AF Geometry Unit 1
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Page 2 of 177 AF Geometry Unit 1 Unit Overview Unit Title Coordinate geometry Duration 11 teaching days Unit Designers Jennifer Tillotson, Sven A. Carlsson IA Period 1
Identify Desired Results Standard Previous Grade Level Standards / Previously Taught & Related Standards Use coordinates to prove simple geometric theorems algebraically. G-GPE.4 Use coordinates to prove simple geometric theorems algebraically. For example, prove that the triangle with vertices (1,5), (7,-1), and (-1,-3) is isosceles.
G-GPE.5 Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line Relate work on parallel lines in G-GPE.5 to work on A-REI.5 in parallel or perpendicular to a given line that passes through a given point). Algebra I involving systems of equations having no solution or infinitely many solutions. G-GPE.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio. G-GPE.7 provides practice with the distance formula and its connection with the Pythagorean theorem. Building on their G-GPE.7 Use coordinates to compute perimeters of polygons and areas of exposure to the Pythagorean theorem in 8th grade as a way of triangles and rectangles, e.g., using the distance formula. calculating distance, students use a rectangular coordinate system to investigate and verify geometric relationships, primarily the G-CO.1 Know precise definitions of [angle, circle,] perpendicular line, properties of special triangles and parallel/perpendicular lines. parallel line, and line segment, based on the undefined notions of point, line, distance along a line, [and distance around a circular arc.] In Units 7 and 8, students will return to the coordinate plane, using the distance formula to examine the equation of a circle (U7) and deepening their understanding of parabolas through exploring the link between the equation of a parabola (focus/directrix) and its graphical behavior (U8).
Page 3 of 177 AF Geometry Unit 1 Enduring Understandings and Big Ideas Grade Level Enduring Understanding What it looks like in this unit Scholars will use their knowledge of the coordinate plane, slope (as developed in Pre-Algebra and Algebra1), and algebraic methods to make Geometric and algebraic procedures are interconnected and build on one statements about geometric figures. Scholars will use algebraic another. evidence to support arguments and classifications involving terminology from Geometry. Scholars will use coordinate geometry formulas (most notably the Two- and three-dimensional objects can be classified, described, and distance and slope formulas) to classify triangles according to the analyzed by their geometric attributes using a variety of strategies, tools, characteristics of their sides. Scholars will use rulers and protractors as and technologies. measuring tools, in the context of informally verifying side and angle congruence. The concept of isometry will be explored with isosceles triangles, as Congruent geometric figures can be mapped onto one another by one or congruent triangles will be provided on and off the coordinate plane, more rigid transformations (isometries). For similar figures, one of the with scholars commenting on the congruence of the triangles (using the transformations need not be an isometry. These transformations can occur distance formula to evaluate the former and tools to investigate the on or off the coordinate plane. latter) – Isosceles, Congruent, Similar Worksheet. Proof is an argument; it is a series of logically valid statements justified by definitions, postulates, and theorems. Writing sound proofs develops Though formal proofs (2-column, paragraph, and flow) may be reasoning and justification skills. introduced later in the course at teacher discretion, scholars will prove statements about triangles in the Cartesian plane using slope and length Effective mathematical arguments involve both concise language and clear as evidence to both confirm and refute given statements about the reasoning, in the form of closely related steps justified by relevant classification of the triangle. evidence. All constructions are based on geometric properties of congruence. Scholars will measure angles and side lengths using tools (ruler, Measurements (both direct and indirect) can be made to describe, compare, protractor) and algebraic formulas (distance), as well as using and make sense of real-life objects. Geometric measurements can be measurements such as slope to draw conclusions about lines and sides. represented in algebraic expressions and equations. Parallel Polygons Worksheet Identify the Narrative This unit reviews and extends scholars’ knowledge of linear equations, largely in the study of parallel and perpendicular lines on the coordinate plane. A heavy emphasis is placed on point-slope form, as fluency with this form will be most useful during their Algebra 2 and Calculus classes. Additionally, scholars will use other properties of coordinate geometry (the midpoint and distance formulas, equations of perpendicular bisectors, etc.) to solve problems. Scholars will be expected to determine which formulae are most appropriate in a given situation and synthesize their knowledge in writing coordinate proofs. Writing proofs requires that scholars apply their knowledge of the properties and classification of shapes in a new way; Page 4 of 177 AF Geometry Unit 1 scholars will move from simply solving questions to articulating their arguments with precise terminology and explicitly stating their reasoning/thought process. Coordinate proof thus deepens scholar knowledge (of both coordinate geometry and the properties of lines) while strengthening their ability to articulate their ideas and support them with evidence.
