[list][*]Subject: Mathematics[br][/*][*]Grade Level: 6th[br][/*][*]Duration: 50'[br][/*][*]Technology setting:[i] t[/i][i]eacher computer with projector, student computers (or tablets, or smartphones)[/i][/*][/list]

Discovery and learning of some of the right triangle properties, the theorem of the median on the hypotenuse, and the theorem of 30[math]^\circ[/math] angle.

After the end of the lesson, students will:[br][list][*]know that in any right triangle, the median on hypotenuse measures a half of hypotenuse.[br][/*][*]know, and explain why in a right triangle, the cathetus opposed to an angle of 30[math]^\circ[/math] measures a half of hypotenuse.[br][/*][*]be able to calculate some unknown elements of a right triangle when hypotenuse or hypotenuse median are given.[/*][/list]

[u]Lesson objectives[/u][br][br]By the end of the lesson, students will be able to:[list][*]indicate and explain that the hypotenuse median divides the right triangle into two isosceles triangles, independent of the measures or position of the right triangle;[/*][*]conclude and retain that the median on hypotenuse measures a half of hypotenuse;[/*][*]conclude and retain that, when an angle of the right triangle measures 30[math]^\circ[/math], the cathetus which opposes it measures a half of hypotenuse;[/*][*]calculate the unknown element from the hypotenuse, half of hypotenuse, the hypotenuse median, cathetus opposed to 30[math]^\circ[/math] angle, etc. when the others are given.[/*][/list][u]Assesment[br][/u][list][*]Discussions and checking the Interactive Worksheet 2 completion;[/*][*]Verifying the homework after the lesson. [/*][/list]

[u]Strategy and the methods: [br][/u][list][*]Investigation methods, using paper triangles models and technology aided modeling, will combine with heuristic and/or structured dialogue;[/*][*]Technology will contribute about a half of the lesson, mostly due to the exercises provided by the Interactive Worksheet 2.[/*][/list][br][u]Moments and activities:[br][br]1) An introductory moment (3')[/u][br]The teacher will welcome the students and provide information about the theme and the use of technology [br][br][u]2) Revising the pre-requisites (7')[/u][br]Students will be invited to discuss and asked about:[br][list][*]the isosceles and equilateral triangle main properties;[/*][*]the mid-segment definition and properties.[/*][/list][br][u]3) The investigatory activities (15')[/u] [br]Students will be supported by paper models and technology to discover the median on hypotenuse theorem and the 30[math]^\circ[/math]theorem. Each student will have access to a packet of paper triangle models (as you can see in the Resources paragraph) and to a digital device with Internet connection and/or GeoGebra.[br][br][u]Investigating by folding the paper models:[br][/u][br][list][*]Take the paper right triangle (the bigger one) and let presume that the right angle vertex is A, and the others are B and C. Pick the right vertex A with a hand and the B vertex with the other hand, and fold the paper such that the vertex B superposes the vertex A. Press gently the paper until is flat. Then do the same with the vertex C, and you should obtain a rectangle shape paper. Then fold the rectangle in two equal parts, by the diagonal formed in the process. Now, you can unfold the paper, and observe it.[/*][*]Fold and unfold the model, and color the lines resulted in the process, which lines are there? Write down the triangle, the names of the important segments observed. We expect students to observe that the two halves of the hypotenuse and the median on hypotenuse are superposing in the process, thus they have the same length. [/*][*]Question: How are the triangles formed by the hypotenuse median with the two halves of the hypotenuse? They are isosceles, and how can we prove that?. [/*][*]Question: After coloring the midsegments, we ask why they are midsegments? Because superposing the vertexes A and B (or A and C), we superpose the two halves of the side, too, and the midsegment is that segment having midpoints as extremities.[/*][*]Question: Each midsegment is a median for each triangle formed by the hypotenuse median, it would be not a height too? Yes, that's why these triangles are isosceles.[/*][*]Take the equilateral triangle and its halves, the little right triangles, and try to superpose them.[/*][*]Question: How the angles of the equilateral triangle measure (we expect 60[math]^\circ[/math])? So, how acutes angles of the two right triangles measure (30[img]data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAoAAAAPCAYAAADd/14OAAAAV0lEQVQoU2NkIBIwEqmOAV3hBAYGhg8MDAwPGBgYFiAbgqwwgYGBYQNUoQIDAwMIH4ApRlYYwMDAcAFqmgNUA4gPBuhWg0wVgCrCaTVef5Hta5ymjkwTAUsaDBArM3svAAAAAElFTkSuQmCC[/img]and 60[img]data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAoAAAAPCAYAAADd/14OAAAAV0lEQVQoU2NkIBIwEqmOAV3hBAYGhg8MDAwPGBgYFiAbgqwwgYGBYQNUoQIDAwMIH4ApRlYYwMDAcAFqmgNUA4gPBuhWg0wVgCrCaTVef5Hta5ymjkwTAUsaDBArM3svAAAAAElFTkSuQmCC[/img])? [/*][*]What can you say about the cathetus and the hypotenuse of the 30[img]data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAoAAAAPCAYAAADd/14OAAAAV0lEQVQoU2NkIBIwEqmOAV3hBAYGhg8MDAwPGBgYFiAbgqwwgYGBYQNUoQIDAwMIH4ApRlYYwMDAcAFqmgNUA4gPBuhWg0wVgCrCaTVef5Hta5ymjkwTAUsaDBArM3svAAAAAElFTkSuQmCC[/img] right triangles? [/*][*]Question: Can we assume that all that we observed and learned about these paper models, will be valid for any other right triangles? The technology will help us to figure out this question.[/*][/list][u][br]Investigating by using the GeoGebra interactive worksheets:[br][br][/u]Students will open the Interactive Worksheet 1 and will verify the properties by changing the dimensions and position of the right triangle. They will use the buttons to visualize the median, the midsegments, and the 30[img]data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAoAAAAPCAYAAADd/14OAAAAV0lEQVQoU2NkIBIwEqmOAV3hBAYGhg8MDAwPGBgYFiAbgqwwgYGBYQNUoQIDAwMIH4ApRlYYwMDAcAFqmgNUA4gPBuhWg0wVgCrCaTVef5Hta5ymjkwTAUsaDBArM3svAAAAAElFTkSuQmCC[/img] right triangles.[br][list][*]How are the measures of the hypotenuse, and hypotenuse median, given any right triangle? Or the cathetus and hypotenuse median, in the 30[img]data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAoAAAAPCAYAAADd/14OAAAAV0lEQVQoU2NkIBIwEqmOAV3hBAYGhg8MDAwPGBgYFiAbgqwwgYGBYQNUoQIDAwMIH4ApRlYYwMDAcAFqmgNUA4gPBuhWg0wVgCrCaTVef5Hta5ymjkwTAUsaDBArM3svAAAAAElFTkSuQmCC[/img] right triangles? [/*][*]Question: Do the congruence depend on the position or measures of the right triangles (no!)?[/*][*]Question: How it can be explained the difference when it is observed (discuss approximation)?[/*][*]What can we conclude?[/*][/list][br][u]4) Resuming, reasoning and writing the demonstration of propositions investigated (12'):[/u][br][br]Students will be invited to reason and write down the logical proofs under the teacher supervision. The following three results will be proven:[br][list][*]Lema: The hypotenuse median and the two halves of hypotenuse determine two isosceles triangles (for example, proving that the midsegments are medians and heights in the same time);[/*][*]Theorem 1: The hypotenuse median measures a half of hypotenuse, in any right triangle;[/*][*]Theorem 2: The cathetus opposed to the 30[img]data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAoAAAAPCAYAAADd/14OAAAAV0lEQVQoU2NkIBIwEqmOAV3hBAYGhg8MDAwPGBgYFiAbgqwwgYGBYQNUoQIDAwMIH4ApRlYYwMDAcAFqmgNUA4gPBuhWg0wVgCrCaTVef5Hta5ymjkwTAUsaDBArM3svAAAAAElFTkSuQmCC[/img] angle measure a half of hypotenuse[/*][/list][br][u]5) Exercising and assessment (10');[/u][br][br]Students will open the Interactive Worksheet 2 and solve the exercises. More exercises can be obtained by moving the dynamic points.[br][br]6) Asking for feedback and giving the homework (3'). [br][br]As a homework, alongside with some exercises provided in their math textbooks, students will have to construct a paper puzzle as a project, using the two big triangles of different colors. These will be cut into four congruent triangles, they will be similar to the bigger ones. Then every little right triangle will be folded as above, and cut into another four triangles, and the process can be repeated. The game will be to reconstitute a big triangle by using the small different colored triangles.

