Unit 6, Lesson 5
Understanding Decay
Algebra 1
5.1
Warm-up
Standards Alignment
Building On
Addressing
Building Toward
HSF-LE.A.2
Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).
Instructional Routines
Notice and WonderMaterials
NoneActivity Narrative
The purpose of this Warm-up is to recognize linear and exponential patterns that involve fractions, which will be useful when students write expressions to represent these patterns in a later activity. While students may notice and wonder many things about these tables, the linear and exponential patterns are the important discussion points.
Launch
Arrange students in groups of 2. Display the two tables for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder about the tables. Give students 1 minute of quiet think time, and then 1 minute to discuss, with their partner, the things they notice and wonder about.
Activity
NoneWhat do you notice? What do you wonder?
Table A
| 0 | 2 |
| 1 | |
| 2 | 5 |
| 3 | |
| 4 | 8 |
Table B
| 0 | 2 |
| 1 | 3 |
| 2 | |
| 3 | |
| 4 |
Student Response
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Building on Student Thinking
Activity Synthesis
Ask students to share the things they noticed and wondered. Record and display their responses, without editing or commentary, for all to see. If possible, record the relevant reasoning on or near the tables. Next, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to observe what is on display and to respectfully ask for clarification, point out contradicting information, or voice any disagreement.
If the way that the values in the tables change from row to row does not come up during the conversation, ask students to discuss this idea.
If no students wonder about whether the values in the two tables will ever agree again, ask, “Will a linear growth and an exponential growth that start with the same value have the same value again at some later point?” Consider posting the question for students to think about throughout the unit.
5.2
Activity
Instructional Routines
MLR8: Discussion SupportsMaterials
NoneActivity Narrative
In this activity, students use the distributive property to rewrite an expression that involves a difference and turn it into an expression that uses only multiplication. They examine and explain a worked example, and then they practice using the method to rewrite expressions.
Launch
Ask students to close their books or devices. Pose this question:
Diego has \$100 and spends of it. How much does he have left?
Give students a minute to think about it, and then ask a few students to share their answer and their reasoning.
MLR8 Discussion Supports. Prior to solving the problems, invite students to make sense of the method and to take turns sharing their understanding with their partner. Listen for and clarify any questions about the context.
Advances: Reading, Representing
Advances: Reading, Representing
Representation: Internalize Comprehension. Use color coding and annotations to highlight connections between representations in a problem. For example, color code steps that are similar in Diego's method and what students share for the additional questions.
Supports accessibility for: Visual-Spatial Processing
Supports accessibility for: Visual-Spatial Processing
5.3
Activity
Instructional Routines
MLR5: Co-Craft QuestionsMaterials
NoneActivity Narrative
In this activity, students examine a situation in which a quantity decreases by repeated multiplication by the same factor. They represent the pattern of decrease with an equation and interpret the different parts of the equation in terms of the context (the depreciation of a car). The definition of growth factor is expanded to accommodate situations involving decrease.
This activity uses the Co-Craft Questions math language routine to advance reading and writing as students make sense of a context and practice generating mathematical questions.
Launch
Arrange students in groups of 2. Introduce the context of depreciation. Use Co-Craft Questions to orient students to the context and to elicit possible mathematical questions.
- Display only the problem stem and related image, without revealing the questions. Give students 1–2 minutes to write a list of mathematical questions that could be asked about the situation, before comparing questions with a partner.
- Invite several partners to share one question with the class, and record their responses. Ask the class to make comparisons among the shared questions and their own. Ask, “What do these questions have in common? How are they different?” Listen for and amplify language related to the learning goal, such as “losing value” and “decrease.”
- Reveal the questions, and give students 1–2 minutes to compare it to their own question and to those of their classmates. Invite students to identify similarities and differences by asking:
- “Which of your questions is most similar to or different from the ones provided? Why?”
Clarify to students that they may complete the table with numerical expressions that show how to find the values rather than with the results of evaluating the expressions.
After groups complete the second question, discuss their solution as a whole class. If needed, remind students of the method for using the distributive property to write . Students should understand that process and the connection that losing one third of its value means that the car is still worth two thirds of its value. When students are comfortable with this understanding, tell them to proceed with the other questions.
Action and Expression: Provide Access for Physical Action. Provide access to tools and assistive technologies such as a graphing calculator.
Supports accessibility for: Visual-Spatial Processing, Conceptual Processing, Organization
Supports accessibility for: Visual-Spatial Processing, Conceptual Processing, Organization
Lesson Synthesis
We studied quantities that decrease or decay by the same factor repeatedly. Use examples from activities in the lesson, or a new example like this one.
Suppose a digital camera worth \$400 loses of its value each year.
- How can we express the value of the camera, in dollars, after one year? After two years?
- Why might it make sense to use only multiplication (instead of subtraction and multiplication) to show the value of the camera over time?
- When we use only multiplication, why doesn’t the show up in the expression?
- What is the value of the camera, in dollars, after years?
Student Lesson Summary
Sometimes a quantity grows by the same factor at regular intervals. For example, a population might double every year. Sometimes a quantity decreases by the same factor at regular intervals. For example, a car might lose one third of its value every year.
Let's look at a situation in which the quantity decreases by the same factor at regular intervals. Suppose a bacteria population starts at 100,000, and of the population dies each day. The population one day later is , which can be written as . The population after one day is of 100,000 or 75,000. The population after two days is . Here are some further values for the bacteria population:
| number of days |
bacteria population |
|---|---|
| 0 | 100,000 |
| 1 | 75,000 (or ) |
| 2 | 56,250 (or , or ) |
| 3 | about 42,188 (or , or ) |
In general, days after the bacteria population was 100,000, the population is given by the equation: with one factor of for each day.
Situations with quantities that decrease exponentially are described with the term exponential decay. The multiplier ( in this case) is still called the growth factor, though sometimes people call it the decay factor instead.