Standards for Mathematical Practice MP.1 Make sense of problems and persevere in Scholars will be required to analyze and decode problems on the coordinate plane in order to solving them understand what they are being asked to do. Coordinate proofs require abstract reasoning skills (the ability to generalize or work with generalizations) applied to coordinate geometry formulas (using quantitative evidence and MP.2 Reason abstractly and quantitatively computation to support their reasoning). Additionally, scholars will use their reasoning and experience with the derivation of formulas to determine which formulas are to be used. See: Midpoints, Perpendicular Bisectors, and Distance WS. With the introduction of coordinate proof, scholars will routinely prove statements about geometric figures and analyze the arguments put forth by others. They will be expected to use coordinate MP.3 Construct viable arguments and critique geometry formulas flexibly, as well as to work with multiple types of proof (two-column, paragraph, the reasoning of others flow, and coordinate). See: Midpoints, Perpendicular Bisectors, and Distance WS. MP.4 Model with mathematics Scholars will use rulers and protractors to measure segments and angles respectively, in contexts MP.5 Use appropriate tools strategically where either (or neither) tool may be most appropriate. See: Parallelogram or not WS Scholars will use precise geometric definitions and terminology when classifying triangles according MP.6 Attend to precision to their sides and when locating points of concurrency of triangles on the coordinate plane. Scholars will articulate not only the meaning but the difference (or equivalence) of expressions such MP.7 Look for and make use of structure as , , or in the distance, midpoint, and slope formulas. MP.8 Look for and express regularity in repeated reasoning
Major Misconceptions & Clarifications Misconception Clarification To show that a triangle is isosceles, slope can be The terms isosceles, equilateral, and scalene all refer to the lengths of the sides of a triangle. used. Slope can be used to determine whether sides are perpendicular or oblique; the slope of a segment is unrelated to its length and thus does not need to be considered when classifying a triangle as isosceles. The importance of reading comprehension should be explicitly addressed – have scholars regularly state in their own words what a question is asking and which ‘trigger words’ in the question prompt tell them which formulas to use. Annotate the question! Page 5 of 177 AF Geometry Unit 1 The coordinates of the midpoint can be used as in The midpoint formula treats whatever points are inserted as endpoints and locates the midpoint the midpoint formula (when finding an endpoint). of the segment between them. The formula cannot directly output endpoints; the formula can only be used indirectly to calculate an endpoint, and this will always require an equation being set up. Scholars should red flag any question where they are given the midpoint, as its coordinates must be handled differently from those of an endpoint. There are several possible methods: scholars can label the midpoint (if given) as ; alternatively, scholars can sketch a rough diagram and label the known points with values/the missing point as (x, y). Build a habit of checking answers for reasonableness. There is no fast way to check if a simplified To quickly check if a simplified radical is correct, find the product of the square of each of the radical is correct. terms and compare to the original radicand. For example, to check that has been simplified correctly, simply multiply by , or 25 x 2. This equals 50, our original radicand, so we have simplified correctly. Similarly, to check , we multiply 16 by 3 and arrive at 48, our original radicand. The perpendicular bisector of a segment passes Each word in the phrase “perpendicular bisector” has meaning. Perpendicular leads to a through one of the endpoints of the segment. consideration of slope, as perpendicular lines are lines with opposite reciprocal slopes. Bisector leads to a calculation of the midpoint, as a bisector passes through the midpoint of the given segment. A consistent theme of the course is this emphasis on vocabulary: what does each word mean? How does that contribute to process? Make sure scholars are annotating the question and using quick sketches to have a visual anchor alongside their work. As always, push scholars to understand the process, rather than to see this as an algorithm to memorize.
