# Finding Parallelogram Vertices

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2 Mathematics Task Suggested Use This mathematics task is intended to encourage the use of mathematical practices. Keep track of ideas, strategies, and questions that you pursue as you work on the task. Also reflect on the mathematical practices you used when working on this task. Given three vertices of a parallelogram at A(1,2), B(4,1), and C(5,3), where can the fourth vertex be located?

3 Student Dialogue Suggested Use The dialogue shows one way that students might engage in the mathematical practices as they work on the mathematics task from this Illustration. Read the student dialogue and identify the ideas, strategies, and questions that the students pursue as they work on the task. Students in this dialogue have just studied the properties and definition of different polygons. They are now working on open-ended problems that require the use of those definitions and properties. Students already have experience plotting points onto the Cartesian plane from previous units. (1) Sam: Well, let s use graph paper and plot the three points we know. [plots on graph paper] Now all we need to do is find one more point that will make a parallelogram. What do we know about parallelograms? (2) Anita: Opposite sides are parallel? (3) Dana: So, let s draw some sides and see if we can t find the fourth vertex. We can connect points A and B to get side AB, and we can connect points B and C to get side BC. (4) Sam: This is starting to look more like a parallelogram! Hmm it looks like the fourth point should be around here [points around the area indicated by the gray circle].

4 Here. What if I kind of draw the lines like this [sketches two shaky lines, one from A and C] (5) Dana: It looks like the lines meet at (2,5). But I can t tell if your lines are quite parallel. It doesn t look like it. Here, let me draw it with a ruler. [erases the sketched lines, plots the point (2,5), and connects the remaining sides with a ruler] (6) Anita: That doesn t look parallel at all! Definitely not a parallelogram! (7) Sam: Yeah, that doesn t look right. Let s try again. But wait! Even if the picture looks right, does that mean it s really a parallelogram? And what if the fourth point is not even on a gridline? (8) Anita: I think there s a way to find the point we want without guessing. If we picture side BC sliding over, we would get the opposite side of the parallelogram. [uses hand to show how the line segment would slide] (9) Dana: You re keeping your hand pointed in the same direction as you move it does that help us? (10) Anita: Yeah, I m trying to keep the line tilted the same way so they re parallel. And we can do the same thing to slide AB up. But it s still hard to see where exactly the fourth point would end up.

5 (11) Sam: We have the graph paper, so why don t we use the grid? To move from B to C, we have to go up 2 and over 1. So let s go up 2 and over 1 from A and see where that gets us. That will make sure the side is the same length and tilted the same way. [students go up 2 and over 1 and get (2,4) and mark the new point D] (12) Anita: They re tilted the same way, so we know that AD is parallel to BC. And they have to be the same length because they cross the same amount of space. Let s check if CD is parallel to BA. [students use the same reasoning to check the remaining two sides of the parallelogram] (13) Sam: Great! We re done! (14) Dana: Wait. We might not be done. I m wondering if we can have more parallelograms. (15) Sam: What do you mean? (16) Dana: Well, I originally connected points A and B to get side AB and points B and C to get side BC, but I could ve drawn AB and AC instead, right? (17) Anita: You re right! We ve only considered one possible parallelogram but there could be more. Let s find those too

