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Practicing What We Preach: Demonstrating Effective

Cooperative Learning PracticesAnnette Leopard and Karen Wells, Monroe Community College

34th Annual AMATYC ConferenceNovember 21, 2008

Washington, DC

This workshop is an expanded version of a workshop presented at the Seaway Section, MAA and NYSMATYC Region 1 2007 Fall Joint Conference, October 20, 2007, Monroe Community College, Rochester, NY.

Annette Leopard and Karen Wells, Monroe Community College, November 21, 2008

Table of Contents

page

Background Notes 4

Cooperative Learning Methods Descriptions

Round Robin 5

Numbered Heads 6

Jigsaw 7

Think/Pair/Share 8

Team/Pair/Solo 9

Three Minute Review 10

Sample Classroom Activities

Round Robin 11

Round Robin Activity for Calculus 12

Numbered Heads Activities 17

Jigsaw for Review – Solving Systems of Linear Equations 18

Jigsaw for Learning Pôlya’s Problem Solving Approach 19

Jigsaw Activity for Finding the Greatest Common Divisor and Least Common Multiple

21

Jigsaw Activity for Calculus – Chain Rule 23

Jigsaw for Learning to Find the Greatest Common Divisor 36

Think/Pair/Share Activities 40

Team/Pair/Solo for Discovery Rules of Exponents 41

Team/Solo Group Quiz on Optimization for Calculus 49

Three Minute Review Sample Activities 54


Group Discovery Activities

Cooperative group activity to discover the rules or tests for divisibility 55

Math for Elementary School Teachers Exploration activity – triangles and angles 57

Group Exploration Activity to Discover the Rules for Corresponding Angles 58

Cooperative Learning – Flash Cards 60

References 61

Background Notes forPracticing What We Preach: Demonstrating Effective Cooperative Learning Practices

Annette Leopard and Karen Wells, Monroe Community College34th Annual AMATYC Conference

November 21, 2008Washington, D.C.

Cooperative Learning – Our Working DefinitionCooperative learning is a kind of collaborative or peer learning which provides structured tasks for small groups characterized by both interdependence among group members and individual accountability.

Cooperative Learning – Why?Collaborative/Cooperative Learning is the first strategy listed in the Active Student Learning section of AMATYC’s Beyond Crossroads Chapter 7 on Instruction that calls on faculty to ‘use a variety of teaching strategies that reflect the results of research to enhance student learning.’ (p. 51) It is not our intention here to review the research and literature on cooperative learning in general and college mathematics instruction in particular. The Mathematics Association of America (MAA) through its Project CLUME (Cooperative Learning in Undergraduate Mathematics Education) produced two MAA Notes issues, numbers 44 and 55, which are an excellent place to begin. Instead, we want to list some of the claims about the benefits of this approach supported both in the literature and our experience with using cooperative learning in our mathematics classrooms.

Increased academic performance Deeper understanding of course materials Increased retention Increased problem solving ability Increased ability to learn independently Increased student satisfaction More positive attitudes about mathematics Increased self-confidence in the ability to do mathematics Increased communication skills


Improved social skills Improved race relations

Characteristics of Cooperative Learning1. Interdependence – Group members must be responsible for each other’s learning.2. Individual Accountability – Students must be held accountable for their contribution to the

group effort; the principal component of a student’s evaluation must be his or her own individual performance.

3. Structured Learning Activities – Students must be provided with clear directions for procedures and criteria for completion of the group work.

Round Robin Brainstorming

1. Students are divided into small groups ( no more than 6).2. One person is chosen to be the recorder. This person will write down everyone’s responses

whether he/she agrees with them.3. A question that has multiple answers is given to the students to think about by themselves for a

few minutes. Then going around the circle (starting with the person sitting next to the recorder) each person gives a response to the question. This procedure continues until all possibilities are exhausted or until the instructor calls time.

4. The teacher may also give each group a set of problems to solve simultaneously that involve several steps. The first student writes down the answer to the first step and passes the problem on to the next student who looks over the answer and then solves the second part. This procedure continues until the whole problem is solved.

5. The group may allow a student to pass on any round but should only allow them to do it once. Otherwise the person may opt to pass on all the rounds, thus not contributing to the group.

Advantages:

1. Generates many examples and ideas.2. Provides an opportunity for students to explore new ideas.

Role of the instructor:

1. Form groups and establish ground rules. 2. Provide the question or problem to solve.3. At the end of the activity the teacher may collect the group’s work for evaluation or call for

groups to share with the class.


Numbered Heads

1. Students are divided into small groups and within their individual groups number themselves from 1to the number of people in the group. Groups need to be the same size otherwise someone within the group will need to have two numbers.

2. The teacher gives each group a problem, issue or question to think about solve, or discuss together.

3. The students are told ahead of time that one number will be called at the end of the group time and that person will be asked to give their group’s response. Therefore everyone in the group must be able to discuss their answer.

4. The teacher calls out a number and on each team the student whose number is called must give their response either verbally or in writing.

Advantages:

1. Promotes positive effect interpersonal relationships.2. Enhances motivation towards learning because everyone is held accountable.


1. Form groups and establish ground rules. 2. Provide the question or problem to solve.3. Choose the number which determines who will give the group’s response. 4. Evaluate the responses according to some criterion.


Jigsaw

The Jigsaw method can be used as a review or teaching strategy. Students begin by dividing into small groups (3 – 6). Each member of the group is assigned one part of the task to become an “expert” in. To learn or review their assigned task, “expert groups” are formed for each task. Within their “experts groups” the students learn or review the material and develop teaching strategies together to use when they go back to their original group. This part can take anywhere from 15 minutes to a couple of class periods depending on whether you are using it as a review or a teaching strategy.

