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Transforming Quadrilaterals and Their Changing Diagonals NCTM Annual Meeting and Exposition Philadelphia, Pennsylvania April 25 – 28, 2012 Charlene Keen, Dauphin County Technical School, Harrisburg, PA Please email for an electronic handout, please email [email protected]. Thanks to Bill Hornung, Hornung’s Hardware, Linglestown, PA, for his help with materials and ideas. Dauphin County Technical School always gives me support. Materials: 8 dowels 8 tubes 4 elastic grommet tabs 4 yarn strands, 2 long and 2 short 2 colors for each model Towel with soap Two magnet clips on each top dowel will hold the models to a magnetic board and make them easier to use. When the yarn strand is pulled, the quadrilateral will change from rectangle to parallelogram OR rhombus to square. As the shifts take place the diagonals will always bisect each other. When the quadrilateral angles are right angles, the diagonals are congruent. You can also discuss what the diagonals do to the quadrilateral angles. I bought the tubing from McMaster-Carr #5234K72 Super soft latex rubber tubing 3/16”ID, 5/16” OD, 1/16” Wall Each kit needs about 7” cut into 8 pieces, each ¾” long. I did make tubes from double knit material cut on bias, hot glued. Two 48” – ¼” dowels for each kit. Each cut 9”, 9”, 11”, and congruent remainder. I made the tabs from leather grommet tape and elastic. I cut the grommet tape halfway between the grommets. I folded the elastic over both sides of the tape and sewed it in place by hand. Directions Parallelogram/ Rectangle Choose 2 pairs of dowels that are different lengths. Slip tabs over both ends of a short dowel. Wipe the end of each dowel with a small amount of soap. Use tubes to connect the dowels to make a parallelogram. Dowels should just touch inside the tubing when it bends. Slip the tabs over the tubes. Center each grommet at the vertex. Lay the parallelogram so that the tabs are closest to you. Use the two longer lengths of yarn. Use two different colors. Tie a length of yarn (different colors) over tubing at each top corner. Make a knot in such a way that you do not have a lot of wasted yarn. To form diagonals, slip the yarn through the grommet in the opposite corner. Hold the free ends together and tie at the bottom. zzzz zzzz zzzz zzzz Tie Tie Knot Rhombus/Square Use the 4 congruent dowel lengths. Use the two shorter yarn lengths. Follow the previous directions. All four sides are the same length. Illuminations http://illuminations.nctm.org Lessons Geometry Key Word: Diagonals to Quadrilaterals Thinkfinity Commercial vendor: GeoLegs and AngLegs Vocational applications www.pde.state.pa.us Go: Math Council BP Math Council Carpentry; HVA Directions to Construct Quadrilaterals using Diagonals and SketchPad Rectangle: The diagonals of a rectangle bisect each other and are the same length. Construct the midpoint of a segment. Construct a circle whose center is the midpoint. The circle contains the segment endpoints. Construct a point on the circle. Construct a LINE through the point and the center. Construct segments through the four consecutive intersection points on the circle. This is the rectangle. This diameter will be the same length as the first segment. Translate by moving either endpoint or the circle point. Show that a square is a special rectangle. Square: The diagonals of a square are perpendicular, congruent and bisect each other. Construct the midpoint of a segment. Construct the perpendicular through the midpoint. Construct a circle whose center is the midpoint. The segment endpoints are on the circle. Construct segments through the four consecutive intersection points on the circle. Rotate using either endpoint to obtain different orientations. Rhombus: The diagonals are perpendicular and bisect each other; may be different lengths. Construct the midpoint of a segment. Construct the perpendicular bisector. Construct a circle such that the center is the midpoint AND the segment endpoints are NOT on the circle. Connect consecutive-- Endpoint, Intersection point, Endpoint, Intersection point--to form a rhombus. Translate by moving either endpoint. Move the indicated circle point to the segment endpoint (making congruent diagonals) to show that a square is a special case of the rhombus. This is the first diagonal. Thus the two diagonals will not be congruent. Parallelogram: The diagonals bisect each other. They may be different lengths. Construct the midpoint of a segment. This is the first diagonal. Construct