Problem-Based Instructional Task Lesson Plan Parallelograms Local Learning Goals: Be able to describe what a parallelogram is Understand and be able to apply properties of parallelograms Title: Parallelograms Grade Level/Course: Integrated 2 (10 th & 11 th ) Estimated Time: One class period Pre-requisite Knowledge: Triangle congruency & similarity Knowledge of bisectors Properties of angles (corresponding angles, supplementary angles, vertical angles, co-interior angles, alternate interior angles, etc.) NCTM Standards: NCTM Content Standards Number & Operations Algebra Geometry Measurement Data Analysis & Probability NCTM Process Standards Problem Solving Reasoning & Proof Communication Connections Representation NCTM Content Standard Goals Geometry NCTM Content Standards: o Analyze properties and determine attributes of two- and three-dimensional objects. o Explore relationships (including congruence and similarity) among classes of two- and three-dimensional geometric objects, make and test conjectures about them, and solve problems involving them. o Establish the validity of geometric conjectures using deduction, prove theorems, and critique arguments made by others. Reasoning and Proof NCTM Content Standards: o Recognize reasoning and proof as fundamental aspects of mathematics. o Make and investigate mathematical conjectures. o Develop and evaluate mathematical arguments and proofs. o Select and use various types of reasoning and methods of proof. Connections NCTM Content Standards: o Recognize and use connections among mathematical ideas. o Understand how mathematical ideas interconnect and build on one another to produce a coherent whole. Materials Needed Audio-visual: n/a Manipulatives: ruler, protractor, graph paper, worksheet of given parallelogram (see Appendix A) Technology/Software: n/a Literature: n/a Other: n/a LAUNCH LESSON DEVELOPMENT Objective: To learn what a parallelogram is and the properties associated with it. Problem-Based Instructional Task Lesson Plan - Parallelograms Page 1 of 7
Prior to this lesson, the students should have explored simple properties of angles through a set of parallel lines or by two single crossed lines, have knowledge of bisectors, as well as basic triangle congruency. This prerequisite knowledge will set the stage for learning and exploring the properties of parallelograms. Once the class is situated, distribute a sheet of graph paper to each student, as well as a ruler and protractor, if needed. Instruct the students to draw a couple of non-similar parallelograms, if they know what they are. Specify that there could be three or four of them. The goal is for students to draw a figure that most represents a parallelogram. This activity should be a diversion from the norm and rather engaging. Promote the students to let their creative juices flow by giving them freedom to draw something that might resemble a parallelogram. Walk around to each student and observe what he/she has drawn thus far. If a student is stuck, encourage them to collaborate with their neighbor. If that doesn t help, ask him/her to think about what the word parallelogram might imply. The name of the quadrilateral gives it away. If more than half of the students have drawn either a rectangle, square, rhombus, or most importantly, a parallelogram after roughly, 5 minutes, then it s time to move on. This assessment shows that the students have an inkling as to what a parallelogram looks like, which was the goal in this first activity. Ask four different students to come up to the board and draw one of the parallelograms that they drew on their paper. Specify to the second, third, and fourth student that they cannot draw a similar figure to what the previous student(s) drew. Draw any missing figures (either a parallelogram, rectangle, square, or a rhombus) on the board. After all four types of parallelograms have been drawn on the board, ask the class how each of them are related. The students should be able to see that each of the figures have two sets of parallel lines. From here, draw a tree diagram on the board depicting how all four types of parallelograms are related to one another, as well as how they are related to other quadrilaterals. An example of the quadrilateral chart is shown below. (Rubenstein, 1995) EXPLORE After the class discussion about how parallelograms relate to other quadrilaterals, the students will be given the opportunity to explore the different properties of parallelograms by being allowed time to measure the sides (to the nearest tenth of an inch) and angles (to the nearest degree) of either the parallelograms they constructed or one that is given. They will be given a ruler and a protractor to help with this task. This is a great opportunity for the teacher to take the time to teach them how to use a protractor if they do not know or remember how. It is also a moment to assess student knowledge of measuring geometric figures. The idea behind this is to help them begin to notice the different properties that parallelograms hold. Guiding questions can be asked to help keep a student on the right track to discovering parallelogram properties (see