MATHEMATICAL GOALS
This lesson unit is intended to help you assess how well students are able to determine if a triangle can be formed from given conditions. It will help you to identify and support students who have difficulty

Using lengths of sides to determine if a triangle can be formed.

Determining when angles and sides create multiple triangles.

Recognizing degenerate cases.
GEORGIA STANDARDS OF EXCELLENCE
This lesson involves mathematical content in the standards from across the grades, with emphasis on:

MGSE.7.G.2 Explore various geometric shapes with given conditions. Focus on creating triangles from three measures of angles and/or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.
Standards for Mathematical Practice
 SMP1 - Make sense of problems and persevere in solving them.

SMP3 - Construct viable arguments and critique the reasoning of others.

SMP4 - Model with mathematics.

SMP5 – Use appropriate tools strategically.
INTRODUCTION
This lesson is structured in the following way:
Before the Lesson,

Students work individually on an assessment task designed to reveal current understandings and difficulties. You then review their work and create questions to improve their understanding.
At the Start of the Lesson,

Display different bridges. Discuss that triangles are used in construction because they are rigid, therefore they provide more strength to a structure. Ask student to identify names of different triangles found in each.

Students try to draw a triangle for each combination that the teacher presents.
During the Lesson,


During the activity, students will work in pairs to construct triangles given three angles and/or sides.

Students will develop rules for determining if a unique triangle, more than one triangle, or no triangle can be formed.
After the Whole-Group Class Discussion,

Student pairs will present their rules written in the activity.

Finally, students complete the post-assessment to demonstrate what they have learned.

MATERIALS REQUIRED
Each individual student will need: Pre-assessment: Building Bridges , Post-assessment: Building Better Bridges , pencil, eraser, white boards, and dry erase marker.
Each small group will need: Straws, pipe cleaners**, protractors or copies of a protractor, chart paper, scissors, building cards, and glue sticks.
**The students will need to be able to model side lengths…this could be done with drawing using a ruler, or the teacher may find items which make manipulatives. Here are some suggested items which could be used (among other things) as models for side lengths as an alternative to drawing with a ruler and protractor.
Use Ruler, Protractor, and
Pencil
Supplies – Modeling Tool Examples
Use Horizontally Striped Use AngLegs ™ (pre-
Pipe Cleaners to mark cut, pre-measured off length. snappable side lengths)
Model by Cutting Paper into appropriate lengths/angles with ruler
Use Technology to Model like “Geometer’s Sketchpad” ™,
“Geogebra” ™ or “Ruler and
Compass Geometry” App ™
Copy Graph Paper, Cut
Out, and Sides can show
Length
Model Using
Horizontally Striped
Straws (Found in Craft
Stores)
Print an alternating stripe pattern, let students cut it in skinny strips (easier to work with when skinny – included in student materials)

TEACHER PREP REQUIRED
The only teacher prep is copying materials and gathering modeling supplies of his/her choice. It will be very important to READ the directions carefully, as some of the answers are a bit tricky.
DIFFERENTIATION IDEA: If you can collect several of the modeling supplies, students can choose their method of modeling. Students will then be differentiating by process.
TIME NEEDED:
For Pre-
Assessment:
15 minutes For Lesson: 80 minutes For Post: 15 minutes Other:
Special Notes about timing: Approximately 15 minutes will be used before the lesson for pre-assessment. 15 minutes for whole class introduction, 45 minutes for collaborative activity, 20 minutes whole class plenary discussion, and a
15 minute post assessment. Timings are approximate and will depend on the needs of the class.

FRAMING FOR THE TEACHER:
This formative assessment lesson will show if students can determine when (if) a triangle will be formed. This lesson builds for the future when students will use Law of Sines to prove the rules they write in this lesson. Applying those rules to identify when unique triangles, multiple triangles, or no triangles are formed is critical in many problems in physics, engineering, statics, etc.
An Angle-Side-Side Discussion : The last problem on the pre- and post-assessment is an extension problem. This concept has connections to 10 th
grade Analytic Geometry (Types of Triangle Congruence) and then later to
PreCalculus (Ambiguous Cases of Law of Sines and Law of Cosines). This type of problem is not specifically mentioned in CCGPS standards for sixth grade, but with the geometric modeling you will have taught, a 6 th
grade student certainly has the skills to figure out how many triangles can be formed with the given conditions. Do not let the fact that this is an extension make you tell students to skip this one. Try it – you’ll be surprised how well your students will understand). This situation is the reason we DO not have “Angle-Side-Side” congruence. (We didn’t abbreviate that for obvious reasons.) If you have never really thought about this before, see the examples below, which actually model the two ambiguous case cards from the collaborative activity.
FRAMING FOR THE STUDENTS:
Say to the students:
This activity will take about ___2 __days for us to complete.
The reason we are doing this is to be sure that you understand triangles before we move on to a new idea.
You will have a chance to work with a partner to correct any misconceptions that you may have. After the partner

work, you will be able to show me what you have learned! IF YOU ARE USING measuring models that have thickness like the alternating stripe tape you could cut out, use the EDGE of the tape on the exact angle, and join the tips/corners of the tape to other tips/corners for more accurate modeling.