Paper models:[br]Each student will receive, preferably, two paper right triangles of different colors, one equilateral triangle together with two right triangles with 30[math]^\circ[/math] angle (being exactly the halves of the equilateral one). A student packet could be produced using two A4 paper sheets of different colors: [url=https://ggbm.at/EbCRpSSN]https://ggbm.at/EbCRpSSN[/url][br][br]Interactive Worksheet 1 "Median on Hypotenuse": [url=https://ggbm.at/cwKxTU7f]https://ggbm.at/cwKxTU7f[/url][br]Interactive Worksheet 2 "Median on Hypotenuse - Exercises": [url=https://ggbm.at/hbpvzeur]https://ggbm.at/hbpvzeur[/url]

Minimizing the technology related problems:[br][list][*]Students need only some short explanation to use the interactive worksheets, and no other previous knowledge to adequately use technology during the lesson[br][/*][*]Ideally, we will use the Internet network, a teacher computer, and projector, and students computers (or tablets, or smartphones), however, we should download the worksheets, for the case that the network doesn't work;[/*][*]The lesson can continue without any technology (using only the paper models prepared previously), more slowly though, so can consider a longer duration, for students to make supplementary drawings and measurements.[/*][/list]

[i]I have implemented this lessons more than once. It had to be adapted function by the students' math level. The students who have not developed yet the formal logic or those just learn slowly would benefit more by the practical moments and could not pay attention when formal proofs are written down. The teacher has to find the right balance between practice and formal, such that to stimulate developing the logical thinking without sacrificing any student time.[/i]

[i]The lesson was easier to conduct in the ICT lab, however, it worked in the usual classroom, using a teacher laptop and students' smartphones. Of course, the interactive worksheets were prepared, and the paper models.[/i]

[i]Yes, as I am aware, at least the minimum objectives were reached.[/i]

[i]They heavily appreciated the investigation using paper models, as well as the technology.[/i]

[i]I would like to have an easier access in the ICT lab or have digital devices in every classroom.[/i]