Lesson 6

Fermi Problems

6.1: How Old Are You? (20 minutes)

Optional activity

Students attempt to calculate their exact age. Because this is a constantly changing quantity, they need to think carefully about how accurately to report the answer. The mathematics involved is multiple unit conversions in the context of time. In addition to the fact that our age is always growing, we may not know with great accuracy when we were born. 

Launch

Arrange students into groups of 4. Ask them to order themselves according to their age: who is the youngest? Who is the oldest? What information do you need to decide?

Tell students that they will be finding their exact age. 

Provide access to calculators. Give students 10 minutes of group work time, followed by whole-class discussion.

Student Facing

What is your exact age at this moment?

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

Students may struggle with what is meant by at this moment in the prompt. Ask them how they interpret this: the question is intended to be flexible so students can interpret at this moment as being when they read the question, or the day of class, or in another way.

Activity Synthesis

Invite students to share answers and discuss difficulties in answering the question. The discussion should include the following points:

  • We can give an estimate, but the question cannot be answered because we probably do not know the exact time when we were born and “at this moment” keeps moving forward.
  • How we express our answer depends on what we know about the time we were born. We probably know the day, but do we know the time? Should we answer to the nearest hour? Minute? Second?
  • Did you take into account leap years?

Ask students how they usually answer the question, “How old are you?” (This will probably be in whole number of years lived.) Why? (Because this whole number communicates enough information for most purposes.) After more than six months have passed since someone’s last birthday, the person still doesn’t normally round up when they report their age, even though this is customary for reporting many other types of measurements.

Speaking: MLR8 Discussion Supports. Use this routine to support whole-class discussion. For each response or observation that is shared, ask students to restate and/or revoice what they heard using precise mathematical language. Consider providing students time to restate what they hear to a partner, before selecting one or two students to share with the class. Ask the original speaker if their peer was accurately able to restate their thinking. Call students' attention to any words or phrases that helped to clarify the original statement. This will provide more students with an opportunity to produce language as they interpret the reasoning of others.
Design Principle(s): Support sense-making

6.2: A Heart Stoppingly Large Number (15 minutes)

Optional activity

This activity focuses on another complex calculation: the number of times our heart has beaten in our lifetime. In addition to all of the precision issues in determining how old we are from the previous activity, there is an additional level of complexity as our heart does not beat at a constant rate. For this situation, we need to make a reasoned estimate, but there is no hope of getting an exact answer. This makes our lack of knowledge of the exact time of our birth (or the exact time when we are solving the problem) unimportant because a few minutes or hours will not significantly impact the answer.

Launch

Arrange students in groups of 2. Tell students they will be investigating how many times their heart has beaten in their lifetime. Ask students to make an estimate and then poll the class. One way to accomplish this is to display a table like this and complete it with the number of students whose estimate is in each range.

estimate for number of heartbeats number of students
around 10,000
around 100,000
around 1,000,000
around 10,000,000
around 100,000,000
around 1,000,000,000
around 10,000,000,000
around 100,000,000,000

Ask students to brainstorm what they need to know or investigate to answer this question. Provide access to stopwatches (if students will estimate their own heartbeats) and calculators. Alternatively, share with them that a normal heart beat range is 60 to 100 beats per minute with higher rates when we exercise and sometimes lower rates at rest.

Give students 10 minutes of partner work time followed by a whole-class discussion.

Conversing: MLR2 Collect and Display. While students work in their groups to measure heart rates and make calculations, circulate and listen to the language students use related to their estimation process (e.g., “rate,” “per,” “constant,” etc.). Record their language on a visual display and update it throughout the remainder of the lesson. Remind students to borrow language from the display as needed. This helps students produce mathematical language as they justify their estimations.
Design Principle(s): Optimize output (for justification); Cultivate conversation

Student Facing

How many times has your heart beat in your lifetime?