7 Mathematical Overview Suggested Use The mathematical overview provides a perspective on 1) how students in the dialogue engaged in the mathematical practices and 2) the mathematical content and its extensions. Read the mathematical overview and reflect on any questions or thoughts it provokes. Commentary on the Student Thinking Mathematical Practice Make sense of problems and persevere in solving them. Construct viable arguments and critique the reasoning of others. Use appropriate tools stratgeically. Evidence Many students experience mathematics as if it is all about following prescribed methods, but this problem has no such pre-determined route to the solution. The students must make sense of the problem and persevere in solving it. They plan out their own response, choose graph paper as an appropriate tool (line 1; MP 5), and monitor and evaluate their progress and change course (MP 1), refining their initial guess and checking their process by thinking about properties of parallelograms (lines 2, 8). They understand the approaches of others and identify correspondences between different approaches (MP 1) when Sam suggests counting using the graph paper in response to Anita s approach of preserving tilt using hand motions (lines 8 12). Students also think about the assumptions they make and consider other possible cases (lines 14 17). Students use the properties of parallelograms to construct their argument (line 2). They build an argument together as they recognize the need to become more precise in justifying the location of the fourth point. The students start by drawing and sketching, but Sam points out a potential flaw in their reasoning by questioning the validity of determining a parallelogram by what looks right (line 7). The students justify their conclusions, communicate them to others, and respond to the arguments of others (MP 3) when Sam adds a numerical observation to Anita s visual one (lines 8 11), and Anita s response includes justification for Sam s contribution (line 12). Finally, at the end of the dialogue, the students have to check their assumptions when Dana realizes they might not be done and the students must consider other possible cases (lines 13 17). The students use graph paper and a ruler to help them solve the problem. However, it is not simply their use of these tools that demonstrates the thinking highlighted by this MP. The students start by plotting points, and then Sam suggests sketching lines to estimate a possible fourth point (line 4). Dana responds by suggesting they use a ruler to help determine whether Sam has really found the fourth point (line 5). The students recognize that drawing the figure using a straightedge is useful, but Sam also realizes a limitation of the tool by arguing that a picture that looks right doesn t necessarily guarantee a parallelogram (line 7). Then the

8 Look for and make use of structure. students use an intuitive precursor to the notion of slope when they count the units up and over from point B to C to determine the length and slant of BC and then reproduce that same length and slant starting at point A to get the final vertex (line 11). For middle school students, this is an inventive and appropriate use of the coordinate plane to match a distance, even when they are not yet able to calculate that distance precisely, and the students demonstrate they recognize the insight to be gained (MP 5) from using graph paper not just to locate points, but to quantify the relationship between points. Students look for and make use of structure. This can be seen when the students use the properties of parallelograms, namely that opposite sides are parallel (and congruent). These properties are used in lines 8 12 when the students are sliding over one side to get the opposite side of the parallelogram, attending to the structure formed by the relationships between the given points and the point they are trying to find. In this process, students are using structure when they see that counting up and over is a reliable way to produce a segment with the same tilt and the same length and, therefore, is a way to determine the fourth vertex of the parallelogram precisely. Commentary on the Mathematics Common misconception: Overreliance of idealized standard shapes This dialogue also serves to illustrate a common tendency to over-rely on idealized images of shapes. Given the points (1,2), (4,1), and (5,3), a natural first tendency is to look for the final vertex in the area of (2,4). Parallelograms are very often depicted like the example on the left, below, and less often shown in general shapes and positions. Idealized standard image of parallelogram Generic parallelograms in general positions Limiting the examples in this case, examples of parallelograms that students see narrows the image they generate and can lead students to depend on the idealized image rather than the concept definition (necessary and sufficient properties). In the problem students explore in this dialogue, idealized images about the shape and orientation of parallelograms make (0,0) and

9 (8,2) less visible and tempting choices. The use of standard idealizations in geometry is problematic with other shapes such as triangles and trapezoids as well. When called upon to think of or sketch or state properties of a triangle, students whose image is isosceles or equilateral are likely to overgeneralize properties and relationships that don t occur in all cases. Extensions to upper grades While this dialogue is targeted to middle school content, the math problem posed would be equally appropriate for Algebra I students, who learn to calculate slope, learn that parallel lines have equal slopes, and might apply that notion to find and check solutions. The problem in this dialogue can be extended to high school geometry students as well. Once students find all three possible locations for the fourth vertex, they will notice that those fourth vertices form a triangle whose sides have midpoints that are the original three vertices of the parallelogram. A rigorous proof of this observation can be done and is the subject of the Illustration titled Proof with Parallelogram Vertices. Evidence of the Content Standards Students in the dialogue draw geometric shapes on the coordinate plane (e.g., lines 3 5, line 11) in an attempt to fulfill the parallelogram condition (7.G.A.2 and 6.G.A.3). Students are able identify and use properties of parallelograms (5.G.B.4) such as opposite sides are parallel (line 2) and congruent (lines 11 and 12). In fact these properties of parallelograms, along with the aid of the coordinate system s grid, allow students to measure the length of parallelogram sides using an up and over technique (line 11) (6.G.A.3).

