[table][tr][td]Subject: [/td][td]Mathematics[/td][/tr][tr][td]Grade Level:[/td][td]8. grade[/td][/tr][tr][td]Duration:[/td][td]about 50 min[/td][/tr][tr][td]Technology Setting:[/td][td][i]Computer and projector for teacher, computers / tablets for students[/i][/td][/tr][/table]

Exploration of the graphical representation of the slope of a line using slope triangles, whose size can be changed dynamically and which can be drawn on both sides of the y-axis.

Students will be able to calculate the slope of a line using a slope triangle.[br]Students will know that the slope of a line does not depend on the size and position of the slope triangle used to calculate it.

[b]Lesson objectives[/b][br][list][*]Students can distinguish between lines that are increasing, decreasing or horizontal.[br][/*][*]Students can create slope triangles of different sizes at different positions on the line.[/*][*]Students can determine the slope of a line by calculating the ratio of the legs of a slope triangle.[br][/*][*]Students can explain that the position of the slope triangle does not affect the calculated slope of the corresponding line.[/*][*]Students can explain that slope triangles of the same line represent the slope of this line independent of their size. [/*][/list][br][b]Assessment[/b][br]You may check the written notes of students in order to assess whether they reached the objectives of the lesson. Additionally, you may administer a written or oral test, asking your students to calculate the slope of a line using different slope triangles.

The students should be familiar with the concept of a standardized slope triangle (horizontal leg of length 1 and vertical leg of length m) indicating the slope m of the corresponding line.

In the first part of the lesson, students will create the applet shown below guided by the teacher. Thus, the teacher is working with a computer and projector, while the students are working on their own digital devices (pc, notebook, tablet).[br][br][u]Option 1: Internet connection for all devices[/u][br]If all devices used by teacher and students have a reliable Internet connection, you may use the online version of the GeoGebra Math Apps for this lesson (see [url=https://www.geogebra.org/apps]www.geogebra.org/apps[/url]). [br][br][u]Option 2: Offline use of devices[/u][br]If there is no reliable internet connection available during the lesson, you may use the GeoGebra Math Apps offline as well. Just [url=https://wiki.geogebra.org/en/Reference%3AGeoGebra_Installation#GeoGebra_Math_Apps]download [/url]and install them on all devices used by teacher and students.[br][br][u]Hints for using the prepared interactive worksheets[/u]: [br][list][*]If your students are well familiar with using the GeoGebra Math Apps, they might be able to create the construction using a prepared worksheet with written instructions, instead of following the instructions of the teacher (see interactive worksheet 1).[/*][*]If your students are not familiar with using the GeoGebra Math apps or if you do not have enough time in order to create the construction together, your students may also use a prepared interactive figure in order to solve the following tasks (see second interactive worksheet 2).[/*][/list]

[b]Construction steps[/b][br][table][br][tr][td]1.[/td][td][icon]/images/ggb/toolbar/mode_slider.png[/icon][/td][td]Create two sliders for m and b. [br][u]Hint[/u]: Use the default values for the slider interval of -5 to 5.[/td][/tr][tr][td]2.[/td][td][/td][td]Enter the equation of a line [code]y = m*x + d[/code] into the Algebra Input and press the Enter key.[/td][/tr][tr][td]3.[/td][td][icon]/images/ggb/toolbar/mode_intersect.png[/icon][/td][td]Create the intersection point of the line and the y-axis.[br][u]Note[/u]: The name of the intersection point is A.[/td][/tr][tr][td]4.[/td][td][icon]/images/ggb/toolbar/mode_parallel.png[/icon][/td][td]Create a line through point A that is parallel to the x-axis.[/td][/tr][br][tr][td]5.[/td][td][icon]/images/ggb/toolbar/mode_pointonobject.png[/icon][/td][td]Create a point B on the parallel line.[br][/td][/tr][br][tr][td]6.[/td][td][icon]/images/ggb/toolbar/mode_orthogonal.png[/icon][/td][td]Create a line through point B that is perpendicular to the x-axis.[/td][/tr][tr][td]7.[/td][td][icon]/images/ggb/toolbar/mode_intersect.png[/icon][/td][td]Intersect these two new lines.[br][u]Note[/u]: The name of the intersection point is C.[/td][/tr][br][tr][td]8.[/td][td][icon]/images/ggb/toolbar/mode_polygon.png[/icon][/td][td]Draw the slope triangle ABC.[/td][/tr][br][tr][td]9.[/td][td][/td][td]Change the names of the two legs of the slope triangle to Δy and Δx.[br][/td][/tr][tr][td]10.[/td][td][/td][td]Show [i]Name & Value[/i] of the legs of the slope triangle by changing the [i]Labeling [/i]setting on tab [i]Basics [/i]of the Properties dialog. [/td][/tr][br][/table][br]

[b]Exercise 1[/b][br][list=1][*]Move point B so that the length of the horizontal leg of the slope triangle is Δx=1. Determine the slope of the line from this standardized slope triangle and write it down in addition to the line's equation.[/*][*]Modify the values of the sliders three times and repeat step 1 for the corresponding lines.[/*][/list][br][b]Discussion 1[/b][br]Discuss the concept of determining the slope of a line using a standardized slope triangle with your students.

[b]Exercise 2[/b][br]Students should work on the following tasks using the interactive figure, as well as paper and pencil: [br][br]Change the values of the sliders m and b, so that you can successively explore the following lines:[br] (1) y = 2x + 1[br] (2) y = 3x - 2[br] (3) y = -x + 2[br][br][u]Task 1[/u][br]For each of the lines above, move point B and write down the values of Δy (length of the vertical leg) and Δx (length of the horizontal leg) for at least four different slope triangles.[br][u]Note[/u]: Two of your slope triangles should be on the right and two on the left side of the y-axis.[br][br][u]Task 2[/u][br]For each of your slope triangles, calculate the ratio of its two legs [math]\frac{\Delta y}{\Delta x}[/math] and compare the four resulting values for each line.[br][br][u]Task 3[/u][br]Compare the calculated values with the slope of the line, which you can determine using a standardized slope triangle. Write down your observations.[br][br][b]Discussion 2[/b][br]What can be determined by the ratio of the two legs of an arbitrary slope triangle? [br]How do the position and size of the slope triangle affect the calculation of the ratio of the legs?[br][br][b]Exercise 3[/b][br][list=1][*]Create three different lines by changing the values of the sliders m and b. For each of these lines, record the equation and calculate the slope using an arbitrary, not standardized slope triangle.[/*][*]Compare your slope values with the parameters m and b of the corresponding equation. Write down your observations.[/*][/list][br][b]Discussion 3[/b][br]Discuss the students' observations of exercise 3. Which parameter of a line's equation determines the slope of the line?

[b]Prior Knowledge[/b][br]Students are well familiar with the basic use of the GeoGebra Math Apps and are able to create the interactive construction described above. If no, they may use the included interactive worksheets providing instructions for the construction steps or the prepared construction.[br][br][b]Using the interactive worksheets offline[/b][br]You may also implement this lesson without a reliable Internet connection, if you prepare either one of the following options prior to your lesson:[br][list][*]Download and install the GeoGebra Math Apps on all devices.[br]OR[/*][*]Download the interactive worksheets prior to the lesson and save them on all student devices.[/*][/list][br]It is also possible for students to solve these tasks using paper and pencil only, in case technical problems prevent them from using their digital device.