When the students go back to their original groups, it is their responsibility to make sure everyone in the group masters the material they will be teaching. As a follow up, quizzes can be given to each person to assess how well they learned the different components.

Advantages:

1. Involves all the students in your classroom.2. Builds a sense of positive interdependence because in order for everyone to succeed, they must

work together and help each other out.3. Everyone is “accountable”.


1. Prepare instructions for the groups and instructional materials for each expert group.2. Circulate around the room, listening to discussions and answering questions only when needed.3. Decide when to call students back into their original groups.4. Assess learning outcomes.


Think/Pair/Share Activity

This is an activity in three stages:

1. Think: The instructor poses a question or problem and gives students time to think about how to answer the question or solve the problem individually. The amount of time would vary, and could even extend over more than one class.

2. Pair: Students pair up and exchange ideas and/or solutions, coming to consensus on an answer or articulating their differences.

3. Share: The instructor may choose to have one, some or all of the pairs share their results with the entire group. Another option is to have pairs share with pairs.

Advantages:

1. Involves all the students in your classroom. 1. Builds a sense of confidence since students don’t have to share their ideas/solutions with the

whole group until they have had an opportunity to share with another student.


1. Pose thought-provoking questions.2. Circulate around the room, listening to discussions and answering questions only when needed.3. Decide when and how to share results.4. Assess learning outcomes.


Team/Pair/Solo Activity

This activity is a strategy for giving students the confidence to tackle difficult problems on their own by first working similar problems with other students. Students are given a problem to solve within a small group of 4 – 6 students. After the small groups solve the problem, two pairs or three pairs (or a triad and a pair if need be) are formed and these new teams solve a problem similar to the problem solved in the larger group. Lastly, individual students are given another similar problem to solve, and this time must solve it on their own.

This activity lends itself well to group quizzes, having both group and individual accountability. Some part of the student’s grade can be based on preparatory homework done outside of class, some part on the group’s problem solution, a part on the pair’s problem solution, and a part on the individual’s problem solution. A bonus could be given to the scores of team members if every member of the team scores at a given criterion level. Alternatively, the group portion of the score could be based part on the group’s solution to the problem and part on the average score of the group members on the pair and individual problem solutions. Research shows that the individual accountability should be a larger portion of the grade than the group grade.

Advantages:

1. Involves all the students in your classroom.2. Builds a sense of positive interdependence because in order for everyone to succeed, they must

work together and help each other out.3. Gives students the confidence to tackle problems they might not otherwise have attempted as

they see their peers successes and learn from them. 4. Everyone is “accountable”.


1. Prepare a set of related problems.2. Circulate around the room, listening to discussions and answering questions only when needed.3. Decide when to move from groups to pairs, from pairs to solo efforts. 4. Evaluate results in a way that holds everyone accountable both individually and as a group. 5. Assess learning outcomes.


Three Minute Review Activity

A quick way to ‘test for understanding’ during a lecture or presentation is to pause to ask for questions, or to ask students to review the main points of what you have been saying. Doing this in pairs or small teams provides an opportunity for those who are too shy to ask questions in the larger group a chance to speak. You might have standing groups, form the groups from those sitting near each other, or use it as an opportunity to have students move around by forming groups of students who are not sitting together. Questions which the group cannot answer can be asked by a spokesperson for the group at the end of three minutes. The teacher might call upon one of the groups to give a synopsis of the material and invite others to add to it, or disagree with it.

Advantages:

1. Involves all the students in your classroom.2. Gives the instructor immediate feedback of the students’ level of understanding. 3. Is an efficient way to be sure that students’ questions are answered.


1. Circulate around the room, listening to discussions and answering questions only when needed.2. Decide when to initiate and when to terminate the activity. 3. Decide when and how to share results with the entire class.


Sample Round Robin Activities

The opening activity in our presentation which solicited from participants the benefits of using Cooperative Learning methods was an example of round robin brainstorming.

For the following activities, I usually divide my class into 6 groups and give them about 5 minutes. Then for the next 5 minutes, (using the round robin strategy), I go around the class and each group gives me one example from their list to write down on the board.

Reviewing Subsets:

Given : A = { 2, 4, 6, 7, 8,} in your group list all the possible subsets.

Finding Additive Inverses:

1. Recall that a + (-a) = 0.

2. In your group list examples of additive inverses.

Reviewing Problem Solving Techniques:

In class we have discusses various problem-solving strategies. Discuss within your groups the different strategies and give an example of each.

(working backwards, solve a similar problem, elimination, make a table, use a variable, draw a picture, guess and check, find a pattern etc)

Reviewing Statistics:

1. Problem: Your group believes more people enjoy vanilla ice cream than chocolate ice cream. To prove this, your group decides to do an experiment. List the different ways you can collect your data along with the pros and cons of each method.

2. Given the following data: 2, 4, 5, 5, 6, 7, 8, 10 use various descriptive statistics to find the center of this data.


Round Robin Activity for Calculus

An example of the way in which I use Round Robin Brainstorming in my Calculus classes is to break the ice while starting student out on task on the very first day. The activity is important review to connect what students already know about lines and slopes to the material we will soon be covering in Calculus and to emphasize slopes as rates of change. It works as an ice breaker to make students comfortable working with each other in class because the material, going back to their first algebra course, is very familiar to them and all students will be able to contribute.

Forming the Groups:

I do not form permanent groups in my small classes (maximum size 30). As the course goes on, I form groups as I need them ‘randomly’ in various ways. I do not take much time forming groups on the first day. I will usually quickly put them in groups of three, with some groups of four when I need to.

Time Allotment for Round Robin:

I distribute the attached handout, Round Robin Review of Lines. I give students about five minutes for each question. I tell them when the first five minutes are up and ask them to go on to question two if they haven’t already (most will have).

