The Size of an Angle on a Clock
10 min
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 the segments forming the angles, and to consider how they affect our perception of the sizes 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 Activity Synthesis.
Launch
- Groups of 2
- Display the two sets of angles.
- “What do you notice? What do you wonder?”
- 1 minute: quiet think time
Teacher Instructions
- “Discuss your thinking with your partner.”
- 1 minute: partner discussion
- Share and record responses.
Student Task
What do you notice? What do you wonder?
Sample Response
- There are two sets of angles. Each set has 3 angles.
- The lengths of the segments that make the angles are different—some are short, some are long.
- In Set 1, the segments seem to be the same length but the angles get larger.
- In Set 2, the segments seem to get shorter.
- Are the angles in the second set getting larger or getting smaller?
- Are the sizes of the angles the same in the two sets?
- Can the last angle (formed by the shortest pair of segments) be measured?
Activity Synthesis (Teacher Notes)
- “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 the first angle 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.
Standards
- 4.MD.5·Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement:
- 4.MD.C.5·Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement:
- 4.MD.5.a·An angle is measured with reference to a circle with its center at the common endpoint of the rays, by considering the fraction of the circular arc between the points where the two rays intersect the circle. An angle that turns through 1/360 of a circle is called a "one-degree angle," and can be used to measure angles.
- 4.MD.C.5.a·An angle is measured with reference to a circle with its center at the common endpoint of the rays, by considering the fraction of the circular arc between the points where the two rays intersect the circle. An angle that turns through 1/360 of a circle is called a “one-degree angle,” and can be used to measure angles.
15 min
20 min
Knowledge Components
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The Size of an Angle on a Clock
10 min
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 the segments forming the angles, and to consider how they affect our perception of the sizes 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 Activity Synthesis.
Launch
- Groups of 2
- Display the two sets of angles.
- “What do you notice? What do you wonder?”
- 1 minute: quiet think time
Teacher Instructions
- “Discuss your thinking with your partner.”
- 1 minute: partner discussion
- Share and record responses.
Student Task
What do you notice? What do you wonder?
Sample Response
- There are two sets of angles. Each set has 3 angles.
- The lengths of the segments that make the angles are different—some are short, some are long.
- In Set 1, the segments seem to be the same length but the angles get larger.
- In Set 2, the segments seem to get shorter.
- Are the angles in the second set getting larger or getting smaller?
- Are the sizes of the angles the same in the two sets?
- Can the last angle (formed by the shortest pair of segments) be measured?
Activity Synthesis (Teacher Notes)
- “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 the first angle 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.
Standards
- 4.MD.5·Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement:
- 4.MD.C.5·Recognize angles as geometric shapes that are formed wherever two rays share a common endpoint, and understand concepts of angle measurement:
- 4.MD.5.a·An angle is measured with reference to a circle with its center at the common endpoint of the rays, by considering the fraction of the circular arc between the points where the two rays intersect the circle. An angle that turns through 1/360 of a circle is called a "one-degree angle," and can be used to measure angles.
- 4.MD.C.5.a·An angle is measured with reference to a circle with its center at the common endpoint of the rays, by considering the fraction of the circular arc between the points where the two rays intersect the circle. An angle that turns through 1/360 of a circle is called a “one-degree angle,” and can be used to measure angles.