Checkpoint - Course 2, Unit 2
Patterns of Location, Shape, and Size

In each unit, there is a final lesson and Checkpoint that helps students summarize the key ideas in the unit. The final Checkpoint will generally be discussed in class, with the teacher facilitating the summarizing, and students making notes in their Math Toolkits (teachers may just refer to this as "notes") of any points they need to remember, adding illustrative examples as needed. If your student is having difficulty with any investigation in this unit, this Checkpoint and the accompanying answers may help you recall the concepts involved, and give you the big picture of what the entire unit is about. If your student has completed the unit, then a version of this should be in his or her notes or toolkit. Students should also have Technology Tips in their toolkits, which may be useful for this unit.

Possible Responses to Unit Summary Checkpoint
In this unit, students investigated how coordinates and matrices can be used to model geometric ideas. The bolded words are vocabulary and concepts your student should be familiar with.
a. Describe several different ways that coordinates are used to model geometric ideas. Illustrate with examples.
Coordinates can be used to model points, lines, shapes, distance, slope, and transformations. Examples: points could be named (2, 1), (-3, 0.5), (a, b), (c, d), (x, y); lines can be created by joining points, such as (2, 1) and (-3, 0.5) and the equation of the line can be written by finding the slope between 2 points, and substituting the slope and a point into the equation y = mx + b; lines can be named y = -2x + 3, y = 4, x = -2.5; shapes can be created by connecting in order (0, 0), (2, 3), (1, 5), (-4, 1), and back to (0, 0); the distance from (2, 3) to (5, -4) is the square root of (5 - 2)2 + (-4 - 3)2, a result that becomes an obvious application of the Pythagorean Theorem if the two points are used to create the hypotenuse of a right triangle; the slope from (a, b) to (c, d) is (d - b)/(c - a), a result familiar to students from their work in Course 1; patterns of change from preimage points (2, 1) and (-3, 0.5) to image points (4, 2) and (-6, 1) can be discerned from the coordinates (size transformation in this case since both x- and y-coordinates are multiplied by the same scale factor, 2). Other kinds of transformations are as follows: translations, reflections, rotations. Each can be discerned from the patterns of change from preimage to image.
b. Describe how to solve a system of two linear equations using a linear combination method. How can this method be interpreted geometrically. Illustrate with an example.
To solve a system of equations, multiply each equation in such a way that either the x-terms or the y-terms add to zero. Add the equations. The resulting equation will be easy to solve for the remaining variable. Use the value of the variable just found to substitute in one of the original equations to find the value of the other variable. Geometrically, such a linear combination gives a horizontal or vertical line. This line intersects the original lines at a point which is on both lines, that is at the point of intersection. For example, suppose we are solving 2x + y = 7 and 3x + 4y = 13. We could focus on either the x- or y-terms, whichever is more convenient. Suppose we decide to focus on the x-terms. We would then choose a common multiple of 2x and 3x, which is 6x, and then multiply the first equation by 3, to get 6x + 3y = 21. This equation is equivalent to the original equation, so geometrically this is the same line. Keeping in mind that the goal is to be able to add the two equations to eliminate the x-terms, we must now multiply the second equation by -2, to get -6x - 8y = -26. We now have the same two lines rewritten as 6x + 3y = 21, and -6x - 8y = -26. Adding these we get another equation which is also true, 0x - 5y = -5, or y = 1, which is a horizontal line. Since this equation was created from the original two equations, its solution is compatible with the solution of these equations. Geometrically this means that the line y = 1 intersects the original equations at the point of intersection. But a point of intersection has two coordinates. We have ascertained that this is (?, 1). To find the x-coordinate, we should substitute y = 1 into either of the original equations. We then get 2x + 1 = 7, so x = 3. The solution for the original system is (3, 1). The intersection point for the lines represented is (3, 1). We could have decided to start by eliminating the y-terms. Sometimes this method leads to no solution, or an infinite number of solutions, depending on the lines you start with. For example, if the system is 2x + y = 7 and 4x + 2y = 11, then the linear combination method will produce 0x + 0y = -3, which is never true. And if the system is 2x + y = 7 and 4x + 2y = 14, then the linear combination method will produce 0x + 0y = 0, which is always true. Interpreting these results geometrically, there is no intersection point for the first case because the two lines are parallel, and in the second case the two lines are identical, so every solution for the first equation is also a solution for the second.
c. Describe several different ways that matrices are used to model geometric ideas. Illustrate with examples.
Matrices may represent
  • the coordinates of the vertices of a polygon. For example, the following matrix represents a triangle joining the points (1, 7), (3, 8), and (5, 11):
  • a transformation. For example, adding the matrix to the above matrix for a triangle produces a new triangle that has been translated 1 right and 5 up, because the 1s in the first row are added to the x-coordinates, while the 5s in the second row are added to the y-coordinates. Multiplying a matrix by a number can also produce a size transformation. For example, the triangle given can be enlarged by a scale factor of 3 by multiplying by . The new triangle will be three times larger and three times further from the origin. Multiplying by a matrix can create a reflected image. For example, to create a reflection over the y-axis, we multiply the original triangle matrix by . In addition, various rotations are possible. Rotation of 90° counterclockwise around the origin and rotation of 45° counterclockwise around the origin are effected by multiplying by and .
  • a linear system of equations. For example, the equations 2x + y = 7 and 3x + 4y = 13 can be rewritten as = . This system can then be solved using the matrix method students investigated in Unit 1, Course 2.
d. How do rigid transformations affect distance? Angle measure? Parallelism of lines? Areas of plane shapes?
Rigid transformations (translations, reflections, rotations) leave the size and shape unaffected; distances, angle measures, slopes, and areas are unchanged.
e. How do size transformations affect distance? Angle measure? Parallelism of lines? Areas of plane shapes? Distances of a point and its images from the center of the transformation.
Size transformations affect distances and areas, but leave angle measures and slopes unaffected. Distances between points within the figure, and from the origin, will be multiplied by the scale factor. Areas will be multiplied by (scale factor)2. Thus, a scale factor of 3 makes all the sides of a preimage triangle 3 times longer, the distance from each vertex to the origin 3 times longer, and the area 9 times bigger.
f. How can animation effects be produced by a graphing calculator?
To create an animation on the calculator, first the original shape is entered as a matrix A with 2 rows and n columns, for n points. Next matrices are entered to create the desired effects, B, C, D, etc. For example, if we want to start with a triangle and have the triangle reflect over the y-axis several times, then we start with a 2 by 3 matrix, and the reflection matrix, and we write a program that
  • draws the original matrix A,
  • erases the drawing,
  • multiplies the original matrix by the reflection matrix B and
  • draws that result, BA,
  • erases the new drawing, and
  • multiplies the BA result by B again,
  • draws that result, and so on. Since the plan involves a repetitive loop of steps, replacing one drawing with a transformation of itself, some simple programming is called for. Students learn how to do this with a straightforward program. They can then create more complex animations, starting with any shape they like and controlling the stages in the animation by creating subprograms that contain as many repetitions as they like.

If you would like to see specific problems from Course 2, Unit 2, a link is provided to Examples of Tasks from Course 2, Unit 2. If you are interested in following up on the Geometry strand in general, then the Scope and Sequence will help you see where different concepts are introduced. On the Geometry page, you can read an explanation of the main geometry concepts as they are developed in all four courses.

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