a circle such that the center is the midpoint AND the segment endpoints are NOT on the circle Construct a point on the circle. Construct the LINE through the circle point and the center. Connect consecutive--Endpoint, Intersection point, Endpoint, Intersection point--to form a parallelogram. Translate by moving the endpoints and the indicated circle point to change the parallogram into a rectangle, rhombus, and square. Isosceles Trapezoid: The diagonals are the same length and the intersection point makes congruent segments above and below. Construct a ray. There are two points shown—the endpoint and a “ray point”. Construct a circle with the ray point as the center. The circle contains the endpoint. Construct a point on the circle. Construct a line through the point and the center. Highlight the RAY and construct a point. Construct a second circle with the same center but a larger diameter. This new larger circle contains the new point on the ray. To form the isosceles trapezoid, connect consecutive intersections of the two small circles and the two large circles. The diagonals are congruent and the center point creates a 1:1 ratio. The best translations use either point on the small circle and the ray-large circle intersection. Rectangles and squares are formed when the first circle overlaps the second. Trapezoid: In general the diagonals are different lengths and the intersection point makes proportional sections. I am using a 1 : 2 ratio. Recently I was able to simplify this just using circles and the lines. Construct a ray. There are two points shown—the endpoint and a “ray point”. Make a new initial point off the ray but close to it. Construct a line through the initial point and the ray point. Hide the initial point Construct a circle using the “ray point” as a center; the circle contains the endpoint of the ray. Click where the circle intersects the LINE. This intersection point will be the center of a new 2nd circle. Construct the second circle. Use the “line-circle intersection” point as the center. The second circle contains the “ray point” (1st circle center). Click where the 2nd circle intersects the line to form an important point. For a smaller circle, carefully move the ray point closer to the ray’s endpoint or start again Highlight the LINE and construct a point on the lower line somewhat close to the ray point. Construct a circle on the lower LINE using the “ray point” as the center; the circle contains the new “line point”. Connect the proper points— Ray endpoint, 2nd top point on line, 1st circle- lower ray intersection, 2nd circle-line intersection. Be careful. The circles have the same radii. You have marked off two congruent lengths on the upper LINE, compared to one length on the upper RAY. Hide the two circles. Check parallelism by copy and pasting the right base and dragging it over the left base OR check that same-side interior angles are supplementary Make a second circle-- the center is the new “line point”; the second circle contains the “ray point”. Again there is a 2:1 ratio on the LINE compared to the RAY. The trapezoid diagonals have 2:1 ratio. The best translations occur by moving the ray endpoint and the first lower line point. When the ray endpoint is moved to the first large circle you have a parallelogram. To form a rhombus, move the ray endpoint around, but on, the circle, until the line and ray are perpendicular. Or onsider displaying the two upper circles with a thin rather than hiding them. Move the endpoint or the first lower line point so that first large circle overlaps first small circle to form a 1 : 1 ratio that again produces the same figures. Move the first lower line point to the “ray point” to make a triangle -- a special trapezoid with top base length of zero. AT THE CONFERENCE I MADE BOTH TRAPEZOIDS USING INTERSECTING LINES. ISOSCELES TRAPEZOIDS: FROM THE INTERSECTION I MADE A SMALL CIRCLE AND A LARGE CIRCLE. THE ISO. TRAP. WAS MADE BY CONNECTING THE INTERSECTIONS THAT THE CIRCLES MADE WITH THE LINES. SMALL ABOVE, SMALL ABOVE, LARGE BELOW, LARGE BELOW. REGULAR TRAPEZOID: FROM LINE INTERSECTION I MADE ONE LARGE AND ONE SMALL CIRCLE. WHERE THE CIRCLES INTERSECTED THE SECOND LINE I MADE AN ADDITIONAL PAIR OF CIRCLES, LARGE AND SMALL THAT WERE CONGRUENT TO THE FIRST PAIR. CONNECT OUTERMOST INTERSECTION POINTS. IF THESE ARE CONFUSING PLEASE EMAIL ME AT [email protected]. Thanks This has thrown off my spacing. There is more below, including the question pages. You could want to separate the page into two as