Geometry Math Tool Unit 4
Geometry Unit 4 Overview: Connecting Algebra and Geometry Through Coordinates
Building on their work with the Pythagorean theorem in 8th grade to find distances, students use a rectangular coordinate system to verify geometric relationships, including properties of special triangles and quadrilaterals and slopes of parallel and perpendicular lines. Students continue their study of quadratics by connecting the geometric and algebraic definitions of the parabola.
Unit 4: Connecting Algebra and Geometry Through Coordinates G.GPE.4Cluster: Use coordinates to prove simple geometric theorems algebraically.
Standard / Suggested Learning Targets / Directly
Aligned / Somewhat Aligned / Not
Aligned
Instructional Notes: This unit has a close connection with the next unit. For example, a curriculum might merge G.GPE.1 and the Unit 5 treatment of G.GPE.4 with the standards in this unit. Reasoning with triangles in this unit is limited to right triangles; e.g., derive the equation for a line through two points using similar right triangles.
G.GPE.4 Use coordinates to prove simple geometric theorems
algebraically. For example, prove or disprove that a figure defined
by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2). / · I can represent the vertices of a figure in the coordinate plane using variables.
· I can connect a property of a figure to the tool needed to verify that property.
· I can use coordinates and the right tool to prove or disprove a claim about a figure.
For example:
· Use slope to determine if sides are parallel, intersecting, or perpendicular.;
· Use the distance formula to determine if sides are congruent or to decide if a point is inside a circle, outside a circle, or on the circle;
· Use the midpoint formula or the distance formula to decide if a side has been bisected. / Content/Skills Included in Textbook
(Include page numbers and comments)
Essential Questions/ Enduring Understandings / Assessment
· How can algebra be useful when expressing geometric properties?
Algebra can be used to efficiently and effectively describe and apply geometric properties. / Assessments align to suggested learning targets.
Directly
Aligned / Somewhat Aligned / Not
Aligned
Check all assessment types that address this standard
Drill and practice
Multiple choice
Short answer (written)
Performance (verbal explanation)
Product / Project
Vocabulary
Side length, vertex, first quadrant, slope, distance, midpoint, parallel, perpendicular, intersecting
Unit 4: Connecting Algebra and Geometry Through Coordinates G.GPE.5
Cluster: Use coordinates to prove simple geometric theorems algebraically.
Standard / Suggested Learning Targets / Directly
Aligned / Somewhat Aligned / Not
Aligned
Instructional Notes: This unit has a close connection with the next unit. Relate work on parallel lines in G.GPE.5 to work on A.REI.5 in High School Algebra I involving systems of equations having no solution or infinitely many solutions.
G.GPE.5 Prove the slope criteria for parallel and perpendicular lines and uses them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). / Parallel Lines:
· I can draw a line on a coordinate plane and translate that line to produce its image.
· I can explain that these lines are parallel since translations preserve angle.
· I can determine slopes of the original line and its image after translation and show they have the same slope using specific examples and general coordinates (x,y).
· I can state that parallel line have the same slope.
· I can determine if lines are parallel using their slop.
· I can write an equation for a line that is parallel to a given line that passes through a given point. / Content/Skills Included in Textbook
(Include page numbers and comments)
Essential Questions/ Enduring Understandings / Assessment
· How can algebra be useful when expressing geometric properties?
Algebra can be used to efficiently and effectively describe and apply geometric properties. / Assessments align to suggested learning targets.
Directly
Aligned / Somewhat Aligned / Not
Aligned
Check all assessment types that address this standard
Drill and practice
Multiple choice
Short answer (written)
Performance (verbal explanation)
Product / Project
Vocabulary
Slope, parallel, coordinate plane, translations, coordinates (x, y)
Unit 4: Connecting Algebra and Geometry Through Coordinates G.GPE.5 continued
Cluster: Use coordinates to prove simple geometric theorems algebraically.
Standard / Suggested Learning Targets / Directly
Aligned / Somewhat Aligned / Not
Aligned
Instructional Notes: This unit has a close connection with the next unit. Relate work on parallel lines in G.GPE.5 to work on A.REI.5 in High School Algebra I involving systems of equations having no solution or infinitely many solutions.
G.GPE.5 Prove the slope criteria for parallel and perpendicular lines and uses them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). / Perpendicular Lines:
· I can draw a line on a coordinate plane and rotate that line 90⁰ to produce a perpendicula image.
· I can determine the slope of the original line and its image after a 90⁰ rotation and show they have the opposite reciprocal slopes using specific examples and genreal coordiantes (x, y)
· I can state that perpendicular lines have the opposite reciprocal slopes.
