The Highly Engaged Math Student

Subject Focus:  Algebra 2 & Geometry

October 29, 2009

2009 NCCTM State Conference

Presenters

Bryan Haywood: Randolph Early College High School

bhaywood@randolph.k12.nc.us

Scott Walker:  Randolph Early College High School

swalker@randolph.k12.nc.us

Presentation Description

Geometry and Algebra 2 hands-on activities, labs and projects that focuses on student engagement while building 21^st century skills.

21^st Century Skills

Critical Thinking

Problem Solving

Troubleshooting

Ability to Apply Knowledge

Mastery of Content Knowledge

Innovative/Creative Thinking

Communication Skills

Ability to Learn and Work in Teams

Relationship Building

Interpersonal/Social Skills

Work Ethic

Decision Making

Flexibility/Adaptability

Global Awareness

Cultural Competency

CEO Quotes

…”I look for someone who asks good questions…We can teach them the technical stuff, but we can’t teach them how to ask good questions—how to think.”   Clay Parker (BOC Edwards)

“I want people who can engage in good discussion—who can look me in the eye and have a give and take.  All of our work is done in teams.  You have to know how to work well with others.  But you also have to know how to engage customers—to find out what their needs are.  If you can’t engage others, then you won’t learn what you need to know.” Clay Parker (BOC Edwards)

“The challenge is this:  How do you do things that haven’t been done before, where you have to rethink or think anew?  It’s not incremental improvement any more.  The markets are changing too fast.”  Ellen Kumata  (Cambria Associates)

“Kids just out of school have an amazing lack of preparedness in general leadership skills and collaborative skills…..They lack the ability to influence.”  Mike Summers  (Global Talent Management at Dell)

…”We change what we do all the time.  I can guarantee the job I hire someone to do will change or may not exist in the future, so this is why adaptability and learning skills are more important than technical skills.”  Clay Parker (BOC Edwards)

“I say to my employees, if you try five things and get all five of them right, you may be failing.  If you try 10 things, and get eight of them right, you’re hero.  You’ll never be blamed for failing to reach a stretch goal, but you will be blamed for not trying.  One of the problems of a large company is risk aversion.  Our challenge is how to create an entrepreneurial culture in a larger organization.”  Mike Chander (Cisco)

“We are routinely surprised at the difficulty some young people have in communicating: verbal skills, written skills, presentation skills.  They have difficulty being clear and concise; it’s hard for them to create focus, energy, and passion around the points they want to make.  If you’re talking to an exec, the first thing you’ll get asked if you haven’t made it perfectly clear in the first 60 seconds of your presentation is, ‘What do you want me to take away from this meeting?’  They don’t know how to answer that question.”  Mike Summers (Dell)

“For businesses it’s no longer enough to create a product that’s reasonably priced and adequately functional.  It must also be beautiful, unique, and meaningful.”  Daniel Pink author of A Whole New Mind.

How would these CEO’s feel about my students?

What can I do to get my students ready to meet this expectation of the workplace?

Whiteboard Activity

This activity is designed to open the line of communication between students.  It can be used at the beginning of the semester to get students involved and encourage classroom talk.  It sets the tone that your class is not an ordinary math class with only direct instruction and mindless note-taking.  Your class is going to engage students and allow them to be an integral part of their education.

How the activity works:

1. Partner students up and have them arrange their chairs where one student can see the board and one cannot.
2. The student who cannot see the board will be the illustrator and the student who can see the picture will give a verbal description of the picture.
3. There are three tiers to this activity

• First, the speaker can see the illustration and give feedback to what is right or wrong but cannot use his/her hands to guide the person drawing.  In this section they can communicate to each other openly.
• Second, the student talking cannot see what is being drawn but they can still have an open line of communication
• Third, the student talking cannot see what is being drawn and the person drawing cannot talk at all.  They can only draw what they here without asking clarifying questions

If you use this activity at the beginning of the year and again at the end of the semester, you will hear the difference in the language the students use to talk about math.  For example, in Geometry if you were to draw a picture of a right triangle and mark some sides congruent, you would hear students say things like, “triangle, square in the corner, lines in middle of side, etc.”  At the end of the semester you would hear things like, “Draw a right triangle, mark the two legs as congruent, etc.”

