Perpendicular & Parallel Lines

The big idea

Parallel lines can most simply be defined as "lines that are side-by-side and never touch, no matter how long we draw them." Perpendicular lines can be defined as "lines that make a 'T.'" Both of these definitions are somewhat incomplete. Lines can be parallel and still not be side-by-side, and perpendicular lines can make an 'L' shape as well as a 'T' shape. Still, introducing those complications is best left for later lessons.

The teacher should draw examples and counter-examples to help students understand both terms: a pair that is not parallel beside a pair that is, and a pair that is not perpendicular beside a pair that is.

Learning objectives

By the end of the lesson, students will be able to:

• Identify parallel lines as lines that are side by side and never touch.
• Identify perpendicular lines as lines that make a "T."
• Draw a line perpendicular to a given line, and check it by sight.
• Draw additional lines parallel to lines already on the page.

Common Core alignment

• CCSS.MATH.CONTENT.4.G.A.1

  Draw points, lines, line segments, rays, angles (right, acute, obtuse), and perpendicular and parallel lines. Identify these in two-dimensional figures.

  Drawing perpendicular and parallel lines and identifying them in two-dimensional figures is exactly the geometric vocabulary this standard names.

Materials

• Pencils (1 per student)
• Starter sheets, page 116 (1 per student)
• Rulers (1 per student)
• Crayons or colored pencils (2 colors per student)
• The completed project, prepared by the teacher before the lesson

The project

Tape a starter sheet at the front of the room and use it to model every step. The size of the sheet makes it a little hard for students in the back to see, but that's not a problem as long as each step is explained clearly.

Ask students what type of lines are already printed on their starter sheets. They are parallel. Explain that you're about to draw a line that is perpendicular to this group, and model it before letting students try. A helpful image: pretend the new line is the path of an arrow shot from a bow, fired directly into the parallel lines, which we can pretend is a wall. To be perpendicular, the arrow has to go straight into the wall.

Now have students try on their own. If a line doesn't look perpendicular, ask them to erase and redraw; they can check by tilting the paper to see whether they've made a "T." Next, model drawing three more lines parallel to that new line, then have students repeat the entire process on the opposite side of the sheet, always after you've modeled it.

Everyone should now have eight lines on the page. Point out that every line is both perpendicular and parallel to others, since each of the three groups of parallel lines is perpendicular to at least one other group. The paper is now divided into four sections. Choose the top-right section to work on first and model drawing two more lines inside it; then let students try.

Finally, give students some artistic freedom: fill each section with any combination of lines they like. The only rules are that the lines must not cross, and every new line must be parallel to lines already drawn, so each section is restricted to just two directions. Drawing a double-sided arrow in each section helps students remember the two allowed directions. When all four sections are filled, students pick two "opposite" (complementary) colors and color the spaces between the lines, alternating colors.

Common student mistakes

• Lines that only look perpendicular. Have students tilt the paper and check for a clean "T." If it isn't one, erase and redraw.
• Letting lines cross inside a section. Within a section, every line must run in one of just two directions and never cross. The double-sided arrow reminder keeps students honest.
• Drifting off-parallel. Free-hand lines slowly fan out. Encourage ruler use and frequent comparison to the lines already drawn.