Thursday: Review of Proportions and Similar Triangles
Kyle Sikorski, Rodney Nelson
Level: Geometry
Materials required:
Markers, paper, pencils. (Stickers?)
Lesson Overview:
Students will use this class to clear up any misconceptions they may have about proportions and similar triangles. By doing what?
Lesson Objectives:
· Students will test their knowledge of the aforementioned material. Analysis level
· Students will defend their answers with mathematically correct explanations. Evaluation level
NYS Standards:
5F. Apply proportions to scale drawings and direct variation.
5I. Use geometric relationships in relevant measurement problems involving geometric concepts.
Anticipatory Set: 2-4 minutes
Before class have two similar triangles drawn on the board. Label the triangles ABC and DEF respectively. Have the board labeled as it is below:
1. 2.
AB= <A=
BC= <B=
AC= <C=
DE= <D=
EF= <E=
DF= <F= Are there values missing here? Are you wanting to use the similar sign: ~ or the congruent sign: ?
What are you expecting for responses?
Ask the students what the two properties of similar triangles are and list them as 1 and 2. Next ask them to come up of (with?) an example on their own by filling in the missing parts of the equations. Tell them drawing a figure is a good idea too (what are the students to do with the triangles on the board?). When they are done have each student fill in their answers on the board individually. Ask each student what their proportion was for filling in the sides. It should be noted that if they draw the figure on the board that the largest side is opposite the largest angle. We are not just looking for the fact that their proportions are correct! Also make sure that the students realize that if all the angles of a triangle are congruent then the triangle will always be (at least) similar.
Developmental Activity: 15-20 Minutes
Do the following worksheet as a class. After doing the required number of questions together (see assessment) question completely together (what?), have the students answer each question one at a time and go over each problem one at a time. This will help you keep pacing so that the assessment can be completed. After the students figure out the answers, let one of them come up to the board and present their answer. Make sure to rotate through the students so that everyone has a chance to answer a few questions.
More is needed. Specifically state what to emphasize in each problem.
Find at three more problems.
Assessment: Time?
You will assess the students through their board work. Use the following rubric with each question out of 3, to figure out the student’s grade for their work in class. They should receive a sticker (will you supply these?) for homework if they get at least a 1/2 average (what does this mean?) for the questions they do. This is assuming each student answers at least 2 of the questions. If the number of students in a group cannot easily divide the number of questions evenly (i.e. so that each student gets to answer the same number of questions) make sure you go over enough questions so that this comes out evenly. It is obvious that some problems are more difficult than others. So when deciding who to give each problem to make sure to split up the question assignments and do not just have one person do #8 and #9 (and #6).
Closure: Time?
Have students answer the following question:
Name:______
Answer the following question:
Can two triangles that are similar can they be congruent? Why or why not?
Reword: Can two similar triangles be congruent?
Name:______
1. Are the following true proportions?
a)
b)
2. Solve for x:
a)
b)
c)
3. If 4 tickets to a show cost $9.00, find the cost of 14 tickets.
4. A house which is assessed for $10,000 pays $300 in taxes. What should the tax be on a house assessed at $15,500?
5. In triangle ABC, angle A = 90º and angle B = 35º. In triangle DEF, angle E = 35º and angle F = 55º. Are the triangles similar? Explain your answer.
6. Rectangle ABCD is similar to rectangle A'B'C'D'. If AB = 4, BC = 8, B'C' is 6
units longer than A'B', find B'C'.
7. A vertical flagpole casts a shadow 12 feet long at the same time that a nearby
vertical post 8 feet (tall? high?) casts a shadow 3 feet long. Find the height of the flagpole.
Explain your answer.
8. Answer the following questions:
a.) Are the triangles similar? Why or why not?
b.) Find AC.
Given angle A and angle A' are 59º.
9. The corresponding sides of two similar regular hexagons are 32 and 8. What is the proportion of their perimeters?
Rubric for each student (Teacher only)
Name:______
Criteria / TotalAttempted to answer the correct question (1pt)
Used correct mathematical notation throughout the problem (1pt)
Correct answer given (1pt)
Name:______
Criteria / TotalAttempted to answer the correct question (1pt)
Used correct mathematical notation throughout the problem (1pt)
Correct answer given (1pt)
Name:______
Criteria / TotalAttempted to answer the correct question (1pt)
Used correct mathematical notation throughout the problem (1pt)
Correct answer given (1pt)
Name:______
Criteria / TotalAttempted to answer the correct question (1pt)
Used correct mathematical notation throughout the problem (1pt)
Correct answer given (1pt)
Name:______Answers
1. Are the following true proportions?
a)
No
b)
Yes
2. Solve for x:
a)
x=28
b)
x=4
c)
x=2rm/n Write the fraction using Equation Editor or do you mean for only m
to be divided by n?
3. If 4 tickets to a show cost $9.00, find the cost of 14 tickets.
$31.50
4. A house which is assessed for $10,000 pays $300 in taxes. What should the tax be on a house assessed at $15,500?
$465
5. In triangle ABC, angle A = 90º and angle B = 35º. In triangle DEF, angle E = 35º and angle F = 55º. Are the triangles similar? Explain your answer.
Yes
90+35+55=180 so all angles are equal
6. Rectangle ABCD is similar to rectangle A'B'C'D'. If AB = 4, BC = 8, B'C' is 6
units longer than A'B', find B'C'.
B’C’=12 Show the work for this one!
7. A vertical flagpole casts a shadow 12 feet long at the same time that a nearby
vertical post 8 feet casts a shadow 3 feet long. Find the height of the flagpole.
Explain your answer.
32 feet
8. Answer the following questions:
a.) Are the triangles similar? Why or why not?
b.) Find AC.
Given angle A and angle A' are 59º.
a) All angles are congruent therefore all sides are proportional
b) AC=12
9. The corresponding sides of two similar regular hexagons are 32 and 8. What is the proportion of their perimeters?
192/48 or 48/192
Name:______Answers
Answer the following question:
Can two triangles that are similar can they be congruent? Why or why not?
Yes. When two triangles are congruent the angles are all equal and the sides are proportional (1/1)
So you mean “can two congruent triangles be similar?”