Download Keeping Triangles in Balance: Discovering Triangle

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Resource ID#: 130874
Keeping Triangles in Balance: Discovering Triangle
Centroid is Concurrent Medians
In this lesson, students identify, analyze, and understand the Triangle Centroid Theorem. Students discover that the centroid is the point of
concurrency for the medians of a triangle and recognize its associated usage with the center of gravity or barycenter. This set of instructional
materials provides the teacher with hands-on activities using technology as well as paper-and-pencil methods.
Subject(s): Mathematics, Science
Grade Level(s): 9, 10, 11, 12
Intended Audience: Educators
Suggested Technology: Document Camera,
Computer for Presenter, Computers for Students,
Interactive Whiteboard, LCD Projector, Overhead
Projector, Microsoft Office, GeoGebra Free Software
(Download the Free GeoGebra Software)
Instructional Time: 45 Minute(s)
Resource supports reading in content area: Yes
Keywords: centroid, triangle, median, concurrent, geometry, algebra, midpoints of a segment, vertex, ratio,
barycenter, center of gravity, similarity, polygon, angle
Resource Collection: FCR-STEMLearn Geometry
ATTACHMENTS
CentroidGeogebraWorksheet.docx
CentroidTriangleConstruction.pptx
QuiaSpecialTrianglesVocabularyQuiz.docx
QuiaPropertiesOfTriangleCentroidActivity.docx
Triangles3totalMedianWorksheet.docx
Triangles6totalMedianFAS.docx
LESSON CONTENT
Lesson Plan Template: General Lesson Plan
Learning Objectives: What should students know and be able to do as a result of this lesson?
1. Students will discover that a centroid is the center of gravity for a triangle.
2. Students will be able to:
a. Identify a median.
b. Construct the median of a triangle.
c. Identify that medians of a triangle are concurrent and this point of concurrency is called the centroid.
d. Measure parts of the median as partitioned by the centroid as having a ratio of 2:1.
e. Determine that the longer part of the median (partitioned by the centroid) is 2/3 the length of the median.
Prior Knowledge: What prior knowledge should students have for this lesson?
Students must:
page 1 of 4 be familiar with the concept of a median.
able to use a ruler/straight-edge for basic length measure and drawing lines.
recognize basic geometric shapes such as triangles, quadrilaterals, and polygons.
identify the median of a triangle as a segment connecting the midpoint of a side to the opposite vertex.
Guiding Questions: What are the guiding questions for this lesson?
1. What is a median in geometry and how can medians be used in real-life?
2. How can medians be found within triangles?
3. How can the medians of a triangle create other smaller triangles, and what relationships do these triangles have to each other?
4. How can we define the centroid of a triangle using vertices, medians, ratios, and concurrency?
5. Can we see any physical properties of the triangle centroid when we balance a triangle on the tip of a pencil?
Teaching Phase: How will the teacher present the concept or skill to students?
1. To start the class with a warm-up, the teacher posts the link to the online Quia quiz (http://www.quia.com/quiz/5383625.html; also attached as a Word document)
for a vocabulary review.
There is a total of five questions with four multiple answers (8 total choices) from an interactive drop-down menu.
The time range for the warm-up is 5 to 10 minutes; this includes the students taking the quiz with an option for a retry, and a review of the five questions.
Review of the quiz is interactive, with students volunteering solutions and explanations.
Note that a short answer paper copy is available in case of computer issues.
2. To begin the lesson, the teacher passes out the "Triangles 2 total Median Worksheet" with three different types of triangles (Acute, Right, and Obtuse) to students
along with a straight-edge ruler.
Students are asked to draw the three medians for each side of the triangle.
This activity has a duration of 5 to 10 minutes.
The teacher walks around the classroom asking students about the properties of the medians.
3. The teacher asks students:
"Did you draw all three medians?"
"What do you notice about the intersection of the three medians?"
"Who has medians that do not intersect?" (No student should respond to this question because all medians intersect inside the triangle.) The teacher tells
students that this point of concurrency is called the centroid.
4. The teacher asks students to measure the median and the two parts as divided by thecentroid.
The teacher then asks students to calculate the ratio of the median segments from the vertex to the intersection to the opposite side.
