Lesson Standards/Objective(s): What mathematical skill(s) and understanding(s) will be developed? Which Mathematical Practices do you expect students to engage in during the lesson?
G.CO.C.10 Prove theorems about triangles, specifically, that the medians of a triangle meet at a point.
MP1: Make sense of problems and persevere in solving them.
MP2: Reason abstractly and quantitatively.
MP3: Construct viable arguments and critique the reasoning of others.
MP5: Use appropriate tools strategically.
MP6: Attend to precision.
Lesson Launch Notes: Exactly how will you use the first five minutes of the lesson?
Provide students patty paper and a ruler. Present them with the following statement:
“A median is defined as a segment with an endpoint on the vertex an angle and the another endpoint on the midpoint of the opposite side. Draw a triangle on your patty paper and construct all three medians. Then, record what you notice about the relationship of the medians in a triangle.”
Possible student responses: The medians intersect at one point. (You can tell students that this is called a point of concurrency, specifically, the centroid of the triangle.) Some students may also realize that the length of the median from the point of concurrency to the vertex is twice the distance of the length of the segment from the point of concurrency to the midpoint of the opposite side. (Look for evidence of MP3.)
Lesson Closure Notes: Exactly what summary activity, questions, and discussion will close the lesson and connect big ideas? List the questions. Provide a foreshadowing of tomorrow.
Have students form groups of 3-4 students and have each group member share one take away from today’s activity.
Lesson Tasks, Problems, and Activities (attach resource sheets): What specific activities, investigations, problems, questions, or tasks will students be working on during the lesson? Be sure to indicate strategic connections to appropriate mathematical practices.
1. Have students share out the conjectures they developed about the medians of a triangle. Then, explain that today they will have the chance to test these conjectures for all types of triangle. (Look for evidence of MP3.)
2. Provide students with access to dynamic geometry software, such as, Geometer’s Sketchpad, Cabri, Jr., or GeoGebra. Have students work in pairs to create a triangle, construct the medians of all three sides, and label. (Teachers can use the Investigating the Centriod of a Triangle Tutorial podcast as a reference.)
3. Give students 10-15 minutes to measure the side lengths and test their conjectures. (Look for evidence of MP2, MP3, and MP5.)
4. Next, have students share out their findings. Teachers should address any misconceptions.
5. Inform students that they will be playing “Two Minutes to Win It!” Provide students with an index card, straightedge, and scissors. Tell students they will have two minutes to draw and cut out a triangle and then figure out how to balance the triangle on a pencil tip. Once they have done it, they must also mark the approximate “balancing point” on the triangle. (Look for evidence of MP6.)
6. After the two minutes are up, give students two minutes to construct the centroid of their triangle, using only their triangle and a straightedge. Then, have them determine the relationship between their “balancing point” and the centroid. (It should be approximately the same point.) Share with students that the centroid is also referred to as the “balancing point” of a triangle.
Evidence of Success: What exactly do I expect students to be able to do by the end of the lesson, and how will I measure student success? That is, deliberate consideration of what performances will convince you (and any outside observer) that your students have developed a deepened and conceptual understanding.
Students should be able to construct the centroid of a triangle and define median.
Students should also be able to use dynamic geometry software to test their conjectures about the centroid of a triangle.
Students should also be able to articulate the properties of the centroid of the triangle.
Notes and Nuances: Vocabulary, connections, anticipated misconceptions (and how they will be addressed), etc.
Vocabulary: points of concurrency, median, centroid, midpoint
Students may have different ways of stating the properties of the centroid of the triangle, but give students opportunities to see the connections between the different ways of stating the same theorem.
Resources: What materials or resources are essential for students to successfully complete the lesson tasks or activities?
Dynamic geometry software
Medians of a Triangle Sketchpad Example
Investigating the Centriod of a Triangle Tutorial
Homework: Exactly what follow-up homework tasks, problems, and/or exercises will be assigned upon the completion of the lesson?
Have students create examples of problems involving finding different segments in a triangle, applying the centriod theorem they investigated. Be sure to have students include an answer key.
Optional Activity: The next day, give students time to solve each other problems as a warm up activity.
Lesson Reflections: How do you know that you were effective? What questions, connected to the lesson standards/objectives and evidence of success, will you use to reflect on the effectiveness of this lesson?
Are students able to identify the point of concurrency of the medians as the centroid?
Are students able to use dynamic geometry software to test their conjectures about the centroid?
Are students able to articulate the theorem about the centroid?
Howard County Public Schools Office of Secondary Mathematics Curricular Projects has licensed this product under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.