# Lesson 14

Coordinate Proof

## 14.1: Which One Doesn’t Belong: Coordinate Quadrilaterals (5 minutes)

### Warm-up

This warm-up prompts students to compare the graphs of four quadrilaterals. It gives students a reason to use language precisely (MP6). It gives the teacher an opportunity to hear how students use terminology and talk about characteristics of the items in comparison to one another.

### Launch

Arrange students in groups of 2–4. Display the graphs for all to see. Give students 1 minute of quiet think time and then time to share their thinking with their small group. In their small groups, ask each student to share their reasoning as to why a particular item does not belong, and together, find at least one reason each item doesn‘t belong.

### Student Facing

Which one doesn’t belong?

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

Ask each group to share one reason why a particular item does not belong. Record and display the responses for all to see. After each response, ask the class if they agree or disagree. Since there is no single correct answer to the question asking which one does not belong, attend to students’ explanations and ensure the reasons given are correct.

During the discussion, ask students to explain the meaning of any terminology they use, such as *equilateral* or *diagonal*. Also, press students on unsubstantiated claims.

If possible, leave the list of responses displayed until the end of class. Students will return to these images during the lesson synthesis.

## 14.2: Name This Quadrilateral (15 minutes)

### Activity

Students are presented with 4 points and asked to fully describe the quadrilateral with those vertices. Slopes will show that all pairs of adjacent sides are perpendicular, making the shape a rectangle. Then students will use the Pythagorean Theorem to calculate both area and perimeter.

Making dynamic geometry software available gives students an opportunity to choose appropriate tools strategically (MP5).

### Launch

Make graph paper available to students who would like to use it.

### Student Facing

A quadrilateral has vertices \((0,0), (4,3), (13,\text-9),\) and \((9,\text-12)\).

- What type of quadrilateral is it? Explain or show your reasoning.
- Find the perimeter of this quadrilateral.
- Find the area of this quadrilateral.

### Student Response

For access, consult one of our IM Certified Partners.

### Student Facing

#### Are you ready for more?

- A parallelogram has vertices \((0,0), (5,1), (2,3)\), and \((7,4)\). Find the area of this parallelogram.
- Consider a general parallelogram with vertices \((0,0),(a,b),(c,d),\) and \((a+c,b+d),\) where \(a,b,c,\) and \(d\) are positive. Write an expression for its area in terms of \(a,b,c,\) and \(d\).

### Student Response

For access, consult one of our IM Certified Partners.

### Anticipated Misconceptions

Some students may state that the quadrilateral is a rectangle simply because it looks like one. Remind these students that we need to back up our reasoning with mathematics. Suggest students review their reference charts for definitions and properties of rectangles.

### Activity Synthesis

Invite students to share their reasoning for each question. Highlight students who carried information from one question to the next, such as recognizing that in a rectangle, opposite sides have equal length, so they only need to calculate 2 distances (rather than all 4).

*Speaking: MLR8 Discussion Supports.*As students share how they know that the quadrilateral is a rectangle, press for details by asking students how they know that the quadrilateral has four right angles. Show concepts multi-modally by displaying the quadrilateral on the coordinate plane and annotating the slopes and lengths of each side. This will help students justify why the quadrilateral is a rectangle and the calculations of the area and perimeter.

*Design Principle(s): Support sense-making; Optimize output (for justification)*

## 14.3: Circular Logic (15 minutes)

### Activity

Students observe that the angle formed by connecting endpoints of a diameter to a third point on the circle appears to be a right angle. They confirm that this is true for a few particular points, then write a conjecture. This conjecture will be generalized in an upcoming unit.

Monitor for different ways that students word the conjecture to highlight during the discussion. Listen for terms such as *diameter*, *endpoints*, *right triangle*, and *right angle*. The term *chord* will be defined in a subsequent unit. It is not necessary to define it here.

This activity works best when each student has access to devices that can use the embedded applet because students will benefit from seeing the relationship in a dynamic way. If students don’t have individual access, projecting the applet would be helpful during the synthesis.

### Launch

Arrange students in groups of 4. Provide access to internet-enabled devices, ideally one per group. Distribute one index card with each internet-enabled device.

*Conversing: MLR2 Collect and Display.*As students work on this activity, listen for and collect the language students use to justify why the angle formed by segments \(BD\) and \(CD\) is a right angle. Write the students’ words and phrases on a visual display. As students review the visual display, ask them to revise and improve how ideas are communicated. For example, a phrase such as, “The lines make 90 degrees because they have opposite slopes” can be improved by restating it as “Segments \(BD\) and \(CD\) are perpendicular because their slopes are opposite reciprocals.” This will help students use the mathematical language necessary to precisely justify why the measure of the angle formed by segments \(BD\) and \(CD\) is 90 degrees.

*Design Principle(s): Optimize output (for justification); Maximize meta-awareness*

### Student Facing

Use the applet to answer the questions.

- First, observe and describe the image:
- What kind of line is \(BC\) in reference to the circle?
- What does the measurement of angle \(BDC\) appear to be?

- Move point \(D\) around. How does the measurement of angle \(BDC\) appear to change? Use a corner of the index card to compare the size of the angle as it moves.
- Move point \(D\) to a location with integer coordinates. Each student in the group should choose a different spot for \(D\). Calculate the slopes of segments \(BC\) and \(BD\). What do your results tell you?
- Now select the button labeled “slopes.” This will show the slopes of \(BD\) and \(BC\) and their product. Move point \(D\) around. What do you notice about the product of the slopes?
- Based on the results, write a conjecture that captures what you are seeing.

