this lesson is adapted from “the thousand dollar...

Marianne Reid Week 9 Culminating Project

The culminating project should include the following:

A five-day plan Specific MMC content expectation(s) as a foundation Outcomes for the mini-unit An Introduction or "Hook" to get students interested in the mini-unit's topic At least one full lesson plan using the Lesson Plan Template (you may use one already developed for an

Integration Assignment) Activities which could be from the course content modules or the social network to support the mini-

unit's instructional focus At least one activity using a graphing calculator or some other technology (e.g. online resources) Problems and solution techniques incorporating multiple representations (verbal, symbolic, tabular, and

graphical) Both formative and summative assessments

My mini-unit will incorporate Algebra for All activities into the Connected Mathematics 2 Grade 8 unit called “Growing, Growing, Growing” to teach students about exponential functions. I will also use some of the assessment resources from Connected Mathematics 2, such as the check-up and the partner quiz.

Formative assessments will include observing student interactions in the classroom, reviewing student work and journal entries, and observing students as they work on the investigations. I will also use a thinking routine called “Noticings & Wonderings” to help the students collect their thoughts and formulate questions about the unit. This will also give me some insight into my students’ thinking processes.

Instruct the students to compile a list of their “Noticings and Wonderings”. They will add to this throughout the unit. The students should make a list of all of the mathematical information and relationships they notice throughout this unit, along with listing anything they wonder about after completing each activity.

Their “noticings” may include:• The shapes or trends of the graphs.• Relationships between the functions, tables, and graphs.• Key concepts from each activity.• Anything they think is important.

Their “wonderings” may include:• I wonder what will happen if …• I wonder what this word means …• I wonder if this pattern will continue …• Any questions they have about the lesson or activities.

Assessments such as the check up, partner quiz, and question bank from “Growing, Growing, Growing” will be used as summative assessments. These assessments will give me an idea of what the students understand, and where I may need to re-teach the class, or intervene with specific students.

This term, I am going to give more check-ups and quizzes at various points throughout the unit in order to get a better gauge of the students’ abilities and understanding.

Strands: A: Algebra and Functions L: Quantitative Literacy and Logic

Standards: A2: Functions A3: Families of Functions L1: Reasoning about Numbers, Systems, AND Quantitative Situations

Topics: A2.1 – Definitions, Representations, and Attributes of Functions A2.3 – Representations of Functions A3.2 – Exponential and Logarithmic Functions L1.2 – Representations and Relationships

Content Expectations: A2.1.3 Represent functions in symbols, graphs, tables, diagrams, or words and translate among representations. A2.3.1 Identify a function as a member of a family of functions based on its symbolic or graphical representation;

recognize that different families of functions have different asymptotic behavior. A2.3.2 Describe the tabular pattern associated with functions having a constant rate of change (linear); or variable

rates of change. A3.2.1 Write the symbolic form and sketch the graph of an exponential function given appropriate information. A.3.2.5 Relate exponential and logarithmic functions to real phenomena, including half-life and doubling time. L1.2.4 Organize and summarize a data set in a table, plot, chart, or spreadsheet; find patterns in a display of data;

understand and critique data displays in the media

Mini-Unit Outcome:

Students will be able to recognize linear and exponential functions represented in tables and graphs. Students will be able to plot different sets of data on the same graph and draw conclusions from the data. Students will be able to make decisions based on data presented in a graphical or tabular format. Students will be able to describe the differences in the graphs of linear and exponential growth and decay functions. Students will be able to sketch the graphs of exponential growth and decay functions. Students will be able to recognize the graphs of linear, exponential growth and exponential decay functions. Students will be able to give real-world examples of exponential growth and decay. Students will be able to use an Excel spreadsheet to enter data, create a scatter plot, and fit a trendline to the data.

Day 1: Introduction to non-linear functions

Introduce the unit on exponential functions by describing the following scenario and asking the students which option they would choose.

You are walking down the street and you see an older gentleman who needs assistance. You stop to lend him a hand, and he is very grateful. This is your lucky day. Unbeknownst to you, this man is a wealthy business man who is doing philanthropic work in your city. Being unfamiliar with the area, he offers you a job as his personal assistant for the next two weeks. He offers to pay you one of the following salaries:

Option 1: You earn $500 each day you work.

Option 2: You earn $100 on the first day, $200 on the second day, $300 on the third day, and so on with your pay increasing by $100 each day you work.

Option 3: You earn $1 on the first day, $2 on the second day, $4 on the third day, and so on with your pay doubling each day you work for 14 days.

Instruct the students to record their choice in their math journals. Also, have the students vote by a show of hands and write the counts for each option on the board. The students will explore these options in today’s lesson.

Introduce the lesson “Choose Wisely”, which focuses on the three scenarios above. One is a linear relationship, one is a quadratic relationship, and one is an exponential relationship. The students will begin by completing a table and graphing these functions.

