# Lesson 2

Points, Lines, Rays, and Segments

## Warm-up: Number Talk: Finding Differences (10 minutes)

### Narrative

This Number Talk encourages students to look for and make use of structure to mentally evaluate a series of subtraction expressions. The numbers preview some benchmark angle measurements students will see in upcoming lessons.

### Launch

• Display one expression.
• “Give me a signal when you have an answer and can explain how you got it.”
• 1 minute: quiet think time

### Activity

• Keep expressions and work displayed.
• Repeat with each expression.

### Student Facing

Find the value of each expression mentally.

• $$90 - 45$$
• $$270 - 45$$
• $$270 - 135$$
• $$360 - 135$$

### Activity Synthesis

• “What do these expressions have in common?” (The first number in each sequence is a multiple of 90.)
• “How did this observation—that the first numbers are all multiples of 90—help you find the value of the differences?”
• “Who can restate _____’s reasoning in a different way?”
• “Did anyone have the same strategy but would explain it differently?”
• “Did anyone approach the expression in a different way?”
• “Does anyone want to add on to _____’s strategy?”

## Activity 1: Card Sort: Who Am I? (20 minutes)

### Narrative

In this activity, students are given cards that contain illustrations, definitions, and descriptions of points, lines, rays, and segments. They sort the cards into groups so that each group describes one of the four geometric figures.

When students sort the cards, they begin to connect the terms to their formal definitions and attributes (MP6).

Students may have trouble making sense of a point having “no size.” It is not necessary to discuss this in depth at this point, but during the synthesis, clarify that a point marks a location, and we need a symbol or a mark to represent it. The symbol or the mark has size, but the location itself doesn’t. It is important that students recognize that points make up lines even though we do not always identify them or label them with a dot. If needed, revisit the isometric grid from the previous activity as a reference.

MLR8 Discussion Supports. Students should take turns finding a match and explaining their reasoning to their partner. Display the following sentence frames for all to see: “I noticed _____, so I matched . . . .” Encourage students to challenge each other when they disagree.
Engagement: Provide Access by Recruiting Interest. Revisit math community norms to prepare students for group work and the whole-class discussion.
Supports accessibility for: Memory, Social-Emotional Functioning

### Required Materials

Materials to Gather

Materials to Copy

• Card Sort: Who Am I?

### Required Preparation

• Create a set of cards from the blackline master for each group of 2–4 students.

### Launch

• Groups of 2–4
• Give each group a set of cards from the blackline master and access to rulers or straightedges.
• Display the list of words and phrases collected during the previous lesson.
• “We used many different words to describe figures. We learned how to identify points, lines, and line segments.”
• Ask students to act out each term with arms and hand gestures. (For example, “Show me a point using only your arms or hands or both.” Students may makes fists to represent points or identify a spot along their arm as a point.)
• “We are going to continue to define points, lines, and line segments in this next activity. You are also going to use the cards to define another figure, a ray.”

### Activity

• “Work together to sort the cards into four groups. The cards in each group should describe a particular geometric figure.”
• “Set aside cards that were hard to place. Be prepared to explain why they were more challenging.”
• Monitor for reasoning students use to sort cards. (Examples: Points are parts of lines so the description of a point will not use the word point in it. Line segments are parts of lines.)

### Student Facing

Your teacher will give you a set of cards that describe or illustrate points, lines, rays, and line segments.

Sort the cards into 4 groups. Each group should represent the attributes or characteristics of one of the geometric figures.

Pause for directions from your teacher before completing the graphic organizer.

### Student Response

Students may determine that the clue “I have no size” is not a match for any of the other clues. Consider asking, “If we had to place this clue with some of the others, which would you place it with and why?”

### Activity Synthesis

• Invite students to discuss their sorting decisions.
• “Which cards did you spend the most time debating as a group?” (“I have no size.” or “My length cannot be measured.”)
• “A point might be tricky to think about. It is often represented by a dot or a circle, which could be large or small. But the point itself cannot be large or small since it only marks a location.”
• “What about a line? Why can it not be measured?” (It just keeps going in both directions so we don’t know where to start or stop measuring.)
• “Why is a ray also impossible to measure?” (There is a starting point for measurement but there is no endpoint.)
• Consider displaying a graphic organizer (as shown in the activity statement) for all to see and placing the cards in the right boxes along the way.
• Ask students to write a sentence and draw an image to represent each figure in the graphic organizer in the student material.

## Activity 2: Make Some Shapes (15 minutes)

### Narrative

The purpose of this activity is for students to use line segments and rays to draw familiar two-dimensional figures, letters, and numerals. Drawing on dot paper helps to reinforce the idea that segments have a defined endpoint on both ends and to distinguish rays from segments. As they describe and compare figures, students use vocabulary from the previous activity.

The activity also enables the teacher to hear the geometric vocabulary students are bringing from earlier grades. Consider displaying a chart with an image of each shape listed in the first problem during the launch.

This activity uses MLR7 Compare and Connect. Advances: representing, conversing.

### Required Materials

Materials to Gather

### Launch

• Groups of 2
• “We are going to use the field of dots to create different figures.”

### Activity

• 3–5 minutes: independent work time

MLR7 Compare and Connect

• “How are your shapes and figures the same? How are they different?”
• 3–5 minutes: partner discussion.
• As students discuss, consider asking:
• “Are the shapes you drew for the first question the same or very different? For example, are your triangles alike? Your trapezoids?” (Some are taller and wider, and others are shorter and narrower.)
• “How many line segments did you use to make your letters or numbers?” (3, 4, 5, 6)
• “Did you make the same letters or numbers? If so, did you have the same reasons for choosing those letters or numbers?” (Some numbers have curves in them and are not as easy to draw.)

### Student Facing

1. Each dot on the grid represents a point. Draw line segments to create:

a triangle
a rhombus

a trapezoid
a hexagon

a pentagon
a rectangle

2. Draw a combination of rays and line segments to create:

an uppercase letter

a number

a lowercase letter

### Activity Synthesis

• “How many possible triangles can we draw on the dot paper? How many possible trapezoids? Hexagons?” (There are countless ways to create and connect a specified number of line segments to make a certain type of shape.)
• “Did anyone not start or end their line segments or rays on a dot? Do the results still count as segments or rays?” (Yes. A line segment stops at 2 endpoints, but the endpoints don’t have to be on a dot of the paper.)
• “How did you distinguish line segments and rays when drawing numbers and letters in the second question?” (Line segments stop on both ends. Rays go on on one end, the one marked with an arrow.)

## Lesson Synthesis

### Lesson Synthesis

Use student examples to reference during the synthesis questions, or invite students to illustrate their explanations for each question.

“Today we learned the meanings of points, lines, line segments, and rays, and we used those geometric parts to create drawings.”

“How might we explain to a new student how lines, rays, and line segments are different?” (A line is straight and goes on forever in both directions. A line segment is a part of a line that ends on both ends. A ray is also a part of a line, but it goes on forever in one direction and ends in the other direction. We can use drawings to show how they are different.)

“Are the dots on the paper we used today the only points that could be in the shapes and figures?” (No. Each shape we drew had many points, not just the ones that were already there.)

Draw a capital A. “The tip of the letter A and the ends of the horizontal segment don’t have any dots. Can we call these parts of the ‘A’ line segments?” (Yes. There doesn’t have to be dots at the end. They have a starting point and an endpoint. Dots are just what we use to label points.)

Draw a capital L. “Is the bottom left corner of the letter L a point? Why or why not?” (Yes. A point is a location. It doesn’t have to be marked by a dot. Any location on the line segments that make up the L are points on that letter.)