# Lesson 7

The Size of Angles on a Clock

## Warm-up: Notice and Wonder: Two Sets of Angles (10 minutes)

### Narrative

Students commonly think that angles formed by longer segments are greater in size than those formed by shorter segments. The purpose of this warm-up is to bring up and address this likely misconception. The diagrams prompt students to observe the lengths of segments forming the angles and consider how they affect our perception of the size of the angles.

While students may notice and wonder many things about these sets of angles, it is important to discuss the relative sizes of the angles in the two sets. Make sure students see that the two sets of angles are identical in size even though the segments that form them seem to suggest otherwise.

Consider using patty paper to demonstrate equal-size angles during the synthesis.

### Required Materials

Materials to Gather

### Launch

• Groups of 2
• Display the two sets of angles.
• “What do you notice? What do you wonder?”
• 1 minute: quiet think time

### Activity

• 1 minute: partner discussion
• Share and record responses.

### Student Facing

What do you notice? What do you wonder?

### Activity Synthesis

• “Are the angles getting larger or getting smaller in the top set? What about in the bottom set?” (They are getting larger as you move from left to right.)
• “How does the first angle in set 1 compare to that in set 2?” (They look very similar and almost the same size.)
• “How does the second angle in set 1 compare to that in set 2?” (They look the same size but the rays that create the angles are longer in set 1 than in set 2.)
• “What about the third angle in each set?” (Set 2 has shorter lines than set 1 but the angle is the same size.)
• “How can we find out if one is larger, smaller, or the same size as the other?” (Measure them by laying them on top of each other.)
• Consider using patty paper to trace corresponding angles in the two sets and show that they are the same size even if the segments in the second set are shorter.
• Highlight that the size of an angle is not determined by the length of the segments that frame it.

## Activity 1: Draw Angles Andre’s Way (15 minutes)

### Narrative

In a previous lesson, students described an angle to a peer so that they might draw the angle accurately without seeing it. Students learned what an angle is and have reasoned about how to describe the size of an angle.

In this activity, students use the features of an analog clock (minute hand, hour hand, and position of numbers) to explain how to draw a given angle. In doing so, they use language related to turning one or both hands on the clock. In future activities, students will relate this idea to turning a ray around a fixed point.

The hands on an actual clock are designed so that moving the minute moves the hour hand. The examples here disregard this constraint. For the purposes of this activity, it is assumed that both hands can be moved freely, or that one hand can be held in place while the other moved freely.

### Required Materials

Materials to Gather

• Groups of 2

### Activity

• “In an earlier lesson, we played a game where you described an angle to your partner to help them draw it without seeing it.”
• “Read Andre’s explanation for how to draw an angle. Try to draw his angle, and then think of another way to describe how to draw the same angle.”
• 3 minutes: independent work time on the first two problems
• “Discuss how you would explain how to draw the other angles.”
• 3–5 minutes: partner discussion and work time
• Monitor for students who clearly describe turning a hand around the clock and who connect the hands to the rays of the angles.

### Student Facing

Andre used the hands of a clock to explain how to draw an angle to his partner.

“Imagine both hands are pointing at the 12. Turn the minute hand so it's pointing at the 3.”

1. Draw Andre’s angle.
2. What is another way to describe how to draw the same angle using the clock?
3. Use Andre’s way to explain how to draw these angles:

1.

2.

3.

### Activity Synthesis

• “Let’s listen to some different directions for how to draw these angles. As you listen, try to sketch the angle and guess which one they are describing how to draw.”
• Invite 2–3 students to share the directions they wrote for drawing one of the angles. Choose students to share in a different order than the angles are presented in the task.
• “Which angle did they describe (a, b, or c) and how do you know?”
• “What words help you picture how to draw the angle?” (turn right, turn left, point at 4)
• “We can describe the size of an angle by explaining how much one ray has turned from the other.”

## Activity 2: Compare Angles on the Clock (20 minutes)

### Narrative

In this activity, students compare the size of angles by thinking in terms of a turn from one ray from the other ray. They continue to use the clock as a tool for reasoning and for talking about “how much” of a turn. This work helps to elicit a need for more formal units and tools for measuring degrees while building the foundation for understanding angle measurement as additive. Students may describe the unit of measurement as “minutes,” “hours,” or other informal names (turn-units). They will learn about degrees as a unit of measurement in the upcoming lessons.

The final question gives students an opportunity to describe the size of an angle in as many ways as they can, using the clockface, benchmark angles, or other informal descriptions (MP6) preparing them for the introduction of a numerical way to measure angles.

### Launch

• Groups of 2
• “Earlier, we used the hands of a clock to describe how to draw an angle.”
• “Now, let’s think about how a clock might help us talk about the size of an angle.”

### Activity

• “Use the clocks to compare the angles formed by each clock’s hands.”
• 3–4 minutes: independent work time
• 2–3 minutes: partner discussion
• Monitor for different ways students compare the same pairs of angles.

### Student Facing

1. Here are some angles formed by the two hands of a clock.

In each pair of angles, which angle is larger? Explain or show your reasoning.

1.

2.

3.

4.

2. How large is this angle?

Describe its size in as many ways as you can.

### Activity Synthesis

• Invite 2–3 previously identified students to share their responses.
• “How did each student see the angle formed by the hands in different ways?” (Some thought about moving the _____ hand toward the _____ hand, like the direction it really moves on the clock. Others thought about the minute hand starting at the same spot as the hour hand and moved it in either direction.)
• “How did this change which angle they thought was larger?” (It changed how far they thought one ray turned.)
• “When we describe an angle to others, we often draw an arc, or part of a circle, between the rays to show which turn we are talking about.”

## Lesson Synthesis

### Lesson Synthesis

“Today we used a clock to compare angles. We described the size of an angle as the amount of turn one ray makes from another ray that is fixed at the shared endpoint.”

“What were some different ways you described ‘how much’ a ray turned from another, or ‘how much‘ bigger or smaller one angle was than another?” (We used the tick marks on the clock. We know they represent minutes on a clock, so we just called them minutes. For some angles, we just used the large numbers to describe the turn. We know those mean the hours on a clock, so we called them hours.)

As needed, prompt students to use examples from the last activity.