Math Dr. Miller - Constructing in Sketchpad (tm) - Due via by Friday, Mar. 18, 2016

Transcription

1 Math Dr. Miller - Constructing in Sketchpad (tm) - Due via by Friday, Mar. 18, 2016 As with our second GSP activity for this course, you will the assignment at the end of this tutorial to me at lyn.miller@sru.edu. Please give your FILE a name that includes your own last name and the phrase GSP3. Let s begin with a brief discussion of some of the remaining tools and menus in Sketchpad. Other Tools: We refer to the icon-based items at the left of the Sketchpad window as tools. You are already familiar with: 1. the Selection Tool - the arrow at the top of the list, that allows you to drag items around the window. Remember that selecting parts of your diagram and then pressing the Delete key on your keyboard will also delete them. By the way, if you delete an object, anything CONSTRUCTED (see later) based on that object will also be deleted, so take care in deleting objects in a construction diagram. (Of course, you can always undo a bad choice with help from the Edit menu at the top of the window, or via the usual Microsoft Control-Z keyboard command.) 2. the Segment Tool - remember that you can adjust this tool to allow you to draw full, infinite lines and also rays: just click and hold on the Segment Tool to see the other options, then slide to the one you want. Of course, you can use that same method to switch back to a segment when you want. 3. the Text Tool - the letter A in the Tool list, it allows you to create a text box anywhere you like in a sketch, and it also controls the display and renaming of labels for objects in your diagram. Here are two more tools: 4. the Circle Tool - this obvious tool draws a perfect circle for you. Experiment with its use. 5. the Polygon Tool - I believe some of you discovered and used this on your own for our earliest Sketchpad assignment in the course. If you did not, experiment with it now - it allows you to draw a polygon complete with interior (necessary for measuring area) with no extra steps. Notice that you can select one of three visual variations with it: just the interior of the polygon, just the outline, or both. The Polygon Tool builds a polygon in order as you draw points around your window, so to close off a polygon, you must click a second time on the very first vertex (point) that you created. Again, experiment briefly with the use of this tool. Some New Menu Items: Across the top of the window are the various MENUS that allow us to do more than simply DRAW pictures in our sketch - Tools always just draw/doodle. 1. File menu: You are of course able to navigate your way through the standard File menu on any Microsoft application. Sketchpad may have a few extra items in this menu, but they re of little interest to us now, EXCEPT for the option to have multiple pages/windows in the same ONE file. I ll tell you how to do that later.

2 2. Edit menu: Sketchpad s Edit menu has several advanced abilities, but for now, just recall from our last tutorial that the Action Buttons submenu under the Edit menu is where we find the options for animating objects. 3. Measure menu: you explored the Measure menu in our first tutorial. 4. Number menu: this is the menu where you can find the Calculator feature, which you used in the first tutorial. Two new menus or parts of them will feature today: 5. Display menu: The Display menu has useful options for making your sketches more visually instructive. There are only three options that we ll have much occasion to use: (a) Line Style - Sketchpad s default for drawing lines, rays, and segments is as solid, fairly thick, and black. Explore the Line Style submenu under the Display menu to change the thickness and solidity (I really like dashed for some tasks) of lines, rays, and segments in a window. Such style changes take effect only on objects you already have selected (using the Selection Tool) and on new objects drawn after you ve made a Line Style change. So once you change settings, every new object you draw satisfies your changes, but older objects don t. If you do want to change the style of an older object, just select that object using the Selection Tool, then make the choices you want from the Display submenu. Those changes will apply to the selected object, AND they will also become your defaults. (b) Color - This submenu applies a little more broadly than Line Style in that you can change the color of ANYTHING in your sketch using the Color submenu - points, text boxes, measurement displays, etc. Its effects on old and new objects behave as described above for Line Style. Experiment. However, to change text, measurement, or label colors, it seems that you actually have to type the text in or display the measurements or labels first, THEN go back and change their colors, even if you do already have a particular color set as the default for your OBJECTS. (c) Hide - In a complicated diagram - especially one you CONSTRUCTED (see below) - you often want to de-clutter in order to expose the most significant qualities. Obviously, one could just delete unwanted aspects of a diagram, but in a construction setting, deleting some objects also deletes everything created based upon them. So deleting in Sketchpad is like deleting your intermediate steps in an algebra problem, and finding that any later steps based on the intermediate ones also get deleted - including your answer! Nobody wants that, so instead, we HIDE unwanted objects. Hiding means they still exist in Sketchpad s memory and in usage in your window, but they just aren t visible (exactly as many upper-level professors want your basic algebra steps to be). The Hide command is easy to use: just use the Selection Tool to choose the objects (including measurements, text, etc.) you don t want to see, and (we re in the Display menu, remember) choose Hide Objects. Notice that if you change your mind, there s also the option to Show All Hidden under the Display menu also, or you can just Control-Z. 6. The Construct menu - It is vital to understand the conceptual difference between DRAWING and CONSTRUCTING in Sketchpad (and later by hand on paper in class). Saying that we are constructing objects means we will create the objects in such a way that their key mathematical features will NOT change when

