Peanut Geometry Activities

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Buck Welch
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1 Peanut Geometry Activities The following set of activities provide the opportunity for students to explore geometric properties using the Wingeom program available free from the Philips Exeter website ( Although this program is not as user-friendly as the commercial program, The Geometer s Sketchpad, teachers and students will be able to discover any appropriate applications. These activities are not in any particular order. Teachers will need to use them at appropriate times within their specific curriculum. One special caution is in order when using the Wingeom program. This program labels points in alphabetical order, but sometimes students may not see the underlying operations. In these instances, the point labels may be different from those included in the activities. Students will need to relabel the directions as appropriate when they encounter point label changes. It is advisable as with all activities that teachers try the activities themselves before making use of them in their classrooms. Dr. Ron Pelfrey Mathematics Specialist Appalachian Rural Systemic Initiative

2 Tangent Segments 1. Use Point Grid to label point A at (0,0) and point B at (2,0). 2. Use Circle Radius-center with center A, and radius AB to draw circle AB. 3. Use Coordinate Input to label point D (6,0). 4. Use Line Tangents to draw tangents to AB at E and F. 5. Use Meas to measure DF and DE. 6. Use Line Segments to draw AF and AE. 7. Use Meas to measure AFD and AED. Question: Write two conjectures about tangent segments. Construct a new circle and point outside of the circle to test your conjectures.

3 Escher Figures 1. Use Shape Triangle SSS to draw a triangle. 2. Use Transf Rotate 32.6 about A to place the triangle in normal position. 3. Use Edit Point delete to remove points B and C. 4. Use View Grid and check box to show grid. 5. Use Point Grid to insert points B (0, -3) and C (6, -6). 6. Use Line Segments to draw B B, BC, and CC. 7. Use Transf Rotate 60 with B as center about B -B-C-C. 8. Use Points Midpoints to label the midpoint of Ac as D. 9. Use Coordinate Input box to label point E (6, -3). 10. Use Line Segments to draw D E and ED. 11. Use Transf Rotate C -D-D about center D, Use Edit Point delete to remove B~. 13. Use Edit Segment delete to remove AB, B C, C D, and DA. (NOTE: Do not remove yet if you want to shade the figure later. Leave until step 16.) 14. Use View Grid axes (off) to remove grid axes. 15. Use Transf Rotate with center B and apply to A-C with angle of 60 ; center B and apply to A-E and an angle of 120 ; center B and apply to A- E& and an angle of Use View Highlights Fill regions to shade/color appropriately. 17. Use CTRL L to remove labels and View Labels Dot mode to remove drag points.

4 Sierpinski Triangle 1. Use Shape Random Triangle to generate a random triangle. 2. Use Meas to find the perimeter (AB + BC + CA) of the triangle. Call this the perimeter of Stage 0 (before any changes are made). 3. Use Point Midpoints to find the midpoints (D, E, and F) of each side. 4. Use Line Segments to draw DE, EF, and DF. 5. Use Meas to find the sum of the perimeters of the three corner triangles (AD + DE + AE + EF + FB + EB + DF + CD + CF). Call this the perimeter at Stage Use Point Midpoints to find the midpoints of each side of the three corner triangles. 7. Use Line Segments to connect the midpoints of each of the three corner triangles to form 9 corner triangles. 8. Use Meas to find the sum of the perimeters of the 9 corner triangles. Compare the total perimeters for each stage and look for patterns. Write any conjectures and explain your reasoning. Test your conjectures by dragging any vertex of the original triangle. You can also create more stages to test your conjectures. Investigate total area using the same stages.

