So many angle relationships!

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Objective

SWBAT analyze data to explore angle relationships and find missing vertical, adjacent, and other angles about a point.

Big Idea

Over the next 4 days, through peer coaching and data analysis, students will work on mastering the concept of missing angles connected by a single point

Do Now/Quiz

30 minutes

Students enter silently according to the daily entrance routine. For the next few days students will be working on finding missing angles about a point. This was a concept many struggled to master and thus took place across 4 days in the scope. Two of those days include quizzes where students enter their answers into Senteo clickers. This allowed us to have instant data to work with within that given class. Two of the other days did not include quizzes, thus students completed Do Now assignments. On a skill as complex and stubborn as this one, I needed to make sure I collected some data each day to inform my next steps and how to improve mastery in different Geometry standards. Below you will find a breakdown of each resource used to collect data at the beginning of class, what common misconceptions showed up, and how I addressed these misconceptions across 4 days.

 The questions within this resource (Day 147 - Do Now) aim to review area and complementary angles. We spent 5 minutes working silently and the remainder of time (15 – 20 mins) reviewing answers and extra examples on white boards.

Many students struggled to write a correct equation for #1, making one of two errors:

  • 7x – 4 = 24

Setting the equations equivalent to each other makes the false assumption that the two angles are equal simply because a line divides them into two angles. These students collectively agreed that if the angles are being split into 2, they must be equal. I asked the following guiding questions:

So when we cut something into two pieces, those pieces are always going to be equal?

This is a great opportunity to use a candy bar and play the “fairness” game where I cut it into two visibly unequal pieces and state, “here, I cut them into two pieces, you can have this one (the smaller piece). Since I cut them into two pieces they must be equal, so this is fair, right?”

  • (7x – 4) + 24 = 180

This was a common misconception for many days. Because we started this geometry unit mostly with supplementary angle relationships as well as the sum of the interior angles of a triangle, many students falsely assume that ALL angles add up to 180 degrees. I spend most of the time over the next four days ensuring that students are asking themselves these initial questions before solving:

  • What are the angle relationships I need to consider in this problem?
  • Are the angles equivalent to each other (i.e. vertical angles)
  • Are the supplementary, complementary or something else?
  • So they share a common point? Do they share a common side (or ray)?

I also made sure to look out for students who were completely lost. These students usually freeze when faced with problems like this and do not understand basic tenets of angle geometry, such as where the angle is located and what is actually being measured. For these students it is extremely important to address these misconceptions early and often throughout this unit. One way to do so it to make sure they are paired with a capable peer coach (in the seating chart) whom I can trust will explain these concepts and continuously remind their partners about basic identification of parts and angle relationships.

Questions #2 and #3 are diagnostic. As I have noticed students struggling to correctly identify the use of the perimeter or area formulas in real word problems (such as #3), I’ve come to wonder if students understand what each of these formulas measures:

  • Perimeter measures the distance around a shape. Area measures the space inside a shape. I give students several examples to gauge and improve understanding:
    • I have a rectangular picture frame and want to put ribbon around the sides. How much ribbon will I need? Is this a perimeter or area problem?
    • I have a rectangular wall in my apartment and want to paint it. How much paint will I need? Is this a perimeter or area problem?

 

On this day students spend the first 30 minutes of class completing a quiz to assess understanding of basic geometric terms.

  • Most questions at the beginning asses knowledge of vocabulary (i.e. complementary, supplementary, adjacent, etc).
  • The results to this quiz indicated that students needed to spend more time identifying and labeling angles using three points as well as more time practicing the use of algebra to find missing angles. For these particular questions, students seem to struggle most when asked to find the value of a variable vs. the value of an angle. Sometimes the value of an angle was not the same as the value of a variable, but instead the value of a term (i.e. problem #7, angle FYE measures 3x – 1 degrees)
  • The bonus questions helped me to identify students who felt confident finding missing angles about a point. These students will become peer coaches during our classwork sections within this topic.

 

Day 149 - Do Now 

This assignment was created and given to students as a response to the results to the most recent quiz, questions #4 – 7. Many students incorrectly answered these questions. Their work and feedback during class indicated that they main issue was identifying angles using three points, with the center letter being the vertex of the angle. Once again, I initially made the error of assuming students knew how to label angles this way. Once I noticed so many students struggling to label and identify correctly, I decided to give this assignment to students. The “make up” included is given to students who need to continue practicing to master this concept, as well as continued practice with corresponding sides in similar triangles.