Follow Up:

I ask for a recorder to volunteer to read the group list for question one and write the forms of equations of lines on the board, inviting other to add others and supplying the names of the forms. I then usually explain that in Calculus we will often want to find an equation of a line given its slope and a point and so I circle the point-slope form of the equation of the line.

I ask for another volunteer to read the group responses to question two. I explain that it is important for them to realize that what makes a line straight is a constant rate of change between any two points. I tell them that we will build on what we know about slopes of lines to be study rates of change of functions for which those rates of change are themselves changing.

I then give the students the attached handout, Review of Lines, and go over that material for their class notes. Depending on how much time is left in class, I will have students find an equation of the line graphed, talk about the signs of the slopes of the lines graphed, and find and interpret the slope in the example problem in their groups, or do it with them as a whole class. I usually take the time to talk about increasing, decreasing and constant functions and the fact that the vertical line does not give us y as a function of x while I am going over the signs of the slopes of the lines graphed. I remind students that they are responsible for more material on lines in the review I have assigned from their text.

MTH 210


Calculus ILeopard

Round Robin Review of LinesInstructions to the Group:

1. Introduce yourselves to each other.2. The person whose name is first in alphabetical order is the recorder.

Question One: List different forms of an equation of a (straight) line in the x-y coordinate plane.1. Beginning with the person to the left of the recorder, give one form of an equation of a line.2. The next person to the left gives another form of an equation of a line and so on until the group

can think of no more forms or I say that time is up. Then move on to question two.3. Recorder: Write down the responses as given below:

Question Two: List everything you know about slopes of lines – different ways to find them, what information they give about the graph of the line, what they tell us about the way that y depends on x, etc.

1. Beginning with the person to the left of the recorder, give one fact about slopes of lines.2. The next person to the left gives another fact and so on until the group can think of no more

facts or I say that time is up. 3. Recorder: Write down the responses as given below:


Review of Lines

MTH 210

Calculus I

Leopard

Find an equation of the line whose graph is given below:

1 2 3 4 5 6 7-1-2-3-4-5-6-7

1

2

3

4

5

-1

-2

-3

-4

-5

x

y

Important Concepts for Calculus:

1. Slope is a rate of change,

A positive slope gives the amount y, the output, increases for each unit increase in x, the input.

A negative slope gives the amount y, the output, decreases for each unit increase in x, the input.


Give the signs of the slopes of the lines graphed above.

2. The units of the slope are y units per x unit; output units per input unit.

Example 1:

C = 1.89 x gives the cost, C, in dollars for x cans of orange juice. The slope, 1.89, is 1.89 dollars per can, or $1.89/can

Example 2:

s = 55t + 10 gives the number of miles, s, a person is from home after t hours if he starts 10 miles away from home (t = 0). The slope is 55, which is 55 miles per hour or 55 mph which tells us how much his distance from home increases for each one hour increase in time.

Example 3:

P = 100 – 5x gives the percent correct, P, on a multiple choice test with 20 questions if x questions are answered incorrectly.


The slope is _____________. It means:

3. Equation of a line in point slope form

If a line has slope m and goes through the point (a,b), an equation of the line is:

Example:

If you travel at an average speed of 45 mph away from home and are 100 miles from home after one hour,

s = 45(t-1) + 100 gives your position, s, in miles from home, after t hours.


Numbered Heads Activity for Solving Word Problems in an Algebra Class

Procedure:

1. Students are divided into small groups between 4 and 6.2. Each student is given a number from 1 to the number of students in the group.3. Each group is given a word problem to solve. Here you have a choice; you can give each

group the same problem or each group a different problem to solve.4. Groups work together to solve the problem.5. When most of the class is done with their group problem, the teacher calls out a number

and the person with that number from each group comes up to the board and writes their group’s solution to the problem.

Word Problem 1: New Hyde Park High School sold 600 tickets to their concert. The number of tickets sold to students was three times the number of tickets sold to nonstudents. How many tickets were sold to students and to nonstudents?

I tend to give the same problem to each group. Sometimes there might be more than one way of solving the problem and it makes for an interesting discussion when the solutions are posted. Also if one of the solutions is wrong it lends itself to a great “teaching moment”.

Numbered Heads Activity for Calculations in Other Bases

1. Each group is given a different problem to solve in base 5. (can be any mathematical operation)

2. After each group has had a chance to solve their problem, a random number is called out. The person having that number comes to the board and writes down their problem and solution.

Numbered Heads Activity for Reviewing Basic Derivative Rules in Calculus

1. Each group spends ten minutes quizzing each other on the basic derivative rules.

2. After ten minutes, a random number is called out. The person having that number stands. The instructor chooses a derivative formula at random for each of the individuals standing, in turn. If the student answers correctly, the student can sit down. The activity continues until all students are seated.


Jigsaw activity to be used as a review

Solving systems of linear equations with two unknown variables

Procedure:

1. Have your students get into groups of three. If you have one or two students who can not form a group of three you can assign them the task of being editors. Their job at the end will be to print up the three different methods used to solve two linear equations with two unknown variables.

2. Everyone in the class is given these two linear equations to solve:

2x – 3y = -2

4x + y = 24

3. Within their groups each person takes one of the following strategies: elimination, substitution or graphing.

4. New “expert groups” will be formed based on one of these strategies. You can have more than one “expert” group mastering each strategy. For example if you have 27 students in your class, this would give you 9 people becoming experts in the elimination strategy. Instead of having all of them working together, you might want 3 groups of 3 mastering the elimination method.

5. After about 10 - 15 minutes, each person goes back to their original group and teaches the other two people in their group their area of “expertise”.

6. It is the responsibility of each member to make sure everyone in the group can do all three methods. Allow between 10 – 15 minutes.