Agree or Disagree

(Whole Class Assessment)

Description:   Each student is responsible for working at least one question on the board.  The entire class will receive the same base score.  At their seats, students can choose to disagree with the problem on the board and correct it for extra points.  This assessment will work best with topics like factoring, radicals, exponents in which the entire assessment will contain the same type of problems. 

Benefits:             -     Allows students to practice the 21^st century skills of TROUBLESHOOTING, MASTERY OF CONTENT KNOWLEDGE, FLEXIBILITY and the ABILITY TO WORK IN TEAMS

• Gives low-performing students an opportunity for a good test grade
• Allows students to do “test corrections” while they are taking the test
• Promotes teamwork instead of competition
• Challenges the students to prepare differently than they would for a normal assessment
• Easy to grade

Drawbacks:              -      Difficult for a student to get a very high score

• Students who are very slow workers may have trouble keeping up
• Assessment may not provide enough information to decide if a student has mastered the content

Steps:

1. Pass out the assessment to every student in the class.  Make sure that there is at least as many problems as there are students in the classroom.  Also, make an answer sheet for students to complete during the assessment.
2. Give the class 10 minutes to decide who will work each problem.  The student’s use of this time is CRITICAL!
3. Allow 5 minutes for students to work their problem at their seat and begin to work each test question in order.
4. One at a time, each student will come to the board (Smartboard works great) and work out each problem.  They must show all work.  At their seats, students can either agree with the answer on the board or disagree.  If they disagree, they can provide what they believe the answer is supposed to be.  They can earn extra points on their individual score by providing the correct answer.
5. After all of the problems have been worked out on the board.  The teacher grades the work to get the score for the class.  This will be the base score.
6. Add points to individual students’ scores for questions they corrected from mistakes made by their classmates.

Assessment Options

21^st Century Skills:  Making Difficult Decisions by weighing Trade-Off Costs, Flexibility, Working in Teams, Working Under Time Constraints/Pressure Situations

• Students are allowed to ask the teacher a question but the teacher will deduct point(s) from his/her score.  This can be beneficial for multi-part problems or on assessments that have very few questions.

• Give students the option of working with a partner or using their notes/book.

• Allow students to use a “cheat sheet” or note card.  This forces students to analyze the material and choose the most important topics from the unit.  In some cases, this option will result in some students studying more for a math test than ever before.

• Give students the option to take the test in multiple-choice format or short answer.  This works well if you use a test generator.

• When assessing in groups of 3 or more, assign groups during the previous class and allow them to work together on a practice test.  You can also give groups a list of topics so they can designate certain topics or problems to individual group members.

• Allow students to work together for only the first 10-15 minutes of the assessment.  This ensures that a student can’t just copy all of their classmates’ answers and they must focus on how to work the problems.

News Reporters

The following chart lists the career saves for Mariano Rivera, pitcher for the New York Yankees. (sports.yahoo.com)

1. What is the independent variable?  Why?______________________________________

____________________________________________________________________________

1. What is the dependent variable?  Why? ________________________________________

________________________________________________________________________________

1. What is the linear regression equation for the data? _______________________________

1. What is the correlation coefficient?  Describe the relationship between the variables.

________________________________________________________________________________

1. What is your interpretation of the slope (complete sentence)? __________________________

____________________________________________________________________________________

____________________________________________________________________________________

1. Use your prediction equation to predict how many saves Mariano Rivera will have in 2010.

_________________________________________________________________________

1. List some of the factors that could influence the validity of your answer for question 6.

_________________________________________________________________________________

1. Write a news report consisting of at least four sentences that uses the results that you found in your investigation.  When you are ready to give your report, raise your hand.

The following chart lists the number of Influenza-like illnesses reported in North Carolina from Sunday, August 23, 2009 through Saturday, October 10, 2009. (epi.state.nc.us)

1. What is the independent variable?  Why?______________________________________

____________________________________________________________________________

1. What is the dependent variable?  Why? ________________________________________

________________________________________________________________________________

1. What is the linear regression equation for the data? _______________________________

1. What is the correlation coefficient?  Describe the relationship between the variables.

________________________________________________________________________________

1. What is your interpretation of the slope (complete sentence)? __________________________

____________________________________________________________________________________

____________________________________________________________________________________

1. Use your prediction equation to predict how many influenza-like illnesses will be reported during the week of Christmas this year.

_____________________________________________________________________________________

1. List some of the factors that could influence the validity of your answer for question 6.

_________________________________________________________________________________

1. Write a news report consisting of at least four sentences that uses the results that you found in your investigation.  When you are ready to give your report, raise your hand.