 

Day 150 - Quiz  ï»¿

On the last day of this series of lessons I give students another quiz. Over the previous 3 days, the term that students have struggled to recognize and understand is vertical angles. This is a concept that comes up in multiple questions on this quiz ranging from:

  • Least complex: basic identification of vertical angles (question #2). Notice that this question also includes a ray that cuts one of the vertical angles, making it a bit more difficult to identify the vertical angles than if this ray were not there. You may choose to remove this ray to assess at a more basic level.
  • Most complex: a more complex example is questions #10. Here, the concept of vertical angles is also linked to angles that add up to 180 degrees. It is a more complex item on this quiz because students must identify an unlabeled angle (no variable or number is given) as part of a trio that add up to 180 degrees because they lie on a straight line. This justification is something students have continued to struggle with. Thus, it is an idea that I continue to spiral when we review the answers to any angle geometry question. I may ask:
    • Why do they add up to 180? Or 90? Or something else?
    • Give a reason using a vocabulary term from your notes to defend this answer

Class Notes

15 minutes

We use C Notes across all 4 days of this series. We review a few examples each day, adding notes and examples needed to practice whatever we are targeting that given day. The following is a breakdown of the major topics reviewed each day along with some questions asked to push understanding:

Day 147

  • Adjacent angles, vertical angles, linear pairs
  • Do adjacent angles have to “touch”? what part of the definition states this?
  • Describe vertical angles. What do we know about vertical angles that could be useful to find missing information?
  • What is the sum of a linear pair? How many angles make up a linear pair? Do linear pairs always add up to the same thing?

 

Day 148

  • Adjacent angles, vertical angles, linear pairs
  • Complementary, supplementary angle examples and definitions (page 2)
  • Always, sometimes, or never? – all supplementary angles are adjacent.
  • Always, sometimes, or never? – all complementary angles are adjacent.

 

Day 149

  • Angles about a point (last item on page 1)
  • If we add all the angles about a point, what is the sum?
  • What if we have a pair of vertical angles, one of which is cut into two pieces… can we assume that both are equivalent? How would we find a missing angle? What information MUST we have to find a missing angle in this example?
  • Correspondence – students copy the angle correspondence definition at the end of these notes off the board

 

Day 150

On this day, after students complete and enter their answers to the quiz they will be given the correct answers and paired up according to ability for the rest of class. Those who earn 70% or above will be paired with students who earn below 70%. They will be asked to choose one quiz question from #5 – bonus. They will have 10 minutes to show the solution for their elected question on a piece of chart paper. The solution must have the following elements:

  • Step by step horizontal and algebraic solutions that use different angle relationships to justify each subsequent step
  • Each of these steps must also include a justification in the form of a vocab term or brief explanation

 

Class Work

15 minutes

The class work each day will range in time, between 15 – 20 minutes. On quiz days it will be on the shorter side. Most of the resources attached aimed to continue the practice of using angle relationships to find missing angles and variables. It may be necessary to group students into 4s, given each group has at least one student who has mastered or is close to mastering the ability to recognize the appropriate angle relationships needed to solve. I make sure to continue cycling through the essential questions and concepts covered in the do now and class notes sections of these lessons.

On day 148, after students have completed the notes section of class, we will be playing a game called “angles bingo”. These sheets are attached as pdf documents. Students will have the opportunity to choose a card from these 5 different formats. I will be using a random word generator like this one to select different terms. I have included the 5 versions of bingo cards used in class as well as a word document that includes a blank card and a list of words used in the random generator. This game provides a fun opportunity to review the key terms within this unit. Students are encouraged to ask for a definition of the word, but I will not be helping them identify whether or not they have this drawing on their cards. 

Closing

10 minutes

On non-quiz days in this series I make sure to collect an exit ticket to continue checking up on progress of these skills. Each student is given a half sheet of paper and one problem from the classwork is selected. Students must submit the full solution to this problem by copying the problem and showing the full solution on their half sheet of paper. I will be using these exit tickets to create groups each day and spread out as many student coaches as possible. Each day that student coaches are asked to help out, I share with the most common misconceptions I notice on the exit tickets and ask them to continue reviewing these ideas within their groups.

Day 147:

Students are asked to complete the example below.

 

The following misconceptions are flagged and shared with student coaches the following day.

  • Students that do not know how to construct an equation, missing addition and equal signs
  • Students who assume everything adds up to 180 degrees

 

Day 149

Students are asked to complete the example below.

 

The following misconceptions are flagged and shared with student coaches the following day.

  • Students who assume everything adds up to 180 degrees
  • Students that do not know how to construct an equation, missing addition and equal signs
  • Students who are not able to identify right angles and how this information can be used to find the missing variables
  • Students who have no idea where to begin! What strategies could you give them so that they’re not frozen in place?
    • List all the things you know
    • List the unknown information
    • List vocabulary terms/concepts you can identify (i.e. vertical angles, supplementary/complementary angles, linear pairs)

 

On quiz days, students are asked to use their half sheet of paper to tell me which was the most difficult quiz question and why. Their explanations must include at least 2 questions they have about any misunderstandings or unknowns that are getting in the way of successfully solving these problems.

Relevant homework assignments are also given to students and attached in this section.