7. If you want, individual quizzes can then be given and the group’s average on the quiz will be taken.

If you had one or two students who did not become “experts’ and were assigned the task “ editors”, after step 6 is concluded have them come up to the board and write the three methods that were used to solve the equations. Then during the next class, distributed those papers.


Jigsaw Activity for Learning Pôlya’s Problem Solving Approach

1. Students will form groups of 4 and be assigned a number 1 – 4.

2. Students will then form “expert groups” of 3 – 4 people by finding other students with the same number they were assigned.

3. Each expert group will be given one of Pôlya’s 4 problem solving steps to learn and discuss ways to teach the members of their original group their problem solving step.

4. By the end of the activity, everyone will know and be able to demonstrate the 4 problem solving steps.

Each group will be given one of these problem solving steps (Long & DeTemple, 2006, p. 9) to learn and teach the other members of their original group.

Step 1: Understand the problem. Many times students have a hard time solving a word problem because they don’t know what the problem is asking. Here are some questions they should ask themselves as they read the problem and before they start to solve the problem:

“Do you understand all the words used in stating the problem? If not, look them up in a dictionary or wherever they can be found.

What are you asked to find or to show? Can you restate the problem in your own words? Could you work out some numerical examples that would help make the problem clear/ Could you think of a picture or a diagram that might help you understand the problem? Is there enough information to enable you to find a solution? Is there extraneous information? What do you really need to know to find a solution”?

Step 2: Devise a plan. Once you fully understand the problem, think of a strategy you can use to solve the problem.

Following is a partial list of strategies: “guess and check look for a pattern eliminate possibilities use symmetry solve an equation draw a picture think of a similar problem you have already solved use a model work backward use a formula”


Step 3: Carry out the plan. During this step, using your plan, you try to solve the problem. Sometimes you might need to discard your plan and try a different one.

Step 4: Look back. This is the step where you analyze what you have done.

Questions to ask yourself:

“ Does my answer make sense? What was the key factor that allowed me to devise an effective plan for solving this problem? Can I think of a simpler strategy for solving this problem? Can I think of a more effective or powerful strategy for solving this problem? Can I think of any other problems for which this plan of attack would be effective”?


Jigsaw Activity for Finding the Greatest Common Divisor and Least Common Multiple

Can be used either as initial instruction or as review.

Find the GCD and LCM for the following pairs of numbers using set intersection, prime power representation, Venn diagrams and the Euclidean Algorithm.

15 and 35

1. Procedure: The students get into groups of 4. If a group has more than 4 members in it, the 5 th member will be assigned the role as “editor” and at the end of the activity, will come up to the board and write the four different methods that were used to find the LCM and GCD.

2. Each person will choose one of the four methods and become an “expert” mastering that technique.

3. New expert groups will be formed based on one of these strategies. Together they will master the techniques and be able to later go back to their original group and teach the others their strategy.

4. After about 10 – 15 minutes, each person goes back to their original group and teaches the other three their strategy.

5. It is the responsibility of each member to make sure everyone can do all 4 strategies.

6. If you want individual quizzes can then be given or you can use the number heads method and have someone from each group come up to the board and write down one of their strategies.

Examples:

Intersection method :

Divisors of factors of 15 = { 1, 3, 5, 15}

Divisors of 35={ 1, 4, 5, 7, 35}

The intersection of both sets ={ 1, 5} The largest natural number is 5. This will be your GCD.

Multiples of 15 = { 15, 30, 45, 60, 75, 90, 105, …}

Multiples of 35 = { 35, 70, 105,…}

The first multiple that is common to both will be your LCM. So 105 is your LCM


Prime factor representation:

Finding the GCD: 15 = 3 * 535 = 5 * 7

Take the lowest power of each prime factor that is common to both of the numbers and multiply them together. In this case it would be 5^1. This will be your GCD.

Finding the LCM:

Take the highest power of each prime factor and multiply them together(3 ^ 1) * ( 5 ^ 1) * ( 7 ^ 1) = 105. This will be your LCM.

Euclidean Method To find the GCD. Divide the smaller number into the larger number. 15/35 = 2 R5. Now take the remainder and divide it into the previous divisor. 5/15 = 3. The divisor that ends up giving you no remainder is your GCD. So 5 is your GCD.

To find the LCM: Multiply your 2 natural numbers and divide the product by your GCD.(15 * 35)/5 = 105.


Jigsaw Activity for Calculus

Practicing What We Preach: Demonstrating Effective Cooperative Education Practices

Annette Leopard and Karen Wells, Monroe Community College

Seaway Section, MAA and NYSMATYC Region 1 2007 Fall Joint Conference

October 20, 2007

Monroe Community College

This is an activity for the first day of working with the Chain Rule.

1. After introducing the Chain Rule I do some examples with power functions, and summarize my work by saying, “The derivative of a function to a power is the power, times the function to the power minus one, times the derivative of the function.” I write this on the board as

where

Now I am ready for the Cooperative Learning Part.2. Divide the class into ‘expert’ groups whose area of expertise will be a particular outer function.

Each expert group of 3 or 4 students will have several derivatives of sine of a function, or e raised to a function power, or natural log of a function to solve along with an example and instructions for completing the expert group stage of the process.

3. After the members of the expert groups have successfully worked all problems assigned to them, they must write in words a statement similar to the one I wrote for power rule problems.

4. Form new groups of 3 or 4 students with at least one of each expert type in each new group. 5. The new groups solve three new problems, one of each type, with each team member teaching

the other team members their area of expertise. 6. Each person in the class solves three new problems individually, to test for understanding.


MTH 210 Jigsaw Class Work The Chain Rule – Derivatives of Composite FunctionsPower Rule examplesLeopard

In general,

The derivative of a function to a power is the power times the function to the power minus one times the derivative of the function.


MTH 210 Calculus I Jigsaw GroupsLeopardThe Chain Rule Expert GroupSine Function

Example 1:

Example 2:

Step One: Find derivatives of the following functions:

1.

