## Investigations and Benchmark Lessons

**Investigations**

"The idea is for students to learn how to draw a conclusion based on data they collect and analyze. As part of the process, they also learn about the content under investigation" (Macguire, Myerowitz, Sampson 2010).

"The idea is for students to learn how to draw a conclusion based on data they collect and analyze. As part of the process, they also learn about the content under investigation" (Macguire, Myerowitz, Sampson 2010).

**Investigation 1: Penny Investigation**

(Link for handout below)

**Overview of the Investigation**

The goal of this investigation is to introduce the students to exponential equations. The students will be dropping pennies, counting either Heads or Tails and discarding the other. They will repeat this process until they have no more pennies left. Without it being obvious, the students will be discovering exponential decay. They will use a TI-83 graphing calculator to run a regression on their data in order to come up with an equation. This investigation will be student lead. A worksheet will be provided in order to guide the discussion if needed. This would be a level 2 inquiry since the students would be coming up with their own data but will need guidance on what a mathematical exponential model would be and how to interpret the statistics on the calculator.

**Objectives**

Students will be able to:

- Observe a pattern in collected data
- Create graphs using technology to model exponential decay
- Explain the difference between the “a” and “b” in y= ab^x
- Run a regression to find a best-fit line
- Differentiate between a mathematical model and a best-fit line

**Alignment with Texas Essential Knowledge and Skills (TEKS)**

This investigation addresses the following TEKS:

- 1.B Use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution

- 1.E Create and use representations to organize, record, and communicate mathematical ideas
- 9.B interpret the meaning of the values of a and b in exponential functions of the form f(x) = ab^x in real-world problems
- 9.C Write exponential functions in the form f(x) = ab^x (where b is a rational number) to describe problems arising from mathematical and real-world situations, including growth and decay
- 9.E Write, using technology, exponential functions that provide a reasonable fit to data and make predictions for real-world problems

**Investigation 2: How Fast Viruses Spread**

**Overview of the Investigation**

Students simulate the spread of a virus such as HIV through a population by "sharing" (but not drinking) the water in a plastic cup with several classmates. Although invisible, the water in a few of the cups has already be tainted with the "virus" (sodium carbonate). After all the students have shared their liquids, the contents of the cups are tested for the virus with phenolphthalein, a chemical that causes a striking color change in the presence of sodium carbonate. Students then set about trying to determine which of their classmates were the ones originally infected with the virus. Each student will keep track of the number of students they interact with, and write their number on their cups at the end of the activity. They will then estimate who is more infected and why.

**Objectives**

Students will be able to:

- Describe how a virus is spread within a population

**Alignment with Texas Essential Knowledge and Skills (TEKS)**

This investigation addresses the following TEKS:

- 5.C describe the roles of DNA, ribonucleic acid (RNA), and environmental factors in cell differentiation; and Supporting Standard
- 5.D recognize that disruptions of the cell cycle lead to diseases such as cancer.

**Investigation 3: Prokaryotic and Eukaryotic Cells**

**Overview of the Investigation**

Students will be paired up and given a microscope. The students will then use cotton swabs to swab a cheek of one of the students, look at the cells under the microscope and draw the cells observed. The day before the group will prepare a.

*E.coli*agar plate for this day. The pair will then make a wet mount of the bacteria and look at it under the microscope and draw the structures. The students will then compare and contrast the different cells.

**Objectives**

Students will be able to:

- Describe the differences between prokaryotic, eukaryotic, and viruses

**Alignment with Texas Essential Knowledge and Skills (TEKS)**

This is investigation addresses the following TEKS:

- 4.C compare the structures of viruses to cells, describe viral reproduction, and describe the role of viruses in causing diseases such as human immunodeficiency virus (HIV) and influenza.

**Investigation 4: Exponential and Logarithmic Functions**

(Link for handout below)

**Overview of the Investigation **

The students will be working in groups and explore a situation with allowances. The students will try and come up with a formula for each situation. They can use excel or their graphing calculators to find a best fit line. With out knowing it, they will be working with logarithmic functions which is the inverse of exponential functions.

**Objectives**

Students will be able to:

- Perform simple manipulations of logarithmic expressions according to the rules
- Move back and forth between logarithmic and exponential forms
- Recognize that exponential and logarithmic functions are mutual inverses

**Alignment with Texas Essential Knowledge and Skills (TEKS)**

This investigation addresses the following TEKS:

- 1.B Use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution

- 1.E Create and use representations to organize, record, and communicate mathematical ideas
- 9.B interpret the meaning of the values of a and b in exponential functions of the form f(x) = ab^x in real-world problems
- 9.E Write, using technology, exponential functions that provide a reasonable fit to data and make predictions for real-world problems

**Handouts For Investigations:**

penny_investigation.docx | |

File Size: | 178 kb |

File Type: | docx |

investigation_3.docx | |

File Size: | 48 kb |

File Type: | docx |

investigation_4.docx | |

File Size: | 97 kb |

File Type: | docx |

**Benchmark Lessons**

During benchmark lessons, " students share their ideas, but, in addition, they build on the thinking of their classmates' responses, take up classmates' ideas, and work with these ideas explicitly to extend the line of thinking that emanates" (Staples and Colonis 2007).