At the end of the lesson, pose the question again and ask the students which option they would choose now, and whether this is a change from their original choice.

Lesson Plan: Choose Wisely – Algebra 1

This lesson is adapted from “The Thousand Dollar Proposition” lesson plan on the Algebra for All social network, posted by Joan Wiersma on March 5, 2010 at 1:35pm in the Project and Lesson Sharing AreaThe link to this lesson is: http://a4a.learnport.org/forum/topics/the-thousand-dollarKey Ideas: Linear / Non-linear Models

Strands: A: Algebra and Functions L: Quantitative Literacy and Logic

Standards: A2: Functions L1: Reasoning about Numbers, Systems, AND Quantitative Situations

Topics: A2.1 – Definitions, Representations, and Attributes of Functions A2.3 – Representations of Function L1.2 – Representations and Relationships

http://a4a.learnport.org/forum/topics/the-thousand-dollar

http://a4a.learnport.org/forum/categories/project-and-lesson-sharing/listForCategory

http://a4a.learnport.org/profile/JoanWiersma

Content Expectations:

A2.1.3 – Represent functions in symbols, graphs, tables, diagrams, or words and translate among representations. A2.3.1 – Identify a function as a member of a family of functions based on its symbolic or graphical representation;

recognize that different families of functions have different asymptotic behavior. A2.3.2 – Describe the tabular pattern associated with functions having a constant rate of change (linear); or variable

rates of change. L1.2.4 – Organize and summarize a data set in a table, plot, chart, or spreadsheet; find patterns in a display of data;

understand and critique data displays in the media

Lesson Outcome: What will the students be able to do at the end of the lesson? Students will be able to recognize linear and non-linear functions represented in tables and graphs. Students will be able to plot different sets of data on the same graph and draw conclusions from the data. Students will be able to make decisions based on data presented in a graphical or tabular format. Students will be able to describe the differences in the graphs of linear and non-linear functions.

Materials and Resources: List all supplies and resources to be used in the lesson, including texts, computers, calculators, software, web-based resources, manipulatives, and art supplies.

“Choose Wisely” activity sheet (one for each student) Calculators Computer with Excel (optional) Excel spreadsheet template

Procedures: Describe the anticipatory set or “hook” to start the lesson, sample questions to students, and activities and tasks to be used in the lesson. The flow of the lesson, step by step, should be described, particularly in relation to what students will be doing.

Students will be presented with a hypothetical scenario and asked to choose between three different payment options. The payment options involve linear, polynomial of degree two and exponential functions. Students need to analyze the situations both numerically (making a table) and graphically. At the end of the activity, students will identify which type of equations (linear, polynomial or exponential) is the best model for each option.

Teacher Prompts:

Before: The teacher will read through the scenario with the students. Ask each student to write down his or her choice prior to doing the activity.

During: Monitor students while they are working, the teacher will ask leading questions as necessary. After: The teacher will be able to informally assess students’ knowledge of linear models and their ability to

distinguish between linear and non-linear models.

What Comes Next?

This is an introductory lesson to non-linear functions. Follow with additional activities and lessons to teach exponential growth and decay and other types of non-linear functions.

Choose Wisely Activity Sheet

You are walking down the street and you see an older gentleman who needs assistance. You stop to lend him a hand, and he is very grateful. This is your lucky day. Unbeknownst to you, this man is a wealthy business man who is doing philanthropic work in your city. Being unfamiliar with the area, he offers you a job as his personal assistant for the next two weeks. He offers to pay you one of the following salaries:

Option 1: You earn $500 each day you work. Option 2: You earn $100 on the first day, $200 on the second day, $300 on the third day, and so on with

your pay increasing by $100 each day you work. Option 3: You earn $1 on the first day, $2 on the second day, $4 on the third day, and so on with your

pay doubling each day you work for 14 days.

Comparing the Options

1. Compute the daily earnings and the cumulative earnings for 14 days and record them in the table below.

Number of Days

Option 1: Earnings per

day

Option 1:Cumulative

earnings

Option 2:Earnings per

day

Option 2:Cumulative

earnings

Option 3:Earnings per day

Option 3:Cumulative

earnings1

2

3

4

5

6

7

8

9

10

11

12

13

14

2. Graph the data on the grid below. Use a different color for each option. Put the number of days on the x-axis (horizontal axis). This is the independent variable. Put the total earnings on the y-axis (vertical axis). This is the dependent variable.

TOTA

L EA

RNIN

GS

1000

2000

5000

4000

6000

7000

8000

9000

3000

0 5 10 15 20 25Number of Days

Analyze the graphs.

a. Explain what these graphs show you about your options.

b. Explain the shape of each graph. What about the situation leads to the shape?