5 Block Design 1. Use Shape Polygon Regular to construct a 6-sided figure (hexagon). 2. Use Transf Rotate and an angle of 30 to rotate the hexagon about A. 3. Use Edit Point Delete to delete points B, C, D, E, and F. 4. Use Line Segments to draw AD, B E, C F. 5. Use Point Intersection Line-Line to label the intersection of AD and B E as point B. 6. Use Edit Segments Delete to delete AB, BC, and BE. 7. Use Transf Translate with a vector of B F to create a translation. 8. Use Transf Again twice to generate a total of four hexagons in row Use Transf Translate with a vector of AE to generate a second row. 10. Use Transf Again twice to generate two more rows. 11. Use View Highlights Fill Regions to appropriate color like sides of each hexagon. 12. Press CTRL L to remove all labels. Describe different shapes in this design. What s the shape of the smallest pieces quilters use in this design? Describe the optical effect of the design. Question 3: Describe any places where you have seen this design before.

6 Transformations 1. Construct the vertices of a flag, e.g., (-4, 4), (-1, 4), (-1.5, 2.5), (-3.5, 2.5), (-3.9, -3). 2. Draw line segments AB, BC, CD, DE, and EA. 3. Use View Highlights Fill Region and select color or pattern to color your flag. 4. Mark two points on the y-axis, e.g., (0, 4) and (0, -4). Draw the segment connecting these two points, e.g., FG. 5. Use Transf Mirror and mirror line FG to show the Reflection of the flag about the y-axis. Compare the translated image to the original figure. How are they different and how are they the same? 6. Using Transf Rotate, an angle of 90, E as center, dilation factor of 1, rotate the flag. Compare the rotated figure to the original figure. How are they the same? 7. Experiment with Transf Translate, Dilate, and Glide-Reflect and compare the results.

7 Slope of a Line 1. Use Point to label two points, A and B. 2. Draw line AB. 3. Use Measurements to find the slope and equation of line AB. Type [slope](a, B) to find the slope. Type [eqn](a, B) to find the equation. Which lines have a positive slope and which have a negative slope? What is the slope of a horizontal line? Question 3: How can you tell a steeper slope from a shallower slope? Question 4: What is the slope of a vertical line? Playing the Slope Game: Play this with a partner. 1. Draw five different random lines on a grid. Write the labels for the five lines on a sheet of paper, e.g., AB, CD, etc. 2. Measure the slopes of the five lines. Write the five slopes on the sheet of paper with the lines, but in random order. Then press CTRL L to hide the labels of the lines. 3. Challenge your partner to match each measured slope with a line. 4. To check your partner s score, show all the point labels by again pressing CTRL L. Award one point for each correctly matched slope. 5. Switch roles and play the game with new lines.

8 Properties of Parallel Lines 1. Construct line AB and point C, not on AB. 2. Construct a line parallel to AB through point C (Use Lines Parallel). 3. Construct line CA. 4. Use Point Seg Division with marks at 1.5 to label additional points on AB, BA, CD, and DC. 5. Use Measurements to measure each of the eight angles. 6. Observe how many of the eight angles appear to be congruent. When two parallel lines are crossed by a transversal, the pairs of angles formed have specific names and properties. The following chart shows one example of each type of angle pair. Fill in the chart with a second angle pair of each type, then state what relationship, if any, you observe between the angles in a pair type. Angle Type Pair 1 Pair 2 Relationship Corresponding FCE and CAB Alternate Interior ECA and CAG Alternate Exterior FCE and HAG Same-side Interior ECA and BAC Same-side Exterior FCD and HAG One of the angle types has more than two pair. Name that angle type in the chart below, and name the third and fourth pairs of angles of that type. Angle Type Pair 3 Pair 4 Relationship 7. Next, you ll investigate the converses of your conjectures. In a new sketch, draw two lines that are not quite parallel. Construct a transversal. 8. Add points of intersection and other labeling points, then measure all eight angles formed by the three lines. Question 3: What conjecture can you make about the angles formed by lines which are not parallel? Question 4: If two lines are crossed by a transversal so that corresponding angles, alternate interior angles, and alternate exterior angles are congruent, what can you say about the lines?

