Finding Side Lengths of Triangles

5 min

Teacher Prep
Setup
Students in groups of 2–4. Display the image for all to see. Students indicate when they have one that does not belong. 1 minute of quiet think time followed by small group and then whole group discussion.

Narrative

This Warm-up prompts students to carefully analyze and compare features of triangles. In making comparisons, students have a reason to use language precisely (MP6). The activity also enables the teacher to hear the terminologies students know and how they talk about characteristics of triangles.

Launch

Arrange students in groups of 2–4. Display the figures for all to see. Give students 1 minute of quiet think time and ask them to indicate when they have noticed three triangles that go together and can explain why. Next, tell students to share their response with their group, and then together find as many sets of three as they can.

Student Task

Which three go together? Why do they go together?

A
First of four triangles.
First of four triangles is a right triangle, legs = 5, 5. hypotenuse = square root 50.

B
Second of four triangles.
Second of four triangles has sides = 2, 5, square root 33.

C
Third of four triangles.
Third of four triangles is a right triangle, legs = 1,8.  

D
Fourth of four triangles.
Fourth of four triangles is a right traingle, legs = 3,4. hypotenuse = 5.

Sample Response

Sample responses:

A, B, and C go together because:

  • They all have 1 side that is an irrational number.
  • They all have 1 side that is written as a square root.

A, B, and D go together because:

  • They all have at least 1 side length that is 5 units.

A, C, and D go together because:

  • They are all right triangles.
  • They all have 1 right angle.

B, C, and D go together because:

  • They are all scalene triangles.
  • They all have 3 different side lengths.
  • They are not isosceles triangles.
Activity Synthesis (Teacher Notes)

Invite each group to share one reason why a particular set of three go together. Record and display the responses for all to see. After each response, ask the class if they agree or disagree. Since there is no single correct answer to the question of which three go together, attend to students’ explanations and ensure the reasons given are correct.

During the discussion, ask students to explain the meaning of any terminology they use, such as irrational, scalene, or isosceles, and to clarify their reasoning as needed. Consider asking:

  • “How do you know . . . ?”
  • “What do you mean by . . . ?”
  • “Can you say that in another way?”
Standards
Building On
  • 5.G.4·Classify two-dimensional figures in a hierarchy based on properties.
  • 5.G.B.4·Classify two-dimensional figures in a hierarchy based on properties.
  • 7.G.A·Draw, construct, and describe geometrical figures and describe the relationships between them.
  • 7.G.A·Draw, construct, and describe geometrical figures and describe the relationships between them.
  • 8.EE.2·Use square root and cube root symbols to represent solutions to equations of the form x² = p and x³ = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.
  • 8.EE.A.2·Use square root and cube root symbols to represent solutions to equations of the form <span class="math">\(x^2 = p\)</span> and <span class="math">\(x^3 = p\)</span>, where <span class="math">\(p\)</span> is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that <span class="math">\(\sqrt{2}\)</span> is irrational.
Building Toward
  • 8.G.6·Explain a proof of the Pythagorean Theorem and its converse.
  • 8.G.B.6·Explain a proof of the Pythagorean Theorem and its converse.

15 min

10 min