3 we drag parts of our Sketch, even though certain mathematically irrelevant aspects can. So for instance, drawing a perfect 90 o angle using the Ray Tool and Measure menu is great, but when you drag parts of your sketch around, likely your angle will change. That s obviously not good if you needed it to demonstrate some quality of, say, squares, and now you ve lost your square. Here s a brief encounter with the distinction between merely drawing and actually constructing in Sketchpad: (a) Using the Ray Tool, draw an angle. Use the Text Tool to name/rename points on your angle so that its name will be CAT. (b) Now use the Ray Tool to draw an additional ray, labeled to have the name AD in the interior of your CAT. We want to turn this newly DRAWN ray into the bisector of CAT. (c) To do so, use the Measure menu atop the Sketch to measure CAD and TAD, displaying those measurements, then drag point D until you achieve m( CAD) = m( TAD). Perfect!...except that if you drag any part of your diagram, AD is very unlikely to remain the bisector of CAT. Since one of Sketchpad s most important teaching features is the ability to drag and dynamically create a variety of similar examples, losing necessary details like congruence among parts of a diagram is NOT a good thing. (d) Now let s construct a ray that will REMAIN the angle bisector of CAT no matter what gets dragged around. Hopefully, it makes sense to you that if you want Sketchpad to construct such a bisector, you simply need to indicate the angle you want to bisect, just as you indicate an angle that you want to measure. So delete your former AD, select the points on CAT in appropriate order, then from the Construct menu, choose Angle Bisector. (e) No measurements will automatically appear (although you can insert them if you wish). Drag parts of your sketch now to observe that your CONSTRUCTED ray really will behave as the bisector no matter what. If you did include extra measurements on your own, you d see that they always demonstrate congruence of the two half-angles the bisector creates. (f) By the way, a nice variation to make your constructed ray stand out a bit is to change its color or its style. The rest of this tutorial will introduce you to the use of other frequently used constructions, then conclude with the assignment that you ll to me. SKETCHPAD CONSTRUCTIONS: Before we begin, understand that just like the Measure menu, the Construct menu will require you to have certain ingredients selected in your window before you can actually construct items that you want. Those pre-selected ingredients are called the Parents, and any item directly constructed from them is called a Child. Deleting a Parent deletes all its Children, but HIDING a Parent does not. 1. Angle Bisector: You ve already practiced constructing an angle bisector. The necessary parents for that construction are three points defining the original angle, chosen in correct order.

