By Christopher Danielson

In Common Core math, fourth grade students begin to expand their view of geometry beyond shapes. They think about lines, angles, and the relationships that they can have with each other.

This activity generates quite a bit more vocabulary. For example, students study parallel lines, which are two lines in a plane that don’t intersect. Students will think about parallel lines as never meeting, or maybe as headed in exactly the same direction.

Students begin to make distinctions between lines (which extend infinitely in either direction), rays (which extend infinitely in only one direction), and line segments (which have finite length and two endpoints). Students should investigate whether the definition of parallel lines applies to line segments. Look at the following figure.

These line segments don't intersect, but they aren't parallel.

These line segments don’t intersect, but they aren’t parallel.

As the figure shows, this definition doesn’t apply — line segments may not meet even if they aren’t parallel — so a slightly different definition is required. You can say that two line segments are parallel if they would never meet, even if extended infinitely.

Students use line and angle relationships such as parallel and perpendicular lines (parallel lines are in the same plane as each other, run in the same direction, and never meet, even if extended infinitely; perpendicular lines meet at right angles), and right and acute angles (a right angle has a measure of 90 degrees; an acute angle has a measure greater than 0 degrees, but less than 90 degrees) to classify and relate shapes to each other.

Finally, students recognize lines of symmetry in shapes. If a shape is drawn on a piece of paper, folding the paper along the line of symmetry causes two halves of the shape to match up with each other perfectly. A square has four lines of symmetry. A non‐square rectangle has only two lines of symmetry, as you can see in this figure.

A square has four lines of symmetry; a non-square rectangle has two.

A square has four lines of symmetry; a non-square rectangle has two.

Although you can connect the corners of a rectangle and get two same‐sized halves, folding along this line won’t make the two halves match up.