2.

3.

4.

5.

6.

Step Two: Check your answers with me.

Step Three: Write in words the procedure to follow to find the derivative of the sine of a function:


MTH 210 Calculus I Jigsaw GroupsLeopardThe Chain Rule Expert GroupSine Functions Answers

1.

4.

5.

6.


MTH 210 Calculus I Jigsaw GroupsLeopardThe Chain Rule Expert GroupExponential Functions

Example 1:

Example 2:


1.

2.

3.

4.

5.

6.



Step Three: Write in words the procedure to follow to find the derivative of e to a function of x power:


MTH 210 Calculus I JigsawLeopardThe Chain Rule Expert GroupExponential Functions Answers

1.

4.

5.

6.


MTH 210 Calculus I JigsawLeopardThe Chain Rule Expert GroupTangent Functions

Example 1:

Example 2:


1.

2.

3.

4.

5.

6.



Step Three: Write in words the procedure to follow to find the derivative of the tangent of a function:


MTH 210 Calculus I JigsawLeopardThe Chain Rule Expert GroupTangent Functions Answers

1.

4.

5.

6.


MTH 210 Calculus I JigsawLeopardThe Chain Rule Group Work

1. The ‘experts’ should explain how to do their problem types to the whole group, using an example problem.

2. The group should solve the problems below together, making sure that everyone in the group can solve every problem type.

3. The group should check their solutions with me and make corrections.4. The group should write in English how to find derivatives of composite functions using the Chain

Rule

5. Obtain the next set of Chain Rule problems from me for each group member to work individually.

6. The team with the highest average score on these individual problems will be declared the champions. In the case of a tie, the team that finished first will be declared the champions.

Find the derivative of the following functions:

1.

2.

3.

4.


5.

MTH 210 Calculus I JigsawLeopardThe Chain Rule Group Work Answers

1.

2.

3.

4.

5.


MTH 210 Calculus I JigsawLeopardThe Chain Rule Individual Efforts

The team with the highest average score for the individual team members on these problems will be declared the champions. In the case of a tie, the team that finished first will be declared the champions.


1.

2.

3.

4.


MTH 210 Calculus I JigsawLeopardThe Chain Rule Individual Efforts Answers

The team with the highest average score for the individual team members on these problems will be declared the champions. In the case of a tie, the team that finished first will be declared the champions.


1.

2.

3.

4.


Jigsaw activity as an instructional strategy to find Greatest Common Divisor or Least Common Multiple

1. Students are divided into groups of 3 and assigned numbers.

2. Student # 1 will become an expert finding the Greatest Common Divisor by Intersection of Sets.

3. Student # 2 will become an expert finding the Greatest Common Divisor by using Prime Power Representation.

4. Student # 3 will become an expert finding the Greatest Common Divisor using the Euclidean Algorithm.

5. Students with the same assigned number will form expert groups of 3 to 4. Together they will learn the technique and decide how they will teach their method to the other people in their original group. What examples they will use, what strategies they will use etc.


Directions for Finding the Greatest Common Divisor or Least Common Multiple

1. Form groups of 3 and decide who will be #1, #2, #3.

2. Once you have been assigned a number, form a group of 3 or 4 with other people with that same number.

3. Each one of you will become an expert in one of the methods used to find the Greatest Common Divisor for two numbers. It is your responsibility, within your expert group, to learn the method and decide how you will teach your method to the other members of your initial group. Think about what teaching strategies you will use, what examples you will you use and how you will know whether they understand the method.

4. You will be given 10 minutes to learn your method, 10 minutes to decide what strategies you will use when you return to your original group and 5 minutes to teach your method to them.

5. At the end of this activity, every person in your group should be able to use all three methods to find the Greatest Common Divisor of any two numbers.

6. As a follow up, everyone in the class will be given the same problem to work on individually using all three methods.


Finding the Greatest Common Divisor or Least Common Multiple

Directions for expert group 1: Using the intersection of sets.

1. Find the set of divisors for each number.2. Find the intersection between the two sets.3. The greatest natural number in your intersection is the G.C.D.

Example: Find the Greatest Common Divisor of 20 and 35.

1. The set of divisors for 20 = {1, 2, 4, 5, 10, 20}2. The set of divisors for 35 = { 1, 5, 7, 35}3. The intersection of both sets = { 1, 5}4. Since 5 is the greatest natural number in the set, the GCD( 20, 35) = 5.



Directions for expert group 2: Using Prime Power Representation.

20 35

5 4 5 7

2 2

20 = • and 35 =

Note for the prime factorization, both numbers must contain the same factor. By adding a factor raised to the zero power, we have not changed the product.

Now take each factor with the lesser exponent and multiply.

= 5. So the GCD( 35, 30) = 5



Directions for expert group 3: Using the Euclidean Algorithm.

1. Start off dividing both numbers. 35/20 = 1 R 152. We continue dividing successive divisors by remainders until we arrive at a nonzero remainder.3. The last nonzero remainder is the desired greatest common divisor.

Example: Find the GCD for 20 and 35

35/20 = 1 R15

20/15 = 1 R 5

5/1 = 5 R0

5 is the GCD because there was no remainder.


Think- Pair – Share Activity for Perimeter and Area

1. First, by yourself, write down different dimensions of quadrilaterals that will give you an area of 64. ( 5 minutes).

2. Then turn to your neighbor, compare and share your answers. If you don’t agree, help each other out until you have come to a consensus. ( 3 minutes)

3. Share your answers with the class ( 3 minutes)

Think – Pair – Share Activity for Statistics

Lurking variables

1. First read the following example and think about what other variables could be contributing to the strong negative correlation. ( 2 - 3 minutes)

2. Then discuss your examples with the person sitting next to you. ( 2 minutes).

3. Share your answers with the class. ( 2- 5 minutes)

It has been shown that there is a strong negative correlation between the average annual income and the record time to run one mile. That is as annual income increases, the time to run a mile decreases.

