[list][list][*]Subject: Mathematics[/*][*]Class: 7th grade[br][/*][*]Duration: 50 minutes[br][/*][*]ICT tools: teacher's computer with projector, student computer, tablets or smartphones.[/*][/list][/list]

[i]Parallelogram - definition and characterization theorems[/i]

[i]At the end of the lesson, students will be able to:[list][*]to identify/recognize a parallelogram;[/*][*]to recognize parallelogram properties in different situations;[/*][*]to apply parallelogram properties to solve problems.[/*][/list][/i]

[i]During the lesson, students will become able to:[br][list][/list][list][*]identify parallelogram according to opposite sides parallelism;[/*][*]know/use that each diagonal divides a parallelogram into two congruent triangles; [/*][*]characterize a parallelogram as a quadrilateral with two pairs of congruent opposite sides (angles);[/*][/list][/i][i][list][*][i]characterize a parallelogram as a quadrilateral with one pair of congruent and parallels opposite sides;[/i][br][/*][*][i][i]characterize a parallelogram as a quadrilateral with pairs of supplementary adjacent angles;[/i][/i][/*][*][i][i][i]characterize a parallelogram as a quadrilateral with diagonals intersect each other in midpoint (bisect each other);[/i][/i][/i][br][/*][*][i][i][i]apply parallelogram properties to solve problems.[/i][/i][/i][/*][/list][/i]

[i]Strategies and methods:[br][br][/i]Lesson objectives will be reach combining several active methods such as modelling (physical and digital models), investigation, problematization, and demonstration. Dialogue and evaluation will be done through heuristic conversation, and individual work, with online or frontal checking.[br][br][i]Lesson moments:[br][br][u]I) Class organizing (3 min.)[/u][/i][br][br]The class is organized, the theme and the working methods are announced, including the use of ICT, the auxiliary materials (parallels of paper)[br][br][u]II) Updating knowledge (5 min.)[br][br][/u]Using the video projector, the teacher draws two parallel lines intersected by a transversal and the pupils will identify the congruent alternate interior angles (exterior and correspondence) and supplementary interior (exterior) on the same side of the transversal angles.[br][br][u]III) Teaching-learning of new knowledge (20 min.)[br][br][/u][list][*]The teacher draws and presents a parallelogram with the video projector, and the pupils write down the definition and draw the parallelogram on the notebooks.[br][/*][*]Announces that the purpose of the lesson is to establish the necessary and sufficient conditions (CNS) for a convex quadrilateral to be a parallelogram, and invites students to manipulate paper patterns, bend them diagonally, split them, and compare triangles obtained. Students discuss and conclude that the two obtained triangles overlap, so they have to be congruent.[br][/*][*]Students open the interactive flyers with the dynamic parallelogram. Students will modify the position and dimensions of the parallelogram and will note all the observed properties, then discuss it with each other and with the teacher. We should leave enough time for students to observe all congruences revealed by the application.[br][/*][*]Systematization of new knowledge, students note on notebooks, lemma and 5 necessary and sufficient condition (NSC):[br][/*][*] 1) Lemma: Quadrilateral [math]\left[ABCD\right][/math] is a parallelogram, if and only if [math]\Delta ABC\equiv\Delta CDA[/math]. [/*][*] 2) Characterization theorems: NSC that a quadrilateral to be a parallelogram is to have: [/*][/list] a) two pairs of congruent opposite sides, or[br] b) one pair of congruent and parallel sides, or[br] c) two pairs of congruent opposite angles, or[br] d) pairs of supplementary adjacent angles, or[br] e) diagonals intersect each other in midpoints (bisect each other).[br][list][*]Lemma and one of the NSC are demonstrated verbally with the help of students who will observe the congruent triangles and/or congruent elements on the interactive tabs; the other NSCs can remain a homework theme.[br][/*][/list][br][i][u]IV) Knowledge consolidation (15 min.)[br][br][/u][/i]There are some problems to solve (see problems 1, 2), insisting on drawing as accurate and suggestive as possible so that students can easily understand the solutions.[br][br][i][u]V) Lesson assesment, retention and transfer (7 min.)[br][br][/u][/i]Through the frontal heuristic dialogue, it is verified that new knowledge is set, then the homework theme (the NSCs that they have solved in the textbooks and some additional issues).

[list][*]GeoGebra Graph Calculator[/*][*]Interactive worksheet Parallelogram (characterization and particularization): [url=https://ggbm.at/Dtx4jbcg]https://ggbm.at/Dtx4jbcg[/url][/*][*]Problem 1: [url=https://ggbm.at/D5f6bUpY]https://ggbm.at/D5f6bUpY[/url] [/*][*]Problem 2: [url=https://ggbm.at/VgTtwdUD]https://ggbm.at/VgTtwdUD[/url][/*][/list]

[i][i][list][*][i][/i][/*][*]An earlier scheduling is ensured for ICT resources to work properly;[/*][*][i]In case that scheduling is not possible, Teacher will draw on blackboard and students could use their smartphones (GeoGebra Graph Calculator is optimized for smartphones);[/i][/*][*][i]If the last one is not possible, then we use longer paper models such that students to understand parallelogram properties without ICT.[br][br][/i][/*][/list][/i][/i]

Problem 1: Show that sides midpoints of any convex quadrilateral are vertices of a parallelogram.[br]Problem 2: Let the parallelogram [ABCD] with center O, M and Q two points on sides [AB] and [CD] such that lines MO and QO intersect opposite sides in points P and N. Show that quadrilateral [MNPQ] is a parallelogram.

Problema 2: Fie M și Q, două puncte variabile, pe laturile [AB], respectiv [CD] ale paralelogramului [ABCD], a. î. dreptele MO și QO intersectează laturile opuse în P, respectiv N. Demonstrați că patrulaterul [MNPQ] este un paralelogram.