12 consider new possibilities in a problem or solutions that we would not otherwise consider. 5. How could a high school student with knowledge of algebra solve this problem? As Dana hints in line 9 of the dialogue, slope plays an important part in this problem. The students in this problem essentially use the coordinate plane and the rise-overrun definition of slope to find the fourth vertex. A high school student with more knowledge of algebra could use the fact that two parallel lines have the same slope and the slope formula, m = y y 2 1, to calculate the coordinates of the fourth x 2 x 1 vertex. This type of reasoning would exhibit high school standard G-GPE.B.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 parallel or perpendicular to a given line that passes through a given point). A high school student may also use the coordinates to calculate the distance between the points to establish that the opposite sides of the parallelogram are congruent. Finally, older students may be able to address Sam s question in line 7: What if the fourth point is not even on a gridline? High school students may explore parallelograms whose vertices don t have integer coordinate values, and may also explore the question: If a parallelogram has three vertices that have integer coordinate values, is it possible for the fourth vertex not to have integer coordinate values? 6. What did using graph paper as opposed to blank paper allow the students to do? How/why can graph paper be considered a tool? Go back to the dialogue to find evidence of how using graph paper contributed to the students thinking as they worked to solve the problem. In this problem, the students use graph paper not only to plot the three given points on the Cartesian plane, but also as a measuring tool. Once the points are plotted, the students are able to count the number of vertical and horizontal units between two points. We can see this in line 11 when Sam measures the rise and run between points B and C and then measures out the same rise and run to get from point A to the fourth possible vertex. For a middle school student with limited or no formal knowledge of algebra, the use of graph paper would have been vital in allowing them to find the exact coordinate of the fourth vertex. 7. What might be the affordances and the risks in providing Geoboards to the students in the dialogue? Using a Geoboard could affect the students thinking in several ways. First, if students start out with the same two lines ( AB and BC ), the Geoboard makes it much easier to find the fourth point of the parallelogram. Students would likely not guess an incorrect point, and students might feel like they are done with the problem much sooner. Geoboards, however, don t allow for the possibility of Sam s insight in line 7 that the fourth point might not even have integer coordinate values. The Geoboard limits vertices to the gridlines, so students are not led to question their assumption that what looks like

14 10. If the points A(0,0), B(5,0), and C(8,3) were the vertices of an isosceles trapezoid, how many possible fourth vertices could you have? Justify your answer. When first looking at this problem, one might be tempted to say that there are three possible locations for the fourth vertex because you can break the problem up into three cases similar to the parallelogram problem. However, unlike the parallelogram problem, not all cases allow for an isosceles trapezoid. Some cases allow for two isosceles trapezoids, some allow for one, and some do not allow for any isosceles trapezoids to be formed. CASE #1: If you connect the vertices and have AB and BC, that means the fourth vertex, point D, must be located at (-3,3) in order to have AD BC or at (8,8) to have AB CD. CASE #2: If you connect the vertices and have AC and BC, the fourth vertex must be at , 73 73, which is approximately equal to (4.23, -.29). This point can be found by calculating the point of intersection between a line parallel to AC passing through point B (since trapezoids have one set of parallel sides) and a circle around point A with a radius of 18 (since isosceles trapezoids have two congruent sides and BC has a length of 18 ). Please note that the second orientation, an isosceles trapezoid with a side parallel to BC, does not work. This can be verified with a sketch on the coordinate plane.

15 CASE #3: If you connect the vertices to have AB and AC, there is no way to make an isosceles trapezoid (using the definition that trapezoids have exactly one pair of parallel sides). Drawing a line parallel to AC and passing through point B, and drawing a circle around point C with a radius of 5 (the length of AB ), you see that the point of intersection, point D, would not form an isosceles trapezoid with the other three given vertices. Please note that the second orientation, an isosceles trapezoid with a side parallel to AB, also does not work. This can be verified with a sketch on the coordinate plane. 11. If the points A(0,0), B(5,0), and C(8,3) were the vertices of a trapezoid (not necessarily isosceles), how many possible fourth vertices could you have? Justify your answer. There are an infinite number of locations where the fourth vertex could be. Once the restriction that the trapezoid must be isosceles is removed, there are an infinite number of places where the fourth vertex may be located along certain lines. For example, if you connect the vertices to have AB and BC, any point on the dotted line parallel to AB (in the case that AB is a base of the trapezoid) and passing through point C whose x- coordinate is less than 8 would work. Note, however, that (3,3) produces a parallelogram, and, depending on your definition of a trapezoid, may or may not be counted as a possible fourth vertex. Similar lines of possible points exist in the other cases considered in problem 10.