Skills and Procedural Knowledge Write the equation of a line given a slope and y-intercept, or given the slope and any point on the line, or given any two points on the line. Identify line segments that are parallel or perpendicular given their slopes, points on the lines, or the equations of the lines. Write the equation of a line parallel or perpendicular to a given line, passing through a given point. Calculate the midpoint or length of a line segment, given its endpoints. Use the Pythagorean Theorem to find missing sides of right triangles. Describe relationships within triangles, and use these relationships to classify the shapes based on their characteristics. Find the area and perimeter of triangles by applying formulas. Find a point on a segment that partitions the segment into a given ratio. Solve a system of linear equations. Write an equation of the perpendicular bisector of a segment, given its endpoints. Use coordinate geometry formulas to prove statements about figures on the coordinate plane (for example, proving that a triangle is/isn’t a right triangle, finding the perimeter of a triangle, etc.)
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Unit Vocabulary Slope: the slope m of a non-vertical line is the ratio of the vertical change (the rise) to the horizontal change (the run) between any two points on the line. The slope of a line can be calculated using the equation . Y-intercept(s): the point(s) where a curve crosses the y-axis. X-intercept(s): the point(s) where a curve crosses the x-axis. Slope-intercept form: a form of a linear equation; , where m is the slope of the line and b is the y-intercept of the line. Standard form: a form of a linear equation, , where A, B, and C are integers and A is positive. Point-slope form: a form of a linear equation, , where is a point on the line and m is the slope of the line. Length of a Segment: the distance between the endpoints of the segment Parallel lines: two lines that do not intersect. Parallel lines have equal slopes, but different y-intercepts. Perpendicular lines: two lines that intersect to form a right (90o) angle. Perpendicular lines have opposite reciprocal slopes. Midpoint: a point that bisects a segment (divides it into two congruent segments). The midpoint of a line segment on the coordinate plane can be calculated by . Perpendicular bisector: A segment or line that is perpendicular to a segment and bisects it. Types of triangles – right (one right angle; two perpendicular sides), scalene (no congruent sides), isosceles (at least two congruent sides), equilateral (all sides congruent). Circumcenter: the point of intersection (concurrency) of the three perpendicular bisectors of the sides of any triangle. The circumcenter is the center of a circle that is circumscribed about the triangle (passing through all three vertices), thus the circumcenter is equidistant from all three vertices. The circumcenter of a triangle may lie inside the triangle (if the triangle is acute), on a side of the triangle (if the triangle is right), or outside the triangle (if the triangle is obtuse). Altitude: the perpendicular line segment from the vertex of a triangle to the opposite side. Scholars should not be exposed to altitudes until the Unit Exam. Incenter: the point of concurrency of the angle bisectors. The incenter is the center of a circle that is inscribed in the triangle (tangent to all three sides). This will not be seen in any problems until Constructions in Unit 3. Median: in a triangle, the median is the segment from the vertex to the midpoint of the opposite side. Centroid: the point of intersection (concurrency) of the three medians of any triangle. The centroid always lies in the interior of a triangle. The centroid is the center of mass of a triangle (the point at which a triangle can be balanced), and it is located 2/3 of the way from the vertex to the midpoint. Orthocenter: the point of intersection (concurrency) of the three altitudes of any triangle. What is interesting about the orthocenter is that if you consider the orthocenter and the three vertices of the triangle (4 points total), any one of those four points is the orthocenter for the other three. The orthocenter of a triangle may lie inside the triangle (if the triangle is acute), on one side of the triangle (if the triangle is right), or outside the triangle (if the triangle is obtuse).
Topics: Page 7 of 177 AF Geometry Unit 1 Topic 1 – Lines, slope and equation of Topic 2 – Segments, length, midpoint, and perpendicular bisector of Topic 3 – Triangles in the Cartesian plane
Aims Sequence A i m / E x i Lesson # MPS t Key Points, Resources and Notes
T i c k e t 1.1 MP.2 S The slope of a line can be calculated by dividing the T1 MP.7 Wchange in y by the change in x (, or equivalently by B using the slope formula (). A If the x- or y-coordinate of a point is unknown, it is T possible to use the slope formula to solve for this missing value by substituting known quantities and i solving the resulting equation. n t Note: this lesson can be shortened to allow for review e of linear equations, i.e. converting between forms or to r slope-intercept form. Teachers may also use this time to p review other selected concepts from Algebra – writing r linear equations given slope and a point, writing linear e equations from graphs, writing equations of vertical and t horizontal lines, etc.