Students are also asked to determine the ratio of the length from the intersecting point to the vertex versus the midpoint on the side.
This duration of this activity is 5 minutes.
5. Lastly, the teacher asks students if the intersection of the medians is always inside the triangle or if it is possible for it to lie outside the triangle.
Students should be able to see that the point of concurrency is always inside the triangle. (Compare the work of other students).
The duration of this activity is 5 minutes.
Guided Practice: What activities or exercises will the students complete with teacher guidance?
1. Students complete a Quia online quiz (http://www.quia.com/quiz/5383625.html; also attached as a Word document) - Special Triangle Vocabulary, consisting of 5
questions, four multiple answers and one true/false as a vocabulary review and warm up.
2. Students complete a worksheet identifying three triangle medians Worksheet using different triangle types.
3. Students complete a GeoGebra construction of a triangle, median, and a centroid with the presentation. The teacher will follow the PowerPoint presentation slides
5-12.
4. Students complete a Quia online geometry activity(http://www.quia.com/pop/587255.html) - Properties Triangle Centroid- 10 multiple answer questions.
5. The teacher distributes cardboard triangles.
Students find the medians, identify the centroid, and then use the centroid to balance the triangle their pencils.
See PowerPoint slide 14.
Note: The teacher will now give the formative assessment "Triangles 6 total Median FAS" to have students find the ratio of the median segmented by the point of
concurrency- centroid.
Independent Practice: What activities or exercises will students complete to reinforce the concepts and skills developed in the
lesson?
Students provide their own example of a triangle, with medians and point of concurrency or centroid. Students sketch it, cut it out of paper, or use GeoGebra to
demonstrate mastery of the centroid of a triangle.
Closure: How will the teacher assist students in organizing the knowledge gained in the lesson?
Students are asked to answer the following questions in a whole-group discussion format:
"What is a median in geometry? How can medians be found in triangles?"
"How can medians be used in real-life?"
Students are asked to describe the similarities and differences between the median of a set of numbers (data set) versus the median in geometry.
Summative Assessment
Students complete the Quia quiz "Properties of Triangle Centroid."
This Quia quiz contains 10 questions with multiple answer selections on the concepts, properties, and vocabulary of a triangle centroid.
Questions are multiple choice via an interactive drop-down menu.
A hard-copy is provided to use as a short answer assessment.
Students have an option to retake the quiz to improve their score.
Formative Assessment
1. Students find the medians of different types of triangles (scalene, isosceles, equilateral, acute, right, and obtuse).
2. There is a total of 6 triangles, and the students have to find the three medians for each triangle.
3. Walking around the classroom and interacting with students during this activity teacher will query students on the properties of the triangle medians and how they
relate to each other.
page 2 of 4 4. Students are asked to find the intersection of the 3 medians. The teacher will emphasize that this intersection is the point of concurrency known as the centroid.
5. The teacher asks students to measure the distance from the centroid to the vertex and compare this to the distance from the centroid to the opposite side for the
same median segment.
6. The teacher will query the students on what ratio is formed from these two distances. Students can respond if they have found a 2:1 ratio representing a fractional
length of 2/3 from the centroid to vertex and 1/3 from the centroid to median.
7. The teacher summarizes how the ratio of the two segments of the median forms a 2:1 ratio and represent 2/3 and 1/3 of the total median length.
Feedback to Students
1. The teacher aides students in distinguishing between the different types of triangles, for example, the difference between an acute and obtuse triangle.
2. The teacher ensures students describe different types of triangles using appropriate vocabulary. For example, triangles are classified as scalene, isosceles, or
equilateral according to side length relationships and acute, right, or obtuse based on angle measures.
If students are not using the proper naming convention for the triangle, the teacher draws a diagram and explains the properties that make up the triangle
definition.
3. The teacher helps students with misconceptions about medians or difficulty of finding the median.
4. Students are provided with positive encouragement and guidance by the teacher to ensure the correct methods and procedures for constructing medians and
thecentroid.
For example if students cannot find the medians, the teacher draws a diagram of a triangle similar to the triangle the student is modeling, then demonstrates
the proper technique.