Olympic Data

The following chart displays the winning Olympic times for the 100m dash since 1928 (www.databaseolympics.com).

Analyze the data in the chart.  Pay particular attention to patterns or gaps in the data, as well as any anomalies.  In your groups, formulate 10 critical thinking questions.  Some of your questions may be open-ended or not have exact answers.  You must include at least 2 questions pertaining to each of the following subjects:  mathematics, history, and science and technology.

After you finish your questions, pair up with another group and discuss the questions and possible answers.

Examples of good critical thinking questions:

• Do you think that the women will ever be faster than the men?  Why or why not?

• Why do you think that the United States won so many gold medals?

• What might have contributed to the women’s winning times increasing in 1992, 1996, and 2000?

• What do think the fastest winning times will be in the future? Justify your answer.

• What is the historical and political significance of the Olympics?

• Do you think that the historical results from other Olympic events will be similar to the 100m dash data?

• What technological breakthroughs and inventions can you think of that have contributed to the results of the winning times?

• Do you think that the Olympics are as significant to today’s culture as they were to past generations?  What role do you think the Olympics will play to future generations?

• Will there ever be a significant increase in the winning Olympic times?  Why or why not?

• What future scientific developments could influence the results of the winning times in future Olympics or cause controversy?

Name: __________________________

Name: __________________________

Name: __________________________

Name: __________________________

RECHS Surveillance

Your group has been placed in charge of maintaining surveillance at RECHS.  Your first objective is to minimize the number of cameras used to survey the second floor of the VT building.  In order to accomplish this task efficiently, follow the rubric below:

• Make a blueprint drawing to scale of the floor plan (each block on the graph paper should represent 2 square feet)
• Label on the blueprint the position of the cameras (cameras can only see 30 ft. and have a sweeping capability of 90 degrees).

You have discovered that items are missing from classrooms.  You go to the video to find that during the time of the crime, the cameras were not recording.  To help find the criminals you need to:

• Install motion detectors along the base of the floors hidden by the floor moldings.  How many feet of wiring will you need?
• You then need to dust the floor for footprints with no space being left undusted.  How many square feet needs to be dusted?
• Next you must dust the walls for fingerprints.  How many square feet needs to be dusted?

You have still yet to find enough evidence to pinpoint a criminal.  A week later you get a call that the VT building is being vandalized.  When you arrive at the location, it appears that the criminal is still in the building.  You need to force the perpetrator out of the building using tear gas.

• How many cans of tear gas will be needed to ensure the violator will be forced out of the building (each can of tear gas covers 200 cubic feet)?

Answer the preceding questions on the blueprint.  Make sure to put all appropriate units for measurements.

Quality Control Experiment

Honors Algebra 2

Group Members:              ______________________________________________

                                          ______________________________________________

                                          ______________________________________________

                                          ______________________________________________

Purpose:  The purpose of this experiment is to investigate the percent error between serving sizes listed in the nutritional guides of fast food restaurants and their actual weights.  This activity will use three different samples of two different menu items from the McDonalds on Hwy 64 in Asheboro, NC.  The items will be weighed and compared to the nutritional guidelines listed on McDonalds.com.  Also, conclusions will be drawn and suggestions will be made on quality control issues that could minimize discrepancies in weights and nutritional value.               

Collecting Data:             

List all Nutritional Information for Your Menu Items from McDonalds.com:

Item #1: ____________________________              Item #2: ____________________________

Calories: __________________________              Calories: __________________________             

Calories from Fat: ____________________              Calories from Fat: ____________________

Total Fat: _________________________              Total Fat: _________________________             

Saturated Fat: ______________________              Saturated Fat: ______________________

Cholesterol: _______________________              Cholesterol: _______________________                           

Sodium: ____________________________              Sodium: ____________________________

Carbohydrates: _____________________              Carbohydrates: _____________________             

Fiber: ______________________________              Fiber: ______________________________

Sugar: ____________________________              Sugar: ____________________________             

Protein: _____________________________              Protein: _____________________________

