Lesson 14
Reasoning about Angles (Part 1)
Warmup: Which One Doesn’t Belong: Time After Time (10 minutes)
Narrative
This warmup prompts students to carefully analyze and compare geometric features of four clock faces. Students may compare the times being represented, but because no numbers are shown, they are likely to compare the hands of the clocks and the angles they form.
In making comparisons, students have a reason to use language precisely (MP6). Teachers have a chance to hear the terminologies students use and how they talk about characteristics of angles.
Launch
 Groups of 2
 Display the image.
 “Pick one that doesn’t belong. Be ready to share why it doesn’t belong.”
Activity
 1 minute: quiet think time
 2–3 minutes: partner discussion
 Record responses.
 As students explain, find opportunities to reinforce the terms “acute” and “obtuse.”
Student Facing
Which one doesn’t belong?
Student Response
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Activity Synthesis
 Consider stating: “Let’s find at least one reason why each one doesn’t belong.”
 Display the following sentences and ask students to complete each sentence with “always,” “sometimes,” or “never:”
 “The hour and minute hands of a clock _____ form one angle.”
 “The hour and minute hands of a clock _____ form two angles.”
 Ask students to explain their choice or use counterexamples to disagree with a classmate’s choice.
Activity 1: Draw a Clock (15 minutes)
Narrative
In an earlier lesson, students had folded paper and used supplemental tools to form and draw some benchmark angles (\(30^\circ\), \(45^\circ\), \(60^\circ\), and so on). In this activity, they apply their ability to measure and draw angles with a protractor to create a reasonably accurate clock face. The measuring and drawing here prepare students to reason about the angles formed by the hands of a clock in the next activity.
Students may notice that lines that give the positions of 1 and 2 on the clock can be extended through the center of the clock to give the positions of 7 and 8. Students who use these observations to create the drawing practice making use of structure (MP7).
The clock that students draw in this activity can be a helpful reference in the next activity.
Advances: Listening, Representing
Required Materials
Materials to Gather
Launch
 Groups of 2
 Give one protractor and a straightedge or a ruler to each student.
Activity
 5 minutes: independent work time
 1–2 minutes: partner discussion
 Monitor for students who:
 measure the angle needed to place each number (one at a time)
 draw the lines to position the numbers 1, 2, 4, and 5 and then try to find a way to mirror them vertically to locate 11, 10, 8, and 7
 extend the lines they drew to find the numbers 1 and 2 and use them to find 7 and 8, and do the same with 3 and 4
Student Facing
Kiran is drawing a clock. He draws a pair of perpendicular lines to find the placement of the numbers 3, 6, 9, and 12 around the circle.
 How many degrees is each angle he has drawn so far? Explain how you know.

Help Kiran find the exact placement of the numbers “1” and “2” on the clock.
 How many new lines does he need to draw?
 What angles should be formed between the two lines he has already drawn and the new ones?
 Draw the lines precisely and place the numbers “1” and “2” on the drawing.
 Measure and draw as many lines as needed to complete the clock drawing so that all the numbers are precisely placed where they should be.
Student Response
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Activity Synthesis
 Display the incomplete clock face. Select students to share their completed drawing and their drawing process. Sequence the presentation in the order shown in the monitoring notes.
 “How did you find the size of the angle formed between the number 1 and 2?” (Divide 90 by 3, or divide 180 by 6, or divide 360 by 12.)
 “Is the angle formed by any two consecutive rays always \(30^\circ\)? How do you know?” (Yes, because the angle is always \(\frac{1}{12}\) of 360.)
Activity 2: Tick Tock (20 minutes)
Narrative
In grade 3, students learned to tell and write time to the nearest minute and measure time intervals in minutes. They understand that moving from one number on the clock to the next means 5 minutes have elapsed. In this activity, students build on those understandings to solve problems about angles formed by the hands of a clock.
Many students would benefit from having a visual reference of a clock as they are solving these problems. Encourage them to use their clock drawing from the previous activity for support.
Some students may try to answer each question by drawing each indicated time and then measuring the angles formed by the hands. Ask them to consider finding the size of the angles by reasoning and without measuring. For example, ask: “What do you know about the angle that is formed when a hand goes from 12 to 3? From 12 to 1?” This encourages students to use the structure of the clock and the equal parts the clock face is divided into by the numbers on the clock (MP7).
This activity uses MLR1 Stronger and Clearer Each Time. Advances: Reading, Writing.
Supports accessibility for: Social Emotional Functioning
Required Materials
Materials to Gather
Launch
 Groups of 4
 Give students access to protractors.
Activity
 5 minutes: independent work time on the first two sets of questions
MLR1 Stronger and Clearer Each Time
 “Share your response to the second question with a partner. Take turns being the speaker and the listener. If you are the speaker, share your ideas and writing so far. If you are the listener, ask questions and give feedback to help your partner improve their work.”
 1–2 minutes: structured partner discussion
 Repeat with 2 different partners (other members of the group).
 “Revise your initial draft based on the feedback you got from your partners.”
 2 minutes: independent work time
 3–4 minutes: quiet work time on the last two sets of questions
Student Facing

What angles are formed by the hour and minute hands of the clock at these times?

6 o’clock

8 o’clock

9 o’clock

11 o’clock

12 o’clock


How many degrees has the minute hand turned when it moves from 2:00 to 2:05?
What about from 2:05 to 2:30? Explain how you know.
 The minute hand of the clock is vertical at 7 p.m. Sometime later, it makes an angle that is \(120^\circ\) from where it was at 7 p.m. What time could it be?

How many degrees does the minute hand turn in:
 10 minutes?
 1 minute?
 4 minutes?
Student Response
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Activity Synthesis
 Display a clock face. Invite students to share their responses to the first question and the last two.
 When discussing the first set of questions, highlight that—except at 12 o’clock— the positions of the hour and minute hands produce two angles—a larger angle and a smaller angle.
 Likewise, when discussing the third question, if no students mentioned that there are two possible times that meet the described constraint, bring it up.
 “How did you find out the number of degrees the minute hand turns in 10 minutes and 1 minute?” (Ten minutes is twice 5 minutes, so it is twice \(30^\circ\), or \(60^\circ\). One minute is 10 minutes divided by 10, so it is \(60^\circ\) divided by 10, which is \(6^\circ\).)
Lesson Synthesis
Lesson Synthesis
“Today we learned about angle measurements on a clock. We looked at the angles formed by the two hands, and we also thought about the number of degrees that a minute hand turns over time.”
“Which is more useful for finding the size of angles on a clock: thinking in terms of number of minutes, the number of 5 minutes, or the numbers 1–12?” (It depends on the situation.)
Display the following images of clocks:
“Does the minute hand on a square clock or an oval clock turn the same number of degrees every minute as it does on a round clock? Explain or show how you know.” (Yes. The minute hand still travels a full turn or \(360^\circ\) in an hour or 60 minutes, so each minute it still travels \(6^\circ\), regardless of the outer shape of the clock or how far away the numbers are spread out from the center point.)
Consider displaying an image of the oval clock showing 12 equal angles. Reinforce the idea that the size of an angle is not determined by the length of segments or rays that form the angle.
“Take 1–2 minutes to add the new words from the past two lessons to your word wall. Share your new entries with a neighbor and add any new ideas you learn from your conversation.”
Cooldown: One Angle at a Time (5 minutes)
CoolDown
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