This opening Warm-up gives students an opportunity to create a pictorial representation of exponential growth. The purpose of this Warm-up is to help students make sense of the situation coming up in the next activity, in which they are not explicitly asked to make a pictorial representation.
Launch
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Activity
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Some bacteria are growing in a dish. Every hour, each bacterium splits into 3 bacteria.
This diagram shows a bacterium at hour 0 and then at hour 1. Draw what has happened at hours 2 and 3.
How many bacteria are there at hours 2 and 3?
Student Response
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Building on Student Thinking
Activity Synthesis
Invite a few students to share their drawings and their observations about how the number of bacteria is increasing.
The goal of this activity is to allow students to explore two growth patterns and to notice that a doubling pattern eventually grows very, very large even with a small starting value. Students may perform a repeated calculation, or they may generalize the process by writing expressions or equations.
Making spreadsheet technology available gives students an opportunity to choose appropriate tools strategically (MP5).
This activity allows students to compare and contrast linear and exponential patterns visually to make sense of the graphs, and to use them to answer questions. They recall that a quantity that grows by adding the same amount at each step forms a line when plotted, and they see an example where a quantity that grows by the same factor at each step forms a curve when plotted.
This activity uses the Stronger and Clearer Each Time math language routine to advance writing, speaking, and listening as students refine mathematical language and ideas.
Launch
Tell students they will now consider what the two offers look like graphically. One of the questions asks students about the vertical intercept. If needed, review the meaning of the term.
Activity
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Lesson Synthesis
We looked at two different situations: One in which the same amount was added each day, and one in which the amount doubled each day. Ask students to reflect on the work done today, either verbally or in writing.
What happened in this lesson that surprised you?
What happened in this lesson that confused you?
What questions do you have about what you learned today?
Student Lesson Summary
When we repeatedly double a positive number, it eventually becomes very large. Let's start with 0.001. The table shows what happens when we begin to double.
number of times doubled
0
1
2
3
4
number
0.001
0.002
0.004
0.008
0.016
If we want to continue this process, it is convenient to use an exponent. For example, the last entry in the table, 0.016, is 0.001 doubled 4 times, or , which can be expressed as .
Even though we started with a very small number, 0.001, we don't have to double it that many times to reach a very large number. For example, if we double it 30 times, represented by , the result is greater than 1 million.
Throughout this unit, we will look at many situations in which quantities grow or decrease by applying the same factor repeatedly.
to access Student Response.
Launch
Provide access to calculators or spreadsheets for every student.
Ask students what stories they may know about a person encountering a wish giver. This situation uses a magical fish, which has historical roots in many cultures including Chinese folklore (Ye Xian (YEE SHEE-ahn)), ancient English fables (The Fish and the Ring), and Haitian fairy tales (Tenzin the Fish).
In many traditions, wish givers are also tricksters that will try to corrupt a wish. Emphasize that being precise in language and explanation is helpful not only in the phrasing of a wish, but also in math class for communication.
Present the situation (either by asking students to read it quietly, or by performing a dramatic reading).
Ask students, “How much money will be in each purse after a week?” (After 7 days, Purse A will contain \$2,400 because it grows by \$200 each day for 7 days. Purse B will contain \$1.28. It doubles each day for 7 days, so it will have , or 128 cents (the 1 here represents the 1 cent in the purse when the purses are offered).) If no students mention using multiplication for repeated addition or an exponent for repeated multiplication, ask how they might use the number 7 to find the worth of each purse. Tell students to enter the amounts into the table in their Task Statement.
In addition to orienting students to the calculations, use this opportunity to point out that the convention here is that the days are numbered the same as the number of times additional money is added. For example, when the purses are initially offered, it is day 0. After the \$200 is added or the money is doubled once, it is day 1. Day 2 is when the money has been added twice.
MLR6 Three Reads. Keep books or devices closed. Display only the problem stem, without revealing the table. “We are going to read this question 3 times.”
After the 1st read, say, “Tell your partner what this situation is about.”
After the 2nd read, say, “List the quantities. What can be counted or measured?”
For the 3rd read, reveal and read the table. Ask, “What are some ways we might get started on this?”
Advances: Reading, Representing
Representation: Internalize Comprehension. Represent the same information through different modalities by using diagrams and tables. Supports accessibility for: Conceptual Processing, Visual-Spatial Processing
Activity
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While walking along a beach, you notice a fish trapped in a small pool. You release the fish into the ocean, but before it swims away, it turns to reward you for your good deed.
It tells you, "Thank you for freeing me from that tide pool! I was getting claustrophobic. You can choose one of these purses as a reward. You cannot add money to the purses, and once you take any money out, the magic is broken and no more money will be added, but you may keep what is inside."
Purse A contains \$1,000 today. If you leave it alone, it will contain \$1,200 tomorrow (by magic). The next day, it will contain \$1,400. This pattern of \$200 additional dollars per day will continue.
Purse B contains 1 penny today. Leave that penny in there, because tomorrow it will (magically) turn into 2 pennies. The next day, there will be 4 pennies. The amount in the purse will continue to double each day.
time passed
0 days
1 day
2 days
7 days
(1 week)
14 days
(2 weeks)
21 days
(3 weeks)
30 days
(1 month)
Purse A
\$1,000
\$1,200
\$1,400
Purse B
\$0.01
\$0.02
\$0.04
Complete the table for the amount of money in each purse for the given number of days after the fish offers the purses. Be prepared to explain your reasoning for each value.
