# Lesson 13

Intersection Points

## 13.1: Which One Doesn’t Belong: Lines and Curves (5 minutes)

### Warm-up

This warm-up prompts students to compare four graphs. 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.

The term tangent line will be defined and explored in a subsequent unit. It is not necessary to define it here.

### 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?

### 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 parabola and intersection. Also, press students on unsubstantiated claims.

Engagement: Develop Effort and Persistence. Encourage and support opportunities for peer interactions. Prior to the whole-class discussion, invite students to share their work with a partner. Display sentence frames to support student conversation, such as: “Item _____ does not belong because . . . .”
Supports accessibility for: Language; Social-emotional skills

## 13.2: Circles and Lines (15 minutes)

### Activity

In this activity, students solve a system of a consisting of a linear equation and a quadratic equation in 2 variables (the equation of a circle) by estimating the solutions on a graph, then verifying the solutions algebraically.

Monitor for students who substitute the points directly into the circle equation and for those who set up the Pythagorean Theorem independent of the circle equation.

Making graphing technology available gives students an opportunity to choose appropriate tools strategically (MP5).

### Launch

Action and Expression: Provide Access for Physical Action. Provide access to tools and assistive technologies such as a graphing calculator or graphing software. Some students may benefit from a checklist or list of steps to be able to use the calculator or software.
Supports accessibility for: Organization; Conceptual processing; Attention

### Student Facing

1. The equation $$(x-3)^2 + (y-2)^2 = 25$$ represents a circle. Graph this circle on the coordinate grid.
2. Graph the line $$y=6$$. At what points does this line appear to intersect the circle?
3. How can you verify that the 2 figures really intersect at these points? Carry out whatever procedure you decide.
4. Graph the line $$y=x-2$$. At what points does this line appear to intersect the circle? Verify that the 2 figures really do intersect at these points.

### Activity Synthesis

Invite previously identified students to share their strategies for verifying that the points are on both the line and the circle. If possible, invite one who used the Pythagorean Theorem and one who substituted the point into the circle equation. Ask students, “Why do both of these methods work?” (They are both basically saying the same thing. We need to verify that the point is 5 units away from the center $$(3,2)$$. Any point that satisfies the circle equation meets this description. The Pythagorean Theorem provides the distance directly.)

Speaking: MLR8 Discussion Supports. Use this routine to support whole-class discussion. As students share their strategies for verifying that the points are on both the line and the circle, ask students to restate what they heard using precise mathematical language. Consider providing students time to restate what they hear to a partner before selecting one or two students to share with the class. Ask the original speaker if their peer was accurately able to restate their thinking. Call students’ attention to any words or phrases that helped to clarify the original explanation, such as, “each point is exactly five units away from the center $$(3,2)$$.” This provides more students with an opportunity to produce language as they interpret the reasoning of others.
Design Principle(s): Support sense-making

## 13.3: Creating Lines (15 minutes)

### Activity

Students combine concepts of parallel and perpendicular lines and circles, and consider possible intersections of circles and lines. Monitor for a variety of strategies for the final question. Students may solve this graphically. To verify their answer, they may rewrite their equation in slope-intercept form.

Making graphing technology available gives students an opportunity to choose appropriate tools strategically (MP5).

### Launch

Representing, Conversing: MLR7 Compare and Connect. Use this routine to prepare students for the whole-class discussion about the intersections of the circle and lines. After students find the intersection points between their lines and the circle, invite them to create a visual display of their work. Then ask students to quietly circulate and observe at least two other visual displays in the room. Give students quiet think time to consider what is the same and what is different about the equations and graphs of their lines. Next, ask students to find a partner to discuss what they noticed. Listen for and amplify the language students use to compare and contrast the point-slope form and slope-intercept form of the last line they graphed.
Design Principle(s): Cultivate conversation

### Student Facing

1. Write an equation representing the circle in the graph.
2. Graph and write equations for each line described:
1. any line parallel to the $$x$$-axis that intersects the circle at 2 points
2. any line perpendicular to the $$x$$-axis that doesn’t intersect the circle
3. the line perpendicular to $$y=\text-\frac13x + 5$$ that intersects the circle at $$(6,8)$$
3. For the last line you graphed, find the second point where the line intersects the circle. Explain or show your reasoning.

### Student Facing

#### Are you ready for more?

1. Graph the equations $$y-3=(x-2)^2$$ and $$y-4=2(x-3)$$ and find their point of intersection.
2. Show that the graph of $$y-3=(x-2)^2$$ and $$y-4=m(x-3)$$ intersect at the point $$(m+1, m^2-2m+4)$$.

### Anticipated Misconceptions

If student graphs aren’t accurate enough to find the second intersection point in the last problem, suggest they use slope triangles to find several points on the line until they find one that is also on the circle.

### Activity Synthesis

If possible, invite a student who rewrote the third equation in slope-intercept form to share their method. If no student did this, ask the class to do so now. Then, ask students:

• “We can rewrite the line as $$y=3x-10$$. What does that tell you about where the circle and line intersect?” (The $$y$$-intercept of the line is $$(0,\text-10)$$. That point is also on the circle, so it’s the second intersection point.)
• “How could you find the exact points where your horizontal line intersects the circle?” (Substitute the particular value of $$y$$ into the circle’s equation and solve for $$x$$.)
• “What is an equation for a line that intersects the circle in exactly 1 point?” (Sample response: $$y=10$$)

## Lesson Synthesis

### Lesson Synthesis

Display this image related to the activity Circles and Lines as well as the 2 equations that follow for all to see:

$$(x-3)^2 + (y-2)^2 = 25$$

$$y=x-2$$

Ask students, “What does each equation represent on the graph?” (The first is the circle; the second is the line.) “What is special about the point $$(7,5)$$ on the graph?” (It is one of the points where the circle and line intersect.) “What is special about the point $$(7,5)$$ with regard to the equations?” (It is a point that makes both equations true.)

## 13.4: Cool-down - Find and Verify (5 minutes)

### Cool-Down

We can graph circles and lines on the same coordinate grid and estimate where they intersect. The image shows the circle $$(x-10)^2+(y+6)^2=169$$ and the line $$y=x-23$$. The 2 figures appear to intersect at the points $$(22,\text-1)$$ and $$(5,\text-18)$$. To verify whether these truly are intersection points, we can check if substituting them into each equation produces true statements.
Let’s test $$(22,\text-1)$$. First, substitute it into the equation for the line. When we do so, we get $$\text-1=22-23$$. This is a true statement, so this point is on the line.
Next, substitute it into the equation for the circle. This is the same as checking to see if the distance from the point to the center is $$\sqrt{169}$$, or 13 units. We get $$(22-10)^2+(\text-1-(\text-6))^2=169$$. Evaluate the left side to get $$144+25=169$$. This is a true statement, so the point $$(22,\text-1)$$ is on the circle. It’s on both the circle and the line, so it must be an intersection point for the 2 figures.