Much like Algebra, Geometry is not in Jamaal’s vocabulary when he is only 4 and 5; but the basic concept of geometry will provide the groundwork for Jamaal to excel in high school geometry down the line. At this stage, Jamaal is learning the names and characteristics for basic shapes like circles, squares and triangles. He can gain geometry concepts by building with Blocks or recreating shapes and patterns with blocks. Doing this, he may realize that a square has four sides and a block looks like a bunch of squares put together. He can associate shapes in his everyday life like triangular shapes form roofs and rectangular shapes form windows. Putting together simple puzzles and playing with tangrams to reconstruct the pieces to look like different animals or shapes is a great way to help develop Jamaal’s spatial sense.

Geometry in early elementary school focuses on the names and characteristics of various plane (2 dimensional) and solid (3 dimensional) shapes. Jamaal will learn that objects have shapes and the shapes have specific names. A box is also called a cube, a ball is a sphere. It is important at this stage to talk about how a shape like a rhombus is similar and different from a square. Jamaal should be able to describe a 2 dimensional object by how many sides or corners it has. He should also be establishing a strong understanding of the differences between “flat” (2 dimensional) and “solid” (3 dimensional) shapes. Hands-on approaches allowing Jamaal to build shapes to mimic everyday objects will develop his spatial sense and provide context for the geometry vocabulary he is learning.

At the end of 2nd grade, Jamaal should be able to solve this geometry problem:

Draw and describe a square, a triangle, and a rectangle, using words like sides and corners to compare them.

What is the difference between a sphere, a cube, and a pyramid?

Having started with a strong foundation in basic geometry vocabulary and descriptions, Jamaal should easily be able to build on that with more specific measurable attributes and more complex shape relationships. He should be able to describe and name shapes like quadrilaterals and parallelograms, using words like angles and rays, as well as be able to draw the shape from its description. This is the time to introduce the characteristics of a circle and the concept of degrees, that there are always 360 degrees in a circle. Folding something round like a coffee filter in half and then in half again will help Jamaal understand the degree of a straight line or a right angle. He should use tools like protractors to help define the degrees of an angle.

He should also be learning to solve basic problems involving a number line with an x/y axis and plotting points. Creating shapes on a grid starts the foundation of understanding ratios. As with all math concepts, it is important that Jamaal knows how this knowledge relates to the real world. Explaining how careers like architecture and construction use geometry every day may sustain Jamaal’s enthusiasm to learn.

At the end of 5th grade, Jamaal should be able to solve this geometry problem:

1. What is the volume, in cubic inches, of a school locker that measures 30” tall, 12” wide, and 8” deep?

2. A swimming pool is 25 feet wide and 40 feet long. Around the pool is a path that is 4 feet wide. What is the area of the path?

At this point, Jamaal should start to learn the formulas needed to solve for attributes like area, volume, and mass. He should know how to find these attributes for regular and irregular shapes by understanding the formula for simple shapes and applying it to the sections of an irregular shape. Understanding these formulas and why they work and getting a grasp on the more complex vocabulary sets the necessary groundwork for geometry work in higher grades. Students who do not acquire these crucial foundational skills often struggle with geometry down the line. Geographical measurement is strongly focused and students should be able to translate area, volume or mass into simple form like 3ft 9in + 4ft 10in = 8ft 7in. Jamaal should understand this translation using whole numbers, rational numbers (integers, fractions, terminating and repeating decimals) and irrational numbers (non-terminating and non-repeating decimals). Jamaal should have a strong understanding of shapes, angles, faces, sides and vertices. Visual demonstration of formulas is a necessity to make the connection between the formula and the deductive and inductive reasoning that is necessary to progress in high school Geometry.

At the end of 8th grade, Jamaal should be able to solve this geometry problem:

A 10-meter ladder is leaning against a building. The bottom of the ladder is 5 meters from the building. How many meters high is the top of the ladder? Round to the nearest tenth.