below). Once each student has successfully measured his/her different parallelograms, the class will be brought back together and given an opportunity to share the different properties they noticed through construction and exploration. It is encouraged that they follow along and take notes on the sheet with the given parallelogram. The properties they discovered will be written on the board by the teacher as a student presents them. This will be a way to bring all of the ideas together into one place, and it will provide the students with a source they can utilize while working on their homework later. The properties the students discovered will then be checked by using the parallelograms that were provided. After this discussion, class will move forward by focusing on the properties of the parallelogram diagonals by having the students recall triangle congruency properties and angle properties. Problem-Based Instructional Task Lesson Plan - Parallelograms Page 2 of 8
On the board, the teacher will have the given parallelogram drawn from the worksheet. This parallelogram will then have its diagonals drawn. Each angle created within the parallelogram will be labeled on the board (1-12). Using properties of angles and triangle congruency properties, the class will be asked to determine the measurement of the different labeled angles. The students will be allowed to discuss their ideas with their partner for a minute. As visual aids, parallel lines with a transversal may be drawn on the side to help them recall the angle properties. The teacher will bring the students back together as a class where they will be asked to state any discoveries they might have made. The students will be guided to discover the property that the diagonals of a parallelogram bisect each other. They will be able to discover this using the congruent angles they found from the angle properties and the triangle congruency properties they previously learned. Once they are successful with this, they will be ready to begin work on the homework assignment. At this time, the students will be allowed to work in pairs and will have the opportunity to ask questions of the teacher to clarify or clear up any information from the lesson they did not understand. Key Ideas Key ideas and important points in the lesson: The different kinds of parallelograms. Opposite sides of parallelograms are congruent. Opposite angles of parallelograms are congruent. Opposite sides of parallelograms are parallel. Consecutive angles in parallelograms are supplementary. Teacher strategies and actions to ensure that all students recognize and understand the key ideas and important points (e.g., ask targeted questions, facilitate minisummary, point out key problems in the lesson, etc.): The quadrilateral chart will help depict how the different kinds of parallelograms are related. From this chart, students should be able to see how different types of parallelograms are related, as well as how they are related to other quadrilaterals. The parallelograms that the students drew should help the students recognize that all parallelograms have two sets of congruent sides. This can be emphasized with the measurements that the students produce after having measured the side lengths of their parallelograms. The parallelograms that the students drew should help the students recognize that all parallelograms have two sets of congruent angles. This can be emphasized with the measurements that the students produce after having measured all of the angles in their parallelograms. Students will notice that a parallelogram, rectangle, square, and rhombus all have two sets of parallel sides. Summarize that those four figures are all related, but it is important to keep with the original nomenclature of each figure so as not to confuse it with anything else. Specify that for the remainder of the lesson, a parallelogram will be depicted as two sets of parallel lines, one of which has a small slant to it. The parallelograms that the students drew at the beginning of class will be useful for this property. Students should notice that every parallelogram contains two sets of supplementary consecutive angles. One key idea that would help the students visualize this property would be to draw the below picture on the board and explain to the students HOW it is know that there are two sets of supplementary consecutive angles in parallelograms. It might be best to take a step back by drawing a set of parallel lines on the board with a transversal through it and explaining why it relates to parallelograms. (Classzone.com) Problem-Based Instructional Task Lesson Plan - Parallelograms Page 3 of 8