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1.2 MP.7 G To write the equation of a line, you need to know the T1 i slope and a point on the line. If that point is not the y- v intercept, you can write the equation by substituting the e slope and point into point-slope form. n If you are given two points on a line, you must first find Page 22 of 177 AF Geometry Unit 1 the slope of the line between the two points using the t slope formula, and then you can use point-slope form to h write the equation of the line. Either point can be used e as the point in point-slope form. Parallel lines have the same slope; perpendicular lines e have negative reciprocal slopes. The product of a q number and its reciprocal is -1; the reciprocal of an u integer is and the reciprocal of is . a t Just because two lines intersect does not mean that they i are perpendicular. o n
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1.3 G Same key points as above, with the following additions: T1 i The slope of a line is the coefficient of the x term when v the equation is in slope-intercept form. To calculate the e slope you need two points on the line. n Perpendicular lines have negative reciprocal slopes, not reciprocal slopes. Page 46 of 177 AF Geometry Unit 1 t h To avoid arithmetic errors, ALWAYS write the formula e first, then substitute values
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1.4 Fl Questions involving variables/unknowns as coefficients T1 ex eg. Find the equation of the line through (4,-5) that is perpendicular to the line . D Additional problems involving linear equations ay e.g. Given the line , give the equation of a line perpendicular, parallel, oblique… Pre-teach/review of solving linear systems, in preparation for 1.9 1.5 MP.1 G The x-coordinate of the midpoint of a line segment is T2 MP.7 i the average of the x coordinates of the endpoints; the v same is true for the y-coordinate. e The midpoint M of a segment with endpoints at (x1, y1) Page 61 of 177 AF Geometry Unit 1
n and (x2, y2) is given by M = . To find a missing endpoint (if given the midpoint and t other endpoint), substitute in known values, set up two w equations (one for the x-coordinate, the other for the y- o coordinate), and solving each equation. Make sure to rewrite the final answer as an ordered pair. e When applying the slope, midpoint, or distance n formulas, it doesn’t matter which point is designated d (x1, y1) or (x2, y2). The reason for this is as follows: p In the slope formula, the magnitude of the difference o between the x- and y-coordinates will be the same – i only the sign will differ. However, because the sign will n change for both the numerator and denominator, the end t result will be the same. s In the midpoint formula, the answer remains the same because addition is commutative. o In the distance formula, the result is the same because . f
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1.6 MP.1 G The distance formula is derived from the Pythagorean T2 i theorem. v When simplifying radicals by hand, show every step to e avoid errors, e.g. . Also check final answer. Page 81 of 177 AF Geometry Unit 1 n To partition a segment, find the change in the x- and y- coordinates of the endpoints, set up a proportion with t the desired fraction, then apply it to each coordinate. w Scholars are actually using similar triangles and the o ratio of similitude when applying this method. The distance formula can be used to verify that this e ratio is correct and the correct coordinate have been n determined. d p o i n t s
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1.7 Fl Further work with partitioning a segment into 1/3, 2/3, 4/5, etc. T2 ex Informally confirming partitions by using a ruler and setting up proportions D ay 1.8 MP.1 G Same key points as above, with the addition of: T3 MP.2 i A scalene triangle has no congruent sides, an isosceles MP.3 v triangle has (at least) two sides congruent, and an MP.6 e equilateral triangle has all sides congruent. n
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1.9 G Annotate the question! The words “perpendicular T2 i bisector” tell you what to do: v Perpendicular implies slope; specifically, find the e negative reciprocal slope. n Bisector implies splitting into two equal pieces; specifically, find the midpoint). t Emphasize to scholars that it is important to understand w the process and not simply try to memorize an o algorithm. Scholars should sketch a quick visual anchor to check the reasonableness of their final answer and to e guard against errors. Clearly label all steps with headers n so that you know what you are calculating d p Note: make sure to teach perpendicular bisectors both o graphically and algebraically – for example, by giving i scholars the graph of a segment and asking them to Page 123 of 177 AF Geometry Unit 1 n write the equation of the perpendicular bisector of the t segment. This can also be useful in having them s visualize their errors, e.g. using an endpoint of the segment instead of the midpoint to write the equation. o f
