2012-13 and 2013-14 Transitional Comprehensive Curriculum
Grade 8
Mathematics
Unit 3: Transversals, Surface Area and Volume
Time Frame: Approximately three weeks
Unit Description
The content of this unit focuses on the properties of the relationships among angles formed by parallel lines; determine surface area and volume of cylinders, spheres and cones.
Student Understandings
Students can apply terms appropriately when discussing the relationships between angles formed by parallel lines. Students can solve problems involving surface area and volume relationships of prisms, cones, spheres and cylinders.
Guiding Questions
1. Can students define and apply the terms adjacent, corresponding, vertical, alternate interior, complementary, and supplementary angles appropriately and use them in discussing figures?
2. Can students determine dimensions of three dimensional figures and apply these dimensions to find surface area and volume?
3. Can students define and apply the terms adjacent, corresponding, vertical, alternate interior, complementary, and supplementary angles appropriately and use them in discussing figures?
4. Can students define and apply the terms adjacent, corresponding, vertical, alternate interior, complementary, and supplementary angles appropriately and use them in discussing figures?
5. Can students apply and interpret the results of surface area and volume considerations applied to prisms, cylinders, and cones?
Unit 3 Grade-Level Expectations (GLEs) and Common Core State Standards (CCSS)
Grade-Level ExpectationsGLE # / GLE Text and Benchmarks
Geometry and Measurement
17. / Determine the volume and surface area of rectangular prisms and cylinders (M-1-M) (G-7-M)24. / Demonstrate conceptual and practical understanding of symmetry, similarity, and congruence and identify similar and congruent figures (G-2-M)
28. / Apply concepts, properties, and relationships of adjacent, corresponding, vertical, alternate interior, complementary, and supplementary angles (G-5-M)
CCSS# / CCSS Text
8.G.4 / Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotation, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.
8.G.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.
8.G.9 / Know the formulas for the volume of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.
8.EE.2 / Use square root and cube root symbols to represent solutions to equations of the form x2 = p and x3=p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that is irrational.
Writing standards for Literacy History/Social Studies, Science and Technical Subjects 6 - 12
1.b / Support claim(s) with logical reasoning and demonstrate an understanding of the topic or text, using credible sources.
Sample Activities
Activity 1: Angle Relationships (CCSS: 8.G.5)
Materials: Vocabulary Self-Awareness Chart BLM, Drawing A, B, and C BLMs and Making Conjectures BLMs; tracing paper or patty paper, paper, pencil
Begin the activity by distributing the Vocabulary Self-Awareness Chart BLM. Students will begin the vocabulary self-awareness (view literacy strategy descriptions). Since students completed a Vocabulary Self-Awareness in Unit 1, a suggestion is to remove the words from the chart and have them write the words in from a list provided on the board. Students should rate their understanding of each of the vocabulary words by placing a plus sign (+) if they are very the word, a check mark (ü) if they are uncertain of the exact meaning and a minus sign if the word is completely new (-) to them. Students will begin the unit by writing what they know about the definition of the words in the list. They will also give what they think is an example of the word. Explain to the students that some of the definitions and examples might be guesses, but it is important that they make an effort to write a definition. As these terms are explained during activities, they will be given an opportunity to revise the definitions and examples. Through this unit, students should develop an understanding of the vocabulary in the chart. Throughout the unit, students should be encouraged to pull out the chart and edit their chart as the vocabulary meanings become more understandable. The repeated use of the vocabulary chart will give the students multiple opportunities to practice and extend their growing understanding of the vocabulary.
Distribute copies of the facilitator page, Drawings A, B, or C BLM and the handout, Angle Relationships. Each table should be given 4 copies of the same drawing, but make sure to assign each of the drawings A, B, or C to at least one table group. Tell participants they will investigate angles that are formed by intersecting two lines with a transversal. A transversal is a line that intersects two or more other lines.
Have participants follow the directions on the BLM. As participants work, check to see that they are comparing angles within each set of four and between the sets of angles.
Drawings A and C have parallel lines cut by a transversal. Congruent angles on drawings A and C are a and d, b and c, e and h, f and g, a and e, b and f, c and g, d and h, c and f, d and e, a and h, and b and g.
In drawing B, the lines cut by the transversal are not parallel. Congruent angles on Drawing B are a and d, b and c, e and h, f and g.
Hand the students the Conjectures page 2 BLM . It is important not to put these statements on the first BLM because the students need to find the congruent angles with their tracings. Make sure that all participants have found these congruent angles.
Have participants state conjectures that they can make about vertical angles. (They are congruent. They are opposite angles formed by the intersection of 2 lines. Two pairs are formed when 2 lines intersect.)
Facilitator Note: Proposition 15 in Euclid’s Elements, Book I, defines vertical angles and then goes on to prove that they are equal. Since they are basing their conclusions on just a few examples, their conclusion is a conjecture.
Ask participants to raise their hands if they found angles a and e; b and f; c and g; and d and h congruent. All tables with Drawings A and C should raise their hands. But tables with drawing B should not.
Have participants color each pair of angles (a and e; b and f; c and g; and d and h) a different color on one of the drawings at their table. Coloring the pairs of angles will help participants see the positions of the angles. Discuss the positions of the angles. Make sure that those who have Drawing B also color these angles. They may not think they need to do anything since they don’t have these as congruent angles. Explain in words the position of the pairs of angles using math vocabulary explained in activity 1.