3. After looking at the data and analyzing the graphs, do you still think you chose the best option? Explain.

4. Determine the best fit equation for each option from the following choices:

Linear: y=mx+b where m=slope and b = y-intercept

Polynomial of degree 2: y=50 x2+50 x∨ y=50 x (x+1) Exponential growth: y=2x – 1Option 1

a. Decide whether option 1 is linear, and if so, find the slope and y-intercept and write the linear equation for option 1.

b. If option 1 is not linear, then pick the best fit equation from the above choices. Try a few points in the non-linear equations given to determine which equation works best for the option 1 data.

c. Use your equation to find the amount you will earn if you choose option 1 and you work for two weeks (14 days). Compare this with your graph.

d. Use the equation to find the amount you would earn if you worked for 3 weeks (21 days).

Option 2





Option 3





Assessment: How will you determine if the students have achieved the learning outcome(s).

Assessment will be informal, by observing the students as they interpret the data, asking questions and listening to the students’ questions and reasoning, examining students’ activity sheets, and by discussing their noticings and wonderings about the activity (see below).

Day 2: Computer lab – Excel extension for the “Choose Wisely” lesson.

Take students to the computer lab and work with them to create an Excel spreadsheet. Go through the steps with the students and have them follow along on their own computers, or provide written instructions and help students individually.

Attachments:

See the “21 days” worksheet in ChooseWiselyExtension.xlsx

Students will: Create an Excel spreadsheet, with seven columns. Label the columns as:

1. Day2. Option 1: Earnings per Day3. Option 1: Cumulative Earnings4. Option 2: Earnings per Day5. Option 2: Cumulative Earnings6. Option 3: Earnings per Day7. Option 3: Cumulative Earnings

Enter data for Day 1 Earnings per Day for each option. Use formulas to compute the data for days 2 through 21. The teacher will guide the students in creating

the formulas. Create a scatter chart in Excel showing all three options. The teacher will guide the students in creating

the chart. Select each data set in the graph, right click somewhere on the line, and choose “Add trendline” Experiment with the “Trend/Regression Type” options to find the best fit for each data set.

Students will learn how to do the following in Excel: Create a table of data using a formula (students will need to come up with a Now-Next formula to

populate the data in the Excel table). Insert a scatter chart and let Excel graph each option. Use Excel to insert a trendline.

Instruct the students to add to their lists of their “Noticings and Wonderings”. After each activity, they should add to their list of noticings, and look to see whether any of their wonderings or questions have been answered. What new things do they notice after completing this activity? What do they understand now that they did not before? What are some new questions or wonderings they have?

Day 2: Classroom – Lesson on exponential growth and decay.

Introduce formulas and example problems for exponential growth and decay. Refer to the link to Regents Exam Prep Center Exponential Growth and Decay lesson:http://www.regentsprep.org/Regents/math/ALGEBRA/AE7/ExpDecayL.htm

Instruct the students work in groups of two to solve the applied exponential growth and decay problems. The link below contains links to the lesson, practice sheets, and a lab.http://www.regentsprep.org/Regents/math/ALGEBRA/AE7/indexAE7.htm

Day 3: Review exponential growth and decay. Follow with the paper folding activity, the “M&M” investigation, and the rhino activity.

Discuss each of the applied exponential growth and decay problems as a class after the student groups have worked out the problems. Ask for volunteers to put the solutions to the problems on the board.

Formative assessment – Assess the students understanding about exponential growth by having them complete the paper folding activity, the “M&M” investigation, and the rhino activity.

Paper folding activity – part 1: The students will fold a sheet of paper in half and record the number of rectangles. Continue folding the paper in half and recording the number of rectangle until it becomes too hard to fold the paper. Plot the number of folds on the x-axis and the number of rectangles on the y-axis. Discuss the shape of the graph.

Paper folding activity – part 2: The students will fold an 8.5 by 11" sheet of paper in half and calculate the area of one rectangle (8.5 x 5.5"). Record the area in a table. Continue folding the paper in half and recording the area of a rectangle. Plot the number of folds on the x-axis and the area of the rectangle on the y-axis. Discuss the shape of the graph. Compare this graph with the graph in part 1.

“M&M” investigation – Investigate exponential decay using M&M' or Skittles. Students should work in pairs to collect data, but they should complete the activity sheets individually.

African Black Rhino population activity – look at the decrease in the black rhino population and make predictions.

Day 4: Review and assessment

Review and discuss the previous day’s activities, the paper folding activity, the “M&M” investigation, and the rhino activity.

