Lesson 3

In this activity, students first sketch the situation and then notice and wonder about their sketches. The purpose of this warm-up is to elicit the idea that equilateral triangles and altitudes have properties we can generalize, which will be useful when students look for patterns in a later activity. While students may notice and wonder many things about their sketch, angle measures and the proportions of side lengths are the important discussion points.

If students have printed workbooks invite them to sketch on scrap paper rather than in their workbooks, as the surrounding information may influence their observations.

Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice and wonder with their partner, followed by a whole-class discussion.

If needed, remind students of the definition of altitude: “An altitude in a triangle is a line segment from a vertex to the opposite side that is perpendicular to that side.”

Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near a displayed sketch. After all responses have been recorded without commentary or editing, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to respectfully disagree, ask for clarification, or point out contradicting information.

In a previous lesson, students proved that all equilateral triangles are similar. In this lesson, they convince themselves that equilateral triangles decomposed into two congruent halves are similar right triangles, and therefore the ratios of their side lengths will be equal.

The first triangle students encounter in this activity is an equilateral triangle with side lengths of two, which means that the shortest side of the 30-60-90 triangles formed is one unit long. Monitor for students who use scale factors to generate side lengths in the triangles they measure as well as students who measure and compute all the ratios.

Students will need a set of equilateral triangles to measure. They could use isometric dot paper, construction tools, or the blackline master.

Making dynamic geometry software available gives students an opportunity to choose appropriate tools strategically (MP5).

Arrange students in groups of 2. Distribute equilateral triangles of several different sizes or the tools to make them to each group.

If students are struggling to organize their thinking, suggest that they make a table. Help students brainstorm what would be good columns to organize their measurements and calculations, such as “short leg length,” “altitude length,” and “ratio of hypotenuse to short leg.”

Ask students what they found for the ratio of the altitude, \(y\), to the short leg, \(x\). Include students who approximated the lengths using a calculator and students who left the length of the altitude as \(\sqrt3\).

Ask students why the altitude in any equilateral triangle seems to be half the side length multiplied by about 1.7 each time. (The altitude is also the median in an equilateral triangle, so the short leg is half the side length. All equilateral triangles are similar to the first one we studied, where the altitude was \(\sqrt3\).)

Display the image and ask students if they agree that both of these things are true:

• The length \(y\) is about 1.7 times \(x\).
• The length \(y\) is exactly \(\sqrt3\) times \(x\).

In the previous activity and summary, students generalized that the altitudes of equilateral triangles are related to half the side length of the triangle (the short leg) by a factor of about 1.7, or exactly \(\sqrt3\).

In this activity, students apply their generalization about the altitudes of equilateral triangles to 30-60-90 triangles. To find the lengths of the unlabeled sides in the second figure, students will need to generalize that a right triangle with one 60 degree angle is a 30-60-90 triangle (because it’s half an equilateral triangle, or because of the Triangle Angle Sum Theorem, or because it’s similar to the triangle in the first figure).

Monitor for different solutions in the third figure. Students will have to use the ratios to find unknown side lengths. Students may use the approximate ratio \(1.7:1:2\) or the exact ratio \(\sqrt3:1:2\).

Make sure all students understand that the three triangles are 30-60-90 right triangles, and represent half of an equilateral triangle, and so are prepared to connect their reasoning from earlier activities to this activity.

Focus on the third figure. Ask students what was different about that figure that might have made the task a bit harder. (There's only one side given and it has a square root in it.) Invite previously selected students to explain their strategies for calculating the lengths of the other two sides.