Exploring Angle Relationships Through Transformations

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SWBAT use transformations to connect angle relationships created by intersecting lines and parallel lines cut by a transversal.

Big Idea

Intersecting lines form angle relationships that can be explored and understood as an application of transformations.

Bellringer (Warm-Up)

15 minutes

I chose to use a strategy called “My Favorite No” during our bellringer today.  I took this strategy from a teacher channel video:   When using this warm up strategy, you give one question on the board and paper supplies to answer the question.  As students finish you gather the papers and sort into two piles – yes (correct answer) or no (incorrect answer).  Then you pick your favorite no that has some really good correct parts but some common problem areas that many students usually have.  You put this favorite no under the document camera and ask students to analyze what is correct first and then what is incorrect.  I chose to ask students to graph the equation y = 1/4x – 2 because our activity today requires students to graph lines.  I printed graph paper from Mathbits and cut it into four small sheets to give to students:.  I chose a favorite no that had a correct y-intercept and positive slope, but not the correct ¼ slope.  The paper also graphed a very short line segment, not a full line across the page.  I gave students time to “think-pair-share” around the questions what is correct about this graph, and then what needs to be corrected.  I wrote the questions on the board and students thought for about 30 seconds and then shared with a partner for 30 seconds and then shared with the class.   



I always allow students to work in pairs that I have pre-assigned so they will have another student who is on the same level mathematically (homogeneously paired) to talk with and discuss the new concept. I begin class with a warm-up of some kind that is short but gets their minds thinking towards the activity of the day and then I give them the clear learning expectations for the day. After this, I begin the activity whole group but mostly allow students to work on their own within their pre-assigned group to complete a certain section of the activity. I even set a timer sometimes for how long they have to work. Here is a video,  Activating students as resources for one another, from my strategy toolkit on how to make students resources for one another through cooperative groups.

Making Experts: While students are working within teams to complete a section of the activity, I am walking about the room formatively assessing their understanding by listening to their conversations, viewing their work, and by asking questions myself to see if students can clearly explain their thinking. While assessing learning, I am also making note of all the different correct approaches that students are taking to complete the work. I ask students to present their thinking during our whole class discussion time (consolidating student thinking) so that the class may listen to multiple methods of completing the work. If struggling students are having difficulties, then I spend even more time with these groups to "create" an expert for the whole group discussion later. To create an expert you need to question the student partnership in such a way that they begin to think about the task in a productive way and begin to understand enough to generate solutions on their own. Here is a short video further explaining how I give feedback to move learning forward and make experts:

Consolidating Thinking (Mini Wrap Up Time) At the end of the designated time, I call the class together and ask students to come to the front of the room and either present their work by writing on the white board or present by putting their papers under the document camera. This time of sharing out helps students to consolidate their learning as we move through the lesson and is usually why I do not spend a long period of time at the end of class wrapping the entire lesson. We hold mini wrap up sessions as we move through the lesson. If you would like to watch a short video explaining my classroom environment for this lesson and all my lessons, 


Beginning The Activity + Homework

35 minutes

When I passed out the lesson, I clarified the goal of the lesson they were about to begin.  This activity will require students to apply transformations (we listed the three transformations on the board: translation, rotation, reflection) to find and understand relationships between angles when line intersect. I let them know the graphing part is intended to review their graphing skills which are so important. I gave everyone tracing paper because I expected them to use the paper to literally compare the sizes of each angle in the pair and then think about the movement it took for the tracing paper to accurately compare angle sizes. If you would like to watch a video of my reasoning for this activity, click on the link below.


Exploring angles video rational

I reminded students that there is a teacher check after the graphing and numbering of the angles directions is completed.  The questions about the angles are very specific and students want to ensure they have a correct graph.  Here is an example of a student graph from one of my classes.


Student paper intersecting graphs

Most students completed the graph and numbered the angles within this first class period. Some students finished and continued into answering the questions, while others had to restart their lines over.  It was a good formative assessment over graphing linear equations to teacher check each student paper.  Many of my students greatly struggled with graphing a fractional slope.  I had a variety of unique graphing methods that are summarized in the following video.


Link to graphing methods video.

At the conclusion of the first class period, I required students to have a correct graph which meant if they received feedback from me on improving their graph; it was up to them to fix the graph before the following class period. 


Math Standards

The first class period really focused around correctly graphing linear equations before students could really use them for a different purpose.  Therefore the math standard of focus to was really 8.F.A.1 Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.  Working in small groups, receiving feedback from both their partner and from me as they worked with many different representations of the equation (algebraic, numeric, graphic) and using different tools from rulers to calculator really utilized the following math practice standards:  MP3 Construct viable arguments and critique the reasoning of others,  MP5 Use appropriate tools strategically, and MP6 Attend to precision (as we discussed initial amount, rate of change, slope, y-intercept, inputs and outputs and how all of these work together in a graph).