Diagonals of parallelograms bisect each other. A parallelogram with its diagonals will be drawn on the board. Each angle within the parallelogram will be labeled, and the students will be provided an opportunity to determine which angles are congruent. They will be guided to use the angle properties and triangle congruency properties in order to discover that the diagonals of a parallelogram bisect each other. An idea that will help the students determine which angles are congruent will be to draw parallel lines with a transversal and have them compare the properties they know about angles there with the angles of the parallelograms. Once they know this, it may also be useful to draw the triangles to the side of the parallelogram. This would be a way for them to see the information they know and that they discovered. Guiding Questions Good questions to ask students: Possible student responses and actions: Possible teacher responses: What will you do? How will you respond? How do you know what you have drawn is a parallelogram? Because these two sides are parallel. Yes, those two sides are parallel. What about the other pair of sides? What can you say about those? Why do you consider this [rectangle, square, or rhombus] a parallelogram? They have two sets of parallel sides. What else about these figures? Can you say anything about their diagonals? Is there another way in which you could measure the length of the side lengths? What if the parallelogram were sitting on a Cartesian plane? You could use the distance formula to determine the side lengths. Besides the side lengths, is there anything else in the parallelogram that you would be able to measure using the distance formula? What ideas or properties of parallel lines have you previously learned in class that might help you? Does it have something to do with angles? Yes, what do we know about alternate interior angles when looking at parallel lines and a transversal? How do you know that these two diagonals bisect each other? Because they cross. Close, but not quite. Yes, they cross, but there is something specific about where they cross at. Misconceptions, Errors, Trouble Spots Possible student misconceptions, errors, or potential trouble spots: Inconsistencies with measuring their parallelograms, especially the given parallelograms. The diagonals bisect the angles of a parallelogram. Teacher questions and actions to resolve misconceptions, errors, or trouble spots: Remind the students to measure to the nearest tenth of an inch. Ask the students if they can use properties of angles and the lengths of the sides and diagonals to show that the diagonals indeed bisect each other. Problem-Based Instructional Task Lesson Plan - Parallelograms Page 4 of 8
Trouble using the protractor. Experiencing difficulties with identifying different angles within a parallelogram. Noticing the differences between a parallelogram, rectangle, square, and a rhombus. Provide examples on how to use a protractor and visually show them the technique. Show them how to extend the sides of a parallelogram in order to see parallel lines with a transversal through them. Emphasis the basic properties that each of these quadrilaterals hold. Allow the students to take note of these differences and how they are similar/different. SUMMARIZE: Before the homework assignment is handed out, the teacher will ask the students questions involving the main points of the lesson. They will need to understand what the properties of parallelograms are in order to do the homework. The properties include the following concepts: opposite sides of parallelograms are parallel; opposite sides of parallelograms are congruent; opposite angles of parallelograms are congruent; consecutive angles of parallelograms are supplementary; and the diagonals of a parallelogram bisect each other. The students will be reminded that they can use the angle properties and triangle congruency properties they previously learned to discover these facts. If they understand this material, they will be ready to begin work on the homework. MODIFY/EXTEND If students have finished before their peers during the launch of the lesson, challenge them to come up with all four types of parallelograms (parallelogram, rectangle, square, and a rhombus). If this has already been accomplished, question the students how each of the parallelograms are related. Specifically, ask them about the properties that each contain and how they are similar or different. For those students who were able to correctly measure the lengths and angles of their drawn and given parallelogram, challenge the students to re-draw a new parallelogram on a Cartesian plane, with the four vertices landing on different coordinate pairs (x, y), and to determine the side lengths by using the distance formula. If they do not know this formula, teach it to them. Ask them to investigate if they know of any other lengths in which they could determine other than the line segments between each coordinate pair (diagonals). If they notice these diagonals bisect each other, challenge them to determine if the midpoint of the diagonals. How do they know that the diagonals bisect each other? CHECKING FOR UNDERSTANDING One of the main indicators that the students understand the properties of a parallelogram are if they recognize that there are two sets of congruent triangles within the parallelogram. From here, they can solve any proof handed to them. Knowing and understanding the essence of triangle congruency is key to understanding the connection to the mathematical concept of parallelograms. Assessment of each student can be determined throughout the