Unit 8 Connecting Algebra and Geometry

Lesson 1

Learning Focus

Find the distance between two points in the coordinate plane.

Find the perimeter of a geometric figure in the coordinate plane.

Lesson Summary

In this lesson, we learned to find the distance between two points. We used the Pythagorean theorem to develop a formula that could be used whenever we need to find the length of a segment between two points. The formula can be applied to find the length of the sides of a geometric figure in the coordinate plane to calculate the perimeter.

Lesson 2

Learning Focus

Locate the midpoint of a segment and a point that divides the segment in a given ratio.

Lesson Summary

In this lesson, we examined strategies for dividing a line segment into two parts that fit a given ratio. One common application of this concept is to find the coordinates of the midpoint of a segment, given the coordinates of the endpoints.

Lesson 3

Learning Focus

Prove slope relationships between parallel lines and perpendicular lines.

Lesson Summary

In this lesson, we used transformations to prove that the slopes of perpendicular lines are negative reciprocals and the slopes of parallel lines are equal. To prove the theorems, we needed to write the lines and points so that they were general enough to cover all cases. When we used a specific point like the origin, we needed to make an argument that the relationship could be extended to any pair of lines that are parallel or perpendicular.

Lesson 4

Learning Focus

Prove quadrilaterals are parallelograms, rectangles, rhombi, or squares using coordinates.

Find the perimeter and area of a quadrilateral on the coordinate plane.

Lesson Summary

In this lesson, we used the distance formula, the midpoint rule, and the properties of slopes of parallel and perpendicular lines to determine if a given set of 4 points on a coordinate plane formed the vertices of a parallelogram, rectangle, rhombus, or square.

Lesson 5

Learning Focus

Represent quantities that have both magnitude and direction using vectors, and examine the arithmetic of vectors.

Lesson Summary

In this lesson, we learned how to represent quantities that have both magnitude and direction, such as a wind blowing at $30 mph$ towards the northeast, as a directed line segment, or vector, on a coordinate grid. We also learned how to add and subtract vector quantities, and examined contexts where vector arithmetic is useful.

Lesson 6

Learning Focus

Examine properties of matrix addition and multiplication.

Lesson Summary

In this lesson, we compared the properties of matrix addition and multiplication with the properties of addition and multiplication of real numbers, such as the associative properties, the commutative properties, and properties of identities and inverses. We found a lot of similarities, and some differences, and learned that we can find matrices that behave like $0$ and $1$ in the real number system.

Lesson 7

Learning Focus

Use matrices to perform geometric transformations on figures.

Lesson Summary

In this lesson, we learned how to use matrix multiplication to rotate the vertices of geometric figures around the origin on the coordinate grid, and to reflect figures across either of the axes.

Lesson 8

Learning Focus

Examine a new strategy for finding the multiplicative inverse of a matrix.

Lesson Summary

In this lesson, we learned a second method for finding the multiplicative inverse of a $2 \times 2$ matrix. We also found ways to determine if a $2 \times 2$ matrix has a multiplicative inverse or not.

Lesson 9

Learning Focus

Use multiplicative inverse matrices to solve systems.

Lesson Summary

In this lesson, we learned a new method for solving systems of equations by representing the system with a matrix equation and using multiplicative inverse matrices to solve the equation. The solution to the matrix equation provides the solutions to the system.

Lesson 10

Learning Focus

Solve $n \times n$ systems of linear equations.

Lesson Summary

In this lesson, we extended the method of solving systems by using matrix equations and the multiplicative inverse to $n \times n$ systems of equations. We used technology to find the multiplicative inverse matrix.