· I can determine if lines ar eperpendicular using their slope.
· I can write an equation for a line that is perpendicular to a given line that passes through a given point. / Content/Skills Included in Textbook
(Include page numbers and comments)
Essential Questions/ Enduring Understandings / Assessment
· How can algebra be useful when expressing geometric properties?
Algebra can be used to efficiently and effectively describe and apply geometric properties. / Assessments align to suggested learning targets.
Directly
Aligned / Somewhat Aligned / Not
Aligned
Check all assessment types that address this standard
Drill and practice
Multiple choice
Short answer (written)
Performance (verbal explanation)
Product / Project
Vocabulary
Slope, parallel, coordinate plane, translations, coordinates (x, y), product, line, linear equation, slope-intercept form, point-slope form
Unit 4: Connecting Algebra and Geometry Through Coordinates G.GPE.6
Cluster: Use coordinates to prove simple geometric theorems algebraically.
Standard / Suggested Learning Targets / Directly
Aligned / Somewhat Aligned / Not
Aligned
Instructional Notes: none
G.GPE.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio. / · I can calculate the point(s) on a directed line segment whose endpoints are (x1,y1) and (x2,y2) that partitions the line segment into a given ratio, r1 to r2 using the formula
x = r2 x1+ r1 x2 r1 +r2 y = r2 y1+ r1 y2 r1 +r2 (e.g., for the directed line segment whose endpoints are (0,0) and (4,3), the point that partitions the segment into a ratio of 3 to 2 can be found
x= 2∙0+3∙43+2 =125 and y=2∙0+3∙33+2 =95 so the point is (12/5, 9/5). (R) / Content/Skills Included in Textbook
(Include page numbers and comments)
Essential Questions/ Enduring Understandings / Assessment
· How can algebra be useful when expressing geometric properties?
Algebra can be used to efficiently and effectively describe and apply geometric properties. / Assessments align to suggested learning targets.
Directly
Aligned / Somewhat Aligned / Not
Aligned
Check all assessment types that address this standard
Drill and practice
Multiple choice
Short answer (written)
Performance (verbal explanation)
Product / Project
Vocabulary
Directed line segment, endpoint, ratio
Unit 4: Connecting Algebra and Geometry Through Coordinates G.GPE.7
Cluster: Use coordinates to prove simple geometric theorems algebraically.
Standard / Suggested Learning Targets / Directly
Aligned / Somewhat Aligned / Not
Aligned
Instructional Notes: G.GPE.7 provides practice with the distance formula and its connection with the Pythagorean theorem.
G.GPE.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.★ / · I can use the coordinates of the vertices of a polygon graphed in the coordiante plane and use the distance formula to compute the perimeter.
· I can use the coordinates of the vertices of triangles and rectangles graphed in the coordiante plane to compute area. / Content/Skills Included in Textbook
(Include page numbers and comments)
Essential Questions/ Enduring Understandings / Assessment
· How can algebra be useful when expressing geometric properties?
Algebra can be used to efficiently and effectively describe and apply geometric properties. / Assessments align to suggested learning targets.
Directly
Aligned / Somewhat Aligned / Not
Aligned
Check all assessment types that address this standard
Drill and practice
Multiple choice
Short answer (written)
Performance (verbal explanation)
Product / Project
Vocabulary
Coordinate plane, coordinates, distance formula, perimeter, polygon, area, triangle, rectangle
Unit 4: Connecting Algebra and Geometry Through Coordinates G.GPE.2
Cluster: Translate between the geometric description and the equation for a conic section.
Standard / Suggested Learning Targets / Directly
Aligned / Somewhat Aligned / Not
Aligned
Instructional Notes: The directrix should be parallel to a coordinate axis.
G.GPE.2 Derive the equation of a parabola given a focus and directrix. / · I can define a parabola.
· I can find the distance from a point on the parabola (x, y) to the directrix.
· I can find the distance from a point on the parabola (x, y to the focus using the distance formula (Pythagoprean Theorem).
· I can equate the two distance expressions for a parabola to write its equation.
· I can identify the focus and directrix of a prabola when given its equation. / Content/Skills Included in Textbook
(Include page numbers and comments)
Enduring Understandings / Essential Questions / Assessment
Algebra can be used to efficiently and effectively describe and apply geometric properties. / · How can algebra be useful when expressing geometric properties? / Assessments align to suggested learning targets.
Directly
Aligned / Somewhat Aligned / Not
Aligned
Check all assessment types that address this standard
Drill and practice
Multiple choice
Short answer (written)
Performance (verbal explanation)
Product / Project
Vocabulary
Parabola, focus, directrix, distance formula, factor, perfect square trinomial
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