2, Unit 2 -
Patterns of Location, Shape, and Size
In the "Patterns of Location, Shape, and Size" unit, students learn to
use coordinates to model points, lines, and geometric shapes, and to
analyze the properties of lines and shapes. In addition, they learn to
write systems of linear equations to model real-life situations, to solve
those systems and interpret the solutions in context. Combining these
concepts with matrix operations from Course 2, Unit 1 and programming
techniques, students learn to model polygons and transformations of polygons
and to investigate the properties of shapes that are preserved under
Ideas from Course 2, Unit 2
- Coordinates for points: (2, 1) is 2 units right of the origin
and 1 unit up.
- Coordinates for lines: Lines can be created by joining points.
The slope of the line joining (2, 1) and (-3, 0.5)
is (1 - 0.5)/(2 + 3) = 0.1. So, the equation
of this line must be y = 0.1x + b.
To find b, substitute (2, 1) into y = mx + b to
get 1 = 0.1(2) + b, so b = 0.8.
So, the equation of the line through the given points is y = 0.1x + 0.8.
- Distance: from (a, b) to (c, d)
is the square root of (a - c)2 + (b - d)2.
- Size transformation: Coordinates of the original shape are
multiplied by the scale factor to produce an image. For example,
a triangle made of (1, 3), (2, 8) and (3, -5) can be
scaled up, using a scale factor of 2, to make another triangle (2, 6),
(4, 16), (6, -10) which has sides twice as long as the original,
and area 4 times as large. In general, the area of the image will be
increased by a factor of a2 if the scale factor is a.
Other kinds of transformations are as follows: translations, reflections, rotations.
Each can be discerned from the patterns of change from coordinates
of preimage to coordinates of image.
- Linear combination method: A way of solving a system of linear
equations symbolically. 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. For example, suppose we are solving 2x + y = 7
and 3x + 4y = 13. We could choose
a common multiple of 2x and 3x, which is 6x, and
then multiply the first equation by 3, to get 6x + 3y = 21.
We must now multiply the second equation by -2, to get -6x - 8y = -26.
We now have the same two equations 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. We have ascertained that the solution 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). Sometimes this method leads to no solution, or an infinite
number of solutions.
- Multiplying the coordinates of a shape by a matrix can create
a congruent 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 about
the origin and rotation of 45° counterclockwise about
the origin are effected by multiplying by and .
Rigid transformations (translations, reflections, rotations)
leave the size and shape unaffected; distances, angle measures, slopes
and areas are unchanged. For example, adding the matrix to
the triangle matrix will
add 2 to every x value and 3 to every y value so the
triangle will move 2 right and 3 up to become the new but congruent