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InterMath Title: Mystery Quadrilateral Problem Statement: Some quadrilaterals have special names because they have some special properties. For example, a rectangle is any quadrilateral with four right angles. Alternately, a rectangle is a parallelogram with 1 right angle (Can you explain why?) A square is "more special" than a rectangle because it has four right angles and four equal sides (so a square is a special rectangle.) Each of the Geometer's Sketchpad files below contains a different quadrilateral. Your goal is to use your knowledge about quadrilateral properties and the Measure menu in Geometer's Sketchpad to determine the MOST specific name of each quadrilateral. Be careful! All of the quadrilaterals will look like squares, but only one of them will actually be a square. Justify each of your responses by including properties of the quadrilateral that make it unique. Mystery Quadrilateral 1 Mystery Quadrilateral 2 Mystery Quadrilateral 3 Mystery Quadrilateral 4 Mystery Quadrilateral 5 Mystery Quadrilateral 6 Problem setup: I am going to look at the properties of each of the mystery quadrilaterals and determine what type of quadrilateral each one is. Because all the quadrilaterals look like square, it is hard to make a predication. I predict that there is at least one rectangle, and one parallelogram. Plans to Solve/Investigate the Problem: I am going to use the GSP to measure the segments, angles, and slope of each mystery quadrilateral to determine the properties of the mystery figure. Then, I will use the properties to classify the figure appropriately. MYSTERY QUADRILATERAL 1 j = 4.17 cm B m ABC = 91.00 Three line segments are congruent. m DAB = 90.00 A One pair of parallel s ides j Slope j = 0.02 Only one right angle This quadrilateral is a trapezoid. BC = 4.17 cm AD = 4.17 cm BC Slope BC = -24.71 Slope AD = -43.48 AD C m BCD = 89.01 m CDA = 89.99 D k Slope k = 0.02 k = 4.25 cm The first quadrilateral is not a square because the there are only three congruent line segments. It’s not a rectangle because it does not have four right angles, or opposite sides are not parallel. This figure only has one pair of parallel sides. According to the slope segment AB and DC are parallel. Since these sides are parallel, this figure is a trapezoid. MYSTERY QUADRILATERAL 2 j = 5.86 cm m ABC = 90.00 B m DAB = 90.00 A j Slope j = 0.06 Slope q = -15.79 s = 5.91 cm q q = 5.91 cm Slope s = -15.79 s C m CDA = 90.00 D r Slope r = 0.06 m BCD = 90.00 r = 5.86 cm Oppos ite sides are congruent. All 4 angles are right angles. There are two pair of parallel lines. This quadrilateral is a rectangle. The second quadrilateral has four right angles, but the sides are not congruent. So, this figure is not a square. According to the slopes, the opposite sides are parallel. Although this figure has four right angles, the segments are not congruent. This quadrilateral must be a rectangle. MYSTERY QUADRILATERAL 3 m ABD = 90.00 Slope CB = -0.34 B CB = 4.71 cm CB m BDC = 90.00 m = 4.71 cm m D Slope m = 2.93 o = 4.71 cm A m CAB = 90.00 Slope o = 2.93 p o p = 4.71 cm C Slope p = -0.34 m DCA = 90.00 There are 4 congruent s ides and 4 right angles . The s lopes indicates that opposite sides are parallel. This quadrilateral is a square. The third quadrilateral has four right angles, and all four sides are congruent. The slopes indicate that the opposites are parallel. This quadrilateral must be a square. MYSTERY QUADRILATERAL 4 m ABC = 91.00 B BA = 3.92 cm BA BC = 3.92 cm Slope BA = 0.82 BC Slope BC = -1.17 m DAB = 89.00 A C m BCD = 89.00 DA = 3.92 cm CD = 3.92 cm DA CD Slope DA = -1.17 Slope CD = 0.82 D m CDA = 91.00 All four s ides are congruent. This quadrilateral has two pairs of congruent angles . The s lope indicates that there are two pairs of parallel lines. This quadrilateral is a rhombus because the s ides are congruent, but the angles are not congruent. The fourth quadrilateral has four congruent sides, and two pair of congruent angles. The slopes indicate that opposite sides are parallel. The opposite angles are also congruent. This quadrilateral is a rhombus. MYSTERY QUADRILATERAL 5 AB = 2.30 in. B m ABC = 90.94 m DAB = 89.06 A AB Slope AB = 0.11 Oppos ite segments are congruent. There are 2 congruent angles . l = 2.25 in. There are 2 pairs of parallel sides. This quadrilateral is a parallelogram. Slope l = -7.92 m = 2.25 in. l Slope m = -7.92 m Slope n = 0.11 C m BCD = 89.06 n m CDA = 90.94 D n = 2.30 in. diagonals form 2congruent triangles diagonals als o bis ect each other The measurements are s hown below. Perimeter CDA = 7.79 in. Perimeter ABC = 7.79 in. Area CDA = 2.59 in2 Area ABC = 2.59 in2 m DAC = 45.15 m ACB = 45.15 m ACD = 43.91 m BAC = 43.91 The fifth quadrilateral has 2 sets of congruence sides and opposites angles. According to the slopes this figure has 2 sets of parallel sides. The diagonals forms 2 congruent triangles and the diagonals bisect each other. This quadrilateral is a parallelogram. MYSTERY QUADRILATERAL 6 AD = 4.27 cm AD D mCDA = 91 mDAB = 88 A Slope AD = 0 Perimeter ADC = 15 cm Area ADC = 9 cm2 Perimeter ABC = 15 cm Slope CD = -6 Area ABC = 9 cm 2 AB CD Slope AB = -5 A CD = 4.27 cm AB = 4.23 cm Slope BC = 0 C mBCD = 88 BC mBAC = 44 B BC = 4.23 cm mDAB+mBAC+mBCD+mCDA = 312 Diagonals are perpendicular Diagonals create 2 congruence triangles According to the s lope, there are a pair of parallel lines segment AB is parallal to BC. Adjacent sides are equal m DAA = 90 m DAC = 90 m AAB = 90 m BAC = 90 The sixth quadrilateral has adjacent sides that are equal in length. The diagonals form 2 congruent triangles and the diagonals perpendicular. I measured the diagonals to see if they formed angles that measured 90°. So I concluded that this quadrilateral is a Kite. I predicted that there was at least one rectangle (2) and one parallelogram (5). Extensions of the Problem: Discuss possible extensions for the problem and explore/investigate at least one of the extensions you discussed. Since the properties of quadrilaterals are difficult and confusing, I thought an extension would be to have the students create a chart to keep in their notebook to use as a guide until their proficient with the characteristics. I inserted a simple chart with the basic quadrilaterals and some of their properties. QUADRILATERAL CHART Quadrilateral Opposite Opposite Opposite Diagonal Diagonals Diagonals Diagonals A All All sides || sides angles forms bisect each are are diagonal angles sides other bisects 2 are angles right Parallelogram x x x x x Rectangle x x x x x x x Rhombus x x x x x x x x Square x x x x x x x x x x Trapezoid Isosceles Trap. x Kite x x GSP Connections: Grades 4-8 M4M2 Students will understand the concept of angle and how to measure angles. a. Use tools, such as a protractor or angle ruler, and other methods, such as paper folding or drawing a diagonal in a square, to measure angles. M4G1 Students will define and identify the characteristics of geometric figure through examination and construction. b. Examine and classify quadrilaterals (including parallelograms, squares, rectangles, trapezoids, and rhombi). d. Compare and contrast the relationship among quadrilaterals. M5G1 Students will understand congruence of geometric figures and the correspondence of their vertices, sides, and angles. M4- Students will solve problems (using appropriate technology) 8P1 a. Build new mathematical knowledge through problem solving b. Make and investigate mathematics and in other contexts. c. Apply and adapt a variety of appropriate strategies to solve problems. d. Monitor and reflect on the process of mathematical problem solving. M4- Students will reason and evaluate mathematical arguments. 8P2 a. Recognize reasoning and proof as fundamental aspects of mathematics. b. Make and investigate mathematical conjectures. c. Develop and evaluate mathematical arguments and proofs. d. Select and use various types of reasoning and methods of proof. Author & Contact Norma Smith norma_smith19@yahoo.com Link(s) to resources, references, lesson plans, and/or other materials http://www.mathsteacher.com.au/year7/ch09_polygons/04_quad/quad.htm http://www.icteachers.co.uk/children/sats/quadrilaterals.htm http://regentsprep.org/Regents/math/quad/LQuad.htm Important Note: You should compose your write-up targeting an audience in mind rather than just the instructor for the course. You are creating a page to publish it on the web.