![]() ![]() ![]() If students have not yet learned the terminology for trapezoids and parallelograms, the teacher can begin by explaining the meaning of those terms. This task is best suited for instruction although it could be adapted for assessment. The only pictures missing here, from this point of view, are those of a rhombus and a general quadrilateral which does not fit into any of the special categories considered here. Interesting and important that these types of quadrilaterals can be distinguished by their lines of symmetry. Since $PQRS$ is a rectangle whose four sides have the same length, it must be a square.This task provides students a chance to experiment with reflections of the plane and their impact on specific types of quadrilaterals. Reflections preserve line segment lengths so Because reflections map line segments to line segments, knowing where the vertices of the rectangles map is enough to determine the reflected image of the rectangle.īelow is a picture of a quadrilateral $PQRS$ with the line $\overleftrightarrow$. Similar reasoning applies to all vertices in the three pictures. Similarly in the picture for part (iii), the vertex of the rectangle below the red line is a little more than $3$ boxes below the red line: so its reflection will be on the same line through the $x$-axis but a little over three boxes above the red line. So its reflection is just under three units below the red line, with the same $x$ coordinate. In the picture for (i), the rectangle vertex above the red line is just under three boxes above the red line. ![]() Reflecting over the $x$-axis does not change the $x$-coordinate of a point but it changes the sign of the $y$-coordinate. Line can be thought of as the $x$-axis of the grid. The case where the length and width of the rectangleįor each of the pictures below, the same type of reasoning applies: the red Also notice that as the length and width of the rectangles become closer to one another, the two vertices are getting closer and closer to the vertices of the original rectangle. The same size and shape as the original rectangle. Notice that the reflected rectangle is, in each case, still a rectangle of This is because the line of reflection passes through two vertices and reflection over a line leaves all points on the line in their original position. The reflected image shares two vertices with the original rectangle. The reflections of each rectangle are pictured below in blue. Seeįor an example of what students might be expected to do in 4th grade. The solution to this task reflects this expectation. While the 8th grade standards do not require students to reason as formally as they will in high school geometry, they are certainly able to reason more formally than they did in 4th grade. Note that students study lines of symmetry in 4th grade, but only informally. However, this choice of grid also makes it easier to reason about the reflections if they understand the descriptions of rigid motions indicated in 8.G.3. Also providing a challenge is the fact that the grids have been drawn so that they are aligned with the diagonal of the rectangles rather than being aligned with the vertical and horizontal directions of the page. This means that students need to approximate and this provides an extra challenge. Note that the vertices of the rectangles in question do not fall exactly at intersections of the horizontal and vertical lines on the grid. The task is intended for instructional purposes and assumes that students know the properties of rigid transformations described in 8.G.1. Note that a line of symmetry can be thought of as a transformation that shows a figure is congruent to itself in a non-trivial way. ![]() This leads naturally to the last part of the question: lines through a diagonal are lines of symmetry for a square (but not a non-square rectangle). Moreover, this "displacement" of the rectangle becomes smaller and smaller as the rectangle becomes closer to being a square. The examples show that the reflected image looks like a rotation of the original rectangle about its center point. In the case of reflecting a rectangle over a diagonal, the reflected image is still a rectangle and it shares two vertices with the original rectangle. The goal of this task is to give students experience applying and reasoning about reflections of geometric figures using their growing understanding of the properties of rigid motions. ![]()
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