[list][*]Subject: Mathematics[br][/*][*]Grade: 9th grade[br][/*][*]Duration: 50 minutes[br][/*][*]ICT tools: [i][i]computer teacher, projector, smartphones (tablets / computers) for students, Internet connection.[/i][/i][i] [i][/i][br][/i][/*][/list]

[justify]Extreme and Monotony of Quadratic Function[/justify]

[justify][/justify]At the end of the lesson, students will be able to:[br][list][*]determine the extreme point and the extreme value of a function of the second degree, specifying if the maximum or minimum;[/*][*]to identify the coordinates of the peak point and the end parable;[/*][*]to characterize the function of the monotony of degree II and indicate the intervals of monotonicity;[/*][*]interpret graphical representation of a function of the second degree (to read graphics).[/*][/list]

[i]Cognitive objectives:[br][/i][list][*]to write the vertex coordinates of the parabola; [i][br][/i][/*][*]to discuss according to [b][i]a [/i][/b]and [math]\Delta[/math] graph shape of the quadratic function; [br][/*][*]to find the minimum/maximum of a quadratic function;[br][/*][*]to read the graph of a quadratic function.[/*][/list][i]Psychomotor objectives: [br][/i][list][*]to show interest in the lesson; [i][br][/i][/*][*]to write legibly in notebooks and on the blackboard.[br][/*][/list][i]Affective objectives:[br][/i][list][*]to participate actively in the lesson;[i][br][/i][/*][*]to develop their interest in maths study using GeoGebra.[br][/*][/list]

[b][i]Didactic methods:[/i][/b][list][*]heuristic conversation, exercise, knowledge transfer, problematization, guided observation. [/*][/list][b]Equipment, software: [br][/b][list][*]PCs, video projector, smartphones, Geogebra, online resources.[/*][/list][b][i]Lesson moments[br][br][/i][/b][u]I) Organizing moment [/u][u](2 min.)[br][/u]The class is organized, presence is done, digital tools are presented, workbooks are distributed, where students will also find GeoGebra applications addresses.[br][u][br]II) Checking homework and previous knowledge (8 min.) [br][/u]The theme is checked verbally, and if the homework cannot be solved, it is solved at the blackboard.[br][br][u]III) Introducing new knowledge (30 min) [br][/u]Using heuristic conversation they will find following conclusions:[br][list][*]Because domain and codomain of quadratic function is [math]\mathbb{R}[/math] we will write function like this: [math]f\left(x\right)=ax^2+bx+c[/math] or [math]y=ax^2+bx+c[/math], where [math]a,b,c\in\mathbb{R},a\ne0[/math][br][/*][/list][list][*]A quadratic function [math]f:\mathbb{R}\longrightarrow\mathbb{R},f\left(x\right)=ax^2+bx+c[/math] is known when we know real numbers [math]a,b,c\in\mathbb{R},a\ne0[/math]. [/*][/list][list][*]We must observe that condition [math]a\ne0[/math] is essential in the way that if [math]a=0[/math] then they obtain a linear function (studied before in 8th grade). [/*][/list][list][*]The name of quadratic function comes from quadratic trinom [math]ax^2+bx+c[/math]. [/*][*]Graph of any quadratic function is a parabola. [/*][/list][br]We introduce extrem points through guided discovery method, facilitated by new technologies integration: students watch the material [url=https://www.geogebra.org/m/D7GWhRbP]https://www.geogebra.org/m/D7GWhRbP[/url] realizing changes of a, b and c and identifyind the graph changes for [math]f:\mathbb{R}\longrightarrow\mathbb{R},f\left(x\right)=ax^2+bx+c[/math]. [br]They will draw following concluzions and will note them in notebooks:[br]o If [math]a>0[/math], the minimum value of function f in [math]\mathbb{R}[/math] is [math]-\frac{\Delta}{4a}=f\left(-\frac{b}{2a}\right)[/math] , and minimum point is [math]-\frac{b}{2a}[/math]. [br][br]o If [math]a<0[/math], the maximum value of function f in [math]\mathbb{R}[/math] is [math]-\frac{\Delta}{4a}=f\left(-\frac{b}{2a}\right)[/math] , and maximum point is [math]-\frac{b}{2a}[/math]. [br][br]Monotony: students watch the function graph in GeoGebra and they will interpretate different graphs of function [math]f:\mathbb{R}\longrightarrow\mathbb{R},f\left(x\right)=ax^2+bx+c,a,b,c\in\mathbb{R},a\ne0[/math] for different values of a, b, c. Then they will note in notebooks:[br][br]o If [math]a>0[/math], then: [math]f[/math] reach the minimum value in point [math]-\frac{b}{2a}[/math]; [br]       [math]f[/math] decrease in [math]\left(-\infty,-\frac{b}{2a}\right)[/math] and increase in [math]\left(-\frac{b}{2a},+\infty\right)[/math]; [br]o If [math]a<0[/math], then: [math]f[/math] reach the maximum vakue in point [math]-\frac{b}{2a}[/math];[br]        [math]f[/math] increase in [math]\left(-\infty,-\frac{b}{2a}\right)[/math] and decrease in[math]\left(-\frac{b}{2a},+\infty\right)[/math]. [br][u][br]IV) Retention and transfer (8 min.) [br][/u]They solve exercices from worksheet and they check with GeoGebra.[u][br][br]V) Homework (2 min.) [/u][br]Exercise 8) from worksheet.[br][br]

[i]Quadratic function graph : [url=https://www.geogebra.org/m/D7GWhRbP]https://www.geogebra.org/m/D7GWhRbP[/url][br][/i]GeoGebra web resources : [url=https://www.geogebra.org/m/bJJb6Vp9]https://www.geogebra.org/m/bJJb6Vp9[/url][br]Web resources: [url=https://ro.wikipedia.org/wiki/Func%C8%9Bie_de_gradul_doi]https://ro.wikipedia.org/wiki/Func%C8%9Bie_de_gradul_doi[/url][br][url=https://prowebdelia.wordpress.com/about/matematica-pe-web/algebra/functia-de-gradul-al-ii-lea-aplicatii/]https://prowebdelia.wordpress.com/about/matematica-pe-web/algebra/functia-de-gradul-al-ii-lea-aplicatii/[/url]

[i][i]Plan B (if we cannot use ICT lab): using students smartphones. In absence of Internet connection we will use powerpoint material.[/i][/i]