Teacher notes
Shapes, measurements and transformations on the coordinate plane
Shape Transformations puts a shape on a four-quadrant grid and shows everything about it at once: the coordinates of every corner, the length of every side and the area, with the formula worked out step by step. Transform it, and the new shape arrives with the same information.
Draw a shape
Choose Circle, Triangle, Quadrilateral or Custom beside the graph, then click grid points. A circle takes two clicks (the center, then a point on the circle), a triangle three, a quadrilateral four, and a custom polygon as many as you set. Until the last point is placed, drag any point to move it. Every point always sits on whole-number coordinates.
Move, turn and resize it
Once the shape is finished, drag inside it to slide it, drag a side to turn it in quarter turns, or drag a corner to grow or shrink it from its center. Click a shape (without dragging) to delete it; deleting an image also removes the images made from it. For a circle, drag inside to move it and drag the point on the circle to change the radius. Everything snaps to whole coordinates, so the numbers stay friendly.
Read the measurements
Side lengths come from the distance formula, shown as √(3² + 4²) = 5. When an area formula needs a length that isn't a side, such as the height of a triangle, the diagonals of a rhombus or the radius of a circle, the tool draws it as a dashed helper line and lists it with a dashed marker. The area panel names the formula it used (½ × b × h, l × w, πr² and so on) and substitutes the actual numbers. Hover over any point or side to light up its row, or hover a row to find it on the graph.
Transform it
The four buttons on the left translate, reflect, rotate or dilate the newest shape. Each opens a short form with a live preview and the coordinate rule, for example (x, y) → (−y, x). Press it again to transform the new shape, building a chain such as ABC → A′B′C′ → A″B″C″. Drag the original and every image follows, which makes it easy to see that the rule works for any starting position. Hover over any point, such as A′, and an arrow joins it to its pair in the shape before and after it (A and A″), with both sets of coordinates lit up.
Projector tip: use the quadrant buttons beside the graph to zoom into one quadrant. The grid squares double in size, which helps students count units from the back of the room. The magnifier on the Measurements panel pops it out at double size as a window you can drag anywhere, and the graph grows to fill the space it leaves.
Frequently asked questions
Why do some scale factors not work?
Every point has to land on whole-number coordinates. A scale factor of ½ centered at the origin only works when every coordinate is even, for example. The dilation form lists the scale factors that work for the current shape and center.
Why can I only reflect over lines with a slope of 1, −1 or 0?
Reflecting over any other sloped line, such as y = 2x, sends grid points to places between grid points. Lines like y = x + 2, y = −x − 1 and y = 3, plus any vertical line x = a, keep every point on whole-number coordinates.
Why does the rotation only go in quarter turns?
Turning a shape by 90°, 180° or 270° about a grid point keeps every corner on the grid, which is what most middle-school and Geometry standards ask students to do on the coordinate plane.
How is the area of an irregular shape found?
Squares, rectangles, parallelograms, trapezoids, rhombuses, kites, triangles and circles use their own formulas. Any other polygon uses the shoelace formula, which finds the area from the coordinates alone. Every step is shown.
Does it work on a touchscreen or interactive whiteboard?
Yes. Points, sides and shapes can be dragged with a finger or stylus, and every button is sized for touch.