Reporting Your Findings

Write a lab report (one per group) that explains the experiment and the results.  Assume the audience to be the general public.  Be sure to include your opinion on whether you think the percent error was acceptable.  Also include any results that you found particularly interesting or surprising. You also need to specify whether the experiment is a viable way to generalize these findings to other restaurants, both McDonalds and other chains. Please support your theories.  You should also offer suggestions on how restaurants can minimize discrepancies between the nutritional information listed in their guide and what is actually served in the restaurant.  Feel free to investigate the Food and Drug Administration’s website at www.fda.gov to see how food labeling and nutritional guidelines are regulated to add merit to your report.  Your report will be shared with the school and possibly the management at McDonalds restaurants in Asheboro.    A list of things that you may consider doing to strengthen the quality of your report includes, but is not limited to:

• interviewing workers or managers at McDonalds or other local fast-food restaurants
• visiting other restaurants such as Chick-fil-a, Taco Bell, Arby’s, Wendy’s, etc… to observe their practices, interview workers about how they guarantee the quality of their products, or conduct further experiments

Quality Control Rubric

___________   - Purpose (15 points)

___________  - Hypotheses (10 points)

___________ - Data (% error 15, weights 10)

___________- Analysis (25 points)

___________ - How can McDonalds improve? (12.5 pts)

__________ - Was the experiment valid? (12.5 points)

__________ - Total

Picture Enlargement Project

• Choose a picture to enlarge.  Its dimensions should be no larger than 3 inches by 5 inches.
• Use a ruler and a pencil to draw horizontal and vertical lines every quarter inch so that it will separate your picture into congruent squares.
• Find an appropriate scale factor that will maximize the area used on your enlargement.  Use standard sized paper or poster board.
• Draw evenly spaced vertical and horizontal lines on your enlargement with a pencil.  Make sure that you have the same number of squares as your original picture.
• Carefully copy the content from each original box into the corresponding box on the enlargement.
• Make sure that you color your project.  It will be graded on appearance and neatness.
• Make sure to fill out your part of the rubric below before turning in.

Name: _______________________________________________________

Dimension of original picture (don’t forget units):___________________

Area of original picture: ________________________________________

Dimension of enlargement (don’t forget units): _____________________

Area of enlargement:___________________________________________

Scale factor from original to enlargement:_________________________

Ratio of Areas:________________________________________________

Rubric

Accuracy – 50 points                                                                      ___________

Neatness and Appearance – 20 points                                          ___________

Color – 10 points                                                                                    ___________

Answering Questions – 20 points                                          ___________

Total                                                                                                                ___________                                                                                                                             

How Much is That Doggy in the Window?

              You are walking by a pet store one day and discover an adorable puppy.  You are also interested in getting a cat and a fish as well.  Unfortunately, the person who is working there is new and can’t tell you how much each one costs.  However, he does have store receipts that show how many of each animal previous customers purchased and their total bill.  Being a brilliant mathematician you quickly set up a system of three equations and begin to solve it.  After a few minutes you discovered the individual prices for a puppy, cat, and a fish.

Here are the receipts.  Use your favorite method to solve the system.

Customer 1:  3p + 2c+ 5f = $292.50

Customer 2:   p + 3c +10f = $270.00

Customer 3:  5p + c + 2f = $320.00

Write the prices you found here:

Puppies =                                          Cats =                                          Fish =

Now it’s your turn!  Come up with your own business that sells three items at three different prices.  Make up a name for your business and a poster advertising your products.  Include on your poster 3 receipts that show different amounts of items and the total cost.  DO NOT INCLUDE THE INDIVIDUAL PRICES ON YOUR POSTER!              On separate sheets of paper, one group member should solve the system using Cramer’s Rule.  A different group member should solve the system using inverse matrices.  The third member of your group should solve the system using elimination.  For extra credit, your group can attempt to solve the system by using augmented matrices and elementary row operations found on page 237 of your textbook.

Business Partners: _______________________________________________________________

Business Name: __________________________________________________________________

Grading Rubric for the group assignment:

Correct answer to the Pet Problem with work shown (20) _____________________

Poster with advertisement of business and 3 receipts (20) ____________________

Solution to system by Cramer’s rule                                           (20) ____________________

Solution to system using Inverse Matrices                            (20) ____________________

Solution to system using Elimination                                           (20) _____________________

