Unit 7 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

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 3

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 four points on a coordinate plane formed the vertices of a parallelogram, rectangle, rhombus, or square.

Lesson 4

Learning Focus

Find the equation of a circle.

Lesson Summary

In this lesson, we derived the equation of a circle. We learned that the equation of a circle describes all the points a given distance from the center. Like the distance formula, it is based on the Pythagorean theorem.

Lesson 5

Learning Focus

Write and graph the equation of a circle.

Find the center and radius of a circle in general form.

Lesson Summary

In this lesson, we learned to write equations of circles in both standard form and general form. We used the process of completing the square to change an equation from standard form to general form.

Lesson 6

Learning Focus

Apply understanding of circles and their equations to new situations.

Lesson Summary

In this lesson, we solved problems about circles that required us to use graphs and formulas such as the Pythagorean theorem, the distance formula, and the midpoint formula. We found it useful to use the equation of the circle to find points on the circle or to determine that a point is not on a circle. Sometimes it was useful to change forms of the equation to find more information about the circle from the equation.

Lesson 7

Learning Focus

Develop a geometric definition for a familiar shape.

Lesson Summary

In this lesson, we learned the geometric definition of a parabola. Much like circles, parabolas are a geometric shape that can be constructed from a definition and as a set of points generated from an equation. In the same way that the defining features of a circle are the center and radius, the defining features of a parabola are the focus and directrix.

Lesson 8

Learning Focus

Write and graph parabolas.

Compare the geometric definition of parabolas with quadratic functions.

Lesson Summary

In this lesson, we learned that the graph of a quadratic function meets the definition of a parabola. We learned to write equations given the focus and directrix and to find the focus and directrix of the parabola when given the equation of a quadratic function.

Lesson 9

Learning Focus

Write equations for parabolas with vertical directrices.

Determine the direction of opening for any parabola.

Lesson Summary

In this lesson, we learned to work with parabolas that have a vertical directrix. We found how to determine if they opened left or right, and how to write an equation of the parabola given a focus and directrix.

Lesson 10

Learning Focus

Understand the definition of an ellipse.

Understand relationships between parts of an ellipse.

Write the equation of an ellipse.

Lesson Summary

In this lesson, we learned to understand the definition of an ellipse. We identified many of the features of an ellipse, including the foci, center, and major and minor axes. We found the equation of an ellipse based on the definition and learned to write the equation in standard form with any center.

Lesson 11

Learning Focus

Compare the equation of an ellipse to the equation of other geometric figures.

Graph hyperbolas.

Write the equation of a hyperbola.

Lesson Summary

In this lesson, we learned about hyperbolas, the last of the conic sections. We learned that the graphs of hyperbolas can open up and down or left and right. The definition of a hyperbola is much like the definition of an ellipse, except that a point on a hyperbola is the difference between the distances to the foci and the graph of an ellipse is the sum of the distances to the foci. This makes the equation of a hyperbola like the equation of an ellipse, except the terms are subtracted rather than added.

Lesson 12

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 from 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 13

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 14

Learning Focus

Model real-world scenarios with vectors, matrices and trigonometry.

Lesson Summary

In this lesson, we used vectors and matrices to represent a real-world context of a plane’s flight path being impacted by the wind. While representing this context, we had to represent vectors in a variety of ways, including directed line segments, horizontal and vertical components, single-column matrices, and magnitude and angle of direction.