Think – Pair – Share Activity for Algebra

We have talked about several different properties for addition of whole numbers:

Closure, associative, identity element and commutative properties. Do these properties hold for integers under addition? Do these properties hold for subtraction under the set of integers?

Think – Pair – Share Activity for Function Notation

Is it true that f(a+b) = f(a) + f(b)? Always? Sometimes? Never? Support your answer with examples.


Team/ Pair/ Solo Activity Involving Exponents

Purpose: Students will discover the different rules for exponents.

Procedure:

1. Students will be divided into groups and together they will discover and write down the rule for multiplying numbers with exponents. Example 2

2. Using the same method they used to discover the rule for multiplying exponents, with a

partner they will discover the rule for dividing exponents. Example:

3. By themselves, using the same method they used when working as a team and then with a partner, they will discover and write down the rule for solving

problems like this (7


Worksheet for Team/ Pair/ Solo Activity Involving Exponents

Step 1: Discover the rule for solving problems in this form: n (x and y are natural numbers for this activity. Later on you can introduce negative integers)

A. What does 6 mean? How many powers of 6 do you have? 6 means 6*6*6

6 means 6*6*6*6*6

So 6 means (6*6*6)*(6*6*6*6*6)

How can we rewrite this as a power ? 6

B. Using the same procedure as above, what is ? ( write your answer as a power of 5)

If your answer is 5 try the next one. If not go back to the beginning and see if you can find your mistake.

C. What is 8 ?

Did you get 8 for an answer?

If not try again.

Do you see a pattern? If not try this one: 4 .

If you do see a pattern, try writing a rule for n .

Now make up a problem of your own and apply your rule. Does it work? If not go back over your work and see if you see a different pattern. Hint, try some more similar problems.


Step 2: Now with a partner and using the same method you used above discover and write down the

rule for solving problems in this form:

What is ? This means = 7*7*7*7*7 =

Using the same procedure try this problem:

What is

So our answer is?

If your answer was 4 try the next one. If not go back and check your work.

Try this one using the same procedure:

Do you see a pattern? If you do go on to the next question. If not make up some problems and solve them.

Write a rule for dividing exponents. Create a new problem and see if your rule works. If it does, go on to #3. If it doesn’t, try some more problems.

Step 3: By yourself using the same method (procedure) you used above, find the rule for solving problems in this form: (n

Example: (5 )

When you have all three rules, look in your book and see if you are right.


Team/Solo Group Quiz on Function Notation and Lines – Tangent Line Preview

MTH 210 Name: ________________________

Calculus I

Group Quiz One Instructions and Grading:

1. Turn in the homework due today. (Homework from the text led them to discover the slope of the tangent line at a point on a quadratic function.)

2. Choose a recorder for your group. This person is responsible for writing up the group submission for this quiz and so should have neat handwriting.

3. Do problems 1 through 6 together. You should finish no later than 10:30.4. Submit the Group Quiz One Group Submission for your group and turn in the individual papers

with the group quiz to receive a solo quiz.5. Work the Solo quiz problems one through three on your own.6. Turn these in at the end of class. If you finish early, you are free to leave.

Grading:

1. Today’s Homework: 5 points

2. Group Submission: 45 points

3. Solo Part 45 points

4. Group’s Solo Average .05*Average %, maximum 5 points


MTH 210 Names of Group Members:Calculus IGroup Quiz One Group Submission

1. Sketch the graph of the function .

1 2 3 4 5 6 7-1-2-3-4-5-6-7

1

2

3

4

5

6

-1

-2

-3

-4

-5

-6

x

y

2. Find and label the point on the graph of corresponding to .

3. Find the slope of the line between the points (2,4) and where . Show work.

3. ___________________

4. Find a formula in terms of for the slope of the line between the points and ( ,) where .

4. ____________________________


5. Simplify the formula for the slope of the line between the points and ( ,

) where and evaluate the simplified formula for equal to zero. This is called

the slope of the tangent line to at the point . Show work.

5. ___________________________

6. Find an equation of the line through the point with slope equal to the value found

in problem 5. Show work. Sketch this line on the graph of This is called the

tangent line to the graph of at the point .

6. __________________________


MTH 210 Name ______________________________

Calculus I

Group Quiz One Solo Part

1. Find the slope of the line between the points ( , ) and ( , ) where . Show work.

1. ____________________________

2. Simplify the formula for the slope of the line between the points ( , ) and ( ,

) where and evaluate the simplified formula for equal to zero. This is

called the slope of the tangent line to at the point ( , ). Show work.

2. ___________________________


3. Find an equation of the line through the point ( , ) with slope equal to the value found

in problem 2. Show work. Sketch this line and the graph of on the same axes below.

This is called the tangent line to the graph of at the point ( , )

3. __________________________

1 2 3 4 5 6 7-1-2-3-4-5-6-7

1

2

3

4

5

6

-1

-2

-3

-4

-5

-6

x

y


Team/Solo Group Quiz on Optimization for Calculus

MTH 210 Name: ________________________

Leopard

Calculus I

Group Quiz Five Instructions and Grading:

1. Choose a recorder for your group. This person is responsible for writing up the group submission for this quiz and so should have neat handwriting.

2. Do problem 1 together. You should take no more than 30 minutes to complete this. Remember – no notes, no texts.

3. Submit the Group Quiz Five Group Submission for your group and turn in the individual working papers with the group quiz to receive a solo quiz for each of you.

4. Work the Solo quiz problem on your own. Remember – no texts, no notes including no notes from the group part of today’s quiz.