ASSESSMENT TASK:
Name of Assessment Task: Building Bridges
Time This Should Take: (15 minutes)
Have the students do this task in class or for homework, a day or more before the formative assessment lesson. This will give you an opportunity to assess the work, and to find out the kinds of difficulties students have with it. You will them be able to target your help more effectively in the follow-up lesson.
Give each student a copy of Building Bridges .
Briefly introduce the task and help the class to understand the problem and its context.
Spend 15 minutes working individually on this task. Read through the task and try to answer it as carefully as you can. Show all your work so that I can understand your reasoning. Don’t worry if you can’t complete everything.
There will be a lesson that should help you understand these concepts better. Your goal is to be able to confidently answer questions similar to these by the end of the next lesson.
Students should do their best to answer these questions, without teacher assistance. It is important that students are allowed to answer the questions on their own so that the results show what students truly do not understand.
Students should not worry too much if they cannot understand or do everything on the pre-assessment, because in the next lesson they will engage in a task which is designed to help them.. Explain to students that by the end of the next lesson, they should expect to be able to answer questions such as these confidently.
This is their goal.

You Will Not “Grade” These!
Collect student responses to the task. It is helpful to read students’ responses with colleagues who are also analyzing student work. Make notes (on your own paper, not on their pre-assessment) about what their work reveals about their current levels of understanding, and their approaches to the task. You will find that the misconceptions reveal themselves and often take similar paths from one student to another, and even from one teacher to another. Some misconceptions seem to arise very organically in students’ thinking. Pair students in the same classes with other students who have similar misconceptions. This will help you to address the issues in fewer steps, since they’ll be together. (Note: Pairs are better than larger groups for FAL’s because both must participate in order to discuss!)
You will begin to construct Socrates-style questions to try and elicit understanding from students. We suggest you write a list of your own questions; however some guiding questions and prompts are also listed below as starting point.
GUIDING QUESTIONS
COMMON ISSUES
Student has a hard time getting started.
Student draws an isosceles triangle incorrectly for part 1.
Student identifies a degenerate triangle as a triangle.
Student adds little or no explanations as to why answers are formed.
SUGGESTED QUESTIONS AND PROMPTS

Can you draw a diagram to show what is being asked?

Can you tell me what an equilateral triangle is?
An isosceles triangle? A scalene triangle?

Here – lets take three straw/paper/anglegs pieces so that the sum of the two smaller sides is the same as the third side. Build the triangle. What do you notice about the triangle you are trying to build? Is it possible to build this triangle?

Can you explain to Tommy why you chose that answer?

SUGGESTED LESSON OUTLINE:
Part 1: Whole-Class Introduction: Time to Allot: ( 15 minutes)
 Show the class the slide “Triangles in Bridges - One”.

Using white boards, have students write down any triangle that they see. Call on students to point out and name each triangle. Most students will see the right triangle and the isosceles triangle. The right triangles are also scalene. The isosceles triangles are also obtuse. Challenge students to define each.
 Show the class the slide “Triangles in Bridges – Two”.

Repeat white board response and discussion. Most students will see the right triangle, the isosceles triangle, and the acute triangle. The right triangles are also scalene. Challenge students to define any new term.
 Show the class the slide “Triangles in Bridges -Three”.

Repeat white board response and discussion. Most students will see the right triangle and the acute triangle.
The triangles are also equilateral triangles – it is easier to see across the top of the bridge. Challenge students to define any new term.

Show the class slide – Can you model this triangle? Have students share their drawing. It is impossible to draw a right equilateral or an obtuse equilateral.

Part 2: Collaborative Activity: Time to Allot: ( 45 minutes)
Collaborative activity: Building Triangles

Assign partners according to responses to the pre-assessment.

Pass out straws, pipe cleaners, protractors or copies of a protractor, chart paper, scissors, building cards, and glue sticks to each pair of students.
 Students will cut the straws to the lengths described on the activity slide. Students will also cut apart the building cards. Each card has a description of a possible triangle. Students should use straws and pipe cleaners and try to assemble the triangle. Decide if the triangle is unique, not possible, or if multiple triangles are possible.

On the chart paper, set up three categories - unique, not possible, or if multiple triangles are possible. Glue the triangle in the correct category.

