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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 Expectations |

|GLE # |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 [pic] 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 (() 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 (AEF. 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

This activity has not changed because it already incorporates the CCSS.

Give student pairs a set of 16 one-inch cubes or centimeter cubes, and ask them to build all possible rectangular solids. Have students count the number of cubes to determine the volume of the solids and count the number of exposed faces to calculate the surface area of each solid built, recording information on the Volume and Surface Area BLM. Have students make sketches of the solids and label the dimensions of the rectangular prisms built.

Have students repeat the exercise using a different number of cubes and record the information in the chart.

Ask students to study their findings and list their observations. Make sure observations include the relationship of the surface area and the shape of the rectangular solid (i.e., the closer to the shape of a cube, the smaller the surface area).

Have students take 8 of their cubes and build a cube. Ask them what the area of one face of the cube would be. (4) Review the square root of a number by asking what the square root of 4 would be. Ask them to give the dimensions of the cube. (2 x 2 x 2) Review the cube root of a number by reminding the students that it is the number that can be multiplied by itself 3 times to get the number or it might be a number with an exponent of 3. It is written 23. If they need to find the length of a side of a cube with a volume of 23 they take the cube root of the number. Therefore, [pic]. This is different than square root because they must have three of the same number that multiply together to find the cubed root. It is written [pic]= 2. Ask the students to show the cube root of 27 and 64. If calculators are used that find the cube root of a number, show students how to use the calculator. Have students list the first 5 cube numbers (1, 8, 27, 64, 125). Have students sketch a cube that has a volume of 216 cubic units by recording the length of the side [pic] and use their calculator to find the cubed root.

Have students respond to the prompt in their math learning logs (view literacy strategy descriptions).

Measurements of rectangular solids can be linear, square and cubic units. These units refer to . . .

Once the students have responded to the prompt, have students share with their responses with their shoulder partner.

Activity 6: Cylinders (GLE: 17; CCSS: 8.G.9)

Materials List: LEAP Reference Sheet BLM, pencil, paper, math learning log, compass, ruler

This activity has not changed because it already incorporates the CCSS.

Explain to the students that they will use SPAWN writing (view literacy strategy descriptions).to answer the question “What If?” SPAWN is an acronym that stands for five categories of writing prompts (Special Powers, Problem Solving, Alternative Viewpoints, What If? and Next). SPAWN prompts can be posed before, during, or after a lesson to foster students’ higher-level understanding of content. The What If? SPAWN writing prompt below was written to stimulate the students’ reflective and critical thinking about surface area and volume.

What if you form a cylinder from a piece of paper by rolling it so it is long and then rolling it so that it is short? Will the volume and surface area remain the same?

Ask students to record their thoughts in their math learning logs (view literacy strategy descriptions). This is informal SPAWN writing and should not be taken as a grade. It is important that the students know the importance of communicating mathematically (give points for completion if necessary). Invite students to share their responses to the What If? prompt and use the students’ SPAWN writing responses for whole class discussion.

Have students create a cylinder from standard [pic] inch x 11 inch paper. Engage the class in a discussion about how to make a cylinder with a circumference of [pic] inches without cutting or tearing the paper. Have students roll paper to form a cylinder shape. Ask what the height of this cylinder is. (The height of the cylinder will be 11 inches). Ask the students to form a cylinder with a circumference of 11 inches made from the same sheet of paper. What will the height be? (8 ½ inches)

Show students the circles formed when the paper is rolled so that the circles have a circumference of 8 ½ inches. Ask them again what the circumferences of the circles are and how they know. (The circumference is formed by the side of the sheet of paper which is 8½ inches long.) Ask students if they remember the formula for the circumference of the circle. (A = 2(r

+2) Then ask how they can find the diameter if the circumference is known. This is a good time to use the LEAP Reference Sheet BLM and have them substitute values into the circumference formula to find the diameter of the cylinder using 3.14 for pi.

Once the diameter’s length is determined, have students use a ruler and a compass to measure the radius needed to construct the circles for the cylinder. Have the students draw circles with their compasses and cut them out with scissors. Before the cylinder is assembled, have students determine the volume of the cylinder. After the cylinder is assembled, have students determine its surface area using the formula on the LEAP Reference Sheet. Have students take out their SPAWN writing notes and read them. Students should make any additions or deletions needed after constructing the cylinder and using the formula.

Have the students determine the volume and surface area of the cylinder with circumference of 11 inches and compare the volumes and surface area to the other cylinder. Refer to the Spawn writing that was used initially and give students time to make any additions or changes so that their notes will be accurate.

Have students determine volumes and surface areas by measuring cylinders with U.S. system units and then repeat the activity using metric measures. Have students make a table of the surface area dimensions given in metric and put the U.S. measurements in the column beside the metric units for comparison. Have students make observations of the comparisons.

Activity 7: Cylinders and Cones (CCSS: 8.G.9)

Materials List: Cylinders and Cones BLM, scissors, tape, paper, pencil, rice, beans or un-popped popcorn

Have students find examples of cones used in everyday life and make a list of these as students recall where they might have seen a cone. The students have previously created a cylinder with an [pic] inch by 11 inch sheet of paper and discussed the circumference. Have them cut out the models for the cone and cylinder on Models of Cylinder and Cone BLM. Have students compare the volumes of the cone and the cylinder by filling the cone with rice (beans, corn kernels) and pouring it into the cylinder until the cylinder is filled. The cone has been created to fit inside the cylinder so that the comparison of volumes can be made. FYI -The cone is 1/3 the volume of the cylinder with the same circumference and height.

