Primary Type: Lesson Plan
Status: Published
This is a resource from CPALMS (www.cpalms.org) where all educators go for bright ideas!
Resource ID#: 131876
Partitioning a Segment
In this lesson, students find the point on a directed line segment between two given points that partitions the segment in a given ratio.
Subject(s): Mathematics
Grade Level(s): 9, 10, 11, 12
Intended Audience: Educators
Suggested Technology: Computer for Presenter,
Computers for Students, Interactive Whiteboard, Basic
Calculators, GeoGebra Free Software (Download the
Free GeoGebra Software)
Instructional Time: 1 Hour(s) 30 Minute(s)
Keywords: partition
Resource Collection: FCR-STEMLearn Geometry
ATTACHMENTS
Partitioning a Segment - Feedback to Students Example Question.docx
Partitioning a Segment - Guided Practice Example Questions.docx
Partitioning a Segment - Worksheet.docx
Partitioning a Segment - Homework.docx
Partitioning a Segment - Quiz.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?
Students will be able to determine:
a point on a directed line segment between two given points (endpoint) and a given ratio
an endpoint when given a ratio, one endpoint and a point on the directed line segment
a ratio when given the endpoints (both) and a point on the directed line segment
Prior Knowledge: What prior knowledge should students have for this lesson?
Students should be able to:
perform operations on rational numbers
create and demonstrate understanding of ratios
plot coordinates on a rectangular coordinate plane
solve linear equations including those with variables as coefficients
Guiding Questions: What are the guiding questions for this lesson?
Why is the labeling of the coordinates and their order important when using the formula for partitioning a segment with a ratio different than 1:1?
How are the two formulas listed here used and what are the similarities and differences between them?
page 1 of 3 Formulas for Partitioning a Segment:
Ratio 1:1 (It is the midpoint of the line segment.)
Midpoint Formula is
Ratio a:b (Partition a segment with a ratio other than 1:1)
Formula is
Teaching Phase: How will the teacher present the concept or skill to students?
The teacher will introduce new vocabulary.
Partition: a part, division, section
Ratio: comparison of two quantities
The teacher will explain the difference between partition with a ratio of 1:1 (determining the midpoint) and partition with a different ratio of a:b (point on direct line
segment).
The teacher will explain how to transform a fraction that lies on a directed line segment into a ratio and vice-versa.
To change a fraction to a ratio, the numerator must be the first number in the ratio (a), and the difference between the denominator and the numerator is the second
number of the ratio (b).
Example: 2/3 is a fraction. It has to be converted to a ratio 2:(3-2). Answer is 2:1
To change a ratio to a fraction, (that is only necessary when you use GeoGebra) the first number in your ratio (a) will be the fraction's numerator, and the sum of both
numbers in your ratio (a+b) will be the fraction's denominator.
Example: 4:7 converted to a fraction will be 4/(4+7). Answer is 4/11
Guided Practice: What activities or exercises will the students complete with teacher guidance?
The teacher will explain on the board six examples:
an example of how to find a point on a directed line segment between two given endpoints with ratio 1:1 (midpoint)
an example of how to find a point on a direct line segment between two given points with a ratio a:b
how to find an endpoint when given a ratio, one endpoint and a point on the directed line segment
how to find a ratio given both endpoints and a point in between on the directed line segment
how to find a partition point using dilation with the free GeoGebra software
All examples from the guided practice are attached. Students will do all of the examples in their notebook to keep as studying material. Problems number 1, number 2
part a and number 3 part b is expected to be answered by students individually. The teacher will create a class discussion about the difference between problem
number 3 part a and problem number 3 part b. Problems number 4 and 5 are more difficult, so the teacher will explain in detail, step by step, how to complete these
problems on the board. To do problem number 6, the teacher will use the interactive whiteboard and GeoGebra software on the computer to show the students how
to find the answers and then to check that they determined the correct answer algebraically.
Partitioning a Segment - Guided Practice Example Questions is attached.
