Illustrative Mathematics Algebra 1, Unit 5.1 - Teachers (2024)

There are some bacteria in a dish. Every hour, each bacterium splits into 3 bacteria.

1. This diagram shows a bacterium in hour 0 and then hour 1. Draw what happens in hours 2 and 3.

   

2. How many bacteria are there in hours 2 and 3?

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).

Provide access to calculators or spreadsheets. Present the situation (either by asking students to read it quietly, or performing a dramatic reading). Before students do any calculating or any other work, poll the class about which option they think is better. Display the results of the poll (the number of students who think Purse A is better and Purse B is better) for all to see.

Reading, Listening, Conversing: MLR6 Three Reads. Use this routine to support reading comprehension of this word problem. Use the first read to orient students to the situation. Ask students to describe what the situation is about without using numbers (as a reward, students make a choice between Purse A and Purse B). Use the second read to identify quantities and relationships. Ask students what can be counted or measured without focusing on the values. Listen for, and amplify, the important quantities that vary in relation to each other in this situation: the amount of money, in dollars, and the amount of time, in days. After the third read, invite students to brainstorm possible strategies to answer the questions. This helps students connect the language in the word problem and the reasoning needed to solve the problem.
Design Principle(s): Support sense-making

Representation: Internalize Comprehension. Represent the same information through different modalities by using diagrams. Encourage students to sketch diagrams that show how the amount of money grows over the first few days. Students may benefit from this visual as they transition to the use of a table or other representation to track growth.
Supports accessibility for: Conceptual processing; Visual-spatial processing

You are walking along a beach and your toe hits something hard. You reach down, grab onto a handle, and pull out a bottle! It is sandy. You start to brush it off with your towel. Poof! A genie appears.

He tells you,"Thank you for freeing me from that bottle! I was getting claustrophobic. You can choose one of these purses as a reward."

• Purse A which contains \$1,000 today. If you leave it alone, it will contain \$1,200 tomorrow (by magic). The next day, it will have \$1,400. This pattern of \$200 additional dollars per day will continue.
• Purse B which 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.

1. How much money will be in each purse after a week? After two weeks?
2. The genie later added that he will let the money in each purse grow for three weeks. How much money will be in each purse then?
3. Which purse contains more money after 30 days?

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 genie stories, 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.

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. 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\(1,\!000 + 200\) for day 1, then \(1,\!000 + 200 + 200\) for day 2, \(1,\!000 + 200 + 200 + 200\) for day 3, and so on. Connect this to writing expressions like \(1,000+200 \boldcdot 3\) and \(1,000+200x\) where \(x\) represents the number of days since the genie appeared.

For Purse B, highlight the fact that after two days it contains \((1 \boldcdot 2) \boldcdot 2\) cents, which is \(2^2\) cents. After three days it contains \((1 \boldcdot 2 )\boldcdot 2 \boldcdot 2\) cents, which is \(2^3\) cents. (Here the day the genie appears is treated as day 0.) 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 \(2^{30}\) cents.

If no students use a spreadsheet, consider demonstrating how it might be helpful in performing calculations like these.

Tell students they will now consider what the two offers look like graphically. One of the questions asks students about the verticalintercept. If needed, review the meaning of the term.

Reading, Conversing, Writing: MLR5 Co-Craft Questions. Before revealing the task, begin by displaying only the graphs that show how the amount of money in the purses changes. Give students 1–2 minutes to write their own mathematical questions about the situation. Invite students to share their questions with the class. Listen for and amplify any questions seeking to determine which purse is the better choice or those that reference specific critical points on the graphs. This will build student understanding of the language of mathematical comparisons and help ensure students interpret the task correctly.
Design Principle(s): Maximize meta-awareness; Cultivate conversation

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.

1. Which graph shows the amount of money in Purse A? Which graph shows the amount of money in Purse B? Explain how you know.
2. Points \(P\) and \(Q\) are labeled on the graph. Explain what they mean in terms of the genie’s offer.
3. What are the coordinates of the vertical intercept for each graph? Explain how you know.
4. When does Purse B become a better choice than Purse A? Explain your reasoning.
5. Knowing what you know now, which purse would you choose? Explain your reasoning.

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.

Invite students to share their responses with the class. Discuss how they interpreted the points on the graphs and the intersection of the two graphs.

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 genie's 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.)

Consider asking students to explain to a partner their response and reasoning to the last question: “Knowing what you know now, which purse would you choose? Why?”

Representation: Internalize Comprehension. Use color coding and annotations to highlight important connections between the representations for each situation (Purse A or Purse B).
Supports accessibility for: Conceptual processing; Visual-spatial processing

We looked at two different situations: one where the same amount was added each day, and one where 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?

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:

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 being doubled 4 times, or \((0.001) \boldcdot 2\boldcdot 2\boldcdot 2\boldcdot 2\), which can be expressed as \((0.001) \boldcdot 2^4\).

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 \((0.001) \boldcdot 2^{30}\), the result is greater than 1,000,000.

Throughout this unit, we will look at many situations where quantities grow or decrease by applying the same factor repeatedly.