Solution to system using Augmented Matrices (10 pts extra credit) ___________

     Total ____________________

Estimating Heights Lab

• In this lab, you will be responsible for estimating the height of 4 different objects outside that you are unable to measure the actual height.
• For each object, you will need to use the three different methods shown.
• For each object, you will need to choose a person to do each of the jobs listed on that page. (Each person in the group will need to do each job once).  The job titles and their responsibilities are on the back of this sheet.
• Each group will be responsible for observing another group perform the operations on object 1.  During this observation you must not talk to the members of the other group.
• In conclusion, you will need to use the data gathered from object 1 to explore further.  (on the large 11 X 17 paper you will do 1-5 for all 3 methods on object 1)

1. During this exploration, you will need to make a scale drawing to model each method of estimation.
2. Briefly explain what the experiment was. (for all three methods)
3. How is your estimation valid?
4. Why does this method work mathematically?
5. What obstacles were you faced with and how did they affect the results of this lab?

• Compare and Contrast the results of the three methods for the exploration of the light pole.
• Fill out the chart below before leaving the classroom.  This data will be vital in your calculations to follow.  Gather the data needed for each calculation while outside them return to the classroom to do your calculation.

Facilitator – This person is overseeing the data that is being collected.  You should be managing the time spent on each method as well as managing the groups’ ability to stay on task.  You are also a helping hand to whoever may need it.

Reference – This is the person you will be using to make reference to height, eye height, shadow length, holding the tangent gauge, distance to object, looking in the mirror and distance to the mirror.

Surveyor – This person should be responsible for finding all the distances.

Recorder – This person should be recording the data that is collected during the experiment.  This person will also do the write up on the object and calculations.

Object #1 __________________

Facilitator: _______________________

Reference: _______________________

Surveyor: ________________________

Recorder: ________________________

Show ALL work and how you set the problem up

Tangent Height Gauge:

Distance from reference to object ____________

Estimated height _____________

Shadows:

Reference shadow _______________

Object’s shadow ________________

Estimated height _______________

Mirror:

Distance from reference to mirror _______________

Distance from mirror to object _______________

Estimated height _______________

Promote Systems of Linear Inequalities with Real-World Problems

State Objective from Algebra 2 Curriculum

2.10 Use systems of 2 or more equations or inequalities to model and solve problems; justify results.  Solve using tables, graphs, matrix operations, and algebraic properties.

Modeling a Product-Mix Problem:  The Lego Factory

The Problem

Suppose that a factory manufactures only tables and chairs and that the profit on one chair is $15 and on one table is $20.  Each chair requires one large piece of stock and two small pieces of stock, that is, one large Lego or other construction block and two small ones.  Each table requires two large and two small pieces of stock.  Figure 1 shows a table and a chair.  Finally, suppose that you have only six large and eight small pieces of stock.  How many chairs and how many tables should you build to maximize profit?

A concrete exploration

The student activity works best if students actually have construction blocks with which to build “tables” and “chairs.”  It is important to allow students an opportunity to see what they can build using the available materials and to determine the profit for each possibility.  Later, during a more abstract exploration of this problem, the abstract concepts can be easily linked to this more concrete example.

              Students might compete in pairs or in small groups to obtain the optimal solution.  Students will quickly see that only four logical possibilities for the optimal solution exist:

(1) Four chairs                                                                        profit of $60

(2) Three chairs and one table                                          profit of $65

(3) Two chairs and two tables                                          profit of $70

(4) Three tables                                                                      profit of $6

They should have no trouble determining the profit for each possibility.  It will then be evident that building 2 chairs and two tables will maximize profit.

An abstract exploration

After students complete their concrete exploration of the Lego problem, it can be linked to a more abstract one that uses the terminology of mathematics and operations research.

The steps in the abstract exploration involve:

1. Defining variables.
2. Identifying any constraints, which will result in a system of inequalities.
3. Graphing the system of constraints to locate a feasible region.
4. Writing the objective function, which is will find the maximum or minimum.
5. Identifying the vertices.
6. Determining which solution within the feasible region is the optimal solution.

For our problem, the variables should be

C = the number of chairs built

T = the number of tables built

Constraints

1C + 2T < 6  (because we can use a maximum of 6 large legos)

2C + 2T < 8  (because we can use a maximum of 8 small legos)

Graph  - Use graphing calculator to assist

Objective Function (for profit)

P = 15C + 20T

Vertices – Use graphing calculator to find these points

(0,0) – profit of $0

(0,3) – profit of $60

(4,0) – profit of $60

(2,2) – profit of $70

Optimal solution

Making two tables and two chairs from the resources given would maximize the profit.  The amount of profit would be $20.