it represents two topics. Vocational Applications Masonry, Carpentry, Building Construction: Lay out a Foundation Determine if Walls are Square Ornamental Horticulture: Lay out a garden Carpentry, Ornamental Horticulture: Determine the Center of a Circle www.pde.state.pa.us Magazine that Lowe’s distributes. Auto Body: Bumper Alignment Auto Mechanics: Tire Alignment Blue Prints, Drafting: Marking the center of a ceiling Go: Math Council BP Math Council Carpentry CHRYSLER 1984 LEBARON UNDERHOOD VIEW All Underhood View Dimensions are Point-to-Point. Holes are Measured to the Closest Edge. Bolts are Measured to Center. Draw diagonals with a straight edge. Quadrilateral Questions This quadrilateral is a ___________________ Use patty paper to compare Sizes of the Opposite angle Same Different Lengths of Opposite Sides Same Different Are the opposite sides parallel? Yes No Questions Diagonals are the same length Yes No Diagonals bisect each other Yes No Diagonals are perpendicular Yes No Diagonals bisect Quadrilateral angles Y / N Other Observations? This quadrilateral is a ___________________ Use patty paper to compare: Sizes of the Opposite angle Same Different Lengths of Opposite Sides Same Different Are the opposite sides parallel? Yes No List another observation: Diagonals are the same length Yes No Diagonals bisect each other Yes No Diagonals are perpendicular Yes No Diagonals bisect Quadrilateral angles Y / N Other Observations? This quadrilateral is a ____________________ Use patty paper to compare: Sizes of the Opposite angle Same Different Lengths of Opposite Sides Same Different Are the opposite sides parallel? Yes No List another observation: Diagonals are the same length Yes No Diagonals bisect each other Yes No Diagonals are perpendicular Yes No Diagonals bisect Quadrilateral angles Y / N Other Observations? This quadrilateral is a ____________________ Use patty paper to compare: Sizes of the Opposite angle Same Different Lengths of Opposite Sides Same Different Are the opposite sides parallel? Yes No List other observations: Diagonals are the same length Yes No Diagonals bisect each other Yes No Diagonals are perpendicular Yes No Diagonals bisect Quadrilateral angles Y / N Other Observations? This quadrilateral is a ____________________ Use patty paper to compare: Sizes of the Opposite angle Same Different Lengths of Opposite Sides Same Different Are the opposite sides parallel? Yes No List other observations: Diagonals are the same length Yes No Diagonals bisect each other Yes No Diagonals are perpendicular Yes No Diagonals bisect Quadrilateral angles Y / N Other Observations? This quadrilateral is a ____________________ Use patty paper to compare: Sizes of the Opposite angle Same Different Lengths of Opposite Sides Same Different Are the opposite sides parallel? Yes No List other observations: Diagonals are the same length Yes No Diagonals bisect each other Yes No Diagonals are perpendicular Yes No Diagonals bisect Quadrilateral angles Y / N Other Observations? This quadrilateral is a ____________________ Use patty paper to compare: Sizes of the Opposite angle Same Different Lengths of Opposite Sides Same Different Are the opposite sides parallel? Yes No List other observations: Diagonals are the same length Yes No Diagonals bisect each other Yes No Diagonals are perpendicular Yes No Diagonals bisect Quadrilateral angles Y / N Other Observations? Use the Quadrilateral kits to further confirm you answers. Use the Quadrilateral kits to further confirm you answers. Diagonals of Quadrilaterals Lab: Materials: Rope: 2 congruent lengths and one different—mark the midpoint of each, Long piece of rope, Tape, Direction/Answer sheet Directions: Students are in groups of 2 or 3. Students lay out the diagonals as described on paper. They tape the ends to the floor. Then they connect the endpoints with the long rope. They write the name of the shape on the handout, by the proper description. DIAGONALS NAME THE SHAPE Same Length Bisect each other Perpendicular Same Length Bisect each other Not Perpendicular Different Length Bisect each other Perpendicular Different Length Bisect each other Not Perpendicular Same Length Top part of each diagonal is same length; they don't bisect Does not matter Different lengths (could be same) Only one is bisected Perpendicular B C Illuminations Diagonals and Quadrilaterals Illuminations, from NCTM, has a worksheet similar to this. There is an interactive applet. A D Diagonals congruent. Top small portion of each diagonal congruent.