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Page 135 of 177 AF Geometry Unit 1 1.10 MP.1 G This is an exciting lesson as it combines ideas from 1.4 T3 i (midpoints), 1.5 (ratios), and 1.8 (perpendicular v bisectors). The points of concurrency in a triangle are of e interest to us because of their unique properties. n The circumcenter of a triangle can be found by finding the intersection point of any two perpendicular bisectors t of the sides of the triangle. This involves writing the h equations of both perpendicular bisectors and solving a e linear system. Similarly, the centroid of a triangle can be found by c calculating the intersection point of any two of the o medians. However, the centroid can also be found by o simply finding the endpoints of one of the medians, r then partitioning that segment into the ratio 2:1 using d the slope partition method of 1.5. i n As always, emphasis should be placed on understanding a how the process for (finding the circumcenter) flows t naturally from the definition, rather than presenting this e aim as a process to be memorized. While scholars are s memorizing that the centroid is located 2/3rd of the way from the vertex to the midpoint, they are not o memorizing an algorithm that spits out the centroid – f they are applying a method learned in 1.5 to a definition (median) learned today. t h NOTE: Do not mention altitudes and orthocenters, as e these terms should be first introduced to students on the Unit Exam. v e r t i c e Page 136 of 177 AF Geometry Unit 1 s
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Page 145 of 177 AF Geometry Unit 1 9 ) ,
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1.11 Fl Areas of triangles in the coordinate plane ex For added difficulty, consider making no side of the triangle horiztonal/vertical. This requires scholars to find the D equations of the lines serving as the height and base, to solve a system to find their point of intersection, and lastly ay to apply the distance formula to determine the height of the triangle. Unit Exam
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1. Find the y-intercept of the line passing through (-4, 2) and (6, 6).
(A)
(B)
(C)
(D)
Page 156 of 177 AF Geometry Unit 1
(E)
2. Which of the following lines is perpendicular to the line and passes through the point (5, -2)?
(A)
(B)
(C)
Page 157 of 177 AF Geometry Unit 1
(D)
(E)
3. Line has a positive slope and passes through the origin. If line is perpendicular to line , which of
the following must be true?
(A) Line k passes through the origin.
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(B) Line k has a positive slope.
(C) Line k has a negative slope.
(D) Line k has a positive x-intercept.
(E) Line k has a negative y-intercept.
4. A triangle has vertices at A (-4, 1), B (-8, -3), and C (-6, -5).
Which of the following is true about triangle ABC?
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(A) is a right triangle because is perpendicular to .
(B) is isosceles triangle because is perpendicular to .
(C) is a right triangle because is perpendicular to .
(D) is isosceles because is perpendicular to .
(E) is not a right triangle.
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5. Which equation below represents the perpendicular bisector of the line segment connecting the
points (3, -1) and (-9, 5)?
(A)
(B)
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(C)
(D)
(E)
6. A line passing through the points (a, -5) and (2, -6) is parallel to the line represented by the
equation . Find the value of a. Show all of your work for full credit.
(A)
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(B)
(C)
(D) 3
7. ΔACD has vertices at A (-2, 4), C (6, 7), and D (-7, -5). Which of the following formulas would be
needed to classify the triangle as scalene, isosceles, or equilateral?
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(A) The midpoint formula
(B) The distance formula
(C) The slope formula
(D) Both (B) and (C)
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(E) (A), (B), and (C)
8. The midpoint of is M (-10, -16). If one endpoint is B (-1, 8), what are the coordinates of the other
endpoint, C?