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

Some students may struggle with simplifying the assumptions needed to solve this problem. Encourage these students by:

  • Telling them to think about how old they are if they have done the previous activity.
  • Telling them the normal rate of heartbeats per minute.
  • Asking them how they can find out about how many times a heart beats in an hour (multiply minute rate by 60), in a day (multiply hourly rate by 24), in a year, and so on.

Activity Synthesis

Invite students to share answers and discuss the difficulties they encountered answering these questions:

  • “How did you estimate how often your heart beats?” (By checking pulse, looking online, etc.)
  • “Is your heart rate always the same?” (No, it is faster when I exercise and slower when I sleep.)
  • “Did you use the information you found for your age?” (No, because I only have an estimate for my pulse or yes because I already had the information available.)

Time permitting, students can be encouraged to check their pulse while resting and then after a short amount of exercise (jumping jacks, running in place, push ups). This gives them an idea of the variability involved in how frequently our heart beats.

The calculation of our number of heartbeats is only an estimate. One way we can indicate this is with the way we report the final answer. An answer of 524,344,566 would not be appropriate because that makes it look like it is an exact answer. An answer of 500,000,000 makes it clear that we are only making an estimate. Alternatively, students might say that the number of heartbeats is between 400,000 and 700,000 with the estimate of 400,000 coming from the low value of 60 beats per minute and the estimate of 700,000 coming from the high value of 100 beats per minute.

6.3: All the Hairs on Your Head (20 minutes)

Optional activity

Students estimate how many hairs they have on their head. Although there is a definite number of hairs on each student’s head, finding this number exactly is not feasible: although it is a much smaller number than the number of heartbeats in a student’s lifetime, it is still too large to count and, even if we could count, we would need to cut all of those hairs to do so! An additional geometric layer of estimation comes into play in this task as students need to estimate the surface area of part of their head. If one or more students in the class are bald or are otherwise sensitive about their hair, consider asking them to estimate the number of hairs on another student’s head.

Launch

Arrange students in groups of 2. Provide access to calculators. Before starting, poll the class to guess how many hairs they have on their head. One way to accomplish this is to display a table like this, and record the number of students with each guess:

number of hairs number of guesses
about 100
about 1,000
about 10,000
about 100,000
about 1,000,000

Next, ask students to brainstorm the information they need to answer the question. Provide the following information below when students ask for it.

Data to keep in reserve and share with students as needed:

  • The number of hairs per square cm varies from person to person, but for this analysis, we can assume approximately 150 hairs per square cm.
  • Students have to be creative in their estimates for the area of their scalp. One possibility is to approximate it with a circle. 600 square centimeters is a good estimate, but students may come up with slightly different estimates.

Provide access to string, tape measures, and calculators. After taking 5 minutes to introduce the activity, give students 10 minutes of partner work time, followed by whole-class discussion.

Action and Expression: Internalize Executive Functions. Provide students with a graphic organizer for recording measurements and calculations of areas of circles to represent hair on head.
Supports accessibility for: Language; Organization

Student Facing

How many strands of hair do you have on your head?

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

Since the scalp is three dimensional, students may not think of approximating the scalp with a circle and may struggle with trying to identify the size of the circle. Consider asking these students:

  • What two-dimensional shape can you use to model the part of your scalp covered with hair? (A circle)
  • How can you estimate the diameter of the circle? (Measuring from front to back or from side to side)
  • How do you estimate the area of the circle using the diameter? (The diameter gives the radius, and then the area can be found once the radius is known.)

Activity Synthesis

Invite students to share their answers and methods. Consider asking the following questions:

  • “Why is it challenging to find the exact number of hairs on your head?” (There are too many to count, and they are too small to count accurately. The number changes as time goes by.)
  • “How can you estimate the number of hairs on your head?” (Estimate the area of the part of the head covered by hair. Try to find how many hairs there are on a small part of the head.)
  • “How did you estimate the area of your head?” (Approximated area with a circle. Covered the head with little pieces of paper (sticky notes) and added up their areas.)

This is a situation where there is a definite number of hairs on our head, but because they are so small (and because they are on our head), we cannot calculate this number exactly. Fortunately, for almost any purpose, an estimate will do.