5. Formative assessment feedback:
If students have difficulty with constructing the medians:
The teacher guides the student to finding the midpoint of the line, and draw a line from the midpoint to the opposite vertex.
If students are having difficulty with finding the centroid
The teacher explains how the three medians intersect which is called the point of concurrency.
If students are having problems forming a 2:1 ratio of the distances from thecentroid to vertex and opposite side
The teacher explains how the distance from the centroid to the vertex is twice that of the distance from the centroid to the opposite side.
If students do not understand that the sum of the two median segments, as delineated by thecentroid, are fractionally represented as 2/3 + 1/3 = 1 or median
(total distance).
The teacher explains that the ratio of the lengths is 2:1 and fractional total 1 and this as 2/3 + 1/3.
ACCOMMODATIONS & RECOMMENDATIONS
Accommodations:
Students will work in pairs, taking turns at the computer and note taking. Students with learning challenges will receive extra help from the teacher. To help students
challenged by these practice exercises, the teacher can provide additional time with less practice exercises.
Extensions:
As an extension to this lesson, there is an activity using GeoGebra for Napoleon's triangle.
1. Launch GeoGebra to create a new worksheet.
a. Construct a triangle of any type.
b. Construct equilateral triangles from each side of the original triangle created in step 2.
c. The centers of the three equilateral triangles (from step 3) form an equilateral triangle.
2. Centroid of a Triangle as shown in the PowerPoint, slides 16/17.
a. Students can manipulate the GeoGebra dynamic worksheet.
b. Students can come to conclusion about ratio of median segments.
c. Students can see how the ratio of the median segments for all medians are congruent.
d. Students, by manipulating the triangle, can understand how the centroid changes.
Suggested Technology: Document Camera, Computer for Presenter, Computers for Students, Interactive Whiteboard, LCD Projector, Overhead Projector, Microsoft
Office, GeoGebra Free Software
Special Materials Needed:
Windows or Apple Personal Computer (PC), Internet connection with Mozilla Firefox or equivalent Web Browser.
GeoGebra loaded and configured properly to run on computers.
Overhead or document projector for teacher visual aid, interactive whiteboard if available for student involvement in the lesson.
Rulers, scissors, and pencils are needed for the construction activities.
Cut-out cardboard triangles to be handed out to students to find centroid balances with a pencil activity.
Further Recommendations:
Prior to the lesson:
1. The classroom teacher can introduce students toGeoGebra before this lesson. This prior introduction to GeoGebra ensures students are comfortable using this
online software application.
In addition to student familiarity with GeoGebra, having students use laptops before lesson provides that these computers are on the Internet working properly
with the GeoGebra software application.
2. The classroom teacher reviews with students solved problems involving triangles. This review includes the midpoint and triangle congruence postulates (SSS, SAS,
ASA, AAS).
3. Additionally teacher ensures students are familiar with the concepts of inductive and deductive thinking, and parallel lines with corresponding, alternate, and
consecutive angles.
page 3 of 4 Additional Information/Instructions
By Author/Submitter
This lesson aligns with the following Standards for Mathematical Practice:
MAFS.K12.MP.1.1 - Make sense of problems and persevere in solving them.
MAFS.K12.MP.2.1 - Reason abstractly and quantitatively.
SOURCE AND ACCESS INFORMATION
Contributed by: Norman Ebsary
Name of Author/Source: Norman Ebsary
District/Organization of Contributor(s): Broward
Access Privileges: Public
License: CPALMS License - no distribution - non commercial
Related Standards
Name
MAFS.912.G-CO.3.10:
SC.912.N.3.5:
Description
Prove theorems about triangles; use theorems about triangles to solve problems. Theorems include: measures of
interior angles of a triangle sum to 180°; triangle inequality theorem; base angles of isosceles triangles are congruent;
the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a
triangle meet at a point.
Describe the function of models in science, and identify the wide range of models used in science.
Remarks/Examples:
Describe how models are used by scientists to explain observations of nature.
Florida Standards Connections: MAFS.K12.MP.4: Model with mathematics.
page 4 of 4