9 Constructing a Perpendicular Bisector 1. Use Point to label two points, A and B. 2. Draw line segment AB. 3. Use Circle to draw circle AB (point A as center and AB as radius). 4. Use Circle to draw circle BA (point B as center and AB as radius). 5. Use Point Intersection Circle-Circle to label the intersection points E and F. 6. Draw segment EF. EF is the perpendicular bisector of segment AB. Without measuring, what can you say about the distances AE and BE and the distances AF and BF? 7. Use Point Intersection Line-Line to label the intersection of AB and EF (as point G). What s special about point G? 8. Hide the circles. Use View Highlights Color line/circle and choose all circles and select White. 9. Use Point Seg Division and Mark at 1.5 on segment EF to label another point on the perpendicular bisector, point G. 10. Measure the distances AG and BG. Question 3: What conjecture can you make about any point on a segment s perpendicular bisector?

10 Angle Bisectors 1. Use Point and Line to draw lines AB and AC to form BAC. 2. Use Line Angles Bisect Old to construct the bisector of BAC. 3. Use Point Seg Division and Mark at 0.5 to label a point, E, on bisector AD. 4. Use Line Perpendiculars General to construct the perpendicular to AB through E; to AC through E. 5. Use Point Intersection Line-line to label the points of intersection of the perpendiculars to each side of BAC, e.g., H on AB and I on AC. Make a conjecture about the distances from point E to each of the angle s two sides. 6. Measure these distances, e.g., EH and EI. 7. Measure BAE and CAE. How do the measures of BAE and CAE compare?

11 Angles Formed by Intersecting Lines 1. Use Point to label four different points, A, B, C, and D, in each of the four quadrants. 2. Use Line Segment to draw segments between the points in the 1 st and 3 rd quadrants; between the 2 nd and 4 th quadrants, e.g., AB and CD. 3. Use Point Intersection Line-Line to mark the intersection point of line segments AB and CD as Point E. 4. Use Meas to measure the four angles, DEB, BEC, CEA, and AED. DEB and CEA are a pair of vertical angles. Name another pair of vertical angles. Write a conjecture about the measures of vertical angles. CEB and DEB are a linear pair because two of their sides form a line. Find and name all the other linear pairs. Write a conjecture about the relationship between angles in a linear pair. Question 3: Redraw the two line segments, AB and CD, in such a way that the linear pairs are congruent. Describe all four angles. Question 4: Redraw the two line segments, AB and CD, to form random linear pairs. Find the measure of CEB using Meas. Find the measures of the other three angles without using Measurements. Use Meas to check your solutions.

12 Duplicating an Angle 1. Use Point to label A, B, and C. 2. Use Line Segment to draw segments AB and AC. Thus, you have constructed CAB. 3. Use Circle Radius-Center with A as center and radius of 2 to draw a circle. 4. Use Point Intersection Mixed to label the intersection points of the circle with each side of the angle, i.e., points E and G. 5. Use Line Segment to draw EG. 6. Use Point to label another point, e.g., in the 3 rd quadrant, point I. 7. Use Circle Radius-Center with I as center and AE as radius to draw circle IJ. 8. Use Line Segment to draw IJ. 9. Use Circle Radius-Center with J as center and EG as radius to draw circle JK. 10. Use Point Intersection Circle-Circle to label the intersection of circles IJ and JK as point L. 11. Use Line Segment to draw IL. 12. Use Meas to measure BAC and JIL. Both should be equal. 13. Explain why this construction works the way it does.

Point A location in geometry. A point has no dimensions without any length, width, or depth. This is represented by a dot and is usually labelled.

Point A location in geometry. A point has no dimensions without any length, width, or depth. This is represented by a dot and is usually labelled. Test Date: November 3, 2016 Format: Scored out of 100 points. 8 Multiple Choice (40) / 8 Short Response (60) Topics: Points, Angles, Linear Objects, and Planes Recognizing the steps and procedures for