Similar Triangles

The big idea

Start by drawing two similar triangles on the board and asking students, "Are these triangles the same?" Try to elicit the response, "they are the same shape but different sizes." Explain further that similar triangles have equal angles but not equal sides. This is different from congruent triangles, which are exactly the same. Drawing examples of similar, congruent, and neither helps make the distinction clear.

Learning objectives

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

• Define similar triangles as having equal angles but not equal sides.
• Tell similar triangles apart from congruent ones.
• Test two triangles for similarity by nesting one inside the other.
• Sort a set of triangles into groups of similar shapes.

Common Core alignment

• CCSS.MATH.CONTENT.4.G.A.2

  Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a specified size. Recognize right triangles as a category, and identify right triangles.

  Telling triangles apart by their attributes is the triangle-classification piece of this grade-4 standard.

• CCSS.MATH.CONTENT.8.G.A.4

  Understand that a two-dimensional figure is similar to another two-dimensional figure if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations.

  The nesting test (same shape, different size, equal angles) is the hands-on entry to the formal definition of similarity this standard introduces.

Materials

• Triangle sheets, pages 125–128 (1 set per student)
• Scissors (1 per student)
• 12" × 18" construction paper (1 sheet per student)
• Glue sticks (1 per student)
• The completed project, prepared by the teacher before the lesson

The project

Students cut out 15 triangles and test them for similarity by placing them inside one another. Show students the two ways this nesting method can demonstrate that a group of triangles is similar. The fifteen shapes form five groups of three. To prevent confusion, students should write their name on the back of each triangle as they cut it out.

This project generates a lot of waste paper, so consider having students keep a "scrap ball" of their crumpled scraps. Before pasting their triangles onto construction paper, students should check with the teacher to make sure their groupings are correct.

Common student mistakes

• Confusing similar with congruent. Similar means same shape, different size; congruent means exactly the same. The nesting test is what separates them.
• Expecting similar triangles to match in size. Equal angles, not equal sides, is the test. A small triangle and a large one can still be similar.
• Grouping by eye instead of nesting. Two triangles are only confirmed similar when one nests cleanly inside the other; have the teacher check the five groups before gluing.