Lesson 3 | Polygons and Algebraic Relationships | 10th Grade Mathematics

Objective

Divide a line segment on a coordinate plane proportionally and identify the point that divides the segment proportionally.

Common Core Standards

Core Standards

The core standards covered in this lesson

G.GPE.B.6 — Find the point on a directed line segment between two given points that partitions the segment in a given ratio.

Expressing Geometric Properties with Equations

G.GPE.B.6 — Find the point on a directed line segment between two given points that partitions the segment in a given ratio.

Foundational Standards

The foundational standards covered in this lesson

7.RP.A.2

Ratios and Proportional Relationships

7.RP.A.2 — Recognize and represent proportional relationships between quantities.

Criteria for Success

The essential concepts students need to demonstrate or understand to achieve the lesson objective

Describe that to divide a directed line (that is not horizontal or vertical), the vertical and horizontal components of the line segment need to be partitioned separately.
Describe that right triangle legs are formed when horizontal and vertical distances are divided in the desired proportion.
Identify the hypotenuse of the partitioned horizontal and vertical distances as one part of the partitioned line segment.
Describe that by partitioning the vertical and horizontal components in the desired proportion, you will be partitioning the segment itself in that proportion.
Verify that a line segment is partitioned in the correct proportion by finding the distance between the endpoints of the partition.
Describe the relationship between the distance formula and the process for partitioning a line segment into a desired proportion.
Name a coordinate that is a certain proportion away from one of the endpoints. Verify this distance algebraically.

Tips for Teachers

Suggestions for teachers to help them teach this lesson

This lesson is an extension of Lesson 2. We are focusing here on the division of a line segment on the coordinate plane rather than the number line (Lesson 2) or using constructions (Unit 3, Lesson 4 and 6).

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Anchor Problems

Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding

Problem 1

Plot the points $${A(-4,5) }$$ and $${B(12,13)}$$.

Find the midpoint of this segment.

References

EngageNY Mathematics Geometry > Module 4 > Topic D > Lesson 12 — Opening Exercise

Geometry > Module 4 > Topic D > Lesson 12 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds. © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US license. Accessed Dec. 2, 2016, 5:15 p.m..

Modified by Fishtank Learning, Inc.

Problem 2

Find the point on the directed line segment from $${C(-2,0) }$$ to $${ D(5,8)}$$ that divides the line segment in the ratio of $${1:3}$$. Show your reasoning. Then, graph the points to show your reasoning using the graph.

Guiding Questions

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References

EngageNY Mathematics Geometry > Module 4 > Topic D > Lesson 12 — Exercise #2

Modified by Fishtank Learning, Inc.

Target Task

A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved

Problem 1

Given points $${ A(3,-5)}$$ and $${B(19,-1)}$$, find the coordinates of point $$C$$ that sit $${{3\over8}}$$ of the way along $${\overline{AB}}$$, closer to $$A$$ than to $$B$$.

References

EngageNY Mathematics Geometry > Module 4 > Topic D > Lesson 12 — Exit Ticket, Question #1

Problem 2

Given points $${ A(3,-5)}$$ and $${ B(19,-1)}$$, find the coordinates of point $$C$$ such that $${CB\over AC}={1\over7}$$.

References

EngageNY Mathematics Geometry > Module 4 > Topic D > Lesson 12 — Exit Ticket, Question #2

Additional Practice

The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.

Dan Meyer's Three-Act Math Best Midpoint — (Focus on: What information do you have in Act 1? What do you think you need to know in order to determine which is the best square? How will these features help you distinguish a square from any other polygon?)
New Visions for Public Schools Seeing Segments — (Focus on Lesson 2)
Open Middle Equidistant Points
Open Middle Equidistant Points 2
GeoGebra CCSS High School: Geometry (Expressing GEO. Prop's w/Equns.) — HSG.GPE.B.6 "Partition Line Segments"
Mathematics Vision Project: Secondary Mathematics Two Module 6: Similarity and Right Triangle Trigonometry — Lesson 6.6
EngageNY Mathematics Geometry > Module 4 > Topic D > Lesson 12

Lesson 2

Lesson 4

Lesson Map

Topic A: Distance on the Coordinate Plane

Use the Pythagorean Theorem to calculate distance on a coordinate plane. Develop a formula for calculating distances.

G.GPE.B.7

Partition horizontal and vertical line segments into equal proportions on a number line.

G.GPE.B.6

Divide a line segment on a coordinate plane proportionally and identify the point that divides the segment proportionally.

G.GPE.B.6

Topic B: Classify Polygons using Slope Criteria and Proportional Line Segments

Algebraically and using the Pythagorean Theorem, determine the slope criteria for perpendicular lines.

G.GPE.B.5 G.SRT.C.8

Describe and apply the slope criteria for parallel lines.

G.GPE.B.5

Identify and create parallelograms, rectangles, rhombuses, and squares on the coordinate plane.

G.GPE.B.4

Algebraically verify midsegment, median, and parallel line relationships in triangles.

G.CO.C.10 G.SRT.B.4

Algebraically verify diagonal relationships in quadrilaterals and parallelograms.

G.CO.C.11 G.GPE.B.4

Topic C: Area and Perimeter On and Off the Coordinate Plane

Calculate and justify the area and perimeter of polygons and composite shapes off the coordinate plane.

G.GPE.B.7

Calculate and justify the area and perimeter of parallelograms and triangles on the coordinate plane.

G.GPE.B.7

Calculate and justify composite and irregular areas on the coordinate plane.

G.GPE.B.7 N.Q.A.3

Describe how the area changes when a figure is scaled.

G.GPE.B.7

Solve area applications by creating and solving quadratic equations.

A.CED.A.1

Describe a polygon on the coordinate plane using a system of inequalities.

A.CED.A.3 G.CO.A.2

Calculate and justify the area and perimeter of polygons on the coordinate plane given a system of inequalities.

A.REI.D.12 G.GPE.B.4 G.GPE.B.7

Polygons and Algebraic Relationships

Objective

Common Core Standards

Core Standards

Expressing Geometric Properties with Equations

Foundational Standards

Ratios and Proportional Relationships

Criteria for Success

Tips for Teachers

Anchor Problems

Problem 1

References

Problem 2

Guiding Questions

References

Target Task

Problem 1

References

Problem 2

References

Additional Practice

Lesson Map

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