Sum of the Angles in Triangles

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Objective

SWBAT determine that the sum of the angles of a triangle equals 180 degrees through a triangle investigation.

Big Idea

Students cut angles in a triangle and rearrange them to discover why the sum is 180 degrees

Do Now

10 minutes

Over the next two days we will be exploring the sum of the angles in triangles. Each day, students enter silently according to the daily entrance routine.  Do Now assignments will be handed at the door.

The first day’s assignment is a diagnostic that will inform me about students’ prior knowledge on the topic of angle and triangle geometry. Students must match the terms (i.e. scalene, isosceles, etc) to the shapes labeled all around the paper. There will be no timer on the board because this assignment is not about speed, it’s about information needed to inform the rest of my lessons.

As I walk around I make the following observations, which are usually common in this grade:

  • Many cannot recall the differences between the triangle names according to lengths (scalene, isosceles).
  • Equilateral triangle seems to be the only exception as most are correctly identifying this triangle.
  • Most also do not appear to know the meaning of supplementary or complementary.
    • Watch out! These are commonly confused terms. We spend a lot of time on multiple days reviewing this and coming up with strategies to remember the difference (i.e. if two angles are supplementary, what do they add up to?)

 

The second day’s assignment will also be a diagnostic to assess how much students were able to grasp about the angles in triangles the previous day as well as assess how much prior knowledge students have about reading symbols in angle geometry (i.e. identifying angles given three letters).

Each day, once there are 2 minutes left in this section I stop students and we review the answers. This should be enough time to gather information that will inform the material I need to review, reteach, or introduce in the following days.

Class Notes

10 minutes

Each day, students will be asked to put their Do Nows away and Class Notes will be distributed. These notes will consist of key definitions of triangle categorization (according to sides and angles) and other key geometric terms

Before class begins, I will need to draw key pictures (unlabeled) on the blackboard.

On the first day pictures include:

  • Isosceles, scalene, equilateral triangles
  • Acute, obtuse, right triangles
  • Straight line
  • Right, acute, obtuse angles

 

Students will then be asked to partner up and read the definitions in the C Notes  to each other. If any questions come up, they must either raise their hands to ask me as I walk around, OR they must write their question down to ask later. On the second day I will need to show all students what mathematic notation looks like for each of the terms listed in the Class Notes.

 

On the second day, I will redraw the pictures included in the Class Notes page:

  • Point
  • Line segment
  • Ray
  • Line
  • Parallel lines
  • Perpendicular lines
  • Angles (focus on how to label)
  • Interior/exterior angle
  • Clockwise
  • Counter clockwise

 

After reviewing the notes and definitions I check for understanding by asking the following questions on both days:

  • Who can show me what a full turn looks like?
  • How many degrees are in a “full turn”?
  • Who can show me what a half turn looks like?
  • How many degrees are in a “half turn”?
  • Why is a straight line a “half turn”? who can show me where the half turn is on this picture on the board?
  • Show me an acute angle with your arms
  • Show me an obtuse angle with your arms
  • Show me a right angle with your arms
  • What is the difference between complementary and supplementary angles?
  • Does anyone remember how many degrees there are in a triangle? Is this true of all triangles? Is it true of all shapes?
    Ending with this last question is a great set up and transition to the investigation on the first day. On the second day, it is yet another way to check for continuous understanding of this concept. 
  • On the second day only: What if you don’t know ANY of the angles and you are only given algebraic expressions to describe all three angles? I give an example of this by drawing it on the board and having students copy it on their paper. 

 

Investigation: Day 1

20 minutes

To begin the investigation on the first day, I distribute the Task  and explain the following to students:

Script: You may remember that the sum of the angles in a triangle is 180 degrees. We also just discussed that a straight like is half a turn, 180 degrees. Here are some questions for you to consider during today's investigation:

    • Is the fact above true for ALL triangles, no matter what type, they all have interior angle sums of180 degrees?
      • Equilateral
      • Isosceles
      • Scalene
    • How can we prove this fact?

 

Today we will be proving the sum of the angles of a triangle. I will show you how to do this using an equilateral triangle and you will complete the rest of the activity on your own with the isosceles and scalene triangles. Then, you will use the facts you proved to answer some questions and world problems. Continue to think about the following questions as you work: (write on the board)

    • Do the angles in ANY triangle add up to 180°? Why?
    • How can you apply what you learn to the math problems you will complete at the end of the investigation?

 

 

The following directions for cutting triangles should also be written on the board

Step 1: Cut around the triangle.

Step 2: Mark each angle with a different color.

Step 3: Cut the angles off along the dotted lines.

Step 4: Tape the angles side by side to create a straight line.

 

Once students begin completing the rest on their own, I will be walking around to help students with the following likely struggles:

  •  Lining up the angles. If students struggle, explain:
    Think if it as a puzzle. I’m going to line up the solid lines of my cut out angles and make sure the dotted “cut out” line is touching no other angle or the straight line. (Facing away from the bottom of the paper)
  • Line up the angles to create a semi-circle. Have students tape their angles on the line provided on their paper.
  • Advice students that it is important to cut as neatly as possible, as it affects the accuracy of the lines created

 

I am also asking some essential question to spiral throughout this activity:

  • What did all the interior angles of this triangle create when we put them together this way? (a line)
  • What kind of angle is a line? What else can we call it? (straight angle)
  • How many degrees are there in a straight angle? (180)

 

After students finish the investigation they must complete the “Try this” problems to check for understanding. If they do not understand how to solve, they must raise their hand and either I or another student will help guide them through the work. This activity, which takes place on the first day and is the focus of the lesson, helps students understand the concept of the sum of the interior angles of a triangle through the use of MP2 – reasoning abstractly and quantitatively. By manipulating the triangles, cutting and pasting, students are “[probing] into the referents of the symbols involved” with interior angles. The linked website is a great resource for understanding and using mathematical practices to push students learning.      

Class Work: Day 2

20 minutes

On the second day of this lesson, students will dive into the classwork after reviewing and answering questions about basic definitions from both days’ notes. As I walk around while students are working in pairs, I bring back the essential questions from the day before and add more for the new material:

  • What did we find in the investigation yesterday? Is the sum of the interior angles of ANY triangle 180 degrees?
  • Did you all notice the angle, side and equivalence symbols used at the top of each page? what do they mean?
  • If we are missing the measure of an angle, how can we use what we know about the other angles to find the missing one?
  • Can anyone think of a real world example of parallel lines/perpendicular lines? i.e. railroads, streets

Once there are 5-7 minutes left in this section, we will stop working in pairs and review the answers. It’s a good idea to choose students during work time to put the solutions on the board in order to facilitate this review. I also make sure to share out the questions I was asking as I walking around with the rest of the class to further check for understanding

Closing

15 minutes

In the class work or investigation sections of each day, I will be identifying any problem I notice many students struggling to solve. This picture must already be drawn on the blackboard for us to review in the closing section. I review by asking students to call on each other and guide me to write the steps for solving on the board. Some key questions to end the class include:

  • How many degrees make up half a turn?
  • How many degrees are there in a triangle?
  • Do you think this is true of all polygons? Why or why not?

 

 

I have students take 5 minutes to free write the answer to this last question on the second day on a half sheet of paper so that I have some feedback on the material or concepts I need to continue to spiral in our future lessons.

 

Homework is distributed by a student helper at the end of e ach lesson and students are dismissed.