Similar Triangles

Home > Lesson Plans > Grade 10 > Similar Triangles

Lesson Rubric
Similar Polygons Notes
Ratio of Triangles Notes
Area and Perimeter Ratio Notes
Student Sample One
Student Sample Two

Name: Donna McGinnis
Grade: 9 / 10
School: Colton-Pierrepont

  1. Title/Context Of Learning Experience
    This learning experience will investigate similar triangles and the ratios of their sides, perimeters and areas.Standards:

    • A.A.26 Solve algebraic proportions in one variable which result in linear and quadratic equations.
    • G.R.1 Use physical objects, diagrams, charts, tables, graphs, symbols, equations, or objects created using technology as representations of mathematical concepts
    • G.R.3 Use representation as a tool for exploring and understanding mathematical ideas.
    • G.R.8 Use mathematics to show and understand mathematical phenomena (e.g., use investigation, discovery, conjecture, reasoning, arguments, justification and proofs to validate that the two base angles of an isosceles triangle are congruent)
    • G.G.45 Investigate, justify, and apply theorems about similar triangles
    • G.G.46 Investigate, justify, and apply theorems about proportional relationships among the segments of the sides of the triangle, given one or more lines parallel to one side of a triangle and intersecting the other two sides of the triangle.

    Prior to this learning experience, the student will need to know:

    • What polygons are, and what corresponding parts means.
    • Measures of corresponding angles formed by parallel lines.
    • How to write a ratio.
    • Perimeter and Area of triangles.
    • How to set up and solve a proportion.
  2. Essential Questions
    • What makes triangles similar?
    • What relationship is there between the ratios of corresponding sides, perimeters and areas of similar triangles?
    • How can the ratios in similar triangles be used?
  3. Assessment Plan
    • This learning experience is mainly assessed through teacher observation and group discussion as the students participate in the presentation of the lesson.
    • Homework assignments are graded on a 4 – point system based on completion, use of mathematical concepts, and neatness. (see rubric)
    • Quiz on similarity.
  4. Procedure
    Day 1 – Similar Polygons

    • Pass out the student note sheet “Similar Polygons”.
    • Bring up the Geometry Workshop website on the Interwrite SchoolPad. http://www.iknowthat.com/com/L3?Area=GeometryWorkbench. I drew the figures needed on separate pages to save time, but you can have students draw their own.
    • Discuss what the word “similar” means to the students, and have students write it on the notes.
    • On the Geometry Workshop website, draw and label a quadrilateral, then copy and paste another one. Click on the second one, and click resize. When the resize boxes come up, hold the shift key down while pulling one of the corner boxes up or down. (This keeps the sides in proportion when resizing.)
    • Have the students identify the corresponding angles, then measure each with the protractor tool from the Geometry Workshop. The angles should be congruent.
    • Have the students identify the corresponding sides, then measure with the Geometry Workshop ruler tool.
    • Ask the students what the ratio of the corresponding sides is for each. They should all have the same ratio. (Or close to it – the ruler tool is picky)
    • Have students fill in the definition of similar, with corresponding angles congruent, and corresponding sides in proportion (have the same ratio).
    • Draw and label two similar rectangles as ABCD and A’B’C’D’. Have a student measure sides AB, AD, and A’B’ with the ruler tool. Using these values, set up and solve a proportion to find A’D’, then have the student measure the actual length to see if it is correct. Have students work out the next problem on the Interwrite.
    • Draw two similar right triangles on the Geometry Workshop labeled and . Have students measure and record with the protractor tool (one pair of angles should not correspond). Discuss whether the two triangles are similar or not. The students should find they only need two pairs of corresponding angles of a triangle to be congruent for the triangles to be similar.
    • Remind students that if two triangles are similar, the sides are also in proportion. Have them solve the last two problems by sketching a diagram on the Interwrite and solving the proportion.
    • Assign homework: p. 434-435 #7-19.

    Day 2 – Ratios in Similar Triangles

    • Go over the homework from the previous day, and answer any questions.
    • Take the students to the computer lab. Pass out the student note sheet “Ratios in Similar Triangles”,and have them bring up the website on the top of the page: http://nlvm.usu.edu/en/nav/frames_asid_279_g_4_t_3.html?open=activities
    • Show the students how to place and move bands on the Geoboard to make shapes, and how to use the “Measure” tool to find perimeter and area. Remind them to count spaces, not pegs for side length, and to measure a slanted side, they can place a band on the side, highlight, and click the “Measure” tool.
    • Review the concept of similarity from the previous day by working through the first two questions about why the triangles are similar with the students.
    • Have students create triangles on the Geoboard. The students will make a set of overlapping triangles with a common vertex and following the sides of the common angle, making the third sides parallel.
    • Once the triangles are made, the students will count and record the lengths of the common sides of each triangle set and then use the “Measure” tool to record the perimeter and area of each triangle.
    • Have the students open a Word document, then use “CTRL-Print Screen” to copy and paste their triangle into the Word document. This takes a screen shot of their work.
    • Have them each create two more sets of triangles, and find the measures, perimeters and areas to fill in the rest of the tables. Make sure that they CTRL-Print Screen each set of triangles they used into the Word document.
    • Looking at the three sets of data, have students find the ratios of the sides, perimeters and areas of each set of triangles. Fill in the table.
    • Have students figure out the relationships between the ratio of the sides and the ratio of the perimeters (they are the same), and the ratio of the areas (the square of the ratio of the sides), and fill in the theorem conclusions on the worksheet. They will hand in their worksheet and triangles as a homework assignment.

    Day 3 – Ratio of Perimeter and Ratio of Area Problems

    • Hand out “Ratios of Perimeters and Ratios of Areas Problems” note sheet, and put it up on the Interwrite.
    • Review Similarity, Ratios of Perimeter and Ratios of Area by working through the front page with the students. Have them fill in the answers on their sheet.
    • Choose a student to use the Interwrite to label the diagram on the back of the page, and work out the ratio of the sides, ratio of the perimeter, and ratio of the area.
    • Choose different students to draw diagrams on the Interwrite to solve the last three word problems.
    • Assign homework: p. 447-448 #1-16. The next day will be p.448-449 #17-30.
  5. Resources
  6. Instructional/Environmental Modifications
  7. Time Required
    • Day 1 – 40 minute class period
    • Day 2 – 40 minute class period
    • Day 3 – 40 minute class period
  8. Reflection
    • I developed this learning experience so that students could discover for themselves that the theorems and concepts in class really work in practice. They seemed to enjoy the use of the websites in class, and had fun with the Geoboard manipulative in the computer lab.
    • This lesson could easily be adapted for use in a computer lab or with a SmartBoard if you don’t have an Interwrite.
    • There was some difficulty with measuring the sides of the triangle on the Geoboard if the sides of the triangle weren’t perpendicular. Instead of having to use the Pythagorean theorem to find the sides of the triangle, Roberta suggested putting a band on the side of the triangle, then highlighting it so the measure tool could be used. Thanks for the help!