TPACK Lesson: Triangles

Title: Triangles

Subject Area: Math II – Geometry

Grade Level: 9-12

Learning Objectives:

Students will be able to identify triangles by their sides and angles.

Students will be able to name vertices of a triangle, as well as name triangles by their

vertices.

Students will examine why the angle measures in any triangle add up to 180° and find

the missing angle(s) in any triangle.

Students will be able to find the length of a midsegment given the base, or find the

length of the base given the midsegment.

Students will be able to make conjectures about isosceles triangles.

Students will be able to prove the Exterior Angle Theorem.

Essential Questions:

When is a triangle equilateral, isosceles or scalene?

When is a triangle acute, obtuse, or right?

What do the angles in a triangle add up to? Justify your answer.

How can you find missing angles? (Which theorems or definitions)

Common Core State Mathematical Standards:

F-BF.1 Write a function that describes a relationship between two quantities.

a. Determine an explicit expression, a recursive process, or steps for calculation from a

context.

G-CO.10 Prove theorems about triangles.

Common Core State Mathematical Practice Standards:

MP1 Make sense of problems and persevere in solving them.

MP4 Model with mathematics.

MP7 Look for and make use of structure.

Technology Standards:

HS.SI.1 Evaluate resources needed to solve a given problem.

HS.TT.1 Use technology and other resources for assigned tasks.

a. HS.TT.1.1 Use appropriate technology tools and other resources to access

information (multi-database search engines, online primary resources, virtual

interviews with content experts).

b. HS.TT.1.2 Use appropriate technology tools and other resources to organize

information (e.g. online note-taking tools, collaborative wikis).

HS.SE.1 Analyze issues and practices of responsible behavior when using resources.

Necessary Materials:

Geometer’s Sketchpad (GSP – teacher only)

Paper (students only)

Pencil (students only)

Mimeo/Smartboard

Classwork

Homework

Prior Knowledge:

Students have knowledge from previous classes on how to solve one- and two-step equations,

and should be familiar with what a triangle is. Students should have also learned about angles

(acute, obtuse, right, etc.)

Time: 90 minutes

Triangles

**This document supplements the GSP file – the purple text in GSP is to be written as the

lesson progresses.

Vocabulary

The following are ways to classify triangles by sides and angles:

Sides:

Equilateral Triangle: all three of its sides are equal in length and all angles have the

same measure

Isosceles Triangle: a triangle with two equal sides

Scalene Triangle: three sides are different in length and has three different angle

measures

Angles:

Equiangular Triangle: a triangle is equilateral if and only if the triangle is equiangular

(equilateral ↔ equiangular)

Right Triangle: has one right angle (90°)

Acute Triangle: all internal angles’ measures are acute (less than 90°)

Obtuse Triangle: has one internal angle whose measure is greater than 90° (obtuse)

Vertex (vertices): the point where two sides meet, usually denoted by the three letters

on the shape that define the angle, with the middle letter being where the angle

actually is (its vertex). E.g. ABC

Ex)

Naming a triangle: a triangle is named by its vertices. In the previous example we had

triangle ABC

((Examples in GSP)) – Lesson questions 1 and 2.

Angle Sum Theorem: the angle measures in any triangle add up to 180°

- Why this works: the three angles of a triangle make a straight line when put side to

side.

- ((Demo and examples)) – Lesson question 3.

Midsegment Theorem: a midsegment of a triangle is parallel to the third side and is half as long.

((Demo and examples)) – Lesson question 4.

Ex) If D is the midpoint of 𝐴𝐶 and E is the midpoint of 𝐵𝐶

Then 𝐴𝐵 II 𝐷𝐸 and 𝐷𝐸=½𝐴𝐵

Isosceles Triangle Theorem: a triangle is isosceles if and only if the angles opposite the

congruent sides are congruent. ((Example)) – Lesson question 5.

Exterior Angle Theorem: the measure of an exterior angle of a triangle is equal to the sum of

the measures of the two opposite interior angles.

I.e. m BCD=mABC+m BAC

A linear pair is a pair of adjacent angles that form a straight line, and are thus

supplementary. The measure of a straight angle is 180°, so a linear pair of angles must

add up to 180°.

o Two angles are supplementary if they add up to 180°.

The midpoint is halfway

between the two end points.

1 2

3

4

Vertical angles are formed when two lines intersect forming four angles; each opposite

pair are vertical angles. Vertical angles are always congruent.

Lesson question 6.

Exterior Angle Theorem Proof:

Statement Reason

1. ABC is a triangle Given

2. m 2+m 3+m 4 = 180° Triangle Sum Theorem

3. 1 and 2 form a linear pair Definition of a linear pair.

4. 1 and 2 are supplementary If two angles form a linear pair, they are supplementary.

5. m 1+m 2 = 180° Definition of supplementary angles.

6. m 1+m 2 = m 2+m 3+m

4

Substitution

7. m 1 = m 3+m 4 Subtraction

((Example))

Lesson question 7.

Kahoot: https://play.kahoot.it/#/k/ee7088d8-722a-4021-9a17-1ec7b92bb49c

Classwork (to be finished for homework if not finished in class):

Students should show all of their work if an equation is involved

Find 1-4 and name the triangle

by its sides and angles.

1+ 3=123°

2+ 3=92°

1+ 2= 4

1+ 2+ 3=180°

Homework:

Keys

Questions for Students

1. If an analog clock is reading 3:05, what type of triangle is formed by the hour and

minute hands?

2. How you would determine the name of the following triangles using their sides and

angles?

