1.1: Which One Doesn't Belong: Reading Clocks (5 minutes)
This warm-up prompts students to compare four clock faces. 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, such as how students describe circular motion or the directionality of the clock hands. The context of a clockface is used throughout this unit, so this warm-up is an opportunity for students to start building familiarity with the context (MP1).
Arrange students in groups of 2–4. Display the images 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 why a particular item does not belong, and together, find at least one reason each item doesn’t belong.
Which one doesn’t belong?
Some students may be unfamiliar with telling time using an analog clock. For these students, display the time shown on each clock image for all to see. Encourage students to discuss with each other the process for telling time with this type of clock so they may be more familiar with it in future lessons.
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 angle, arc, or arc length. Also, press students on unsubstantiated claims.
1.2: Around and Around (15 minutes)
The goal of this activity is to get students thinking about an input-output relationship whose outputs repeat at regular intervals. This leads to defining period and naming the functions that represent these types of relationships as periodic functions. Clocks provide a familiar repeating context for students to reason about. Several relationships between time, angles created by clock hands, and the height of the end of a clock hand will be explored in this unit.
Monitor for students who use clear language to describe the repetition in the height of the ladybug. Students will refine their language regarding periodic functions throughout the unit, so here their language can be less formal and more context-based. For example, in reflecting on the differences and similarities between the motion of the second hand and the motion of the minute hand, students can use the context to reason about period and amplitude informally without using those terms.
Display a blank clock face, such as the one shown here, for all to see throughout the activity.
Ask students to read the situation and first problem. Give students quiet work time and then time to share their work with a partner. Invite students to share their answers and reasoning for each of the 4 times before asking them to continue with the rest of the task.
Supports accessibility for: Language; Social-emotional skills
A ladybug lands on the end of a clock’s second hand when the hand is pointing straight up. The second hand is 1 foot long and when it rotates and points directly to the right, the ladybug is 10 feet above the ground.
- How far above the ground is the ladybug after 0, 30, 45, and 60 seconds have passed?
Pause here for a class discussion.
- Estimate how far above the ground the ladybug is after 10, 20, and 40 seconds. Be prepared to explain your reasoning.
- If the ladybug stays on the second hand, describe how its distance from the ground will change over the next minute. What about the minute after that?
- At exactly 3:15, the ladybug flies from the second hand to the minute hand, which is 9 inches long.
- How far off the ground is the ladybug now?
- At what time will the ladybug be at that height again if it stays on the minute hand? Be prepared to explain your reasoning.
The purpose of this discussion is for students to share their observations about the height of a ladybug on the clock hands over time and introduce students to the idea of period and periodic functions.
Select previously identified students to share their responses. If no students notice how the heights of the ladybug on the second hand are the same for times such as 20 and 40 seconds after 0 seconds, point this out and invite students to identify other times where the ladybug would be the same height over the ground. An important takeaway here is that the heights repeat every minute. Some students may also notice the symmetric nature of the heights from maximum to minimum and back to maximum, but it is okay to not focus on this aspect yet since it will be a focus of future lessons.
Tell students that the type of function represented by the height of the ladybug on the end of the clock hand over time can be described by a periodic function. Periodic functions are ones in which the values of the function repeat at regular intervals. An important feature of a periodic function is the period, which is the length of the interval the function repeats. In this situation, we say that the second hand has a period of 60 seconds. Movement around and around a circle is one type of periodic function and we’ll explore more types throughout the unit.
Conclude the discussion by inviting students to describe how the motion of the ladybug on the minute hand is the same or different from the motion of the ladybug on the second hand. Possible responses to highlight include:
- The period of the height of the ladybug on the minute hand is 60 minutes.
- The ladybug travels vertically between a maximum of 10.75 feet to a minimum of 9.25 feet.
Design Principle(s): Maximize meta-awareness; Support sense-making
1.3: Where is the Point? (15 minutes)
Building on their work in the previous activity, the goal of this activity is for students to make connections between points on a circle, the coordinates of those points if the circle is centered at the origin, and how right triangles and the Pythagorean Theorem can help determine unknown information. While the focus in this activity is on using the Pythagorean Theorem, in future lessons, students will incorporate the trigonometric work they learned in a previous course and use cosine, sine, tangent, and the Pythagorean Theorem to determine the location of specific points on a circle.
The idea that we can choose to overlay the coordinate plane to reason about the location of a point on a circle is part of reasoning abstractly and quantitatively (MP2). By adding the familiar structure, we can use a greater number of mathematical tools to determine information about a situation.
Monitor for students with clear explanations about why point \(S\) could be in 2 different places on the circle, particularly students who have created a sketch to help them think about the situation.