4 2. Point on Object: often we ll need to have an arbitrary point on a given object (like a particular line, circle, etc.) that is able to be dragged around ON that object, but NOT off of it. The necessary parent for this construction is the circle, line, segment, or ray that you want the point to be on. (a) Use the Circle Tool to DRAW a circle, leaving it selected. Now use the Construct menu to CONSTRUCT a point on the circle. Drag the circle, and observe that the point stays on it. Drag the point, and notice that nothing else in the diagram changes, except that the point can slide around on the circle. 3. Midpoint: Obviously, this choice constructs the midpoint of a segment. The parent here must be ONLY the segment. Do NOT, NOT, NOT select the endpoints also, for then the construction won t be allowed. Practice this. 4. Segment: Constructed segments behave more usefully than drawn ones. The parents here are simply the endpoints you desire to connect. Practice this construction. 5. Intersection: One of the most important options in the entire software package! Often, the points of intersection of certain segments, lines, rays, and/or circles (Sketchpad lumps these four items into the heading Path Objects ) are vital for conceptual understanding. The parents of this command are the TWO path objects whose intersection you desire. If there are multiple points of intersection, Sketchpad will construct them all. (a) Use the Circle Tool to DRAW two overlapping circles, then construct their intersections. Notice what happens as you drag the parent circles to change their sizes, positions, and whether they intersect at all. Now notice what happens when you drag a child intersection point instead. Dragging a CHILD changes nothing about PARENTS except their location. (b) Repeat the task with a line and a circle. 6. Perpendicular Line: The parents needed for this construction are a line you want to be perpendicular to and a point either on or off that parent line that you want your child line to go through. (a) Practice this one first with a parent line and parent point OFF that line, then with a parent point ON your parent line. (b) This construction is necessary for constructing perpendicular bisectors: first construct the midpoint of your desired segment, then use that segment and the midpoint as your parents. Practice constructing a few midpoints of segments that you ve drawn. 7. Circle by Center and Point: Constructing circles with specific features is the best way to construct congruent lengths, it turns out, since we all know that the radii of a given circle are always congruence to each other. The parents for this construction are of course two points, where the first one you select will be the circle s center, and the second is where the circle will pass through. Practice this construction as well.

5 ASSIGNMENT GSP #3: I want you to turn in this assignment as ONE file with TWO pages. I ll tell you below how to add the second page to your work when it s time for that. 1. (a) Open a new Sketch (this is an option under the File menu). (b) DRAW a triangle, labeled with vertices O, L, D, and CONSTRUCT the midpoints of its three sides, labeling them N, E, W. (I don t care which goes where.) (c) Now CONSTRUCT the segments joining those three midpoints, making a new, smaller triangle inside your original one. Color the sides of this inner triangle with some distinctive color. (d) Measure the areas of OLD and of NEW, and display these measurements meaningfully. (e) Use the Calculator (under the Number menu) to evaluate and display the ratio of the two areas. (Again, I don t care which order you use.) (f) Finally, ANIMATE point D, labeling the button as A Little Animation. (g) Put your name and the title Midpoint Triangle in a caption box somewhere in your window. Save this file as something like (your last name)-gsp3. 2. I want the next task to be on a new PAGE in the file you just named. Go to the File menu and click on Document Options. A little pop-up window will open, covering part of your Sketchpad window. You want to Add Page, so choose that option in your pop-up, and then you want it to be a Blank Page. (a) If you want, you can rename your pages by using the pop-up window also. (If it closed, just reopen it using the Document Options submenu.) Where the box for Page Name appears - initially, it merely has the page number as the page name as well, - just type in your new name. (b) Good choices here would be for page 1 to be named Midpoint Triangle and page 2 to be named Inscribed Triangle. 3. (a) So on your NEW page, again DRAW a triangle, labeled A, B, C for a change. (b) CONSTRUCT the angle bisectors of its three angles - these three rays should all intersect in a single point. (c) CONSTRUCT the point of intersection of two of the bisectors (the third is irrelevant, since we ll later prove that it passes through that same intersection point). Label the intersection point I. (d) Now HIDE all three angle bisectors. (e) CONSTRUCT a line through I that is perpendicular to one of ABC s sides. (f) Now construct the point where that perpendicular line meets the side, and hide the perpendicular. What you ve just done is locate a point on ABC whose distance to I is minimal - i.e., you ve found the closest point on the triangle to point I. (g) CONSTRUCT the circle with center I that passes through this closest point. Give the circle a distinctive color or dashed outline. (h) Notice that the circle is tangent to all three sides of the triangle! (i) (No need to animate here, but you can test your constructions by dragging parts of your diagram yourself to make sure they behave.) (j) Put your name and the title Inscribed Circle somewhere meaningful. 4. me your 2-page file by the end of the day on Friday, Mar. 18, 2016.