3. Find the missing angle measure.

4. If 𝐶𝐴 measures 18cm, what is the measure of its midsegment?

5. If ∠CAB is 30° (the vertex angle), find the base angles.

6. How would you prove the Exterior Angle Theorem?

a. What other theorems, postulates or definitions might you use?

7. Find angles 1-4 and name the triangle by its sides and angles.

a. Explain how you found each angle.

b. Are there different approaches to solving this problem?

In addition to the Kahoot questions.

Instructional Design Plan (IDP)

Content Describe: content here

F-BF.1 Write a function that describes a relationship between two

quantities.

a. Determine an explicit expression, a recursive process, or steps for

calculation from a context.

G-CO.10 Prove theorems about triangles.

Describe: Standards of Mathematical Practice

MP1 Make sense of problems and persevere in solving them.

MP4 Model with mathematics.

MP7 Look for and make use of structure.

Pedagogy 1. Describe the instructional strategy (method) appropriate for the content,

the learning environment, and students. This is what the teacher will

plan and implement.

This lesson implements direct, whole group instruction using technology

integrating active participation.

The teacher will also use formative assessment and independent practice.

2. Describe what the learner will be able to do, say, write, calculate, or solve

as the learning objective.

Students will be able to identify triangles by their sides and angles.

Students will be able to name vertices of a triangle, as well as name

triangles by their vertices.

Students will examine why the angle measures in any triangle add up

to 180° and find the missing angle(s) in any triangle.

Students will be able to find the length of a midsegment given the

base, or find the length of the base given the midsegment.

Students will be able to make conjectures about isosceles triangles.

Students will be able to prove the Exterior Angle Theorem.

3. Describe how creative thinking, critical thinking, or innovative problem

solving is reflected in the content.

The problems that students are asked to solve require them to thinking

critically about how they would reach the answer. Students are not given

information in the same way every time, they must be critical of what is

given and develop a way to solve the problem. They must incorporate new

knowledge with prior knowledge in order to come to the right answer.

Technology 1. Describe the technology.

For this lesson I used Geometer’s Sketchpad (GSP). GSP is a dynamic

geometry software that allows students to see relationships among figures

or variables in real time. It enables the teacher to construct and

manipulate the triangles to show students that, for example, an isosceles

triangle isn’t always going to look the same way; i.e. the base (congruent)

angles won’t always be on the bottom of the figure.

I also used Kahoot, which is an interactive formative classroom assessment

tool. It is presented to students as a game, but can be used as a quiz, as it

allows the teacher to see who missed which questions.

2. Describe how the technology enhances the lesson, transforms content,

and/or supports pedagogy.

The use of Geometer’s Sketchpad enhances the lesson because it allows

the students and teacher to collectively explore multiple cases. With ease,

the teacher can create a blank page, construct a new triangle, or simply

change an old triangle by dragging points for students to investigate

concepts. GSP also makes it effortless to calculate sums or measures of

angles, which would not be the case without this technology. Kahoot

enhances the lesson by allowing the teacher to formatively assess students

in a way that students see as enjoyable. It is also quicker (typically) to

administer and much quicker to grade.

GSP transforms the content of the lesson by making it easier for students

to confirm conjectures or visualize the proof and how it works. This is done

through the use of multiple representations.

The pedagogy is supported by Geometer’s Sketchpad because it allows the

teacher to provide an interactive lecture while showing students the many

ways that GSP can be used. This type of engaged pedagogy enhances

student learning. Kahoot supports the pedagogy by enabling the teacher

to do a formative assessment on the spot. The teacher can then assess

student understanding of the content with ease.

3. Describe how the technology affects students’ thinking processes.

The use of Geometer’s Sketchpad enables students to see more than just a

static image of a triangle or angles. Students are able to visualize the

properties of and differences among triangles, as well as the theorems.

This can be very powerful for students that are skeptical; e.g. the teacher

can construct a triangle and use the angle measure, calculate and drag

features to show that the measure of all three angles will always equal

180°.

The game-like structure of Kahoot makes students think on their feet and

utilize newly-acquired knowledge without the fear of getting the wrong

answer.

Reflection I think that using Geometer’s Sketchpad and Kahoot certainly enhanced the

lesson and were the right technologies to use. The teacher is able to

dynamically explore triangles with the students by manipulating various

constructions. This allows students to see multiple representations at once,

cementing the concepts they are learning. GSP also allows the teacher to

construct a triangle and measure its sides and angles quickly, saving time for

other activities. It also enables students to visualize properties and theorems

by using demonstrations in conjunction with action buttons. It would take a

lot of time to find side lengths and angle measures to prove theorems and

properties to students. The use of Kahoot allowed the teacher to give a timed

formative assessment without the added stress on students. The teacher is

able to see students’ answers on each question and find students’

misconceptions, errors, and gaps in knowledge. Without these technologies,

the lesson would have to be broken up into two lessons in order to cover the

same amount of material and examples. Also, if a standard paper-and-pencil

quiz was given to students, it would take a lot longer for students to complete.

References

Kuta Software:

Triangle angle sum worksheet - http://cdn.kutasoftware.com/Worksheets/Geo/4-

Angles%20in%20a%20Triangle.pdf

Exterior Angle Theorem - http://cdn.kutasoftware.com/Worksheets/Geo/4-

The%20Exterior%20Angle%20Theorem.pdf

Mrs. Roach from Bear Creek High School, Lakewood, CO

4-5 Enrichment -

http://meroach.wikispaces.com/file/view/4.5+Isosceles+and+Equilateral+Triangles+WS

+solutions.pdf