Independent Practice: What activities or exercises will students complete to reinforce the concepts and skills developed in the
lesson?
Independent practice will be a home-learning handout (attached) with five questions addressing the lesson objectives. After the students do the five problems using
the formulas they have to redo problems 1 and 2 using GeoGebra software as a tool to check the answer that they attained algebraically. To use GeoGebra they have
specific instructions on the homework worksheet attached and also they could refer to problem number six in the guided practice, which is in their notebooks where
the teacher explained how to use GeoGebra.
Partitioning a Segment - Homework is attached.
Closure: How will the teacher assist students in organizing the knowledge gained in the lesson?
The teacher reminds students to organize their work in steps as follows:
1. Write the Midpoint Formula
Or
The formula to find the coordinates of a point that lies along the directed line segment and partitions the segment in the ratio of a to b
2. Label points and ratio
3. Substitute values
4. Solve
Summative Assessment
Students will complete a partitioning line segment quiz that covers the three objectives of this lesson.
Partition a Segment - Quiz is attached.
page 2 of 3 Formative Assessment
The teacher will monitor and assist the students while they complete a partitioning line segment worksheet. The teacher will verbally ask the students questions such
as,
How do you know when they're asking you to find the midpoint or any other point that partitions the line segment?. Answer: Keywords for to find midpoint: halfway,
midway, or ratio 1:1. Keywords for any other point: any ratio a:b, different from 1:1.
What information do you need to find a point that partitions a line segment? Answer: The coordinates of the two endpoints and the ratio that partitions the line
segment.
What information do you need to find the endpoint of the line segment? Answer: One of the endpoint coordinates, the coordinates of the point that partitions the
line in a specific ratio and its ratio.
What information do you need to find the ratio? Answer: The coordinates of the two endpoints and the coordinates of the point that partitions the line.
Partitioning a Segment - Worksheet is attached.
Feedback to Students
A selected student will solve one problem with parts a and b on the board to demonstrate to the class why ordering and labeling the coordinates is important to get
the correct answer. After the student gets the answer on the board, the teacher will ask the class to look at answers letter a and letter b to identify whether the
answers are the same or different. The answers are different because the order of the coordinates matter. The teacher will also ask "What would happen if the ratio
is 1:1 (midpoint)?" To find the midpoint the order of the coordinates does not matter.
Partitioning a Segment - Feedback to Students Example Question is attached.
ACCOMMODATIONS & RECOMMENDATIONS
Accommodations:
Students should be working in cooperative learning groups to maximize student learning.
Provide visual aides by drawing the directed line segments.
Check progress and provide timely feedback on regular basis.
Work one-on-one with student, if needed.
Increase wait time for responses.
Allow extended time to complete quiz.
Suggested Technology: Computer for Presenter, Computers for Students, Interactive Whiteboard, Basic Calculators, GeoGebra Free Software
Special Materials Needed:
Graph paper, straightedge, and computer/tablet with GeoGebra software.
Further Recommendations:
The teacher should reinforce that when using formulas, students need ratios to find the point that partitions the line, and when using GeoGebra they need to convert
the ratio to a fraction. Review how to convert one to the other. It is very important that when the students are solving partitioning line segment problems that the
teacher emphasises the direction of the vector. That means, identify which one is the starting point and the end point of the line segment. A common mistake is to not
label or confuse the points and thus arrive at an incorrect answer.
Additional Information/Instructions
By Author/Submitter
This lesson aligns with the following Standards for Mathematical Practice:
MAFS.K12.MP.2.1 - Reason abstractly and quantitatively
MAFS.K12.MP.5.1 - Use appropriate tools strategically
SOURCE AND ACCESS INFORMATION
Contributed by: Carmen Concha
Name of Author/Source: Carmen Concha
District/Organization of Contributor(s): Miami-Dade
Access Privileges: Public
License: CPALMS License - no distribution - non commercial
Related Standards
Name
MAFS.912.G-GPE.2.6:
Description
Find the point on a directed line segment between two given points that partitions the segment in a given ratio.
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