Student Response
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Building on Student Thinking
Some students may confuse the units for the two purses. Remind them that Purse B contains pennies (and thus cents rather than dollars).
Due to prior experience with stories in which wishes or magical items are offered, students may be suspicious of the offers. Ask them to show their mathematical reasoning, rather than basing their choice of purses on their suspicions or other considerations, such as how to carry a purse containing over 2 million pennies.
Activity Synthesis
Focus the discussion on how students found the amounts of money in the two purses after many days. Invite students who performed recursive calculations and those who wrote expressions or equations to share their reasoning. Record and display students' reasoning for all to see. Seeing the repeated reasoning will support students' generalization work later. For example, the amount in Purse A will be for day 1, then for day 2, for day 3, and so on. Connect this to writing expressions like and where represents the number of days since the fish offered the purses.
For Purse B, highlight the fact that after two days it contains cents, which is cents. After three days it contains cents, which is cents. This is a good opportunity to emphasize the meaning of exponential notation. Exponential notation is particularly useful for expressing the amount in Purse B on day 30, which is cents.
Conclude the activity by polling the class about which purse they would choose.
If no students use a spreadsheet, consider demonstrating how it might be helpful in performing calculations like these.
Here are graphs showing how the amount of money in the purses changes. Remember Purse A starts with \$1,000 and grows by \$200 each day. Purse B starts with \$0.01 and doubles each day. Point P has the coordinates , and point Q has the coordinates .
Graph of two functions, origin O. Horizontal axis, number of days, scale 0 to 25, by 1’s Vertical axis, amount in dollars, scale 0 to 40,000, by 10,000’s. The functions are discrete. Purse A is red and has these points: (0 comma 1,000), (1 comma 800), (2 comma 1,400), (3 comma 1,600), (4 comma 1,800), (5 comma 2,000), (6 comma 2,200), (7 comma 2,400), (8 comma 2,600), (9 comma 2,800), (10 comma 3,000), (11 comma 3,200), (12 comma 3,400), (13 comma 3,600), (14 comma 3,800), (15 comma 4,000), (16 comma 4,200), (17 comma 4,400), (18 comma 4,600), (19 comma 4,800), (20 comma 12 point 2), (21 comma 5,200), (22 comma 5,400), (23 comma 5,600), (24 comma 5,800), (25 comma 6,000), (26 comma 6,200), (27 comma 6,400) and (28 comma 6,600). Purse B is blue and has these points: (0 comma point zero 1), (1 comma point zero 2), (2 comma point zero 4), (3 comma point zero 8), (4 comma point 16), (5 comma point 32), (6 comma point 64), (7 comma 1 point 28 ), (8 comma 2 point 56), (9 comma 5 point 12), (10 comma 10 point 24), (11 comma 20 point 48), (12 comma 40 point 96), (13 comma 81 point 92), (14 comma 163 point 84), (15 comma 327 point 68), (16 comma 655 point 36), (17 comma 1,310 point 72), (18 comma 2,621 point 44), (19 comma 5,242 point 88), (20 comma 10,485 point 76), (21 comma 20,971 point 52).
Which graph shows the amount of money in Purse A? Which graph shows the amount of money in Purse B? Explain how you know.
Points and are labeled on the graph. Explain what they mean in terms of the fish's offer.
What are the coordinates of the vertical intercept for each graph? Explain how you know.
When does Purse B become a better choice than Purse A? Explain your reasoning.
Knowing what you know now, which purse would you choose? Explain your reasoning.
Student Response
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Building on Student Thinking
Due to the exponential nature of the problem, the scale for the vertical axis makes it difficult to accurately estimate vertical coordinates of points from the graph. Remind students that they know how the value of each purse is calculated, so they may use calculations similar to those in the previous activity to compute actual dollar values.
If color versions of the materials are not available, it may be difficult to determine which points correspond with which purse. Encourage students to imagine connecting the dots (perhaps using a ruler for Purse A) to distinguish the two sets of points.
Activity Synthesis
To highlight the power of using graphs to illustrate what is happening in a situation, ask “You have used calculations and graphs to compare the offers. Did using graphs help? Did they add new insights to the offers?” (The graphs give a clear visual representation of how the amount of money in each purse grows. It is easier to tell when Purse B becomes a better option than Purse A, and how different the two values are at different points in time. They de-emphasize the actual numbers and, so, give a very clean comparison of the two situations.)
Use Stronger and Clearer Each Time to give students an opportunity to revise and refine their response to “Knowing what you know now, which purse would you choose? Why?” In this structured pairing strategy, students bring their first draft response into conversations with 2–3 different partners. They take turns being the speaker and the listener. As the speaker, students share their initial ideas and read their first draft. As the listener, students ask questions and give feedback that will help clarify and strengthen their partner's ideas and writing.
If time allows, display these prompts for feedback:
“_____ makes sense, but what do you mean when you say. . . ?”
“Can you describe that another way?”
“How do you know . . . ? What else do you know is true?”
Close the partner conversations, and give students 3–5 minutes to revise their first draft. Encourage students to incorporate any good ideas and words that they got from their partners to make their next draft stronger and clearer.
As time allows, invite students to compare their first and final drafts. Select 2–3 students to share how their drafts changed and why they made the changes they did.
Representation: Internalize Comprehension. Use color coding and annotations to highlight connections between representations in a problem. For example, color code connections between the options. Supports accessibility for: Visual-Spatial Processing
Standards Alignment
Building On
6.EE.A.1
Write and evaluate numerical expressions involving whole-number exponents.
Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies.
Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.
Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.
For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.
Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.
