for exactly this reason.

The Quadrilateral Hierarchy

Categories nest inside one another, and a property of a category belongs to every shape beneath it.

                    Quadrilateral
                    (4 sides)
                         |
          ┌──────────────┼──────────────┐
     Trapezoid      Parallelogram      Kite
   (≥1 pair of     (2 pairs of      (2 pairs of
    parallel        parallel          adjacent
    sides)          sides)          equal sides)
                         |
                ┌────────┴────────┐
           Rectangle          Rhombus
        (4 right angles)   (4 equal sides)
                └────────┬────────┘
                      Square
              (4 right angles AND
                4 equal sides)

Reading the hierarchy correctly:

The inheritance rule that TABE tests. Because opposite sides of a parallelogram are parallel and congruent, and every rectangle, rhombus, and square is a parallelogram, all of them have opposite sides parallel and congruent. Properties flow downward through the hierarchy, never upward.

Always, sometimes, never

Partitioning Shapes into Equal Parts

Standards 2.G.3 and 3.G.2 ask candidates to partition shapes into parts with equal areas and express each part as a unit fraction of the whole.

A rectangle divided into 4 parts of equal area: each part is $\frac{1}{4}$ of the rectangle's area.

Two facts learners commonly miss:

1. Equal area does not require equal shape. A square can be divided into four equal-area parts as four vertical strips, four horizontal strips, four small squares, or four triangles meeting at the center. All are quarters.
2. The unit fraction names the part, not the number of cuts. Three cuts across a rectangle create four parts, each $\frac{1}{4}$.

Naming multiple parts

If a circle is partitioned into 8 equal sectors and 3 are shaded, the shaded region is $\frac{3}{8}$ of the circle. This is where the geometry standard hands off directly to Chapter 3's fraction work.

Partitioning with area attached

A rectangular garden measures 12 ft by 9 ft and is divided into 6 equal plots. What is the area of each plot?

Total area: $12 \times 9 = 108$ sq ft. Each plot is $\frac{1}{6}$ of the whole: $108 \div 6 = \mathbf{18}$ square feet.

Notice that the problem never says what shape the plots are, and it does not matter — equal area is the only requirement.

Figures on the Coordinate Plane

Level M places classification on a grid: given four vertices, identify the quadrilateral.

Vertices at $(1, 1)$, $(6, 1)$, $(6, 4)$, $(1, 4)$.

• Bottom side: from $x = 1$ to $x = 6$, length 5.
• Right side: from $y = 1$ to $y = 4$, length 3.
• Opposite sides are equal and all corners are right angles.
• It is a rectangle, not a square, because $5 \ne 3$.

Compute side lengths by subtracting coordinates along horizontal and vertical edges. If all four sides come out equal and the corners are square, it is a square.