lesson as the teacher observes the students drawing, asks the students thought-provoking questions while the students are measuring their parallelograms related to possible properties that parallelograms could possess, as well as through classroom guided discussion. The students will be provided time to begin working on their homework at which point the teacher will have the time to observe student work and determine which students understand the material and which students need more direction. At the end of the lesson, ask the students if they can think of any situation in real life in which parallelograms would come in handy. One such example would be playing billiards. If they ever wanted to become an amazing billiard player, it would be wise to study the angles and parallelograms existent in the game of billiards. Question the students if they found this lesson interesting and what the most significant thing that they learned. How would they go out and apply parallelograms to their everyday life? REFLECTION On Thursday, October 3 rd, Ben Hellman and I co-taught a lesson on parallelograms to two separate class periods. We met up the night before and discussed how we wanted to teach our lesson. Mr. Gassman sent us the homework covering parallelograms a day or two earlier so that we knew how much the students needed to know and so we used the homework as a guideline as to how to structure the lesson. I was surprised to see three proofs on the Problem-Based Instructional Task Lesson Plan - Parallelograms Page 5 of 8
last two pages, especially since the students would have had only a day working with parallelograms. We did our best to create a lesson that allowed the students to discover the properties of parallelograms for themselves so that they had a solid understanding of how the properties related to the homework. Ben and I were very confident of the material in our lesson. At the beginning of the class period, we instructed the students to draw what they thought a parallelogram was on a sheet of graph paper. In the first section, three students drew a combination of parallelograms and one trapezoid. Four students did the same thing in the second section. Ben and I conversed to the side that this would help lead a nice discussion as to what a parallelogram was and was not. I had only assumed students would forget to draw a rhombus, not that students would draw a trapezoid. When I chose students to come up to the board to draw their figures, I unambiguously chose one student that had drawn a trapezoid on their paper to draw his/her figure of choice on the board and miraculously, he/she drew a trapezoid. This led to a nice discussion about what was considered a parallelogram and how it related to other quadrilaterals. At the end of the day, I asked Mr. Gassman if there was anything that he would have changed in our lesson or something that we could have done better. It took him a minute, but he soon realized that he would have changed the structure of the quadrilateral chart that I had drawn on the board. Instead of following the chart in the lesson plan, I accidentally drew something like this: parallelogram rectangle square rhombus I m glad that he pointed this minor mistake out because it made me realize how conscious I need to be with definitions and charts. Students take this information for granted and if it is presented incorrectly, they are bound to be confused later on. Another interesting thing I noticed was that many students had a difficult time measuring the angles of the parallelogram using the protractor. There were two reasons for this. First, the parallelogram was free-handed, which allowed the variance of human error to increase. Secondly, the students did not know how or did not remember how to use a protractor. Ben and I walked around and reminded them how to use it and they caught on very quickly. Lastly, one student in the first section noticed that the parallelogram was made up of a bunch of triangles and that each of the angles within the triangle added up to 180. This was a KEY POINT for understanding how triangles play a role in parallelograms. Since this student was familiar with all of the parallelogram properties AND he noticed this one unique fact, I m confident that he would have gathered that there are two sets of congruent triangles in every parallelogram. Compared to the other students, he was already thinking strategically about parallelograms and the unique qualities about them. I was very proud of the student, as well as myself and Ben, for we had taught the lesson to him. Our lesson must have been successful! Problem-Based Instructional Task Lesson Plan - Parallelograms Page 6 of 8
Works Cited Classzone.com. Proving Quadrilaterals are Parallelograms. Retrieved from http://www.nexuslearning.net/books/ml-geometry/chapter6/ml%20geometry%206-3%20proving%20quadrilaterals%20are%20parallelograms.pdf Rubenstein, R. N., Craine, T. V., Butts, T. R. (1995). Integrated Mathematics 2. Boston: McDougal Littell. Problem-Based Instructional Task Lesson Plan - Parallelograms Page 7 of 8
Appendix A Problem-Based Instructional Task Lesson Plan - Parallelograms Page 8 of 8