5. Turn the Solo quiz in at the end of class. If you finish early, you are free to leave.

Grading:

1. Group Submission: 45 points

2. Solo Part 45 points

3. Group’s Solo Average .10*Average %, maximum 10 points


MTH 210 Name ___________________________

Calculus I

Group Quiz Five Working paper

Determine the dimensions of the rectangular solid with a square base of maximum volume if the surface area is 300 square inches. Use Calculus methods to find and justify your answer. Include units in your answer.

1. Side of base _________

Height __________


MTH 210 Names:

Calculus I

Group Quiz Five Group Submission

Determine the dimensions of the rectangular solid with a square base of maximum volume if the surface area is 300 square inches. Use Calculus methods to find and justify your answer. Include units in your answer.

Side of base _________

Height __________


MTH 210 Name ___________________________

Calculus I

Group Quiz Five Solo Submission

A farmer needs to enclose a rectangular pasture alongside a river. The pasture must have an area of 200 square meters. What are the dimensions that will minimize the amount of fencing he needs if no fencing is needed alongside the river? Show all work and justify your answer using Calculus techniques learned in class. Put final answers on lines provided. Remember to include units.

side parallel to the river _________________

side perpendicular to the river _____________


Follow – Up to Optimization Team/Solo Quiz

MTH 210 Calculus ILeopard

Optional:

You can retake the solo submission part of group quiz 5 on optimization if you work and turn in problems 20 and 34 from section 4.7 of the textbook by Wednesday morning at 8 a.m. If a classmate who scored 44 or 45 on the solo submission helps you with problems 20 and 34 to prepare you for the retake, that classmate will receive one-ninth of your score on the re-take as bonus points (with a maximum of 5 bonus points per person.) You must record the name of the student who helped you when you turn in problems 20 and 34.


Three Minute Review Activity for Statistics

1. Discuss the pros and cons for using histograms verses dot plots or stem and leaf graphs.

2. Discuss what type of graphs you can use with different types of data (qualitative and quantitative)

3. Discuss the pros and cons of using the mean, mode and median to describe the center of your data.

Three Minute Review Activity for Properties of Addition and Multiplication.

Discuss the properties we have talked about and give an example of each.

Three Minute Review Activity for Conic Sections

Give the general formula for each and identify the parameters in each formula:

1. parabola

2. ellipse

3. hyperbola


Group Discovery Activity: Cooperative group activity to discover the rules or tests for divisibility

Many times our students have a difficult time determining the divisors of a natural number. However knowing some of the various divisibility tests might help them with a starting point in trying to simplified fractions.

I divide the students into groups of 2 or 3 and let them work through this activity and have them discover the various tests. I usually give them about 15 minutes. Then we come back together as a whole class and I have them share their conclusions.

1. Tell me are these numbers divisible by 2: 46 53 35 88 100046

What can you conclude? What type of numbers are divisible by 2?

2. Tell me if these numbers are divisible by 3: 30 45 90 126 110 107 141201

Add up the digits of each number that was divisible by 3 and write them down (30) ( 45) (90) (126) (110) (107) (141201)

What conclusions can you draw?

3. Tell me if these numbers are divisible by 4: 80 283624 348 290 165 492

Look at the last two digits of each number above… Are the last two digits of the Number divisible by 4? (80) (283624) (348) (290) (165) (492)

What conclusions can you draw?

4. Tell me if the following numbers are divisible by 5: 38 345 600 847 200045 92 9277866600

Look at the last digit of the numbers that can be divided by 5. What conclusions can you draw?


5. Tell me if the following numbers are divisible by 6: 23 2742 336 642 777 891

Look at the numbers that you said were divisible by 6. Are they also divisible by 2 and 3?What conclusions can you make concerning whether a number is divisible by 6?6. Tell me if the following numbers are divisible by 7: 87 94 315 6118 192 88109

Looking at the numbers that you said were divisible by 7:A. Look at the numbers that were divisible by 7.B. Take each number and choose the last digit in those numbers.C. Double and subtract the last digit in your number from the rest of the digits in that number.D. Is your number divisible by 7?

What conclusions can you make?

7. Divisibility test just for 11:

1. Are these numbers divisible by 11?308 17479 561 876

2. Looking at the numbers that were divisible by 11:Add up the digits in the odd positions and add up the digits in the even positions. Subtract the two sums. Is your difference divisible by 11?

3. What conclusion can you make?

4. Is this number divisible by 11? No calculators!8193246781053476109

Try this number to see if it is divisible by 7, 11 or 13: 1927643001548

8. Tell me if these numbers are divisible by 9:

1430784 156 27 298 270 10, 020, 006 1009

b. For those numbers that you said were divisible by 9 add up their digits and find the sum. Is the sum of their digits divisible by 9?

c. What can you conclude?


Math for Elementary School Teachers Exploration activity – triangles and angles

Working with a partner – draw 2 different triangles.

A. How are your 2 different triangles different? What features did you change?

B. Measure the angles of each of your triangles and add the measurements up. What do you notice about the sum of the angles in a triangle?

Therefore the sum of the measures of the angles in any triangle is___________.

C. Looking at each triangle separately, measure the sides of each triangle. Now still looking at each triangle separately, add up the measurements of two sides of your triangle and compare the sum to your third side measurement. Do this for all three sides. Compare side A + side B to side C, side B + side C to side A and side A + side C to side B. What do you notice?

Now do the same thing with the other triangle. What do you notice?

Suppose you were asked to make a triangle with sides 4, 4, and 10 units long. Do you think you could do it? Explain your answer. Keep in mind the goal is not to try to build the triangle, but to predict the outcome.

Come up with a rule that describes when three lengths will make a triangle and when they will not. Write down the rule in your own words.


Group Exploration Activity to Discover the Rules for Corresponding Angles.

Students will together either in groups or with a partner to discover the rules for corresponding angles and alternate interior angles.