Each partner group compares with another pair. Pairs will try to write rules for the patterns that they found.
During both Collaborative Activities, the Teacher has 3 tasks:
 Circulate to students’ whose errors you noted from the pre-assessment and support their reasoning with your guiding questions.

Circulate to other students also to support their reason in the same way.

Make a note of student approaches for the summary (plenary discussion). Some students have interesting and novel solutions!

Part 3: Plenary (Summary) Discussion: Time to Allot: ( 20 minutes)
 Were you able to construct a triangle for this card?
 Can you move the pieces around to form a different triangle?

What caused the triangle to be impossible to make?

What must be true about the angles of a triangle?
 What must be true about the sides of a triangle?
 If we have all the angles the same, would it be a unique triangle, impossible triangle, or multiple triangles?

What would be true about the multiple triangles?

Display the cards and have pairs of students explain where they placed each card. Remember to question why the placement was made. Allow others to challenge or support the placement.

The teacher should scribe (or script) the key points and recognize specific students.
NOTE: “Scribing” helps to increase student buy-in and participation. When a student answers your question, write the student’s name on the board and scribe his/her response quickly. You will find that students volunteer more often when they know you will scribe their responses – this practice will keep the
discussions lively and active!

Part 4: Improving Solutions to the Assessment Time to Allot: ( 15 minutes)
Task
The Shell MAP Centre advises handing students their original assessment tasks back to guide their responses to their new Post-Assessment. In practice, some teachers find that students mindlessly transfer incorrect answers from their
Pre- to their Post-Assessment, assuming that no “X” mark means that it must have been right. Until students become accustomed to UNGRADED FORMATIVE assessments, they may naturally do this. Teachers often report success by displaying a list of the guiding questions to keep in mind while they improve their solutions.

Return student pre-assessments.

If you did not mark pre-assessments with questions, display a list of questions on the board.

Distribute post-assessments and have student spend approximately 15 minutes completing.
Note: If you are running short of time, this post-assessment can be done the following day.

Triangles are effective tools for architecture and are used in the design of bridges and other structures as they provide strength and stability. The use of triangles in building structures predates the wheel.
Problem 1: You are an architect that has been assigned to complete a bridge design that was started by another member of your firm. You cannot change the work he has completed. His design uses repeating isosceles triangles with two sides having lengths of 6 feet.
Which of the following lengths will you choose as the third side: 10 feet, 12 feet, or 14 feet?
Make sure to model and explain why you made your choice.
The 14-foot side is too big. You can see that if you line up the two 6-foot sides, they can’t stretch to the end of the 14foot side even when they are totally flat. It’s not 14. The 12-foot side is too big also. Lying flat, they reach the end, but that doesn’t make a triangle. To be able to “pitch” in the middle, the third side cannot be equal to the sum of the other two sides. The 10-foot side works because it makes a triangle.
Problem 2: Your young niece is playing with a building set. She is trying to make a bridge like the one you just designed. She has chosen the following pieces for her triangles: red (3 cm), yellow (5 cm) and green (10 cm).
Do you think she will be able to build a triangle based bridge with these pieces? Explain and show your answer.
The correct answer is NO. 3+5 is 8 and that is no more than the third side of 10.
Problem 3: You are completing the base of a bridge over a stream in your backyard. The first board you use is
4 feet long and makes a 30°with the base board. You have a second board that is 3 feet long. How many distinct triangles can be formed? Explain your answer.
There are two different ways to form a triangle with these conditions: 1) by swinging the 3foot side away from the 30◦ angle past the altitude, and 2) by swinging the the 3-foot side toward the 30◦ angle.

Triangles are effective tools for architecture and are used in the design of bridges and other structures as they provide strength and stability. The use of triangles in building structures predates the wheel.
Problem 1: You are an architect that has been assigned to complete a bridge design that was started by your supervisor. You do not want to change the work he has completed. His design uses repeating scalene triangles with two sides having lengths of 6 feet and 8 feet.
Which of the following lengths will you choose as the third side: 12 feet, 14 feet, or 16 feet?
Make sure to model and explain why you made your choice.
The correct answer is 12 feet. The third side must be less than 6+8 or 14 and more than 8-6 or 2.
Problem 2: Your young nephew is playing with a building set. He is trying to make a bridge like the one you just designed. She has chosen the following pieces for her triangles: 2 red (3 cm) and 1 yellow (5 cm).
Do you think she will be able to build a triangle based bridge with these pieces? Model and xxplain your answer.
The correct answer is YES. 3+3 is 6 which is more than the third side of 5.
Problem 3: You are completing the base of a bridge over a stream in your backyard. The first board you use is 4 feet long and makes a 30°with the base board. You have a second board that is 3 feet long.
How many distinct triangles can be formed? Explain your answer.
The correct answer is one triangle. The 5 foot side can only be drawn away from the 30-degree angle, because the side is longer than the 4-foot side, and therefore wouldn’t touch the base board.