16 Possible Responses to Student Discussion Questions 1. What does Sam do in line 11 of the dialogue to measure the side of the parallelogram? Will this method always work to measure the side of a parallelogram? Sam measures BC by counting the vertical units ( up ) and horizontal units ( over ) between points B and C. By counting, Sam is able to figure out a way to describe how long and slanted BC is. This method will always work as long as the vertices are on the gridlines (i.e., have integer coordinate values). The method does work in general, but counting is not precise if the points don t have integer coordinate values. Note that Sam does not find the length of BC, however. The up and over method is enough to produce a line of the same length, but to find the length takes at least one more calculation. 2. In line 11, why does Sam move up 2 units and over 1 unit from point A? Sam is trying to make the side opposite to BC be the same length and slanted the same way as BC. In line 8 of the dialogue, Anita visualizes BC sliding over to form the opposite side of the parallelogram, and this is what Sam is trying to draw.

17 3. In line 2, Anita identifies a property of parallelograms that the students use to find the fourth vertex in the dialogue. What is the property? What are other properties of parallelograms that the students could also use? A parallelogram is a quadrilateral that can be defined from several perspectives. The students use the property that the opposite sides in a parallelogram are parallel. The students could also use the property that the opposite sides of a parallelogram are congruent (same length), or that the opposite angles of a parallelogram are congruent (same measure). 4. What assumptions do the students make in the beginning of their problem-solving process that Dana discovers in lines 14 and 16? How does making the assumptions affect their problem-solving process? The students assume that the parallelogram has AB and BC. By connecting the vertices in this way, they limit themselves to only one place where the fourth vertex can be. The students realize this at the end of the dialogue and must now consider the other parallelograms with the three given vertices: the parallelogram with AC and BC and the parallelogram with AB and AC. Note that making the assumption is not a bad thing, and is even necessary as a starting place for the problem-solving process. Possible Responses to Related Mathematics Tasks 1. Find the coordinates of all other possible fourth vertices of the parallelogram described in the problem. The possible fourth vertices are (2,4), (0,0), and (8,2). 2. How do you know you have found all the possible fourth vertices? Justify your answer. Initially, we are only given three vertices of the parallelogram. Like in the dialogue, we can connect the vertices to start drawing the parallelogram. Considering all the different possible scenarios, we can connect the vertices to get AB and BC (case #1), AC and BC (case #2), or AB and AC (case #3). Another way to justify why there are only three possible points for the fourth vertex is to think about opposite corners of a parallelogram. The fourth vertex of the parallelogram must be opposite some other vertex. Since we have three vertices to choose from, that must mean the fourth vertex may be in one of three places. 3. Given points (2,1), (6,1), and (2,-2), how many parallelograms can you make? How many rectangles? Justify your answer. As in the previous problem there are three possible points that could be the fourth vertex of this parallelogram. The possible fourth vertices are (6,4), (6,-2), and (-2,-2). However, given the three points as the corners of a rectangle, there is only one possible fourth

18 vertex at (6,-2). The reason for this is that not only must the shape made by the four vertices fit the definition of a parallelogram, but it must also have all 90º angles. Out of the three possible parallelogram fourth vertices, only (6,-2) produces a parallelogram with all 90º angles. 4. Name three points that cannot be the vertices of a parallelogram, no matter where the fourth point is. What must be true about these three points? Any three collinear points cannot be the vertices of a parallelogram, no matter where the fourth point is. As long as the three points are not collinear, any three points can make up three of the four vertices of a parallelogram. 5. Four keys are required to open a treasure vault. Your ripped map shows the location of three of the keys. You know the four keys are located at the corners of a parallelogram. Where could the fourth key be? To solve this problem, students have to use MP 4 (model with mathematics) to label the points and find a way to talk about the location of the fourth key (again, there are three possible locations).

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