(A) (, )
(B) (, )
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(C) (, )
(D) (, )
(E) (8, 32)
9. A quadrilateral is a shape with four sides. In the quadrilateral GRIT below, two triangles are
formed by constructing .
D.a.a) Prove that both triangles are right triangles.
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(b) Prove that is not equilateral.
10. Write the equation of the line that passes through the point P (1, -2) and is parallel to
the line shown in the diagram below.
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11. The map below shows a straight local highway that runs between two towns in Massachusetts.
Highway planners want to build a rest stop somewhere between the two towns.
If the lead planner wants the rest stop to be 1/3 of the way from Ashton to Bedford, at what
coordinates will the town be located?
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12. Consider below.
(a) Find the perimeter of the
triangle.
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(b) Find the coordinates of the centroid.
(c) Your friend Barack wants to find the area of the triangle. He says to you, “Let’s say is the
base. AB is 7 and from B to O is 5 units up, which is the height, so the area should be about
2 ½ x 7 x 5 or 17.5 units .” Do you agree or disagree? Explain your reasoning.
Consider the triangle ABC with vertices A(1,0), B(5,8), and C(10,3).
13. An altitude of a triangle is a segment drawn from a vertex to the opposite side that is also perpendicular to the side.
Page 170 of 177 AF Geometry Unit 1 a. Write the equation of the altitude to side .
b. Write the equation of the altitude to side .
c. The orthocenter is the point of concurrency of the altitudes. In other words, the orthocenter is where the altitudes of a triangle intersect. Find the coordinates of the orthocenter of .
d. Your friend Barack says to you, “The perpendicular bisector of is parallel to the altitude to side .” Decide whether he is correct or incorrect and then prove your answer.
Unit Exam Scoring Guide
Question Correct Answer Points Point Breakdown 1 A 2 1 – Linear equation with slope of 2/5 1 – Answer 2 D 2 1 – Indicate desired slope is 4 or 1 – Answer 3 C 2 2 – Answer 4 C 2 2 – Answer 5 D 3 1 – Correct midpoint (-3,2) (lose ½ point for each incorrect coordinate) 1 –Indicate the desired slope is 2 1 – Answer 6 D 3 1 – Use given points to calculate slope ; simplification not necessary 1 – Equate the slopes of both lines 1 – Answer 7 C 2 2 – Answer 8 C 2 1 – Clear indication of method used, be it graphical or algebraic 1 – Answer Page 171 of 177 AF Geometry Unit 1 9a is a right triangle 3 1 – Calculate slopes as -2 and ½ (do not need to simplify). because segments 1 – Conclude perpendicularity by comparing slopes of adjacent sides GT and TI are 1 – Answer perpendicular. We know the segments Note: if scholar calculates area of a larger rectangle then subtracts area of triangles, are perpendicular score as follows: because their slopes 1 – Correct area for larger rectangle are opposite 1 – Correct area for at least 3 of 4 triangles reciprocals; and . 1 - Answer Similarly, since and , we know that , so is right. 9b GT = and TI =3, so 2 1 – Argument: if all sides are not congruent, then the triangle is not equilateral. . In an equilateral 1 – Evidence that two sides are not congruent through a distance computation (correct triangle, all sides are or incorrect). congruent, so is not an equilateral triangle because its two sides GT and TI are not congruent. 10 3 1 – Indicate the slope of the given line is 1 – Scholar uses parallel lines have equal slopes 1 – Consistent Answer (for eligible scholars) Eligibility: scholar indicates negative slope for the graphed line
Note: For example, a scholar who incorrectly identifies the slope of the graphed line as , then goes on to give the consistent answer or correctly simplifies to , will earn a 0-1-1, or 2 points total. 11 5 1 – Calculate the slope between Ashton and Bedford 1 – Find correct slope of 1 – Multiply and by 1 – Add and to coordinates of Ashton 1 – Answer
Note: If scholars using proportions, can earn the first four points through correctly set- up proportions Page 172 of 177 AF Geometry Unit 1 and .
Note: If scholar gives an answer of , the point 1/3rd of the way from Bedford to Ashton, earns a maximum of 1-1-1-0-0, or 3 points total.