Arrange students in groups of 2. Display the clock with point \(P\) at the 2 for all to see. Ask, “What do you need to find the location of the point \(P\) marked on the clock?”
After a brief quiet think time, invite students to share their ideas. Students may suggest things like the height of the clock off the ground, the radius of the clock, a ruler, and so on. A key idea here is that in order to say where the point is, we need something to measure from and some type of scale to measure with.
Display a new image like the one given here where the clock is centered at the origin with a radius of 5 units. Invite students to work with a partner to determine how they can calculate the \(y\)-coordinate of the point \(P\).
After a brief work think time, select students to share their solutions, recording their reasoning for all to see on or near the image. While students may reason about the \(y\)-coordinate in many ways, focus on those who recognize that we can use a right triangle and the Pythagorean Theorem to identify the value of the \(y\)-coordinate. If no students suggest doing so, draw in the right triangle with hypotenuse 5, known side length 3, and right angle on the horizontal axis and then invite students to consider again how they could determine the \(y\)-coordinate.
Supports accessibility for: Visual-spatial processing; Conceptual processing
What is the radius of the circle?
If \(Q\) has a \(y\)-coordinate of -4, what is the \(x\)-coordinate?
If \(B\) has a \(y\)-coordinate of 4, what is the \(x\)-coordinate?
- A circle centered at \((0,0)\) has a radius of 10. Point \(S\) on the circle has an \(x\)-coordinate of 6. What is the \(y\)-coordinate of point \(S\)? Explain or show your reasoning.
Are you ready for more?
- How many times a day do the minute hand and the hour hand on a clock point in the same direction?
- At what times do they point in the same direction?
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Some students may be unsure where to begin working with the given information. Encourage these students to draw in the right triangle, recalling the problem discussed during the launch of the activity.
The purpose of this discussion is for students to share how they calculated the unknown values. Highlight students who drew in right triangles as a strategy. If time allows, pair partners up into groups of 4 to first share strategies with each other before selecting students to share their responses, including any visuals made, with the class.
For the last question, an important takeaway for students is that without more information, point \(S\) could be in 1 of two places on the circle since there are two quadrants where the \(x\)-value is positive. This repeating feature of coordinates on a circle is one students will work with more in the future and connects to the periodic nature of trigonometric functions.
Design Principle(s): Support sense-making; Cultivate conversation
The purpose of this discussion is for students to reflect on circular motion and the \((x,y)\) coordinates of a point on a circle that is centered at the origin.
Invite students to consider what is true about the \(x\)- and \(y\)-coordinates of the ladybug if the clock is centered at \((0,0)\). In particular, invite students to consider how the values of the coordinates change as time passes. After a brief quiet think time, invite students to share their responses, recording them for all to see. Here are some things students may notice:
- The values of the coordinates of the ladybug repeat each time they make a full circle: every minute for the second hand and every hour for the minute hand.
- We could use right triangles and the Pythagorean Theorem to figure out the \(x\)-coordinate of the ladybug if we knew the \(y\)-coordinate (the height) at that time.
- The \(y\)-coordinate of the ladybug was the same at times like 10 seconds and 50 seconds after 0. Except for straight up and down, every point on the clock has a “matching” point on the opposite side.
- The \(x\)-coordinate of the ladybug was the same at times like 10 seconds and 20 seconds after 0. Except for straight left and right, every point on the clock has a “matching” point on the opposite side.
If students do not mention all the points on the list, there is no need to bring them up at this time. They will have more opportunities to consider these ideas in future lessons.
If time allows, invite students to propose other situations we could model with a function whose outputs repeat. For example, the height of a person on a Ferris wheel, phases of the Moon, frequency of a sound wave, or Earth’s distance from the Sun.
1.4: Cool-down - Two Particular Points (5 minutes)
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Student Lesson Summary
Consider the height of the end of a second hand on a clock over a full minute. It starts pointing up, then rotates to point down, then rotates until it is pointing straight up again. This motion repeats once every minute.
If we imagine the clock centered at \((0,0)\) on the coordinate plane, then we can study the movement of the end of the second hand by thinking about its \((x,y)\) coordinates on the plane. Over one minute, the \(y\)-coordinate starts at its highest value (when the hand is pointing up), decreases to its lowest value (when the hand is pointing down), and then returns to its highest value. This happens once every minute that passes.
While we have worked with many types of functions, such as rational or exponential, none of them are characterized by output values that repeat over and over again, so we can’t use them to model the height of the end of the second hand. This means we need to use a new type of function. A function whose values repeat at regular intervals is called a periodic function, and the length of the interval at which a periodic function repeats is called the period. We will study several types of periodic functions in this unit.