Summary
In this lesson, students will discover how to determine if any three side lengths make a triangle through a GeoGebra activity. Then, students will formalize their knowledge of the Converse of the Pythagorean Theorem before applying their knowledge through a Card Matching activity and a real-world scenario.
Essential Question(s)
How do I use side lengths to determine if a triangle is acute, obtuse, or right?
Snapshot
Engage
Students look at diagrams of attic rafters and use their prior knowledge to try to determine the third length of a triangle.
Explore
Students work with partners to complete a GeoGebra activity and discover a pattern to help determine when three sides make a triangle.
Explain
Students participate in a whole-class discussion to formalize their findings and introduce mathematical inequalities and equations to classify triangles as acute, right, or obtuse.
Extend
Students work in groups to complete a Card Matching activity to apply their new knowledge.
Evaluate
Students are posed with a construction task to create triangles out of lengths of lumber to demonstrate their understanding.
Materials
Lesson Slides (attached)
Is It a Triangle? handout (attached; 1 per student; printed front only)
Is It a Triangle? (Sample Response) document (attached; for teacher use)
Three Sides Card Sort Mat (attached; 1 per group; printed front only)
Three Sides Sorting Cards (attached; half page per group; printed front only)
Three Sides Card Sort Teacher Resource document (attached; for teacher use)
Student devices with internet access (1 per student)
Scratch paper (3 per student) or personal whiteboard
Sandwich bag (1 per card set)
Scientific calculators
AngLegs (optional)
Engage
10 Minute(s)
Use the attached Lesson Slides to guide the lesson. Review the essential question and learning objectives on slides 3 and 4.
Place students into groups of 3–4, transition to slide 5, ask students to to get out a piece of paper, and introduce the Think-Pair-Share strategy. Preview the activity with the students by explaining that they will be given a problem on the next two slides and they must use prior knowledge to find the side marked with a question mark. They will be given time to independently find an answer or write questions they have on their paper before discussing the picture with their group. Lastly, have groups share with the class what they decided during their conversation.
Show slide 6 and give individual students about 2 minutes to think about their answer. Next, ask groups to turn and talk with each other about the displayed image and come to a decision as a group on the answer. After a couple of minutes, ask groups to volunteer to share their thoughts with the class.
After two or three groups have shared, move to slide 7 and repeat the process again with the second image.
Ask students to set the work they did for slides 6 and 7 aside to revisit later in the lesson.
Explore
10 Minute(s)
Have students find a new partner and give each student a copy of the Is It a Triangle? Handout. Show slide 11 and provide students with the link to the GeoGebra activity: geogebra.org/m/tgwg6tnj.
Explain to students that centuries ago, Egyptians used knotted cords (ropes with knots indicating certain lengths) to measure distances and construct triangles. These ancient surveyors used stretched rope to ensure that measurements were consistent. This activity lets you imagine that you are a new surveyor in ancient Egypt and have been given knotted ropes to test. They will get to find out if each rope is useful for creating triangles.
This interactive GeoGebra activity gives students 3 line segments to move to attempt to create triangles. Guide students to move points B, C, and/or D, as point A is fixed. When points A and D align, the enclosed region will turn blue, indicating that those three segments create a triangle.
While students are working with partners to complete the handout, walk around the classroom to keep students on task. This is a time in the lesson for students to discover patterns and relationships. Allow students time for productive struggle. You will clear up any misconceptions later in the Explain phase. Use the attached Is it a Triangle? (Sample Response) document for possible student answers. Encourage students to use a pencil to complete this handout in case they need to correct answers in the Explain phase.
Allow students to work together for at least 10 minutes. The information students record on this handout will be used in the Explain phase.
Explain
15 Minute(s)
Move to slide 12 and begin a whole class conversation about what they found in the Explore phase. The purpose of this class discussion is for students to interpret their findings and lead the class in learning while the teacher steps in when necessary to formalize the knowledge. Have groups volunteer to explain which sets were and were not triangles and what influenced their decision. Probe the rest of the class by asking who agrees with the explanations, and if the class has any questions for the answering group.
Transition to slide 13 and ask students to look at the 4th column on the first table to find a pattern that could explain how set 1, 2, 3 is not a triangle and set 3, 4, 5 is a triangle. Show slide 14-15 to confirm the Notation on their table and introduce the inequality that must be true for three lengths to make a triangle.
Ask students to write this inequality in the box under “What algebra can help us calculate this?”
Move to slide 16 and provide students a chance to share which sets were right triangles and how they knew.
Move through slides 17-19 to reveal which set was a right triangle, how to determine this algebraically, and complete an example to formalize their knowledge.
Ask students to write the equation/inequality that proves three lengths make a right triangle in the table at the bottom of the page under “What Algebra can help us calculate this?”
Repeat this same process for slides 20-23 for acute triangles.
Repeat this same process for slides 24-27 for obtuse triangles.
Extend
15 Minute(s)
Have students find a partner, then transition to slide 28 and introduce the Card Matching strategy. Give each pair a copy of the attached Three Sides Card Sort Mat and Three Sides Sorting Cards.
Instruct students to use their new knowledge to sort each set of side lengths into the box that represents the type of triangle it is: acute, right, obtuse, or not a triangle. Remind students that they must determine if the three lengths make a triangle before they classify the triangle.
As pairs work together, rotate around the room to keep students on task and clear up misconceptions as necessary. Utilize the attached Three Sides Card Sort Teacher Resource document to review the correct placement of the cards.
As groups complete the card sort, review their answers and point out any cards that are in the wrong category and ask students to look back at their Is It a Triangle? Handout and make corrections before putting away the activity.
Evaluate
10 Minute(s)
Show slide 29 and remind students of the building task from the beginning of the lesson. Explain that in construction this sort of planning for attic supports is important to providing clients with the kinds of roofs they prefer. Construction workers have to plan carefully because one wrong measurement can ruin their lumber supply and lead to waste and lost money.
Give the students the following scenario: You are a building and have been given 4 pieces of lumber to cut into 3 segments. The lumber is 8 feet long and the client would like to see options for an acute, right, and obtuse triangle. Ask the class if it is possible to have leftover lumber in construction, and if that idea can be extended to this task?
After giving students about 10 minutes to decide on their lengths and write the answers with all necessary work shown on a piece of paper, use already established classroom procedures to turn in the work.
Resources
K20 Center. (n.d.). Chain notes. Strategies. https://learn.k20center.ou.edu/strategy/52
K20 Center. (n.d.). Card matching. Strategies. https://learn.k20center.ou.edu/strategy/1837
K20 Center. (n.d.). GeoGebra. Tech tools. https://learn.k20center.ou.edu/tech-tool/2352
K20 Center. (n.d.). Think-Pair-Share. Strategies. https://learn.k20center.ou.edu/strategy/139