12a units 2 1 – Correct value for either AO or BO 1 – Answer 12b (5,2) 3 1 – calculate two correct median equations (y=1.33x-4.67, y=0.17x+1.17, y=-x+7) 1 – Solve linear system formed by scholar’s median equations 1 – Answer 12c 2 1 – Answer (Barack is incorrect) 1 – In the area formula, the base and height must be perpendicular. Barack is incorrect.
When calculating
the area, the base
and height must be
perpendicular. If
Page 173 of 177 AF Geometry Unit 1
is used as the
height, then Barack
must use a
horizontal distance
as the height, not a
vertical distance.
13a 3 1 – Find the slope of as 1/3 1 – Find the opposite reciprocal of their 1 – Answer
Note: incorrect final answer earns a maximum of 0-1-0, or 1 point total 13b 3 1 – Find the slope of as Page 174 of 177 AF Geometry Unit 1 1 – Find the opposite reciprocal of their 1 – Answer
Note: incorrect final answer earns a maximum of 0-1-0, or 1 point total 13c (6,5) 3 2 – attempt to solve a linear system composed of linear equations from (a) and (b) using either Substitution or Linear Combination 1 – Answer
Note: 1/3 if correct point of intersection is found graphically with no evidence that scholar checked the point of intersection in both equations.
Finding the correct point of intersection from a labeled graph with evidence of scholar checking the solution in both equations earns 3/3. 13d Barack is correct. 2 1 – Answer The slopes of 1 – Evidence parallel lines are equal. The slope of the altitude is , and the slope of the perpendicular bisector is the opposite reciprocal of 1/3, which is -3 as well.
Appendix A: Tasks
Page 175 of 177 AF Geometry Unit 1 Performance Assessments Triangles inscribed in a circle (link: Suppose A = (−1, 0) and B = (1, 0) are points in the Illustrative Mathematics) coordinate plane as pictured at right.
Suppose C = (x, y) is a third point in the coordinate plane, and that C does not lie on the x-axis. Show that if is a right angle, then . Solve the above equation to show that if is a right angle, then C (x, y) must lie on .
Getting to know Perpendicular Bisectors Consider the line segment with endpoints A (-1, 2) and B (3, -4). a. Write the equation of the perpendicular bisector of . Call this line . b. Prove that the point (0, 5) is not on the line . c. List one point on the line l that is in the first quadrant and one point on l that is in the third quadrant. d. Find a new point that is on the line l (it may lie in any quadrant). Name the point P. Prove that P is equidistant from A and B. e. Is the point F (2017, 1342) equidistant from points A and B? What does this tell you about the point F with respect to the line l ?
Applications of the Circumcenter Your family is considering moving to a new home. The diagram shows the locations of the office and factory where your parents work, and the location of your school. The three locations form a triangle. Your family wants to live at a point that is equidistant from each location. Describe how you could use the diagram to find a point that fits this criteria, citing any formulas, postulates, or theorems you use.
Critiquing vague language
Taken from Exeter Math 1, page 47 Working with Standard Form
Taken from Exeter Math 1, Page 50
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Appendix B: Teacher Background Knowledge
Important formulas, postulates, and theorems Slope formula: Forms of linear Slope-intercept form: equations Standard form: Point-slope form: Parallel lines have the same slope. Perpendicular lines have opposite (negative) reciprocal slopes. Slopes of lines on the Any line parallel to the x-axis has a slope of zero. coordinate plane Any line parallel to the y-axis has an undefined slope. Midpoint formula: Coordinate geometry Distance formula: formulas Centroid formula: If teaching to scholars, relate this formula to the centroid as center of mass.
Questions The Teacher Should Be Able to Answer
1. Why are the slopes of perpendicular lines opposite reciprocals? or Why is the product of the slopes of perpendicular lines equal to -1? While scholars have already learned that the product of the slopes of perpendicular lines is negative one, they may not be clear on why this is so. It is quick to explain and stems from taking a segment with a given slope, say , rotating it 900 about an endpoint, showing that the horizontal and vertical components have thus switched, and concluding that the new slope is .
See the following explanations for a quick refresher: Central Oregon Community College Drexel’s Ask Dr. Math
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