For a QUICK walkaround CHECK…look for the words (scrambled) “BUNKED(unique),
FIST-RAY(impossible), and PHLOX(multiple)!)
Two right angles Sides 2, 6, and 6 Sides 4, 4, and 6
Unique Unique Any side lengths
Multiple With the pieces shown, only the 3-4-5 would work, so most students will put unique.
Challenge them to see if others can be made using different side lengths. If students see 6-8-
10, their answer would be multiple.
Two obtuse angles
Any side lengths
Not possible
Sides 2, 4, and 8
Not possible
Sides 4, 5, and 8
Unique
All sides are 2
One right angle
Not possible
Angles 60 ⁰ , 70 ⁰ , and 80 ⁰
Not possible – angles are over 180 ⁰
Angles 55 ⁰ , 55 ⁰ , and 60 ⁰
Not possible – angles are less than 180 ⁰
Sides 4, 6, and 8
Unique
Sides 2, 2, and 5
Not possible
Sides 4 and 6
Angle 40 ⁰
Multiple – angle may be between sides or nonincluded (limits may be due to straw pieces, but creative students will see to cut the pieces or use longer ones by laying out the angles.
Angle 40 ⁰ which is adjacent to a side of length 6, but not adjacent to a side of length 5. Multiple
(TWO) – angle may be between sides or nonincluded. (See “angle-side-side” discussion)
Sides 2, 4, and 6
Not possible – explain degenerate triangles as situations when a+b=c
All angles 60 ⁰
Multiple – angles are exactly 180 ⁰ so sides are proportional (limits may be due to straw pieces, but creative students will see to cut the pieces or use longer ones by laying out the angles.
Angles 55 ⁰ , 60 ⁰ , and 65 ⁰
Multiple – angles are exactly 180 ⁰ so sides are proportional (limits may be due to straw pieces, but creative students will see to cut the pieces or use longer ones by laying out the angles.
Sides 3 and 4
One included right angle
Unique
Angle 40 ⁰ which is adjacent to a side of length 5, but not adjacent to a side of length 6. Unique – because 6 longer than 5, so it won’t “swing” to the other side of the altitude and still be inside the triangle.

Triangles are effective tools for architecture and are used in the design of bridges and other structures as they provide strength and stability. The use of triangles in building structures predates the wheel.
Problem 1: You are an architect that has been assigned to complete a bridge design that was started by another member of your firm. You cannot change the work he has completed. His design uses repeating isosceles triangles with two sides having lengths of 6 feet.
Which of the following lengths will you choose as the third side?
10 feet 12 feet 14 feet
Make sure to explain why you made your choice.
Problem 2: Your young niece is playing with a building set. She is also trying to make a bridge. She has chosen the following pieces for her triangles: red (3 cm), yellow (5 cm) and green (10 cm).
Do you think she will be able to build a triangle based bridge with these pieces? Explain your answer.
Problem 3: You are completing the base of a bridge over a stream in your backyard. The first board you use is
4 feet long and makes a 30°with the base board. You have a second board that is 3 feet long.
How many distinct triangles can be formed? Explain your answer.

Triangles are effective tools for architecture and are used in the design of bridges and other structures as they provide strength and stability. The use of triangles in building structures predates the wheel.
Problem 1: You are an architect that has been assigned to complete a bridge design that was started by your supervisor. You do not want to change the work he has completed. His design uses repeating scalene triangles with two sides having lengths of 6 feet and 8 feet.
Which of the following lengths will you choose as the third side?
12 feet 14 feet 16 feet
Make sure to explain why you made your choice.
Problem 2: Your young nephew is playing with a building set. He is also trying to make a bridge. He has chosen the following pieces for his triangles: 2 red (3 cm) and 1 yellow (5 cm).
Do you think she will be able to build a triangle based bridge with these pieces? Explain your answer.
Problem 3: You are completing the base of a bridge over a stream in your backyard. The first board you use is 4 feet long and makes a 30°with the base board. You have a second board that is 5 feet long.
How many distinct triangles can be formed? Explain your answer.

Activity: Protractor Master and “stripes” page (optional – free materials for modeling).
“Stripes” page…cut laterally across the stripes to make a unit bar for measuring.

Activity: Building Cards
Angle 40 o which is adjacent to a side of length 6, but not adjacent to a side of length 5.

Activity: Building Cards
o
o
o
o
o
o
o
⁰
o
o
o
D
Angle 40 o which is adjacent to a side of length 5, but not adjacent to a side of length 6.


