Instruct the students to add to their lists of their “Noticings and Wonderings”. After each activity, they should add to their list of noticings, and look to see whether any of their wonderings or questions have been answered. What new things do they notice after completing this activity? What do they understand now that they did not before? What are some new questions or wonderings they have? What do the paper folding activity, the “M&M” investigation, and the rhino activity have in common? How are the graphs similar, and how are they different.

http://www.regentsprep.org/Regents/math/ALGEBRA/AE7/indexAE7.htm

http://www.regentsprep.org/Regents/math/ALGEBRA/AE7/ExpDecayL.htm

Summative assessment – the students will work on problems from the “Check-up” and “Question Bank” assessments from “Growing, Growing, Growing” in the Connected Mathematics 2 Grade 8 curriculum.

I am looking for the students’ ability to:

Recognize situations in which one variable is an exponential function of another variable Recognize the connections between exponential equations and growth patterns in tables and

graphs of those equations Compare exponential and linear relationships

Day 5: “How to Survive a Zombie Attack” investigation

Investigate exponential growth in a hypothetical zombie attack situation. Walk around and observe the students working on the activity to determine what they know, what questions they have, and where they are still struggling.

How to Survive a Zombie Attack (understanding exponential growth and decay). - Algebra for All

Instruct the students to add to their lists of their “Noticings and Wonderings” after completing this activity.

Have any of their wonderings or questions been answered? Did they notice anything new? What do they understand now that they did not before? What are some new questions or wonderings they have? What does the “How to Survive a Zombie Attack” investigation have in common with the previous

activities and investigations? What is different about this activity?

Web resources:

“The Thousand Dollar Proposition” lesson plan on the Algebra for All social network:


Regents Exam Prep Center Exponential Growth and Decay lesson:


Regents Exam Prep Center Exponential Growth and Decay lesson, practice sheets, and lab.


PBS Paper Folding, Rhinos, and M&M's® (Exponential Models) investigations:

http://www.pbs.org/teachers/mathline/lessonplans/hsmp/rhinos/rhinos_procedure.shtm

Algebra for All investigation – How to Survive a Zombie attack:

How to Survive a Zombie Attack (understanding exponential growth and decay). - Algebra for All

http://a4a.learnport.org/forum/topics/how-to-survive-a-zombie-attack

http://www.pbs.org/teachers/mathline/lessonplans/hsmp/rhinos/rhinos_procedure.shtm




http://a4a.learnport.org/forum/topics/how-to-survive-a-zombie-attack

Noticing and Wondering article:

http://mathforum.org/articles/communicator2010.html

Other resources used:

Connected Mathematics 2 Grade 8

Mathematics BackgroundThe basic goal in Growing, Growing, Growing is for students to learn to recognize situations, data patterns, and graphs that are modeled by exponential equations and to use tables, graphs, and equations to answer questions about exponential patterns. This unit is designed to introduce the topic and to give students a sound, intuitive foundation on which to build later.

Investigation 1: Exponential GrowthIn Investigation 1, students explore situations that involve repeated doubling, tripling, and quadrupling. Students are introduced to one of the essential features of many exponential patterns: rapid growth.Students make and study tables and graphs for exponential situations, describe the patterns they see, and write equations for them, looking for a general form of an exponential equation. Students also compare and contrast linear and exponential patterns of growth.

Investigation 2: Examining Growth PatternsInvestigation 2 focuses on exponential relationships with y-intercepts greater than 1. The standard form of an exponential equation isy=a ∙bx. Whenx=0, the equation becomes y=a since b0= 1. Thus a, the coefficient of the exponential term, generally indicates the initial value of the exponentially growing quantity. This initial value is the y-value corresponding to x=0, or the y-intercept. Each problem in the investigation presents information about an exponential pattern in a different form—in a verbal description, in an equation, and as a graph—helping students develop flexibility in moving among representations.

Investigation 3: Growth Factors and Growth RatesIn Investigation 3, students study non-whole-number growth factors other than 1 and relate these growth factors to growth rates. As an example, consider money invested at 6% annual interest. To find the amount of money for a given year, multiply the amount from the previous year by 1.06.The growth factor in this case is 1.06, while the growth rate is 6% (or 0.06). Students also explore how the growth rate and the initial value affect the growth pattern.

Investigation 4: Exponential DecayInvestigation 4 introduces students to exponential decay—patterns of change that exhibit successive, non-constant decreases rather than increases. These decreasing relationships are generated by repeated multiplication by factors between 0 and 1, called decay factors. Strategies for finding decay factors and initial population and for representing decay patterns are similar to those used for exponential growth patterns.

Investigation 5: Patterns With ExponentsInvestigation 5 develops rules for operating with exponents. Students examine patterns among the ones digits of powers and use these patterns to predict ones digits for powers that would be tedious to find directly. Then, they look for relationships among numbers written in exponential form. This leads to the rules for operating on numerical expressions with exponents. Finally, students use graphing calculators to study the effects of the values of a and b on the graph ofy=a(bx ).

http://mathforum.org/articles/communicator2010.html

this lesson is adapted from “the thousand dollar...

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