Discovering What it Means to be Similar Triangles

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Objective

SWBAT understand angle relationships with similar triangles, discover correct methods of enlarging similar triangles and then apply this to dilating figures.

Big Idea

Imagine teaching similarity, angle-angle similarity for triangles, and dilations all at once! It’s a three for one deal!

Opening Bellringer Activity

15 minutes

My state has testing next week and I still need to teach volume.  Volume is a supporting standard (additional cluster of standards), so I have chosen to introduce this concept through short bellringer lessons each day this week for about 15 minutes each.  Yesterday I introduced volume of cones through a water demonstration that links the formula for cones to the formula for cylinders.  Today, I asked students to find the volume of four different cones and one cylinder.  In the first group of cones I drew on the board, I gave them one cone where the actually height was given along with a diameter length.  The second example labeled the radius and the slant height along the outside of the cone.  As predicted, all students used the slant height as height for the second example.  I brought out my cone and used a pencil to demonstrate the difference in location of height and slant height.  Then I drew both on the diagram on the board and asked them to find the real height.  As I walked about the room it was amazing to see how many students very quickly realized Pythagorean Theorem was the tool they needed and explained it to other students as they worked through the problem.  After several minutes I asked one group to present their thinking and solution to the second problem.  Then I gave them two more on the board similar in design to the first two and I walked about the room checking every student’s work. 

 

Throughout the bellringer, I used several strategies I always try to bring into my classroom environment daily.  These strategies include allowing students to be resources for one another, providing feedback myself that moves student learning forward, and students showing ownership of their own learning as we use mini wrap ups to discuss work.  Below are links to short videos explaining each of these.

 

Activating students as resources for one another

 

Providing feedback that moves learning forward

 

Activating students as owners of their own learning

 

Mini-Wrap Up Strategy Explained

 

The Main Exploration of the Lesson + Homework

40 minutes

Have students put all tangrams back in the bags and pass them in as they will not need them for the remainder of the lesson.  I demonstrate how we will be creating our own tangrams using centimeter graph paper.  Read question 3 aloud, or ask a student to read it out loud and then demonstrate how the centimeter paper makes measuring and cutting tangrams so easy.  Stress to students they must use a ruler to draw a straight hypotenuse and cut accurately for this activity to work as intended.  I have a short video with these and more helpful tips on cutting tangrams linked here. 

Centimeter paper is located on the website mathbits.com and is really easy to print and copy, here is the link MathBits. 

After your demonstration of the drawing, cutting, and coloring the edges, pass out the supplies and ask students to work through questions three through seven.  I usually write my directions for which questions to work through on the board and set a timer for a reasonable limit such as 10 minutes since cutting is involved.  As students work, move about the room formatively assessing students, providing feedback that moves learning forward, and keeping students focused.  Your goal today is to answer all the questions that require cutting and comparing tangrams.  If you cannot finish all the questions up to number 12, then you will need to make plans for how students will store triangles until the following day.  

Select a group or groups to present answers to questions 3 – 6 under the document camera and be sure to use a set of tangrams with color to show up (MP3).  Allow time for questions from other students.  This mini wrap up time is very important.  Script more notes on the board, “Adding length to sides does not create similar triangles.”

 

Give students time to work through questions seven through nine.  Provide feedback then select a group to present the mini-wrap up.  If time allows, repeat this process for questions 10 – 12 so that the hands on materials are finished for this activity.  Make sure to script notes on the board as each investigation is discussed.

 

Assign questions 13 and 14 for homework as an extension of the activity today.

 

Standards Addressed

The application of congruence through movement to stack triangles and compare corresponding angles brings back the definition of congruence through movement and begins the understanding that when corresponding angles are congruent, triangles are similar.  The standard in eighth grade math that includes angle-angle similarity of triangles is:

 

CCSS.Math.Content.8.G.A.5 Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.

 

This activity extends from similar triangles towards understanding that dilation is the process of creating similar triangles and dilations involve a scale factor.  The activity does not explicitly say dilation; therefore on day two, it is up to you to make this fact direct with students. 

 

The math practice standards used throughout this activity arise from students working in cooperative groups to discuss and record observations (MP3 Construct viable arguments and critique the reasoning of others) that arise from hands-on exploration ( MP5 Use appropriate tools strategically) in an effort to make generalizations about all similar triangles ( MP7 Look for and make use of structure.)