1. Using the drawing above measure D and B. What do you notice?

Measure angle C and measure angle A. What do you notice?

Measure angle 3 and measure angle 7. Angle 3 is under angle C and angle 7 is under angle A. What do you notice?

Measure angle 4 which is under angle D and measure angle 8 which is under angle B. What do you notice?

The above pair of angles are called corresponding angles. The property of corresponding – angles state:

1. When lines 1 and 2 appear to be parallel and are cut by a transversal then the corresponding angles _______________________________.

2. If two lines in the plane are cut by a transversal and some pair of corresponding angles has the same measure, then the lines are parallel.

Note: corresponding angles will have the same measurements when any two lines ( l and m) are intersected at two points by a transversal line t.


Now measure 3 and measure B. What do you notice? Are they congruent? Give a reason for your answer.

Now measure 4 and measure A. What do you notice? Are they congruent? Give a reason for your answer.

The above pairs of angles are called alternate interior angles. The Alternate – Interior – Angles Theorem states: Two lines cut by a transversal are parallel if and only if a pair of alternate interior angles is congruent.

Vertical angles.

Students will work in groups or with a partner to discover the rules for vertical angles and supplementary angles..

Given: 1. With your protractor measure each of the angles and record your measurements below.

2. What do you notice about the measurements of the angles opposite each other?

A pair of nonadjacent angles formed by two interesting lines is called vertical angles. The Vertical – Angles Theorem states: Vertical angles have the ________measurement.

Measure 2 and measure 3. What do you notice?

Measure 1 and measure 4. What do you notice?

Two angles that have a common side and non overlapping interiors are called adjacent angles. Here we have supplementary adjacent angles. Supplementary adjacent angles add up to 180 degrees.


Cooperative Learning – Flashcards

I have found that students need instruction in how to memorize definitions, rules and formulas. I require them to bring 3X5 cards to every class. If there is a definition, formula, or rule that requires memorization, I have students make a flash card with the information. In Calculus class, for example, they have flashcards for the definition of the derivative, continuity and so on and for derivative and integration rules. If a student forgets a definition, rule or formula from a previous class or even previous course, I have the student make a flash card. For example, my students in Calculus class make flashcards when they have to ask me to explain an algebra or trig step they have forgotten how to do. When I introduce definitions or new rules I give students five minutes in pairs to practice with flash cards. Before class begins, while I am taking attendance and handing back papers I have students work in pairs on flashcards. Students who were reluctant at first made the cards so that they would be ready for the pair activity. They have reported that using flashcards seemed corny to them at first, but works. I repeat 5 minute mastery quizzes on things such as derivative rules at the beginning of class until the majority of the class scores 85% or above. Students work to quiz each other before class begins to be ready for these quizzes.


References

Bell, James. Improving Student Learning and College Teaching. Retrieved August 31, 2007 from

http://classweb.howardcc.edu/jbell/learning/cooperative_learning.htm

Blair, Richelle (Rikki) (editor) (2006). Beyond Crossroads: Implementing Mathematics Standards in the

First Two Years of College. Memphis, TN: American Mathematical Association of Two-Year Colleges.

Cooper, Jim, Randall Mueck (1990). Student Involvement in Learning: Cooperative Learning and College

Instruction. Journal on Excellence in College Teaching, I, 68 – 76.

Dubinsky, Ed, David Mathews, Barbara E. Reynolds (editors) (1997). Readings in Cooperative

Learning for Undergraduate Mathematics. MAA Notes, Number 44. Washington, D.C.: The

Mathematical Association of America.

Fortney, Dan, Ashley Kerstetter Cooperative Learning Strategies. Retrieved September 18, 2007

from http://courses.ed.asu.edu/clark/CoopLearn/Index.htm

Kennesaw State University Education Technology Training Center. Cooperative Learning. Retrieved

August 13, 2007, from http://edtech.kennesaw.edu/intech/cooperativelearning.htm

Long, Calvin T., Duane W. DeTemple (2006). Mathematical Reasoning for Elementary Teachers( fourth

edition). Pearson Addison Wesley.

Millis, Barbara J. (1991) Making Cooperative Learning Work.

Workshop, Annual Fall Teaching Conference, University at Buffalo.

Millis, Barbara J. Helping Faculty Build Learning Communities Through Cooperative Groups. In Hilsen,

(Editor). (1990). To Improve the Academy: Helping Faculty Build Learning Communities, 10,

43 – 58. Stillwater OK: New Forums Press.

Millis, Barbara J. Fulfilling the Promise of the “Seven Principles” Through Cooperative Learning: An


http://edtech.kennesaw.edu/intech/cooperativelearning.htm

http://courses.ed.asu.edu/clark/CoopLearn/Index.htm

http://classweb.howardcc.edu/jbell/learning/cooperative_learning.htm

Action Agenda for the University Classroom. (1991) . Journal on Excellence in College Teaching,

Vol. 2. Oxford, Ohio: Miami University.

Nastasi, Bonnie K., Douglas H. Clements. (1991). Research on Cooperative Learning: Implications for

Practice. School Psychology Review, Volume 20, No. 1. , 110 – 131.

Rogers, Elizabeth C., Barbara E. Reynolds, Neil A. Davidson, Anthony D. Thomas (editors) (2001).

Cooperative Learning in Undergraduate Mathematics: Issues That Matter and Strategies That

Work, MAA Notes, Number 55. Washington, D.C.: The Mathematical Association of America.

Slavin, Robert E. Research on Cooperative Learning: Consensus and Controversy (1989-90). Educational

Leadership, 47, (4): 52 – 55.

Sutton, Gail O. Cooperative Learning Works in Mathematics. (1992). The Mathematics Teacher, Vol.85,

No.1, pgs 63 – 66.


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