Have students work in groups of four to make a graphic organizer (view literacy strategy descriptions) comparing the relationship of the cone to the cylinder. Possible example at the right shows a Venn Diagram comparison. Have groups share their graphic organizer with at least one other group, writing questions on post-its as to the relationships that are not understood. Address the questions either by group or as whole class to ensure that students can understand the relationship.

Activity 8: Comparing Cones: (CCSS: 8.G.9)

Materials List: Cones BLM, ruler, pencil, paper, scissors

Begin this activity with the SQPL literacy strategy (view literacy strategy descriptions) by taping a diagram or the cut-out of the comparing cones BLM on the board. The SQPL strategy is used to engage the students in purposeful learning as they develop questions that they would like to have answered about the statement posted.

The strategy begins as the teacher posts a statement about the lesson that is either true or false, but not outrageous. The statement is, “The greater a cone’s height, the greater the volume.”

Have pairs of students develop 2 – 3 questions that they would like to have answered about the statement. Once the students have had time to develop their questions, have students from each group share one question. Continue this until all groups have posted at least one question and two if there are enough questions developed.

Distribute the Comparing Cones BLM and the Model for Cone BLM. Have students cut out the circle leaving the points so that they can use them for the activity.

Have students cut along the radius and form a cone by moving point L so that it lies on top of point A.

Teacher Note: Students will begin to make connections at this point to the relationship of height and the volume, they should begin to see that the circumference is decreasing as the height is increasing and the volume is decreasing.

Instruct students to measure the diameter, circumference, and height of the cone and to record the measures on the Cones BLM. Have students calculate the volume of the cone. Next, have students form a second cone by sliding point L so that it lies on top of point B. Each time a new sized cone is formed, have students record the diameter, circumference and height. After the students have completed forming cones by moving point L to at least 5 different locations, have them find the volume of each of the cones and develop a conjecture about how the change in circumference affects the volume in their math learning logs (view literacy strategy descriptions). Students have discussed the idea that ( is an irrational number, so have them use the symbol for pi in their volume measurement. This measurement should give them enough information to develop the conjecture. Until this point, ( has been given the approximate value 3.14 for the value of (. In 8th grade with the CCSS, students developed an understanding of irrational numbers and since ( is irrational, sometimes they will use the approximation (3.14 or [pic] ) for the value of pi and other times they will use the symbol ‘(’ in their answer.

Once the students have determined the volume of at least 5 cones formed by the cut-out, have them look back at their list of questions and determine if each of the questions can be answered. Discuss these questions and answers as a whole class to ensure that all students are developing an understanding of the relationship of the height, circumference and volume of cones made from the same size circle.

As closure, ask students to explain whether the SQPL statement was true or false and justify their choice.

Activity 9: Spheres: (CCSS: 8.G.9)

Materials List: Spheres BLM, balls of various size (enough for one ball per group of 4 students), paper, pencil

Ask students to work with a shoulder partner and think about which measurements will be necessary when finding the volume of a sphere. Students will probably mention ( because of the circle used with the cone and cylinder in activities in this unit. Distribute Sphere BLM, and ask students to predict the volume of the ball that their group was given to use. Have students record their predictions on the BLM. Discuss predictions and then tell them that the formula for finding the volume of spheres is [pic]. Ask students to work with their group and determine a method to find the radius of their group’s sphere (ball). Students may use the circumference formula and determine diameter or radius this way or some students have stood books at 90( angles to the floor and place the ball in between the books and then measured the distance between the books. Have students find the volume and record the volume using the symbol ( rather than using 3.14 for the value.

Have students go back to their prediction for the volume of their group’s sphere.

Instruct the students to complete the BLM with practice finding the volume of spheres. Discuss results and, as an exit ticket, have students explain what measurements are needed to find the volume of a sphere and hand it to you as they leave class.

Sample Assessment

Performance assessments can be used to ascertain student achievement.

General Assessments

• Provide the student with unlined paper and rulers. The student will design a stained-glass window to show understanding of the terms midpoint, bisector, perpendicular bisector, symmetry, similar, complementary, supplementary, vertical angles, corresponding angles, and congruent angles. The student will label the different components of his/her stained-glass window to ensure that examples have been included for each of the vocabulary words from the unit. The student will present his/her stained-glass sketch to his/her group and justify examples to the group members. The teacher will provide the student with a rubric to self-assess his/her work prior to presentations and teacher evaluation.

• Provide the student with several right triangles that have a missing side measure. The student will find the lengths of the missing sides.

• Whenever possible, create extensions to an activity by increasing the difficulty or by asking “what if” questions.

• Have the students to produce a portfolio containing samples of experiments and activities.

• Have the students to create a scale drawing. A rubric that assesses the appropriateness of the scale factor, as well as the accuracy of the drawing, will be used to determine student understanding.

Activity-Specific Assessments

• Activity 1: Provide the student with a sketch of a picnic table with crossed legs. In a math learning log entry, the student will explain the relationships of the angles formed by the legs of the table.

• Activity 1: Provide the student with a list of vocabulary (complementary angles, supplementary angles, vertical angles, adjacent angles, corresponding angles, corresponding angles) used in the unit.. Have the student use diagrams to illustrate each of the vocabulary words and write the meaning in his/herown words.

• Activity 9: Provide the student with a circumference and a volume; have him/her work to develop a cylinder that with the given volume.

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