Benchmark Instructional Guide
Connecting Benchmarks/Horizontal Alignment
Terms from the K-12 Glossary
Purpose and Instructional Strategies
In grade 8 and Algebra 1, students used coordinate systems to study lines and the find distances
between points. In Geometry, students expand their knowledge of coordinate geometry to further
study lines and distances and relate them to classifying geometric figure. In later courses,
coordinates will be used to study a variety of figures, including conic sections and shapes that
can be studied using polar coordinates.
- Instruction includes the connection of the Pythagorean Theorem (as was used in grade 8)
to the distance formula. It is important that students not depend on just the memorization
of the distance formula.
- Instruction includes discussing the convenience of answering with exact values (e.g., the
simplest radical form) or with approximations (e.g., rounding to the nearest tenth or
hundredth). It is also important to explore the consequences of rounding partial answers
on the accuracy or precision of the final answer, especially when working in real-world
- In this benchmark, instruction is related to circles, triangles and quadrilaterals, their
definitions and properties. It may be helpful to review these definitions and properties
and the different types of triangles and quadrilaterals as it was part of instruction within
- Instruction includes determining when slope criteria may be necessary.
- For example, when classifying triangles and quadrilaterals or finding side lengths,
the slope criteria may be needed.
- For example, determining sides of equal measures will decide if a triangle is
isosceles or if a quadrilateral is a rhombus.
- For example, the slope criteria for parallel lines may help when deciding if a
quadrilateral is a parallelogram.
- For example, the slope criteria for perpendicular lines may help when deciding if
a triangle is right or if a quadrilateral is a rectangle.
- Explore with the students different approaches for the same goal.
- For example, given a parallelogram they can determine if it is a rectangle using
the slope criteria to identify right angles or using the distance formula (or the
Pythagorean Theorem) to identify if the diagonals are congruent.
- Instruction includes opportunities for students to find the coordinates of missing vertices
of a triangle or quadrilateral using coordinate geometry and applying definitions,
properties, or theorems.
- For example, when finding the coordinates of P such that PQRS is a rhombus
(given the coordinates of Q, R and S), guide the students to plot the points on the
coordinate plane and make a conjecture about the location of P. Have students
determine if their conjectures are true. Additionally, have students discuss the
definitions or properties they may use in each case
Common Misconceptions or Errors
- Students may use imprecise methods or incomplete definitions to classify figures.
Instructional Task 1 (MTR.2.1, MTR.4.1)
- Part A. What are the coordinates of P if PQRS is a right triangle and Q(−1, 2) and R(3, 0)?
- Part B. Show that PQ2+ QR2 = PR2.
- Part C. Compare your right triangle with a partner.
Instructional Task 2 (MTR.3.1)
- Three vertices of quadrilateral PQRS are at the points Q(−2, 1), R(3,−1) and S(−2,−3).
- Part A. What are possible coordinates of P if PQRS is a parallelogram?
- Part B. Show that PR bisects QS.
- Part C. Justify that PQRS is a parallelogram.
Instructional Task 3 (MTR.3.1, MTR.4.1)
- Coordinates for three two-dimensional figures are given.
Figure A (2,3), (3,−4), (3,−2)
Figure B (3,3), (2, −1), (−2,0), (−1,4)
Figure C (−2,3), (−3,1), (0,−4), (3,2)
- Part A. Plot the points on the coordinate plane.
- Part B. Write a conjecture about the specific name of each two-dimensional figure. What
would you need to determine your conjectures are true?
- Part C. Classify each figure.
Instructional Item 1
- Points A (0,2) and B (2,0) are endpoints of segment AB, the side of quadrilateral ABCD.
List possible coordinates for points C and D if quadrilateral ABCD is a rhombus, not a
Instructional Item 2
- Given quadrilateral ABCD with vertices (−3,−4), (1,5), (5,3), and (5, −8), respectively,
classify the type of quadrilateral.
*The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.