Ask participants to state which angles are alternate interior angles. (c and f, e and d)
Have participants answer the question on the slide. (Pairs of alternate interior angles are congruent.)
Ask participants to raise their hands if they found these pairs of angles congruent. All tables with Drawings A and C should raise their hands. But tables with drawing B should not.
Have participants follow the directions on the slide. The different colors will help participants see the positions of the angles.
Discuss the locations of the pairs of angles. This will lead to the definitions.
Ask participants to state which angles are alternate interior angles. (c and f, e and d)
Ask participants to share their answer to question 15. (Pairs of alternate interior angles are congruent.)
Have students take out their vocabulary awareness chart and make any revisions that can be made after today’s lesson. As closure, ask students to complete an exit card and explain the difference in alternate exterior angles and corresponding angles from drawing A.
Activity 2: Triangles and Transversals (CCSS: 8.G.5)
Materials: Triangles and Transversals BLM, pencil, paper, scissors, ruler or other straight edge
Divide the students into groups of four. Have groups fold a sheet of paper into thirds and cut a triangle so that there are three copies of the triangle that are congruent. Instruct the students to place the triangles on their desk or table so that the base of the three triangles forms a line (see diagram). Before beginning the discussion strategy, encourage students to place a straight-edge along the top vertices of the triangles and justify why this straight-edge forms a parallel line with the base of the three triangles. Each student must give one justification about angle measures, and go around the group giving a different justification. They may have one opportunity to “pass” but must give at least one response on the second round. After groups have had time to come up with some ideas, have the groups share one idea with the class and write it on the board for all to see until all justifications have been stated. Once the students have given their justifications, have them mark the angles that are equal by using one color for each of the angles that have the same measure.
Ask students to determine the measure of the angle (?) between the red and yellow angle. This angle will be congruent to the blue angle. Red + Yellow + ? = 180 degrees, straight line.
Distribute Triangles and Transversal BLM and have the students complete these questions independently and then review their responses with their shoulder partner.
Activity 3: Angle-Angle similarity (CCSS: 8.G.5)
Materials: Grid Paper BLM, Angle Similarity BLM, colored pencils
Have students plot points A(2,6); B(5,4) and C(3,9) to form triangle one and plot points A(2, 6), E(8, 2) and F(4,12) to form DAEF. Have the students outline triangle ABC in one color and DEF in a second color to make it easier to compare the triangles. Ask students to state how these triangles are related (similar triangles). Have students identify the corresponding angles and justification as to why the angles are corresponding. Ask the students what conjecture they could write about the angle measurements in similar triangles. Challenge the students to prove that their conjectures are true by sketching at least two more sets of similar triangles and justifying their conjectures. Corresponding angles are congruent. Sides are proportional.
Distribute Angle BLM and give students time to test their conjectures with the triangles given.
Lead the class discussion so that it is clear that if two angles of two triangles are congruent, then the triangles are similar.
Activity 4: The Net! (GLEs: 17)
Materials list: Triangular Prism BLM, Right-Triangular Prism BLM, tape, scissors, rulers, pencils, paper
This activity has changed minimally because it already incorporates this CCSS.
Using a shoe box from home and one other rectangular prism box, have students discuss the number and location of faces, vertices and edges. Have measurements of the boxes used for modeling written on the boxes and the board. Lead a review about how these measures are involved when finding surface area of a rectangular prism.
Provide students with the Triangular Prism BLM and have them fold and tape it together to form a triangular prism. The Triangular Prism BLM is an equilateral triangular prism. If time is a factor, have students cut out and tape together these nets at home the day before this activity begins. Ask students to determine the number of faces, edges, and vertices.
Label all lengths with l, w, and h, trace each length with one color, width with a second color, and height with a third color. It will not matter which is which, but it is important to have the three colors and to make sure that whichever edge is the length, all other edges that represent the length are the same color. This should be true for the width and height edges, also. Have students find the area of one face of the prism and write mathematically how to find the area of the face. Provide rulers for measuring lengths so that the groups can find the areas. Make sure the students realize that this is not a right triangle and that they have to find the height of the equilateral triangles. These triangles are located at either end of the center rectangle region of the net and the students will discover that they can fold it in congruent parts to find the height of the triangles. This is also a good time for a discussion about congruency. Challenge the students to find the surface area of the entire net. Students should be prepared to explain the method used to find the surface area. Write the different methods on the board and have students compare these different methods. Students should prove that these different methods are equal. Make comparisons of these methods and the formula used on the LEAP Reference Sheet.
Next, provide students with the Right Triangular Prism BLM and have students construct the prism by appropriately folding and taping it together. Determine faces, edges, and vertices. Have students discuss shapes that make up each face of the triangular prism. Determine a method of finding the area of each face. Have students identify faces, edges, and vertices. Have students use the Pythagorean Theorem to determine the area of the right triangular ends of the prism as they find the surface area of the right-triangular prism. Have students share methods by putting different methods on the board for discussion.
To extend this activity, have students construct a triangular prism at home and bring it to school so that the class can arrange the prisms in order from least to greatest volume and/or surface area.
Activity 5: Volume and Surface Area (GLE: 17; CCSS: 8.G.9)
Materials List: Volume and Surface Area BLM, 16 cubes for each pair of students, paper, pencil, calculators, math learning log