### Student Response

For access, consult one of our IM Certified Partners.

### Launch

Arrange students in groups of 4.

*Conversing: MLR2 Collect and Display.*As students work on this activity, listen for and collect the language students use to justify why the angle formed by segments \(BD\) and \(CD\) is a right angle. Write the students’ words and phrases on a visual display. As students review the visual display, ask them to revise and improve how ideas are communicated. For example, a phrase such as, “The lines make 90 degrees because they have opposite slopes” can be improved by restating it as “Segments \(BD\) and \(CD\) are perpendicular because their slopes are opposite reciprocals.” This will help students use the mathematical language necessary to precisely justify why the measure of the angle formed by segments \(BD\) and \(CD\) is 90 degrees.

*Design Principle(s): Optimize output (for justification); Maximize meta-awareness*

### Student Facing

The image shows a circle with several points plotted on the circle.

- What kind of segment is \(BC\) in reference to the circle?
- Choose one of the plotted points on the circle and call it \(D\). Each student in the group should choose a different point. Draw segments \(BD\) and \(DC\). What does the measure of angle \(BDC\) appear to be?
- Calculate the slopes of segments \(BD\) and \(DC\). What do your results tell you?
- Compare your results to those of others in your group. What did they find?
- Based on your group’s results, write a conjecture that captures what you are seeing.

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

Ask previously identified groups to share their conjectures. Challenge students to decide whether all the conjectures are saying the same thing, and if any of the conjectures need to be refined. Encourage students to respectfully disagree, ask for clarification, or point out contradicting information.

There are many ways to correctly word the conjecture. As students work to refine the wording, be sure that the word *diameter *is included, that it’s clear that 2 points are at the endpoints of the diameter, and that the third point is clearly somewhere else on the circle. Students may describe the result as 2 segments that form a right angle or 3 segments that form a right triangle.

Once the conjecture is finalized, ask students if they have proven it (they have not; they’ve just shown it’s true for a few particular cases). Tell students that we will look at a more general case of this assertion in an upcoming unit.

*Action and Expression: Develop Expression and Communication.*Invite students to talk about their ideas with a partner. Display sentence frames, such as: “How did you develop your conjecture?” and “I chose this conjecture because. . . .” to support students when they explain their ideas.

*Supports accessibility for: Language; Organization*

## Lesson Synthesis

### Lesson Synthesis

Display the images from the warm-up again.

Arrange students in groups of 2–4. Invite students to each choose a different quadrilateral and then prove as many things about the quadrilateral as they can and to calculate their quadrilateral’s area and perimeter. Suggest that they refer to the list of their responses from the warm-up for ideas of properties to prove.

After a few minutes of quiet work time, invite students to share some of their properties and reasons with their group. Ask the other group members to listen and critique their reasoning. Repeat as time allows.

Sample responses:

- Quadrilaterals A and D are squares. For Quadrilateral A, the slopes show the sides are perpendicular, and the Pythagorean Theorem shows the sides are all congruent. For Quadrilateral D, the sides are aligned with the coordinate grid lines, so it is easy to see the 90-degree angles and side measurements.
- Quadrilateral B is a rhombus because its sides are congruent. It is not a square because the sides aren’t perpendicular.
- Quadrilateral C is a rectangle. Its adjacent sides are perpendicular.
- The diagonals in Quadrilaterals A, B, and D are perpendicular.
- The diagonals in Quadrilaterals A, C, and D are congruent.

area (square units) | perimeter (units) | |
---|---|---|

A | 8 | \(4\sqrt{8}\) or about 11.3 |

B | 4 | \(4\sqrt{5}\) or about 8.9 |

C | 4 | \(2(\sqrt{2}+\sqrt{8})\) or about 8.5 |

D |
4 | 8 |

## 14.4: Cool-down - Name That Quadrilateral (5 minutes)

### Cool-Down

For access, consult one of our IM Certified Partners.

## Student Lesson Summary

### Student Facing

What kind of shape is quadrilateral \(ABCD\)? It looks like it might be a rhombus. To check, we can calculate the length of each side. Using the Pythagorean Theorem, we find that the lengths of segments \(AB\) and \(CD\) are \(\sqrt{45}\) units, and the lengths of segments \(BC\) and \(DA\) are \(\sqrt{37}\) units. All side lengths are between 6 and 7 units long, but they are not exactly the same. So our calculations show that \(ABCD\) is not really a rhombus, even though at first glance we might think it is.

We did just show that two pairs of opposite sides of \(ABCD\) are congruent. This means that \(ABCD\) must be a parallelogram. Checking slopes confirms this. Sides \(AB\) and \(CD\) each have slope \(\frac12\). Sides \(BC\) and \(DA\) each have slope 6.

Can we find the area of triangle \(EFG\)? That seems tricky, because we don’t know the height of the triangle using \(EG\) as the base. However, angle \(EFG\) seems like it could be a right angle. In that case, we could use sides \(EF\) and \(FG\) as the base and height.

To see if \(EFG\) is a right angle, we can calculate slopes. The slope of \(EF\) is \(\frac86\) or \(\frac43\), and the slope of \(FG\) is \(\text-\frac{3}{4}\). Since the slopes are opposite reciprocals, the segments are perpendicular and angle \(EFG\) is indeed a right angle. This means that we can think of \(EF\) as the base and \(FG\) as the height. The length of \(EF\) is 10 units and the length of \(FG\) is 5 units. So the area of triangle \(EFG\) is 25 square units because \(\frac12 \boldcdot 10 \boldcdot 5 = 25\).