Session 2 - Similarity, Right Triangles, and Trigonometry

Warm-Up Activity & Directions:

Warm-Up: Warm up - Solving Equivalent Fractions

1. Have students go to the warm-up activity worksheet.
2. This activity reviews finding missing numerators or denominators from equivalent fractions.
3. Since this is a review activity,  most students will be able to do this fairly independently.  
4. For facilitator information,  here is how you can prompt students and remind them how to solve these problems. 
5. Method 1 to solve: Students can use equivalent fraction knowledge (multiplying numerators and denominators by the same factor) to find the missing measurement.
6. eg: multiply the fraction  by a factor of 2: 12x 22=x4 to get x=2
7. Method 2 to solve: Students can multiply both fractions by the same factor as the denominator to simplify.
8. eg: multiply the fraction by a factor of 2: 12 x 4=x4 x 4 to get 42=x to get  x=2

Lesson Activity Model (20 - 25 minutes)

Activity: Using Proofs Activity

Directions for Giving the Task:

1. This activity reviews the skills needed to be able to do proofs in an applied manner.
2. This activity will provide practice in understanding geometric relationships and statements while having students confirm these claims through solving and sharing their thinking. 
3. Students will demonstrate knowledge of the standards for this activity and also validate their solutions and understanding. 
4. Students will be asked to use proofs in different ways through drawing diagrams, using logic, explaining the relationships, using theorems, and completing logical steps. 
5. If students need support,  they can reference the Congruent & Similar Triangles presentation.
6. Students will determine if the triangles are congruent or similar, and explain how they might know that.
7. Students will find the lengths of the missing sides of various triangles. 
8. Students will be asked to use strategies to approach word problems and provide models of their work. 
9. Have students complete the questions asked on the worksheet.

Progress Monitoring

1. Fill out this Progress Monitoring chart to determine how well students understand the concepts.  
2. If students are struggling with these activities, please review the pre-assessment activities with them: Pythagorean Theorem Activity, Congruent & Similar Triangle Practice & Answers, Similarity Triangle Activity, Similarity Versus Congruence Triangles & Answers

Collaborative Activity #1: (15 - 20 minutes)

Activity:  Height of a Building

Focus: MP3: Construct viable arguments and critique the reasoning of others and MP4: Model with mathematics

Note: This activity is from our Pre-Lesson Knowledge Check in Step 2. The activity is used in this step so that the concepts can now be formally taught and/or reviewed. Some students may have little or some developing knowledge of this concept, so the review will help solidify the content.

Directions:

1. To solve this problem you need to set up the ratio of the sides according to their similarity and solve for the unknown. Since the building height and its shadow are similar to the triangle of the pole and its shadow, we set the proportion of the building height: shadow and pole height: shadow
2. The ratio will be set up as  equivalent fractions: h12=53
3. Students will solve for h. If they use equivalent fractions they will see that h=20.
4. Bailey would like to know the height of a building.
5. He gets Balin to hold up a 5-foot pole near the building and measure the length of its shadow. 
6. The shadow of the pole is 3 feet long, and the shadow of the building is 12 feet long.
7. Use your knowledge of similar triangles and proportions to find the height of the building.

Progress Monitoring

Directions:

1. While students are working, you will want to be watching how they approach this task. 
2. When they feel ready to share, look at their answers/responses. Use the suggestions below to help decide if they “got it” or are still struggling with this skill.  
3. Students should create equivalent triangles and use equivalent fractions or cross multiplication to arrive at the answer. 
4. The answer: The answer to this problem is 20 feet.