**Benchmark Lesson 1: Exponential Equations**

(Link for handout below)

**Overview of the lesson**

After the student’s complete the Penny Investigation, they will be introduced to more detailed characteristics of exponential functions. During the lesson, the student’s will be exploring with different exponential functions and their graphs to determine how the “a” and the “b” in y=abx affects the graph. Students will choose their values based on restrictions given by the teacher. The students will then do a Think-Pair-Share in groups in order to compare answers and compare values. Once the Think-Pair-Share is complete, the class will be brought together to share student’s discoveries and why they are true.

**Objectives**

Students will be able to:

- Determine how different values of a and b affect the graph of y=ab^x
- Differentiate which specific range of values of a and b correspond with growth and decay

**Alignment with Texas Essential Knowledge and Skills (TEKS)**

This lesson addresses the following TEKS:

- 9.A determine the domain and range of exponential functions of the form f(x) = ab^x and represent the domain and range using inequalities;

- 9.B interpret the meaning of the values of a and b in exponential functions of the form f(x) = ab^x in real-world problems;
- 9.D graph exponential functions that model growth and decay and identify key features, including y-intercept

**Benchmark Lesson 2: Viruses**

**Overview of the lesson**

In groups, the members will imagine that they are part of a team of scientists assigned to stop a local weed epidemic by genetically engineering a virus that will target a local pest plant. Students should describe how such a way of using a virus, while useful in some ways, could create dangers to the environment. The students will be challenged to suggest ways of safeguarding against such dangers.

**Objectives**

Students will be able to:

- Describe a virus as an infectious organism that reproduces within the cells of an infected host.
- Explain a virus is not alive until it enters the cells of a living plant or animal.
- Describe virus contains genetic information wrapped in a protein coat.
- Understand that a virus mutates to ensures its own survival by making itself unrecognizable to immune systems and vaccines.

**Alignment with Texas Essential Knowledge and Skills (TEKS)**

This lesson addresses the following TEKS:

- 5.D recognize that disruptions of the cell cycle lead to diseases

**Benchmark Lesson 3: Lytic and Lysogenic Cycle**

(Link for handout below)

**Overview of the lesson**

In this lesson, the students look at two visuals on the computer: A Lytic cycle, and a Lysogenic cycle. They will be asked to compare these two cycles and to identify the role of the host cell in viral reproduction. After the students finish their assignment, they will come together in a student led review and discuss what they discovered in this activity.

**Objectives**

Students will be able to:

- Describe the process of how viruses infect host cells
- Explain how DNA replicates itself through mitosis in a virus cell

**Alignment with Texas Essential Knowledge and Skills (TEKS)**

This lesson addresses the following TEKS:

- 4.A compare and contrast prokaryotic and eukaryotic cells;
- 4.C compare the structures of viruses to cells, describe viral reproduction, and describe the role of viruses in causing diseases such as human immunodeficiency virus (HIV) and influenza.

**Benchmark Lesson 4: Exponential and Logarithmic Functions**

**Overview of the Lesson **

The students will be given an exponential function which models a cells population growth. They will be provided with the starting population and the population at the end of the experiment. Their task is to decide how long it took for the cell to reach that population. They will work in groups and do a think-pair-share in order to solve the problem. The students will then come together as a class and discuss what they discovered. Then an interactive lesson on exponential functions and their inverses will be conducted.

**Objectives**

Students will be able to:

- Differentiate which specific range of values of a and b correspond with growth and decay
- Understand how and why logarithmic and exponential functions are related

**Alignment with Texas Essential Knowledge and Skills (TEKS)**

This lesson addresses the following TEKS:

- 9.B interpret the meaning of the values of a and b in exponential functions of the form f(x) = ab^x in real-world problems;
- 9.D graph exponential functions that model growth and decay and identify key features, including y-intercept

**Benchmark Lesson 5: Ro Values**

**Overview of the Lesson**

The students will research Ro values of an illness of their choice and try to incorporate the value into exponential equations. They will up with there own population and time frame and explore what happens. They will then briefly present to the class what they learned.

**Objectives**

Students will be able to:

- Understand how the Ro value of an illness can help make the infectious rate of the virus visible

**Alignment with Texas Essential Knowledge and Skills (TEKS)**

This lesson addresses the following TEKS:

- 1.E communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate;
- 9.B interpret the meaning of the values of a and b in exponential functions of the form f(x) = abx in real-world problems;
- 9.C write exponential functions in the form f(x) = abx (where b is a rational number) to describe problems arising from mathematical and real-world situations, including growth and decay;
- 9.D graph exponential functions that model growth and decay and identify key features, including y-intercept
- 9.E write, using technology, exponential functions that provide a reasonable fit to data and make predictions for real-world problems.

**Handouts For Benchmark Lessons:**

exponential_graphs.png | |

File Size: | 86 kb |

File Type: | png |

benchmark_lesson_3.docx | |

File Size: | 69 kb |

File Type: | docx |

benchmark_lesson_4.docx | |

File Size: | 45 kb |

File Type: | docx |